ECONOMETRICS - University of Wisconsin–Madisonbhansen/econometrics/Econometrics2008.pdf ·...
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ECONOMETRICS Bruce E. Hansen c 2000, 2008 1 University of Wisconsin www.ssc.wisc.edu/~bhansen This Revision: January 17, 2008 Comments Welcome 1 This manuscript may be printed and reproduced for individual or instructional use, but may not be printed for commercial purposes.
Transcript of ECONOMETRICS - University of Wisconsin–Madisonbhansen/econometrics/Econometrics2008.pdf ·...
ECONOMETRICS
Bruce E. Hansenc 2000, 20081
University of Wisconsin
www.ssc.wisc.edu/~bhansen
This Revision: January 17, 2008Comments Welcome
1This manuscript may be printed and reproduced for individual or instructional use, but may not beprinted for commercial purposes.
Contents
1 Introduction 11.1 Economic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Observational Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Economic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Regression and Projection 32.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Conditional Density and Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Regression Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Best Linear Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Least Squares Estimation 133.1 Random Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Normal Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Model in Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Residual Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 Bias and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.9 Gauss-Markov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.10 Semiparametric E¢ ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.11 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.12 In�uential Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.13 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Inference 314.1 Sampling Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Alternative Covariance Matrix Estimators . . . . . . . . . . . . . . . . . . . . . . . . 384.6 Functions of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.7 t tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.8 Con�dence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.9 Wald Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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4.10 F Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.11 Normal Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.12 Semiparametric E¢ ciency in the Projection Model . . . . . . . . . . . . . . . . . . . 464.13 Semiparametric E¢ ciency in the Homoskedastic Regression Model . . . . . . . . . . 484.14 Problems with Tests of NonLinear Hypotheses . . . . . . . . . . . . . . . . . . . . . 494.15 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.16 Estimating a Wage Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.17 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Additional Regression Topics 635.1 Generalized Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Testing for Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Forecast Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 NonLinear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5 Least Absolute Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.6 Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.7 Testing for Omitted NonLinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.8 Omitted Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.9 Irrelevant Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.10 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.11 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 The Bootstrap 816.1 De�nition of the Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 The Empirical Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3 Nonparametric Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.4 Bootstrap Estimation of Bias and Variance . . . . . . . . . . . . . . . . . . . . . . . 836.5 Percentile Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.6 Percentile-t Equal-Tailed Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.7 Symmetric Percentile-t Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.8 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.9 One-Sided Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.10 Symmetric Two-Sided Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.11 Percentile Con�dence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.12 Bootstrap Methods for Regression Models . . . . . . . . . . . . . . . . . . . . . . . . 916.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Generalized Method of Moments 947.1 Overidenti�ed Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.2 GMM Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.3 Distribution of GMM Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.4 Estimation of the E¢ cient Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . 967.5 GMM: The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.6 Over-Identi�cation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.7 Hypothesis Testing: The Distance Statistic . . . . . . . . . . . . . . . . . . . . . . . 987.8 Conditional Moment Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.9 Bootstrap GMM Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
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8 Empirical Likelihood 1058.1 Non-Parametric Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2 Asymptotic Distribution of EL Estimator . . . . . . . . . . . . . . . . . . . . . . . . 1078.3 Overidentifying Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.4 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.5 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.6 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9 Endogeneity 1129.1 Instrumental Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139.2 Reduced Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149.3 Identi�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.5 Special Cases: IV and 2SLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.6 Bekker Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179.7 Identi�cation Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
10 Univariate Time Series 12210.1 Stationarity and Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12210.2 Autoregressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12310.3 Stationarity of AR(1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.4 Lag Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.5 Stationarity of AR(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12510.7 Asymptotic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.8 Bootstrap for Autoregressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.9 Trend Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.10Testing for Omitted Serial Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.11Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.12Autoregressive Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.13Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11 Multivariate Time Series 13211.1 Vector Autoregressions (VARs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13211.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.3 Restricted VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.4 Single Equation from a VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.5 Testing for Omitted Serial Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 13411.6 Selection of Lag Length in an VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13411.7 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13511.8 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13511.9 Cointegrated VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
12 Limited Dependent Variables 13712.1 Binary Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13712.2 Count Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13812.3 Censored Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13912.4 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
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13 Panel Data 14213.1 Individual-E¤ects Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14213.2 Fixed E¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14213.3 Dynamic Panel Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
14 Nonparametrics 14514.1 Kernel Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14514.2 Asymptotic MSE for Kernel Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A Matrix Algebra 150A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150A.2 Matrix Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.5 Rank and Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.6 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154A.7 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.8 Positive De�niteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.9 Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156A.10 Kronecker Products and the Vec Operator . . . . . . . . . . . . . . . . . . . . . . . . 156A.11 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B Probability 159B.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159B.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.3 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.4 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162B.5 Common Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.6 Multivariate Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165B.7 Conditional Distributions and Expectation . . . . . . . . . . . . . . . . . . . . . . . . 167B.8 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.9 Normal and Related Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
C Asymptotic Theory 173C.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C.2 Weak Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174C.3 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176C.4 Asymptotic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
D Maximum Likelihood 178
E Numerical Optimization 182E.1 Grid Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182E.2 Gradient Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182E.3 Derivative-Free Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
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Chapter 1
Introduction
Econometrics is the study of estimation and inference for economic models using economicdata. Econometric theory concerns the study and development of tools and methods for appliedeconometric applications. Applied econometrics concerns the application of these tools to economicdata.
1.1 Economic Data
An econometric study requires data for analysis. The quality of the study will be largelydetermined by the data available. There are three major types of economic data sets: cross-sectional,time-series, and panel. They are distinguished by the dependence structure across observations.
Cross-sectional data sets are characterized by mutually independent observations. Surveys area typical source for cross-sectional data. The individuals surveyed may be persons, households, orcorporations.
Time-series data is indexed by time. Typical examples include macroeconomic aggregates,prices and interest rates. This type of data is characterized by serial dependence.
Panel data combines elements of cross-section and time-series. These data sets consist surveysof a set of individuals, repeated over time. Each individual (person, household or corporation) issurveyed on multiple occasions.
1.2 Observational Data
A common econometric question is to quantify the impact of one set of variables on anothervariable. For example, a concern in labor economics is the returns to schooling � the change inearnings induced by increasing a worker�s education, holding other variables constant. Anotherissue of interest is the earnings gap between men and women.
Ideally, we would use experimental data to answer these questions. To measure the returnsto schooling, an experiment might randomly divide children into groups, mandate di¤erent levelsof education to the di¤erent groups, and then follow the children�s wage path as they mature andenter the labor force. The di¤erences between the groups could be attributed to the di¤erent levelsof education. However, experiments such as this are infeasible, even immoral!
Instead, most economic data is observational. To continue the above example, what we observe(through data collection) is the level of a person�s education and their wage. We can measure thejoint distribution of these variables, and assess the joint dependence. But we cannot infer causality,as we are not able to manipulate one variable to see the direct e¤ect on the other. For example,a person�s level of education is (at least partially) determined by that person�s choices and theirachievement in education. These factors are likely to be a¤ected by their personal abilities andattitudes towards work. The fact that a person is highly educated suggests a high level of ability.This is an alternative explanation for an observed positive correlation between educational levels
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and wages. High ability individuals do better in school, and therefore choose to attain higher levelsof education, and their high ability is the fundamental reason for their high wages. The point isthat multiple explanations are consistent with a positive correlation between schooling levels andeducation. Knowledge of the joint distibution cannot distinguish between these explanations.
This discussion means that causality cannot be infered from observational data alone. Causalinference requires identi�cation, and this is based on strong assumptions. We will return to adiscussion of some of these issues in Chapter 9.
1.3 Economic Data
Fortunately for economists, the development of the internet has provided a convenient forum fordissemination of economic data. Many large-scale economic datasets are available without chargefrom governmental agencies. An excellent starting point is the Resources for Economists DataLinks, available at http://rfe.wustl.edu/Data/index.html.
Some other excellent data sources are listed below.Bureau of Labor Statistics: http://www.bls.gov/Federal Reserve Bank of St. Louis: http://research.stlouisfed.org/fred2/Board of Governors of the Federal Reserve System: http://www.federalreserve.gov/releases/National Bureau of Economic Research: http://www.nber.org/US Census: http://www.census.gov/econ/www/Current Population Survey (CPS): http://www.bls.census.gov/cps/cpsmain.htmSurvey of Income and Program Participation (SIPP): http://www.sipp.census.gov/sipp/Panel Study of Income Dynamics (PSID): http://psidonline.isr.umich.edu/U.S. Bureau of Economic Analysis: http://www.bea.doc.gov/CompuStat: http://www.compustat.com/www/International Financial Statistics (IFS): http://ifs.apdi.net/imf/
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Chapter 2
Regression and Projection
2.1 Variables
The most commonly applied econometric tool is regression. This is used when the goal is toquantify the impact of one set of variables (the regressors, conditioning variable, or covariates)on another variable (the dependent variable). We let y denote the dependent variable and(x1; x2; :::; xk) denote the k regressors. It is convenient to write the set of regressors as a vector inRk :
x =
1CCCA : (2.1)
Following mathematical convention, real numbers (elements of the real line R) are written usinglower case italics such as y, and vectors (elements of Rk) by lower case bold italics such as x: Uppercase bold italics such as X will be used for matrices.
The random variables (y;x) have a distribution F which we call the population. This �popu-lation�is in�nitely large. This abstraction can be a source of confusion as it does not correspond toa physical population in the real world. The distribution F is unknown, and the goal of statisticalinference is to learn about features of F from the sample.
For most of our analysis it is unimportant whether the observations y and x come from con-tinuous or discrete distributions. For example, many regressors in econometric practice are binary,taking on only the values 0 and 1. Binary variables are often called dummy variables.
2.2 Conditional Density and Mean
To study how the distribution of y varies with the variables x in the population, we start withf (y j x) ; the conditional density of y given x:
To illustrate, Figure 2.1 displays the density1 of hourly wages for men and women, from thepopulation of white non-military wage earners with a college degree and 10-15 years of potentialwork experience. These are conditional density functions �the density of hourly wages conditionalon race, gender, education and experience. The two density curves show the e¤ect of gender on thedistribution of wages, holding the other variables constant.
While it is easy to observe that the two densities are unequal, it is useful to have numericalmeasures of the di¤erence. An important summary measure is the conditional mean
m (x) = E (y j x) =Z 1
�1yf (y j x) dy: (2.2)
1These are nonparametric density estimates using a Gaussian kernel with the bandwidth selected by cross-validation. See Chapter 14. The data are from the 2004 Current Population Survey.
3
Figure 2.1: Wage Densities for White College Grads with 10-15 Years Work Experience
In general, m (x) can take any form, and exists so long as E jyj <1: In the example presented inFigure 2.1, the mean wage for men is $27.22, and that for women is $20.73. These are indicated inFigure 2.1 by the arrows drawn to the x-axis.
Take a closer look at the density functions displayed in Figure 2.1. You can see that the right tailof the density is much thicker than the left tail. These are asymmetric (skewed) densities, which isa common feature of wage distributions. When a distribution is skewed, the mean is not necessarilya good summary of the central tendency. In this context it is often convenient to transform thedata by taking the (natural) logarithm. Figure 2.2 shows the density of log hourly wages for thesame population, with mean log hourly wages drawn in with the arrows. The di¤erence in the logmean wage between men and women is 0.30, which implies a 30% average wage di¤erence for thispopulation. This is a more robust measure of the typical wage gap between men and women thanthe di¤erence in the untransformed wage means. For this reason, wage regressions typically use logwages as a dependent variable rather than the level of wages.
The comparisons in Figures 2.1 and 2.2 are facilitated by the fact that the control variable(gender) is binary. When the distribution of the control variable takes on multiple values oris continuous, then comparisons become more complicated. To illustrate, Figure 2.3 displays ascatter plot2 of log wages against education levels. Assuming for simplicity that this is the truejoint distribution, the solid line displays the conditional expectation of log wages varying witheducation. The conditional expectation function is close to linear; the dashed line is a linearprojection approximation which will be discussed in the Section 2.6. The main point to be learnedfrom Figure 2.3 is that the conditional expectation describes the central tendency of the conditionaldistribution. Of particular interest to graduate students may be the observation that di¤erencebetween a B.A. and a Ph.D. degree in mean log hourly wages is 0.36, implying an average 36%di¤erence in wage levels.
2White non-military male wage earners with 10-15 years of potential work experience.
4
Figure 2.2: Log Wage Densities
2.3 Regression Equation
The regression error e is de�ned to be the di¤erence between y and its conditional mean (2.2)evaluated at the observed value of x:
e = y �m(x):
By construction, this yields the formula
y = m(x) + e: (2.3)
Theorem 2.3.1 Properties of the regression error e:
1. E (e j x) = 0:
2. E(e) = 0:
3. E (h(x)e) = 0 for any function h (�) :
4. E(xe) = 0:
To show the �rst statement, by the de�nition of e and the linearity of conditional expectations,
E (e j x) = E ((y �m(x)) j x)= E (y j x)� E (m(x) j x)= m(x)�m(x)= 0:
Proofs of the remaining parts of the Theorem are left as an exercise.The equations
y = m(x) + e
E (e j x) = 0:
5
Figure 2.3: Conditional Mean of Wages Given Education
are often stated jointly as the regression framework. It is important to understand that this is aframework, not a model, because no restrictions have been placed on the joint distribution of thedata. These equations hold true by de�nition. A regression model imposes further restrictions onthe permissible class of regression functions m (x) :
The conditional mean has the property of being the the best predictor of y in the sense ofachieving the lowest mean squared error. To see this, let g (x) be an arbitrary predictor of y givenx: The expected squared error using this prediction function is
E (y � g (x))2 = E(e+m (x)� g (x))2
= Ee2 + 2E (e (m (x)� g (x))) + E (m (x)� g (x))2
= Ee2 + E(m (x)� g (x))2
� Ee2
where the third equality uses Theorem 2.3.1.3. The right-hand-side is minimized by setting g (x) =m (x) : Thus the mean squared error is minimized by the conditional mean.
2.4 Conditional Variance
While the conditional mean is a good measure of the location of a conditional distribution,it does not provide information about the spread of the distribution. A common measure of thedispersion is the conditional variance
�2(x) = var (y j x) = E�e2 j x
�:
Generally, �2 (x) is a non-trivial function of x and can take any form subject to the restriction thatit is non-negative. The conditional standard deviation is its square root �(x) =
p�2(x):
In the special case where �2(x) is a constant and independent of x so that
E�e2 j x
�= �2 (2.4)
we say that the error e is homoskedastic. In the general case where �2(x) depends on x we saythat the error e is heteroskedastic.
6
Some textbooks describe heteroskedasticity as the case where �the variance of e varies acrossobservations�. This concept is less helpful than de�ning heteroskedasticity as the dependence ofthe conditional variance �2 (x) on the observables x:
As an example, take the conditional wage densities displayed in Figure 2.1. The conditionalstandard deviation for men is 12.1 and that for women is 10.5. So while men have higher averagewages, they are also somewhat more dispersed. Thus the wage distribution is heteroskedastic �itsvairance depends on observables.
2.5 Linear Regression
An important special case of (2.3) is when the conditional mean function m (x) is linear in x(or linear in functions of x): Notationally it is convenient to augment the regressor vector x bylisting the number �1�as an element. We call this the �constant�or �intercept�. Equivalently, weassume that x1 = 1, where x1 is the �rst element of the vector x de�ned in (2.1). Thus (2.1) hasbeen rede�ned as the k � 1 vector
x =
1CCCA : (2.5)
When m(x) is linear in x; we can write the mean equation as
m(x) = x0� = �1 + x2i�2 + � � �+ xki�k (2.6)
where
� =
0B@ �1...�k
1CA (2.7)
is a k � 1 parameter vector.In this case (2.3) can be written as
y = x0� + e (2.8)
E (e j x) = 0: (2.9)
Equation (2.8) is called the linear regression model,An important special case is homoskedastic linear regression model
y = x0� + e (2.10)
E (e j x) = 0 (2.11)
E�e2 j x
�= �2: (2.12)
2.6 Best Linear Predictor
While the conditional mean m(x) = E (y j x) is the best predictor of y among all functions ofx; its functional form is typically unknown, and the linear assumption of the previous section isempirically unlikely to be accurate. Instead, it is more realistic to view the linear speci�cation (2.6)as an approximation. We derive an appropriate approximation in this section.
A linear predictor for y given x is any linear function of the form x0� where � 2 Rk. If � isselected arbitrarily then x0� will likely be a poor predictor for y: We can measure the accuracy ofthe predictor by the expected squared prediction error
S(�) = E�y � x0�
�2= Ey2 � 2�0E (xy) + �E
�xx0
��
7
(or mean squared error (MSE)). There is a unique � which minimizes this expression. The corre-sponding linear predictor x0� has the property that it has the lowest expected squared predictionerror S(�) in the class of all linear predictors. We de�ne this as the best linear predictor of ygiven x, and this de�nes the linear projection model.
The MSE can be written out as
S(�) = Ey2 � 2�0E (xy) + �E�xx0
��
which is a quadratic function of �: The �rst-order condition for minimization (from Appendix A.9)is
0 =@
@�S(�) = �2E (xy) + 2E
�xx0��:
Solving for � we �nd� =
�E�xx0���1
E (xy) : (2.13)
This is the unique best linear predictor.It is worth taking the time to understand the notation involved in this expression. E (xx0) is a
k� k matrix and E (xy) is a k� 1 column vector. Therefore, alternative expressions such as E(xy)E(xx0)
or E (xy) (E (xx0))�1 are incoherent and incorrect. Appendix A provides a comprehensive reviewof matrix notation and operation.
The vector (2.13) exits and is unique as long as the k�k matrix Q = E(xx0) is invertible. Thematrix Q plays an important role in least-squares theory so we will discuss some of its propertiesin detail. Observe that for any non-zero � 2 Rk;
�0Q� = E��0xx0�
�= E
��0x
�2 � 0so Q by construction is positive semi-de�nite. It is invertible if and only if it is positive de�nite,which requires that for all non-zero �; E (�0x)2 > 0: Equivalently, there cannot exist a non-zerovector � such that �0x = 0 identically. This occurs when redundant variables are included in x:In order for � to be uniquely de�ned, this situation must be excluded.
Given the de�nition of � in (2.13), x0� is the best linear predictor for y: The error is
e = y � x0�: (2.14)
Notice that the error e from the linear prediction equation is equal to the error from the regressionequation when (and only when) the conditional mean is linear in x; otherwise they are distinct.
Rewriting, we obtain a decomposition of y into linear predictor and error
y = x0� + e: (2.15)
This completes the derivation of the model. We call x0� alternatively the best linear predictor of ygiven x; or the linear projection of y onto x: In general we call equation (2.15) the linear projectionmodel.
We now summarize the assumptions necessary for its derivation and list the implications inTheorem 2.6.1.
Assumption 2.6.1
1. x contains an intercept;
2. Ey2 <1;
3. Ex2j <1 for j = 1; :::; k:
4. Q = E(xx0) is invertible.
8
Theorem 2.6.1 Under Assumption 2.6.1, (2.13) and (2.14) are well de�ned. Furthermore,
E�e2�<1 (2.16)
E (xe) = 0 (2.17)
andE (e) = 0: (2.18)
A complete proof of Theorem (2.6.1) is presented in Section 2.7.It is useful to note that the facts that E (xe) = 0 and E (e) = 0 means that the variables x and
e are uncorrelated.The two equations (2.15) and (2.17) summarize the linear projection model. Let�s compare
it with the linear regression model (2.8)-(2.9). Since from Theorem 2.3.1.4 we know that theregression error has the property E (xe) = 0; it follows that linear regression is a special case of theprojection model. However, the converse is not true as the projection error does not necessarilysatisfy E (e j x) = 0:
To see this in a simple example, suppose we take a normally distributed random variablex � N(0; 1) and set y = x2: Consider the linear projection of y on x and an intercept:
� =
�1 E (x)
E (x) E�x2� ��1� E (y)
E (xy)
�=
�1 E (x)
E (x) E�x2� ��1� E
�x2�
E�x3� �
=
�10
�Thus the linear projection equation takes the form
y = �1 + x�2 + e
where �1 = 1, �2 = 0 and e = y � 1 = x2 � 1: Observe that E (e) = E�x2�� 1 = 0 and
E (xe) = E�x3�� E (e) = 0; yet E (e j x) = x2 � 1 6= 0: In this simple example e is a deterministic
function of x; yet e and x are uncorrelated!The conditions listed in Assumption 2.6.1 are weak. The �nite second moment Assumptions
2.6.1.2 and 2.6.1.3 are called regularity conditions. Assumption 2.6.1.4 is required to ensure that� is uniquely de�ned. Assumption 2.6.1.1 is employed to guarantee that (2.18) holds.
We have shown that under mild regularity conditions for any pair (y;x) we can de�ne a linearequation (2.15) with the properties listed in Theorem 2.6.1. No additional assumptions are required.However, it is important to not misinterpret the generality of this statement. The linear equation(2.15) is de�ned by the de�nition of the best linear predictor and the coe¢ cient de�nition (2.13).In contrast, in many economic models the parameter � may be de�ned within the model. In thiscase (2.13) may not hold and the implications of Theorem 2.6.1 may be false. These structuralmodels require alternative estimation methods, and are discussed in Chapter 9.
Returning to the joint distribution displayed in Figure 2.3, the dashed line is projection of logwages onto education. In this example the linear predictor is a close approximation to the condi-tional mean. In other cases the two may be quite di¤erent. Figure 2.4 displays the relationship3
between mean log hourly wages and labor market experience. The solid line is the conditionalmean, and the straight dashed line is the linear projection. In this case the linear projection is apoor approximation to the conditional mean. It over-predicts wages for young and old workers, andunder-predicts for the rest. Most importantly, it misses the strong downturn in expected wages forthose above 35 years work experience (equivalently, for those over 53 in age).
3 In the population of Caucasian non-military male wage earners with 12 years of education.
9
Figure 2.4: Hourly Wage as a Function of Experience
This defect in the best linear predictor can be partially corrected through a careful selection ofregressors. In the example just presented, we can augment the regressor vector x to include bothexperience and experience2: The best linear predictor of log wages given these two variables canbe called a quadratic projection, since the resulting function is quadratic in experience: Other thanthe rede�nition of the regressor vector, there are no changes in our methods or analysis. In Figure2.4 we display as well the quadratic projection. In this example it is a much better approximationto the conditional mean than the linear projection.
Another defect of linear projection is that it is sensitive to the marginal distribution of theregressors when the conditional mean is non-linear. We illustrate the issue in Figure 2.5 for aconstructed4 joint distribution of y and x. The solid line is the non-linear conditional mean ofy given x: The data are divided in two �Group 1 and Group 2 �which have di¤erent marginaldistributions for the regressor x; and Group 1 has a lower mean value of x than Group 2. Theseparate linear projections of y on x for these two groups are displayed in the Figure by the dashedlines. These two projections are distinct approximations to the conditional mean. A defect withlinear projection is that it leads to the incorrect conclusion that the e¤ect of x on y is di¤erent forindividuals in the two Groups. This conclusion is incorrect because in fact there is no di¤erence inthe conditional mean function. The apparant di¤erence is a by-product of a linear approximationto a non-linear mean, combined with di¤erent marginal distributions for the conditioning variables.
2.7 Technical Proofs
Proof of Theorem 2.6.1. We �rst show that the moments E (xy) and E (xx0) are �nite and wellde�ned. First, it is useful to note that Assumption 2.6.1.3 implies that
E kxk2 = E�x0x�=
kXj=1
Ex2j <1: (2.19)
4The xi in Group 1 are N(2; 1) and those in Group 2 are N(4; 1); and the conditional distriubtion of y given x isN(m(x); 1) where m(x) = 2x� x2=6:
10
Figure 2.5: Conditional Mean and Two Linear Projections
Note that for j = 1; :::; k; by the Cauchy-Schwarz Inequality (C.3) and Assumptions 2.6.1.2 and2.6.1.3
E jxjyj ��Ex2j
�1=2 �Ey2
�1=2<1:
Thus the elements in the vector E (xy) are well de�ned and �nite. Next, note that the jl�th elementof E (xx0) is E (xjxl) : Observe that
E jxjxlj ��Ex2j
�1=2 �Ex2l
�1=2<1
under Assumption 2.6.1.3. Thus all elements of the matrix E (xx0) are �nite.Equation (2.13) states that � = (E (xx0))�1 E (xy) which is well de�ned since (E (xx0))�1 exists
under Assumption 2.6.1.4. It follows that e = y � x0� as de�ned in (2.14) is also well de�ned.Note the Schwarz Inequality (A.7) implies (x0�)2 � kxk2 k�k2 and therefore combined with
(2.19) we see thatE�x0�
�2 � E kxk2 k�k2 <1: (2.20)
Using Minkowski�s Inequality (C.5), Assumption 2.6.1.2 and (2.20) we �nd�E�e2��1=2
=�E�y � x0�
�2�1=2�
�Ey2
�1=2+�E�x0�
�2�1=2< 1
establishing (2.16).An application of the Cauchy-Schwarz Inequality (C.3) shows that for any j
E jxjej ��Ex2j
�1=2 �Ee2�1=2
<1
and therefore the elements in the vector E (xe) are well de�ned and �nite.Using the de�nitions (2.14) and (2.13), and the matrix properties that AA�1 = I and Ia = a;
E (xe) = E�x�y � x0�
��= E(xy)� E
�xx0� �E�xx0���1
E (xy) = 0:
Finally, equation (2.18) follows from (2.17) and Assumption 2.6.1.1. �
11
2.8 Exercises
1. Prove parts 2, 3 and 4 of Theorem 2.3.1.
2. Suppose that the random variables y and x only take the values 0 and 1, and have thefollowing joint probability distribution
x = 0 x = 1
y = 0 .1 .2y = 1 .4 .3
Find E (y j x) ; E�y2 j x
�and var (y j x) for x = 0 and x = 1:
3. Suppose that y is discrete-valued, taking values only on the non-negative integers, and theconditional distribution of y given x is Poisson:
P (y = k j x) = exp (�x0�) (x0�)j
j!; j = 0; 1; 2; :::
Compute E (y j x) and var (y j x) : Does this justify a linear regression model of the formy = x0� + e?
Hint: If P (y = j) = exp(��)�jj! ; then Ey = � and var(y) = �:
4. Let x and y have the joint density f (x; y) = 32
�x2 + y2
�on 0 � x � 1; 0 � y � 1: Compute
the coe¢ cients of the best linear predictor y = �1 + �2x+ e: Compute the conditional meanm(x) = E (y j x) : Are they di¤erent?
5. Take the bivariate linear projection model
y = �1 + �2x+ e
E (e) = 0
E (xe) = 0
De�ne �y = Ey; �x = Ex; �2x = var(x); �2y = var(y) and �xy = cov(x; y): Show that�2 = �xy=�
2x and �1 = �y � �2�x:
6. True or False. If y = x� + e; x 2 R; and E (e j x) = 0; then E�x2e�= 0:
7. True or False. If y = x0� + e and E (e j x) = 0; then e is independent of x:
8. True or False. If y = x0� + e, E (e j x) = 0; and E�e2 j x
�= �2; a constant, then e is
independent of x:
9. True or False. If y = x� + e; x 2 R; and E (xiei) = 0; then E�x2e�= 0:
10. True or False. If y = x0� + e and E(xe) = 0; then E (e j x) = 0:
11. Let x be a random variable with � = Ex and �2 = var(x): De�ne
g�x j �; �2
�=
�x� �
(x� �)2 � �2�:
Show that Eg (x j m; s) = 0 if and only if m = � and s = �2:
12
Chapter 3
Least Squares Estimation
3.1 Random Sample
In a typical application, an econometrician has a set of observed measurements on a set ofvariables for a group of individuals. These individuals may be persons, households, �rms or othereconomic agents. We call this information the data, dataset, or sample. We denote the numberof individuals in the dataset by the natural number n, and call this the number of observations.Each observation consists of a set of measurements on a list of variables. Ideally all variables aremeasured for all observations, but in some cases some variables are not measured, and describethese variables as unobserved or missing. The variables include the dependent variable y 2 R andthe regressor x 2 Rk:
We use the index i to indicate the i�th individual in the dataset. The observation for the i�thindividual will be written as the pair (yi;xi) : yi is the observed value of y for individual i and xiis the observed value of x for the same individual.
If the data is cross-sectional (each observation is a di¤erent individual) it is often reasonable toassume the observations are mutually independent. This means that the pair (yi;xi) is independentof (yj ;xj) for i 6= j. (Sometimes the label �independent� is misconstrued. It is not a statementabout the relationship between yi and xi:) Furthermore, if the data is randomly gathered, it isreasonable to model each observation as a random draw from the same probability distribution. Inthis case we say that the data are independent and identically distributed or iid. We callthis a random sample.
Assumption 3.1.1 The observations (yi;xi) i = 1; :::; n; are mutually independent across obser-vations i and identically distributed.
In Chapter 2 we derived and discussed the best linear predictor of y given x for a pair of randomvariables (y;x) 2 R�Rk; and called this the linear projection model. Applied to observations froma random sample the linear projection model takes the form
yi = x0i� + ei (3.1)
E (xiei) = 0 (3.2)
� =�E�xix
0i
���1E (xiyi) : (3.3)
This chapter explores estimation and inference in this context.In Sections 3.8 and 3.9, we narrow the focus to the linear regression model, but for most of the
chapter we retain the broader focus on the projection model.
13
3.2 Estimation
Equation (3.3) writes the projection coe¢ cient � as an explicit function of population momentsE (xiyi) and E (xix0i) : Their moment estimators are the sample moments
bE (xiyi) =1
n
nXi=1
xiyi
bE �xix0i� =1
n
nXi=1
xix0i:
The moment estimator of � replaces the population moments in (3.3) with the sample moments:
� =�bE �xix0i���1 bE (xiyi)
=
1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiyi
!
=
nXi=1
xix0i
!�1 nXi=1
xiyi
!(3.4)
Another way to derive � is as follows. Observe that (3.2) can be written in the parametric formg (�) = E (xi (yi � x0i�)) = 0: The function g (�) can be estimated by
g (�) =1
n
nXi=1
xi�yi � x0i�
�:
This is a set of k equations which are linear in �: The estimator � is the value which jointly setsthese equations equal to zero:
0 = g���
(3.5)
=1
n
nXi=1
xi
�yi � x0i�
�=
1
n
nXi=1
xiyi �1
n
nXi=1
xix0i�
whose solution is (3.4).To illustrate, consider the data used to generate Figure 2.3. These are white male wage earners
from the March 2004 Current Population Survey, excluding military, with 10-15 years of potentialwork experience. This sample has 988 observations. Let yi be log wages and xi be an interceptand years of education. Then
1
n
nXi=1
xiyi =
�2:9542:40
�1
n
nXi=1
xix0i =
�1 14:14
14:14 205:83
�:
14
Thus
� =
�1 14:14
14:14 205:83
��1�2:9542:40
�
=
�34:94 �2: 40�2: 40 0:170
��2:9542:40
�
=
�1: 3130:128
�:
We often write the estimated equation using the format
\log(Wage) = 1:313 + 0:128 Education:
An interpretation of the estimated equation is that each year of education is associated with a12.8% increase in mean wages.
3.3 Least Squares
Least squares is another classic motivation for the estimator (3.4). De�ne the sum-of-squarederrors (SSE) function
Sn(�) =
nXi=1
�yi � x0i�
�2=
nXi=1
y2i � 2�0nXi=1
xiyi + �0nXi=1
xix0i�:
This is a quadratic function of �: To visualize this function, Figure 3.1 displays an example sum-of-squared errors function Sn(�) for the case k = 2:
The Ordinary Least Squares (OLS) estimator is the value of � which minimizes Sn(�):Matrix calculus (see Appendix A.9) gives the �rst-order conditions for minimization:
0 =@
@�Sn(�)
= �2nXi=1
xiyi + 2nXi=1
xix0i�
whose solution is (3.4). Following convention we will call � the OLS estimator of �:As a by-product of OLS estimation, we de�ne the predicted value
yi = x0i�
and the residual
ei = yi � yi= yi � x0i�:
Note that yi = yi + ei: It is important to understand the distinction between the error ei and theresidual ei: The error is unobservable while the residual is a by-product of estimation. These twovariables are frequently mislabeled, which can cause confusion.
Equation (3.5) implies that1
n
nXi=1
xiei = 0:
15
Figure 3.1: Sum-of-Squared Errors Function
Since xi contains a constant, one implication is that
1
n
nXi=1
ei = 0:
Thus the residuals have a sample mean of zero and the sample correlation between the regressorsand the residual is zero. These are algebraic results, and hold true for all linear regression estimates.
The error variance �2 = Ee2i is also a parameter of interest. It measures the variation in the�unexplained�part of the regression. Its method of moments estimator is the sample average ofthe squared residuals
�2 =1
n
nXi=1
e2i : (3.6)
An alternative estimator uses the formula
s2 =1
n� k
nXi=1
e2i : (3.7)
A justi�cation for the latter choice will be provided in Section 3.8.A measure of the explained variation relative to the total variation is the coe¢ cient of de-
termination or R-squared.
R2 =
Pni=1 (yi � y)
2Pni=1 (yi � y)
2 = 1��2
�2y
where
�2y =1
n
nXi=1
(yi � y)2
16
is the sample variance of yi: The R2 is frequently mislabeled as a measure of ��t�. It is aninappropriate label as the value of R2 does not help interpret the parameter estimates � or teststatistics concerning �. Instead, it should be viewed as an estimator of the population parameter
�2 =var (x0i�)
var(yi)= 1� �
2
�2y
where �2y = var(yi): An alternative estimator of �2 proposed by Theil called �R-bar-squared�is
R2= 1� s2
~�2y
where
~�2y =1
n� 1
nXi=1
(yi � y)2 :
Theil�s estimator R2is a ratio of adjusted variance estimators, and therefore is expected to be a
better estimator of �2 than the unadjusted estimator R2:
3.4 Normal Regression Model
Another motivation for the least-squares estimator can be obtained from the normal regressionmodel. This is the linear regression model with the additional assumption that the error ei isindependent of xi and has the distribution N
�0; �2
�: This is a parametric model, where likelihood
methods can be used for estimation, testing, and distribution theory.The log-likelihood function for the normal regression model is
logL(�; �2) =
nXi=1
log
1
(2��2)1=2exp
�� 1
2�2�yi � x0i�
�2�!
= �n2log�2��2
�� 1
2�2Sn(�)
The MLE (�; �2) maximize logL(�; �2): Since logL(�; �2) is a function of � only through the sumof squared errors Sn(�); maximizing the likelihood is identical to minimizing Sn(�). Hence theMLE for � equals the OLS estimator:
Plugging � into the log-likelihood we obtain
logL��; �2
�= �n
2log�2��2
�� 1
2�2
nXi=1
e2i :
Maximization with respect to �2 yields the �rst-order condition
@
@�2logL
��; �2
�= � n
2�2+
1
2��2�2 nX
i=1
e2i = 0:
Solving for �2 yields the method of moments estimator (3.6). Thus the MLE��; �2
�for the normal
regression model are identical to the method of moment estimators. Due to this equivalence, theOLS estimator � is frequently referred to as the Gaussian MLE.
17
3.5 Model in Matrix Notation
For many purposes, including computation, it is convenient to write the model and statistics inmatrix notation. The linear equation (2.15) is a system of n equations, one for each observation.We can stack these n equations together as
y1 = x01� + e1
y2 = x02� + e2...
yn = x0n� + en:
Now de�ne
y =
1CCCA ; X =
1CCCA ; e =
1CCCA :Observe that y and e are n� 1 vectors, and X is an n� k matrix. Then the system of n equationscan be compactly written in the single equation
y =X� + e:
Sample sums can also be written in matrix notation. For example
nXi=1
xix0i = X 0X
nXi=1
xiyi = X 0y:
Thus the estimator (3.4), residual vector, and sample error variance can be written as
� =�X 0X
��1 �X 0y
�e = y �X��2 = n�1e0e:
A useful result is obtained by inserting y =X� + e into the formula for � to obtain
� =�X 0X
��1 �X 0 (X� + e)
�=
�X 0X
��1X 0X� +
�X 0X
��1 �X 0e
�= � +
�X 0X
��1X 0e: (3.8)
3.6 Projection Matrices
De�ne the matricesP =X
�X 0X
��1X 0
and
M = In �X�X 0X
��1X 0
= In � P
18
where In is the n � n identity matrix. P and M are called projection matrices due to theproperty that for any matrix Z which can be written as Z =X� for some matrix � (we say thatZ lies in the range space of X); then
PZ = PX� =X�X 0X
��1X 0X� =X� = Z
andMZ = (In � P )Z = Z � PZ = Z �Z = 0:
As an important example of this property, partition the matrix X into two matrices X1 andX2 so that
X = [X1 X2] :
Then PX1 = X1 and MX1 = 0: It follows that MX = 0 and MP = 0; so M and P areorthogonal.
The matrices P andM are symmetric and idempotent1. To see that P is symmetric,
P 0 =�X�X 0X
��1X 0�0
=�X 0�0 ��X 0X
��1�0(X)0
= X��X 0X
�0��1X 0
= X�(X)0
�X 0�0��1X 0
= P :
To establish that it is idempotent,
PP =�X�X 0X
��1X 0��X�X 0X
��1X 0�
= X�X 0X
��1X 0X
�X 0X
��1X 0
= X�X 0X
��1X 0
= P :
Similarly,M 0 = (In � P )0 = In � P =M
and
MM = M (In � P )= M �MP
= M;
sinceMP = 0:Another useful property is that
trP = k (3.9)
trM = n� k (3.10)
1A matrix P is symmetric if P 0 = P : A matrix P is idempotent if PP = P: See Appendix A.8
19
(See Appendix A.4 for de�nition and properties of the trace operator.) To show (3.9) and (3.10),
trP = tr�X�X 0X
��1X 0�
= tr��X 0X
��1X 0X
�= tr (Ik)
= k;
andtrM = tr (In � P ) = tr (In)� tr (P ) = n� k:
Given the de�nitions of P andM ; observe that
y =X� =X�X 0X
��1X 0y = Py
ande = y �X� = y � Py =My: (3.11)
Furthermore, since y =X� + e andMX = 0; then
e =M (X� + e) =Me: (3.12)
Another way of writing (3.11) is
y = (P +M)y = Py +My = y + e:
This decomposition is orthogonal, that is
y0e = (Py)0 (My) = y0PMy = 0:
3.7 Residual Regression
PartitionX = [X1 X2]
and
� =
��1�2
�:
Then the regression model can be rewritten as
y =X1�1 +X2�2 + e: (3.13)
Observe that the OLS estimator of � = (�01;�02)0 can be obtained by regression of y on X = [X1
X2]: OLS estimation can be written as
y =X1�1 +X2�2 + e (3.14)
Suppose that we are primarily interested in �2; not in �1; so we are only interested in obtainingthe OLS sub-component �2: In this section we derive an alternative expression for �2 which doesnot involve estimation of the full model.
De�neM1 = In �X1
�X 01X1
��1X 01:
Recalling the de�nitionM = I �X (X 0X)�1X 0; observe that X 01M1 = 0 and thus
M1M =M �X1
�X 01X1
��1X 01M =M :
20
It follows thatM1e =M1Me =Me = e:
Using this result, if we premultiply (3.14) byM1 we obtain
M1y = M1X1�1 +M1X2�2 +M1e
= M1X2�2 + e (3.15)
the second equality since M1X1 = 0. Premultiplying by X 02 and recalling that X
02e = 0; we
obtainX 02M1y = X
02M1X2�2 +X
02e =X
02M1X2�2:
Solving,�2 =
�X 02M1X2
��1 �X 02M1y
�an alternative expression for �2:
Now, de�ne
~X2 = M1X2 (3.16)
~y = M1y; (3.17)
the least-squares residuals from the regression of X2 and y; respectively, on the matrix X1 only.Since the matrixM1 is idempotent,M1 =M1M1 and thus
�2 =�X 02M1X2
��1 �X 02M1y
�=
�X 02M1M1X2
��1 �X 02M1M1y
�=
�~X02~X2
��1 �~X02~y�:
This shows that �2 can be calculated by the OLS regression of ~y on ~X2: This technique is calledresidual regression.
Furthermore, using the de�nitions (3.16) and (3.17), expression (3.15) can be equivalently writ-ten as
~y = ~X2�2 + e:
Since �2 is precisely the OLS coe¢ cient from a regression of ~y on ~X2; this shows that the residualvector from this regression is e, numerically the same residual as from the joint regression (3.14).We have proven the following theorem.
Theorem 3.7.1 (Frisch-Waugh-Lovell). In the model (3.13), the OLS estimator of �2 and theOLS residuals e may be equivalently computed by either the OLS regression (3.14) or via the fol-lowing algorithm:
1. Regress y on X1; obtain residuals ~y;
2. Regress X2 on X1; obtain residuals ~X2;
3. Regress ~y on ~X2; obtain OLS estimates �2 and residuals e:
In some contexts, the FWL theorem can be used to speed computation, but in most casesthere is little computational advantage to using the two-step algorithm. Rather, the primary useis theoretical.
A common application of the FWL theorem, which you may have seen in an introductoryeconometrics course, is the demeaning formula for regression. Partition X = [X1 X2] whereX1 = � is a vector of ones, and X2 is the vector of observed regressors. In this case,
M1 = I � ���0���1
�0:
21
Observe that
~X2 = M1X2
= X2 � ���0���1
�X2
= X2 �X2
and
~y = M1y
= y � ���0���1
�0y
= y � y;
which are �demeaned�. The FWL theorem says that �2 is the OLS estimate from a regression ofyi � y on x2i � x2 :
�2 =
nXi=1
(x2i � x2) (x2i � x2)0!�1 nX
i=1
(x2i � x2) (yi � y)!:
Thus the OLS estimator for the slope coe¢ cients is a regression with demeaned data.
3.8 Bias and Variance
In this and the following section we consider the special case of the linear regression model(2.8)-(2.9). In this section we derive the small sample conditional mean and variance of the OLSestimator.
By the independence of the observations and (2.9), observe that
E (e jX) =
0BB@...
E (ei jX)...
1CCA =
0BB@...
E (ei j xi)...
1CCA = 0: (3.18)
Using (3.8), the properties of conditional expectations, and (3.18), we can calculate
E�� � � jX
�= E
��X 0X
��1X 0e jX
�=
�X 0X
��1X 0E (e jX)
= 0:
We have shown thatE�� jX
�= � (3.19)
which implies
E���= �
and thus the OLS estimator � is unbiased for �:Next, for any random vector Z de�ne the covariance matrix
var(Z) = E (Z � EZ) (Z � EZ)0
= EZZ 0 � (EZ) (EZ)0
and for any pair (Z;X) de�ne the conditional convariance matrix
var(Z jX) = E�(Z � E (Z jX)) (Z � E (Z jX))0 jX
�:
22
Then given (3.19) we see that
var�� jX
�= E
��� � �
��� � �
�0jX�
=�X 0X
��1X 0DX
�X 0X
��1where
D = E�ee0 jX
�:
The i�th diagonal element of D is
E�e2i jX
�= E
�e2i j xi
�= �2i
while the ij0th o¤-diagonal element of D is
E (eiej jX) = E (ei j xi) E (ej j xj) = 0:Thus D is a diagonal matrix with i�th diagonal element �2i :
D = diag��21; :::; �
2n
�=
0BBB@�21 0 � � � 00 �22 � � � 0...
.... . .
...0 0 � � � �2n
1CCCA : (3.20)
It is useful to note that
X 0DX =
nXi=1
xix0i�2i ;
a weighted version of X 0X.In the special case of the linear homoskedastic regression model, �2i = �2 and we have the
simpli�cations D = In�2, X 0DX =X 0X�2; and
var�� jX
�=�X 0X
��1�2:
We now calculate the �nite sample bias of the method of moments estimator �2 for �2: From(3.12), the properties of projection matrices and the trace operator observe that
�2 =1
ne0e =
1
ne0MMe =
1
ne0Me =
1
ntr�e0Me
�=1
ntr�Mee0
�:
Then
E��2 jX
�=
1
ntr�E�Mee0 jX
��=
1
ntr�ME
�ee0 jX
��=
1
ntr (MD) : (3.21)
Adding the assumption of conditional homoskedasticity E�e2i j xi
�= �2; then D = In�
2 so thissimpli�es to
E��2 jX
�=
1
ntr�M�2
�= �2
�n� kn
�;
the �nal equality by (3.10). Thus �2 is biased towards zero. As an alternative, the estimator s2
de�ned in (3.7) is unbiased for �2 by this calculation. This is the justi�cation for the commonpreference of s2 over �2 in empirical practice. It is important to remember, however, that thisestimator is only unbiased in the special case of the homoskedastic linear regression model. It isnot unbiased in the absence of homoskedasticity or in the projection model.
23
3.9 Gauss-Markov Theorem
In this section we restrict attention to the homoskedastic linear regression model, which is(2.10)-(2.12). Now consider the class of estimators of � which are linear functions of the vector y;and thus can be written as
~� = A0y
where A is an n � k function of X. The least-squares estimator is the special case obtained bysetting A =X(X 0X)�1:What is the best choice of A? The Gauss-Markov theorem, which we nowpresent, says that the least-squares estimator is the best choice, as it yields the smallest varianceamong all unbiased linear estimators.
By a calculation similar to those of the previous section,
E�~� jX
�= A0X�;
so ~� is unbiased if (and only if) A0X = Ik: In this case, we can write
~� = A0y = A (X� + e) = � +Ae:
Thus since var (e jX) = In�2 under homoskedasticity,
var�~� j X
�= A0 var (e jX)A = A0A�2:
The �best�linear estimator is obtained by �nding the matrix A for which this variance is thesmallest in the positive de�nite sense. The Gauss-Markov theorem is the famous solution to thisproblem.
Theorem 3.9.1 Gauss-Markov. In the homoskedastic linear regression model, the best (minimum-variance) unbiased linear estimator is OLS.
The Gauss-Markov theorem is an e¢ ciency justi�cation for the least-squares estimator, but itis quite limited in scope. Not only has the class of models been restricted to homoskedastic linearregressions, the class of potential estimators has been restricted to linear unbiased estimators. Thislatter restriction is particularly unsatisfactory as the theorem leaves open the possibility that a non-linear or biased estimator could have lower mean squared error than the least-squares estimator.
3.10 Semiparametric E¢ ciency
In the previous section we presented the Gauss-Markov theorem as a limited e¢ ciency justi�-cation for the least-squares estimator. A broader justi�cation is provided in Chamberlain (1987),who established that in the projection model the OLS estimator has the smallest asymptotic mean-squared error among feasible estimators. This property is called semiparametric e¢ ciency, andis a strong justi�cation for the least-squares estimator. We discuss the intuition behind his resultin this section.
Suppose that the joint distribution of (yi;xi) is discrete. That is, for �nite r;
P�yi = � j ; xi = �j
�= �j ; j = 1; :::; r
for some constants � j ; �j ; and �j : Assume that the � j and �j are known, but the �j are unknown.(We know the values yi and xi can take, but we don�t know the probabilities.)
In this discrete setting, the de�nition (3.3) can be rewritten as
� =
0@ rXj=1
�j�j�0j
1A�10@ rXj=1
�j�j� j
1A (3.22)
24
Thus � is a function of (�1; :::; �r) :As the data are multinomial, the maximum likelihood estimator (MLE) is
�j =1
n
nXi=1
1 (yi = � j) 1�xi = �j
�for j = 1; :::; r; where 1 (�) is the indicator function. That is, �j is the percentage of the observationswhich fall in each category. The MLE �mle for � is then the analog of (3.22) with the parameters�j replaced by the estimates �j :
�mle =
0@ rXj=1
�j�j�0j
1A�10@ rXj=1
�j�j� j
1A :Substituting in the expressions for �j ,
rXj=1
�j�j�0j =
rXj=1
1
n
nXi=1
1 (yi = � j) 1�xi = �j
��j�
0j
=1
n
nXi=1
rXj=1
1 (yi = � j) 1�xi = �j
�xix
0i
=1
n
nXi=1
xix0i
and
rXj=1
�j�j� j =rXj=1
1
n
nXi=1
1 (yi = � j) 1�xi = �j
��j� j
=1
n
nXi=1
rXj=1
1 (yi = � j) 1�xi = �j
�xiyi
=1
n
nXi=1
xiyi:
Thus
�mle =
1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiyi
!= �ols:
In other words, if the data have a discrete distribution, the maximum likelihood estimator isidentical to the OLS estimator. Since this is a regular parametric model the MLE is asymptoticallye¢ cient (see Appendix D). It follows that the OLS estimator is asymptotically e¢ cient.
The hard part of the argument (which was rigorously developed in Chamberlain�s paper, but wedo not present it here) is the extension to the case of continuously-distributed data:The intuitionis that all continuous distributions can be arbitrarily well approximated by some multinomialdistribution, and for any multinomial distriubtion the moment estimator is asymptotically e¢ cient.Formalizing this intuition using a rigorous mathematical argument, Chamberlain proved that theOLS estimator (3.4) is asymptotically semiparametrically e¢ cient for the parameter � de�ned in(2.13) for the class of models satisfying Assumption 2.6.1.
25
3.11 Multicollinearity
If rank(X 0X) < k; then � is not de�ned2. This is called strict multicollinearity. Thishappens when the columns of X are linearly dependent, i.e., there is some � 6= 0 such thatX� = 0: Most commonly, this arises when sets of regressors are included which are identicallyrelated. For example, if X includes both the logs of two prices and the log of the relative prices,log(p1); log(p2) and log(p1=p2): When this happens, the applied researcher quickly discovers theerror as the statistical software will be unable to construct (X 0X)�1: Since the error is discoveredquickly, this is rarely a problem for applied econometric practice.
The more relevant issue is near multicollinearity, which is often called �multicollinearity�forbrevity. This is the situation when the X 0X matrix is near singular, when the columns of X areclose to linearly dependent. This de�nition is not precise, because we have not said what it meansfor a matrix to be �near singular�. This is one di¢ culty with the de�nition and interpretation ofmulticollinearity.
One implication of near singularity of matrices is that the numerical reliability of the calculationsis reduced. In extreme cases it is possible that the reported calculations will be in error.
A more relevant implication of near multicollinearity is that individual coe¢ cient estimates willbe imprecise. We can see this most simply in a homoskedastic linear regression model with tworegressors
yi = x1i�1 + x2i�2 + ei;
and1
nX 0X =
�1 �� 1
�In this case
var�� jX
�=�2
n
�1 �� 1
��1=
�2
n (1� �2)
�1 ���� 1
�:
The correlation � indexes collinearity, since as � approaches 1 the matrix becomes singular. We cansee the e¤ect of collinearity on precision by observing that the asymptotic variance of a coe¢ cientestimate �2
�1� �2
��1 approaches in�nity as � approaches 1. Thus the more �collinear� are theregressors, the worse the precision of the individual coe¢ cient estimates.
What is happening is that when the regressors are highly dependent, it is statistically di¢ cultto disentangle the impact of �1 from that of �2: As a consequence, the precision of individualestimates are reduced.
3.12 In�uential Observations
The i�th observation is in�uential on the least-squares estimate if the deletion of the observationfrom the sample results in a meaningful change in �. To investigate the possibility of in�uentialobservations, we de�ne the leave-one-out estimator as that obtained from the sample excluding thei�th observation. The leave-one-out OLS estimator of � is
�(�i) =�X 0(�i)X(�i)
��1X(�i)y(�i) (3.23)
where X(�i) and y(�i) are the data matrices omitting the i�th row. A convenient alternativeexpression (derived in Section 3.13) is
�(�i) = � � (1� hi)�1�X 0X
��1xiei (3.24)
wherehi = x
0i
�X 0X
��1xi
2See Appendix A.5 for the de�ntion of the rank of a matrix.
26
is the i�th diagonal element of the projection matrix X (X 0X)�1X 0:We can also de�ne the leave-one-out residual
ei;�i = yi � x0i�(�i) = (1� hi)�1 ei: (3.25)
A simple comparison yields that
ei � ei;�i = (1� hi)�1 hiei: (3.26)
As we can see, the change in the coe¢ cient estimate by deletion of the i�th observation dependscritically on the magnitude of hi: The hi take values in [0; 1] and sum to k: If the i�th observationhas a large value of hi; then this observation is a leverage point and has the potential to bean in�uential observation. Investigations into the presence of in�uential observations can plot thevalues of (3.26), which is considerably more informative than plots of the uncorrected residuals ei:
3.13 Technical Proofs
Proof of Theorem 3.9.1. Let A be any n � k function of X such that A0X = Ik: Thevariance of the least-squares estimator is (X 0X)�1 �2 and that of A0y is A0A�2: It is su¢ cient toshow that the di¤erence A0A� (X 0X)�1 is positive semi-de�nite. Set C = A�X (X 0X)�1 : Notethat X 0C = 0: Then we calculate that
A0A��X 0X
��1=
�C +X
�X 0X
��1�0 �C +X
�X 0X
��1�� �X 0X��1
= C 0C +C 0X�X 0X
��1+�X 0X
��1X 0C
+�X 0X
��1X 0X
�X 0X
��1 � �X 0X��1
= C 0C
The matrix C 0C is positive semi-de�nite (see Appendix A.7) as required. �
Proof of Equation (3.24). Equation (A.2) in Appendix A.5 states that for nonsingular Aand vector b �
A� bb0��1
= A�1 +�1� b0A�1b
��1A�1bb0A�1:
This implies �X 0X � xix0i
��1=�X 0X
��1+ (1� hi)�1
�X 0X
��1xix
0i
�X 0X
��1and thus
�(�i) =�X 0X � xix0i
��1 �X 0y � xiyi
�=
�X 0X
��1X 0y �
�X 0X
��1xiyi
+(1� hi)�1�X 0X
��1xix
0i
�X 0X
��1 �X 0y � xiyi
�= � �
�X 0X
��1xiyi + (1� hi)�1
�X 0X
��1xi
�x0i� � hiyi
�= � � (1� hi)�1
�X 0X
��1xi
�(1� hi) yi � x0i� + hiyi
�= � � (1� hi)�1
�X 0X
��1xiei
the third equality making the substitutions � = (X 0X)�1X 0y and hi = x0i (X0X)�1 xi; and the
remainder collecting terms. �
27
3.14 Exercises
1. Let y be a random variable with � = Ey and �2 = var(y): De�ne
g�y; �; �2
�=
�y � �
(y � �)2 � �2�:
Let (�; �2) be the values such that gn(�; �2) = 0 where gn(m; s) = n�1
Pni=1 g
�yi; �; �
2�:
Show that � and �2 are the sample mean and variance.
2. Consider the OLS regression of the n � 1 vector y on the n � k matrix X. Consider analternative set of regressors Z = XC; where C is a k � k non-singular matrix. Thus, eachcolumn of Z is a mixture of some of the columns of X: Compare the OLS estimates andresiduals from the regression of y on X to the OLS estimates from the regression of y on Z:
3. Let e be the OLS residual from a regression of y on X = [X1 X2]. Find X 02e:
4. Let e be the OLS residual from a regression of y on X: Find the OLS coe¢ cient estimatefrom a regression of e on X:
5. Let y =X(X 0X)�1X 0y: Find the OLS coe¢ cient estimate from a regression of y on X:
6. Prove that R2 is the square of the simple correlation between y and y:
7. Explain the di¤erence between 1n
Pni=1 xix
0i and E (xix
0i) :
8. Let �n = (X 0nXn)
�1X 0nyn denote the OLS estimate when yn is n � 1 and Xn is n � k.
A new observation (yn+1;xn+1) becomes available. Prove that the OLS estimate computedusing this additional observation is
�n+1 = �n +1
1 + x0n+1 (X0nXn)
�1xn+1
�X 0nXn
��1xn+1
�yn+1 � x0n+1�n
�:
9. True or False. If yi = xi� + ei, xi 2 R; E(ei j xi) = 0; and ei is the OLS residual from theregression of yi on xi; then
Pni=1 x
2i ei = 0:
10. A dummy variable takes on only the values 0 and 1. It is used for categorical data, such asan individual�s gender. Let d1 and d2 be vectors of 1�s and 0�s, with the i0th element of d1equaling 1 and that of d2 equaling 0 if the person is a man, and the reverse if the person isa woman. Suppose that there are n1 men and n2 women in the sample. Consider the threeregressions
y = �+ d1�1 + d2�2 + e (3.27)
y = d1�1 + d2�2 + e (3.28)
y = �+ d1�+ e (3.29)
(a) Can all three regressions (3.27), (3.28), and (3.29) be estimated by OLS? Explain if not.
(b) Compare regressions (3.28) and (3.29). Is one more general than the other? Explain therelationship between the parameters in (3.28) and (3.29).
(c) Compute �0d1 and �0d2; where � is an n� 1 is a vector of ones.(d) Letting � = (�1 �2)
0; write equation (3.28) as y = X� + e: Consider the assumptionE(xiei) = 0. Is there any content to this assumption in this setting?
28
11. Let d1 and d2 be de�ned as in the previous exercise.
(a) In the OLS regressiony = d1 1 + d2 2 + u;
show that 1 is sample mean of the dependent variable among the men of the sample(y1), and that 2 is the sample mean among the women (y2).
(b) Describe in words the transformations
y� = y � d1y1 � d2y2X� = X � d1X1 � d2X2:
(c) Compare ~� from the OLS regresion
y� =X�~� + ~e
with � from the OLS regression
y = d1�1 + d2�2 +X� + e:
12. The data �le cps85.dat contains a random sample of 528 individuals from the 1985 Cur-rent Population Survey by the U.S. Census Bureau. The �le contains observations on ninevariables, listed in the �le cps85.pdf.
V1 = education (in years)V2 = region of residence (coded 1 if South, 0 otherwise)V3 = (coded 1 if nonwhite and non-Hispanic, 0 otherwise)V4 = (coded 1 if Hispanic, 0 otherwise)V5 = gender (coded 1 if female, 0 otherwise)V6 = marital status (coded 1 if married, 0 otherwise)V7 = potential labor market experience (in years)V8 = union status (coded 1 if in union job, 0 otherwise)V9 = hourly wage (in dollars)
Estimate a regression of wage yi on education x1i, experience x2i, and experienced-squaredx3i = x
22i (and a constant). Report the OLS estimates.
Let ei be the OLS residual and yi the predicted value from the regression. Numericallycalculate the following:
(a)Pni=1 ei
(b)Pni=1 x1iei
(c)Pni=1 x2iei
(d)Pni=1 x
21iei
(e)Pni=1 x
22iei
(f)Pni=1 yiei
(g)Pni=1 e
2i
(h) R2
Are these calculations consistent with the theoretical properties of OLS? Explain.
29
13. Use the data from the previous problem, restimate the slope on education using the residualregression approach. Regress yi on (1; x2i; x22i), regress x1i on (1; x2i; x
22i), and regress the
residuals on the residuals. Report the estimate from this regression. Does it equal the valuefrom the �rst OLS regression? Explain.
In the second-stage residual regression, (the regression of the residuals on the residuals),calculate the equation R2 and sum of squared errors. Do they equal the values from theinitial OLS regression? Explain.
30
Chapter 4
Inference
4.1 Sampling Distribution
The least-squares estimator is a random vector, since it is a function of the random data, andtherefore has a sampling distribution. In general, its distribution is a complicated function of thejoint distribution of (yi;xi) and the sample size n:
Figure 4.1: Sampling Density of �2
To illustrate the possibilities in one example, let yi and xi be drawn from the joint density
f(x; y) =1
2�xyexp
��12(log y � log x)2
�exp
��12(log x)2
�and let �2 be the slope coe¢ cient estimate computed on observations from this joint density. Usingsimulation methods, the density function of �2 was computed and plotted in Figure 4.1 for samplesizes of n = 25; n = 100 and n = 800: The vertical line marks the true value of the projectioncoe¢ cient.
From the �gure we can see that the density functions are dispersed and highly non-normal. Asthe sample size increases the density becomes more concentrated about the population coe¢ cient.To characterize the sampling distribution more fully, we will use the methods of asymptotic ap-
31
proximation. A review of the most important tools in asymptotic theory is contained in AppendixC.
4.2 Consistency
As discussed in Section 4.1, the OLS estimator � is has a statistical distribution which isunknown. Asymptotic (large sample) methods approximate sampling distributions based on thelimiting experiment that the sample size n tends to in�nity. A preliminary step in this approachis the demonstration that estimators converge in probability to the true parameters as the samplesize gets large. This is illustrated in Figure 4.1 by the fact that the sampling densities become moreconcentrated as n gets larger.
This derivation is based on three key components. First, the OLS estimator can be written asa continuous function of a set of sample moments. Second, the weak law of large numbers (WLLN,Theorem C.2.1) shows that sample moments converge in probability to population moments. Andthird, the continuous mapping theorem (Theorem C.4.1) states that continuous functions preserveconvergence in probability. We now explain each step.
First, observe that the OLS estimator
� =
1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiyi
!
is a function of the sample moments 1n
Pni=1 xix
0i and
1n
Pni=1 xiyi:
Second, the WLLN states that for any iid random variable ui such that E juij <1;
1
n
nXi=1
uip�! E (ui)
as n!1: We want to apply the WLLN to the elements of the sample moments 1n
Pni=1 xix
0i and
1n
Pni=1 xiyi: This is okay since these are sample averages of the random variables xjixli and xjyi.
The WLLN says that for any j and l;
1
n
nXi=1
xjixlip�! E (xjixli) ;
and1
n
nXi=1
xjiyip�! E (xjiyi) :
Since this holds for all elements in the matrix 1n
Pni=1 xix
0i and the vector
1n
Pni=1 xiyi; it follows
that1
n
nXi=1
xix0i
p�! E�xix
0i
�= Q (4.1)
and1
n
nXi=1
xiyip�! E (xiyi) : (4.2)
The continuous mapping theorem then implies that
� =
1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiyi
!p�!�E�xix
0i
���1(E (xiyi))
= �: (4.3)
32
We have shown that �p�! �: In words, the OLS estimator converges in probability to the true
parameter vector � as the sample size n gets large.For a slightly di¤erent demonstration of this result, recall that (3.8) implies that
� � � = 1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiei
!: (4.4)
The WLLN and (2.17) imply1
n
nXi=1
xieip�! E (xiei) = 0: (4.5)
Therefore
� � � = 1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiei
!p�! Q�10
= 0
which is the same as �p�! �.
Theorem 4.2.1 Under Assumptions 2.6.1 and 3.1.1, �p�! � as n!1:
For further details on the proof see Section 4.17.Theorem 4.2.1 states that the OLS estimator � converges in probability to � as n diverges
to positive in�nity. When an estimator converges in probability to the true value as the samplesize diverges, we say that the estimator is consistent. This is a good property for an estimatorto possess. It means that for any given joint distribution of (yi;xi) ; there is a sample size nsu¢ ciently large such that the estimator � will be arbitrarily close to the true value � with highprobability. Consistency is also an important preliminary step in establishing other importantasymptotic approximations.
We can similarly show that the estimators �2 and s2 are consistent for �2:
Theorem 4.2.2 Under Assumptions 2.6.1 and 3.1.1, �2p�! �2 and s2
p�! �2 as n!1:
The proof is given in Section 4.17.One implication of this theorem is that multiple estimators can be consistent for the sample
population parameter. While �2 and s2 are unequal in any given application, they are close invalue when n is very large.
4.3 Asymptotic Normality
We started this Chapter discussing the need for an approximation to the distribution of the OLSestimator �: In the previous section we showed that � converges in probability to �. This is a useful�rst step, but in itself does not provide a useful approximation to the distribution of the estimator.In this section we derive an approximation typically called the asymptotic distribution of theestimator.
The derivation starts by writing the estimator as a function of sample moments. One of themoments must be written as a sum of zero-mean random vectors and normalized so that the centrallimit theory can be applied. The steps are as follows.
33
Take equation (4.4) and multiply it bypn: This yields the expression
pn�� � �
�=
1
n
nXi=1
xix0i
!�1 1pn
nXi=1
xiei
!: (4.6)
This shows that the normalized and centered estimatorpn�� � �
�is a function of the sample
average 1n
Pni=1 xix
0i and the normalized sample average
1pn
Pni=1 xiei: Furthermore, the latter has
mean zero so the central limit theorem (CLT) applies. Recall, the CLT (Theorem C.3.1) statesthat if ui 2 Rk is iid, Eui = 0 and Eu2ji <1 for j = 1; :::; k; then as n!1
1pn
nXi=1
uid�! N
�0;E
�uiu
0i
��:
For our application, ui = xiei which is iid and mean zero since E (ui) = E (xiei) = 0:We calculatethat E (uiu0i) = E
�xix
0ie2i
�and de�ne this matrix by . By the CLT we conclude
1pn
nXi=1
xieid�! N(0;) : (4.7)
where = E
�xix
0ie2i
�:
Then using (4.6), (4.1), and (4.7),
pn�� � �
�d�! Q�1N(0;)
= N�0;Q�1Q�1
�:
where the �nal equality follows from Theorem B.9.1A formal statement of this result requires the following strengthening of the moment conditions.
Assumption 4.3.1 In addition to Assumption 2.6.1, E�e4i�<1 and for j = 1; :::; k; Ex4ji <1:
Theorem 4.3.1 Under Assumptions 3.1.1 and 4.3.1, as n!1pn�� � �
�d�! N(0;V )
whereV = Q�1Q�1:
Further details of the proof are given in Section 4.17.
As V is the variance of the asymptotic distribution ofpn�� � �
�; V is often referred to as
the asymptotic covariance matrix of �: The expression V = Q�1Q�1 is called a sandwichform.
Theorem 4.3.1 states that the sampling distribution of the least-squares estimator, after rescal-ing, is approximately normal when the sample size n is su¢ ciently large. This holds true for all jointdistibutions of (yi;xi) which satisfy the conditions of Assumption 4.3.1. However, for any �xed nthe sampling distribution of � can be arbitrarily far from the normal distribution. In Figure 4.1we have already seen a simple example where the least-squares estimate is quite asymmetric andnon-normal even for reasonably large sample sizes.
34
There is a special case where and V simplify. We say that ei is aHomoskedastic ProjectionError when
cov(xix0i; e
2i ) = 0: (4.8)
Condition (4.8) holds when xi and ei are independent, but this is not a necessary condition. Under(4.8) the asymptotic variance formulas simplify as
= E�xix
0i
�E�e2i�= Q�2 (4.9)
V = Q�1Q�1 = Q�1�2 � V 0 (4.10)
In (4.10) we de�ne V 0 = Q�1�2 whether (4.8) is true or false. When (4.8) is true then V = V 0;otherwise V 6= V 0: We call V 0 the homoskedastic covariance matrix.
The asymptotic distribution of Theorem 4.3.1 is commonly used to approximate the �nite
sample distribution ofpn�� � �
�: The approximation may be poor when n is small. How large
should n be in order for the approximation to be useful? Unfortunately, there is no simple answerto this reasonable question. The trouble is that no matter how large is the sample size, thenormal approximation is arbitrarily poor for some data distribution satisfying the assumptions.We illustrate this problem using a simulation. Let yi = �0 + �1xi + ei where xi is N(0; 1) ; andei is independent of xi with the Double Pareto density f(e) = �
2 jej���1 ; jej � 1: If � > 2 the
error ei has zero mean and variance �=(� � 2): As � approaches 2, however, its variance divergesto in�nity. In this context the normalized least-squares slope estimator
qn��2�
��2 � �2
�has the
N(0; 1) asymptotic distibution for any � > 2. In Figure 4.2 we display the �nite sample densities
of the normalized estimatorqn��2�
��2 � �2
�; setting n = 100 and varying the parameter �.
For � = 3:0 the density is very close to the N(0; 1) density. As � diminishes the density changessigni�cantly, concentrating most of the probability mass around zero.
Figure 4.2: Density of Normalized OLS estimator
Another example is shown in Figure 4.3. Here the model is yi = �1 + ei where
ei =uki � E
�uki��
E�u2ki���E�uki��2�1=2
35
and ui � N(0; 1): We show the sampling distribution ofpn��1 � �1
�setting n = 100; for k = 1;
4, 6 and 8. As k increases, the sampling distribution becomes highly skewed and non-normal. Thelesson from Figures 4.2 and 4.3 is that the N(0; 1) asymptotic approximation is never guaranteedto be accurate.
Figure 4.3: Sampling distribution
4.4 Covariance Matrix Estimation
Let
Q =1
n
nXi=1
xix0i
be the method of moments estimator for Q: The homoskedastic covariance matrix V 0 = Q�1�2 istypically estimated by
V0= Q
�1s2: (4.11)
Since Qp�! Q and s2
p�! �2 (see (4.1) and Theorem 4.2.1) it is clear that V0 p�! V 0: The
estimator �2 may also be substituted for s2 in (4.11) without changing this result.To estimate V = Q�1Q�1; we need an estimate of = E
�xix
0ie2i
�: The MME estimator is
=1
n
nXi=1
xix0ie2i (4.12)
where ei are the OLS residuals. The estimator of V is then
V = Q�1Q
�1:
This estimator was introduced to the econometrics literature by White (1980):
Theorem 4.4.1 Under Assumptions 3.1.1 and 4.3.1, as n!1; p�! and Vp�! V :
36
The estimator V0was the dominate covariance estimator used before 1980, and was still the
standard choice for much empirical work done in the early 1980s. The methods switched duringthe late 1980s and early 1990s, so that by the late 1990s the White estimate V emerged as thestandard covariance matrix estimator. When reading and reporting applied work, it is importantto pay attention to the distinction between V
0and V , as it is not always clear which has been
computed. When V is used rather than the traditional choice V0;many authors will state that their
�standard errors have been corrected for heteroskedasticity�, or that they use a �heteroskedasticity-robust covariance matrix estimator�, or that they use the �White formula�, the �Eicker-Whiteformula�, the �Huber formula�, the �Huber-White formula�or the �GMM covariance matrix�. Inmost cases, these all mean the same thing.
The variance estimator V is an estimate of the variance of the asymptotic distribution of �.A more easily interpretable measure of spread is its square root � the standard deviation. Thismotivates the de�nition of a standard error.
De�nition 4.4.1 A standard error s(�) for an estimator � is an estimate of the standard devi-ation of the distribution of �:
When � is scalar, and V is an estimator of the variance ofpn�� � �
�; we set s(�) = n�1=2
pV :
When � is a vector, we focus on individual elements of � one-at-a-time, vis., �j ; j = 0; 1; :::; k:Thus
s(�j) = n�1=2
qV jj :
Generically, standard errors are not unique, as there may be more than one estimator of thevariance of the estimator. It is therefore important to understand what formula and method isused by an author when studying their work. It is also important to understand that a particularstandard error may be relevant under one set of model assumptions, but not under another set ofassumptions, just as any other estimator.
From a computational standpoint, the standard method to calculate the standard errors is to�rst calculate n�1V , then take the diagonal elements, and then the square roots.
To illustrate, we return to the log wage regression of Section 3.2. We calculate that s2 = 0:20and
=
�0:199 2:802:80 40:6
�:
Therefore the two covariance matrix estimates are
V0=
�1 14:14
14:14 205:83
��10:20 =
�6:98 �0:480�0:480 :039
�and
V =
�1 14:14
14:14 205:83
��1�:199 2:802:80 40:6
��1 14:14
14:14 205:83
��1=
�7:20 �0:493�0:493 0:035
�:
In this case the two estimates are quite similar. The (White) standard errors for �0 arep7:2=988 =
:085 and that for �1 isp:035=988 = :006:We can write the estimated equation with standard errors
using the format
\log(Wage) = 1:313(:085)
+ 0:128(:006)
Education:
37
4.5 Alternative Covariance Matrix Estimators
MacKinnon and White (1985) suggested a small-sample corrected version of V based on thejackknife principle. Recall from Section 3.12 the de�nition of �(�i) as the least-squares estimatorwith the i�th observation deleted. From equation (3.13) of Efron (1982), the jackknife estimator ofthe variance matrix for � is
V�= (n� 1)
nXi=1
��(�i) � ��
���(�i) � ��
�0(4.13)
where
�� =1
n
nXi=1
�(�i):
Using formula (3.24), you can show that
V�=
�n� 1n
�Q�1�Q�1
(4.14)
where
�=1
n
nXi=1
(1� hi)�2 xix0ie2i � 1
n
nXi=1
(1� hi)�1 xiei
! 1
n
nXi=1
(1� hi)�1 xiei
!0and hi = x0i (X
0X)�1 xi: MacKinnon and White (1985) present numerical (simulation) evidencethat V
�works better than V as an estimator of V . They also suggest that the scaling factor
(n� 1)=n in (4.14) can be omitted.Andrews (1991) suggested a similar estimator based on cross-validation, which is de�ned by re-
placing the OLS residual ei in (4.12) with the leave-one-out estimator ei;�i = (1� hi)�1 ei presentedin (3.25). Using this substitution, Andrews�proposed estimator is
V��= Q
�1��Q�1
where
��=1
n
nXi=1
(1� hi)�2 xix0ie2i :
It is similar to the MacKinnon-White estimator V�; but omits the mean correction. Andrews
(1991) argues that simulation evidence indicates that V��is an improvement on V
�.
4.6 Functions of Parameters
Sometimes we are interested in some lower-dimensional function of the parameter vector � =(�1; :::; �k): For example, we may be interested in a single coe¢ cient �j or a ratio �j=�l: In thesecases we can write the parameter of interest as a function of �: Let h : Rk ! Rq denote thisfunction and let
� = h(�)
denote the parameter of interest. The estimate of � is
� = h(�):
What is an appropriate standard error for �? Assume that h(�) is di¤erentiable at the truevalue of �: By a �rst-order Taylor series approximation:
h(�) ' h(�) +H 0�
�� � �
�:
38
where
H� =@
@�h(�) k � q:
Thus
pn�� � �
�=pn�h(�)� h(�)
�'H 0
�
pn�� � �
�d�!H 0
� N(0;V )
= N (0;V �) : (4.15)
whereV � =H
0�V H�:
If V is the estimated covariance matrix for �; then the natural estimate for the variance of � is
V � = H0�V H�
where
H� =@
@�h(�):
In many cases, the function h(�) is linear:
h(�) = R0�
for some k � q matrix R: In this case, H� = R and H� = R; so V � = R0V R:
For example, if R is a �selector matrix�
R =
�I0
�so that if � = (�1;�2); then � = R
0� = �1 and
V � =�I 0
�V
�I0
�= V 11;
the upper-left block of V :When q = 1 (so h(�) is real-valued), the standard error for � is the square root of n�1V�; that
is, s(�) = n�1=2qH0�V H�:
4.7 t tests
Let � = h(�) : Rk ! R be any parameter of interest (for example, � could be a single elementof �), � its estimate and s(�) its asymptotic standard error. Consider the statistic
tn(�) =� � �s(�)
(4.16)
which di¤erent writers alternatively label as t-statistic, a z-statistic or a studentized statistic. Wewon�t be making such distinctions and will typically refer to tn(�) as a t-statistic. We also oftensuppress the parameter dependence, writing it as tn: The t-statistic is a simple function of theestimate, its standard error, and the parameter.
39
Theorem 4.7.1 tn(�)d�! N(0; 1)
Thus the asymptotic distribution of the t-ratio tn(�) is the standard normal. Since this dis-tribution does not depend on the parameters, we say that tn(�) is asymptotically pivotal. Inspecial cases (such as the normal regression model, see Section 3.4), the statistic tn has an exact tdistribution, and is therefore exactly free of unknowns. In this case, we say that tn is exactly piv-otal. In general, however, pivotal statistics are unavailable and so we must rely on asymptoticallypivotal statistics.
The t-test is routinely used to test hypotheses on �. A simple null and composite hypothesistakes the form
H0 : � = �0
H1 : � 6= �0
where �0 is some pre-speci�ed value. A t-test rejects H0 in favor of H1 when jtn(�0)j is large. By�large�we mean that the observed value of the t-statistic would be unlikely if H0 were true.
Formally, we �rst pick an asymptotic signi�cance level �. We then �nd z�=2; the upper �=2quantile of the standard normal distribution which has the property that if Z � N(0; 1) then
P�jZj > z�=2
�= �:
For example, z:025 = 1:96 and z:05 = 1:645: A test of asymptotic signi�cance � rejects H0 ifjtnj > z�=2: Otherwise the test does not reject, or �accepts�H0:
The asymptotic signi�cance level is � because Theorem 4.7.1 implies that
P (reject H0 j H0 true) = P�jtnj > z�=2 j � = �0
�! P
�jZj > z�=2
�= �:
The rejection/acceptance dichotomy is associated with the Neyman-Pearson approach to hypothesistesting.
While there is no objective scienti�c basis for choice of signi�cance level �; the comon practiceis to set � = :05 or 5%. This implies a critical value of z:025 = 1:96 � 2. When jtnj > 2 it iscommon to say that the t-statistic is statistically signi�cant. and if jtnj < 2 it is common tosay that the t-statistic is statistically insigni�cant. It is helpful to remember that this is simplya simple way of saying �Using a t-test, the hypothesis that � = �0 can [cannot] be rejected at theasymptotic 5% level.�
A related statistic is the asymptotic p-value, which can be interpreted as a measure of theevidence against the null hypothesis. The asymptotic p-value of the statistic tn is
pn = p(tn)
where p(t) is the tail probability function
p(t) = P (jZj > jtj) = 2 (1� �(jtj)) :
The closer the p-value is to zero, the stronger the evidence is against H0: Signi�cance tests can bededuced directly from the p-value since for any �; pn < � if and only if jtnj > z�=2:
If the p-value pn is small (close to zero) then the evidence against H0 is strong. In a sense,p-values and hypothesis tests are equivalent since pn < � if and only if jtnj > z�=2. Thus anequivalent statement of a Neyman-Pearson test is to reject at the �% level if and only if pn < �:The p-value is more general, however, in that the reader is allowed to pick the level of signi�cance�, in contrast to Neyman-Pearson rejection/acceptance reporting where the researcher picks thesigni�cance level.
40
Another helpful observation is that the p-value function has simply made a unit-free transforma-
tion of the test statistic. That is, under H0; pnd�! U[0; 1]; so the �unusualness�of the test statistic
can be compared to the easy-to-understand uniform distribution, regardless of the complication ofthe distribution of the original test statistic. To see this fact, note that the asymptotic distributionof jtnj is F (x) = 1� p(x): Thus
P (1� pn � u) = P (1� p(tn) � u)= P (F (tn) � u)= P
�jtnj � F�1(u)
�! F
�F�1(u)
�= u;
establishing that 1� pnd�! U[0; 1]; from which it follows that pn
d�! U[0; 1]:
4.8 Con�dence Intervals
A con�dence interval Cn is an interval estimate of � 2 R: It is a function of the data and henceis random. It is designed to cover � with high probability. Either � 2 Cn or � =2 Cn: The coverageprobability is P(� 2 Cn).
We typically cannot calculate the exact coverage probability P(� 2 Cn): However we often cancalculate the asymptotic coverage probability limn!1 P(� 2 Cn): We say that Cn has asymptotic(1� �)% coverage for � if P(� 2 Cn)! 1� � as n!1:
A good method for construction of a con�dence interval is the collection of parameter valueswhich are not rejected by a statistical test. The t-test of the previous section rejects H0 : � = �0 ifjtn(�)j > z�=2 where tn(�) is the t-statistic (4.16) and z�=2 is the upper �=2 quantile of the standardnormal distribution. A con�dence interval is then constructed as the values of � for which this testdoes not reject:
Cn =�� : jtn(�)j � z�=2
=
(� : �z�=2 �
� � �s(�)
� z�=2
)=
h� � z�=2s(�); � + z�=2s(�)
i: (4.17)
While there is no hard-and-fast guideline for choosing the coverage probability 1� �; the mostcommon professional choice is 95%, or � = :05: This corresponds to selecting the con�dence intervalh� � 1:96s(�)
i�h� � 2s(�)
i: Thus values of � within two standard errors of the estimated � are
considered �reasonable�candidates for the true value �; and values of � outside two standard errorsof the estimated � are considered unlikely or unreasonable candidates for the true value.
The interval has been constructed so that as n!1;
P (� 2 Cn) = P�jtn(�)j � z�=2
�! P
�jZj � z�=2
�= 1� �:
and Cn is an asymptotic (1� �)% con�dence interval.
4.9 Wald Tests
Sometimes � = h(�) is a q � 1 vector, and it is desired to test the joint restrictions simultane-ously. In this case the t-statistic approach does not work. We have the null and alternative
H0 : � = �0
H1 : � 6= �0:
41
The natural estimate of � is � = h(�) and has asymptotic covariance matrix estimate
V � = H0�V H�
where
H� =@
@�h(�):
The Wald statistic for H0 against H1 is
Wn = n�� � �0
�0V�1�
�� � �0
�= n
�h(�)� �0
�0 �H0�V H�
��1 �h(�)� �0
�: (4.18)
When h is a linear function of �; h(�) = R0�; then the Wald statistic takes the form
Wn = n�R0� � �0
�0 �R0V R
��1 �R0� � �0
�:
The delta method (4.15) showed thatpn�� � �
�d�! Z � N(0;V �) ; and Theorem 4.4.1
showed that Vp�! V : Furthermore, H�(�) is a continuous function of �; so by the continuous
mapping theorem, H�(�)p�!H�: Thus V � = H
0�V H�
p�!H 0�V H� = V � > 0 if H� has full
rank q: Hence
Wn = n�� � �0
�0V�1�
�� � �0
�d�! Z 0V �1
� Z = �2q ;
by Theorem B.9.3: We have established:
Theorem 4.9.1 Under H0 and Assumption 4.3.1, if rank(H�) = q; then Wnd�! �2q ; a chi-square
random variable with q degrees of freedom.
An asymptotic Wald test rejects H0 in favor of H1 if Wn exceeds �2q(�); the upper-� quantileof the �2q distribution. For example, �
21(:05) = 3:84 = z
2:025: The Wald test fails to reject if Wn is
less than �2q(�): As with t-tests, it is conventional to describe a Wald test as �signi�cant� if Wn
exceeds the 5% critical value.Notice that the asymptotic distribution in Theorem 4.9.1 depends solely on q �the number of
restrictions being tested. It does not depend on k �the number of parameters estimated.The asymptotic p-value for Wn is pn = p(Wn); where p(x) = P
��2q � x
�is the tail probability
function of the �2q distribution. The Wald test rejects at the �% level if and only if pn < �; andpn is asymptotically U[0; 1] under H0: In reporting applied work it is good practice to report thep-value.
4.10 F Tests
Take the linear modely =X1�1 +X2�2 + e
where X1 is n� k1 and X2 is n� k2 and k = k1 + k2: The null hypothesis is
H0 : �2 = 0:
42
In this case, � = �2; and there are q = k2 restrictions. Also h(�) = R0� is linear with R =
�0I
�a selector matrix. We know that the Wald statistic takes the form
Wn = n�0V�1� �
= n�02
�R0V R
��1�2:
Now suppose that covariance matrix is computed under the assumption of homoskedasticity, sothat V is replaced with V
0= �2
�n�1X 0X
��1: We de�ne the �homoskedastic�Wald statistic
W 0n = n�
0V0�1� �
= n�02
�R0V
0R��1
�2:
What we show in this section is that this Wald statistic can be written very simply using theformula
W 0n = n
�~�2 � �2
�2
�(4.19)
where~�2 =
1
n~e0~e; ~e = y �X1
~�1; ~�1 =�X 01X1
��1X 01y
are from OLS of y on X1; and
�2 =1
ne0e; e = y �X�; � =
�X 0X
��1X 0y
are from OLS of y on X = (X1;X2):The elegant feature about (4.19) is that it is directly computable from the standard output
from two simple OLS regressions, as the sum of squared errors is a typical output from statisticalpackages. This statistic is typically reported as an �F-statistic�which is de�ned as
Fn =n� kn
W 0n
k2=
�~�2 � �2
�=k2
�2=(n� k):
While it should be emphasized that equality (4.19) only holds if V0= �2
�n�1X 0X
��1; still this
formula often �nds good use in reading applied papers. Because of this connection we call (4.19)the F form of the Wald statistic. (We can also call W 0
n a homoskedastic form of the Wald statistic.)We now derive expression (4.19). First, note that by partitioned matrix inversion (A.3)
R0 �X 0X��1
R = R0�X 01X1 X 0
1X2
X 02X1 X 0
2X2
��1R =
�X 02M1X2
��1whereM1 = I �X1(X
01X1)
�1X 01: Thus�
R0V0R��1
= ��2n�1�R0 �X 0X
��1R��1
= ��2n�1�X 02M1X2
�and
W 0n = n�
02
�R0V
0R��1
�2
=�02 (X
02M1X2) �2�2
:
43
To simplify this expression further, note that if we regress y on X1 alone, the residual is~e = M1y: Now consider the residual regression of ~e on ~X2 = M1X2: By the FWL theorem,~e =X2�2 + e and ~X
02e = 0: Thus
~e0~e =�~X2�2 + e
�0 �~X2�2 + e
�= �
02~X02~X2�2 + e
0e
= �02X
02M1X2�2 + e
0e;
or alternatively,�02X
02M1X2�2 = ~e
0~e� e0e:
Also, since�2 = n�1e0e
we conclude that
W 0n = n
�~e0~e� e0ee0e
�= n
�~�2 � �2
�2
�;
as claimed.In many statistical packages, when an OLS regression is estimated, an �F-statistic�is reported.
This is
Fn =
�~�2y � �2
�= (k � 1)
�2=(n� k):
where~�2y =
1
n(y � y)0 (y � y)
is the sample variance of yi; equivalently the residual variance from an intercept-only model. Thisspecial F statistic is testing the hypothesis that all slope coe¢ cients (all coe¢ cients other than theintercept) are zero. This was a popular statistic in the early days of econometric reporting, whensample sizes were very small and researchers wanted to know if there was �any explanatory power�to their regression. This is rarely an issue today, as sample sizes are typically su¢ ciently large thatthis F statistic is highly �signi�cant�. While there are special cases where this F statistic is useful,these cases are atypical.
4.11 Normal Regression Model
As an alternative to asymptotic distribution theory, there is an exact distribution theory avail-able for the normal linear regression model introduced in Section 3.4. The modelling assumptionthat the error ei is independent of xi and N
�0; �2
�can be used to calculate a set of exact distribution
results.In particular, under the normality assumption the error vector e is independent of X and
has distribution N�0; In�
2�. Since linear functions of normals are also normal, this implies that
conditional on X�� � �e
�=
�(X 0X)�1X 0
M
�e � N
�0;
��2 (X 0X)�1 0
0 �2M
��where M = I �X (X 0X)�1X 0: Since uncorrelated normal variables are independent, it followsthat � is independent of any function of the OLS residuals including the estimated error variances2:
44
The spectral decomposition ofM yields
M =H
�In�k 00 0
�H 0
(see equation (A.5)) where H 0H = In: Let u = ��1H 0e � N(0;H 0H) � N(0; In) : Then(n� k) s2
�2=
1
�2e0e
=1
�2e0Me
=1
�2e0H
�In�k 00 0
�H 0e
= u0�In�k 00 0
�u
� �2n�k;
a chi-square distribution with n� k degrees of freedom. Furthermore, if standard errors are calcu-lated using the homoskedastic formula (4.11)
�j � �js(�j)
=�j � �j
s
rh(X 0X)�1
ijj
�N
�0; �2
h(X 0X)�1
ijj
�q
�2
n�k�2n�k
rh(X 0X)�1
ijj
=N(0; 1)q
�2n�kn�k
� tn�k
a t distribution with n� k degrees of freedom.We summarize these �ndings
Theorem 4.11.1 If ei is independent of xi and distributed N�0; �2
�; and standard errors are
calculated using the homoskedastic formula (4.11) then
� � � N�0; �2 (X 0X)�1
�� (n�k)s2
�2� �2n�k;
� �j��js(�j)
� tn�k
In Theorem 4.3.1 and Theorem 4.7.1 we showed that in large samples, � and t are approximatelynormally distributed. In contrast, Theorem 4.11.1 shows that under the strong assumption ofnormality, � has an exact normal distribution and tn has an exact t distribution. As inference(con�dence intervals) is based on the t-ratio, the notable distinction is between the N(0; 1) andtn�k distributions. The critical values are quite close if n� k � 30; so as a practical matter it doesnot matter which distribution is used. (Unless the sample size is unreasonably small.)
Now let us partition � = (�1;�2) and consider tests of the linear restriction
H0 : �2 = 0
H1 : �2 6= 0
In the context of parametric models, a good testing procedure is based on the likelihood ratio sta-tistic, which is twice the di¤erence in the log-likelihood function evaluated under the null and alter-native hypotheses. The estimator under the alternative is the unrestricted estimator (�1; �2; �
2)discussed above. The Gaussian log-likelihood at these estimates is
logL(�1; �2; �2) = �n
2log�2��2
�� 1
2�2e0e
= �n2log��2�� n2log (2�)� n
2:
45
The MLE of the model under the null hypothesis is (~�1;0; ~�2) where ~�1 is the OLS estimate from
a regression of yi on x1i only, with residual variance ~�2: The log-likelihood of this model is
logL(~�1;0; ~�2) = �n
2log�~�2�� n2log (2�)� n
2:
The LR statistic for H0 is
LRn = 2�logL(�1; �2; �
2)� logL(~�1;0; ~�2)�
= n�log�~�2�� log
��2��
= n log
�~�2
�2
�:
By a �rst-order Taylor series approximation
LRn = n log
�1 +
~�2
�2� 1�' n
�~�2
�2� 1�=W 0
n :
the homoskedastic Wald statistic. This shows that the two statistics �LRn and W 0n �will be
numerically close. It also shows that the homoskedastic Wald statistic for linear hypotheses canalso be interpreted as an appropriate likelihood ratio test under normality.
4.12 Semiparametric E¢ ciency in the Projection Model
In this section we return to the question of semiparametric e¢ ciency as raised in Section 3.10.There we presented the intuition behind Chamberlain�s demonstration of the asymptotic e¢ ciencyof the least-squares estimator. In this section we provide an alternative demonstration. It is basedon the rich but technically challenging theory of semiparametric e¢ ciency bounds. An excellentaccessible review has been provided by Newey (1990).
Our treatment covers what is known as the smooth function model, which includes the projectionmodel as a special case. Let z 2 Rm be a random vector with �nite mean � = Ez and �nite variancematrix � = var (z) ; and let z1; :::;zn be an iid sample from this distribution. The parameter ofinterest is � = g (�) where g (�) is a continuously di¤erentiable function. The standard momentestimator for � is the sample mean � = n�1
Pni=1 zi and that for � is � = g (�) : This setting
includes the least-squares estimator for the projection model y = x0�+ e by letting z be the vectorwith elements xje and xjxl for all j � k and l � k:
The sample mean has the asymptotic distributionpn (�� �) d�! N(0;�) : Applying the Delta
Method (Theorem C.4.3), we see that the moment estimator � has the asymptotic distributionpn�� � �
�d�! N(0;V ) where V = @
@�0g (�)�@@�g (�)
0 : We want to know if � is the best
feasible estimator. Is there another estimator with a smaller asymptotic variance? While it seemsintuitively unlikely that another estimator could have a smaller asymptotic variance than �; howdo we know that this is not the case?
To show that the answer is not immediately obvious, it might be helpful to review a set-ting where the sample mean is ine¢ cient. Suppose that z 2 R has the density f (z j �) =2�1=2 exp
�� jz � �j
p2�: Since var (z) = 1 we see that the sample mean satis�es
pn (�� �) d�!
N(0; 1). In this model the maximum likelihood estimator (MLE) ~� for � is di¤erent than the sam-ple mean (and happens to be the sample median). Recall from the theory of maximum likelhood
that the MLE satis�espn (~�� �) d�! N(0; I0) where I0 =
�ES2�
��1 and S� = @@� log f (z j �) =
�p2 sgn (z � �) is the score. We can calculate that ES2� = 2 and thus conclude that
pn (~�� �) d�!
N(0; 1=2) : The asymptotic variance of the MLE is half that of the sample mean. In this settingthe sample mean is ine¢ cient.
46
But the question at hand is whether or not the sample mean is e¢ cient when the form of thedistribution is unknown. We call this setting semiparametric as the parameter of interest (themean) is �nite dimensional while the remaining features of the distribution are unspeci�ed. In thesemiparametric context an estimator is called semiparametrically e¢ cient if it has the smallestasymptotic variance among all semiparametric estimators.
The mathematical trick is to reduce the semiparametric model to a set of parametric �submod-els�. The classic Cramer-Rao variance bound can be found for each parametric submodel. Thevariance bound for the semiparametric model (the union of the submodels) is then computed bytaking the supremum of the individual variance bounds.
Formally, suppose that the true density of z is the unknown function f(z). A parametricsubmodel for z is a density f (z j �) which is a smooth function of a parameter � 2 Rm and thereis some �0 such that f (z j �0) = f(z): This means that the submodel class passes through the truedensity, so the submodel is a true model. Since each submodel is parametric we can calculate itsCramer-Rao bound. By the Cramer-Rao theorem no estimator (and in particular no semiparametricestimator) has an asymptotic variance smaller than this bound. This comparison is true for allsubmodels, so the asymptotic variance of any semiparametric estimator cannot be smaller than theCramer-Rao bound for any parametric submodel. The semiparametric asymptotic variancebound (which is sometimes called the semiparametric e¢ ciency bound) is the supremum of theCramer-Rao bounds from all conceivable submodels. It is a lower bound for the asymptotic varianceof any semiparametric estimator. If the asymptotic variance of a speci�c semiparametric estimatorequals this bound we say that the estimator is semiparametrically e¢ cient.
For many statistical problems it is quite challenging to calculate the semiparametric variancebound. However the solution is straightforward in the present setting. As the semiparametricvariance bound cannot be smaller than the Cramer-Rao bound for any submodel, and cannot belarger than the asymptotic variance of any feasible semiparametric estimator, it follows that if theasymptotic variance of a feasible semiparametric estimator equals the Cramer-Rao bound for at leastone submodel, then this is the semiparametric asymptotic variance bound, and the aforementionedfeasible semiparametric estimator must be semiparametrically e¢ cient. In these cases, it is su¢ cientto construct a parametric submodel for which the Cramer-Rao bound (equivalently, the asymptoticvariance of the MLE) equals that of a known semiparametric estimator.
We now show this for the moment estimator � = g (�) discussed above. As � has asymptoticvariance V ; our goal is to �nd a parametric submodel whose Cramer-Rao bound for estimation of� is V : The solution involves creating a tilted version of the true density. Consider the parametricsubmodel
f (z j �) = f(z)�1 + �0��1 (z � �)
�(4.20)
where f(z) is the true density and � = Ez: Note thatZf (z j �) dz =
Zf(z)dz + �0��1
Zf(z) (z � �) dz = 1
and for all � close to zero f (z j �) � 0: Thus f (z j �) is a valid density function. It is a parametricsubmodel since f (z j �0) = f(z) when �0 = 0: This parametric submodel has the mean
�(�) =
Zzf (z j �) dz
=
Zzf(z)dz +
Zf(z)z (z � �)0��1�dz
= �+ �
and parameter of interest � (�) = g (�+ �) both which are smooth functions of �:Since
@
@�log f (z j �) = @
@�log�1 + �0��1 (z � �)
�=
��1 (z � �)1 + �0��1 (z � �)
47
it follows that the score function for � is
s =@
@�log f (z j �0) = ��1 (z � �) : (4.21)
By classic theory the asymptotic variance of the MLE � for � is the Cramer-Rao bound (E (ss0))�1 =���1E
�(z � �) (z � �)0
���1
��1= �: The MLE for � is �(�) = g
��+ �
�which by the delta
method has asymptotic variance V = @@�0g (�)�
@@�g (�)
0 ; which is identical to the asymptotic
variance of the moment estimator �: This shows that moment estimators are semiparametricallye¢ cient, and this includes the OLS estimator in the projection model. We have established thefollowing theorem.
Theorem 4.12.1 Under Assumptions 3.1.1 and 4.3.1, the semiparametric variance bound for es-timation of � is V = Q�1Q�1; and the OLS estimator is semiparametrically e¢ cient.
4.13 Semiparametric E¢ ciency in the Homoskedastic RegressionModel
In Section 3.9 we presented the Gauss-Markov theorem, which stated that in the homoskedasticregression model (2.10)-(2.12), in the class of linear unbiased estimators the one with the smallestvariance is least-squares. As we noted in that section, the restriction to linear unbiased estimatorsis unsatisfactory as it leaves open the possibility that an alternative (non-linear) estimator couldhave a smaller asymptotic variance. In Sections 3.10 and 4.12 we showed that the OLS estimatoris e¢ cient in the projection model, but this does not address the question of whether or not OLSis e¢ cient in the homoskedastic regression model (2.10)-(2.12). In this section we return to thequestion of e¢ cient estimation in this model using the theory of semiparametric variance boundsas presented in the previous section.
Recall that in the homoskedastic regression model (2.10)-(2.12) the asymptotic variance ofthe OLS estimator � for � is V 0 = Q�1�2: Therefore, as described in the previous section, itis su¢ cient to �nd a parametric submodel whose Cramer-Rao bound for estimation of � is V 0:This would establish that V 0 is the semiparametric variance bound and the OLS estimator � issemiparametrically e¢ cient for �:
Let the joint density of y and x be written as f (y;x) = f1 (y j x) f2 (x) ; the product of theconditional density of y given x; and the marginal density of x. Now consider the parametricsubmodel
f (y;x j �) = f1 (y j x)�1 +
�y � x0�
� �x0��=�2�f2 (x) : (4.22)
You can check that in this submodel, the marginal density of x is f2 (x) ; and the conditional densityof y given x is f1 (y j x)
�1 + (y � x0�) (x0�) =�2
�: To see that the latter is a valid conditional
density, observe that the regression assumption implies thatRyf1 (y j x) dy = x0� and thereforeZ
f1 (y j x)�1 +
�y � x0�
� �x0��=�2�dy =
Zf1 (y j x) dy +
Zf1 (y j x)
�y � x0�
�dy�x0��=�2
= 1:
48
In this parametric submodel the conditional mean of y given x is
E� (y j x) =
Zyf1 (y j x)
�1 +
�y � x0�
� �x0��=�2�dy
=
Zyf1 (y j x) dy +
Zyf1 (y j x)
�y � x0�
� �x0��=�2dy
=
Zyf1 (y j x) dy +
Z �y � x0�
�2f1 (y j x)
�x0��=�2dy
+
Z �y � x0�
�f1 (y j x) dy
�x0�
� �x0��=�2
= x0 (� + �) ;
using the homoskedasticity assumption thatR(y � x0�)2 f1 (y j x) dy = �2: This means that in
this parametric submodel, the conditional mean is linear in x and the regression coe¢ cient is� (�) = � + �:
We now calculate the score for estimation of �: Since
@
@�log f (y;x j �) = @
@�log�1 +
�y � x0�
� �x0��=�2�=
x (y � x0�) =�21 + (y � x0�) (x0�) =�2
the score is
s =@
@�log f (y;x j �0) = xe=�2:
The Cramer-Rao bound for estimation of � (and therefore � (�) as well) is�E�ss0���1
=���4E
�(xe) (xe)0
���1= �2Q�1 = V 0:
We have shown that there is a parametric submodel (4.22) whose Cramer-Rao bound for estimationof � is identical to the asymptotic variance of the least-squares estimator, which therefore is thesemiparametric variance bound.
Theorem 4.13.1 In the homoskedastic regression model (2.10)-(2.12), the semiparametric vari-ance bound for estimation of � is V 0 = �2Q�1 and the OLS estimator is semiparametricallye¢ cient.
This result is similar to the Gauss-Markov theorem, in that it asserts the e¢ ciency of the least-squares estimator in the context of the homoskedastic regression model. The di¤erence is that theGauss-Markov theorem states that OLS has the smallest variance among the set of unbiased linearestimators, while Theorem 4.13.1 states that OLS has the smallest asymptotic variance amongregular estimators. This is a much more powerful statement.
4.14 Problems with Tests of NonLinear Hypotheses
While the t and Wald tests work well when the hypothesis is a linear restriction on �; theycan work quite poorly when the restrictions are nonlinear. This can be seen by a simple exampleintroduced by Lafontaine and White (1986). Take the model
yi = � + ei
ei � N(0; �2)
and consider the hypothesisH0 : � = 1:
49
Let � and �2 be the sample mean and variance of yi: Then the standard Wald test for H0 is
Wn = n
�� � 1
�2�2
:
Now notice that H0 is equivalent to the hypothesis
H0(r) : �r = 1
for any positive integer r: Letting h(�) = �r; and noting H� = r�r�1; we �nd that the standardWald test for H0(r) is
Wn(r) = n
��r � 1
�2�2r2�
2r�2 :
While the hypothesis �r = 1 is una¤ected by the choice of r; the statistic Wn(r) varies with r: Thisis an unfortunate feature of the Wald statistic.
To demonstrate this e¤ect, we have plotted in Figure 4.4 the Wald statistic Wn(r) as a functionof r; setting n=�2 = 10: The increasing solid line is for the case � = 0:8: The decreasing dashedline is for the case � = 1:6: It is easy to see that in each case there are values of r for which thetest statistic is signi�cant relative to asymptotic critical values, while there are other values of rfor which the test statistic is insigni�cant. This is distressing since the choice of r is arbitrary andirrelevant to the actual hypothesis.
Figure 4.4: Wald Statistic as a function of s
Our �rst-order asymptotic theory is not useful to help pick r; as Wn(r)d�! �21 under H0 for
any r: This is a context where Monte Carlo simulation can be quite useful as a tool to studyand compare the exact distributions of statistical procedures in �nite samples. The method usesrandom simulation to create an arti�cial dataset to apply the statistical tools of interest. Thisproduces random draws from the sampling distribution of interest. Through repetition, features ofthis distribution can be calculated.
In the present context of the Wald statistic, one feature of importance is the Type I errorof the test using the asymptotic 5% critical value 3.84 � the probability of a false rejection,P (Wn(r) > 3:84 j � = 1) : Given the simplicity of the model, this probability depends only on r; n;
50
and �2: In Table 2.1 we report the results of a Monte Carlo simulation where we vary these threeparameters. The value of r is varied from 1 to 10, n is varied among 20, 100 and 500, and � isvaried among 1 and 3. Table 4.1 reports the simulation estimate of the Type I error probabilityfrom 50,000 random samples. Each row of the table corresponds to a di¤erent value of r �and thuscorresponds to a particular choice of test statistic. The second through seventh columns contain theType I error probabilities for di¤erent combinations of n and �. These probabilities are calculatedas the percentage of the 50,000 simulated Wald statistics Wn(r) which are larger than 3.84. Thenull hypothesis �r = 1 is true, so these probabilities are Type I error.
To interpret the table, remember that the ideal Type I error probability is 5% (.05) withdeviations indicating distortion. Typically, Type I error rates between 3% and 8% are consideredreasonable. Error rates above 10% are considered excessive. Rates above 20% are unacceptable.When comparing statistical procedures, we compare the rates row by row, looking for tests forwhich rejection rates are close to 5% and rarely fall outside of the 3%-8% range. For this particularexample the only test which meets this criterion is the conventional Wn = Wn(1) test. Any otherchoice of r leads to a test with unacceptable Type I error probabilities.
In Table 4.1 you can also see the impact of variation in sample size. In each case, the Type Ierror probability improves towards 5% as the sample size n increases. There is, however, no magicchoice of n for which all tests perform uniformly well. Test performance deteriorates as r increases,which is not surprising given the dependence of Wn(r) on r as shown in Figure 4.4.
Table 4.1Type I error Probability of Asymptotic 5% Wn(r) Test
� = 1 � = 3
r n = 20 n = 100 n = 500 n = 20 n = 100 n = 500
1 .06 .05 .05 .07 .05 .052 .08 .06 .05 .15 .08 .063 .10 .06 .05 .21 .12 .074 .13 .07 .06 .25 .15 .085 .15 .08 .06 .28 .18 .106 .17 .09 .06 .30 .20 .117 .19 .10 .06 .31 .22 .138 .20 .12 .07 .33 .24 .149 .22 .13 .07 .34 .25 .1510 .23 .14 .08 .35 .26 .16
Note: Rejection frequencies from 50,000 simulated random samples
In this example it is not surprising that the choice r = 1 yields the best test statistic. Otherchoices are arbitrary and would not be used in practice. While this is clear in this particularexample, in other examples natural choices are not always obvious and the best choices may in factappear counter-intuitive at �rst.
This point can be illustrated through another example which is similar to one developed inGregory and Veall (1985). Take the model
yi = �0 + x1i�1 + x2i�2 + ei (4.23)
E (xiei) = 0
and the hypothesis
H0 :�1�2= r
where r is a known constant. Equivalently, de�ne � = �1=�2; so the hypothesis can be stated asH0 : � = r:
51
Let � = (�0; �1; �2) be the least-squares estimates of (4.23), let V be an estimate of theasymptotic variance matrix for � and set � = �1=�2. De�ne
H1 =
0BBBBBBBBB@
0
1
�2
� �1�2
2
1CCCCCCCCCAso that the standard error for � is s(�) =
�n�1H
01V H1
�1=2: In this case a t-statistic for H0 is
t1n =
��1�2� r�
s(�):
An alternative statistic can be constructed through reformulating the null hypothesis as
H0 : �1 � r�2 = 0:
A t-statistic based on this formulation of the hypothesis is
t2n =
��1 � r�2
�2�n�1H 0
2V H2
�1=2 :where
H2 =
0@ 01�r
1A :To compare t1n and t2n we perform another simple Monte Carlo simulation. We let x1i and x2i
be mutually independent N(0; 1) variables, ei be an independent N(0; �2) draw with � = 3, andnormalize �0 = 0 and �1 = 1: This leaves �2 as a free parameter, along with sample size n: Wevary �2 among :1; .25, .50, .75, and 1.0 and n among 100 and 500:
Table 4.2Type I error Probability of Asymptotic 5% t-tests
n = 100 n = 500
P (tn < �1:645) P (tn > 1:645) P (tn < �1:645) P (tn > 1:645)
�2 t1n t2n t1n t2n t1n t2n t1n t2n.10 .47 .06 .00 .06 .28 .05 .00 .05.25 .26 .06 .00 .06 .15 .05 .00 .05.50 .15 .06 .00 .06 .10 .05 .00 .05.75 .12 .06 .00 .06 .09 .05 .00 .051.00 .10 .06 .00 .06 .07 .05 .02 .05
The one-sided Type I error probabilities P (tn < �1:645) and P (tn > 1:645) are calculated from50,000 simulated samples. The results are presented in Table 4.2. Ideally, the entries in the tableshould be 0.05. However, the rejection rates for the t1n statistic diverge greatly from this value,especially for small values of �2: The left tail probabilities P (t1n < �1:645) greatly exceed 5%,while the right tail probabilities P (t1n > 1:645) are close to zero in most cases. In contrast, the
52
rejection rates for the linear t2n statistic are invariant to the value of �2; and are close to theideal 5% rate for both sample sizes. The implication of Table 4.2 is that the two t-ratios havedramatically di¤erent sampling behavior.
The common message from both examples is that Wald statistics are sensitive to the algebraicformulation of the null hypothesis. In all cases, if the hypothesis can be expressed as a linearrestriction on the model parameters, this formulation should be used. If no linear formulation isfeasible, then the �most linear�formulation should be selected (as suggested by the theory of Parkand Phillips (1988)), and alternatives to asymptotic critical values should be considered. It is alsoprudent to consider alternative tests to the Wald statistic, such as the GMM distance statisticwhich will be presented in Section 7.7 (as advocated by Hansen (2006)).
4.15 Monte Carlo Simulation
In the previous section we introduced the method of Monte Carlo simulation to illustrate thesmall sample problems with tests of nonlinear hypotheses. In this section we describe the methodin more detail.
Recall, our data consist of observations (yi;xi) which are random draws from a populationdistribution F: Let � be a parameter and let Tn = Tn ((y1;x1) ; :::; (yn;xn) ;�) be a statistic ofinterest, for example an estimator � or a t-statistic (� � �)=s(�): The exact distribution of Tn is
Gn(u; F ) = P (Tn � u j F ) :
While the asymptotic distribution of Tn might be known, the exact (�nite sample) distribution Gnis generally unknown.
Monte Carlo simulation uses numerical simulation to compute Gn(u; F ) for selected choices of F:This is useful to investigate the performance of the statistic Tn in reasonable situations and samplesizes. The basic idea is that for any given F; the distribution function Gn(u; F ) can be calculatednumerically through simulation. The name Monte Carlo derives from the famous Mediterraneangambling resort where games of chance are played.
The method of Monte Carlo is quite simple to describe. The researcher chooses F (the dis-tribution of the data) and the sample size n. A �true� value of � is implied by this choice, orequivalently the value � is selected directly by the researcher which implies restrictions on F .
Then the following experiment is conducted
� n independent random pairs (y�i ;x�i ) ; i = 1; :::; n; are drawn from the distribution F using
the computer�s random number generator.
� The statistic Tn = Tn ((y�1;x�1) ; :::; (y�n;x�n) ;�) is calculated on this pseudo data.
For step 1, most computer packages have built-in procedures for generating U[0; 1] and N(0; 1)random numbers, and from these most random variables can be constructed. (For example, achi-square can be generated by sums of squares of normals.)
For step 2, it is important that the statistic be evaluated at the �true�value of � correspondingto the choice of F:
The above experiment creates one random draw from the distribution Gn(u; F ): This is oneobservation from an unknown distribution. Clearly, from one observation very little can be said.So the researcher repeats the experiment B times, where B is a large number. Typically, we setB = 1000 or B = 5000: We will discuss this choice later.
Notationally, let the b0th experiment result in the draw Tnb; b = 1; :::; B: These results are stored.They constitute a random sample of size B from the distribution of Gn(u; F ) = P (Tnb � u) =P (Tn � u j F ) :
From a random sample, we can estimate any feature of interest using (typically) a method ofmoments estimator. For example:
53
Suppose we are interested in the bias, mean-squared error (MSE), or variance of the distributionof � � �: We then set Tn = � � �; run the above experiment, and calculate
\Bias(�) =1
B
BXb=1
Tnb =1
B
BXb=1
�b � �
\MSE(�) =1
B
BXb=1
(Tnb)2 =
1
B
BXb=1
��b � �
�\var(�) = \MSE(�)�
�\Bias(�)
�2Suppose we are interested in the Type I error associated with an asymptotic 5% two-sided t-test.
We would then set Tn =���� � ���� =s(�) and calculate
P =1
B
BXb=1
1 (Tnb � 1:96) ; (4.24)
the percentage of the simulated t-ratios which exceed the asymptotic 5% critical value.Suppose we are interested in the 5% and 95% quantile of Tn = �:We then compute the 5% and
95% sample quantiles of the sample fTnbg: The �% sample quantile is a number q� such that �% ofthe sample are less than q�: A simple way to compute sample quantiles is to sort the sample fTnbgfrom low to high. Then q� is the N�th number in this ordered sequence, where N = (B + 1)�: Itis therefore convenient to pick B so that N is an integer. For example, if we set B = 999; then the5% sample quantile is 50�th sorted value and the 95% sample quantile is the 950�th sorted value.
The typical purpose of a Monte Carlo simulation is to investigate the performance of a statisticalprocedure (estimator or test) in realistic settings. Generally, the performance will depend on n andF: In many cases, an estimator or test may perform wonderfully for some values, and poorly forothers. It is therefore useful to conduct a variety of experiments, for a selection of choices of n andF:
As discussed above, the researcher must select the number of experiments, B: Often this iscalled the number of replications. Quite simply, a larger B results in more precise estimates ofthe features of interest of Gn; but requires more computational time. In practice, therefore, thechoice of B is often guided by the computational demands of the statistical procedure. Since theresults of a Monte Carlo experiment are estimates computed from a random sample of size B; itis straightforward to calculate standard errors for any quantity of interest. If the standard error istoo large to make a reliable inference, then B will have to be increased.
In particular, it is simple to make inferences about rejection probabilities from statistical tests,such as the percentage estimate reported in (4.24). The random variable 1 (Tnb � 1:96) is iidBernoulli, equalling 1 with probability p = E1 (Tnb � 1:96) : The average (4.24) is therefore anunbiased estimator of p with standard error s (p) =
pp (1� p) =B. As p is unknown, this may be
approximated by replacing p with p or with an hypothesized value. For example, if we are assessingan asymptotic 5% test, then we can set s (p) =
p(:05) (:95) =B ' :22=
pB: Hence, standard errors
for B = 100; 1000, and 5000, are, respectively, s (p) = :022; :007; and .003.
4.16 Estimating a Wage Equation
We again return to our wage equation. We use the sample of wage earners from the March 2004Current Population Survey, excluding military. For the dependent variable we use the natural logof wages so that coe¢ cients may be interpreted as semi-elasticities. For regressors we include yearsof education, potential work experience, experience squared, and dummy variable indicators forthe following: married, female, union member, immigrant, hispanic, and non-white. Furthermore,
54
we included a dummy variable for state of residence (including the District of Columbia, this adds50 regressors). The available sample is 18,808 so the parameter estimates are quite precise andreported in Table 4.1, excluding the coe¢ cients on the state dummy variables.
Table 4.1 displays the parameter estimates in a standard format. The Table clearly states theestimation method (OLS), the dependent variable (log(Wage)), and the regressors are clearly la-beled. Parameter estimates are both reported for the coe¢ cients of interest (the coe¢ cients on thestate dummy variables are omitted) and standard errors are reported for all reported coe¢ cientestimates. In addition to the coe¢ cient estimates, the table also reports the estimated error stan-dard deviation; the sample size and the regression R2: These are useful summary measures of �twhich aid readers.
Table 4.1OLS Estimates of Linear Equation for Log(Wage)
� s(�)Intercept 1.027 .032Education .101 .002Experience .033 .001Experience2 �:00057 .00002Married .102 .008Female �:232 .007Union Member .097 .010Immigrant �:121 .013Hispanic �:102 .014Non-White �:070 .010� .4877Sample Size 18,808R2 .34
Note: Equation also includes state dummy variables.
As a general rule, it is best to always report standard errors along with parameter estimates(as done in Table 4.1). This allows readers to assess the precision of the parameter estimates, andform con�dence intervals and t-tests on individual coe¢ cients if desired. For example, if you areinterested in the di¤erence in mean wages between men and women, you can read from the tablethat the estimated coe¢ cient on the Female dummy variable is �0:232; implying a mean wagedi¤erence of 23%. To assess the precision, you can see that the standard error for this coe¢ cientestimate is 0.007. This implies a 95% asymptotic con�dence interval for the coe¢ cient estimate of[�:246;�:218]. This means that we have estimated the di¤erence in mean wages between men andwomen to lie between 22% and 25%. I interpret this as a precise estimate because there is not animportant di¤erence between the lower and upper bound.
Instead of reporting standard errors, some empirical researchers report t-ratios for each pa-rameter estimate. �t-ratios� are t-statistics which test the hypothesis that the coe¢ cient equalszero. An example is reported in Table 4.2. In this example, all the t-ratios are highly signi�cant,ranging in magnitude from 9.3 to 50. What we learn from these statistics is that these coe¢ cientsare non-zero, but not much more. In a sample of this size this �nding is rather uninteresting;consequently the reporting of t-ratios is a waste of space. Again consider the male-female wagedi¤erence. Table 4.2 reports taht the t-ratio is 33, enabling us to reject the hypothesis that thecoe¢ cient is zero. But how precise is the reported estimate of a wage gap of 23%? It is hard toassess from a quick reading of Table 4.2 Standard errors are much more useful, for they enable forquick and easy assessment of the degree of estimation uncertainty.
55
Table 4.2OLS Estimates of Linear Equation for Log(Wage)
Improper Reporting: t-ratios replacing standard errors
� tIntercept 1.027 32Education .101 50Experience .033 33Experience2 �:00057 28Married .102 12.8Female �:232 33Union Member .097 9.7Immigrant �:121 9:3Hispanic �:102 7:3Non-White �:070 7
Returning to the estimated wage equation, one might question whether or not the state dummyvariables are relevant. Computing the Wald statistic (4.18) that the state coe¢ cients are jointlyzero, we �ndWn = 550: Alternatively, re-estimating the model with the 50 state dummies excluded,the restricted standard deviation estimate is ~� = :4945: The F form of the Wald statistic (4.19) is
Wn = n
�~�2
�2� 1�= 18; 808
�:49452
:48772� 1�= 528:
Notice that the two statistics are close, but not equal. Using either statistic the hypothesis is easilyrejected, as the 1% critical value for the �250 distribution is 76.
Another interesting question which can be addressed from these estimates is the maximal impactof experience on mean wages. Ignoring the other coe¢ cients, we can write this e¤ect as
log(Wage) = �2Experience+ �3Experience2 + � � �
Our question is: At which level of experience � do workers achieve the highest wage? In thisquadratic model, if �2 > 0 and �3 < 0 the solution is
� = � �22�3
:
From Table 4.1 we �nd the point estimate
� = � �2
2�3= 28:69:
Using the Delta Method, we can calculate a standard error of s(�) = :40; implying a 95% con�denceinterval of [27:9; 29:5]:
However, this is a poor choice, as the coverage probability of this con�dence interval is one minusthe Type I error of the hypothesis test based on the t-test. In the previous section we discoveredthat such t-tests had very poor Type I error rates. Instead, we found better Type I error rates byreformulating the hypothesis as a linear restriction. These t-statistics take the form
tn(�) =�2 + 2�3��h0�V h�
�1=2
56
where
h� =
�12�
�and V is the covariance matrix for (�2 �3):
In the present context we are interested in forming a con�dence interval, not testing a hypothesis,so we have to go one step further. Our desired con�dence interval will be the set of parameter values� which are not rejected by the hypothesis test. This is the set of � such that jtn(�)j � 1:96: Sincetn(�) is a non-linear function of �; there is not a simple expression for this set, but it can be foundnumerically quite easily. This set is [27:0; 29:5]: Notice that the upper end of the con�dence intervalis the same as that from the delta method, but the lower end is substantially lower.
4.17 Technical Proofs
Proof of Theorem 4.2.1. In order to apply the WLLN to the sample moments 1nPni=1 xix
0i;
1n
Pni=1 xiyi; or
1n
Pni=1 xiei we need to verify that the random variables xjixli; xjyi and xjei are iid
with �nite absolute �rst moments. These moment conditions are covered by Assumptions 2.6.1 andTheorem 2.6.1. Assumption 3.1.1 states that the observations (yi;xi) are mutually independentand identically distributed. It follows that any function zi = h (yi;xi) is also iid. This includes ei;xjixli; xjyi and xjei: We have veri�ed that the random variables xjixli; xjyi and xjei are iid with�nite absolute �rst moments and therefore the conditions for the WLLN hold. Equations (4.1),(4.2) and (4.5) are therefore valid.
The �nal step of the proof is the application of the continuous mapping theory to obtain (4.3).To fully understand this applicationwe now walk through its application in more detail. Using (4.4)we can write
� � � =
1
n
nXi=1
xix0i
!�1 1
n
nXi=1
xiyi
!
= g
1
n
nXi=1
xix0i;1
n
nXi=1
xiyi
!
where g (A; b) = A�1b is a function of A and b: This function is a continuous function of A andb at all values of the arguments such that A�1 exists. Assumption 2.6.1.4 implies that Q�1 existsand thus g (A; b) is continuous at A = Q: Hence by the continuous mapping theorem (TheoremC.4.1),
� � � = g 1
n
nXi=1
xix0i;1
n
nXi=1
xiyi
!p�! g (Q;E (xiyi))
= E�xix
0i
��1E (xiyi)
= �
�
Proof of Theorem 4.2.2. Note that
ei = yi � x0i�= ei + x
0i� � x0i�
= ei � x0i�� � �
�:
57
Thuse2i = e
2i � 2eix0i
�� � �
�+�� � �
�0xix
0i
�� � �
�(4.25)
and
�2 =1
n
nXi=1
e2i
=1
n
nXi=1
e2i � 2 1
n
nXi=1
eix0i
!�� � �
�+�� � �
�0 1n
nXi=1
xix0i
!�� � �
�p�! �2
the last line using the WLLN, (4.1), (4.5) and Theorem (4.2.1). Thus �2 is consistent for �2:Finally, since n=(n� k)! 1 as n!1; it follows that
s2 =
�n
n� k
��2
p�! �2:
�
Proof of Theorem 4.3.1. The discussion preceeding the statement of the theorem, togetherwith the proof of Theorem 4.2.1, covered all points of the derivation except for the demonstrationthat the elements of the vector xiei have �nite second moments, which we now show. For anyj = 1; :::; k; by the Cauchy-Schwarz Inequality (C.3) and Assumption 4.3.1
E jxjieij2 = E��x2jie2i �� � �Ex4ji�1=2 �Ee4i �1=2 <1: (4.26)
�
Proof of Theorem 4.4.1. We now show p�! ; from which it follows that V
p�! V asn!1: Using (4.25)
=1
n
nXi=1
xix0ie2i
=1
n
nXi=1
xix0ie2i �
2
n
nXi=1
xix0i
�� � �
�0xiei +
1
n
nXi=1
xix0i
��� � �
�0xi
�2: (4.27)
We now examine each k � k sum on the right-hand-side of (4.27) in turn.Take the �rst term on the right-hand-side of (4.27). The jl�th element of xix0ie
2i is xjixlie
2i .
Using the Cauchy-Schwarz Inequality (C.3) twice and Assumption 4.3.1,
E��xjixlie2i �� �
�Ex2jix
2li
�1=2 �Ee4i�1=2
��Ex4ji
�1=4 �Ex4li
�1=4 �Ee4i�1=2
:
Since this expectation is �nite, we can apply the WLLN (Theorem C.2.1) to �nd that
1
n
nXi=1
xix0ie2i
p�! E�xix
0ie2i
�= :
Now take the second term on the right-hand-side of (4.27). Applying the Triangle Inequality(A.9) to the matrix Euclidean norm, the Matrix Schwarz Inequality (A.8), equation (A.6) and the
58
Schwarz Inequality (A.7) 2nnXi=1
xix0i
�� � �
�0xiei
� 2
n
nXi=1
xix0i �� � ��0 xiei � 2
n
nXi=1
xix0i ������ � ��0 xi���� jeij�
2
n
nXi=1
kxik3 jeij! � � � : (4.28)
Using Holder�s inequality (C.4) and Assumption 4.3.1,
E�kxik3 jeij
���E kxik4
�3=4 �E��e4i ���1=4 <1:
By the WLLN1
n
nXi=1
kxik3 jeijp�! E
�kxik3 jeij
�<1:
Since ��� p�! 0 it follows that (4.28) converges in probability to zero. This shows that the secondterm on the right-hand-side of (4.27) converges in probability to zero.
We now take the third term in (4.27). Again by the Triangle Inequality, the Matrix SchwarzInequality, (A.6) and the Schwarz Inequality 1n
nXi=1
xix0i
��� � �
�0xi
�2 � 1
n
nXi=1
xix0i ��� � ��0 xi�2� 1
n
nXi=1
kxik4 � � �
p�! 0
the �nal convergence since � � � p�! 0 and 1n
Pni=1 kxik
4 p�! E kxik4 < 1 under Assumption4.3.1. This shows that the third term on the right-hand-side of (4.27) converges in probability tozero.
Considering the three terms on the right-hand-side of (4.27), we have shown that the �rstterm converges in probability to , and the second and third converge in probability to zero. Weconclude that �! as claimed. �
Proof of Theorem 4.7.1. By (4.15)
tn(�) =� � �s(�)
=
pn�� � �
�qV�
d�! N(0; V�)pV�
= N(0; 1)
�
59
4.18 Exercises
For exercises 1-4, the following de�nition is used. In the model y = X� + e; the least-squaresestimate of � subject to the restriction h(�) = 0 is
~� = argminh(�)=0
Sn(�)
Sn(�) = (y �X�)0 (y �X�) :
That is, ~� minimizes the sum of squared errors Sn(�) over all � such that the restriction holds.
1. In the model y = X1�1 +X2�2 + e; show that the least-squares estimate of � = (�1;�2)subject to the constraint that �2 = 0 is the OLS regression of y on X1:
2. In the model y = X1�1 +X2�2 + e; show that the least-squares estimate of � = (�1;�2);subject to the constraint that �1 = c (where c is some given vector) is simply the OLSregression of y �X1c on X2:
3. In the model y = X1�1 +X2�2 + e; with X1 and X2 each n � k; �nd the least-squaresestimate of � = (�1;�2); subject to the constraint that �1 = ��2:
4. Take the model y =X�+e with the restriction R0� = r where R is a known k�s matrix, ris a known s� 1 vector, 0 < s < k; and rank(R) = s: Explain why ~� solves the minimizationof the Lagrangian
L(�;�) = 1
2Sn(�) + �
0 �R0� � r�
where � is s� 1:
(a) Show that the solution is
~� = � ��X 0X
��1RhR0 �X 0X
��1Ri�1 �
R0� � r�
� =hR0 �X 0X
��1Ri�1 �
R0� � r�
where� =
�X 0X
��1X 0y
is the unconstrained OLS estimator.
(b) Verify that R0~� = r:
(c) Show that if R0� = r is true, then
~� � � =�Ik �
�X 0X
��1RhR0 �X 0X
��1Ri�1
R0��X 0X
��1X 0e:
(d) Under the standard assumptions plus R0� = r; �nd the asymptotic distribution ofpn�~� � �
�as n!1:
(e) Find an appropriate formula to calculate standard errors for the elements of ~�:
5. You have two independent samples (y1;X1) and (y2;X2) which satisfy y1 =X1�1+e1 andy2 = X2�2 + e2; where E (x1ie1i) = 0 and E (x2ie2i) = 0; and both X1 and X2 have kcolumns. Let �1 and �2 be the OLS estimates of �1 and �2. For simplicity, you may assumethat both samples have the same number of observations n:
60
(a) Find the asymptotic distribution ofpn���2 � �1
�� (�2 � �1)
�as n!1:
(b) Find an appropriate test statistic for H0 : �2 = �1:
(c) Find the asymptotic distribution of this statistic under H0:
6. The model is
yi = x0i� + ei
E (xiei) = 0
= E�xix
0ie2i
�:
(a) Find the method of moments estimators��;
�for (�;) :
(b) In this model, are��;
�e¢ cient estimators of (�;)?
(c) If so, in what sense are they e¢ cient?
7. Take the model yi = x01i�1 + x02i�2 + ei with Exiei = 0: Suppose that �1 is estimated
by regressing yi on x1i only. Find the probability limit of this estimator. In general, is itconsistent for �1? If not, under what conditions is this estimator consistent for �1?
8. Verify that equation (4.13) equals (4.14) as claimed in Section 4.5.
9. Prove that if an additional regressor Xk+1 is added to X; Theil�s adjusted R2increases if
and only if jtk+1j > 1; where tk+1 = �k+1=s(�k+1) is the t-ratio for �k+1 and
s(�k+1) =�s2[(X 0X)�1]k+1;k+1
�1=2is the homoskedasticity-formula standard error.
10. Let y be n � 1; X be n � k (rank k): y = X� + e with E(xiei) = 0: De�ne the ridgeregression estimator
� =
nXi=1
xix0i + �Ik
!�1 nXi=1
xiyi
!where � > 0 is a �xed constant. Find the probability limit of � as n ! 1: Is � consistentfor �?
11. Of the variables (y�i ; yi;xi) only the pair (yi;xi) are observed. In this case, we say that y�i is
a latent variable. Suppose
y�i = x0i� + ei
E (xiei) = 0
yi = y�i + ui
where ui is a measurement error satisfying
E (xiui) = 0
E (y�i ui) = 0
Let � denote the OLS coe¢ cient from the regression of yi on xi:
(a) Is � the coe¢ cient from the linear projection of yi on xi?
(b) Is � consistent for � as n!1?
61
(c) Find the asymptotic distribution ofpn�� � �
�as n!1:
12. The data set invest.dat contains data on 565 U.S. �rms extracted from Compustat for theyear 1987. The variables, in order, are
� Ii Investment to Capital Ratio (multiplied by 100).
� Qi Total Market Value to Asset Ratio (Tobin�s Q).
� Ci Cash Flow to Asset Ratio.
� Di Long Term Debt to Asset Ratio.
The �ow variables are annual sums for 1987. The stock variables are beginning of year.
(a) Estimate a linear regression of Ii on the other variables. Calculate appropriate standarderrors.
(b) Calculate asymptotic con�dence intervals for the coe¢ cients.
(c) This regression is related to Tobin�s q theory of investment, which suggests that invest-ment should be predicted solely by Qi: Thus the coe¢ cient on Qi should be positive andthe others should be zero. Test the joint hypothesis that the coe¢ cients on Ci and Diare zero. Test the hypothesis that the coe¢ cient on Qi is zero. Are the results consistentwith the predictions of the theory?
(d) Now try a non-linear (quadratic) speci�cation. Regress Ii on Qi; Ci, Di; Q2i ; C2i , D
2i ;
QiCi; QiDi; CiDi: Test the joint hypothesis that the six interaction and quadratic coef-�cients are zero.
13. In a paper in 1963, Marc Nerlove analyzed a cost function for 145 American electric companies.(The problem is discussed in Example 8.3 of Greene, section 1.7 of Hayashi, and the empiricalexercise in Chapter 1 of Hayashi). The data �le nerlov.dat contains his data. The variablesare described on page 77 of Hayashi. Nerlov was interested in estimating a cost function:TC = f(Q;PL; PF; PK):
(a) First estimate an unrestricted Cobb-Douglass speci�cation
log TCi = �1 + �2 logQi + �3 logPLi + �4 logPKi + �5 logPFi + ei: (4.29)
Report parameter estimates and standard errors. You should obtain the same OLSestimates as in Hayashi�s equation (1.7.7), but your standard errors may di¤er.
(b) Using a Wald statistic, test the hypothesis H0 : �3 + �4 + �5 = 1:
(c) Estimate (4.29) by least-squares imposing this restriction by substitution. Report yourparameter estimates and standard errors.
(d) Estimate (4.29) subject to �3 + �4 + �5 = 1 using the restricted least-squares estimatorfrom problem 4. Do you obtain the same estimates as in part (c)?
.
62
Chapter 5
Additional Regression Topics
5.1 Generalized Least Squares
In the projection model, we know that the least-squares estimator is semi-parametrically e¢ cientfor the projection coe¢ cient. However, in the linear regression model
yi = x0i� + ei
E (ei j xi) = 0;
the least-squares estimator is ine¢ cient. The theory of Chamberlain (1987) can be used to showthat in this model the semiparametric e¢ ciency bound is obtained by the Generalized LeastSquares (GLS) estimator
~� =�X 0D�1X
��1 �X 0D�1y
�(5.1)
where D = diagf�21; :::; �2ng and �2i = �2(xi) = E�e2i j xi
�. The GLS estimator is sometimes called
the Aitken estimator. The GLS estimator (5.1) is infeasible since the matrix D is unknown. Afeasible GLS (FGLS) estimator replaces the unknown D with an estimate D = diagf�21; :::; �2ng:We now discuss this estimation problem.
Suppose that we model the conditional variance using the parametric form
�2i = �0 + z01i�1
= �0zi;
where z1i is some q � 1 function of xi: Typically, z1i are squares (and perhaps levels) of some (orall) elements of xi: Often the functional form is kept simple for parsimony.
Let �i = e2i : Then
E (�i j xi) = �0 + z01i�1and we have the regression equation
�i = �0 + z01i�1 + �i (5.2)
E (�i j xi) = 0:
This regression error �i is generally heteroskedastic and has the conditional variance
var (�i j xi) = var�e2i j xi
�= E
��e2i � E
�e2i j xi
��2 j xi�= E
�e4i j xi
���E�e2i j xi
��2:
Suppose ei (and thus �i) were observed. Then we could estimate � by OLS:
� =�Z 0Z
��1Z 0�
p�! �
63
and pn (���) d�! N(0;V �)
whereV � =
�E�ziz
0i
���1E�ziz
0i�2i
� �E�ziz
0i
���1: (5.3)
While ei is not observed, we have the OLS residual ei = yi � x0i� = ei � x0i(� � �): Thus
�i � � � �i= e2i � e2i= �2eix0i
�� � �
�+ (� � �)0xix0i(� � �):
And then
1pn
nXi=1
zi�i =�2n
nXi=1
zieix0i
pn�� � �
�+1
n
nXi=1
zi(� � �)0xix0i(� � �)pn
p�! 0
Let~� =
�Z 0Z
��1Z 0� (5.4)
be from OLS regression of �i on zi: Then
pn (~���) =
pn (���) +
�n�1Z 0Z
��1n�1=2Z 0�
d�! N(0;V �) (5.5)
Thus the fact that �i is replaced with �i is asymptotically irrelevant. We may call (5.4) the skedasticregression, as it is estimating the conditional variance of the regression of yi on xi: We have shownthat � is consistently estimated by a simple procedure, and hence we can estimate �2i = z
0i� by
~�2i = ~�0zi: (5.6)
Suppose that ~�2i > 0 for all i: Then set
~D = diagf~�21; :::; ~�2ng
and~� =
�X 0 ~D
�1X��1
X 0 ~D�1y:
This is the feasible GLS, or FGLS, estimator of �: Since there is not a unique speci�cation forthe conditional variance the FGLS estimator is not unique, and will depend on the model (andestimation method) for the skedastic regression.
One typical problem with implementation of FGLS estimation is that in the linear speci�cation(5.2), there is no guarantee that ~�2i > 0 for all i: If ~�2i < 0 for some i; then the FGLS estimatoris not well de�ned. Furthermore, if ~�2i � 0 for some i then the FGLS estimator will force theregression equation through the point (yi;xi); which is typically undesirable. This suggests thatthere is a need to bound the estimated variances away from zero. A trimming rule might makesense:
�2i = max[~�2i ; �
2]
for some �2 > 0:It is possible to show that if the skedastic regression is correctly speci�ed, then FGLS is asymp-
totically equivalent to GLS. As the proof is tricky, we just state the result without proof.
64
Theorem 5.1.1 If the skedastic regression is correctly speci�ed,pn�~�GLS � ~�FGLS
�p�! 0;
and thus pn�~�FGLS � �
�d�! N(0;V ) ;
whereV =
�E���2i xix
0i
���1:
Examining the asymptotic distribution of Theorem 5.1.1, the natural estimator of the asymp-totic variance of � is
~V0=
1
n
nXi=1
~��2i xix0i
!�1=
�1
nX 0 ~D
�1X
��1:
which is consistent for V as n!1: This estimator ~V 0is appropriate when the skedastic regression
(5.2) is correctly speci�ed.It may be the case that �0zi is only an approximation to the true conditional variance �2i =
E(e2i j xi). In this case we interpret �0zi as a linear projection of e2i on zi: ~� should perhaps becalled a quasi-FGLS estimator of �: Its asymptotic variance is not that given in Theorem 5.1.1.Instead,
V =�E���0zi
��1xix
0i
���1 �E���0zi
��2�2ixix
0i
���E���0zi
��1xix
0i
���1:
V takes a sandwich form similar to the covariance matrix of the OLS estimator. Unless �2i = �0zi,
~V0is inconsistent for V .An appropriate solution is to use a White-type estimator in place of ~V
0: This may be written
as
~V =
1
n
nXi=1
~��2i xix0i
!�1 1
n
nXi=1
~��4i e2ixix
0i
! 1
n
nXi=1
~��2i xix0i
!�1= n
�X 0 ~D
�1X��1 �
X 0 ~D�1D ~D
�1X��X 0 ~D
�1X��1
where D = diagfe21; :::; e2ng: This is estimator is robust to misspeci�cation of the conditional vari-ance, and was proposed by Cragg (1992).
In the linear regression model, FGLS is asymptotically superior to OLS. Why then do we notexclusively estimate regression models by FGLS? This is a good question. There are three reasons.
First, FGLS estimation depends on speci�cation and estimation of the skedastic regression.Since the form of the skedastic regression is unknown, and it may be estimated with considerableerror, the estimated conditional variances may contain more noise than information about the trueconditional variances. In this case, FGLS will do worse than OLS in practice.
Second, individual estimated conditional variances may be negative, and this requires trimmingto solve. This introduces an element of arbitrariness which is unsettling to empirical researchers.
Third, OLS is a robust estimator of the parameter vector. It is consistent not only in theregression model, but also under the assumptions of linear projection. The GLS and FGLS esti-mators, on the other hand, require the assumption of a correct conditional mean. If the equationof interest is a linear projection, and not a conditional mean, then the OLS and FGLS estimatorswill converge in probability to di¤erent limits, as they will be estimating two di¤erent projections.And the FGLS probability limit will depend on the particular function selected for the skedasticregression. The point is that the e¢ ciency gains from FGLS are built on the stronger assumptionof a correct conditional mean, and the cost is a loss of robustness to misspeci�cation.
65
5.2 Testing for Heteroskedasticity
The hypothesis of homoskedasticity is that E�e2i j xi
�= �2, or equivalently that
H0 : �1 = 0
in the regression (5.2). We may therefore test this hypothesis by the estimation (5.4) and con-structing a Wald statistic. At this point it is typical to impose the stronger assumption that ei isindependent of xi; in which case �i is independent of xi and the asymptotic variance (5.3) for ~�simpli�es to
V� =�E�ziz
0i
���1E��2i�: (5.7)
Hence the standard test of H0 is a classic F (or Wald) test for exclusion of all regressors fromthe skedastic regression (5.4). The asymptotic distribution (5.5) and the asymptotic variance (5.7)under independence show that this test has an asymptotic chi-square distribution.
Theorem 5.2.1 Under H0 and ei independent of xi; the Wald test of H0 is asymptotically �2q :
Most tests for heteroskedasticity take this basic form. The main di¤erences between popular�tests�are which transformations of xi enter zi: Motivated by the form of the asymptotic varianceof the OLS estimator �; White (1980) proposed that the test for heteroskedasticity be based onsetting zi to equal all non-redundant elements of xi; its squares, and all cross-products. Breusch-Pagan (1979) proposed what might appear to be a distinct test, but the only di¤erence is that theyallowed for general choice of zi; and replaced E
��2i�with 2�4 which holds when ei is N
�0; �2
�: If
this simpli�cation is replaced by the standard formula (under independence of the error), the twotests coincide.
It is important not to misuse tests for heteroskedasticity. It should not be used to determinewhether to estimate a regression equation by OLS or FGLS, nor to determine whether classic orWhite standard errors should be reported. Hypothesis tests are not designed for these purposes.Rather, tests for heteroskedasticity should be used to answer the scienti�c question of whether ornot the conditional variance is a function of the regressors. If this question is not of economicinterest, then there is no value in conducting a test for heteorskedasticity.
5.3 Forecast Intervals
In the linear regression model the conditional mean of yi given xi = x is
m(x) = E (yi j xi = x) = x0�:
In some cases, we want to estimate m(x) at a particular point x: Notice that this is a (linear)function of �: Letting h(�) = x0� and � = h(�); we see that m(x) = � = x0� and H� = x; so
s(�) =pn�1x0V x: Thus an asymptotic 95% con�dence interval for m(x) ish
x0� � 2pn�1x0V x
i:
It is interesting to observe that if this is viewed as a function of x; the width of the con�dence setis dependent on x:
For a given value of xi = x; we may want to forecast (guess) yi out-of-sample. A reasonablerule is the conditional mean m(x) as it is the mean-square-minimizing forecast. A point forecast isthe estimated conditional mean m(x) = x0�. We would also like a measure of uncertainty for theforecast.
66
The forecast error is ei = yi � m(x) = ei � x0�� � �
�: As the out-of-sample error ei is
independent of the in-sample estimate �; this has variance
Ee2i = E�e2i j xi = x
�+ x0E
�� � �
��� � �
�0x
= �2(x) + n�1x0V x:
Assuming E�e2i j xi
�= �2; the natural estimate of this variance is �2 + n�1x0V x; so a standard
error for the forecast is s(x) =p�2 + n�1x0V x: Notice that this is di¤erent from the standard
error for the conditional mean. If we have an estimate of the conditional variance function, e.g.
~�2(x) = ~�0z from (5.6), then the forecast standard error is s(x) =q~�2(x) + n�1x0V x
It would appear natural to conclude that an asymptotic 95% forecast interval for yi ishx0� � 2s(x)
i;
but this turns out to be incorrect. In general, the validity of an asymptotic con�dence interval isbased on the asymptotic normality of the studentized ratio. In the present case, this would requirethe asymptotic normality of the ratio
ei � x0�� � �
�s(x)
:
But no such asymptotic approximation can be made. The only special exception is the case whereei has the exact distribution N(0; �2); which is generally invalid.
To get an accurate forecast interval, we need to estimate the conditional distribution of ei givenxi = x; which is a much more di¢ cult task. Given the di¢ culty, many applied forecasters focus
on the simple approximate intervalhx0� � 2s(x)
i:
5.4 NonLinear Least Squares
In some cases we might use a parametric regression function m (x;�) = E (yi j xi = x) whichis a non-linear function of the parameters �: We describe this setting as non-linear regression.Examples of nonlinear regression functions include
m (x;�) = �1 + �2x
1 + �3x
m (x;�) = �1 + �2x�3
m (x;�) = �1 + �2 exp(�3x)
m (x;�) = G(x0�); G known
m (x;�) = �01x1 +��02x1
��
�x2 � �3�4
�m (x;�) = �1 + �2x+ �3 (x� �4) 1 (x > �4)m (x;�) =
��01x1
�1 (x2 < �3) +
��02x1
�1 (x2 > �3)
In the �rst �ve examples, m (x;�) is (generically) di¤erentiable in the parameters �: In the �naltwo examples, m is not di¤erentiable with respect to �4 and �3 which alters some of the analysis.When it exists, let
m� (x;�) =@
@�m (x;�) :
Nonlinear regression is frequently adopted because the functional form m (x;�) is suggestedby an economic model. In other cases, it is adopted as a �exible approximation to an unknownregression function.
67
The least squares estimator � minimizes the sum-of-squared-errors
Sn(�) =nXi=1
(yi �m (xi;�))2 :
When the regression function is nonlinear, we call this the nonlinear least squares (NLLS)estimator. The NLLS residuals are ei = yi �m
�xi; �
�:
One motivation for the choice of NLLS as the estimation method is that the parameter � is thesolution to the population problem min� E (yi �m (xi;�))2
Since sum-of-squared-errors function Sn(�) is not quadratic, � must be found by numericalmethods. See Appendix E. When m(x;�) is di¤erentiable, then the FOC for minimization are
0 =nXi=1
m�
�xi; �
�ei: (5.8)
Theorem 5.4.1 If the model is identi�ed and m (x;�) is di¤erentiable with respect to �,
pn�� � �0
�d�! N(0;V )
V =�E�m�im
0�i
���1 �E�m�im
0�ie
2i
�� �E�m�im
0�i
���1where m�i =m�(xi;�0):
Based on Theorem 5.4.1, an estimate of the asymptotic variance V is
V =
1
n
nXi=1
m�im0�i
!�1 1
n
nXi=1
m�im0�ie
2i
! 1
n
nXi=1
m�im0�i
!�1
where m�i =m�(xi; �) and ei = yi �m(xi; �):Identi�cation is often tricky in nonlinear regression models. Suppose that
m(xi;�) = �01zi + �
02xi( )
where xi ( ) is a function of xi and the unknown parameter : Examples include xi ( ) = x i ;xi ( ) = exp ( xi) ; and xi ( ) = xi1 (g (xi) > ). The model is linear when �2 = 0; and this isoften a useful hypothesis (sub-model) to consider. Thus we want to test
H0 : �2 = 0:
However, under H0, the model isyi = �
01zi + ei
and both �2 and have dropped out. This means that under H0; is not identi�ed. This rendersthe distribution theory presented in the previous section invalid. Thus when the truth is that�2 = 0; the parameter estimates are not asymptotically normally distributed. Furthermore, testsof H0 do not have asymptotic normal or chi-square distributions.
The asymptotic theory of such tests have been worked out by Andrews and Ploberger (1994)and B. Hansen (1996). In particular, Hansen shows how to use simulation (similar to the bootstrap)to construct the asymptotic critical values (or p-values) in a given application.
68
5.5 Least Absolute Deviations
We stated that a conventional goal in econometrics is estimation of impact of variation in xion the central tendency of yi: We have discussed projections and conditional means, but these arenot the only measures of central tendency. An alternative good measure is the conditional median.
To recall the de�nition and properties of the median, let y be a continuous random variable.The median �0 = med(y) is the value such that P(y � �0) = P (y � �0) = :5: Two useful factsabout the median are that
�0 = argmin�
E jy � �j (5.9)
andE sgn (y � �0) = 0
where
sgn (u) =
�1 if u � 0�1 if u < 0
is the sign function.These facts and de�nitions motivate three estimators of �: The �rst de�nition is the 50th
empirical quantile. The second is the value which minimizes 1nPni=1 jyi � �j ; and the third de�nition
is the solution to the moment equation 1n
Pni=1 sgn (yi � �) : These distinctions are illusory, however,
as these estimators are indeed identical.Now let�s consider the conditional median of y given a random vector x: Let m(x) = med (y j x)
denote the conditional median of y given x: The linear median regression model takes the form
yi = x0i� + ei
med (ei j xi) = 0
In this model, the linear function med (yi j xi = x) = x0� is the conditional median function, andthe substantive assumption is that the median function is linear in x:
Conditional analogs of the facts about the median are
� P(yi � x0�0 j xi = x) = P(yi > x0� j xi = x) = :5
� E (sgn (ei) j xi) = 0
� E (xi sgn (ei)) = 0
� �0 = min� E jyi � x0i�j
These facts motivate the following estimator. Let
LADn(�) =1
n
nXi=1
��yi � x0i���be the average of absolute deviations. The least absolute deviations (LAD) estimator of �minimizes this function
� = argmin�
LADn(�)
Equivalently, it is a solution to the moment condition
1
n
nXi=1
xi sgn�yi � x0i�
�= 0: (5.10)
The LAD estimator has the asymptotic distribution
69
Theorem 5.5.1pn�� � �0
�d�! N(0;V ) ; where
V =1
4
�E�xix
0if (0 j xi)
���1 �Exix
0i
� �E�xix
0if (0 j xi)
���1and f (e j x) is the conditional density of ei given xi = x:
The variance of the asymptotic distribution inversely depends on f (0 j x) ; the conditionaldensity of the error at its median. When f (0 j x) is large, then there are many innovations nearto the median, and this improves estimation of the median. In the special case where the error isindependent of xi; then f (0 j x) = f (0) and the asymptotic variance simpli�es
V =(Exix
0i)�1
4f (0)2(5.11)
This simpli�cation is similar to the simpli�cation of the asymptotic covariance of the OLS estimatorunder homoskedasticity.
Computation of standard error for LAD estimates typically is based on equation (5.11). Themain di¢ culty is the estimation of f(0); the height of the error density at its median. This can bedone with kernel estimation techniques. See Chapter 14. While a complete proof of Theorem 5.5.1is advanced, we provide a sketch here for completeness.
5.6 Quantile Regression
The method of quantile regression has become quite popular in recent econometric practice.For � 2 [0; 1] the ��th quantile Q� of a random variable with distribution function F (u) is de�nedas
Q� = inf fu : F (u) � �g
When F (u) is continuous and strictly monotonic, then F (Q� ) = � ; so you can think of the quantileas the inverse of the distribution function. The quantile Q� is the value such that � (percent) ofthe mass of the distribution is less than Q� : The median is the special case � = :5:
The following alternative representation is useful. If the random variable U has ��th quantileQ� ; then
Q� = argmin�
E�� (U � �) : (5.12)
where �� (q) is the piecewise linear function
�� (q) =
��q (1� �) q < 0
q� q � 0 (5.13)
= q (� � 1 (q < 0)) :
This generalizes representation (5.9) for the median to all quantiles.For the random variables (yi;xi) with conditional distribution function F (y j x) the conditional
quantile function q� (x) isQ� (x) = inf fy : F (y j x) � �g :
Again, when F (y j x) is continuous and strictly monotonic in y, then F (Q� (x) j x) = � : For�xed � ; the quantile regression function q� (x) describes how the ��th quantile of the conditionaldistribution varies with the regressors.
As functions of x; the quantile regression functions can take any shape. However for computa-tional convenience it is typical to assume that they are (approximately) linear in x (after suitable
70
transformations). This linear speci�cation assumes that Q� (x) = �0�x where the coe¢ cients ��vary across the quantiles � : We then have the linear quantile regression model
yi = x0i�� + ei
where ei is the error de�ned to be the di¤erence between yi and its ��th conditional quantile x0i�� :By construction, the ��th conditional quantile of ei is zero, otherwise its properties are unspeci�edwithout further restrictions.
Given the representation (5.12), the quantile regression estimator �� for �� solves the mini-mization problem
�� = argmin�
S�n(�)
where
S�n(�) =1
n
nXi=1
���yi � x0i�
�and �� (q) is de�ned in (5.13).
Since the quanitle regression criterion function S�n(�) does not have an algebraic solution,numerical methods are necessary for its minimization. Furthermore, since it has discontinuousderivatives, conventional Newton-type optimization methods are inappropriate. Fortunately, fastlinear programming methods have been developed for this problem, and are widely available.
An asymptotic distribution theory for the quantile regression estimator can be derived usingsimilar arguments as those for the LAD estimator in Theorem 5.5.1.
Theorem 5.6.1pn��� � ��
�d�! N(0;V � ) ; where
V � = � (1� �)�E�xix
0if (0 j xi)
���1 �Exix
0i
� �E�xix
0if (0 j xi)
���1and f (e j x) is the conditional density of ei given xi = x:
In general, the asymptotic variance depends on the conditional density of the quantile regressionerror. When the error ei is independent of xi; then f (0 j xi) = f (0) ; the unconditional density ofei at 0, and we have the simpli�cation
V � =� (1� �)f (0)2
�E�xix
0i
���1:
A recent monograph on the details of quantile regression is Koenker (2005).
5.7 Testing for Omitted NonLinearity
If the goal is to estimate the conditional expectation E (yi j xi) ; it is useful to have a generaltest of the adequacy of the speci�cation.
One simple test for neglected nonlinearity is to add nonlinear functions of the regressors to theregression, and test their signi�cance using a Wald test. Thus, if the model yi = x0i�+ ei has been�t by OLS, let zi = h(xi) denote functions of xi which are not linear functions of xi (perhapssquares of non-binary regressors) and then �t yi = x0i~�+z
0i~ +~ei by OLS, and form a Wald statistic
for = 0:Another popular approach is the RESET test proposed by Ramsey (1969). The null model is
yi = x0i� + ei
71
which is estimated by OLS, yielding predicted values yi = x0i�: Now let
zi =
0B@ y2i...ymi
1CAbe an (m� 1)-vector of powers of yi: Then run the auxiliary regression
yi = x0i~� + z0i~ + ~ei (5.14)
by OLS, and form the Wald statistic Wn for = 0: It is easy (although somewhat tedious) to
show that under the null hypothesis, Wnd�! �2m�1: Thus the null is rejected at the �% level if Wn
exceeds the upper �% tail critical value of the �2m�1 distribution.To implement the test, m must be selected in advance. Typically, small values such as m = 2,
3, or 4 seem to work best.The RESET test appears to work well as a test of functional form against a wide range of
smooth alternatives. It is particularly powerful at detecting single-index models of the form
yi = G(x0i�) + ei
where G(�) is a smooth �link�function. To see why this is the case, note that (5.14) may be writtenas
yi = x0i~� +
�x0i�
�2~ 1 +
�x0i�
�3~ 2 + � � �
�x0i�
�m~ m�1 + ~ei
which has essentially approximated G(�) by a m�th order polynomial.
5.8 Omitted Variables
Let the regressors be partitioned as
xi =
�x1ix2i
�:
Suppose we are interested in the coe¢ cient on x1i alone in the regression of yi on the full set xi:We can write the model as
yi = x01i�1 + x02i�2 + ei (5.15)
E (xiei) = 0
where the parameter of interest is �1:Now suppose that instead of estimating equation (5.15) by least-squares, we regress yi on x1i
only. This is estimation of the equation
yi = x01i 1 + ui (5.16)
E (x1iui) = 0
Notice that we have written the coe¢ cient on x1i as 1 rather than �1 and the error as ui ratherthan ei: This is because the model being estimated is di¤erent than (5.15). Goldberger (1991) calls(5.15) the long regression and (5.16) the short regression to emphasize the distinction.
Typically, �1 6= 1, except in special cases. To see this, we calculate
1 =�E�x1ix
01i
���1E (x1iyi)
=�E�x1ix
01i
���1E�x1i�x01i�1 + x
02i�2 + ei
��= �1 +
�E�x1ix
01i
���1E�x1ix
02i
��2
= �1 + ��2
72
where� =
�E�x1ix
01i
���1E�x1ix
02i
�is the coe¢ cient from a regression of x2i on x1i:
Observe that 1 6= �1 unless � = 0 or �2 = 0: Thus the short and long regressions have thesame coe¢ cient on x1i only under one of two conditions. First, the regression of x2i on x1i yieldsa set of zero coe¢ cients (they are uncorrelated), or second, the coe¢ cient on x2i in (5.15) is zero.In general, least-squares estimation of (5.16) is an estimate of 1 = �1 +��2 rather than �1: Thedi¤erence ��2 is known as omitted variable bias. It is the consequence of omission of a relevantcorrelated variable.
To avoid omitted variables bias the standard advice is to include potentially relevant variablesin the estimated model. By construction, the general model will be free of the omitted variablesproblem. Typically there are limits, as many desired variables are not available in a given dataset.In this case, the possibility of omitted variables bias should be acknowledged and discussed in thecourse of an empirical investigation.
5.9 Irrelevant Variables
In the model
yi = x01i�1 + x02i�2 + ei
E (xiei) = 0;
x2i is �irrelevant�if �1 is the parameter of interest and �2 = 0: One estimator of �1 is to regressyi on x1i alone, ~�1 = (X
01X1)
�1(X 0
1y) : Another is to regress yi on x1i and x2i jointly, yielding(�1; �2): Under which conditions is �1 or ~�1 superior?
It is easy to see that both estimators are consistent for �1: However, they will (typically) havedi¤erent asymptotic variances.
The comparison between the two estimators is straightforward when the error is conditionallyhomoskedastic E
�e2i j xi
�= �2: In this case
limn!1
n var( ~�1) =�Ex1ix
01i
��1�2 = Q�111 �
2;
say, and
limn!1
n var(�1) =�Ex1ix
01i � Ex1ix02i
�Ex2ix
02i
��1Ex2ix
01i
��1�2 =
�Q11 �Q12Q�122 Q21
��1�2;
say. If Q12 = 0 (so the variables are orthogonal) then these two variance matrices equal, and thetwo estimators have equal asymptotic e¢ ciency. Otherwise, since Q12Q
�122 Q21 > 0; then Q11 >
Q11 �Q12Q�122 Q21; and consequently
Q�111 �2 <
�Q11 �Q12Q�122 Q21
��1�2:
This means that ~�1 has a lower asymptotic variance matrix than �1:We conclude that the inclusionof irrelevant variable reduces estimation e¢ ciency if these variables are correlated with the relevantvariables.
For example, take the model yi = �0 + �1xi + ei and suppose that �0 = 0: Let �1 be theestimate of �1 from the unconstrained model, and ~�1 be the estimate under the constraint �0 = 0:(The least-squares estimate with the intercept omitted.). Let Exi = �; and E (xi � �)2 = �2x: Thenunder (4.8),
limn!1
n var( ~�1) =�2
�2x + �2
73
while
limn!1
n var( �1) =�2
�2x:
When � 6= 0; we see that ~�1 has a lower asymptotic variance.However, this result can be reversed when the error is conditionally heteroskedastic. In the
absence of the homoskedasticity assumption, there is no clear ranking of the e¢ ciency of therestricted estimator ~�1 versus the unrestricted estimator.
5.10 Model Selection
In earlier sections we discussed the costs and bene�ts of inclusion/exclusion of variables. Howdoes a researcher go about selecting an econometric speci�cation, when economic theory does notprovide complete guidance? This is the question of model selection. It is important that the modelselection question be well-posed. For example, the question: �What is the right model for y?�is not well-posed, because it does not make clear the conditioning set. In contrast, the question,�Which subset of (x1; :::; xK) enters the regression function E (yi j x1i = x1; :::; xKi = xK)?�is wellposed.
In many cases the problem of model selection can be reduced to the comparison of two nestedmodels, as the larger problem can be written as a sequence of such comparisons. We thus considerthe question of the inclusion of X2 in the linear regression
y =X1�1 +X2�2 + e;
where X1 is n� k1 and X2 is n� k2: This is equivalent to the comparison of the two models
M1 : y =X1�1 + e; E (e jX1;X2) = 0
M2 : y =X1�1 +X2�2 + e; E (e jX1;X2) = 0:
Note thatM1 �M2: To be concrete, we say thatM2 is true if �2 6= 0:To �x notation, models 1 and 2 are estimated by OLS, with residual vectors e1 and e2; estimated
variances �21 and �22; etc., respectively. To simplify some of the statistical discussion, we will on
occasion use the homoskedasticity assumption E�e2i j x1i;x2i
�= �2:
A model selection procedure is a data-dependent rule which selects one of the two models. Wecan write this as cM. There are many possible desirable properties for a model selection procedure.One useful property is consistency, that it selects the true model with probability one if the sampleis su¢ ciently large. A model selection procedure is consistent if
P� cM =M1 j M1
�! 1
P� cM =M2 j M2
�! 1
However, this rule only makes sense when the true model is �nite dimensional. If the truth isin�nite dimensional, it is more appropriate to view model selection as determining the best �nitesample approximation.
A common approach to model selection is to base the decision on a statistical test such asthe Wald Wn: The model selection rule is as follows. For some critical level �; let c� satisfyP��2k2 > c�
�= �: Then selectM1 if Wn � c�; else selectM2.
A major problem with this approach is that the critical level � is indeterminate. The reasoningwhich helps guide the choice of � in hypothesis testing (controlling Type I error) is not relevant for
model selection. That is, if � is set to be a small number, then P� cM =M1 j M1
�� 1 � � but
P� cM =M2 j M2
�could vary dramatically, depending on the sample size, etc. Another problem is
74
that if � is held �xed, then this model selection procedure is inconsistent, as P� cM =M1 j M1
�!
1� � < 1:Another common approach to model selection is to use a selection criterion. One popular choice
is the Akaike Information Criterion (AIC). The AIC under normality for model m is
AICm = log��2m�+ 2
kmn: (5.17)
where �2m is the variance estimate for model m; and km is the number of coe¢ cients in the model.The AIC can be derived as an estimate of the KullbackLeibler information distance K(M) =E (log f(y jX)� log f(y jX;M)) between the true density and the model density. The rule isto select M1 if AIC1 < AIC2; else select M2: AIC selection is inconsistent, as the rule tends toover�t. Indeed, since underM1;
LRn = n�log �21 � log �22
�'Wn
d�! �2k2 ; (5.18)
then
P� cM =M1 j M1
�= P(AIC1 < AIC2 j M1)
= P
�log(�21) + 2
k1n< log(�22) + 2
k1 + k2n
j M1
�= P(LRn < 2k2 j M1)
! P��2k2 < 2k2
�< 1:
While many criterions similar to the AIC have been proposed, the most popular is one proposedby Schwarz based on Bayesian arguments. His criterion, known as the BIC, is
BICm = log��2m�+ log(n)
kmn: (5.19)
Since log(n) > 2 (if n > 8); the BIC places a larger penalty than the AIC on the number ofestimated parameters and is more parsimonious.
In contrast to the AIC, BIC model selection is consistent. Indeed, since (5.18) holds underM1;
LRnlog(n)
p�! 0;
so
P� cM =M1 j M1
�= P(BIC1 < BIC2 j M1)
= P (LRn < log(n)k2 j M1)
= P
�LRnlog(n)
< k2 j M1
�! P (0 < k2) = 1:
Also underM2; one can show thatLRnlog(n)
p�!1;
thus
P� cM =M2 j M2
�= P
�LRnlog(n)
> k2 j M2
�! 1:
75
We have discussed model selection between two models. The methods extend readily to theissue of selection among multiple regressors. The general problem is the model
yi = �1x1i + �2x2i + � � �+ �KxKi + ei; E (ei j xi) = 0
and the question is which subset of the coe¢ cients are non-zero (equivalently, which regressorsenter the regression).
There are two leading cases: ordered regressors and unordered.In the ordered case, the models are
M1 : �1 6= 0; �2 = �3 = � � � = �K = 0M2 : �1 6= 0; �2 6= 0; �3 = � � � = �K = 0
...
MK : �1 6= 0; �2 6= 0; : : : ; �K 6= 0:
which are nested. The AIC selection criteria estimates the K models by OLS, stores the residualvariance �2 for each model, and then selects the model with the lowest AIC (5.17). Similarly forthe BIC, selecting based on (5.19).
In the unordered case, a model consists of any possible subset of the regressors fx1i; :::; xKig;and the AIC or BIC in principle can be implemented by estimating all possible subset models.However, there are 2K such models, which can be a very large number. For example, 210 = 1024;and 220 = 1; 048; 576: In the latter case, a full-blown implementation of the BIC selection criterionwould seem computationally prohibitive.
5.11 Technical Proofs
Proof of Theorem 5.4.1 (Sketch). First, it must be shown that �p�! �0: This can be
done using arguments for optimization estimators, but we won�t cover that argument here. Since�
p�! �0; � is close to �0 for n large, so the minimization of Sn(�) only needs to be examined for� close to �0: Let
y0i = ei +m0�i�0:
For � close to the true value �0; by a �rst-order Taylor series approximation,
m (xi;�) ' m (xi;�0) +m0�i (� � �0) :
Thus
yi �m (xi;�) ' (ei +m (xi;�0))��m (xi;�0) +m
0�i (� � �0)
�= ei �m0
�i (� � �0)= y0i �m0
�i�:
Hence the sum of squared errors function is
Sn(�) =nXi=1
(yi �m (xi;�))2 'nXi=1
�y0i �m0
�i��2
and the right-hand-side is the SSE function for a linear regression of y0i on m�i: Thus the NLLSestimator � has the same asymptotic distribution as the (infeasible) OLS regression of y0i on m�i;which is that stated in the theorem. �
Proof of Theorem 5.5.1: The �rst step is to show that �p�! �0:We sketch the main argument.
For any �xed �; by the WLLN, LADn(�)p�! E jyi � x0i�j : Furthermore, it can be shown that this
76
convergence is uniform in �: It follows that �; the minimizer of LADn(�); converges in probabilityto �0; the minimizer of E jyi � x0i�j.
Since sgn (a) = 1�2�1 (a � 0) ; (5.10) is equivalent to gn(�) = 0; where gn(�) = n�1Pni=1 gi(�)
and gi(�) = xi (1� 2 � 1 (yi � x0i�)) : Let g(�) = Egi(�). We need three preliminary results. First,by the central limit theorem (Theorem C.3.1)
pn (gn(�0)� g(�0)) = �n�1=2
nXi=1
gi(�0)d�! N
�0;Exix
0i
�since Egi(�0)gi(�0)
0 = Exix0i: Second using the law of iterated expectations and the chain rule ofdi¤erentiation,
@
@�0g(�) =
@
@�0Exi
�1� 2 � 1
�yi � x0i�
��= �2 @
@�0E�xiE
�1�ei � x0i� � x0i�0
�j xi��
= �2 @@�0
E
"xi
Z x0i��x0i�0
�1f (e j xi) de
#= �2E
�xix
0if�x0i� � x0i�0 j xi
��so
@
@�0g(�0) = �2E
�xix
0if (0 j xi)
�:
Third, by a Taylor series expansion and the fact g(�0) = 0
g(�) ' @
@�0g(�0)
�� � �0
�:
Together
pn�� � �0
�'�@
@�0g(�0)
��1png(�)
=��2E
�xix
0if (0 j xi)
���1pn�g(�)� gn(�)
�' 1
2
�E�xix
0if (0 j xi)
���1pn (gn(�0)� g(�0))
d�! 1
2
�E�xix
0if (0 j xi)
���1N�0;Exix
0i
�= N(0;V ) :
The third line follows from an asymptotic empirical process argument and the fact that �p�! �0.
�
77
5.12 Exercises
1. For any predictor g(xi) for yi; the mean absolute error (MAE) is
E jyi � g(xi)j :
Show that the function g(x) which minimizes the MAE is the conditional median m (x) =med(yi j xi):
2. De�neg(u) = � � 1 (u < 0)
where 1 (�) is the indicator function (takes the value 1 if the argument is true, else equalszero). Let � satisfy Eg(yi � �) = 0: Is � a quantile of the distribution of yi?
3. Verify equation (5.12).
4. In the homoskedastic regression model y = X� + e with E(ei j xi) = 0 and E(e2i j xi) = �2;suppose � is the OLS estimate of � with covariance matrix V ; based on a sample of size n:Let �2 be the estimate of �2: You wish to forecast an out-of-sample value of yn+1 given thatxn+1 = x: Thus the available information is the sample (y;X); the estimates (�; V ; �2), theresiduals e; and the out-of-sample value of the regressors, xn+1:
(a) Find a point forecast of yn+1:
(b) Find an estimate of the variance of this forecast.
5. In a linear model
y =X� + e; E(e jX) = 0; var (e jX) = �2
with known, the GLS estimator is
~� =�X 0�1X
��1 �X 0�1y
�:
the residual vector is e = y �X~�; and an estimate of �2 is
s2 =1
n� k e0�1e:
(a) Why is this a reasonable estimator for �2?
(b) Prove that e =M1e; whereM1 = I �X�X 0�1X
��1X 0�1:
(c) Prove thatM 01
�1M1 = �1 ��1X
�X 0�1X
��1X 0�1:
6. Let (yi;xi) be a random sample with E(y jX) =X�: Consider the Weighted Least Squares(WLS) estimator of �
~� =�X 0WX
��1 �X 0Wy
�where W = diag (w1; :::; wn) and wi = x�2ji , where xji is one of the xi:
(a) In which contexts would ~� be a good estimator?
(b) Using your intuition, in which situations would you expect that ~� would perform betterthan OLS?
7. Suppose that yi = g(xi;�) + ei with E (ei j xi) = 0; � is the NLLS estimator, and V is the
estimate of var���: You are interested in the conditional mean function E (yi j xi = x) =
g(x) at some x: Find an asymptotic 95% con�dence interval for g(x):
78
8. The model is
yi = xi� + ei
E (ei j xi) = 0
where xi 2 R: Consider the two estimators
� =
Pni=1 xiyiPni=1 x
2i
~� =1
n
nXi=1
yixi:
(a) Under the stated assumptions, are both estimators consistent for �?
(b) Are there conditions under which either estimator is e¢ cient?
9. In Chapter 6, Exercise 13, you estimated a cost function on a cross-section of electric com-panies. The equation you estimated was
log TCi = �1 + �2 logQi + �3 logPLi + �4 logPKi + �5 logPFi + ei: (5.20)
(a) Following Nerlove, add the variable (logQi)2 to the regression. Do so. Assess the meritsof this new speci�cation using (i) a hypothesis test; (ii) AIC criterion; (iii) BIC criterion.Do you agree with this modi�cation?
(b) Now try a non-linear speci�cation. Consider model (5.20) plus the extra term �6zi;where
zi = logQi (1 + exp (� (logQi � �7)))�1 :
In addition, impose the restriction �3 + �4 + �5 = 1: This model is called a smooththreshold model. For values of logQi much below �7; the variable logQi has a regressionslope of �2: For values much above �7; the regression slope is �2 + �6; and the modelimposes a smooth transition between these regimes. The model is non-linear because ofthe parameter �7:The model works best when �7 is selected so that several values (in this example, atleast 10 to 15) of logQi are both below and above �7: Examine the data and pick anappropriate range for �7:
(c) Estimate the model by non-linear least squares. I recommend the concentration method:Pick 10 (or more or you like) values of �7 in this range. For each value of �7; calculatezi and estimate the model by OLS. Record the sum of squared errors, and �nd the valueof �7 for which the sum of squared errors is minimized.
(d) Calculate standard errors for all the parameters (�1; :::; �7).
10. The data �le cps78.dat contains 550 observations on 20 variables taken from the May 1978current population survey. Variables are listed in the �le cps78.pdf. The goal of the exerciseis to estimate a model for the log of earnings (variable LNWAGE) as a function of theconditioning variables.
(a) Start by an OLS regression of LNWAGE on the other variables. Report coe¢ cientestimates and standard errors.
(b) Consider augmenting the model by squares and/or cross-products of the conditioningvariables. Estimate your selected model and report the results.
79
(c) Are there any variables which seem to be unimportant as a determinant of wages? Youmay re-estimate the model without these variables, if desired.
(d) Test whether the error variance is di¤erent for men and women. Interpret.
(e) Test whether the error variance is di¤erent for whites and nonwhites. Interpret.
(f) Construct a model for the conditional variance. Estimate such a model, test for generalheteroskedasticity and report the results.
(g) Using this model for the conditional variance, re-estimate the model from part (c) usingFGLS. Report the results.
(h) Do the OLS and FGLS estimates di¤er greatly? Note any interesting di¤erences.
(i) Compare the estimated standard errors. Note any interesting di¤erences.
80
Chapter 6
The Bootstrap
6.1 De�nition of the Bootstrap
Let F denote a distribution function for the population of observations (yi;xi) : Let
Tn = Tn ((y1;x1) ; :::; (yn;xn) ; F )
be a statistic of interest, for example an estimator � or a t-statistic�� � �
�=s(�): Note that we
write Tn as possibly a function of F . For example, the t-statistic is a function of the parameter �which itself is a function of F:
The exact CDF of Tn when the data are sampled from the distribution F is
Gn(u; F ) = P(Tn � u j F )
In general, Gn(u; F ) depends on F; meaning that G changes as F changes.Ideally, inference would be based on Gn(u; F ). This is generally impossible since F is unknown.Asymptotic inference is based on approximating Gn(u; F ) with G(u; F ) = limn!1Gn(u; F ):
When G(u; F ) = G(u) does not depend on F; we say that Tn is asymptotically pivotal and use thedistribution function G(u) for inferential purposes.
In a seminal contribution, Efron (1979) proposed the bootstrap, which makes a di¤erent ap-proximation. The unknown F is replaced by a consistent estimate Fn (one choice is discussed inthe next section). Plugged into Gn(u; F ) we obtain
G�n(u) = Gn(u; Fn): (6.1)
We call G�n the bootstrap distribution. Bootstrap inference is based on G�n(u):
Let (y�i ;x�i ) denote random variables with the distribution Fn: A random sample from this dis-
tribution is called the bootstrap data. The statistic T �n = Tn ((y�1;x
�1) ; :::; (y
�n;x
�n) ; Fn) constructed
on this sample is a random variable with distribution G�n: That is, P(T�n � u) = G�n(u): We call T �n
the bootstrap statistic: The distribution of T �n is identical to that of Tn when the true CDF of Fnrather than F:
The bootstrap distribution is itself random, as it depends on the sample through the estimatorFn:
In the next sections we describe computation of the bootstrap distribution.
6.2 The Empirical Distribution Function
Recall that F (y;x) = P (yi � y;xi � x) = E (1 (yi � y) 1 (xi � x)) ; where 1(�) is the indicatorfunction. This is a population moment. The method of moments estimator is the corresponding
81
sample moment:
Fn (y;x) =1
n
nXi=1
1 (yi � y) 1 (xi � x) : (6.2)
Fn (y;x) is called the empirical distribution function (EDF). Fn is a nonparametric estimate of F:Note that while F may be either discrete or continuous, Fn is by construction a step function.
The EDF is a consistent estimator of the CDF. To see this, note that for any (y;x); 1 (yi � y) 1 (xi � x)is an iid random variable with expectation F (y;x): Thus by theWLLN (Theorem C.2.1), Fn (y;x)
p�!F (y;x) : Furthermore, by the CLT (Theorem C.3.1),
pn (Fn (y;x)� F (y;x))
d�! N(0; F (y;x) (1� F (y;x))) :
To see the e¤ect of sample size on the EDF, in the Figure below, I have plotted the EDF andtrue CDF for three random samples of size n = 25; 50, 100, and 500. The random draws are fromthe N(0; 1) distribution. For n = 25; the EDF is only a crude approximation to the CDF, but theapproximation appears to improve for the large n. In general, as the sample size gets larger, theEDF step function gets uniformly close to the true CDF.
Figure 6.1: Empirical Distribution Functions
The EDF is a valid discrete probability distribution which puts probability mass 1=n at eachpair (yi;xi), i = 1; :::; n: Notationally, it is helpful to think of a random pair (y�i ;x
�i ) with the
distribution Fn: That is,P(y�i � y;x�i � x) = Fn(y;x):
We can easily calculate the moments of functions of (y�i ;x�i ) :
Eh (y�i ;x�i ) =
Zh(y;x)dFn(y;x)
=
nXi=1
h (yi;xi) P (y�i = yi;x
�i = xi)
=1
n
nXi=1
h (yi;xi) ;
the empirical sample average.
82
6.3 Nonparametric Bootstrap
The nonparametric bootstrap is obtained when the bootstrap distribution (6.1) is de�nedusing the EDF (6.2) as the estimate Fn of F:
Since the EDF Fn is a multinomial (with n support points), in principle the distributionG�n couldbe calculated by direct methods. However, as there are 2n possible samples f(y�1;x�1) ; :::; (y�n;x�n)g;such a calculation is computationally infeasible. The popular alternative is to use simulation to ap-proximate the distribution. The algorithm is identical to our discussion of Monte Carlo simulation,with the following points of clari�cation:
� The sample size n used for the simulation is the same as the sample size.
� The random vectors (y�i ;x�i ) are drawn randomly from the empirical distribution. This is
equivalent to sampling a pair (yi;xi) randomly from the sample.
The bootstrap statistic T �n = Tn ((y�1;x
�1) ; :::; (y
�n;x
�n) ; Fn) is calculated for each bootstrap sam-
ple. This is repeated B times. B is known as the number of bootstrap replications. A theoryfor the determination of the number of bootstrap replications B has been developed by Andrewsand Buchinsky (2000). It is desirable for B to be large, so long as the computational costs arereasonable. B = 1000 typically su¢ ces.
When the statistic Tn is a function of F; it is typically through dependence on a parameter.
For example, the t-ratio�� � �
�=s(�) depends on �: As the bootstrap statistic replaces F with
Fn; it similarly replaces � with �n; the value of � implied by Fn: Typically �n = �; the parameterestimate. (When in doubt use �:)
Sampling from the EDF is particularly easy. Since Fn is a discrete probability distributionputting probability mass 1=n at each sample point, sampling from the EDF is equivalent to randomsampling a pair (yi;xi) from the observed data with replacement. In consequence, a bootstrapsample f(y�1;x�1) ; :::; (y�n;x�n)g will necessarily have some ties and multiple values, which is generallynot a problem.
6.4 Bootstrap Estimation of Bias and Variance
The bias of � is �n = E(� � �0): Let Tn(�) = � � �: Then �n = E(Tn(�0)): The bootstrapcounterparts are �
�= �((y�1;x
�1) ; :::; (y
�n;x
�n)) and T
�n = �
� � �n = �� � �: The bootstrap estimate
of �n is��n = E(T
�n):
If this is calculated by the simulation described in the previous section, the estimate of ��n is
��n =1
B
BXb=1
T �nb
=1
B
BXb=1
��b � �
= �� � �:
If � is biased, it might be desirable to construct a biased-corrected estimator (one with reducedbias). Ideally, this would be
~� = � � �n;
83
but �n is unknown. The (estimated) bootstrap biased-corrected estimator is
~��= � � ��n= � � (�� � �)
= 2� � ��:
Note, in particular, that the biased-corrected estimator is not ��: Intuitively, the bootstrap makes
the following experiment. Suppose that � is the truth. Then what is the average value of �
calculated from such samples? The answer is ��: If this is lower than �; this suggests that the
estimator is downward-biased, so a biased-corrected estimator of � should be larger than �; and the
best guess is the di¤erence between � and ��: Similarly if �
�is higher than �; then the estimator is
upward-biased and the biased-corrected estimator should be lower than �.Let Tn = �: The variance of � is
Vn = E(Tn � ETn)2:
Let T �n = ��: It has variance
V �n = E(T�n � ET �n)2:
The simulation estimate is
V �n =1
B
BXb=1
���b � �
��2:
A bootstrap standard error for � is the square root of the bootstrap estimate of variance,
s�(�) =qV �n :
While this standard error may be calculated and reported, it is not clear if it is useful. Theprimary use of asymptotic standard errors is to construct asymptotic con�dence intervals, which arebased on the asymptotic normal approximation to the t-ratio. However, the use of the bootstrappresumes that such asymptotic approximations might be poor, in which case the normal approxi-mation is suspected. It appears superior to calculate bootstrap con�dence intervals, and we turnto this next.
6.5 Percentile Intervals
For a distribution function Gn(u; F ); let qn(�; F ) denote its quantile function. This is thefunction which solves
Gn(qn(�; F ); F ) = �:
[When Gn(u; F ) is discrete, qn(�; F ) may be non-unique, but we will ignore such complications.]Let qn(�) = qn(�; F0) denote the quantile function of the true sampling distribution, and q�n(�) =qn(�; Fn) denote the quantile function of the bootstrap distribution. Note that this function willchange depending on the underlying statistic Tn whose distribution is Gn:
Let Tn = �; an estimate of a parameter of interest. In (1� �)% of samples, � lies in the region[qn(�=2); qn(1� �=2)]: This motivates a con�dence interval proposed by Efron:
C1 = [q�n(�=2); q�n(1� �=2)]:
This is often called the percentile con�dence interval.Computationally, the quantile q�n(�) is estimated by q
�n(�); the ��th sample quantile of the
simulated statistics fT �n1; :::; T �nBg; as discussed in the section on Monte Carlo simulation. The(1� �)% Efron percentile interval is then [q�n(�=2); q�n(1� �=2)]:
84
The interval C1 is a popular bootstrap con�dence interval often used in empirical practice.This is because it is easy to compute, simple to motivate, was popularized by Efron early in thehistory of the bootstrap, and also has the feature that it is translation invariant. That is, if wede�ne � = f(�) as the parameter of interest for a monotonic function f; then percentile methodapplied to this problem will produce the con�dence interval [f(q�n(�=2)); f(q�n(1� �=2))]; whichis a naturally good property.
However, as we show now, C1 is in a deep sense very poorly motivated.It will be useful if we introduce an alternative de�nition C1. Let Tn(�) = � � � and let qn(�)
be the quantile function of its distribution. (These are the original quantiles, with � subtracted.)Then C1 can alternatively be written as
C1 = [� + q�n(�=2); � + q�n(1� �=2)]:
This is a bootstrap estimate of the �ideal�con�dence interval
C01 = [� + qn(�=2); � + qn(1� �=2)]:
The latter has coverage probability
P��0 2 C01
�= P
�� + qn(�=2) � �0 � � + qn(1� �=2)
�= P
��qn(1� �=2) � � � �0 � �qn(�=2)
�= Gn(�qn(�=2); F0)�Gn(�qn(1� �=2); F0)
which generally is not 1��! There is one important exception. If ���0 has a symmetric distribution,then Gn(�u; F0) = 1�Gn(u; F0); so
P��0 2 C01
�= Gn(�qn(�=2); F0)�Gn(�qn(1� �=2); F0)= (1�Gn(qn(�=2); F0))� (1�Gn(qn(1� �=2); F0))
=�1� �
2
���1�
�1� �
2
��= 1� �
and this idealized con�dence interval is accurate. Therefore, C01 and C1 are designed for the casethat � has a symmetric distribution about �0:
When � does not have a symmetric distribution, C1 may perform quite poorly.However, by the translation invariance argument presented above, it also follows that if there
exists some monotonic transformation f(�) such that f(�) is symmetrically distributed about f(�0);then the idealized percentile bootstrap method will be accurate.
Based on these arguments, many argue that the percentile interval should not be used unlessthe sampling distribution is close to unbiased and symmetric.
The problems with the percentile method can be circumvented by an alternative method.Let Tn(�) = � � �. Then
1� � = P(qn(�=2) � Tn(�0) � qn(1� �=2))
= P�� � qn(1� �=2) � �0 � � � qn(�=2)
�;
so an exact (1� �)% con�dence interval for �0 would be
C02 = [� � qn(1� �=2); � � qn(�=2)]:
This motivates a bootstrap analog
C2 = [� � q�n(1� �=2); � � q�n(�=2)]:
85
Notice that generally this is very di¤erent from the Efron interval C1! They coincide in the specialcase that G�n(u) is symmetric about �; but otherwise they di¤er.
Computationally, this interval can be estimated from a bootstrap simulation by sorting the
bootstrap statistics T �n =��� � �
�; which are centered at the sample estimate �: These are sorted
to yield the quantile estimates q�n(:025) and q�n(:975): The 95% con�dence interval is then [� �
q�n(:975); � � q�n(:025)]:This con�dence interval is discussed in most theoretical treatments of the bootstrap, but is not
widely used in practice.
6.6 Percentile-t Equal-Tailed Interval
Suppose we want to test H0 : � = �0 against H1 : � < �0 at size �: We would set Tn(�) =�� � �
�=s(�) and reject H0 in favor of H1 if Tn(�0) < c; where c would be selected so that
P (Tn(�0) < c) = �:
Thus c = qn(�): Since this is unknown, a bootstrap test replaces qn(�) with the bootstrap estimateq�n(�); and the test rejects if Tn(�0) < q
�n(�):
Similarly, if the alternative is H1 : � > �0; the bootstrap test rejects if Tn(�0) > q�n(1� �):Computationally, these critical values can be estimated from a bootstrap simulation by sorting
the bootstrap t-statistics T �n =��� � �
�=s(�
�): Note, and this is important, that the bootstrap test
statistic is centered at the estimate �; and the standard error s(��) is calculated on the bootstrap
sample. These t-statistics are sorted to �nd the estimated quantiles q�n(�) and/or q�n(1� �):
Let Tn(�) =�� � �
�=s(�). Then taking the intersection of two one-sided intervals,
1� � = P(qn(�=2) � Tn(�0) � qn(1� �=2))
= P�qn(�=2) �
�� � �0
�=s(�) � qn(1� �=2)
�= P
�� � s(�)qn(1� �=2) � �0 � � � s(�)qn(�=2)
�;
so an exact (1� �)% con�dence interval for �0 would be
C03 = [� � s(�)qn(1� �=2); � � s(�)qn(�=2)]:
This motivates a bootstrap analog
C3 = [� � s(�)q�n(1� �=2); � � s(�)q�n(�=2)]:
This is often called a percentile-t con�dence interval. It is equal-tailed or central since the probabilitythat �0 is below the left endpoint approximately equals the probability that �0 is above the rightendpoint, each �=2:
Computationally, this is based on the critical values from the one-sided hypothesis tests, dis-cussed above.
6.7 Symmetric Percentile-t Intervals
Suppose we want to test H0 : � = �0 against H1 : � 6= �0 at size �: We would set Tn(�) =�� � �
�=s(�) and reject H0 in favor of H1 if jTn(�0)j > c; where c would be selected so that
P (jTn(�0)j > c) = �:
86
Note that
P (jTn(�0)j < c) = P (�c < Tn(�0) < c)= Gn(c)�Gn(�c)� Gn(c);
which is a symmetric distribution function. The ideal critical value c = qn(�) solves the equation
Gn(qn(�)) = 1� �:
Equivalently, qn(�) is the 1� � quantile of the distribution of jTn(�0)j :The bootstrap estimate is q�n(�); the 1 � � quantile of the distribution of jT �n j ; or the number
which solves the equation
G�n(q
�n(�)) = G
�n(q
�n(�))�G�n(�q�n(�)) = 1� �:
Computationally, q�n(�) is estimated from a bootstrap simulation by sorting the bootstrap t-
statistics jT �n j =����� � ���� =s(��); and taking the upper �% quantile. The bootstrap test rejects if
jTn(�0)j > q�n(�):Let
C4 = [� � s(�)q�n(�); � + s(�)q�n(�)];
where q�n(�) is the bootstrap critical value for a two-sided hypothesis test. C4 is called the symmetricpercentile-t interval. It is designed to work well since
P (�0 2 C4) = P�� � s(�)q�n(�) � �0 � � + s(�)q�n(�)
�= P(jTn(�0)j < q�n(�))' P (jTn(�0)j < qn(�))= 1� �:
If � is a vector, then to test H0 : � = �0 against H1 : � 6= �0 at size �; we would use a Waldstatistic
Wn(�) = n�� � �
�0V�1�
�� � �
�or some other asymptotically chi-square statistic. Thus here Tn(�) =Wn(�): The ideal test rejectsif Wn � qn(�); where qn(�) is the (1� �)% quantile of the distribution of Wn: The bootstrap testrejects if Wn � q�n(�); where q�n(�) is the (1� �)% quantile of the distribution of
W �n = n
��� � �
�0V��1�
��� � �
�:
Computationally, the critical value q�n(�) is found as the quantile from simulated values of W �n :
Note in the simulation that the Wald statistic is a quadratic form in��� � �
�; not
��� � �0
�:
[This is a typical mistake made by practitioners.]
6.8 Asymptotic Expansions
Let Tn 2 R be a statistic such that
Tnd�! N(0; �2): (6.3)
In some cases, such as when Tn is a t-ratio, then �2 = 1: In other cases �2 is unknown. Equivalently,writing Tn � Gn(u; F ) then
limn!1
Gn(u; F ) = ��u�
�;
87
orGn(u; F ) = �
�u�
�+ o (1) : (6.4)
While (6.4) says that Gn converges to ��u�
�as n ! 1; it says nothing, however, about the rate
of convergence; or the size of the divergence for any particular sample size n: A better asymptoticapproximation may be obtained through an asymptotic expansion.
The following notation will be helpful. Let an be a sequence.
De�nition 6.8.1 an = o(1) if an ! 0 as n!1
De�nition 6.8.2 an = O(1) if janj is uniformly bounded.
De�nition 6.8.3 an = o(n�r) if nr janj ! 0 as n!1.
Basically, an = O(n�r) if it declines to zero like n�r:We say that a function g(u) is even if g(�u) = g(u); and a function h(u) is odd if h(�u) = �h(u):
The derivative of an even function is odd, and vice-versa.
Theorem 6.8.1 Under regularity conditions and (6.3),
Gn(u; F ) = ��u�
�+
1
n1=2g1(u; F ) +
1
ng2(u; F ) +O(n
�3=2)
uniformly over u; where g1 is an even function of u; and g2 is an odd function of u: Moreover, g1and g2 are di¤erentiable functions of u and continuous in F relative to the supremum norm on thespace of distribution functions.
We can interpret Theorem 6.8.1 as follows. First, Gn(u; F ) converges to the normal limit atrate n1=2: To a second order of approximation,
Gn(u; F ) � ��u�
�+ n�1=2g1(u; F ):
Since the derivative of g1 is odd, the density function is skewed. To a third order of approximation,
Gn(u; F ) � ��u�
�+ n�1=2g1(u; F ) + n
�1g2(u; F )
which adds a symmetric non-normal component to the approximate density (for example, addingleptokurtosis).
6.9 One-Sided Tests
Using the expansion of Theorem 6.8.1, we can assess the accuracy of one-sided hypothesis testsand con�dence regions based on an asymptotically normal t-ratio Tn. An asymptotic test is basedon �(u):
To the second order, the exact distribution is
P (Tn < u) = Gn(u; F0) = �(u) +1
n1=2g1(u; F0) +O(n
�1)
since � = 1: The di¤erence is
�(u)�Gn(u; F0) =1
n1=2g1(u; F0) +O(n
�1)
= O(n�1=2);
so the order of the error is O(n�1=2):
88
A bootstrap test is based on G�n(u); which from Theorem 6.8.1 has the expansion
G�n(u) = Gn(u; Fn) = �(u) +1
n1=2g1(u; Fn) +O(n
�1):
Because �(u) appears in both expansions, the di¤erence between the bootstrap distribution andthe true distribution is
G�n(u)�Gn(u; F0) =1
n1=2(g1(u; Fn)� g1(u; F0)) +O(n�1):
Since Fn converges to F at ratepn; and g1 is continuous with respect to F; the di¤erence
(g1(u; Fn)� g1(u; F0)) converges to 0 at ratepn: Heuristically,
g1(u; Fn)� g1(u; F0) � @
@Fg1(u; F0) (Fn � F0)
= O(n�1=2);
The �derivative� @@F g1(u; F ) is only heuristic, as F is a function. We conclude that
G�n(u)�Gn(u; F0) = O(n�1);
orP (T �n � u) = P (Tn � u) +O(n�1);
which is an improved rate of convergence over the asymptotic test (which converged at rateO(n�1=2)). This rate can be used to show that one-tailed bootstrap inference based on the t-ratio achieves a so-called asymptotic re�nement �the Type I error of the test converges at a fasterrate than an analogous asymptotic test.
6.10 Symmetric Two-Sided Tests
If a random variable y has distribution function H(u) = P(y � u); then the random variablejyj has distribution function
H(u) = H(u)�H(�u)since
P (jyj � u) = P (�u � y � u)= P (y � u)� P (y � �u)= H(u)�H(�u):
For example, if Z � N(0; 1); then jZj has distribution function
�(u) = �(u)� �(�u) = 2�(u)� 1:
Similarly, if Tn has exact distribution Gn(u; F ); then jTnj has the distribution function
Gn(u; F ) = Gn(u; F )�Gn(�u; F ):
A two-sided hypothesis test rejects H0 for large values of jTnj : Since Tnd�! Z; then jTnj
d�!jZj � �: Thus asymptotic critical values are taken from the � distribution, and exact critical valuesare taken from the Gn(u; F0) distribution. From Theorem 6.8.1, we can calculate that
Gn(u; F ) = Gn(u; F )�Gn(�u; F )
=
��(u) +
1
n1=2g1(u; F ) +
1
ng2(u; F )
����(�u) + 1
n1=2g1(�u; F ) +
1
ng2(�u; F )
�+O(n�3=2)
= �(u) +2
ng2(u; F ) +O(n
�3=2); (6.5)
89
where the simpli�cations are because g1 is even and g2 is odd. Hence the di¤erence between theasymptotic distribution and the exact distribution is
�(u)�Gn(u; F0) =2
ng2(u; F0) +O(n
�3=2) = O(n�1):
The order of the error is O(n�1):Interestingly, the asymptotic two-sided test has a better coverage rate than the asymptotic
one-sided test. This is because the �rst term in the asymptotic expansion, g1; is an even function,meaning that the errors in the two directions exactly cancel out.
Applying (6.5) to the bootstrap distribution, we �nd
G�n(u) = Gn(u; Fn) = �(u) +
2
ng2(u; Fn) +O(n
�3=2):
Thus the di¤erence between the bootstrap and exact distributions is
G�n(u)�Gn(u; F0) =
2
n(g2(u; Fn)� g2(u; F0)) +O(n�3=2)
= O(n�3=2);
the last equality because Fn converges to F0 at ratepn; and g2 is continuous in F: Another way
of writing this isP (jT �n j < u) = P (jTnj < u) +O(n�3=2)
so the error from using the bootstrap distribution (relative to the true unknown distribution) isO(n�3=2): This is in contrast to the use of the asymptotic distribution, whose error is O(n�1): Thusa two-sided bootstrap test also achieves an asymptotic re�nement, similar to a one-sided test.
A reader might get confused between the two simultaneous e¤ects. Two-sided tests have betterrates of convergence than the one-sided tests, and bootstrap tests have better rates of convergencethan asymptotic tests.
The analysis shows that there may be a trade-o¤ between one-sided and two-sided tests. Two-sided tests will have more accurate size (Reported Type I error), but one-sided tests might havemore power against alternatives of interest. Con�dence intervals based on the bootstrap can beasymmetric if based on one-sided tests (equal-tailed intervals) and can therefore be more informativeand have smaller length than symmetric intervals. Therefore, the choice between symmetric andequal-tailed con�dence intervals is unclear, and needs to be determined on a case-by-case basis.
6.11 Percentile Con�dence Intervals
To evaluate the coverage rate of the percentile interval, set Tn =pn�� � �0
�: We know that
Tnd�! N(0; V ); which is not pivotal, as it depends on the unknown V: Theorem 6.8.1 shows that
a �rst-order approximation
Gn(u; F ) = ��u�
�+O(n�1=2);
where � =pV ; and for the bootstrap
G�n(u) = Gn(u; Fn) = ��u�
�+O(n�1=2);
where � = V (Fn) is the bootstrap estimate of �: The di¤erence is
G�n(u)�Gn(u; F0) = ��u�
�� �
�u�
�+O(n�1=2)
= ���u�
� u�(� � �) +O(n�1=2)
= O(n�1=2)
90
Hence the order of the error is O(n�1=2):The good news is that the percentile-type methods (if appropriately used) can yield
pn-
convergent asymptotic inference. Yet these methods do not require the calculation of standarderrors! This means that in contexts where standard errors are not available or are di¢ cult tocalculate, the percentile bootstrap methods provide an attractive inference method.
The bad news is that the rate of convergence is disappointing. It is no better than the rateobtained from an asymptotic one-sided con�dence region. Therefore if standard errors are available,it is unclear if there are any bene�ts from using the percentile bootstrap over simple asymptoticmethods.
Based on these arguments, the theoretical literature (e.g. Hall, 1992, Horowitz, 2001) tends toadvocate the use of the percentile-t bootstrap methods rather than percentile methods.
6.12 Bootstrap Methods for Regression Models
The bootstrap methods we have discussed have set G�n(u) = Gn(u; Fn); where Fn is the EDF.Any other consistent estimate of F may be used to de�ne a feasible bootstrap estimator. Theadvantage of the EDF is that it is fully nonparametric, it imposes no conditions, and works innearly any context. But since it is fully nonparametric, it may be ine¢ cient in contexts wheremore is known about F:We discuss some bootstrap methods appropriate for the case of a regressionmodel where
yi = x0i� + ei
E (ei j xi) = 0:
The non-parametric bootstrap distribution resamples the observations (y�i ;x�i ) from the EDF, which
implies
y�i = x�0i � + e�i
E (x�i e�i ) = 0
but generallyE (e�i j x�i ) 6= 0:
The the bootstrap distribution does not impose the regression assumption, and is thus an ine¢ cientestimator of the true distribution (when in fact the regression assumption is true.)
One approach to this problem is to impose the very strong assumption that the error "i isindependent of the regressor xi: The advantage is that in this case it is straightforward to con-struct bootstrap distributions. The disadvantage is that the bootstrap distribution may be a poorapproximation when the error is not independent of the regressors.
To impose independence, it is su¢ cient to sample the x�i and e�i independently, and then create
y�i = x�0i � + e�i : There are di¤erent ways to impose independence. A non-parametric method
is to sample the bootstrap errors e�i randomly from the OLS residuals fe1; :::; eng: A parametricmethod is to generate the bootstrap errors e�i from a parametric distribution, such as the normale�i � N(0; �2):
For the regressors x�i , a nonparametric method is to sample the x�i randomly from the EDF
or sample values fx1; :::;xng: A parametric method is to sample x�i from an estimated parametricdistribution. A third approach sets x�i = xi: This is equivalent to treating the regressors as �xedin repeated samples. If this is done, then all inferential statements are made conditionally on theobserved values of the regressors, which is a valid statistical approach. It does not really matter,however, whether or not the xi are really ��xed�or random.
The methods discussed above are unattractive for most applications in econometrics becausethey impose the stringent assumption that xi and ei are independent. Typically what is desirableis to impose only the regression condition E (ei j xi) = 0: Unfortunately this is a harder problem.
91
One proposal which imposes the regression condition without independence is the Wild Boot-strap. The idea is to construct a conditional distribution for e�i so that
E (e�i j xi) = 0
E�e�2i j xi
�= e2i
E�e�3i j xi
�= e3i :
A conditional distribution with these features will preserve the main important features of the data.This can be achieved using a two-point distribution of the form
P
e�i =
1 +
p5
2
!ei
!=
p5� 12p5
P
e�i =
1�
p5
2
!ei
!=
p5 + 1
2p5
For each xi; you sample e�i using this two-point distribution.
6.13 Exercises
1. Let Fn(x) denote the EDF of a random sample. Show that
pn (Fn(x)� F0(x))
d�! N(0; F0(x) (1� F0(x))) :
2. Take a random sample fy1; :::; yng with � = Eyi and �2 = var (yi) : Let the statistic of interestbe the sample mean Tn = yn: Find the population moments ETn and var (Tn) : Let fy�1; :::; y�ngbe a random sample from the empirical distribution function and let T �n = y
�n be its sample
mean. Find the bootstrap moments ET �n and var (T�n) :
3. Consider the following bootstrap procedure for a regression of yi on xi: Let � denote the OLSestimator from the regression of y on X, and e = y �X� the OLS residuals.
(a) Draw a random vector (x�; e�) from the pair f(xi; ei) : i = 1; :::; ng : That is, draw arandom integer i0 from [1; 2; :::; n]; and set x� = xi0 and e� = ei0 . Set y� = x�0� + e�:Draw (with replacement) n such vectors, creating a random bootstrap data set (y�;X�):
(b) Regress y� on X�; yielding OLS estimates ��and any other statistic of interest.
Show that this bootstrap procedure is (numerically) identical to the non-parametric boot-strap.
4. Consider the following bootstrap procedure. Using the non-parametric bootstrap, generatebootstrap samples, calculate the estimate �
�on these samples and then calculate
T �n = (�� � �)=s(�);
where s(�) is the standard error in the original data. Let q�n(:05) and q�n(:95) denote the 5%
and 95% quantiles of T �n , and de�ne the bootstrap con�dence interval
C =h� � s(�)q�n(:95); � � s(�)q�n(:05)
i:
Show that C exactly equals the Alternative percentile interval (not the percentile-t interval).
92
5. You want to test H0 : � = 0 against H1 : � > 0: The test for H0 is to reject if Tn = �=s(�) > cwhere c is picked so that Type I error is �: You do this as follows. Using the non-parametricbootstrap, you generate bootstrap samples, calculate the estimates �
�on these samples and
then calculateT �n = �
�=s(�
�):
Let q�n(:95) denote the 95% quantile of T �n . You replace c with q�n(:95); and thus reject H0 if
Tn = �=s(�) > q�n(:95): What is wrong with this procedure?
6. Suppose that in an application, � = 1:2 and s(�) = :2: Using the non-parametric bootstrap,1000 samples are generated from the bootstrap distribution, and �
�is calculated on each
sample. The ��are sorted, and the 2.5% and 97.5% quantiles of the �
�are .75 and 1.3,
respectively.
(a) Report the 95% Efron Percentile interval for �:
(b) Report the 95% Alternative Percentile interval for �:
(c) With the given information, can you report the 95% Percentile-t interval for �?
7. The data�le hprice1.dat contains data on house prices (sales), with variables listed in the�le hprice1.pdf. Estimate a linear regression of price on the number of bedrooms, lot size,size of house, and the colonial dummy. Calculate 95% con�dence intervals for the regressioncoe¢ cients using both the asymptotic normal approximation and the percentile-t bootstrap.
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Chapter 7
Generalized Method of Moments
7.1 Overidenti�ed Linear Model
Consider the linear model
yi = x0i� + ei
= x01i�1 + x02i�2 + ei
E (xiei) = 0
where x1i is k � 1 and x2 is r � 1 with ` = k + r: We know that without further restrictions, anasymptotically e¢ cient estimator of � is the OLS estimator. Now suppose that we are given theinformation that �2 = 0: Now we can write the model as
yi = x01i�1 + ei
E (xiei) = 0:
In this case, how should �1 be estimated? One method is OLS regression of yi on x1i alone. Thismethod, however, is not necessarily e¢ cient, as there are ` restrictions in E (xiei) = 0; while �1 isof dimension k < `. This situation is called overidenti�ed. There are ` � k = r more momentrestrictions than free parameters. We call r the number of overidentifying restrictions.
This is a special case of a more general class of moment condition models. Let g(y; z;x;�) bean `� 1 function of a k � 1 parameter � with ` � k such that
Eg(yi;zi;xi;�0) = 0 (7.1)
where �0 is the true value of �: In our previous example, g(y;x;�) = x�(y�x01�): In econometrics,this class of models are called moment condition models. In the statistics literature, these areknown as estimating equations.
As an important special case we will devote special attention to linear moment condition models,which can be written as
yi = z0i� + ei
E (xiei) = 0:
where the dimensions of zi and xi are k � 1 and ` � 1 , with ` � k: If k = ` the model is justidenti�ed, otherwise it is overidenti�ed. The variables zi may be components and functions ofxi; but this is not required. This model falls in the class (7.1) by setting
g(y; z;x;�0) = x�(y � z0�) (7.2)
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7.2 GMM Estimator
De�ne the sample analog of (7.2)
gn(�) =1
n
nXi=1
gi(�) =1
n
nXi=1
xi�yi � z0i�
�=1
n
�X 0y �X 0Z�
�: (7.3)
The method of moments estimator for � is de�ned as the parameter value which sets gn(�) = 0.This is generally not possible when ` > k; are there are more equations than free parameters. Theidea of the generalized method of moments (GMM) is to de�ne an estimator which sets gn(�)�close�to zero.
For some `� ` weight matrix W n > 0; let
Jn(�) = n � gn(�)0W n gn(�):
This is a non-negative measure of the �length�of the vector gn(�): For example, ifW n = I; then,Jn(�) = n � gn(�)0gn(�) = n � kgn(�)k2 ; the square of the Euclidean length. The GMM estimatorminimizes Jn (�).
De�nition 1 �GMM = argmin�
Jn (�) :
Note that if k = `; then gn(�) = 0; and the GMM estimator is the method of momentsestimator: The �rst order conditions for the GMM estimator are
0 =@
@�Jn(�)
= 2@
@�gn(�)
0W ngn(�)
= �2�1
nZ 0X
�W n
�1
nX 0�y �Z�
��so
2�Z 0X
�W n
�X 0Z
�� = 2
�Z 0X
�W n
�X 0y
�which establishes the following.
Proposition 7.2.1
�GMM =��Z 0X
�W n
�X 0Z
���1 �Z 0X
�W n
�X 0y
�:
While the estimator depends onW n; the dependence is only up to scale, for ifW n is replacedby cW n for some c > 0; �GMM does not change.
7.3 Distribution of GMM Estimator
Assume that W np�!W > 0: Let
Q = E�xiz
0i
�and
= E�xix
0ie2i
�= E
�gig
0i
�;
where gi = xiei: Then �1
nZ 0X
�W n
�1
nX 0Z
�p�! Q0WQ
and �1
nZ 0X
�W n
�1pnX 0e
�d�! Q0WN(0;) :
We conclude:
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Theorem 7.3.1pn�� � �
�d�! N(0;V ) ; where
V =�Q0WQ
��1 �Q0WWQ
� �Q0WQ
��1:
In general, GMM estimators are asymptotically normal with �sandwich form�asymptotic vari-ances.
The optimal weight matrix W 0 is one which minimizes V : This turns out to be W 0 = �1:The proof is left as an exercise. This yields the e¢ cient GMM estimator:
� =�Z 0X�1X 0Z
��1Z 0X�1X 0y:
Thus we have
Theorem 7.3.2 For the e¢ cient GMM estimator,pn�� � �
�d�! N
�0;�Q0�1Q
��1�:
W 0 = �1 is not known in practice, but it can be estimated consistently. For anyW n
p�!W 0;we still call � the e¢ cient GMM estimator, as it has the same asymptotic distribution.
By �e¢ cient�, we mean that this estimator has the smallest asymptotic variance in the classof GMM estimators with this set of moment conditions. This is a weak concept of optimality, aswe are only considering alternative weight matrices W n: However, it turns out that the GMMestimator is semiparametrically e¢ cient, as shown by Gary Chamberlain (1987).
If it is known that E (gi(�)) = 0; and this is all that is known, this is a semi-parametricproblem, as the distribution of the data is unknown. Chamberlain showed that in this context,no semiparametric estimator (one which is consistent globally for the class of models considered)can have a smaller asymptotic variance than
�G0�1G
��1where G = E @
@�0gi(�): Since the GMM
estimator has this asymptotic variance, it is semiparametrically e¢ cient.This result shows that in the linear model, no estimator has greater asymptotic e¢ ciency than
the e¢ cient linear GMM estimator. No estimator can do better (in this �rst-order asymptoticsense), without imposing additional assumptions.
7.4 Estimation of the E¢ cient Weight Matrix
Given any weight matrix W n > 0; the GMM estimator � is consistent yet ine¢ cient. Forexample, we can set W n = I`: In the linear model, a better choice is W n = (X 0X)�1 : Givenany such �rst-step estimator, we can de�ne the residuals ei = yi � z0i� and moment equationsgi = xiei = g(yi;zi;xi; �): Construct
gn = gn(�) =1
n
nXi=1
gi;
g�i = gi � gn;
and de�ne
W n =
1
n
nXi=1
g�i g�0i
!�1=
1
n
nXi=1
gig0i � gng0n
!�1: (7.4)
Then W np�! �1 =W 0; and GMM using W n as the weight matrix is asymptotically e¢ cient.
A common alternative choice is to set
W n =
1
n
nXi=1
gig0i
!�1
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which uses the uncentered moment conditions. Since Egi = 0; these two estimators are asymptot-ically equivalent under the hypothesis of correct speci�cation. However, Alastair Hall (2000) hasshown that the uncentered estimator is a poor choice. When constructing hypothesis tests, underthe alternative hypothesis the moment conditions are violated, i.e. Egi 6= 0; so the uncenteredestimator will contain an undesirable bias term and the power of the test will be adversely a¤ected.A simple solution is to use the centered moment conditions to construct the weight matrix, as in(7.4) above.
Here is a simple way to compute the e¢ cient GMM estimator for the linear model. First, setW n = (X 0X)�1, estimate � using this weight matrix, and construct the residual ei = yi � z0i�:Then set gi = xiei; and let g be the associated n� ` matrix: Then the e¢ cient GMM estimator is
� =�Z 0X
�g0g � ngng0n
��1X 0Z
��1Z 0X
�g0g � ngng0n
��1X 0y:
In most cases, when we say �GMM�, we actually mean �e¢ cient GMM�. There is little point inusing an ine¢ cient GMM estimator when the e¢ cient estimator is easy to compute.
An estimator of the asymptotic variance of � can be seen from the above formula. Set
V = n�Z 0X
�g0g � ngng0n
��1X 0Z
��1:
Asymptotic standard errors are given by the square roots of the diagonal elements of V :There is an important alternative to the two-step GMM estimator just described. Instead, we
can let the weight matrix be considered as a function of �: The criterion function is then
J(�) = n � gn(�)0 1
n
nXi=1
g�i (�)g�i (�)
0
!�1gn(�):
whereg�i (�) = gi(�)� gn(�)
The � which minimizes this function is called the continuously-updated GMM estimator, andwas introduced by L. Hansen, Heaton and Yaron (1996).
The estimator appears to have some better properties than traditional GMM, but can be nu-merically tricky to obtain in some cases. This is a current area of research in econometrics.
7.5 GMM: The General Case
In its most general form, GMM applies whenever an economic or statistical model implies the`� 1 moment condition
E (gi(�)) = 0:
Often, this is all that is known. Identi�cation requires l � k = dim(�): The GMM estimatorminimizes
J(�) = n � gn(�)0W n gn(�)
where
gn(�) =1
n
nXi=1
gi(�)
and
W n =
1
n
nXi=1
gig0i � gng0n
!�1;
with gi = gi(~�) constructed using a preliminary consistent estimator ~�, perhaps obtained by �rstsettingW n = I: Since the GMM estimator depends upon the �rst-stage estimator, often the weightmatrix W n is updated, and then � recomputed. This estimator can be iterated if needed.
97
Theorem 7.5.1 Under general regularity conditions,pn�� � �
�d�! N
�0;�G0�1G
��1�; where
= (E (gig0i))�1 and G = E @
@�0gi(�): The variance of � may be estimated by
�G0�1G��1
where
= n�1Pi g�i g�0i and G = n�1
Pi@@�0gi(�):
The general theory of GMM estimation and testing was exposited by L. Hansen (1982).
7.6 Over-Identi�cation Test
Overidenti�ed models (` > k) are special in the sense that there may not be a parameter value� such that the moment condition
Eg(yi;zi;xi;�) = 0
holds. Thus the model �the overidentifying restrictions �are testable.For example, take the linear model yi = �01x1i+�
02x2i+ei with E (x1iei) = 0 and E (x2iei) = 0:
It is possible that �2 = 0; so that the linear equation may be written as yi = �01x1i + ei: However,
it is possible that �2 6= 0; and in this case it would be impossible to �nd a value of �1 so thatboth E (x1i (yi � x01i�1)) = 0 and E (x2i (yi � x01i�1)) = 0 hold simultaneously. In this sense anexclusion restriction can be seen as an overidentifying restriction.
Note that gnp�! Egi; and thus gn can be used to assess whether or not the hypothesis that
Egi = 0 is true or not. The criterion function at the parameter estimates is
J = n g0nW ngn
= n2g0n�g0g � ngng0n
��1gn:
is a quadratic form in gn; and is thus a natural test statistic for H0 : Egi = 0:
Theorem 7.6.1 (Sargan-Hansen). Under the hypothesis of correct speci�cation, and if the weightmatrix is asymptotically e¢ cient,
J = J(�)d�! �2`�k:
The proof of the theorem is left as an exercise. This result was established by Sargan (1958)for a specialized case, and by L. Hansen (1982) for the general case.
The degrees of freedom of the asymptotic distribution are the number of overidentifying restric-tions. If the statistic J exceeds the chi-square critical value, we can reject the model. Based onthis information alone, it is unclear what is wrong, but it is typically cause for concern. The GMMoveridenti�cation test is a very useful by-product of the GMM methodology, and it is advisable toreport the statistic J whenever GMM is the estimation method.
When over-identi�ed models are estimated by GMM, it is customary to report the J statisticas a general test of model adequacy.
7.7 Hypothesis Testing: The Distance Statistic
We described before how to construct estimates of the asymptotic covariance matrix of theGMM estimates. These may be used to construct Wald tests of statistical hypotheses.
If the hypothesis is non-linear, a better approach is to directly use the GMM criterion function.This is sometimes called the GMM Distance statistic, and sometimes called a LR-like statistic (theLR is for likelihood-ratio). The idea was �rst put forward by Newey and West (1987).
For a given weight matrix W n; the GMM criterion function is
J(�) = n � gn(�)0W n gn(�)
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For h : Rk ! Rr; the hypothesis is
H0 : h(�) = 0:
The estimates under H1 are� = argmin
�J(�)
and those under H0 are~� = argmin
h(�)=0J(�):
The two minimizing criterion functions are J(�) and J(~�): The GMM distance statistic is thedi¤erence
D = J(~�)� J(�):
Proposition 7.7.1 If the same weight matrix W n is used for both null and alternative,
1. D � 0
2. D d�! �2r
3. If h is linear in �; then D equals the Wald statistic.
If h is non-linear, the Wald statistic can work quite poorly. In contrast, current evidencesuggests that the D statistic appears to have quite good sampling properties, and is the preferredtest statistic.
Newey and West (1987) suggested to use the same weight matrix W n for both null and alter-native, as this ensures that D � 0. This reasoning is not compelling, however, and some currentresearch suggests that this restriction is not necessary for good performance of the test.
This test shares the useful feature of LR tests in that it is a natural by-product of the compu-tation of alternative models.
7.8 Conditional Moment Restrictions
In many contexts, the model implies more than an unconditional moment restriction of the formEgi(�) = 0: It implies a conditional moment restriction of the form
E (ei(�) j xi) = 0
where ei(�) is some s� 1 function of the observation and the parameters. In many cases, s = 1.It turns out that this conditional moment restriction is much more powerful, and restrictive,
than the unconditional moment restriction discussed above.Our linear model yi = z0i� + ei with instruments xi falls into this class under the stronger
assumption E (ei j xi) = 0: Then ei(�) = yi � z0i�:It is also helpful to realize that conventional regression models also fall into this class, except
that in this case zi = xi: For example, in linear regression, ei(�) = yi � x0i�, while in a nonlinearregression model ei(�) = yi � g(xi;�): In a joint model of the conditional mean and variance
ei (�; ) =
8<:yi � x0i�
(yi � x0i�)2 � f (xi)0
:
Here s = 2:Given a conditional moment restriction, an unconditional moment restriction can always be
constructed. That is for any ` � 1 function � (xi;�) ; we can set gi(�) = � (xi;�) ei(�) which
99
satis�es Egi(�) = 0 and hence de�nes a GMM estimator. The obvious problem is that the class offunctions � is in�nite. Which should be selected?
This is equivalent to the problem of selection of the best instruments. If xi 2 R is a validinstrument satisfying E (ei j xi) = 0; then xi; x2i ; x
3i ; :::; etc., are all valid instruments. Which
should be used?One solution is to construct an in�nite list of potent instruments, and then use the �rst k
instruments. How is k to be determined? This is an area of theory still under development. Arecent study of this problem is Donald and Newey (2001).
Another approach is to construct the optimal instrument. The form was uncovered by Cham-berlain (1987). Take the case s = 1: Let
Ri = E
�@
@�ei(�) j xi
�and
�2i = E�ei(�)
2 j xi�:
Then the �optimal instrument�isAi = ���2i Ri
so the optimal moment isgi(�) = Aiei(�):
Setting gi (�) to be this choice (which is k�1; so is just-identi�ed) yields the best GMM estimatorpossible.
In practice, Ai is unknown, but its form does help us think about construction of optimalinstruments.
In the linear model ei(�) = yi � z0i�; note that
Ri = �E (zi j xi)
and�2i = E
�e2i j xi
�;
soAi = �
�2i E (zi j xi) :
In the case of linear regression, zi = xi; so Ai = ��2i xi: Hence e¢ cient GMM is GLS, as wediscussed earlier in the course.
In the case of endogenous variables, note that the e¢ cient instrumentAi involves the estimationof the conditional mean of zi given xi: In other words, to get the best instrument for zi; we need thebest conditional mean model for zi given xi, not just an arbitrary linear projection. The e¢ cientinstrument is also inversely proportional to the conditional variance of ei: This is the same as theGLS estimator; namely that improved e¢ ciency can be obtained if the observations are weightedinversely to the conditional variance of the errors.
7.9 Bootstrap GMM Inference
Let � be the 2SLS or GMM estimator of �. Using the EDF of (yi;xi;zi), we can apply thebootstrap methods discussed in Chapter 6 to compute estimates of the bias and variance of �;and construct con�dence intervals for �; identically as in the regression model. However, cautionshould be applied when interpreting such results.
A straightforward application of the nonparametric bootstrap works in the sense of consistentlyachieving the �rst-order asymptotic distribution. This has been shown by Hahn (1996). However,it fails to achieve an asymptotic re�nement when the model is over-identi�ed, jeopardizing the
100
theoretical justi�cation for percentile-t methods. Furthermore, the bootstrap applied J test willyield the wrong answer.
The problem is that in the sample, � is the �true�value and yet gn(�) 6= 0: Thus according torandom variables (y�i ;x
�i ;z
�i ) drawn from the EDF Fn;
E�gi
����= gn(�) 6= 0:
This means that (y�i ;x�i ;z
�i ) do not satisfy the same moment conditions as the population distrib-
ution.A correction suggested by Hall and Horowitz (1996) can solve the problem. Given the bootstrap
sample (y�;X�;Z�); de�ne the bootstrap GMM criterion
J�(�) = n ��g�n(�)� gn(�)
�0W �
n
�g�n(�)� gn(�)
�where gn(�) is from the in-sample data, not from the bootstrap data.
Let ��minimize J�(�); and de�ne all statistics and tests accordingly. In the linear model, this
implies that the bootstrap estimator is
��=�Z�0X�W �
nX�0Z�
��1 �Z�0X�W �
n
�X�0y� �X 0e
��:
where e = y �Z� are the in-sample residuals. The bootstrap J statistic is J�(��):Brown and Newey (2002) have an alternative solution. They note that we can sample from
the observations with the empirical likelihood probabilities pi described in Chapter 8. SincePni=1 pigi
���= 0; this sampling scheme preserves the moment conditions of the model, so no
recentering or adjustments is needed. Brown and Newey argue that this bootstrap procedure willbe more e¢ cient than the Hall-Horowitz GMM bootstrap.
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7.10 Exercises
1. Take the model
yi = x0i� + ei
E (xiei) = 0
e2i = z0i + �iE (zi�i) = 0:
Find the method of moments estimators��;
�for (�; ) :
2. Take the single equation
y = Z� + e
E (e jX) = 0
Assume E�e2i j xi
�= �2: Show that if � is estimated by GMM with weight matrix W n =
(X 0X)�1 ; thenpn�� � �
�d�! N
�0; �2
�Q0M�1Q
��1�where Q = E(xiz
0i) andM = E(xix
0i) :
3. Take the model yi = z0i� + ei with E (xiei) = 0: Let ei = yi � z0i� where � is consistent for� (e.g. a GMM estimator with arbitrary weight matrix). De�ne the estimate of the optimalGMM weight matrix
W n =
1
n
nXi=1
xix0ie2i
!�1:
Show that W np�! �1 where = E
�xix
0ie2i
�:
4. In the linear model estimated by GMM with general weight matrix W ; the asymptotic vari-ance of �GMM is
V =�Q0WQ
��1Q0WWQ
�Q0WQ
��1(a) Let V 0 be this matrix when W = �1: Show that V 0 =
�Q0�1Q
��1:
(b) We want to show that for any W ; V � V 0 is positive semi-de�nite (for then V 0 is thesmaller possible covariance matrix and W = �1 is the e¢ cient weight matrix). To dothis, start by �nding matrices A and B such that V = A0A and V 0 = B
0B:
(c) Show that B0A = B0B and therefore that B0 (A�B) = 0:(d) Use the expressions V = A0A; A = B + (A�B) ; and B0 (A�B) = 0 to show
that V � V 0:
5. The equation of interest is
yi = g(xi;�) + ei
E (ziei) = 0:
The observed data is (yi;xi;zi). zi is `� 1 and � is k � 1; ` � k: Show how to construct ane¢ cient GMM estimator for �.
102
6. In the linear model y = X� + e with E(xiei) = 0; the Generalized Method of Moments(GMM) criterion function for � is de�ned as
Jn(�) =1
n(y �X�)0X�1n X 0 (y �X�) (7.5)
where ei are the OLS residuals and n = 1n
Pni=1 xix
0ie2i : The GMM estimator of �; subject
to the restriction h(�) = 0; is de�ned as
~� = argminh(�)=0
Jn(�):
The GMM test statistic (the distance statistic) of the hypothesis h(�) = 0 is
D = Jn(~�) = minh(�)=0
Jn(�): (7.6)
(a) Show that you can rewrite Jn(�) in (7.5) as
Jn(�) =�� � �
�0V�1n
�� � �
�where
V n =�X 0X
��1 nXi=1
xix0ie2i
!�X 0X
��1:
(b) Now focus on linear restrictions: h(�) = R0� � r: Thus
~� = argminR0��r=0
Jn(�)
and hence R0~� = r: De�ne the Lagrangian L(�;�) = 12Jn(�) +�
0 (R0� � r) where � iss� 1: Show that the minimizer is
~� = � � V nR�R0nV R
��1 �R0� � r
�� =
�R0nV R
��1 �R0� � r
�:
(c) Show that if R0� = r thenpn�~� � �
�d�! N(0;V R) where
V R = V � V R�R0V R
��1R0V :
(d) Show that in this setting, the distance statistic D in (7.6) equals the Wald statistic.
7. Take the linear model
yi = z0i� + ei
E (xiei) = 0:
and consider the GMM estimator � of �: Let
Jn = ngn(�)0�1gn(�)
denote the test of overidentifying restrictions. Show that Jnd�! �2`�k as n ! 1 by demon-
strating each of the following:
103
(a) Since > 0; we can write �1 = CC 0 and = C 0�1C�1
(b) Jn = n�C 0gn(�)
�0 �C 0C
��1C 0gn(�)
(c) C 0gn(�) =DnC0gn(�0) where
Dn = I` �C 0�1
nX 0Z
���1
nZ 0X
��1�1
nX 0Z
���1� 1nZ 0X
��1C 0�1
gn(�0) =1
nX 0e:
(d) Dnp�! I` �R (R0R)�1R0 where R = C 0E (xiz0i)
(e) n1=2C 0gn(�0)d�! Z � N(0; I`)
(f) Jnd�! Z 0
�I` �R (R0R)�1R0
�Z
(g) Z 0�I` �R (R0R)�1R0
�Z � �2`�k:
Hint: I` �R (R0R)�1R0 is a projection matrix..
104
Chapter 8
Empirical Likelihood
8.1 Non-Parametric Likelihood
An alternative to GMM is empirical likelihood. The idea is due to Art Owen (1988, 2001) andhas been extended to moment condition models by Qin and Lawless (1994). It is a non-parametricanalog of likelihood estimation.
The idea is to construct a multinomial distribution F (p1; :::; pn) which places probability piat each observation. To be a valid multinomial distribution, these probabilities must satisfy therequirements that pi � 0 and
nXi=1
pi = 1: (8.1)
Since each observation is observed once in the sample, the log-likelihood function for this multino-mial distribution is
logL (p1; :::; pn) =
nXi=1
log(pi): (8.2)
First let us consider a just-identi�ed model. In this case the moment condition places noadditional restrictions on the multinomial distribution. The maximum likelihood estimators of theprobabilities (p1; :::; pn) are those which maximize the log-likelihood subject to the constraint (8.1).This is equivalent to maximizing
nXi=1
log(pi)� �
nXi=1
pi � 1!
where � is a Lagrange multiplier. The n �rst order conditions are 0 = p�1i ��: Combined with theconstraint (8.1) we �nd that the MLE is pi = n�1 yielding the log-likelihood �n log(n):
Now consider the case of an overidenti�ed model with moment condition
Egi(�0) = 0
where g is `� 1 and � is k� 1 and for simplicity we write gi(�) = g(yi;zi;xi;�): The multinomialdistribution which places probability pi at each observation (yi;xi;zi) will satisfy this condition ifand only if
nXi=1
pigi(�) = 0 (8.3)
The empirical likelihood estimator is the value of � which maximizes the multinomial log-likelihood (8.2) subject to the restrictions (8.1) and (8.3).
105
The Lagrangian for this maximization problem is
L (�; p1; :::; pn;�; �) =nXi=1
log(pi)� �
nXi=1
pi � 1!� n�0
nXi=1
pigi (�)
where � and � are Lagrange multipliers. The �rst-order-conditions of L with respect to pi, �; and� are
1
pi= �+ n�0gi (�)
nXi=1
pi = 1
nXi=1
pigi (�) = 0:
Multiplying the �rst equation by pi, summing over i; and using the second and third equations, we�nd � = n and
pi =1
n�1 + �0gi (�)
� :Substituting into L we �nd
R (�;�) = �n log (n)�nXi=1
log�1 + �0gi (�)
�: (8.4)
For given �; the Lagrange multiplier �(�) minimizes R (�;�) :
�(�) = argmin�
R(�;�): (8.5)
This minimization problem is the dual of the constrained maximization problem. The solution(when it exists) is well de�ned since R(�;�) is a convex function of �: The solution cannot beobtained explicitly, but must be obtained numerically (see section 6.5). This yields the (pro�le)empirical log-likelihood function for �.
R(�) = R(�;�(�))
= �n log (n)�nXi=1
log�1 + �(�)0gi (�)
�The EL estimate � is the value which maximizes R(�); or equivalently minimizes its negative
� = argmin�
[�R(�)] (8.6)
Numerical methods are required for calculation of � (see Section 8.5).As a by-product of estimation, we also obtain the Lagrange multiplier � = �(�); probabilities
pi =1
n�1 + �
0gi
���� :
and maximized empirical likelihood
R(�) =nXi=1
log (pi) : (8.7)
106
8.2 Asymptotic Distribution of EL Estimator
De�ne
Gi (�) =@
@�0gi (�) (8.8)
G = EGi (�0)
= E�gi (�0) gi (�0)
0�and
V =�G0�1G
��1(8.9)
V � = �G�G0�1G
��1G0 (8.10)
For example, in the linear model, Gi (�) = �xiz0i; G = �E (xiz0i), and = E�xix
0ie2i
�:
Theorem 8.2.1 Under regularity conditions,
pn�� � �0
�d�! N(0;V )
pn�
d�! �1N(0;V �)
where V and V � are de�ned in (8.9) and (8.10), andpn�� � �0
�and
pn� are asymptotically
independent.
The proof is given in Section 8.6.The theorem shows that asymptotic variance V for � is the same as for e¢ cient GMM. Thus
the EL estimator is asymptotically e¢ cient.Chamberlain (1987) showed that V is the semiparametric e¢ ciency bound for � in the overi-
denti�ed moment condition model. This means that no consistent estimator for this class of modelscan have a lower asymptotic variance than V . Since the EL estimator achieves this bound, it is anasymptotically e¢ cient estimator for �.
8.3 Overidentifying Restrictions
In a parametric likelihood context, tests are based on the di¤erence in the log likelihood func-tions. The same statistic can be constructed for empirical likelihood. Twice the di¤erence betweenthe unrestricted empirical log-likelihood �n log (n) and the maximized empirical log-likelihood forthe model (8.7) is
LRn =nXi=1
2 log�1 + �
0gi
����: (8.11)
Theorem 8.3.1 If Egi(�0) = 0 then LRnd�! �2`�k:
The proof is given in Section 8.6.The EL overidenti�cation test is similar to the GMM overidenti�cation test. They are asymp-
totically �rst-order equivalent, and have the same interpretation. The overidenti�cation test is avery useful by-product of EL estimation, and it is advisable to report the statistic LRn wheneverEL is the estimation method.
107
8.4 Testing
Let the maintained model beEgi(�) = 0 (8.12)
where g is ` � 1 and � is k � 1: By �maintained� we mean that the overidentfying restrictionscontained in (8.12) are assumed to hold and are not being challenged (at least for the test discussedin this section). The hypothesis of interest is
h(�) = 0:
where h : Rk ! Ra: The restricted EL estimator and likelihood are the values which solve
~� = argmaxh(�)=0
R(�)
R(~�) = maxh(�)=0
R(�):
Fundamentally, the restricted EL estimator ~� is simply an EL estimator with `�k+a overidentifyingrestrictions, so there is no fundamental change in the distribution theory for ~� relative to �: To testthe hypothesis h(�) while maintaining (8.12), the simple overidentifying restrictions test (8.11) isnot appropriate. Instead we use the di¤erence in log-likelihoods:
LRn = 2�R(�)�R(~�)
�:
This test statistic is a natural analog of the GMM distance statistic.
Theorem 8.4.1 Under (8.12) and H0 : h(�) = 0; LRnd�! �2a:
The proof of this result is more challenging and is omitted.
8.5 Numerical Computation
Gauss code which implements the methods discussed below can be found at
http://www.ssc.wisc.edu/~bhansen/progs/elike.prc
DerivativesThe numerical calculations depend on derivatives of the dual likelihood function (8.4). De�ne
g�i (�;�) =gi (�)�
1 + �0gi (�)�
G�i (�;�) =
Gi (�)0 �
1 + �0gi (�)
The �rst derivatives of (8.4) are
R� =@
@�R (�;�) = �
nXi=1
g�i (�;�)
R� =@
@�R (�;�) = �
nXi=1
G�i (�;�) :
108
The second derivatives are
R�� =@2
@�@�0R (�;�) =
nXi=1
g�i (�;�) g�i (�;�)
0
R�� =@2
@�@�0R (�;�) =
nXi=1
�g�i (�;�)G
�i (�;�)
0 � Gi (�)
1 + �0gi (�)
�
R�� =@2
@�@�0R (�;�) =
nXi=1
0@G�i (�;�)G
�i (�;�)
0 �@2
@�@�0�gi (�)
0 ��
1 + �0gi (�)
1AInner LoopThe so-called �inner loop� solves (8.5) for given �: The modi�ed Newton method takes a
quadratic approximation to Rn (�;�) yielding the iteration rule
�j+1 = �j � � (R�� (�;�j))�1R� (�;�j) : (8.13)
where � > 0 is a scalar steplength (to be discussed next). The starting value �1 can be set to thezero vector. The iteration (8.13) is continued until the gradient R� (�;�j) is smaller than someprespeci�ed tolerance.
E¢ cient convergence requires a good choice of steplength �: One method uses the followingquadratic approximation. Set �0 = 0; �1 = 1
2 and �2 = 1: For p = 0; 1; 2; set
�p = �j � �p (R�� (�;�j))�1R� (�;�j))
Rp = R (�;�p)
A quadratic function can be �t exactly through these three points. The value of � which minimizesthis quadratic is
� =R2 + 3R0 � 4R14R2 + 4R0 � 8R1
:
yielding the steplength to be plugged into (8.13).A complication is that � must be constrained so that 0 � pi � 1 which holds if
n�1 + �0gi (�)
�� 1 (8.14)
for all i: If (8.14) fails, the stepsize � needs to be decreased.Outer LoopThe outer loop is the minimization (8.6). This can be done by the modi�ed Newton method
described in the previous section. The gradient for (8.6) is
R� =@
@�R(�) =
@
@�R(�;�) = R� + �
0�R� = R�
since R� (�;�) = 0 at � = �(�); where
�� =@
@�0�(�) = �R�1
��R��;
the second equality following from the implicit function theorem applied to R� (�;�(�)) = 0:The Hessian for (8.6) is
R�� = � @
@�@�0R(�)
= � @
@�0�R� (�;�(�)) + �
0�R� (�;�(�))
�= �
�R�� (�;�(�)) +R
0���� + �
0�R�� + �
0�R����
�= R0
��R�1��R�� �R��:
109
It is not guaranteed that R�� > 0: If not, the eigenvalues of R�� should be adjusted so that allare positive. The Newton iteration rule is
�j+1 = �j � �R�1��R�
where � is a scalar stepsize, and the rule is iterated until convergence.
8.6 Technical Proofs
Proof of Theorem 8.2.1. (�; �) jointly solve
0 =@
@�R(�;�) = �
nXi=1
gi
���
�1 + �
0gi
���� (8.15)
0 =@
@�R(�;�) = �
nXi=1
Gi
���0�
1 + �0gi
��� : (8.16)
Let Gn =1n
Pni=1Gi (�0) ; gn =
1n
Pni=1 gi (�0) and n =
1n
Pni=1 gi (�0) gi (�0)
0 :Expanding (8.16) around � = �0 and � = �0 = 0 yields
0 ' G0n
��� �0
�: (8.17)
Expanding (8.15) around � = �0 and � = �0 = 0 yields
0 ' �gn �Gn
�� � �0
�+n� (8.18)
Premultiplying by G0n
�1n and using (8.17) yields
0 ' �G0n
�1n gn �G0
n�1n Gn
�� � �0
�+G0
n�1n n�
= �G0n
�1n gn �G0
n�1n Gn
�� � �0
�Solving for � and using the WLLN and CLT yields
pn�� � �0
�' �
�G0n
�1n Gn
��1G0n
�1n
pngn (8.19)
d�!�G0�1G
��1G0�1N(0;)
= N (0;V )
Solving (8.18) for � and using (8.19) yields
pn� ' �1n
�I �Gn
�G0n
�1n Gn
��1G0n
�1n
�pngn (8.20)
d�! �1�I �G
�G0�1G
��1G0�1
�N(0;)
= �1N(0;V �)
Furthermore, since
G0�I ��1G
�G0�1G
��1G0�= 0
pn�� � �0
�and
pn� are asymptotically uncorrelated and hence independent. �
110
Proof of Theorem 8.3.1. First, by a Taylor expansion, (8.19), and (8.20),
1pn
nXi=1
gi
���
'pn�gn +Gn
�� � �0
��'
�I �Gn
�G0n
�1n Gn
��1G0n
�1n
�pngn
' npn�:
Second, since log(1 + u) ' u� u2=2 for u small,
LRn =nXi=1
2 log�1 + �
0gi
����
' 2�0nXi=1
gi
���� �0
nXi=1
gi
���gi
���0�
' n�0n�d�! N(0;V �)
0�1N(0;V �)
= �2`�k
where the proof of the �nal equality is left as an exercise. �
111
Chapter 9
Endogeneity
We say that there is endogeneity in the linear model y = z0i� + ei if � is the parameter ofinterest and E(ziei) 6= 0: This cannot happen if � is de�ned by linear projection, so requires astructural interpretation. The coe¢ cient � must have meaning separately from the de�nition of aconditional mean or linear projection.
Example: Measurement error in the regressor. Suppose that (yi;x�i ) are joint randomvariables, E(yi j x�i ) = x�0i � is linear, � is the parameter of interest, and x�i is not observed. Insteadwe observe xi = x�i + ui where ui is an k � 1 measurement error, independent of yi and x�i : Then
yi = x�0i � + ei
= (xi � ui)0 � + ei= x0i� + vi
wherevi = ei � u0i�:
The problem is that
E (xivi) = E�(x�i + ui)
�ei � u0i�
��= �E
�uiu
0i
�� 6= 0
if � 6= 0 and E (uiu0i) 6= 0: It follows that if � is the OLS estimator, then
�p�! �� = � �
�E�xix
0i
���1E�uiu
0i
�� 6= �:
This is called measurement error bias.Example: Supply and Demand. The variables qi and pi (quantity and price) are determined
jointly by the demand equationqi = ��1pi + e1i
and the supply equationqi = �2pi + e2i:
Assume that ei =�e1ie2i
�is iid, Eei = 0; �1 + �2 = 1 and Eeie
0i = I2 (the latter for simplicity).
The question is, if we regress qi on pi; what happens?It is helpful to solve for qi and pi in terms of the errors. In matrix notation,�
1 �11 ��2
��qipi
�=
�e1ie2i
�
112
so �qipi
�=
�1 �11 ��2
��1�e1ie2i
�=
��2 �11 �1
��e1ie2i
�=
��2e1i + �1e2i(e1i � e2i)
�:
The projection of qi on pi yields
qi = ��pi + "i
E (pi"i) = 0
where
�� =E(piqi)
E�p2i� = �2 � �1
2
Hence if it is estimated by OLS, �p�! ��; which does not equal either �1 or �2: This is called
simultaneous equations bias.
9.1 Instrumental Variables
Let the equation of interest beyi = z
0i� + ei (9.1)
where zi is k�1; and assume that E(ziei) 6= 0 so there is endogeneity. We call (9.1) the structuralequation. In matrix notation, this can be written as
y = Z� + e: (9.2)
Any solution to the problem of endogeneity requires additional information which we call in-struments.
De�nition 9.1.1 The `� 1 random vector xi is an instrumental variable for (9.1) if E (xiei) = 0:
In a typical set-up, some regressors in zi will be uncorrelated with ei (for example, at least theintercept). Thus we make the partition
zi =
�z1iz2i
�k1k2
(9.3)
where E(z1iei) = 0 yet E(z2iei) 6= 0: We call z1i exogenous and z2i endogenous. By the abovede�nition, z1i is an instrumental variable for (9.1); so should be included in xi: So we have thepartition
xi =
�z1ix2i
�k1`2
(9.4)
where z1i = x1i are the included exogenous variables, and x2i are the excluded exogenousvariables. That is x2i are variables which could be included in the equation for yi (in the sensethat they are uncorrelated with ei) yet can be excluded, as they would have true zero coe¢ cientsin the equation.
The model is just-identi�ed if ` = k (i.e., if `2 = k2) and over-identi�ed if ` > k (i.e., if`2 > k2):
We have noted that any solution to the problem of endogeneity requires instruments. This doesnot mean that valid instruments actually exist.
113
9.2 Reduced Form
The reduced form relationship between the variables or �regressors�zi and the instruments xiis found by linear projection. Let
� = E�xix
0i
��1E�xiz
0i
�be the `� k matrix of coe¢ cients from a projection of zi on xi; and de�ne
ui = zi � �0xi
as the projection error. Then the reduced form linear relationship between zi and xi is
zi = �0xi + ui: (9.5)
In matrix notation, we can write (9.5) as
Z =X�+U (9.6)
where U is n� k:By construction,
E(xiu0i) = 0;
so (9.5) is a projection and can be estimated by OLS:
Z = X� + u
� =�X 0X
��1 �X 0Z
�:
Substituting (9:6) into (9.2), we �nd
y = (X�+U)� + e
= X�+ v; (9.7)
where� = �� (9.8)
andv = U� + e:
Observe thatE (xivi) = E
�xiu
0i
�� + E(xiei) = 0:
Thus (9.7) is a projection equation and may be estimated by OLS. This is
y = X�+ v;
� =�X 0X
��1 �X 0y
�The equation (9.7) is the reduced form for y: (9.6) and (9.7) together are the reduced form
equations for the system
y = X�+ v
Z = X�+U :
As we showed above, OLS yields the reduced-form estimates��; �
�
114
9.3 Identi�cation
The structural parameter � relates to (�;�) through (9.8). The parameter � is identi�ed,meaning that it can be recovered from the reduced form, if
rank (�) = k: (9.9)
Assume that (9.9) holds. If ` = k; then � = ��1�: If ` > k; then for any W > 0; � =(�0W�)
�1�0W�:
If (9.9) is not satis�ed, then � cannot be recovered from (�;�) : Note that a necessary (althoughnot su¢ cient) condition for (9.9) is ` � k:
Since X and Z have the common variables X1; we can rewrite some of the expressions. Using(9.3) and (9.4) to make the matrix partitions X = [X1;X2] and Z = [X1;Z2] ; we can partition� as
� =
��11 �12�21 �22
�=
�I �120 �22
�(9.6) can be rewritten as
Z1 = X1
Z2 = X1�12 +X2�22 +U2: (9.10)
� is identi�ed if rank(�) = k; which is true if and only if rank(�22) = k2 (by the upper-diagonalstructure of �): Thus the key to identi�cation of the model rests on the `2�k2 matrix �22 in (9.10).
9.4 Estimation
The model can be written as
yi = z0i� + ei
E (xiei) = 0
or
Egi (�) = 0
gi (�) = xi�yi � z0i�
�:
This is a moment condition model. Appropriate estimators include GMM and EL. The estimatorsand distribution theory developed in those Chapter 8 and 9 directly apply. Recall that the GMMestimator, for given weight matrix W n; is
� =�Z 0XW nX
0Z��1
Z 0XW nX0y:
9.5 Special Cases: IV and 2SLS
If the model is just-identi�ed, so that k = `; then the formula for GMM simpli�es. We �nd that
� =�Z 0XW nX
0Z��1
Z 0XW nX0y
=�X 0Z
��1W�1
n
�Z 0X
��1Z 0XW nX
0y
=�X 0Z
��1X 0y
115
This estimator is often called the instrumental variables estimator (IV) of �; where X is usedas an instrument for Z: Observe that the weight matrixW n has disappeared. In the just-identi�edcase, the weight matrix places no role. This is also the MME estimator of �; and the EL estimator.Another interpretation stems from the fact that since � = ��1�; we can construct the IndirectLeast Squares (ILS) estimator:
� = ��1�
=��X 0X
��1 �X 0Z
���1 ��X 0X
��1 �X 0y
��=
�X 0Z
��1 �X 0X
� �X 0X
��1 �X 0y
�=
�X 0Z
��1 �X 0y
�:
which again is the IV estimator.
Recall that the optimal weight matrix is an estimate of the inverse of = E�xix
0ie2i
�: In the
special case that E�e2i j xi
�= �2 (homoskedasticity), then = E(xix
0i)�
2 / E (xix0i) suggestingthe weight matrix W n = (X
0X)�1 : Using this choice, the GMM estimator equals
�2SLS =�Z 0X
�X 0X
��1X 0Z
��1Z 0X
�X 0X
��1X 0y
This is called the two-stage-least squares (2SLS) estimator. It was originally proposed by Theil(1953) and Basmann (1957), and is the classic estimator for linear equations with instruments.Under the homoskedasticity assumption, the 2SLS estimator is e¢ cient GMM, but otherwise it isine¢ cient.
It is useful to observe that writing
P = X�X 0X
��1X 0
Z = PZ =X�
then
� =�Z 0PZ
��1Z 0Py
=�Z0Z��1
Z0y:
The source of the �two-stage�name is since it can be computed as follows
� First regress Z on X; vis., � = (X 0X)�1 (X 0Z) and Z =X� = PZ:
� Second, regress y on Z; vis., � =�Z0Z��1
Z0y:
It is useful to scrutinize the projection Z: Recall, Z = [Z1;Z2] and X = [Z1;X2]: Then
Z =hZ1; Z2
i= [PZ1;PZ2]
= [Z1;PZ2]
=hZ1; Z2
i;
since Z1 lies in the span of X: Thus in the second stage, we regress y on Z1 and Z2: So only theendogenous variables Z2 are replaced by their �tted values:
Z2 =X1�12 +X2�22:
116
9.6 Bekker Asymptotics
Bekker (1994) used an alternative asymptotic framework to analyze the �nite-sample bias inthe 2SLS estimator. Here we present a simpli�ed version of one of his results. In our notation, themodel is
y = Z� + e (9.11)
Z = X�+U (9.12)
� = (e;U)
E (� jX) = 0
E��0� jX
�= S
As before, X is n� l so there are l instruments.First, let�s analyze the approximate bias of OLS applied to (9.11). Using (9.12),
E
�1
nZ 0e
�= E(ziei) = �
0E (xiei) + E (uiei) = s21
and
E
�1
nZ 0Z
�= E
�ziz
0i
�= �0E
�xix
0i
��+ E
�uix
0i
��+ �0E
�xiu
0i
�+ E
�uiu
0i
�= �0Q�+ S22
where Q = E(xix0i) : Hence by a �rst-order approximation
E��OLS � �
��
�E
�1
nZ 0Z
���1E
�1
nZ 0e
�=
��0Q�+ S22
��1s21 (9.13)
which is zero only when s21 = 0 (when Z is exogenous).We now derive a similar result for the 2SLS estimator.
�2SLS =�Z 0PZ
��1 �Z 0Py
�:
Let P = X (X 0X)�1X 0. By the spectral decomposition of an idempotent matrix, P = H�H 0
where � = diag (I l;0) : Let Q =H 0�S�1=2 which satis�es EQ0Q = In and partition Q = (q01 Q02)
where q1 is l � 1: Hence
E
�1
n�0P� jX
�=
1
nS1=20E
�Q0�Q jX
�S1=2
=1
nS1=20E
�1
nq01q1
�S1=2
=l
nS1=20S1=2
= �S
where
� =l
n:
Using (9.12) and this result,
1
nE�Z 0Pe
�=1
nE��0X 0e
�+1
nE�U 0Pe
�= �s21;
117
and
1
nE�Z 0PZ
�= �0E
�xix
0i
��+ �0E (xiui) + E
�uix
0i
��+
1
nE�U 0PU
�= �0Q�+ �S22:
Together
E��2SLS � �
��
�E
�1
nZ 0PZ
���1E
�1
nZ 0Pe
�= �
��0Q�+ �S22
��1s21: (9.14)
In general this is non-zero, except when s21 = 0 (when Z is exogenous). It is also close to zerowhen � = 0. Bekker (1994) pointed out that it also has the reverse implication �that when � = l=nis large, the bias in the 2SLS estimator will be large. Indeed as � ! 1; the expression in (9.14)approaches that in (9.13), indicating that the bias in 2SLS approaches that of OLS as the numberof instruments increases.
Bekker (1994) showed further that under the alternative asymptotic approximation that � is�xed as n ! 1 (so that the number of instruments goes to in�nity proportionately with samplesize) then the expression in (9.14) is the probability limit of �2SLS � �
9.7 Identi�cation Failure
Recall the reduced form equation
Z2 =X1�12 +X2�22 +U2:
The parameter � fails to be identi�ed if �22 has de�cient rank. The consequences of identi�cationfailure for inference are quite severe.
Take the simplest case where k = l = 1 (so there is no X1): Then the model may be written as
yi = zi� + ei
zi = xi + ui
and �22 = = E(xizi) =Ex2i : We see that � is identi�ed if and only if 6= 0; which occurs
when E (zixi) 6= 0. Thus identi�cation hinges on the existence of correlation between the excludedexogenous variable and the included endogenous variable.
Suppose this condition fails, so E (zixi) = 0: Then by the CLT
1pn
nXi=1
xieid�! N1 � N
�0;E
�x2i e
2i
��(9.15)
1pn
nXi=1
xizi =1pn
nXi=1
xiuid�! N2 � N
�0;E
�x2iu
2i
��(9.16)
therefore
� � � =1pn
Pni=1 xiei
1pn
Pni=1 xizi
d�! N1N2
� Cauchy;
since the ratio of two normals is Cauchy. This is particularly nasty, as the Cauchy distributiondoes not have a �nite mean. This result carries over to more general settings, and was examinedby Phillips (1989) and Choi and Phillips (1992).
118
Suppose that identi�cation does not completely fail, but is weak. This occurs when �22 is fullrank, but small. This can be handled in an asymptotic analysis by modeling it as local-to-zero, viz
�22 = n�1=2C;
where C is a full rank matrix. The n�1=2 is picked because it provides just the right balancing toallow a rich distribution theory.
To see the consequences, once again take the simple case k = l = 1: Here, the instrument xi isweak for zi if
= n�1=2c:
Then (9.15) is una¤ected, but (9.16) instead takes the form
1pn
nXi=1
xizi =1pn
nXi=1
x2i +1pn
nXi=1
xiui
=1
n
nXi=1
x2i c+1pn
nXi=1
xiui
d�! Qc+N2
therefore
� � � d�! N1Qc+N2
:
As in the case of complete identi�cation failure, we �nd that � is inconsistent for � and theasymptotic distribution of � is non-normal. In addition, standard test statistics have non-standarddistributions, meaning that inferences about parameters of interest can be misleading.
The distribution theory for this model was developed by Staiger and Stock (1997) and extendedto nonlinear GMM estimation by Stock and Wright (2000). Further results on testing were obtainedby Wang and Zivot (1998).
The bottom line is that it is highly desirable to avoid identi�cation failure. Once again, theequation to focus on is the reduced form
Z2 =X1�12 +X2�22 +U2
and identi�cation requires rank(�22) = k2: If k2 = 1; this requires �22 6= 0; which is straightforwardto assess using a hypothesis test on the reduced form. Therefore in the case of k2 = 1 (one RHSendogenous variable), one constructive recommendation is to explicitly estimate the reduced formequation for Z2; construct the test of �22 = 0, and at a minimum check that the test rejectsH0 : �22 = 0.
When k2 > 1; �22 6= 0 is not su¢ cient for identi�cation. It is not even su¢ cient that eachcolumn of �22 is non-zero (each column corresponds to a distinct endogenous variable in X2): Sowhile a minimal check is to test that each columns of �22 is non-zero, this cannot be interpretedas de�nitive proof that �22 has full rank. Unfortunately, tests of de�cient rank are di¢ cult toimplement. In any event, it appears reasonable to explicitly estimate and report the reduced formequations for X2; and attempt to assess the likelihood that �22 has de�cient rank.
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9.8 Exercises
1. Consider the single equation model
yi = zi� + ei;
where yi and zi are both real-valued (1� 1). Let � denote the IV estimator of � using as aninstrument a dummy variable di (takes only the values 0 and 1). Find a simple expressionfor the IV estimator in this context.
2. In the linear model
yi = z0i� + ei
E (ei j zi) = 0
suppose �2i = E�e2i j zi
�is known. Show that the GLS estimator of � can be written as an
IV estimator using some instrument xi: (Find an expression for xi:)
3. Take the linear modely = Z� + e:
Let the OLS estimator for � be � and the OLS residual be e = y �Z�.Let the IV estimator for � using some instrumentX be ~� and the IV residual be ~e = y�Z~�.If X is indeed endogeneous, will IV ��t�better than OLS, in the sense that ~e0~e < e0e; atleast in large samples?
4. The reduced form between the regressors zi and instruments xi takes the form
zi = �0xi + ui
orZ =X�+U
where zi is k� 1; xi is l� 1; Z is n� k; X is n� l; U is n� k; and � is l� k: The parameter� is de�ned by the population moment condition
E�xiu
0i
�= 0
Show that the method of moments estimator for � is � = (X 0X)�1 (X 0Z) :
5. In the structural model
y = Z� + e
Z = X�+U
with � l� k; l � k; we claim that � is identi�ed (can be recovered from the reduced form) ifrank(�) = k: Explain why this is true. That is, show that if rank(�) < k then � cannot beidenti�ed.
6. Take the linear model
yi = xi� + ei
E (ei j xi) = 0:
where xi and � are 1� 1:
120
(a) Show that E (xiei) = 0 and E�x2i ei
�= 0: Is zi = (xi x2i )
0 a valid instrumental variablefor estimation of �?
(b) De�ne the 2SLS estimator of �; using zi as an instrument for xi: How does this di¤erfrom OLS?
(c) Find the e¢ cient GMM estimator of � based on the moment condition
E (zi (yi � xi�)) = 0:
Does this di¤er from 2SLS and/or OLS?
7. Suppose that price and quantity are determined by the intersection of the linear demand andsupply curves
Demand : Q = a0 + a1P + a2Y + e1
Supply : Q = b0 + b1P + b2W + e2
where income (Y ) and wage (W ) are determined outside the market. In this model, are theparameters identi�ed?
8. The data �le card.dat is taken from David Card �Using Geographic Variation in CollegeProximity to Estimate the Return to Schooling�in Aspects of Labour Market Behavior (1995).There are 2215 observations with 29 variables, listed in card.pdf. We want to estimate a wageequation
log(Wage) = �0 + �1Educ+ �2Exper + �3Exper2 + �4South+ �5Black + e
where Educ = Eduation (Years) Exper = Experience (Years), and South and Black areregional and racial dummy variables.
(a) Estimate the model by OLS. Report estimates and standard errors.
(b) Now treat Education as endogenous, and the remaining variables as exogenous. Estimatethe model by 2SLS, using the instrument near4, a dummy indicating that the observationlives near a 4-year college. Report estimates and standard errors.
(c) Re-estimate by 2SLS (report estimates and standard errors) adding three additionalinstruments: near2 (a dummy indicating that the observation lives near a 2-year college),fatheduc (the education, in years, of the father) and motheduc (the education, in years,of the mother).
(d) Re-estimate the model by e¢ cient GMM. I suggest that you use the 2SLS estimates asthe �rst-step to get the weight matrix, and then calculate the GMM estimator from thisweight matrix without further iteration. Report the estimates and standard errors.
(e) Calculate and report the J statistic for overidenti�cation.
(f) Discuss your �ndings..
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Chapter 10
Univariate Time Series
A time series yt is a process observed in sequence over time, t = 1; :::; T . To indicate thedependence on time, we adopt new notation, and use the subscript t to denote the individualobservation, and T to denote the number of observations.
Because of the sequential nature of time series, we expect that yt and yt�1 are not independent,so classical assumptions are not valid.
We can separate time series into two categories: univariate (yt 2 R is scalar); and multivariate(yt 2 Rm is vector-valued). The primary model for univariate time series is autoregressions (ARs).The primary model for multivariate time series is vector autoregressions (VARs).
10.1 Stationarity and Ergodicity
De�nition 10.1.1 fytg is covariance (weakly) stationary if
E(yt) = �
is independent of t; andcov (yt; yt�k) = (k)
is independent of t for all k: (k) is called the autocovariance function.
�(k) = (k)= (0) = corr(yt; yt�k)
is the autocorrelation function.
De�nition 10.1.2 fytg is strictly stationary if the joint distribution of (yt; :::; yt�k) is indepen-dent of t for all k:
De�nition 10.1.3 A stationary time series is ergodic if (k)! 0 as k !1.
The following two theorems are essential to the analysis of stationary time series. There proofsare rather di¢ cult, however.
Theorem 10.1.1 If yt is strictly stationary and ergodic and xt = f(yt; yt�1; :::) is a random vari-able, then xt is strictly stationary and ergodic.
Theorem 10.1.2 (Ergodic Theorem). If yt is strictly stationary and ergodic and E jytj <1; thenas T !1;
1
T
TXt=1
ytp�! E(yt):
122
This allows us to consistently estimate parameters using time-series moments:The sample mean:
� =1
T
TXt=1
yt
The sample autocovariance
(k) =1
T
TXt=1
(yt � �) (yt�k � �) :
The sample autocorrelation
�(k) = (k)
(0):
Theorem 10.1.3 If yt is strictly stationary and ergodic and Ey2t <1; then as T !1;
1. �p�! E(yt);
2. (k)p�! (k);
3. �(k)p�! �(k):
A proof is given in Section 10.13.
10.2 Autoregressions
In time-series, the series f:::; y1; y2; :::; yT ; :::g are jointly random. We consider the conditionalexpectation
E (yt j Ft�1)
where Ft�1 = fyt�1; yt�2; :::g is the past history of the series.An autoregressive (AR) model speci�es that only a �nite number of past lags matter:
E (yt j Ft�1) = E (yt j yt�1; :::; yt�k) :
A linear AR model (the most common type used in practice) speci�es linearity:
E (yt j Ft�1) = �+ �1yt�1 + �2yt�1 + � � �+ �kyt�k:
Lettinget = yt � E (yt j Ft�1) ;
then we have the autoregressive model
yt = �+ �1yt�1 + �2yt�1 + � � �+ �kyt�k + etE (et j Ft�1) = 0:
The last property de�nes a special time-series process.
De�nition 10.2.1 et is a martingale di¤erence sequence (MDS) if E (et j Ft�1) = 0:
123
Regression errors are naturally a MDS. Some time-series processes may be a MDS as a conse-quence of optimizing behavior. For example, some versions of the life-cycle hypothesis imply thateither changes in consumption, or consumption growth rates, should be a MDS. Most asset pricingmodels imply that asset returns should be the sum of a constant plus a MDS.
The MDS property for the regression error plays the same role in a time-series regression asdoes the conditional mean-zero property for the regression error in a cross-section regression. Infact, it is even more important in the time-series context, as it is di¢ cult to derive distributiontheories without this property.
A useful property of a MDS is that et is uncorrelated with any function of the lagged informationFt�1: Thus for k > 0; E (yt�ket) = 0:
10.3 Stationarity of AR(1) Process
A mean-zero AR(1) isyt = �yt�1 + et:
Assume that et is iid, E(et) = 0 and Ee2t = �2 <1:
By back-substitution, we �nd
yt = et + �et�1 + �2et�2 + :::
=
1Xk=0
�ket�k:
Loosely speaking, this series converges if the sequence �ket�k gets small as k ! 1: This occurswhen j�j < 1:
Theorem 10.3.1 If j�j < 1 then yt is strictly stationary and ergodic.
We can compute the moments of yt using the in�nite sum:
Eyt =
1Xk=0
�kE (et�k) = 0
var(yt) =
1Xk=0
�2k var (et�k) =�2
1� �2 :
If the equation for yt has an intercept, the above results are unchanged, except that the meanof yt can be computed from the relationship
Eyt = �+ �Eyt�1;
and solving for Eyt = Eyt�1 we �nd Eyt = �=(1� �):
10.4 Lag Operator
An algebraic construct which is useful for the analysis of autoregressive models is the lag oper-ator.
De�nition 10.4.1 The lag operator L satis�es Lyt = yt�1:
124
De�ning L2 = LL; we see that L2yt = Lyt�1 = yt�2: In general, Lkyt = yt�k:The AR(1) model can be written in the format
yt � �yt�1 + et
or(1� �L) yt�1 = et:
The operator �(L) = (1 � �L) is a polynomial in the operator L: We say that the root of thepolynomial is 1=�; since �(z) = 0 when z = 1=�: We call �(L) the autoregressive polynomial of yt.
From Theorem 10.3.1, an AR(1) is stationary i¤ j�j < 1: Note that an equivalent way to saythis is that an AR(1) is stationary i¤ the root of the autoregressive polynomial is larger than one(in absolute value).
10.5 Stationarity of AR(k)
The AR(k) model isyt = �1yt�1 + �2yt�2 + � � �+ �kyt�k + et:
Using the lag operator,yt � �1Lyt � �2L2yt � � � � � �kLkyt = et;
or�(L)yt = et
where�(L) = 1� �1L� �2L2 � � � � � �kLk:
We call �(L) the autoregressive polynomial of yt:The Fundamental Theorem of Algebra says that any polynomial can be factored as
�(z) =�1� ��11 z
� �1� ��12 z
�� � ��1� ��1k z
�where the �1; :::; �k are the complex roots of �(z); which satisfy �(�j) = 0:
We know that an AR(1) is stationary i¤ the absolute value of the root of its autoregressivepolynomial is larger than one. For an AR(k), the requirement is that all roots are larger than one.Let j�j denote the modulus of a complex number �:
Theorem 10.5.1 The AR(k) is strictly stationary and ergodic if and only if j�j j > 1 for all j:
One way of stating this is that �All roots lie outside the unit circle.�If one of the roots equals 1, we say that �(L); and hence yt; �has a unit root�. This is a special
case of non-stationarity, and is of great interest in applied time series.
10.6 Estimation
Let
xt =�1 yt�1 yt�2 � � � yt�k
�0� =
�� �1 �2 � � � �k
�0:
Then the model can be written asyt = x
0t� + et:
The OLS estimator is� =
�X 0X
��1X 0y:
125
To study �; it is helpful to de�ne the process ut = xtet: Note that ut is a MDS, since
E (ut j Ft�1) = E (xtet j Ft�1) = xtE (et j Ft�1) = 0:
By Theorem 10.1.1, it is also strictly stationary and ergodic. Thus
1
T
TXt=1
xtet =1
T
TXt=1
utp�! E (ut) = 0: (10.1)
The vector xt is strictly stationary and ergodic, and by Theorem 10.1.1, so is xtx0t: Thus by theErgodic Theorem,
1
T
TXt=1
xtx0t
p�! E�xtx
0t
�= Q:
Combined with (10.1) and the continuous mapping theorem, we see that
� = � +
1
T
TXt=1
xtx0t
!�1 1
T
TXt=1
xtet
!p�! Q�10 = 0:
We have shown the following:
Theorem 10.6.1 If the AR(k) process yt is strictly stationary and ergodic and Ey2t < 1; then�
p�! � as T !1:
10.7 Asymptotic Distribution
Theorem 10.7.1 MDS CLT. If ut is a strictly stationary and ergodic MDS and E (utu0t) = <1; then as T !1;
1pT
TXt=1
utd�! N(0;) :
Since xtet is a MDS, we can apply Theorem 10.7.1 to see that
1pT
TXt=1
xtetd�! N(0;) ;
where = E(xtx
0te2t ):
Theorem 10.7.2 If the AR(k) process yt is strictly stationary and ergodic and Ey4t <1; then asT !1; p
T�� � �
�d�! N
�0;Q�1Q�1
�:
This is identical in form to the asymptotic distribution of OLS in cross-section regression. Theimplication is that asymptotic inference is the same. In particular, the asymptotic covariancematrix is estimated just as in the cross-section case.
126
10.8 Bootstrap for Autoregressions
In the non-parametric bootstrap, we constructed the bootstrap sample by randomly resamplingfrom the data values fyt;xtg: This creates an iid bootstrap sample. Clearly, this cannot work in atime-series application, as this imposes inappropriate independence.
Brie�y, there are two popular methods to implement bootstrap resampling for time-series data.
Method 1: Model-Based (Parametric) Bootstrap.
1. Estimate � and residuals et:
2. Fix an initial condition (y�k+1; y�k+2; :::; y0):
3. Simulate iid draws e�i from the empirical distribution of the residuals fe1; :::; eT g:
4. Create the bootstrap series y�t by the recursive formula
y�t = �+ �1y�t�1 + �2y
�t�2 + � � �+ �ky�t�k + e�t :
This construction imposes homoskedasticity on the errors e�i ; which may be di¤erent than theproperties of the actual ei: It also presumes that the AR(k) structure is the truth.
Method 2: Block Resampling
1. Divide the sample into T=m blocks of length m:
2. Resample complete blocks. For each simulated sample, draw T=m blocks.
3. Paste the blocks together to create the bootstrap time-series y�t :
4. This allows for arbitrary stationary serial correlation, heteroskedasticity, and for model-misspeci�cation.
5. The results may be sensitive to the block length, and the way that the data are partitionedinto blocks.
6. May not work well in small samples.
10.9 Trend Stationarity
yt = �0 + �1t+ St (10.2)
St = �1St�1 + �2St�2 + � � �+ �kSt�l + et; (10.3)
oryt = �0 + �1t+ �1yt�1 + �2yt�1 + � � �+ �kyt�k + et: (10.4)
There are two essentially equivalent ways to estimate the autoregressive parameters (�1; :::; �k):
� You can estimate (10.4) by OLS.
� You can estimate (10.2)-(10.3) sequentially by OLS. That is, �rst estimate (10.2), get theresidual St; and then perform regression (10.3) replacing St with St: This procedure is some-times called Detrending.
127
The reason why these two procedures are (essentially) the same is the Frisch-Waugh-Lovelltheorem.
Seasonal E¤ects
There are three popular methods to deal with seasonal data.
� Include dummy variables for each season. This presumes that �seasonality�does not changeover the sample.
� Use �seasonally adjusted� data. The seasonal factor is typically estimated by a two-sidedweighted average of the data for that season in neighboring years. Thus the seasonallyadjusted data is a ��ltered�series. This is a �exible approach which can extract a wide rangeof seasonal factors. The seasonal adjustment, however, also alters the time-series correlationsof the data.
� First apply a seasonal di¤erencing operator. If s is the number of seasons (typically s = 4 ors = 12);
�syt = yt � yt�s;or the season-to-season change. The series �syt is clearly free of seasonality. But the long-runtrend is also eliminated, and perhaps this was of relevance.
10.10 Testing for Omitted Serial Correlation
For simplicity, let the null hypothesis be an AR(1):
yt = �+ �yt�1 + ut: (10.5)
We are interested in the question if the error ut is serially correlated. We model this as an AR(1):
ut = �ut�1 + et (10.6)
with et a MDS. The hypothesis of no omitted serial correlation is
H0 : � = 0
H1 : � 6= 0:
We want to test H0 against H1:To combine (10.5) and (10.6), we take (10.5) and lag the equation once:
yt�1 = �+ �yt�2 + ut�1:
We then multiply this by � and subtract from (10.5), to �nd
yt � �yt�1 = �� ��+ �yt�1 � ��yt�1 + ut � �ut�1;
oryt = �(1� �) + (�+ �) yt�1 � ��yt�2 + et = AR(2):
Thus under H0; yt is an AR(1), and under H1 it is an AR(2). H0 may be expressed as the restrictionthat the coe¢ cient on yt�2 is zero.
An appropriate test of H0 against H1 is therefore a Wald test that the coe¢ cient on yt�2 is zero.(A simple exclusion test).
In general, if the null hypothesis is that yt is an AR(k), and the alternative is that the error is anAR(m), this is the same as saying that under the alternative yt is an AR(k+m), and this is equivalentto the restriction that the coe¢ cients on yt�k�1; :::; yt�k�m are jointly zero. An appropriate test isthe Wald test of this restriction.
128
10.11 Model Selection
What is the appropriate choice of k in practice? This is a problem of model selection.One approach to model selection is to choose k based on a Wald tests.Another is to minimize the AIC or BIC information criterion, e.g.
AIC(k) = log �2(k) +2k
T;
where �2(k) is the estimated residual variance from an AR(k)One ambiguity in de�ning the AIC criterion is that the sample available for estimation changes
as k changes. (If you increase k; you need more initial conditions.) This can induce strangebehavior in the AIC. The best remedy is to �x a upper value k; and then reserve the �rst k asinitial conditions, and then estimate the models AR(1), AR(2), ..., AR(k) on this (uni�ed) sample.
10.12 Autoregressive Unit Roots
The AR(k) model is
�(L)yt = �+ et
�(L) = 1� �1L� � � � � �kLk:
As we discussed before, yt has a unit root when �(1) = 0; or
�1 + �2 + � � �+ �k = 1:
In this case, yt is non-stationary. The ergodic theorem and MDS CLT do not apply, and teststatistics are asymptotically non-normal.
A helpful way to write the equation is the so-called Dickey-Fuller reparameterization:
�yt = �+ �0yt�1 + �1�yt�1 + � � �+ �k�1�yt�(k�1) + et: (10.7)
These models are equivalent linear transformations of one another. The DF parameterizationis convenient because the parameter �0 summarizes the information about the unit root, since�(1) = ��0: To see this, observe that the lag polynomial for the yt computed from (10.7) is
(1� L)� �0L� �1(L� L2)� � � � � �k�1(Lk�1 � Lk)
But this must equal �(L); as the models are equivalent. Thus
�(1) = (1� 1)� �0 � (1� 1)� � � � � (1� 1) = ��0:
Hence, the hypothesis of a unit root in yt can be stated as
H0 : �0 = 0:
Note that the model is stationary if �0 < 0: So the natural alternative is
H1 : �0 < 0:
Under H0; the model for yt is
�yt = �+ �1�yt�1 + � � �+ �k�1�yt�(k�1) + et;
which is an AR(k-1) in the �rst-di¤erence �yt: Thus if yt has a (single) unit root, then �yt is astationary AR process. Because of this property, we say that if yt is non-stationary but �dyt isstationary, then yt is �integrated of order d�; or I(d): Thus a time series with unit root is I(1):
129
Since �0 is the parameter of a linear regression, the natural test statistic is the t-statistic forH0 from OLS estimation of (10.7). Indeed, this is the most popular unit root test, and is called theAugmented Dickey-Fuller (ADF) test for a unit root.
It would seem natural to assess the signi�cance of the ADF statistic using the normal table.However, under H0; yt is non-stationary, so conventional normal asymptotics are invalid. Analternative asymptotic framework has been developed to deal with non-stationary data. We do nothave the time to develop this theory in detail, but simply assert the main results.
Theorem 10.12.1 (Dickey-Fuller Theorem). Assume �0 = 0: As T !1;
T �0d�! (1� �1 � �2 � � � � � �k�1)DF�
ADF =�0s(�0)
! DFt:
The limit distributions DF� and DFt are non-normal. They are skewed to the left, and havenegative means.
The �rst result states that �0 converges to its true value (of zero) at rate T; rather than theconventional rate of T 1=2: This is called a �super-consistent�rate of convergence.
The second result states that the t-statistic for �0 converges to a limit distribution which isnon-normal, but does not depend on the parameters �: This distribution has been extensivelytabulated, and may be used for testing the hypothesis H0: Note: The standard error s(�0) is theconventional (�homoskedastic�) standard error. But the theorem does not require an assumptionof homoskedasticity. Thus the Dickey-Fuller test is robust to heteroskedasticity.
Since the alternative hypothesis is one-sided, the ADF test rejects H0 in favor of H1 whenADF < c; where c is the critical value from the ADF table. If the test rejects H0; this means thatthe evidence points to yt being stationary. If the test does not reject H0; a common conclusion isthat the data suggests that yt is non-stationary. This is not really a correct conclusion, however.All we can say is that there is insu¢ cient evidence to conclude whether the data are stationary ornot.
We have described the test for the setting of with an intercept. Another popular setting includesas well a linear time trend. This model is
�yt = �1 + �2t+ �0yt�1 + �1�yt�1 + � � �+ �k�1�yt�(k�1) + et: (10.8)
This is natural when the alternative hypothesis is that the series is stationary about a linear timetrend. If the series has a linear trend (e.g. GDP, Stock Prices), then the series itself is non-stationary, but it may be stationary around the linear time trend. In this context, it is a silly wasteof time to �t an AR model to the level of the series without a time trend, as the AR model cannotconceivably describe this data. The natural solution is to include a time trend in the �tted OLSequation. When conducting the ADF test, this means that it is computed as the t-ratio for �0 fromOLS estimation of (10.8).
If a time trend is included, the test procedure is the same, but di¤erent critical values arerequired. The ADF test has a di¤erent distribution when the time trend has been included, and adi¤erent table should be consulted.
Most texts include as well the critical values for the extreme polar case where the intercept hasbeen omitted from the model. These are included for completeness (from a pedagogical perspective)but have no relevance for empirical practice where intercepts are always included
130
10.13 Technical Proofs
Proof of Theorem 10.1.3. Part (1) is a direct consequence of the Ergodic theorem. For Part(2), note that
(k) =1
T
TXt=1
(yt � �) (yt�k � �)
=1
T
TXt=1
ytyt�k �1
T
TXt=1
yt��1
T
TXt=1
yt�k�+ �2:
By Theorem 10.1.1 above, the sequence ytyt�k is strictly stationary and ergodic, and it has a �nitemean by the assumption that Ey2t <1: Thus an application of the Ergodic Theorem yields
1
T
TXt=1
ytyt�kp�! E(ytyt�k):
Thus (k)
p�! E(ytyt�k)� �2 � �2 + �2 = E(ytyt�k)� �2 = (k):
Part (3) follows by the continuous mapping theorem: �(k) = (k)= (0)p�! (k)= (0) = �(k): �
.
131
Chapter 11
Multivariate Time Series
A multivariate time series yt is a vector process m�1. Let Ft�1 = (yt�1;yt�2; :::) be all laggedinformation at time t: The typical goal is to �nd the conditional expectation E (yt j Ft�1) : Notethat since yt is a vector, this conditional expectation is also a vector.
11.1 Vector Autoregressions (VARs)
A VAR model speci�es that the conditional mean is a function of only a �nite number of lags:
E (yt j Ft�1) = E�yt j yt�1; :::;yt�k
�:
A linear VAR speci�es that this conditional mean is linear in the arguments:
E�yt j yt�1; :::;yt�k
�= a0 +A1yt�1 +A2yt�2 + � � �Akyt�k:
Observe that a0 is m� 1,and each of A1 through Ak are m�m matrices.De�ning the m� 1 regression error
et = yt � E (yt j Ft�1) ;
we have the VAR model
yt = a0 +A1yt�1 +A2yt�2 + � � �Akyt�k + etE (et j Ft�1) = 0:
Alternatively, de�ning the mk + 1 vector
xt =
0BBBBB@1yt�1yt�2...
yt�k
1CCCCCAand the m� (mk + 1) matrix
A =�a0 A1 A2 � � � Ak
�;
thenyt = Axt + et:
The VAR model is a system of m equations. One way to write this is to let a0j be the jth rowof A. Then the VAR system can be written as the equations
Yjt = a0jxt + ejt:
Unrestricted VARs were introduced to econometrics by Sims (1980).
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11.2 Estimation
Consider the moment conditionsE (xtejt) = 0;
j = 1; :::;m: These are implied by the VAR model, either as a regression, or as a linear projection.The GMM estimator corresponding to these moment conditions is equation-by-equation OLS
aj = (X0X)�1X 0yj :
An alternative way to compute this is as follows. Note that
a0j = y0jX(X
0X)�1:
And if we stack these to create the estimate A; we �nd
A =
0BBB@y01y02...
y0m+1
1CCCAX(X 0X)�1
= Y 0X(X 0X)�1;
whereY =
�y1 y2 � � � ym
�the T �m matrix of the stacked y0t:
This (system) estimator is known as the SUR (Seemingly Unrelated Regressions) estimator,and was originally derived by Zellner (1962)
11.3 Restricted VARs
The unrestricted VAR is a system of m equations, each with the same set of regressors. Arestricted VAR imposes restrictions on the system. For example, some regressors may be excludedfrom some of the equations. Restrictions may be imposed on individual equations, or across equa-tions. The GMM framework gives a convenient method to impose such restrictions on estimation.
11.4 Single Equation from a VAR
Often, we are only interested in a single equation out of a VAR system. This takes the form
yjt = a0jxt + et;
and xt consists of lagged values of yjt and the other y0lts: In this case, it is convenient to re-de�nethe variables. Let yt = yjt; and zt be the other variables. Let et = ejt and � = aj : Then the singleequation takes the form
yt = x0t� + et; (11.1)
andxt =
h�1 yt�1 � � � yt�k z0t�1 � � � z0t�k
�0i:
This is just a conventional regression with time series data.
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11.5 Testing for Omitted Serial Correlation
Consider the problem of testing for omitted serial correlation in equation (11.1). Suppose thatet is an AR(1). Then
yt = x0t� + et
et = �et�1 + ut (11.2)
E (ut j Ft�1) = 0:
Then the null and alternative are
H0 : � = 0 H1 : � 6= 0:
Take the equation yt = x0t� + et; and subtract o¤ the equation once lagged multiplied by �; to get
yt � �yt�1 =�x0t� + et
�� �
�x0t�1� + et�1
�= x0t� � �xt�1� + et � �et�1;
oryt = �yt�1 + x
0t� + x
0t�1 + ut; (11.3)
which is a valid regression model.So testing H0 versus H1 is equivalent to testing for the signi�cance of adding (yt�1;xt�1) to
the regression. This can be done by a Wald test. We see that an appropriate, general, and simpleway to test for omitted serial correlation is to test the signi�cance of extra lagged values of thedependent variable and regressors.
You may have heard of the Durbin-Watson test for omitted serial correlation, which once wasvery popular, and is still routinely reported by conventional regression packages. The DW test isappropriate only when regression yt = x0t�+ et is not dynamic (has no lagged values on the RHS),and et is iid N(0; �2): Otherwise it is invalid.
Another interesting fact is that (11.2) is a special case of (11.3), under the restriction = ���:This restriction, which is called a common factor restriction, may be tested if desired. If valid,the model (11.2) may be estimated by iterated GLS. (A simple version of this estimator is calledCochrane-Orcutt.) Since the common factor restriction appears arbitrary, and is typically rejectedempirically, direct estimation of (11.2) is uncommon in recent applications.
11.6 Selection of Lag Length in an VAR
If you want a data-dependent rule to pick the lag length k in a VAR, you may either use a testing-based approach (using, for example, the Wald statistic), or an information criterion approach. Theformula for the AIC and BIC are
AIC(k) = log det�(k)
�+ 2
p
T
BIC(k) = log det�(k)
�+p log(T )
T
(k) =1
T
TXt=1
et(k)et(k)0
p = m(km+ 1)
where p is the number of parameters in the model, and et(k) is the OLS residual vector from themodel with k lags. The log determinant is the criterion from the multivariate normal likelihood.
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11.7 Granger Causality
Partition the data vector into (yt;zt): De�ne the two information sets
F1t =�yt;yt�1;yt�2; :::
�F2t =
�yt;zt;yt�1;zt�1;yt�2;zt�2; ; :::
�The information set F1t is generated only by the history of yt; and the information set F2t isgenerated by both yt and zt: The latter has more information.
We say that zt does not Granger-cause yt if
E (yt j F1;t�1) = E (yt j F2;t�1) :
That is, conditional on information in lagged yt; lagged zt does not help to forecast yt: If thiscondition does not hold, then we say that zt Granger-causes yt:
The reason why we call this �Granger Causality�rather than �causality�is because this is nota physical or structure de�nition of causality. If zt is some sort of forecast of the future, such as afutures price, then zt may help to forecast yt even though it does not �cause�yt: This de�nitionof causality was developed by Granger (1969) and Sims (1972).
In a linear VAR, the equation for yt is
yt = �+ �1yt�1 + � � �+ �kyt�k + z0t�1 1 + � � �+ z0t�k k + et:
In this equation, zt does not Granger-cause yt if and only if
H0 : 1 = 2 = � � � = k = 0:
This may be tested using an exclusion (Wald) test.This idea can be applied to blocks of variables. That is, yt and/or zt can be vectors. The
hypothesis can be tested by using the appropriate multivariate Wald test.If it is found that zt does not Granger-cause yt; then we deduce that our time-series model of
E (yt j Ft�1) does not require the use of zt: Note, however, that zt may still be useful to explainother features of yt; such as the conditional variance.
11.8 Cointegration
The idea of cointegration is due to Granger (1981), and was articulated in detail by Engle andGranger (1987).
De�nition 11.8.1 The m � 1 series yt is cointegrated if yt is I(1) yet there exists �; m � r, ofrank r; such that zt = �0yt is I(0): The r vectors in � are called the cointegrating vectors.
If the series yt is not cointegrated, then r = 0: If r = m; then yt is I(0): For 0 < r < m; yt isI(1) and cointegrated.
In some cases, it may be believed that � is known a priori. Often, � = (1 � 1)0: For example,if yt is a pair of interest rates, then � = (1 � 1)0 speci�es that the spread (the di¤erence inreturns) is stationary. If y = (log(Consumption) log(Income))0; then � = (1 � 1)0 speci�esthat log(Consumption=Income) is stationary.
In other cases, � may not be known.If yt is cointegrated with a single cointegrating vector (r = 1); then it turns out that � can
be consistently estimated by an OLS regression of one component of yt on the others. Thus yt =(Y1t; Y2t) and � = (�1 �2) and normalize �1 = 1: Then �2 = (y
02y2)
�1y02y1p�! �2: Furthermore
this estimation is super-consistent: T (�2 � �2)d�! Limit; as �rst shown by Stock (1987). This
135
is not, in general, a good method to estimate �; but it is useful in the construction of alternativeestimators and tests.
We are often interested in testing the hypothesis of no cointegration:
H0 : r = 0
H1 : r > 0:
Suppose that � is known, so zt = �0yt is known. Then under H0 zt is I(1); yet under H1 zt isI(0): Thus H0 can be tested using a univariate ADF test on zt:
When � is unknown, Engle and Granger (1987) suggested using an ADF test on the estimatedresidual zt = �
0yt; from OLS of y1t on y2t: Their justi�cation was Stock�s result that � is super-
consistent under H1: Under H0; however, � is not consistent, so the ADF critical values are notappropriate. The asymptotic distribution was worked out by Phillips and Ouliaris (1990).
When the data have time trends, it may be necessary to include a time trend in the estimatedcointegrating regression. Whether or not the time trend is included, the asymptotic distribution ofthe test is a¤ected by the presence of the time trend. The asymptotic distribution was worked outin B. Hansen (1992).
11.9 Cointegrated VARs
We can write a VAR as
A(L)yt = et
A(L) = I �A1L�A2L2 � � � � �AkLk
or alternatively as�yt = �yt�1 +D(L)�yt�1 + et
where
� = �A(1)= �I +A1 +A2 + � � �+Ak:
Theorem 11.9.1 (Granger Representation Theorem). yt is cointegrated with m� r � if and onlyif rank(�) = r and � = ��0 where � is m� r, rank (�) = r:
Thus cointegration imposes a restriction upon the parameters of a VAR. The restricted modelcan be written as
�yt = ��0yt�1 +D(L)�yt�1 + et
�yt = �zt�1 +D(L)�yt�1 + et:
If � is known, this can be estimated by OLS of �yt on zt�1 and the lags of �yt:If � is unknown, then estimation is done by �reduced rank regression�, which is least-squares
subject to the stated restriction. Equivalently, this is the MLE of the restricted parameters underthe assumption that et is iid N(0;):
One di¢ culty is that � is not identi�ed without normalization. When r = 1; we typically justnormalize one element to equal unity. When r > 1; this does not work, and di¤erent authors haveadopted di¤erent identi�cation schemes.
In the context of a cointegrated VAR estimated by reduced rank regression, it is simple to testfor cointegration by testing the rank of�: These tests are constructed as likelihood ratio (LR) tests.As they were discovered by Johansen (1988, 1991, 1995), they are typically called the �JohansenMax and Trace� tests. Their asymptotic distributions are non-standard, and are similar to theDickey-Fuller distributions.
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Chapter 12
Limited Dependent Variables
A �limited dependent variable�y is one which takes a �limited�set of values. The most commoncases are
� Binary: y 2 f0; 1g
� Multinomial: y 2 f0; 1; 2; :::; kg
� Integer: y 2 f0; 1; 2; :::g
� Censored: y 2 R+
The traditional approach to the estimation of limited dependent variable (LDV) models isparametric maximum likelihood. A parametric model is constructed, allowing the construction ofthe likelihood function. A more modern approach is semi-parametric, eliminating the dependenceon a parametric distributional assumption. We will discuss only the �rst (parametric) approach,due to time constraints. They still constitute the majority of LDV applications. If, however, youwere to write a thesis involving LDV estimation, you would be advised to consider employing asemi-parametric estimation approach.
For the parametric approach, estimation is by MLE. A major practical issue is construction ofthe likelihood function.
12.1 Binary Choice
The dependent variable yi 2 f0; 1g: This represents a Yes/No outcome. Given some regressorsxi; the goal is to describe P (yi = 1 j xi) ; as this is the full conditional distribution.
The linear probability model speci�es that
P (yi = 1 j xi) = x0i�:
As P (yi = 1 j xi) = E (yi j xi) ; this yields the regression: yi = x0i�+ ei which can be estimated byOLS. However, the linear probability model does not impose the restriction that 0 � P (yi j xi) � 1:Even so estimation of a linear probability model is a useful starting point for subsequent analysis.
The standard alternative is to use a function of the form
P (yi = 1 j xi) = F�x0i�
�where F (�) is a known CDF, typically assumed to be symmetric about zero, so that F (u) =1� F (�u): The two standard choices for F are
� Logistic: F (u) = (1 + e�u)�1 :
137
� Normal: F (u) = �(u):
If F is logistic, we call this the logit model, and if F is normal, we call this the probit model.This model is identical to the latent variable model
y�i = x0i� + ei
ei � F (�)
yi =
�1 if y�i > 00 otherwise
:
For then
P (yi = 1 j xi) = P (y�i > 0 j xi)= P
�x0i� + ei > 0 j xi
�= P
�ei > �x0i� j xi
�= 1� F
��x0i�
�= F
�x0i�
�:
Estimation is by maximum likelihood. To construct the likelihood, we need the conditionaldistribution of an individual observation. Recall that if y is Bernoulli, such that P(y = 1) = p andP(y = 0) = 1� p, then we can write the density of y as
f(y) = py(1� p)1�y; y = 0; 1:
In the Binary choice model, yi is conditionally Bernoulli with P (yi = 1 j xi) = pi = F (x0i�) : Thusthe conditional density is
f (yi j xi) = pyii (1� pi)1�yi
= F�x0i�
�yi (1� F �x0i��)1�yi :Hence the log-likelihood function is
logL(�) =
nXi=1
log f(yi j xi)
=
nXi=1
log�F�x0i�
�yi (1� F �x0i��)1�yi�=
nXi=1
�yi logF
�x0i�
�+ (1� yi) log(1� F
�x0i�
�)�
=Xyi=1
logF�x0i�
�+Xyi=0
log(1� F�x0i�
�):
The MLE � is the value of � which maximizes logL(�): Standard errors and test statistics arecomputed by asymptotic approximations. Details of such calculations are left to more advancedcourses.
12.2 Count Data
If y 2 f0; 1; 2; :::g; a typical approach is to employ Poisson regression. This model speci�es that
P (yi = k j xi) =exp (��i)�ki
k!; k = 0; 1; 2; :::
�i = exp(x0i�):
138
The conditional density is the Poisson with parameter �i: The functional form for �i has beenpicked to ensure that �i > 0.
The log-likelihood function is
logL(�) =nXi=1
log f(yi j xi) =nXi=1
�� exp(x0i�) + yix0i� � log(yi!)
�:
The MLE is the value � which maximizes logL(�):Since
E (yi j xi) = �i = exp(x0i�)
is the conditional mean, this motivates the label Poisson �regression.�Also observe that the model implies that
var (yi j xi) = �i = exp(x0i�);
so the model imposes the restriction that the conditional mean and variance of yi are the same.This may be considered restrictive. A generalization is the negative binomial.
12.3 Censored Data
The idea of �censoring� is that some data above or below a threshold are mis-reported at thethreshold. Thus the model is that there is some latent process y�i with unbounded support, but weobserve only
yi =
�y�i if y�i � 00 if y�i < 0
: (12.1)
(This is written for the case of the threshold being zero, any known value can substitute.) Theobserved data yi therefore come from a mixed continuous/discrete distribution.
Censored models are typically applied when the data set has a meaningful proportion (say 5%or higher) of data at the boundary of the sample support. The censoring process may be explicitin data collection, or it may be a by-product of economic constraints.
An example of a data collection censoring is top-coding of income. In surveys, incomes abovea threshold are typically reported at the threshold.
The �rst censored regression model was developed by Tobin (1958) to explain consumption ofdurable goods. Tobin observed that for many households, the consumption level (purchases) in aparticular period was zero. He proposed the latent variable model
y�i = x0i� + ei
ei � iid N(0; �2)
with the observed variable yi generated by the censoring equation (12.1). This model (now calledthe Tobit) speci�es that the latent (or ideal) value of consumption may be negative (the householdwould prefer to sell than buy). All that is reported is that the household purchased zero units ofthe good.
The naive approach to estimate � is to regress yi on xi. This does not work because regressionestimates E (yi j xi) ; not E (y�i j xi) = x0i�; and the latter is of interest. Thus OLS will be biasedfor the parameter of interest �:
[Note: it is still possible to estimate E (yi j xi) by LS techniques. The Tobit framework postu-lates that this is not inherently interesting, that the parameter of � is de�ned by an alternativestatistical structure.]
139
Consistent estimation will be achieved by the MLE. To construct the likelihood, observe thatthe probability of being censored is
P (yi = 0 j xi) = P (y�i < 0 j xi)= P
�x0i� + ei < 0 j xi
�= P
�ei�< �x
0i�
�j xi�
= �
��x
0i�
�
�:
The conditional distribution function above zero is Gaussian:
P (yi = y j xi) =Z y
0��1�
�z � x0i��
�dz; y > 0:
Therefore, the density function can be written as
f (y j xi) = ���x
0i�
�
�1(y=0) ���1�
�z � x0i��
��1(y>0);
where 1 (�) is the indicator function.Hence the log-likelihood is a mixture of the probit and the normal:
logL(�) =nXi=1
log f(yi j xi)
=Xyi=0
log �
��x
0i�
�
�+Xyi>0
log
���1�
�yi � x0i�
�
��:
The MLE is the value � which maximizes logL(�):
12.4 Sample Selection
The problem of sample selection arises when the sample is a non-random selection of potentialobservations. This occurs when the observed data is systematically di¤erent from the populationof interest. For example, if you ask for volunteers for an experiment, and they wish to extrapolatethe e¤ects of the experiment on a general population, you should worry that the people whovolunteer may be systematically di¤erent from the general population. This has great relevance forthe evaluation of anti-poverty and job-training programs, where the goal is to assess the e¤ect of�training�on the general population, not just on the volunteers.
A simple sample selection model can be written as the latent model
yi = x0i� + e1i
Ti = 1�z0i + e0i > 0
�where 1 (�) is the indicator function. The dependent variable yi is observed if (and only if) Ti = 1:Else it is unobserved.
For example, yi could be a wage, which can be observed only if a person is employed. Theequation for Ti is an equation specifying the probability that the person is employed.
The model is often completed by specifying that the errors are jointly normal�e0ie1i
�� N
�0;
�1 �� �2
��:
140
It is presumed that we observe fxi;zi; Tig for all observations.Under the normality assumption,
e1i = �e0i + vi;
where vi is independent of e0i � N(0; 1): A useful fact about the standard normal distribution isthat
E (e0i j e0i > �x) = �(x) =�(x)
�(x);
and the function �(x) is called the inverse Mills ratio.The naive estimator of � is OLS regression of yi on xi for those observations for which yi is
available. The problem is that this is equivalent to conditioning on the event fTi = 1g: However,
E (e1i j Ti = 1;zi) = E�e1i j fe0i > �z0i g;zi
�= �E
�e0i j fe0i > �z0i g;zi
�+ E
�vi j fe0i > �z0i g;zi
�= ��
�z0i �;
which is non-zero. Thuse1i = ��
�z0i �+ ui;
whereE (ui j Ti = 1;zi) = 0:
Henceyi = x
0i� + ��
�z0i �+ ui (12.2)
is a valid regression equation for the observations for which Ti = 1:Heckman (1979) observed that we could consistently estimate � and � from this equation, if
were known. It is unknown, but also can be consistently estimated by a Probit model for selection.The �Heckit�estimator is thus calculated as follows
� Estimate from a Probit, using regressors zi: The binary dependent variable is Ti:
� Estimate��; �
�from OLS of yi on xi and �(z0i ):
� The OLS standard errors will be incorrect, as this is a two-step estimator. They can becorrected using a more complicated formula. Or, alternatively, by viewing the Probit/OLSestimation equations as a large joint GMM problem.
The Heckit estimator is frequently used to deal with problems of sample selection. However,the estimator is built on the assumption of normality, and the estimator can be quite sensitiveto this assumption. Some modern econometric research is exploring how to relax the normalityassumption.
The estimator can also work quite poorly if � (z0i ) does not have much in-sample variation.This can happen if the Probit equation does not �explain�much about the selection choice. Anotherpotential problem is that if zi = xi; then � (z0i ) can be highly collinear with xi; so the secondstep OLS estimator will not be able to precisely estimate �: Based this observation, it is typicallyrecommended to �nd a valid exclusion restriction: a variable should be in zi which is not in xi: Ifthis is valid, it will ensure that � (z0i ) is not collinear with xi; and hence improve the second stageestimator�s precision.
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Chapter 13
Panel Data
A panel is a set of observations on individuals, collected over time. An observation is the pairfyit;xitg; where the i subscript denotes the individual, and the t subscript denotes time. A panelmay be balanced :
fyit;xitg : t = 1; :::; T ; i = 1; :::; n;
or unbalanced :fyit;xitg : For i = 1; :::; n; t = ti; :::; ti:
13.1 Individual-E¤ects Model
The standard panel data speci�cation is that there is an individual-speci�c e¤ect which enterslinearly in the regression
yit = x0it� + ui + eit:
The typical maintained assumptions are that the individuals i are mutually independent, that uiand eit are independent, that eit is iid across individuals and time, and that eit is uncorrelated withxit:
OLS of yit on xit is called pooled estimation. It is consistent if
E (xitui) = 0 (13.1)
If this condition fails, then OLS is inconsistent. (13.1) fails if the individual-speci�c unobservede¤ect ui is correlated with the observed explanatory variables xit: This is often believed to beplausible if ui is an omitted variable.
If (13.1) is true, however, OLS can be improved upon via a GLS technique. In either event,OLS appears a poor estimation choice.
Condition (13.1) is called the random e¤ects hypothesis. It is a strong assumption, and mostapplied researchers try to avoid its use.
13.2 Fixed E¤ects
This is the most common technique for estimation of non-dynamic linear panel regressions.The motivation is to allow ui to be arbitrary, and have arbitrary correlated with xi: The goal
is to eliminate ui from the estimator, and thus achieve invariance.There are several derivations of the estimator.First, let
dij =
8<:1 if i = j
0 else;
142
and
di =
0B@ di1...din
1CA ;an n� 1 dummy vector with a �1�in the i0th place. Let
u =
0B@ u1...un
1CA :Then note that
ui = d0iu;
andyit = x
0it� + d
0iu+ eit: (13.2)
Observe thatE (eit j xit;di) = 0;
so (13.2) is a valid regression, with di as a regressor along with xi:
OLS on (13.2) yields estimator��; u
�: Conventional inference applies.
Observe that
� This is generally consistent.
� If xit contains an intercept, it will be collinear with di; so the intercept is typically omittedfrom xit:
� Any regressor in xit which is constant over time for all individuals (e.g., their gender) will becollinear with di; so will have to be omitted.
� There are n+ k regression parameters, which is quite large as typically n is very large.
Computationally, you do not want to actually implement conventional OLS estimation, as theparameter space is too large. OLS estimation of � proceeds by the FWL theorem. Stacking theobservations together:
y =X� +Du+ e;
then by the FWL theorem,
� =�X 0 (I � PD)X
��1 �X 0 (I � PD)y
�=
�X�0X���1 �X�0y�
�;
where
y� = y �D(D0D)�1D0y
X� = X �D(D0D)�1D0X:
Since the regression of yit on di is a regression onto individual-speci�c dummies, the predicted valuefrom these regressions is the individual speci�c mean yi; and the residual is the demean value
y�it = yit � yi:
The �xed e¤ects estimator � is OLS of y�it on x�it, the dependent variable and regressors in deviation-
from-mean form.
143
Another derivation of the estimator is to take the equation
yit = x0it� + ui + eit;
and then take individual-speci�c means by taking the average for the i0th individual:
1
Ti
tiXt=ti
yit =1
Ti
tiXt=ti
x0it� + ui +1
Ti
tiXt=ti
eit
oryi = x
0i� + ui + ei:
Subtracting, we �ndy�it = x
�0it� + e
�it;
which is free of the individual-e¤ect ui:
13.3 Dynamic Panel Regression
A dynamic panel regression has a lagged dependent variable
yit = �yit�1 + x0it� + ui + eit: (13.3)
This is a model suitable for studying dynamic behavior of individual agents.Unfortunately, the �xed e¤ects estimator is inconsistent, at least if T is held �nite as n ! 1:
This is because the sample mean of yit�1 is correlated with that of eit:The standard approach to estimate a dynamic panel is to combine �rst-di¤erencing with IV or
GMM. Taking �rst-di¤erences of (13.3) eliminates the individual-speci�c e¤ect:
�yit = ��yit�1 +�x0it� +�eit: (13.4)
However, if eit is iid, then it will be correlated with �yit�1 :
E (�yit�1�eit) = E ((yit�1 � yit�2) (eit � eit�1)) = �E (yit�1eit�1) = ��2e:
So OLS on (13.4) will be inconsistent.But if there are valid instruments, then IV or GMM can be used to estimate the equation.
Typically, we use lags of the dependent variable, two periods back, as yt�2 is uncorrelated with�eit: Thus values of yit�k; k � 2, are valid instruments.
Hence a valid estimator of � and � is to estimate (13.4) by IV using yt�2 as an instrument for�yt�1 (which is just identi�ed). Alternatively, GMM using yt�2 and yt�3 as instruments (which isoveridenti�ed, but loses a time-series observation).
A more sophisticated GMM estimator recognizes that for time-periods later in the sample, thereare more instruments available, so the instrument list should be di¤erent for each equation. This isconveniently organized by the GMM principle, as this enables the moments from the di¤erent time-periods to be stacked together to create a list of all the moment conditions. A simple applicationof GMM yields the parameter estimates and standard errors.
144
Chapter 14
Nonparametrics
14.1 Kernel Density Estimation
Let X be a random variable with continuous distribution F (x) and density f(x) = ddxF (x):
The goal is to estimate f(x) from a random sample (X1; :::; Xng While F (x) can be estimated bythe EDF F (x) = n�1
Pni=1 1 (Xi � x) ; we cannot de�ne d
dx F (x) since F (x) is a step function. Thestandard nonparametric method to estimate f(x) is based on smoothing using a kernel.
While we are typically interested in estimating the entire function f(x); we can simply focuson the problem where x is a speci�c �xed number, and then see how the method generalizes toestimating the entire function.
De�nition 2 K(u) is a second-order kernel function if it is a symmetric zero-mean densityfunction.
Three common choices for kernels include the Gaussian
K(u) =1p2�exp
��u
2
2
�the Epanechnikov
K(u) =
�34
�1� u2
�; juj � 1
0 juj > 1and the Biweight or Quartic
K(u) =
�1516
�1� u2
�2; juj � 1
0 juj > 1
In practice, the choice between these three rarely makes a meaningful di¤erence in the estimates.The kernel functions are used to smooth the data. The amount of smoothing is controlled by
the bandwidth h > 0. LetKh(u) =
1
hK�uh
�:
be the kernel K rescaled by the bandwidth h: The kernel density estimator of f(x) is
f(x) =1
n
nXi=1
Kh (Xi � x) :
This estimator is the average of a set of weights. If a large number of the observations Xi are nearx; then the weights are relatively large and f(x) is larger. Conversely, if only a few Xi are near x;then the weights are small and f(x) is small. The bandwidth h controls the meaning of �near�.
145
Interestingly, f(x) is a valid density. That is, f(x) � 0 for all x; andZ 1
�1f(x)dx =
Z 1
�1
1
n
nXi=1
Kh (Xi � x) dx =1
n
nXi=1
Z 1
�1Kh (Xi � x) dx =
1
n
nXi=1
Z 1
�1K (u) du = 1
where the second-to-last equality makes the change-of-variables u = (Xi � x)=h:We can also calculate the moments of the density f(x): The mean isZ 1
�1xf(x)dx =
1
n
nXi=1
Z 1
�1xKh (Xi � x) dx
=1
n
nXi=1
Z 1
�1(Xi + uh)K (u) du
=1
n
nXi=1
Xi
Z 1
�1K (u) du+
1
n
nXi=1
h
Z 1
�1uK (u) du
=1
n
nXi=1
Xi
the sample mean of the Xi; where the second-to-last equality used the change-of-variables u =(Xi � x)=h which has Jacobian h:
The second moment of the estimated density isZ 1
�1x2f(x)dx =
1
n
nXi=1
Z 1
�1x2Kh (Xi � x) dx
=1
n
nXi=1
Z 1
�1(Xi + uh)
2K (u) du
=1
n
nXi=1
X2i +
2
n
nXi=1
Xih
Z 1
�1K(u)du+
1
n
nXi=1
h2Z 1
�1u2K (u) du
=1
n
nXi=1
X2i + h
2�2K
where
�2K =
Z 1
�1u2K (u) du
is the variance of the kernel. It follows that the variance of the density f(x) isZ 1
�1x2f(x)dx�
�Z 1
�1xf(x)dx
�2=
1
n
nXi=1
X2i + h
2�2K � 1
n
nXi=1
Xi
!2= �2 + h2�2K
Thus the variance of the estimated density is in�ated by the factor h2�2K relative to the samplemoment.
14.2 Asymptotic MSE for Kernel Estimates
For �xed x and bandwidth h observe that
EKh (X � x) =Z 1
�1Kh (z � x) f(z)dz =
Z 1
�1Kh (uh) f(x+ hu)hdu =
Z 1
�1K (u) f(x+ hu)du
146
The second equality uses the change-of variables u = (z�x)=h: The last expression shows that theexpected value is an average of f(z) locally about x:
This integral (typically) is not analytically solvable, so we approximate it using a second orderTaylor expansion of f(x+ hu) in the argument hu about hu = 0; which is valid as h! 0: Thus
f (x+ hu) ' f(x) + f 0(x)hu+ 12f 00(x)h2u2
and therefore
EKh (X � x) 'Z 1
�1K (u)
�f(x) + f 0(x)hu+
1
2f 00(x)h2u2
�du
= f(x)
Z 1
�1K (u) du+ f 0(x)h
Z 1
�1K (u)udu+
1
2f 00(x)h2
Z 1
�1K (u)u2du
= f(x) +1
2f 00(x)h2�2K :
The bias of f(x) is then
Bias(x) = Ef(x)� f(x) = 1
n
nXi=1
EKh (Xi � x)� f(x) =1
2f 00(x)h2�2K :
We see that the bias of f(x) at x depends on the second derivative f 00(x): The sharper the derivative,the greater the bias. Intuitively, the estimator f(x) smooths data local to Xi = x; so is estimatinga smoothed version of f(x): The bias results from this smoothing, and is larger the greater thecurvature in f(x):
We now examine the variance of f(x): Since it is an average of iid random variables, using�rst-order Taylor approximations and the fact that n�1 is of smaller order than (nh)�1
var (x) =1
nvar (Kh (Xi � x))
=1
nEKh (Xi � x)2 �
1
n(EKh (Xi � x))2
' 1
nh2
Z 1
�1K
�z � xh
�2f(z)dz � 1
nf(x)2
=1
nh
Z 1
�1K (u)2 f (x+ hu) du
' f (x)
nh
Z 1
�1K (u)2 du
=f (x)R(K)
nh:
where R(K) =R1�1K (u)
2 du is called the roughness of K:Together, the asymptotic mean-squared error (AMSE) for �xed x is the sum of the approximate
squared bias and approximate variance
AMSEh(x) =1
4f 00(x)2h4�4K +
f (x)R(K)
nh:
A global measure of precision is the asymptotic mean integrated squared error (AMISE)
AMISEh =
ZAMSEh(x)dx =
h4�4KR(f00)
4+R(K)
nh: (14.1)
147
where R(f 00) =R(f 00(x))2 dx is the roughness of f 00: Notice that the �rst term (the squared bias)
is increasing in h and the second term (the variance) is decreasing in nh: Thus for the AMISE todecline with n; we need h ! 0 but nh ! 1: That is, h must tend to zero, but at a slower ratethan n�1:
Equation (14.1) is an asymptotic approximation to the MSE. We de�ne the asymptoticallyoptimal bandwidth h0 as the value which minimizes this approximate MSE. That is,
h0 = argminh
AMISEh
It can be found by solving the �rst order condition
d
dhAMISEh = h
3�4KR(f00)� R(K)
nh2= 0
yielding
h0 =
�R(K)
�4KR(f00)
�1=5n�1=2: (14.2)
This solution takes the form h0 = cn�1=5 where c is a function of K and f; but not of n: We
thus say that the optimal bandwidth is of order O(n�1=5): Note that this h declines to zero, but ata very slow rate.
In practice, how should the bandwidth be selected? This is a di¢ cult problem, and there is alarge and continuing literature on the subject. The asymptotically optimal choice given in (14.2)depends on R(K); �2K ; and R(f
00): The �rst two are determined by the kernel function. Theirvalues for the three functions introduced in the previous section are given here.
K �2K =R1�1 u
2K (u) du R(K) =R1�1K (u)
2 du
Gaussian 1 1/(2p�)
Epanechnikov 1=5 1=5Biweight 1=7 5=7
An obvious di¢ culty is that R(f 00) is unknown. A classic simple solution proposed by Silverman(1986)has come to be known as the reference bandwidth or Silverman�s Rule-of-Thumb. Ituses formula (14.2) but replaces R(f 00) with ��5R(�00); where � is the N(0; 1) distribution and �2 isan estimate of �2 = var(X): This choice for h gives an optimal rule when f(x) is normal, and givesa nearly optimal rule when f(x) is close to normal. The downside is that if the density is very farfrom normal, the rule-of-thumb h can be quite ine¢ cient. We can calculate that R(�00) = 3= (8
p�) :
Together with the above table, we �nd the reference rules for the three kernel functions introducedearlier.
Gaussian Kernel: hrule = 1:06�n�1=5
Epanechnikov Kernel: hrule = 2:34�n�1=5
Biweight (Quartic) Kernel: hrule = 2:78�n�1=5
Unless you delve more deeply into kernel estimation methods the rule-of-thumb bandwidth isa good practical bandwidth choice, perhaps adjusted by visual inspection of the resulting estimatef(x): There are other approaches, but implementation can be delicate. I now discuss some of thesechoices. The plug-in approach is to estimate R(f 00) in a �rst step, and then plug this estimate intothe formula (14.2). This is more treacherous than may �rst appear, as the optimal h for estimationof the roughness R(f 00) is quite di¤erent than the optimal h for estimation of f(x): However, thereare modern versions of this estimator work well, in particular the iterative method of Sheatherand Jones (1991). Another popular choice for selection of h is cross-validation. This works byconstructing an estimate of the MISE using leave-one-out estimators. There are some desirableproperties of cross-validation bandwidths, but they are also known to converge very slowly to the
148
optimal values. They are also quite ill-behaved when the data has some discretization (as is commonin economics), in which case the cross-validation rule can sometimes select very small bandwidthsleading to dramatically undersmoothed estimates. Fortunately there are remedies, which are knownas smoothed cross-validation which is a close cousin of the bootstrap.
149
Appendix A
Matrix Algebra
A.1 Notation
A scalar a is a single number.A vector a is a k � 1 list of numbers, typically arranged in a column. We write this as
a =
1CCCAEquivalently, a vector a is an element of Euclidean k space, written as a 2 Rk: If k = 1 then a isa scalar.
A matrix A is a k � r rectangular array of numbers, written as
A =
26664a11 a12 � � � a1ra21 a22 � � � a2r...
......
ak1 ak2 � � � akr
37775By convention aij refers to the element in the i0th row and j0th column of A: If r = 1 then A is acolumn vector. If k = 1 then A is a row vector. If r = k = 1; then A is a scalar.
A standard convention (which we will follow in this text whenever possible) is to denote scalarsby lower-case italics (a); vectors by lower-case bold italics (a); and matrices by upper-case bolditalics (A): Sometimes a matrix A is denoted by the symbol (aij):
A matrix can be written as a set of column vectors or as a set of row vectors. That is,
A =�a1 a2 � � � ar
�=
26664�1�2...�k
37775where
ai =
26664a1ia2i...aki
37775are column vectors and
�j =�aj1 aj2 � � � ajr
�150
are row vectors.The transpose of a matrix, denoted A0; is obtained by �ipping the matrix on its diagonal.
Thus
A0 =
26664a11 a21 � � � ak1a12 a22 � � � ak2...
......
a1r a2r � � � akr
37775Alternatively, letting B = A0; then bij = aji. Note that if A is k � r, then A0 is r � k: If a is ak � 1 vector, then a0 is a 1� k row vector. An alternative notation for the transpose of A is A>:
A matrix is square if k = r: A square matrix is symmetric if A = A0; which requires aij = aji:A square matrix is diagonal if the o¤-diagonal elements are all zero, so that aij = 0 if i 6= j: Asquare matrix is upper (lower) diagonal if all elements below (above) the diagonal equal zero.
An important diagonal matrix is the identity matrix, which has ones on the diagonal. Thek � k identity matrix is denoted as
Ik =
266641 0 � � � 00 1 � � � 0......
...0 0 � � � 1
37775 :A partitioned matrix takes the form
A =
26664A11 A12 � � � A1rA21 A22 � � � A2r...
......
Ak1 Ak2 � � � Akr
37775where the Aij denote matrices, vectors and/or scalars.
A.2 Matrix Addition
If the matrices A = (aij) and B = (bij) are of the same order, we de�ne the sum
A+B = (aij + bij) :
Matrix addition follows the communtative and associative laws:
A+B = B +A
A+ (B +C) = (A+B) +C:
A.3 Matrix Multiplication
If A is k � r and c is real, we de�ne their product as
Ac = cA = (aijc) :
If a and b are both k � 1; then their inner product is
a0b = a1b1 + a2b2 + � � �+ akbk =kXj=1
ajbj :
Note that a0b = b0a: We say that two vectors a and b are orthogonal if a0b = 0:
151
If A is k � r and B is r � s; so that the number of columns of A equals the number of rowsof B; we say that A and B are conformable. In this event the matrix product AB is de�ned.Writing A as a set of row vectors and B as a set of column vectors (each of length r); then thematrix product is de�ned as
AB =
26664a01a02...a0k
37775 � b1 b2 � � � bs�
=
26664a01b1 a01b2 � � � a01bsa02b1 a02b2 � � � a02bs...
......
a0kb1 a0kb2 � � � a0kbs
37775 :Matrix multiplication is not communicative: in general AB 6= BA: However, it is associative
and distributive:
A (BC) = (AB)C
A (B +C) = AB +AC
An alternative way to write the matrix product is to use matrix partitions. For example,
AB =
�A11 A12A21 A22
� �B11 B12
B21 B22
�
=
�A11B11 +A12B21 A11B12 +A12B22
A21B11 +A22B21 A21B12 +A22B22
�:
As another example,
AB =�A1 A2 � � � Ar
�26664B1
B2...Br
37775= A1B1 +A2B2 + � � �+ArBr
=
rXj=1
AjBj
An important property of the identity matrix is that if A is k�r; then AIr = A and IkA = A:The k � r matrix A, r � k, is called orthogonal if A0A = Ir:
A.4 Trace
The trace of a k � k square matrix A is the sum of its diagonal elements
tr (A) =
kXi=1
aii:
Some straightforward properties for square matrices A and B and real c are
tr (cA) = c tr (A)
tr�A0�= tr (A)
tr (A+B) = tr (A) + tr (B)
tr (Ik) = k:
152
Also, for k � r A and r � k B we have
tr (AB) = tr (BA) :
Indeed,
tr (AB) = tr
26664a01b1 a01b2 � � � a01bka02b1 a02b2 � � � a02bk...
......
a0kb1 a0kb2 � � � a0kbk
37775=
kXi=1
a0ibi
=kXi=1
b0iai
= tr (BA) :
A.5 Rank and Inverse
The rank of the k � r matrix (r � k)
A =�a1 a2 � � � ar
�is the number of linearly independent columns aj ; and is written as rank (A) : We say that A hasfull rank if rank (A) = r:
A square k � k matrix A is said to be nonsingular if it is has full rank, e.g. rank (A) = k:This means that there is no k � 1 c 6= 0 such that Ac = 0:
If a square k � k matrix A is nonsingular then there exists a unique matrix k � k matrix A�1called the inverse of A which satis�es
AA�1 = A�1A = Ik:
For non-singular A and C; some important properties include
AA�1 = A�1A = Ik�A�1
�0=
�A0��1
(AC)�1 = C�1A�1
(A+C)�1 = A�1�A�1 +C�1��1C�1
A�1 � (A+C)�1 = A�1�A�1 +C�1�A�1
Also, if A is an orthogonal matrix, then A�1 = A:Another useful result for non-singular A is
(A+BCD)�1 = A�1 �A�1BC�C +CDA�1BC
��1CDA�1: (A.1)
In particular, for C = �1; B = b and D = b0 for vector b we �nd�A� bb0
��1= A�1 +
�1� b0A�1b
��1A�1bb0A�1: (A.2)
153
The following fact about inverting partitioned matrices is quite useful. If A � BD�1C andD �CA�1B are non-singular, then�
A BC D
��1=
" �A�BD�1C
��1 ��A�BD�1C
��1BD�1
��D �CA�1B
��1CA�1
�D �CA�1B
��1#: (A.3)
Even if a matrix A does not possess an inverse, we can still de�ne the generalized inverseA� as a matrix which satis�es
AA�A = A: (A.4)
The matrix A� is not necessarily unique. TheMoore-Penrose generalized inverse A� satis�es(A.4) plus the following three conditions
A�AA� = A�
AA� is symmetric
A�A is symmetric
For any matrix A; the Moore-Penrose generalized inverse A� exists and is unique.
A.6 Determinant
The determinant is a measure of the volume of a square matrix.While the determinant is widely used, its precise de�nition is rarely needed. However, we present
the de�nition here for completeness. Let A = (aij) be a general k � k matrix . Let � = (j1; :::; jk)denote a permutation of (1; :::; k) : There are k! such permutations. There is a unique count of thenumber of inversions of the indices of such permutations (relative to the natural order (1; :::; k) ;and let "� = +1 if this count is even and "� = �1 if the count is odd. Then the determinant of Ais de�ned as
detA =X�
"�a1j1a2j2 � � � akjk :
For example, if A is 2 � 2; then the two permutations of (1; 2) are (1; 2) and (2; 1) ; for which"(1;2) = 1 and "(2;1) = �1. Thus
detA = "(1;2)a11a22 + "(2;1)a21a12
= a11a22 � a12a21:
Some properties include
� det (A) = det (A0)
� det (cA) = ck detA
� det (AB) = (detA) (detB)
� det�A�1
�= (detA)�1
� det�A BC D
�= (detD) det
�A�BD�1C
�if detD 6= 0
� detA 6= 0 if and only if A is nonsingular.
� If A is triangular (upper or lower), then detA =Qki=1 aii
� If A is orthogonal, then detA = �1
154
A.7 Eigenvalues
The characteristic equation of a square matrix A is
det (A� �Ik) = 0:
The left side is a polynomial of degree k in � so it has exactly k roots, which are not necessarilydistinct and may be real or complex. They are called the latent roots or characteristic roots oreigenvalues of A. If �i is an eigenvalue of A; then A� �iIk is singular so there exists a non-zerovector hi such that
(A� �iIk)hi = 0:
The vector hi is called a latent vector or characteristic vector or eigenvector of A corre-sponding to �i:
We now state some useful properties. Let �i and hi, i = 1; :::; k denote the k eigenvalues andeigenvectors of a square matrix A: Let � be a diagonal matrix with the characteristic roots in thediagonal, and let H = [h1 � � �hk]:
� det(A) =Qki=1 �i
� tr(A) =Pki=1 �i
� A is non-singular if and only if all its characteristic roots are non-zero.
� If A has distinct characteristic roots, there exists a nonsingular matrix P such that A =P�1�P and PAP�1 = �.
� If A is symmetric, then A = H�H 0 and H 0AH = �; and the characteristic roots are allreal. A =H�H 0 is called the spectral decomposition of a matrix.
� The characteristic roots of A�1 are ��11 ; ��12 ; ..., �
�1k :
A.8 Positive De�niteness
We say that a symmetric square matrix A is positive semi-de�nite if for all c 6= 0; c0Ac � 0:This is written as A � 0: We say that A is positive de�nite if for all c 6= 0; c0Ac > 0: This iswritten as A > 0:
Some properties include:
� If A = G0G for some matrix G; then A is positive semi-de�nite. (For any c 6= 0; c0Ac =�0� � 0 where � = Gc:) If G has full rank, then A is positive de�nite.
� If A is positive de�nite, then A is non-singular and A�1 exists. Furthermore, A�1 > 0:
� A > 0 if and only if it is symmetric and all its characteristic roots are positive.
� If A > 0 we can �nd a matrix B such that A = BB0: We call B a matrix square rootof A: The matrix B need not be unique. One way to construct B is to use the spectraldecomposition A =H�H 0 where � is diagonal, and then set B =H�1=2:
A square matrix A is idempotent if AA = A: If A is idempotent and symmetric then all itscharacteristic roots equal either zero or one and is thus positive semi-de�nite. To see this, note thatwe can write A = H�H 0 where H is orthogonal and � contains the (real) characteristic roots.Then
A = AA =H�H 0H�H 0 =H�2H 0:
155
By the uniqueness of the characteristic roots, we deduce that �2 = � and �2i = �i for i = 1; :::; k:Hence they must equal either 0 or 1. It follows that the spectral decomposition of idempotent Atakes the form
A =H
�In�k 00 0
�H 0 (A.5)
with H 0H = In. Additionally, tr(A) = rank(A):
A.9 Matrix Calculus
Let x = (x1; :::; xk) be k � 1 and g(x) = g(x1; :::; xk) : Rk ! R: The vector derivative is
@
@xg (x) =
0B@@@x1g (x)...
@@xkg (x)
1CAand
@
@x0g (x) =
�@@x1g (x) � � � @
@xkg (x)
�:
Some properties are now summarized.
� @@x (a
0x) = @@x (x
0a) = a
� @@x0 (Ax) = A
� @@x (x
0Ax) = (A+A0)x
� @2
@x@x0 (x0Ax) = A+A0
A.10 Kronecker Products and the Vec Operator
Let A = [a1 a2 � � � an] be m� n: The vec of A; denoted by vec (A) ; is the mn� 1 vector
vec (A) =
1CCCA :Let A = (aij) be an m � n matrix and let B be any matrix. The Kronecker product of A
and B; denoted AB; is the matrix
AB =
26664a11B a12B a1nBa21B a22B � � � a2nB...
......
am1B am2B � � � amnB
37775 :Some important properties are now summarized. These results hold for matrices for which allmatrix multiplications are conformable.
� (A+B)C = AC +B C
� (AB) (C D) = AC BD
� A (B C) = (AB) C
156
� (AB)0 = A0 B0
� tr (AB) = tr (A) tr (B)
� If A is m�m and B is n� n; det(AB) = (det (A))n (det (B))m
� (AB)�1 = A�1 B�1
� If A > 0 and B > 0 then AB > 0
� vec (ABC) = (C 0 A) vec (B)
� tr (ABCD) = vec (D0)0 (C 0 A) vec (B)
A.11 Vector and Matrix Norms
The Euclidean norm of an m� 1 vector a is
kak =�a0a�1=2
=
mXi=1
a2i
!1=2:
The Euclidean norm of an m� n matrix A is
kAk = kvec (A)k= tr
�A0A
�1=2=
0@ mXi=1
nXj=1
a2ij
1A1=2 :A useful calculation is for any m� 1 vectors a and b, ab0 = kak kbk
and in particular aa0 = kak2 (A.6)
Some useful inequalities are now given:Schwarz Inequality: For any m� 1 vectors a and b,��a0b�� � kak kbk : (A.7)
Schwarz Matrix Inequality: For any m� n matrices A and B; A0B � kAk kBk : (A.8)
Triangle Inequality: For any m� n matrices A and B;
kA+Bk � kAk+ kBk : (A.9)
Proof of Schwarz Inequality: First, suppose that kbk = 0: Then b = 0 and both ja0bj = 0and kak kbk = 0 so the inequality is true. Second, suppose that kbk > 0 and de�ne c = a �b�b0b��1
b0a: Since c is a vector, c0c � 0: Thus
0 � c0c = a0a��a0b�2=�b0b�:
157
Rearranging, this implies that �a0b�2 � �a0a� �b0b� :
Taking the square root of each side yields the result. �
Proof of Schwarz Matrix Inequality: Partition A = [a1; :::;an] and B = [b1; :::; bn]. Thenby partitioned matrix multiplication, the de�nition of the matrix Euclidean norm and the Schwarzinequality
A0B =
a01b1 a01b2 � � �a02b1 a02b2 � � �...
.... . .
�
ka1k kb1k ka1k kb2k � � �ka2k kb1k ka2k kb2k � � �
......
. . .
=
0@ nXi=1
nXj=1
kaik2 kbjk21A1=2
=
nXi=1
kaik2!1=2 nX
i=1
kbik2!1=2
=
0@ nXi=1
mXj=1
a2ji
1A1=20@ nXi=1
mXj=1
kbjik21A1=2
= kAk kBk
Proof of Triangle Inequality: Let a = vec (A) and b = vec (B) . Then by the de�nition ofthe matrix norm and the Schwarz Inequality
kA+Bk2 = ka+ bk2
= a0a+ 2a0b+ b0b
� a0a+ 2��a0b��+ b0b
� kak2 + 2 kak kbk+ kbk2
= (kak+ kbk)2
= (kAk+ kBk)2
�
158
Appendix B
Probability
B.1 Foundations
The set S of all possible outcomes of an experiment is called the sample space for the exper-iment. Take the simple example of tossing a coin. There are two outcomes, heads and tails, sowe can write S = fH;Tg: If two coins are tossed in sequence, we can write the four outcomes asS = fHH;HT; TH; TTg:
An event A is any collection of possible outcomes of an experiment. An event is a subset of S;including S itself and the null set ;: Continuing the two coin example, one event is A = fHH;HTg;the event that the �rst coin is heads. We say that A and B are disjoint or mutually exclusiveif A \ B = ;: For example, the sets fHH;HTg and fTHg are disjoint. Furthermore, if the setsA1; A2; ::: are pairwise disjoint and [1i=1Ai = S; then the collection A1; A2; ::: is called a partitionof S:
The following are elementary set operations:Union: A [B = fx : x 2 A or x 2 Bg:Intersection: A \B = fx : x 2 A and x 2 Bg:Complement: Ac = fx : x =2 Ag:The following are useful properties of set operations.Communtatitivity: A [B = B [A; A \B = B \A:Associativity: A [ (B [ C) = (A [B) [ C; A \ (B \ C) = (A \B) \ C:Distributive Laws: A\ (B [ C) = (A \B)[ (A \ C) ; A[ (B \ C) = (A [B)\ (A [ C) :DeMorgan�s Laws: (A [B)c = Ac \Bc; (A \B)c = Ac [Bc:A probability function assigns probabilities (numbers between 0 and 1) to events A in S:
This is straightforward when S is countable; when S is uncountable we must be somewhat morecareful: A set B is called a sigma algebra (or Borel �eld) if ; 2 B , A 2 B implies Ac 2 B, andA1; A2; ::: 2 B implies [1i=1Ai 2 B. A simple example is f;; Sg which is known as the trivial sigmaalgebra. For any sample space S; let B be the smallest sigma algebra which contains all of the opensets in S: When S is countable, B is simply the collection of all subsets of S; including ; and S:When S is the real line, then B is the collection of all open and closed intervals. We call B thesigma algebra associated with S: We only de�ne probabilities for events contained in B.
We now can give the axiomatic de�nition of probability. Given S and B, a probability functionP satis�es P(S) = 1; P(A) � 0 for all A 2 B, and if A1; A2; ::: 2 B are pairwise disjoint, thenP ([1i=1Ai) =
P1i=1 P(Ai):
Some important properties of the probability function include the following
� P (;) = 0
� P(A) � 1
� P (Ac) = 1� P(A)
159
� P (B \Ac) = P(B)� P(A \B)
� P (A [B) = P(A) + P(B)� P(A \B)
� If A � B then P(A) � P(B)
� Bonferroni�s Inequality: P(A \B) � P(A) + P(B)� 1
� Boole�s Inequality: P (A [B) � P(A) + P(B)
For some elementary probability models, it is useful to have simple rules to count the numberof objects in a set. These counting rules are facilitated by using the binomial coe¢ cients which arede�ned for nonnegative integers n and r; n � r; as�
n
r
�=
n!
r! (n� r)! :
When counting the number of objects in a set, there are two important distinctions. Countingmay be with replacement or without replacement. Counting may be ordered or unordered.For example, consider a lottery where you pick six numbers from the set 1, 2, ..., 49. This selection iswithout replacement if you are not allowed to select the same number twice, and is with replacementif this is allowed. Counting is ordered or not depending on whether the sequential order of thenumbers is relevant to winning the lottery. Depending on these two distinctions, we have fourexpressions for the number of objects (possible arrangements) of size r from n objects.
Without WithReplacement Replacement
Ordered n!(n�r)! nr
Unordered�nr
� �n+r�1r
�In the lottery example, if counting is unordered and without replacement, the number of po-
tential combinations is�496
�= 13; 983; 816.
If P(B) > 0 the conditional probability of the event A given the event B is
P (A j B) = P (A \B)P(B)
:
For any B; the conditional probability function is a valid probability function where S has beenreplaced by B: Rearranging the de�nition, we can write
P(A \B) = P (A j B) P(B)
which is often quite useful. We can say that the occurrence of B has no information about thelikelihood of event A when P (A j B) = P(A); in which case we �nd
P(A \B) = P (A) P(B) (B.1)
We say that the events A and B are statistically independent when (B.1) holds. Furthermore,we say that the collection of events A1; :::; Ak are mutually independent when for any subsetfAi : i 2 Ig;
P
\i2IAi
!=Yi2IP (Ai) :
Theorem 3 (Bayes�Rule). For any set B and any partition A1; A2; ::: of the sample space, thenfor each i = 1; 2; :::
P (Ai j B) =P (B j Ai) P(Ai)P1j=1 P (B j Aj) P(Aj)
160
B.2 Random Variables
A random variable X is a function from a sample space S into the real line. This induces anew sample space �the real line �and a new probability function on the real line. Typically, wedenote random variables by uppercase letters such as X; and use lower case letters such as x forpotential values and realized values. (This is in contrast to the notation adopted for most of thetextbook.) For a random variable X we de�ne its cumulative distribution function (CDF) as
F (x) = P (X � x) : (B.2)
Sometimes we write this as FX(x) to denote that it is the CDF of X: A function F (x) is a CDF ifand only if the following three properties hold:
1. limx!�1 F (x) = 0 and limx!1 F (x) = 1
2. F (x) is nondecreasing in x
3. F (x) is right-continuous
We say that the random variable X is discrete if F (x) is a step function. In the latter case,the range of X consists of a countable set of real numbers �1; :::; � r: The probability function forX takes the form
P (X = � j) = �j ; j = 1; :::; r (B.3)
where 0 � �j � 1 andPrj=1 �j = 1.
We say that the random variable X is continuous if F (x) is continuous in x: In this case P(X =�) = 0 for all � 2 R so the representation (B.3) is unavailable. Instead, we represent the relativeprobabilities by the probability density function (PDF)
f(x) =d
dxF (x)
so that
F (x) =
Z x
�1f(u)du
and
P (a � X � b) =Z b
af(u)du:
These expressions only make sense if F (x) is di¤erentiable. While there are examples of continuousrandom variables which do not possess a PDF, these cases are unusual and are typically ignored.
A function f(x) is a PDF if and only if f(x) � 0 for all x 2 R andR1�1 f(x)dx:
B.3 Expectation
For any measurable real function g; we de�ne the mean or expectation Eg(X) as follows. IfX is discrete,
Eg(X) =
rXj=1
g(� j)�j ;
and if X is continuous
Eg(X) =
Z 1
�1g(x)f(x)dx:
The latter is well de�ned and �nite ifZ 1
�1jg(x)j f(x)dx <1: (B.4)
161
If (B.4) does not hold, evaluate
I1 =
Zg(x)>0
g(x)f(x)dx
I2 = �Zg(x)<0
g(x)f(x)dx
If I1 = 1 and I2 < 1 then we de�ne Eg(X) = 1: If I1 < 1 and I2 = 1 then we de�neEg(X) = �1: If both I1 =1 and I2 =1 then Eg(X) is unde�ned.
Since E (a+ bX) = a+ bEX; we say that expectation is a linear operator.For m > 0; we de�ne the m0th moment of X as EXm and the m0th central moment as
E (X � EX)m :Two special moments are the mean � = EX and variance �2 = E(X � �)2 = EX2 � �2: We
call � =p�2 the standard deviation of X: We can also write �2 = var(X). For example, this
allows the convenient expression var(a+ bX) = b2 var(X):The moment generating function (MGF) of X is
M(�) = E exp (�X) :
The MGF does not necessarily exist. However, when it does and E jXjm <1 then
dm
d�mM(�)
�����=0
= E(Xm)
which is why it is called the moment generating function.More generally, the characteristic function (CF) of X is
C(�) = E exp (i�X)
where i =p�1 is the imaginary unit. The CF always exists, and when E jXjm <1
dm
d�mC(�)
�����=0
= imE (Xm) :
The Lp norm, p � 1; of the random variable X is
kXkp = (E jXjp)1=p :
B.4 Gamma Function
The gamma function is de�ned for � > 0 as
�(�) =
Z 1
0x��1 exp (�x) :
It satis�es the property�(1 + �) = �(�)�
so for positive integers n;�(n) = (n� 1)!
Special values include� (1) = 1
and
�
�1
2
�= �1=2:
Sterling�s formula is an expansion for the its logarithm
log �(�) =1
2log(2�) +
��� 1
2
�log�� z + 1
12�� 1
360�3+
1
1260�5+ � � �
162
B.5 Common Distributions
For reference, we now list some important discrete distribution function.Bernoulli
P (X = x) = px(1� p)1�x; x = 0; 1; 0 � p � 1EX = p
var(X) = p(1� p)
Binomial
P (X = x) =
�n
x
�px (1� p)n�x ; x = 0; 1; :::; n; 0 � p � 1
EX = np
var(X) = np(1� p)
Geometric
P (X = x) = p(1� p)x�1; x = 1; 2; :::; 0 � p � 1
EX =1
p
var(X) =1� pp2
Multinomial
P (X1 = x1; X2 = x2; :::; Xm = xm) =n!
x1!x2! � � �xm!px11 p
x22 � � � pxmm ;
x1 + � � �+ xm = n;
p1 + � � �+ pm = 1
EXi = pi
var(Xi) = npi(1� pi)cov (Xi; Xj) = �npipj
Negative Binomial
P (X = x) =� (r + x)
x!� (r)pr(1� p)x�1; x = 0; 1; 2; :::; 0 � p � 1
EX =r (1� p)
p
var(X) =r (1� p)p2
Poisson
P (X = x) =exp (��)�x
x!; x = 0; 1; 2; :::; � > 0
EX = �
var(X) = �
We now list some important continuous distributions.
163
Beta
f(x) =�(�+ �)
�(�)�(�)x��1(1� x)��1; 0 � x � 1; � > 0; � > 0
� =�
�+ �
var(X) =��
(�+ � + 1) (�+ �)2
Cauchy
f(x) =1
� (1 + x2); �1 < x <1
EX = 1var(X) = 1
ExponentiaI
f(x) =1
�exp
�x�
�; 0 � x <1; � > 0
EX = �
var(X) = �2
Logistic
f(x) =exp (�x)
(1 + exp (�x))2; �1 < x <1;
EX = 0
var(X) =�2
3
Lognormal
f(x) =1p2��x
exp
�(log x� �)
2
2�2
!; 0 � x <1; � > 0
EX = exp��+ �2=2
�var(X) = exp
�2�+ 2�2
�� exp
�2�+ �2
�Pareto
f(x) =���
x�+1; � � x <1; � > 0; � > 0
EX =��
� � 1 ; � > 1
var(X) =��2
(� � 1)2 (� � 2); � > 2
Uniform
f(x) =1
b� a; a � x � b
EX =a+ b
2
var(X) =(b� a)2
12
164
Weibull
f(x) =
�x �1 exp
��x
�
�; 0 � x <1; > 0; � > 0
EX = �1= �
�1 +
1
�var(X) = �2=
��
�1 +
2
�� �2
�1 +
1
��Gamma
f(x) =1
�(�)��x��1 exp
��x�
�; 0 � x <1; � > 0; � > 0
EX = ��
var(X) = ��2
Chi-Square
f(x) =1
�(r=2)2r=2xr=2�1 exp
��x2
�; 0 � x <1; r > 0
EX = r
var(X) = 2r
Normal
f(x) =1p2��
exp
�(x� �)
2
2�2
!; �1 < x <1; �1 < � <1; �2 > 0
EX = �
var(X) = �2
Student t
f(x) =��r+12
�pr��
�r2
� �1 + x2r
��( r+12 ); �1 < x <1; r > 0
EX = 0 if r > 1
var(X) =r
r � 2 if r > 2
B.6 Multivariate Random Variables
A pair of bivariate random variables (X;Y ) is a function from the sample space into R2: Thejoint CDF of (X;Y ) is
F (x; y) = P (X � x; Y � y) :
If F is continuous, the joint probability density function is
f(x; y) =@2
@x@yF (x; y):
For a Borel measurable set A 2 R2;
P ((X < Y ) 2 A) =Z Z
Af(x; y)dxdy
165
For any measurable function g(x; y);
Eg(X;Y ) =
Z 1
�1
Z 1
�1g(x; y)f(x; y)dxdy:
The marginal distribution of X is
FX(x) = P(X � x)= lim
y!1F (x; y)
=
Z x
�1
Z 1
�1f(x; y)dydx
so the marginal density of X is
fX(x) =d
dxFX(x) =
Z 1
�1f(x; y)dy:
Similarly, the marginal density of Y is
fY (y) =
Z 1
�1f(x; y)dx:
The random variables X and Y are de�ned to be independent if f(x; y) = fX(x)fY (y):Furthermore, X and Y are independent if and only if there exist functions g(x) and h(y) such thatf(x; y) = g(x)h(y):
If X and Y are independent, then
E (g(X)h(Y )) =
Z Zg(x)h(y)f(y; x)dydx
=
Z Zg(x)h(y)fY (y)fX(x)dydx
=
Zg(x)fX(x)dx
Zh(y)fY (y)dy
= Eg (X) Eh (Y ) : (B.5)
if the expectations exist. For example, if X and Y are independent then
E(XY ) = EXEY:
Another implication of (B.5) is that if X and Y are independent and Z = X + Y; then
MZ(�) = E exp (� (X + Y ))
= E (exp (�X) exp (�Y ))
= E exp��0X
�Eexp
��0Y
�= MX(�)MY (�): (B.6)
The covariance between X and Y is
cov(X;Y ) = �XY = E((X � EX) (Y � EY )) = EXY � EXEY:
The correlation between X and Y is
corr (X;Y ) = �XY =�XY�x�Y
:
166
The Cauchy-Schwarz Inequality implies that j�XY j � 1: The correlation is a measure of lineardependence, free of units of measurement.
If X and Y are independent, then �XY = 0 and �XY = 0: The reverse, however, is not true.For example, if EX = 0 and EX3 = 0, then cov(X;X2) = 0:
A useful fact is that
var (X + Y ) = var(X) + var(Y ) + 2 cov(X;Y ):
An implication is that if X and Y are independent, then
var (X + Y ) = var(X) + var(Y );
the variance of the sum is the sum of the variances.A k�1 random vectorX = (X1; :::; Xk)
0 is a function from S to Rk: Let x = (x1; :::; xk)0 denotea vector in Rk: (In this Appendix, we use bold to denote vectors. Bold capitals X are randomvectors and bold lower case x are nonrandom vectors. Again, this is in distinction to the notationused in the bulk of the text) The vector X has the distribution and density functions
F (x) = P(X � x)
f(x) =@k
@x1 � � � @xkF (x):
For a measurable function g : Rk ! Rs; we de�ne the expectation
Eg(X) =
ZRkg(x)f(x)dx
where the symbol dx denotes dx1 � � � dxk: In particular, we have the k � 1 multivariate mean
� = EX
and k � k covariance matrix
� = E�(X � �) (X � �)0
�= EXX 0 � ��0
If the elements of X are mutually independent, then � is a diagonal matrix and
var
kXi=1
Xi
!=
kXi=1
var (Xi)
B.7 Conditional Distributions and Expectation
The conditional density of Y given X = x is de�ned as
fY jX (y j x) =f(x; y)
fX(x)
167
if fX(x) > 0: One way to derive this expression from the de�nition of conditional probability is
fY jX (y j x) =@
@ylim"!0
P (Y � y j x �X � x+ ")
=@
@ylim"!0
P (fY � yg \ fx �X � x+ "g)P(x � X � x+ ")
=@
@ylim"!0
F (x+ "; y)� F (x; y)FX(x+ ")� FX(x)
=@
@ylim"!0
@@xF (x+ "; y)
fX(x+ ")
=
@2
@x@yF (x; y)
fX(x)
=f(x; y)
fX(x):
The conditional mean or conditional expectation is the function
m(x) = E (Y jX = x) =
Z 1
�1yfY jX (y j x) dy:
The conditional mean m(x) is a function, meaning that when X equals x; then the expected valueof Y is m(x):
Similarly, we de�ne the conditional variance of Y given X = x as
�2(x) = var (Y jX = x)
= E�(Y �m(x))2 jX = x
�= E
�Y 2 j X = x
��m(x)2:
Evaluated at x =X; the conditional mean m(X) and conditional variance �2(X) are randomvariables, functions of X: We write this as E(Y j X) = m(X) and var (Y jX) = �2(X): Forexample, if E (Y jX = x) = �+ �0x; then E (Y jX) = �+ �0X; a transformation of X:
The following are important facts about conditional expectations.Simple Law of Iterated Expectations:
E (E (Y jX)) = E (Y ) (B.7)
Proof :
E (E (Y jX)) = E (m(X))
=
Z 1
�1m(x)fX(x)dx
=
Z 1
�1
Z 1
�1yfY jX (y j x) fX(x)dydx
=
Z 1
�1
Z 1
�1yf (y;x) dydx
= E(Y ):
Law of Iterated Expectations:
E (E (Y jX;Z) jX) = E (Y jX) (B.8)
168
Conditioning Theorem. For any function g(x);
E (g(X)Y jX) = g (X) E (Y jX) (B.9)
Proof : Let
h(x) = E (g(X)Y jX = x)
=
Z 1
�1g(x)yfY jX (y j x) dy
= g(x)
Z 1
�1yfY jX (y j x) dy
= g(x)m(x)
where m(x) = E (Y jX = x) : Thus h(X) = g(X)m(X), which is the same as E (g(X)Y jX) =g (X) E (Y jX) :
B.8 Transformations
Suppose that X 2 Rk with continuous distribution function FX(x) and density fX(x): LetY = g(X) where g(x) : Rk ! Rk is one-to-one, di¤erentiable, and invertible. Let h(y) denote theinverse of g(x). The Jacobian is
J(y) = det
�@
@y0h(y)
�:
Consider the univariate case k = 1: If g(x) is an increasing function, then g(X) � Y if and onlyif X � h(Y ); so the distribution function of Y is
FY (y) = P (g(X) � y)= P (X � h(Y ))= FX (h(Y )) :
Taking the derivative, the density of Y is
fY (y) =d
dyFY (y) = fX (h(Y ))
d
dyh(y):
If g(x) is a decreasing function, then g(X) � Y if and only if X � h(Y ); so
FY (y) = P (g(X) � y)= 1� P (X � h(Y ))= 1� FX (h(Y ))
and the density of Y is
fY (y) = �fX (h(Y ))d
dyh(y):
We can write these two cases jointly as
fY (y) = fX (h(Y )) jJ(y)j : (B.10)
This is known as the change-of-variables formula. This same formula (B.10) holds for k > 1; butits justi�cation requires deeper results from analysis.
As one example, take the case X � U [0; 1] and Y = � log(X). Here, g(x) = � log(x) andh(y) = exp(�y) so the Jacobian is J(y) = � exp(y): As the range of X is [0; 1]; that for Y is [0,1):Since fX (x) = 1 for 0 � x � 1 (B.10) shows that
fY (y) = exp(�y); 0 � y � 1;
an exponential density.
169
B.9 Normal and Related Distributions
The standard normal density is
�(x) =1p2�exp
��x
2
2
�; �1 < x <1:
It is conventional to write X � N(0; 1) ; and to denote the standard normal density function by�(x) and its distribution function by �(x): The latter has no closed-form solution. The normaldensity has all moments �nite. Since it is symmetric about zero all odd moments are zero. Byiterated integration by parts, we can also show that EX2 = 1 and EX4 = 3: In fact, for any positiveinteger m, EX2m = (2m� 1)!! = (2m� 1) � (2m� 3) � � � 1: Thus EX4 = 3; EX6 = 15; EX8 = 105;and EX10 = 945:
If Z is standard normal and X = � + �Z; then using the change-of-variables formula, X hasdensity
f(x) =1p2��
exp
�(x� �)
2
2�2
!; �1 < x <1:
which is the univariate normal density. The mean and variance of the distribution are � and�2; and it is conventional to write X � N
��; �2
�.
For x 2 Rk; the multivariate normal density is
f(x) =1
(2�)k=2 det (�)1=2exp
��(x� �)
0��1 (x� �)2
�; x 2 Rk:
The mean and covariance matrix of the distribution are � and �; and it is conventional to writeX � N(�;�).
The MGF and CF of the multivariate normal are exp��0�+ �0��=2
�and exp
�i�0�� �0��=2
�;
respectively.If X 2 Rk is multivariate normal and the elements of X are mutually uncorrelated, then
� = diagf�2jg is a diagonal matrix. In this case the density function can be written as
f(x) =1
(2�)k=2 �1 � � ��kexp
� (x1 � �1)2 =�21 + � � �+ (xk � �k)
2 =�2k2
!!
=
kYj=1
1
(2�)1=2 �jexp
��xj � �j
�22�2j
!
which is the product of marginal univariate normal densities. This shows that if X is multivariatenormal with uncorrelated elements, then they are mutually independent.
Theorem B.9.1 If X � N(�;�) and Y = a + BX with B an invertible matrix, then Y �N(a+B�;B�B0) :
Theorem B.9.2 Let X � N(0; Ir) : Then Q = X 0X is distributed chi-square with r degrees offreedom, written �2r.
Theorem B.9.3 If Z � N(0;A) with A > 0; q � q; then Z 0A�1Z � �2q :
Theorem B.9.4 Let Z � N(0; 1) and Q � �2r be independent. Then Tr = Z=pQ=r is distributed
as student�s t with r degrees of freedom.
170
Proof of Theorem B.9.1. By the change-of-variables formula, the density of Y = a+BX is
f(y) =1
(2�)k=2 det (�Y )1=2exp
�(y � �Y )0��1Y (y � �Y )
2
!; y 2 Rk:
where �Y = a+B� and�Y = B�B0; where we used the fact that det (B�B0)1=2 = det (�)1=2 det (B) :
�
Proof of Theorem B.9.2. First, suppose a random variable Q is distributed chi-square with rdegrees of freedom. It has the MGF
Eexp (tQ) =
Z 1
0
1
��r2
�2r=2
xr=2�1 exp (tx) exp (�x=2) dy = (1� 2t)�r=2
where the second equality uses the fact thatR10 ya�1 exp (�by) dy = b�a�(a); which can be found
by applying change-of-variables to the gamma function. Our goal is to calculate the MGF ofQ =X 0X and show that it equals (1� 2t)�r=2 ; which will establish that Q � �2r .
Note that we can write Q = X 0X =Prj=1 Z
2j where the Zj are independent N(0; 1) : The
distribution of each of the Z2j is
P�Z2j � y
�= 2P (0 � Zj �
py)
= 2
Z py
0
1p2�exp
��x
2
2
�dx
=
Z y
0
1
��12
�21=2
s�1=2 exp��s2
�ds
using the change�of-variables s = x2 and the fact ��12
�=p�: Thus the density of Z2j is
f1(x) =1
��12
�21=2
x�1=2 exp��x2
�which is the �21 and by our above calculation has the MGF of Eexp
�tZ2j
�= (1� 2t)�1=2 :
Since the Z2j are mutually independent, (B.6) implies that the MGF of Q =Prj=1 Z
2j ish
(1� 2t)�1=2ir= (1� 2t)�r=2 ; which is the MGF of the �2r density as desired: �
Proof of Theorem B.9.3. The fact that A > 0 means that we can write A = CC 0 where C isnon-singular. Then A�1 = C�10C�1 and
C�1Z � N�0;C�1AC�10� = N �0;C�1CC 0C�10� = N(0; Iq) :
ThusZ 0A�1Z = Z 0C�10C�1Z =
�C�1Z
�0 �C�1Z
�� �2q :
�
Proof of Theorem B.9.4. Using the simple law of iterated expectations, Tr has distribution
171
function
F (x) = P
ZpQ=r
� x!
= E
(Z � x
rQ
r
)
= E
"P
Z � x
rQ
rj Q!#
= E�
x
rQ
r
!
Thus its density is
f (x) = Ed
dx�
x
rQ
r
!
= E
�
x
rQ
r
!rQ
r
!
=
Z 1
0
�1p2�exp
��qx
2
2r
��rq
r
1
��r2
�2r=2
qr=2�1 exp (�q=2)!dq
=��r+12
�pr��
�r2
� �1 + x2r
��( r+12 )which is that of the student t with r degrees of freedom. �
172
Appendix C
Asymptotic Theory
C.1 Inequalities
The following inequalities are frequently used in asymptotic distribution theory.
Jensen�s Inequality. If g(�) : R ! R is convex, then for any random variable x for whichE jxj <1 and E jg (x)j <1;
g(E(x)) � E (g (x)) : (C.1)
Expectation Inequality. For any random variable x for which E jxj <1;
jE(x)j � E jxj : (C.2)
Cauchy-Schwarz Inequality. For any random m� n matrices X and Y;
E X 0Y
� �E kXk2�1=2 �E kY k2�1=2 : (C.3)
Holder�s Inequality. If p > 1 and q > 1 and 1p +
1q = 1; then for any random m� n matrices X
and Y;E X 0Y
� (E kXkp)1=p (E kY kq)1=q : (C.4)
Minkowski�s Inequality. For any random m� n matrices X and Y;
(E kX + Y kp)1=p � (E kXkp)1=p + (E kY kp)1=p (C.5)
Markov�s Inequality. For any random vector x and non-negative function g(x) � 0;
P(g(x) > �) � ��1Eg(x): (C.6)
Proof of Jensen�s Inequality. Let a + bu be the tangent line to g(u) at u = Ex: Since g(u) isconvex, tangent lines lie below it. So for all u; g(u) � a+ bu yet g(Ex) = a+ bEx since the curveis tangent at Ex: Applying expectations, Eg(x) � a+ bEx = g(Ex); as stated. �
Proof of Expecation Inequality. Follows from an application of Jensen�s Inequality, noting thatthe function g(u) = juj is convex. �
Proof of Holder�s Inequality. Since 1p +
1q = 1 an application of Jensen�s Inequality shows that
for any real a and b
exp
�1
pa+
1
qb
�� 1
pexp (a) +
1
qexp (b) :
173
Setting u = exp (a) and v = exp (b) this implies
u1=pv1=q � u
p+v
q
and this inequality holds for any u > 0 and v > 0:Set u = kXkp =E kXkp and v = kY kq =E kY kq : Note that Eu = Ev = 1: By the matrix Schwarz
Inequality (A.8), kX 0Y k � kXk kY k. Thus
E kX 0Y k(E kXkp)1=p (E kY kq)1=q
� E (kXk kY k)(E kXkp)1=p (E kY kq)1=q
= E�u1=pv1=q
�� E
�u
p+v
q
�=
1
p+1
q= 1;
which is (C.4). �
Proof of Minkowski�s Inequality. Note that by rewriting, using the triangle inequality (A.9),and then Holder�s Inequality to the two expectations
E kX + Y kp = E�kX + Y k kX + Y kp�1
�� E
�kXk kX + Y kp�1
�+ E
�kY k kX + Y kp�1
�� (E kXkp)1=p E
�kX + Y kq(p�1)
�1=q+ (E kY kp)1=p E
�kX + Y kq(p�1)
�1=q=
�(E kXkp)1=p + (E kY kp)1=p
�E (kX + Y kp)(p�1)=p
where the second equality picks q to satisfy 1=p+1=q = 1; and the �nal equality uses this fact to makethe substitution q = p=(p�1) and then collects terms. Dividing both sides by E (kX + Y kp)(p�1)=p ;we obtain (C.5). �
Proof of Markov�s Inequality. Let f denote the density function of x: Then
P (g(x) � �) =
Zfug(u)��g
f(u)du
�Zfug(u)��g
g(u)
�f(u)du
� ��1Z 1
�1g(u)f(u)du
= ��1E (g(x))
the �rst inequality using the region of integration fg(u) > �g: �
C.2 Weak Law of Large Numbers
Let zn 2 R be a random variable. We say that zn converges in probability to z as n!1;denoted zn
p�! z as n!1; if for all � > 0;
limn!1
P (jzn � zj > �) = 0:
174
This is a probabilistic way of generalizing the mathematical de�nition of a limit. The limit z maybe a constant or may be random.
If Zn 2 Rk�r is a matrix, we say that Znp�! Z as n!1 if each element of Zn converges in
probability to the corresponding element of Z:The WLLN shows that sample averages converge in probability to the population average.
Theorem C.2.1 Weak Law of Large Numbers (WLLN). If xi 2 R is iid and E jxij < 1;then as n!1
xn =1
n
nXi=1
xip�! E(xi):
Proof: Without loss of generality, we can set E(xi) = 0 by recentering xi on its expectation.We need to show that for all � > 0 and � > 0 there is some N < 1 so that for all n � N;
P (jxnj > �) � �: Fix � and �: Set " = ��=3: Pick C <1 large enough so that
E (jxij 1 (jxij > C)) � " (C.7)
(where 1 (�) is the indicator function) which is possible since E jxij <1: De�ne the random vectors
wi = xi1 (jxij � C)� E (xi1 (jxij � C))zi = xi1 (jxij > C)� E (xi1 (jxij > C)) :
By the Triangle Inequality (A.9), the Expectation Inequality (C.2), and (C.7),
E jznj = E
����� 1nnXi=1
zi
������ 1
n
nXi=1
E jzij
= E jzij� E jxij 1 (jxij > C) + jE (xi1 (jxij > C))j� 2E jxij 1 (jxij > C)� 2": (C.8)
By Jensen�s Inequality (C.1), the fact that the wi are iid and mean zero, and the bound jwij � 2C;
(E jwnj)2 � Ew2n
=Ew2in
� 4C2
n� "2 (C.9)
the �nal inequality holding for n � 4C2="2 = 36C2=�2�2.Finally, by Markov�s Inequality (C.6), the fact that xn = wn+ zn; the triangle inequality, (C.8)
and (C.9),
P (jxnj > �) �E jxnj�
� E jwnj+ E jznj�
� 3"
�= �;
the equality by the de�nition of ": We have shown that for any � > 0 and � > 0 then for alln � 36C2=�2�2; P (jxnj > �) � �; as needed. �
175
C.3 Convergence in Distribution
Let zn be a random vector with distribution Fn(u) = P (zn � u) : We say that zn convergesin distribution to z as n ! 1, denoted zn
d�! z; where z has distribution F (u) = P (z � u) ;if for all u at which F (u) is continuous, Fn(u)! F (u) as n!1:
Theorem C.3.1 Central Limit Theorem (CLT). If xi 2 Rk is iid and Ex2ji < 1 for j =1; :::; k; then as n!1
pn (xn � �) =
1pn
nXi=1
(xi � �)d�! N(0;) :
where � = Exi and = E(xi � �) (xi � �)0 :
Proof: The moment bound Ex2ji <1 is su¢ cient to guarantee that the elements of � and V arewell de�ned and �nite. Without loss of generality, it is su¢ cient to consider the case � = 0 and = Ik:
For � 2 Rk; let C (�) = E exp�i�0xi
�denote the characteristic function of xi and set c (�) =
logC(�). Then observe
@
@�C(�) = iE
�xi exp
�i�0xi
��@2
@�@�0C(�) = i2E
�xix
0i exp
�i�0xi
��so when evaluated at � = 0
C(0) = 1
@
@�C(0) = iE (xi) = 0
@2
@�@�0C(0) = �E
�xix
0i
�= �Ik:
Furthermore,
c�(�) =@
@�c(�) = C(�)�1
@
@�C(�)
c��(�) =@2
@�@�0c(�) = C(�)�1
@2
@�@�0C(�)� C(�)�2 @
@�C (�)
@
@�0C(�)
so when evaluated at � = 0
c(0) = 0
c�(0) = 0
c��(0) = �Ik:
By a second-order Taylor series expansion of c(�) about � = 0;
c(�) = c(0) + c�(0)0�+
1
2�0c��(�
�)� =1
2�0c��(�
�)� (C.10)
where �� lies on the line segment joining 0 and �:
176
We now compute Cn(�) = E exp�i�0pnxn
�the characteristic function of
pnxn: By the prop-
erties of the exponential function, the independence of the xi; the de�nition of c(�) and (C.10)
logCn(�) = log E exp
i1pn
nXi=1
�0xi
!
= log EnYi=1
exp
�i1pn�0xi
�
= lognYi=1
Eexp
�i1pn�0xi
�= nc
��pn
�=
1
2�0c��(�n)�
where �n ! 0 lies on the line segment joining 0 and �=pn: Since c��(�n) ! c��(0) = �Ik; we
see that as n!1;Cn(�)! exp
��12�0�
�the characteristic function of the N(0; Ik) distribution. This is su¢ cient to establish the theorem.�
C.4 Asymptotic Transformations
Theorem C.4.1 Continuous Mapping Theorem 1 (CMT). If znp�! c as n ! 1 and g (�)
is continuous at c; then g(zn)p�! g(c) as n!1.
Proof: Since g is continuous at c; for all " > 0 we can �nd a � > 0 such that if kzn � ck < �then jg (zn)� g (c)j � ": Recall that A � B implies P(A) � P(B): Thus P (jg (zn)� g (c)j � ") �P (kzn � ck < �)! 1 as n!1 by the assumption that zn
p�! c: Hence g(zn)p�! g(c) as n!1.
Theorem C.4.2 Continuous Mapping Theorem 2. If znd�! z as n ! 1 and g (�) is
continuous; then g(zn)d�! g(z) as n!1.
Theorem C.4.3 Delta Method: Ifpn (�n � �0)
d�! N(0;�) ; where � is m�1 and � is m�m;and g(�) : Rm ! Rk; k � m; then
pn (g (�n)� g(�0))
d�! N�0; g��g
0�
�where g�(�) = @
@�0g(�) and g� = g�(�0):
Proof : By a vector Taylor series expansion, for each element of g;
gj(�n) = gj(�0) + gj�(��jn) (�n � �0)
where ��nj lies on the line segment between �n and �0 and therefore converges in probability to �0:
It follows that ajn = gj�(��jn)� gj�p�! 0: Stacking across elements of g; we �nd
pn (g (�n)� g(�0)) = (g� + an)
pn (�n � �0)
d�! g� N(0;�) = N�0; g��g
0�
�:
177
Appendix D
Maximum Likelihood
If the distribution of yi is F (y;�) where F is a known distribution function and � 2 � is anunknown m� 1 vector, we say that the distribution is parametric and that � is the parameterof the distribution F: The space � is the set of permissible value for �: In this setting the methodof maximum likelihood is the appropriate technique for estimation and inference on �:
If the distribution F is continuous then the density of yi can be written as f(y;�) and the jointdensity of a random sample (y1; :::;yn) is
fn (y1; :::;yn;�) =nYi=1
f (yi;�) :
The likelihood of the sample is this joint density evaluated at the observed sample values, viewedas a function of �. The log-likelihood function is its natural log
logL(�) =nXi=1
log f (yi;�) :
If the distribution F is discrete, the likelihood and log-likelihood are constructed by setting f (y;�) =P (yi = y;�) :
De�ne the Hessian
H = �E @2
@�@�0log f (yi;�0) (D.1)
and the outer product matrix
= E
�@
@�log f (yi;�0)
@
@�log f (yi;�0)
0�: (D.2)
Two important features of the likelihood are
Theorem D.0.4@
@�E log f (yi;�)
�����=�0
= 0 (D.3)
H = � I0 (D.4)
The matrix I0 is called the information, and the equality (D.4) is often called the informationmatrix equality.
Theorem D.0.5 Cramer-Rao Lower Bound. If ~� is an unbiased estimator of � 2 R; thenvar(~�) � (nI0)�1 :
178
The Cramer-Rao Theorem gives a lower bound for estimation. However, the restriction tounbiased estimators means that the theorem has little direct relevance for �nite sample e¢ ciency.
Themaximum likelihood estimator orMLE � is the parameter value which maximizes thelikelihood (equivalently, which maximizes the log-likelihood). We can write this as
� = argmax�2�
logL(�):
In some simple cases, we can �nd an explicit expression for � as a function of the data, but thesecases are rare. More typically, the MLE � must be found by numerical methods.
Why do we believe that the MLE � is estimating the parameter �? Observe that when stan-dardized, the log-likelihood is a sample average
1
nlogL(�) =
1
n
nXi=1
log f (yi;�)p�! E log f (yi;�) :
As the MLE � maximizes the left-hand-side, we can see that it is an estimator of the maximizer ofthe right-hand-side. The �rst-order condition for the latter problem is
0 =@
@�E log f (yi;�)
which holds at � = �0 by (D.3). In fact, under conventional regularity conditions, � is consistentfor this value, �
p�! �0 as n!1:
Theorem D.0.6 Under regularity conditions,pn�� � �0
�d�! N
�0; I�10
�.
Thus in large samples, the approximate variance of the MLE is (nI0)�1 which is the Cramer-
Rao lower bound. Thus in large samples the MLE has approximately the best possible variance.Therefore the MLE is called asymptotically e¢ cient.
Typically, to estimate the asymptotic variance of the MLE we use an estimate based on theHessian formula (D.1) bH = � 1
n
nXi=1
@2
@�@�0log f
�yi; �
�(D.5)
We then set bI�10 = bH�1: Asymptotic standard errors for � are then the square roots of the diagonal
elements of n�1bI�10 :Sometimes a parametric density function f(y;�) is used to approximate the true unknown
density f(y); but it is not literally believed that the model f(y;�) is necessarily the true density.In this case, we refer to logL(�) as a quasi-likelihood and the its maximizer � as a quasi-mleor QMLE.
In this case there is not a �true�value of the parameter �: Instead we de�ne the pseudo-truevalue �0 as the maximizer of
E log f (yi;�) =
Zf (y) log f (y;�) dy
which is the same as the minimizer of
KLIC =
Zf (y) log
�f(y)
f (y;�)
�dy
the Kullback-Leibler information distance between the true density f(y) and the parametric densityf(y;�): Thus the QMLE �0 is the value which makes the parametric density �closest�to the truevalue according to this measure of distance. The QMLE is consistent for the pseudo-true value, but
179
has a di¤erent covariance matrix than in the pure MLE case, since the information matrix equality(D.4) does not hold. A minor adjustment to Theorem (D.0.6) yields the asymptotic distribution ofthe QMLE: p
n�� � �0
�d�! N(0;V ) ; V =H�1H�1
The moment estimator for V isV = bH�1
bH�1
where bH is given in (D.5) and
=1
n
nXi=1
@
@�log f
�yi; �
� @
@�log f
�yi; �
�0:
Asymptotic standard errors (sometimes called qmle standard errors) are then the square roots ofthe diagonal elements of n�1V :
Proof of Theorem D.0.4. To see (D.3);
@
@�E log f (yi;�)
�����=�0
=@
@�
Zlog f (y;�) f (y;�0) dy
�����=�0
=
Z@
@�f (y;�)
f (y;�0)
f (y;�)dy
�����=�0
=@
@�
Zf (y;�) dy
�����=�0
=@
@�1
�����=�0
= 0:
Similarly, we can show that
E
@2
@�@�0f (yi;�0)
f (yi;�0)
!= 0:
By direction computation,
@2
@�@�0log f (yi;�0) =
@2
@�@�0f (yi;�0)
f (yi;�0)�
@@�f (yi;�0)
@@�f (yi;�0)
0
f (yi;�0)2
=@2
@�@�0f (yi;�0)
f (yi;�0)� @
@�log f (yi;�0)
@
@�log f (yi;�0)
0 :
Taking expectations yields (D.4). �
Proof of Theorem D.0.5. Let Y = (y1; :::;yn) be the sample, and set
S =@
@�log fn (Y ; �0) =
nXi=1
@
@�log f (yi; �0)
which by Theorem (D.0.4) has mean zero and variance nH: Write the estimator ~� = ~� (Y ) as afunction of the data. Since ~� is unbiased for any �;
� = E~� =
Z~� (Y ) f (Y ; �) dY :
Di¤erentiating with respect to � and evaluating at �0 yields
1 =
Z~� (Y )
@
@�f (Y ; �) dY =
Z~� (Y )
@
@�log f (Y ; �) f (Y ; �0) dY = E
�~�S�:
180
By the Cauchy-Schwarz inequality
1 =���E�~�S����2 � var (S) var�~��
sovar�~��� 1
var (S)=
1
nH :
�
Proof of Theorem D.0.6 Taking the �rst-order condition for maximization of logL(�), andmaking a �rst-order Taylor series expansion,
0 =@
@�logL(�)
�����=�
=nXi=1
@
@�log f
�yi; �
�'
nXi=1
@
@�log f (yi;�0) +
nXi=1
@2
@�@�0log f (yi;�n)
�� � �0
�;
where �n lies on a line segment joining � and �0: (Technically, the speci�c value of �n varies byrow in this expansion.) Rewriting this equation, we �nd
�� � �0
�=
�
nXi=1
@2
@�@�0log f (yi;�n)
!�1 nXi=1
@
@�log f (yi;�0)
!:
Since @@� log f (yi;�0) is mean-zero with covariance matrix ; an application of the CLT yields
1pn
nXi=1
@
@�log f (yi;�0)
d�! N(0;) :
The analysis of the sample Hessian is somewhat more complicated due to the presence of �n: LetH(�) = � @2
@�@�0log f (yi;�) : If it is continuous in �; then since �n
p�! �0 we �nd H(�n)p�! H
and so
� 1n
nXi=1
@2
@�@�0log f (yi;�n) =
1
n
nXi=1
�� @2
@�@�0log f (yi;�n)�H(�n)
�+H(�n)
p�!H
by an application of a uniform WLLN. Together,
pn�� � �0
�d�!H�1N(0;) = N
�0;H�1H�1� = N �0;H�1� ;
the �nal equality using Theorem D.0.4 . �
181
Appendix E
Numerical Optimization
Many econometric estimators are de�ned by an optimization problem of the form
� = argmin�2�
Q(�) (E.1)
where the parameter is � 2 � � Rm and the criterion function is Q(�) : � ! R: For exampleNLLS, GLS, MLE and GMM estimators take this form. In most cases, Q(�) can be computedfor given �; but � is not available in closed form. In this case, numerical methods are required toobtain �:
E.1 Grid Search
Many optimization problems are either one dimensional (m = 1) or involve one-dimensionaloptimization as a sub-problem (for example, a line search). In this context grid search may beemployed.
Grid Search. Let � = [a; b] be an interval. Pick some " > 0 and set G = (b � a)=" to bethe number of gridpoints. Construct an equally spaced grid on the region [a; b] with G gridpoints,which is {�(j) = a + j(b � a)=G : j = 0; :::; Gg. At each point evaluate the criterion functionand �nd the gridpoint which yields the smallest value of the criterion, which is �(|) where | =argmin0�j�GQ(�(j)): This value � (|) is the gridpoint estimate of �: If the grid is su¢ ciently �ne to
capture small oscillations in Q(�); the approximation error is bounded by "; that is,����(|)� ���� � ":
Plots of Q(�(j)) against �(j) can help diagnose errors in grid selection. This method is quite robustbut potentially costly:
Two-Step Grid Search. The gridsearch method can be re�ned by a two-step execution. Foran error bound of " pick G so that G2 = (b � a)=" For the �rst step de�ne an equally spacedgrid on the region [a; b] with G gridpoints, which is {�(j) = a + j(b � a)=G : j = 0; :::; Gg:At each point evaluate the criterion function and let | = argmin0�j�GQ(�(j)). For the secondstep de�ne an equally spaced grid on [�(|� 1);�(|+ 1)] with G gridpoints, which is {�0(k) =�(| � 1) + 2k(b � a)=G2 : k = 0; :::; Gg: Let k = argmin0�k�GQ(�
0(k)): The estimate of � is
��k�. The advantage of the two-step method over a one-step grid search is that the number of
function evaluations has been reduced from (b�a)=" to 2p(b� a)=" which can be substantial. The
disadvantage is that if the function Q(�) is irregular, the �rst-step grid may not bracket � whichthus would be missed.
E.2 Gradient Methods
Gradient Methods are iterative methods which produce a sequence �i : i = 1; 2; ::: whichare designed to converge to �: All require the choice of a starting value �1; and all require the
182
computation of the gradient of Q(�)
g(�) =@
@�Q(�)
and some require the Hessian
H(�) = @2
@�@�0Q(�):
If the functions g(�) andH(�) are not analytically available, they can be calculated numerically.Take the j0th element of g(�): Let �j be the j0th unit vector (zeros everywhere except for a one inthe j0th row). Then for " small
gj(�) 'Q(� + �j")�Q(�)
":
Similarly,
gjk(�) 'Q(� + �j"+ �k")�Q(� + �k")�Q(� + �j") +Q(�)
"2
In many cases, numerical derivatives can work well but can be computationally costly relative toanalytic derivatives. In some cases, however, numerical derivatives can be quite unstable.
Most gradient methods are a variant of Newton�s method which is based on a quadraticapproximation. By a Taylor�s expansion for � close to �
0 = g(�) ' g(�) +H(�)�� � �
�which implies
� = � �H(�)�1g(�):
This suggests the iteration rule�i+1 = �i �H(�i)�1g(�i):
whereOne problem with Newton�s method is that it will send the iterations in the wrong direction if
H(�i) is not positive de�nite. One modi�cation to prevent this possibility is quadratic hill-climbingwhich sets
�i+1 = �i � (H(�i) + �iIm)�1 g(�i):
where �i is set just above the smallest eigenvalue of H(�i) if H(�) is not positive de�nite.Another productive modi�cation is to add a scalar steplength �i: In this case the iteration
rule takes the form�i+1 = �i �Digi�i (E.2)
where gi = g(�i) and Di = H(�i)�1 for Newton�s method and Di = (H(�i) + �iIm)�1 forquadratic hill-climbing.
Allowing the steplength to be a free parameter allows for a line search, a one-dimensionaloptimization. To pick �i write the criterion function as a function of �
Q(�) = Q(�i +Digi�)
a one-dimensional optimization problem. There are two common methods to perform a line search.A quadratic approximation evaluates the �rst and second derivatives of Q(�) with respect to�; and picks �i as the value minimizing this approximation. The half-step method considers thesequence � = 1; 1/2, 1/4, 1/8, ... . Each value in the sequence is considered and the criterionQ(�i +Digi�) evaluated. If the criterion has improved over Q(�i), use this value, otherwise moveto the next element in the sequence.
183
Newton�s method does not perform well if Q(�) is irregular, and it can be quite computationallycostly if H(�) is not analytically available. These problems have motivated alternative choices forthe weight matrix Di: These methods are called Quasi-Newton methods. Two popular methodsare do to Davidson-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS).
Let
�gi = gi � gi�1��i = �i � �i�1
and : The DFP method sets
Di =Di�1 +��i��
0i
��0i�gi+Di�1�gi�g
0iDi�1
�g0iDi�1�gi:
The BFGS methods sets
Di =Di�1 +��i��
0i
��0i�gi� ��i��
0i�
��0i�gi�2�g0iDi�1�gi +
��i�g0iDi�1
��0i�gi+Di�1�gi��
0i
��0i�gi:
For any of the gradient methods, the iterations continue until the sequence has converged insome sense. This can be de�ned by examining whether j�i � �i�1j ; jQ (�i)�Q (�i�1)j or jg(�i)jhas become small.
E.3 Derivative-Free Methods
All gradient methods can be quite poor in locating the global minimum when Q(�) has severallocal minima. Furthermore, the methods are not well de�ned when Q(�) is non-di¤erentiable. Inthese cases, alternative optimization methods are required. One example is the simplex methodof Nelder-Mead (1965).
A more recent innovation is the method of simulated annealing (SA). For a review see Go¤e,Ferrier, and Rodgers (1994). The SA method is a sophisticated random search. Like the gradientmethods, it relies on an iterative sequence. At each iteration, a random variable is drawn andadded to the current value of the parameter. If the resulting criterion is decreased, this new valueis accepted. If the criterion is increased, it may still be accepted depending on the extent of theincrease and another randomization. The latter property is needed to keep the algorithm fromselecting a local minimum. As the iterations continue, the variance of the random innovations isshrunk. The SA algorithm stops when a large number of iterations is unable to improve the criterion.The SA method has been found to be successful at locating global minima. The downside is thatit can take considerable computer time to execute.
184
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