Contents

List of Figures xvii

Preface xix

1 Finite-Sample Properties of OLS

1.1 The Classical Linear Regression Model

The Linearity Assumption Matrix Notation The Strict Exogeneity Assumption Implications of Strict Exogeneity Strict Exogeneity in Time-Series Models Other Assumptions of the Model The Classical Regression Model for Random Samples "Fixed" Regressors

1.2 The Algebra of Least Squares OLS Minimizes the Sum of Squared Residuals Normal Equations Two Expressions for the OLS Estimator More Concepts and Algebra Influential Analysis (optional) A Note on the Computation of OLS Estimates

1.3 Finite-Sample Properties of OLS Finite-Sample Distribution of b Finite-Sample Properties of s2

Estimate of Var(b 1 X) 1.4 Hypothesis Testing under Normality

Normally Distributed Error Terms Testing Hypotheses about Individual Regression Coefficients Decision Rule for the t-Test Confidence Interval

vi Contents

p-Value 3 8 Linear Hypotheses 39 The F-Test 40 A More Convenient Expression for F 42 t versus F 43 An Example of a Test Statistic Whose Distribution Depends on X 45

1.5 Relation to Maximum Likelihood 47 The Maximum Likelihood Principle 47 Conditional versus Unconditional Likelihood 47 The Log Likelihood for the Regression Model 48 ML via Concentrated Likelihood 48 Cramer-Rao Bound for the Classical Regression Model 49 The F-Test as a Likelihood Ratio Test 52 Quasi-Maximum Likelihood 53

1.6 Generalized Least Squares (GLS) 54 Consequence of Relaxing Assumption 1.4 55 Efficient Estimation with Known V 55 A Special Case: Weighted Least Squares (WLS) 58 Limiting Nature of GLS 58

1.7 Application: Returns to Scale in Electricity Supply 60 The Electricity Supply Industry 60 The Data 60 Why Do We Need Econometrics? 61 The Cobb-Douglas Technology 62 How Do We Know Things Are Cobh-Douglas? 63 Are the OLS Assumptions Satisfied? 64 Restricted Least Squares 65 Testing the Homogeneity of the Cost Function 65 Detour: A Cautionary Note on R~ 67 Testing Constant Returns to Scale 67 Importance of Plotting Residuals 68 Subsequent Developments 68

Problem Set 7 1 Answers to Selected Questions 84

Large-Sample Theory 88

2.1 Review of Limit Theorems for Sequences of Random Variables 88 Various Modes of Convergence 89 Three Useful Results 92

Contents vii

Viewing Estimators as Sequences of Random Variables Laws of Large Numbers and Central Limit Theorems

2.2 Fundamental Concepts in Time-Series Analysis Need for Ergodic Stationarity Various Classes of Stochastic Processes Different Formulation of Lack of Serial Dependence The CLT for Ergodic Stationary Martingale Differences Sequences

2.3 Large-Sample Distribution of the OLS Estimator The Model Asymptotic Distribution of the OLS Estimator s2 IS Consistent

2.4 Hypothesis Testing Testing Linear Hypotheses The Test Is Consistent Asymptotic Power Testing Nonlinear Hypotheses

2.5 Estimating E(E?x~x;) Consistently Using Residuals for the Errors Data Matrix Representation of S Finite-Sample Considerations

2.6 Implications of Conditional Homoskedasticity Conditional versus Unconditional Homoskedasticity Reduction to Finite-Sample Formulas Large-Sample Distribution of t and F Statistics Variations of Asymptotic Tests under Conditional

Homoskedasticity 2.7 Testing Conditional Homoskedasticity 2.8 Estimation with Parameterized Conditional Heteroskedasticity

(optional) The Functional Form WLS with Known a Regression of e? on zi Provides a Consistent Estimate of a WLS with Estimated a OLS versus WLS

2.9 Least Squares Projection Optimally Predicting the Value of the Dependent Variable Best Linear Predictor OLS Consistently Estimates the Projection Coefficients

Contents

2.10 Testing for Serial Correlation Box-Pierce and Ljung-Box Sample Autocorrelations Calculated from Residuals Testing with Predetermined, but Not Strictly Exogenous,

Regressors An Auxiliary Regression-Based Test

2.1 1 Application: Rational Expectations Econometrics The Efficient Market Hypotheses Testable Implications Testing for Serial Correlation Is the Nominal Interest Rate the Optimal Predictor? R, Is Not Strictly Exogenous Subsequent Developments

2.12 Time Regressions The Asymptotic Distribution of the OLS Estimator Hypothesis Testing for Time Regressions

Appendix 2.A: Asymptotics with Fixed Regressors Appendix 2.B: Proof of Proposition 2.10 Problem Set Answers to Selected Questions

3 Single-Equation GMM 3.1 Endogeneity Bias: Working's Example

A Simultaneous Equations Model of Market Equilibrium Endogeneity Bias Observable Supply Shifters

3.2 More Examples A Simple Macroeconometric Model Errors-in-Variables Production Function

3.3 The General Formulation Regressors and Instruments Identification Order Condition for Identification The Assumption for Asymptotic Normality

3.4 Generalized Method of Moments Defined

Method of Moments Generalized Method of Moments Sampling Error

Contents

3.5 Large-Sample Properties of GMM Asymptotic Distribution of the GMM Estimator Estimation of Error Variance Hypothesis Testing Estimation of S Efficient GMM Estimator Asymptotic Power Small-Sample Properties

3.6 Testing Overidentifying Restrictions Testing Subsets of Orthogonality Conditions

3.7 Hypothesis Testing by the Likelihood-Ratio Principle

The LR Statistic for the Regression Model Variable Addition Test (optional)

3.8 Implications of Conditional Homoskedasticity Efficient GMM Becomes 2SLS J Becomes Sargan's Statistic Small-Sample Properties of 2SLS Alternative Derivations of 2SLS When Regressors Are Predetermined Testing a Subset of Orthogonality Conditions Testing Conditional Homoskedasticity

Testing for Serial Correlation 3.9 Application: Returns from Schooling

The NLS-Y Data The Semi-Log Wage Equation Omitted Variable Bias IQ as the Measure of Ability Errors-in-Variables 2SLS to Correct for the Bias Subsequent Developments

Problem Set Answers to Selected Questions

4 Multiple-Equation GMM 4.1 The Multiple-Equation Model

Linearity Stationarity and Ergodicity Orthogonality Conditions Identification

Contents

The Assumption for Asymptotic Normality Connection to the "Complete" System of Simultaneous Equations

4.2 Multiple-Equation GMM Defined 4.3 Large-Sample Theory 4.4 Single-Equation versus Multiple-Equation Estimation

When Are They "Equivalent"? Joint Estimation Can Be Hazardous

4.5 Special Cases of Multiple-Equation GMM: FIVE, 3SLS, and SUR Conditional Homoskedasticity Full-Information Instrumental Variables Efficient (FIVE) Three-Stage Least Squares (3SLS) Seemingly Unrelated Regressions (SUR) SUR versus OLS

4.6 Common Coefficients The Model with Common Coefficients The GMM Estimator Imposing Conditional Homoskedasticity Pooled OLS Beautifying the Formulas The Restriction That Isn't

4.7 Application: Interrelated Factor Demands The Translog Cost Function Factor Shares Substitution Elasticities Properties of Cost Functions Stochastic Specifications The Nature of Restrictions Multivariate Regression Subject to Cross-Equation Restrictions Which Equation to Delete? Results

Problem Set Answers to Selected Questions

5 Panel Data 5.1 The Error-Components Model

Error Components Group Means A Reparameterization

5.2 The Fixed-Effects Estimator

Contents

The Formula Large-Sample Properties Digression: When rli Is Spherical Random Effects versus Fixed Effects Relaxing Conditional Homoskedasticity

5.3 Unbalanced Panels (optional) "Zeroing Out" Missing Observations Zeroing Out versus Compression No Selectivity Bias

5.4 Application: International Differences in Growth Rates Derivation of the Estimation Equation Appending the Error Term Treatment of cri

Consistent Estimation of Speed of Convergence Appendix 5.A: Distribution of Hausman Statistic Problem Set Answers to Selected Questions

6 Serial Correlation 6.1 Modeling Serial Correlation: Linear Processes

MA(oo) as a Mean Square Limit Filters Inverting Lag Polynomials

6.2 ARMA Processes AR(1) and Its MA(oo) Representation Autocovariances of AR(1) AR(p) and Its MA(oo) Representation

ARMA(p7 q) ARMA(p, q) with Common Roots Invertibility Autocovariance-Generating Function and the Spectrum

6.3 Vector Processes 6.4 Estimating Autoregressions

Estimation of AR(1) Estimation of AR(p) Choice of Lag Length Estimation of VARs Estimation of ARMA(p, q)

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6.5 Asymptotics for Sample Means of Serially Correlated Processes LLN for Covariance-Stationary Processes Two Central Limit Theorems Multivariate Extension

6.6 Incorporating Serial Correlation in GMM The Model and Asymptotic Results Estimating S When Autocovariances Vanish after Finite Lags Using Kernels to Estimate S VARHAC

6.7 Estimation under Conditional Homoskedasticity (Optional) Kernel-Based Estimation of S under Conditional Homoskedasticity Data Matrix Representation of Estimated Long-Run Variance Relation to GLS

6.8 Application: Forward Exchange Rates as Optimal Predictors The Market Efficiency Hypothesis Testing Whether the Unconditional Mean Is Zero Regression Tests

Problem Set Answers to Selected Questions

7 Extremum Estimators 7.1 Extremum Estimators

"Measurability" of (? Two Classes of Extremum Estimators Maximum Likelihood (ML) Conditional Maximum Likelihood Invariance of ML Nonlinear Least Squares (NLS) Linear and Nonlinear GMM

7.2 Consistency Two Consistency Theorems for Extremum Estimators Consistency of M-Estimators Concavity after Reparameterization Identification in NLS and ML Consistency of GMM

7.3 Asymptotic Normality Asymptotic Normality of M-Estimators Consistent Asymptotic Variance Estimation Asymptotic Normality of Conditional ML

Contents

Two Examples Asymptotic Normality of GMM GMM versus ML Expressing the Sampling Error in a Common Format

7.4 Hypothesis Testing The Null Hypothesis The Working Assumptions The Wald Statistic The Lagrange Multiplier (LM) Statistic The Likelihood Ratio (LR) Statistic Summary of the Trinity

7.5 Numerical Optimization Newton-Raphson Gauss-Newton Writing Newton-Raphson and Gauss-Newton in a Common

Format Equations Nonlinear in Parameters Only

Problem Set Answers to Selected Questions

8 Examples of Maximum Likelihood 8.1 Qualitative Response (QR) Models

Score and Hessian for Observation t Consistency Asymptotic Normality

8.2 Truncated Regression Models The Model Truncated Distributions The Likelihood Function Reparameterizing the Likelihood Function Verifying Consistency and Asymptotic Normality Recovering Original Parameters

8.3 Censored Regression (Tobit) Models Tobit Likelihood Function Reparameterization

8.4 Multivariate Regressions The Multivariate Regression Model Restated The Likelihood Function Maximizing the Likelihood Function

X ~ V Contents

Consistency and Asymptotic Normality 525 8.5 FIML 526

The Multiple-Equation Model with Common Instruments Restated 526 The Complete System of Simultaneous Equations 529 Relationship between ( r o , Bo) and J0 530 The FIML Likelihood Function 53 1 The FIML Concentrated Likelihood Function 532 Testing Overidentifying Restrictions 533 Properties of the FIML Estimator 533 ML Estimation of the SUR Model 535

8.6 LIML 538 LIML Defined 538 Computation of LIML 540 LIML versus 2SLS 542

8.7 Serially Correlated Observations 543 Two Questions 543 Unconditional ML for Dependent Observations 545 ML Estimation of AR(1) Processes 546 Conditional ML Estimation of AR(1) Processes 547 Conditional ML Estimation of AR(p) and VAR(p) Processes 549

Problem Set 55 1

9 Unit-Root Econometrics 9.1 Modeling Trends

Integrated Processes Why Is It Important to Know if the Process Is I(1)? Which Should Be Taken as the Null, I(0) or 1(1)? Other Approaches to Modeling Trends

9.2 Tools for Unit-Root Econometrics Linear I(0) Processes Approximating I(1) by a Random Walk Relation to ARMA Models The Wiener Process A Useful Lemma

9.3 Dickey-Fuller Tests The AR(1) Model Deriving the Limiting Distribution under the I(1) Null Incorporating the Intercept Incorporating Time Trend

Contents

9.4 Augmented Dickey-Fuller Tests The Augmented Autoregression Limiting Distribution of the OLS Estimator Deriving Test Statistics Testing Hypotheses about 5 What to Do When p Is Unknown? A Suggestion for the Choice of pm(T) Including the Intercept in the Regression Incorporating Time Trend Summary of the DF and ADF Tests and Other Unit-Root Tests

9.5 Which Unit-Root Test to Use? Local-to-Unity Asymptotics Small-Sample Properties

9.6 Application: Purchasing Power Parity The Embarrassing Resiliency of the Random Walk Model?

Problem Set Answers to Selected Questions

10 Cointegration 10.1 Cointegrated Systems

Linear Vector I(0) and 1(1) Processes The Beveridge-Nelson Decomposition Cointegration Defined

10.2 Alternative Representations of Cointegrated Systems Phillips's Triangular Representation VAR and Cointegration The Vector Error-Correction Model (VECM) Johansen's ML Procedure

10.3 Testing the Null of No Cointegration Spurious Regressions The Residual-Based Test for Cointegration Testing the Null of Cointegration

10.4 Inference on Cointegrating Vectors The SOLS Estimator The Bivariate Example Continuing with the Bivariate Example Allowing for Serial Correlation General Case Other Estimators and Finite-Sample Properties

Contents

10.5 Application: The Demand for Money in the United States The Data (m - p, y, R ) as a Cointegrated System DOLS Unstable Money Demand?

Problem Set

Appendix A: Partitioned Matrices and Kronecker Products Addition and Multiplication of Partitioned Matrices Inverting Partitioned Matrices

Index