Eco No Metric ModelING

8/3/2019 Eco No Metric ModelING

1/38

Econometric ModelingEconometric Modeling

Research MethodsResearch Methods

Professor Lawrence W. LanProfessor Lawrence W. LanEmail:Email: [email protected]@mdu.edu.tw

http://140.116.6.5/mdu/http://140.116.6.5/mdu/

Institute of ManagementInstitute of Management
mailto:[email protected]:[email protected]://140.116.6.5/mdu/http://140.116.6.5/mdu/http://140.116.6.5/mdu/mailto:[email protected]


2/38

OutlineOutline

Overview

Single-equation Regression Models

Simultaneous-equation RegressionModels

Time-Series Models


3/38

OverviewOverview

Objectives

Model building

Types of models

Criteria of a good model

Data

Desirable properties of estimators

Methods of estimation

Software packages and books


4/38

ObjectivesObjectives

Empirical verification of the theories in business,economics, management and related disciplines isbecoming increasingly quantitative.

Econometrics, oreconomic measurement, is a socialscience in which the tools of economic theory,mathematical statistics are applied to the analysis ofeconomic phenomena.

Focus on models that can be expressed in equation formand relating variables quantitatively.

Data are used to estimate the parameters of theequations, and the theoretical relationships are testedstatistically.

Used forpolicy analysis and forecasting.


5/38

Model BuildingModel Building

Model building is a science and art, which

serves for policy analysis and forecasting.

science: consists of a set of quantitative tools

used to construct and test mathematical

representations of the real world problems.

art: consists of intuitive judgments that occur

during the modeling process. No clear-cutrules for making these judgments.


6/38

Types of Models (1/4)Types of Models (1/4)

Time-series models

Examine the past behavior of a time series in

order to infer something about its future

behavior, without knowing about the causalrelationships that affect the variable we are

trying to forecast.

Deterministic models (e.g. linearextrapolation) vs. stochastic models (e.g.

ARIMA, SARIMA).


7/38


Single-equation models

With causal relationships (based on

underlying theory) in which the variable (Y)

under study is explained by a single function(linear or nonlinear) of a number of variables

(Xs)

Y: explained or dependent variable Xs: explanatory or independent variables


8/38


Simultaneous-equation models (or multi-

equation simulation models)

With causal relationships (based on

underlying theory) in which the dependent

variables (Ys) under study are related to each

other as well as to a set of equations (linear or

nonlinear) with a number of explanatoryvariables (Xs)


9/38


Combination of time-series and regression

models

Single-input vs. multiple-input transfer function

models

Linear vs. rational transfer functions

Simultaneous-equation transfer functions

Transfer functions with interventions oroutliers


10/38

Criteria of a Good ModelCriteria of a Good Model

Parsimony

Identifiability

Goodness of fit Theoretical consistency

Predictive power


11/38

DataData

Sample data: the set of observations from themeasurement of variables, which may comefrom any number of sources and in a variety offorms.

Time-series data: describe the movement of anyvariable over time.

Cross-section data: describe the activities of anyindividual or group at a given point in time.

Pooled data: a combination of time-series andcross-section data, also known as panel data,longitudinal or micropanel data.


12/38

Desirable Properties of EstimatorsDesirable Properties of Estimators

Unbiased: the mean or expected value of anestimator is equal to the true value.

Efficient (best): the variance of an estimator is

smaller than any other ones. Minimum mean square error(MSE): to trade offbias and variance. MSE is equal to the square ofthe bias and the variance of the estimator.

Consistent: the probability limit of an estimatorgets close to the true value. It is a large-sampleor asymptotic property.


13/38

Methods of EstimationMethods of Estimation

Ordinary least squares (OLS)

Maximum likelihood (ML)

Weighted least squares (WLS)

Generalized least squares (GLS)

Instrumental variable (IV)

Two-stage least squares (2SLS)

Indirect least squares (ILS)

Three-stage least squares (3SLS)


14/38

Software Packages and BooksSoftware Packages and Books

LIMDEP: single-equation and

simultaneous-equation regression models

SCA: time series models

Textbooks (1) Damodar Gujarati, Essentials of Econometrics, 2nd

ed. McGraw-Hill, 1999.

(2) Robert S. Pindyck and Daniel L. Rubinfeld,Econometric Models and Economic Forecasts, 4th ed.

McGraw-Hill, 1997.


15/38

Single-equation Regression ModelsSingle-equation Regression Models

Assumptions

Best Linear Unbiased Estimation (BLUE)

Hypothesis testing Violations for assumptions 1 ~ 5

Forecasting


16/38

AssumptionsAssumptions

A1: (i) The relationship between Y and X is trulyexistent and correctly specified. (ii) Xs arenonstochastic variables whose values are fixed.(iii) Xs are not linearly correlated.

A2: The error term has zero expected value forall observations.

A3: The error term has constant variance for all

observations A4: The error terms are statistically independent.

A5: The error term is normally distributed.


17/38

Best Linear Unbiased EstimationBest Linear Unbiased Estimation

Gauss-Markov (GM) Theorem: Given

assumptions 1, 2, 3, and 4, the estimation of the

regression parameters by least squares (OLS)

method are the best (most efficient) linearunbiased estimators. (BLUE)

GM theorem applies only to linear estimators

where the estimators can be written as a

weighted average of the individual observationson Y.


18/38

Hypothesis TestingHypothesis Testing

Normal, Chi-square, t, and F distributions

Goodness of fit

Testing the regression coefficients (singleequation)

Testing the regression equation (joint

equations) Testing for structural stability or

transferability of regression models


19/38

A1(i) Violation -- Specification ErrorA1(i) Violation -- Specification Error

Omitting irrelevant variables biased and

inconsistent estimators

Inclusion of irrelevant variables

unbiased but inefficient estimators

Incorrect functional form (nonlinearities,

structural changes) biased and

inconsistent estimators


20/38

A1(ii) Violation Xs Correlated with ErrorA1(ii) Violation Xs Correlated with Error

OLS leads to biased and inconsistent estimators

Criteria of good instrumental (proxy) variables

Instrumental-variables estimation consistent,

but no guarantee for unbiased or uniqueestimators

Two-stage least squares (2SLS) estimation

optimal instrumental variable, unique consistentestimators


21/38

A1(iii) Violation -- MulticollinearityA1(iii) Violation -- Multicollinearity

Perfect collinearity between any of Xsno solution will exist

Near or imperfect multicollinearity large

standard error of OLS estimators or widerconfidence intervals; high R2 but fewsignificant t values; wrong signs forregression coefficients; difficulty inexplaining or assessing the individualcontribution of Xs to Y.


22/38

Detection of MulticollinearityDetection of Multicollinearity

Testing the significance of R-i2 from the various

auxiliary regressions. F=[R-i2/(k-1)]/[(1-R-i

2)/(n-k)],

where n=number of observations, k=number of

explanatory variables including the intercept.Check if F-value is significantly different from zero. If yes

(F-value > F-table), X-i and Xi are significantly collinear

with each other.

Variance inflation factor (VIF = 1/(1-R-i2): VIF=1representing no collinearity; if VIF>10 then high degree of

multicollinearity


23/38

A2 Violation Measurement Error in YA2 Violation Measurement Error in Y

OLS will result in biased intercept;

however, the estimated slope parameters

are still unbiased and consistent.

Correction for the dependent variable


24/38

A3 Violation -- HeteroscedasticityA3 Violation -- Heteroscedasticity

It happens mostly for cross-sectional data;

sometimes for time-series data.

OLS will lead to inefficient estimation, but still

unbiased. Can be corrected by weighted least squares

(WLS) method

Detection: Goldfeld-Quandt test, Breusch-Pagantest, White test, Park-Glejser test, Bartlett test,

Peak test, Spearmans rank correlation test, etc.


25/38

A4 Violation -- AutocorrelationA4 Violation -- Autocorrelation

It happens mostly for time-series data;sometimes for cross-sectional data.

OLS will lead to inefficient estimation, but still

unbiased. Can be corrected by generalized least squares(GLS) method

Detection: Durbin-Watson test, runs test. (For

lagged dependent variable, DW2 even whenserial correlation, do not use DW test, use h testor t test instead)


26/38

A5 Violation Non-normalityA5 Violation Non-normality

Chi-square, t, F tests are not valid;however, these tests are still valid for largesample.

Detection: Shapiro-Wilk test, Anderson-Darling test, Jarque-Bera (JB) test.JB=(n/6)[S2 + (K-3)2/4] where n=samplesize, K=kurtosis, S=skewness. (Fornormal, K=3, S=0) JB~ Chi-squaredistribution with 2 d.f.


27/38

ForecastingForecasting

Ex post vs. ex ante forecast

Unconditional forecasting

Conditional forecasting

Evaluation of ex post forecast errors

means: root-mean-square error, root-mean-square

percent error, mean error, mean percent error, mean

absolute error, mean absolute percent error, Theils

inequality coefficient

variances: Akaike information criterion (AIC), Schwarz

information criterion (SIC)


28/38

Simultaneous-equationsSimultaneous-equations

Regression ModelsRegression Models

Simultaneous-equation models

Seemly unrelated equation models

Identification problem


29/38

Simultaneous-equations ModelsSimultaneous-equations Models

Endogenous variables exist on both sides of theequations

Structural model vs. reduced form model

OLS will lead to biased and inconsistentestimation; indirect least squares (ILS) methodcan be used to obtain consistent estimation

Three-stage least squares (3SLS) method will

result in consistent estimation 3SLS often performs better than 2SLS in terms

of estimation efficiency


30/38

Seemly Unrelated Equation ModelsSeemly Unrelated Equation Models

Endogenous variables appear only on the

left hand side of equations

OLS usually results in unbiased but

inefficient estimation

Generalized least squares (GLS) method

is used to improve the efficiency Zellner

method


31/38

Identification ProblemIdentification Problem

Unidentified vs. identified (over identified

and exactly identified)

Order condition

Rank condition


32/38

Time-series ModelsTime-series Models

Time-series data

Univariate time series models

Box-Jenkins modeling approach Transfer function models


33/38

Time-series DataTime-series Data

Yt: A sequence of data observed at equally

spaced time interval

Stationary vs. non-stationary time series

Homogeneous vs. non-homogeneous time

series

Seasonal vs. non-seasonal time series


34/38

Univariate Time Series ModelsUnivariate Time Series Models

Types of models: white noise model,autoregressive (AR) models, moving-average(MA) models, autoregressive-moving average(ARMA) models, integrated autoregressive-

moving average (ARIMA) models, seasonalARIMA models

Model identification: MA(q) sampleautocorrelation function (ACF) cuts off; AR(p)sample partial autocorrelation function (PACF)cuts off; ARMA(p,q) both ACF and PACF dieout


35/38

Box-Jenkins Modeling ApproachBox-Jenkins Modeling Approach

Tentative model identification (p, q) extended

sample autocorrelation function (EACF)

Estimation (maximum likelihood estimation

conditional or exact) Diagnostic checking (t, R2, Q tests, sample ACF

of residuals, residual plots, outlier analysis)

Application (using minimum mean squared errorforecasts)


36/38

Transfer Function ModelsTransfer Function Models

Single input (X) vs. multiple input (Xs) models Linear transfer function (LTF) vs. rational

transfer function (RTF) models Model identification (variables to be used; b, s, r

for each input variable using corner tablemethod; ARMA model for the noise)

Model estimation: maximum likelihoodestimation (conditional or exact)

Diagnostic checking: cross correlation function(CCF)

Forecasting: simultaneous forecasting


37/38

Simultaneous Transfer FunctionSimultaneous Transfer Function

(STF) Models(STF) Models

Purposes (to facilitate forecasting andstructural analysis of a system, and toimprove forecast accuracy)

Yt and Xt can be both endogenousvariables in the system

Use LTF method for model identification,

FIML for estimation, CCM (crosscorrelation matrices) for diagnosticchecking, simultaneous forecasting


38/38

Transfer Function Models withTransfer Function Models with

Interventions or OutliersInterventions or Outliers

Additive Outlier (AO)

Level Shift (LS)

Temporary Change (TC) Innovational Outlier (IO)

Intervention models

Eco No Metric ModelING

Documents

Transcript of Eco No Metric ModelING