Transcription
ECONOMETRICSBRUCE E. HANSEN
233466810Introduction1.1What is Econometrics? . . . . . . . . . . . .1.2The Probability Approach to Econometrics1.3Econometric Terms . . . . . . . . . . . . . .1.4Observational Data . . . . . . . . . . . . . .1.5Standard Data Structures . . . . . . . . . . .1.6Econometric Software . . . . . . . . . . . .1.7Replication . . . . . . . . . . . . . . . . . . .1.8Data Files for Textbook . . . . . . . . . . . .1.9Reading the Manuscript . . . . . . . . . . .I Regression211Conditional Expectation and Projection2.1Introduction . . . . . . . . . . . . . . . . .2.2The Distribution of Wages . . . . . . . . .2.3Conditional Expectation . . . . . . . . . .2.4Logs and Percentages . . . . . . . . . . . .2.5Conditional Expectation Function . . . .2.6Continuous Variables . . . . . . . . . . . .2.7Law of Iterated Expectations . . . . . . . .2.8CEF Error . . . . . . . . . . . . . . . . . . .2.9Intercept-Only Model . . . . . . . . . . . .2.10 Regression Variance . . . . . . . . . . . . .2.11 Best Predictor . . . . . . . . . . . . . . . .2.12 Conditional Variance . . . . . . . . . . . .2.13 Homoskedasticity and Heteroskedasticity2.14 Regression Derivative . . . . . . . . . . . .2.15 Linear CEF . . . . . . . . . . . . . . . . . .2.16 Linear CEF with Nonlinear Effects . . . .2.17 Linear CEF with Dummy Variables . . . .2.18 Best Linear Predictor . . . . . . . . . . . .ii.12121214161719202224242526282930313134
292.302.312.322.332.3434iiiIllustrations of Best Linear Predictor . . . . . . . . . . . . . .Linear Predictor Error Variance . . . . . . . . . . . . . . . . .Regression Coefficients . . . . . . . . . . . . . . . . . . . . . .Regression Sub-Vectors . . . . . . . . . . . . . . . . . . . . . .Coefficient Decomposition . . . . . . . . . . . . . . . . . . . .Omitted Variable Bias . . . . . . . . . . . . . . . . . . . . . . .Best Linear Approximation . . . . . . . . . . . . . . . . . . . .Regression to the Mean . . . . . . . . . . . . . . . . . . . . . .Reverse Regression . . . . . . . . . . . . . . . . . . . . . . . .Limitations of the Best Linear Projection . . . . . . . . . . .Random Coefficient Model . . . . . . . . . . . . . . . . . . . .Causal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .Existence and Uniqueness of the Conditional Expectation* .Identification* . . . . . . . . . . . . . . . . . . . . . . . . . . .Technical Proofs* . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70717273747576777779808183848787889093Least Squares Regression4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .989898The Algebra of Least Squares3.1Introduction . . . . . . . . . . . . . . . . . . . . . .3.2Samples . . . . . . . . . . . . . . . . . . . . . . . . .3.3Moment Estimators . . . . . . . . . . . . . . . . . .3.4Least Squares Estimator . . . . . . . . . . . . . . .3.5Solving for Least Squares with One Regressor . . .3.6Solving for Least Squares with Multiple Regressors3.7Illustration . . . . . . . . . . . . . . . . . . . . . . .3.8Least Squares Residuals . . . . . . . . . . . . . . . .3.9Demeaned Regressors . . . . . . . . . . . . . . . .3.10 Model in Matrix Notation . . . . . . . . . . . . . . .3.11 Projection Matrix . . . . . . . . . . . . . . . . . . .3.12 Annihilator Matrix . . . . . . . . . . . . . . . . . . .3.13 Estimation of Error Variance . . . . . . . . . . . . .3.14 Analysis of Variance . . . . . . . . . . . . . . . . . .3.15 Projections . . . . . . . . . . . . . . . . . . . . . . .3.16 Regression Components . . . . . . . . . . . . . . .3.17 Regression Components (Alternative Derivation)*3.18 Residual Regression . . . . . . . . . . . . . . . . . .3.19 Leverage Values . . . . . . . . . . . . . . . . . . . .3.20 Leave-One-Out Regression . . . . . . . . . . . . . .3.21 Influential Observations . . . . . . . . . . . . . . .3.22 CPS Data Set . . . . . . . . . . . . . . . . . . . . . .3.23 Numerical Computation . . . . . . . . . . . . . . .3.24 Collinearity Errors . . . . . . . . . . . . . . . . . . .3.25 Programming . . . . . . . . . . . . . . . . . . . . . .3.26 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
le Mean . . . . . . . . . . . . . . . . . . . . . . . . . .Linear Regression Model . . . . . . . . . . . . . . . . . . .Expectation of Least Squares Estimator . . . . . . . . . .Variance of Least Squares Estimator . . . . . . . . . . . .Unconditional Moments . . . . . . . . . . . . . . . . . . .Gauss-Markov Theorem . . . . . . . . . . . . . . . . . . .Generalized Least Squares . . . . . . . . . . . . . . . . . .Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . .Estimation of Error Variance . . . . . . . . . . . . . . . . .Mean-Square Forecast Error . . . . . . . . . . . . . . . . .Covariance Matrix Estimation Under HomoskedasticityCovariance Matrix Estimation Under HeteroskedasticityStandard Errors . . . . . . . . . . . . . . . . . . . . . . . .Estimation with Sparse Dummy Variables . . . . . . . . .Computation . . . . . . . . . . . . . . . . . . . . . . . . . .Measures of Fit . . . . . . . . . . . . . . . . . . . . . . . . .Empirical Example . . . . . . . . . . . . . . . . . . . . . .Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . .Clustered Sampling . . . . . . . . . . . . . . . . . . . . . .Inference with Clustered Samples . . . . . . . . . . . . . .At What Level to Cluster? . . . . . . . . . . . . . . . . . . .Technical Proofs* . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .Normal Regression5.1Introduction . . . . . . . . . . . . . . . . . . . . .5.2The Normal Distribution . . . . . . . . . . . . . .5.3Multivariate Normal Distribution . . . . . . . . .5.4Joint Normality and Linear Regression . . . . . .5.5Normal Regression Model . . . . . . . . . . . . .5.6Distribution of OLS Coefficient Vector . . . . . .5.7Distribution of OLS Residual Vector . . . . . . .5.8Distribution of Variance Estimator . . . . . . . .5.9t-statistic . . . . . . . . . . . . . . . . . . . . . . .5.10 Confidence Intervals for Regression Coefficients5.11 Confidence Intervals for Error Variance . . . . .5.12 t Test . . . . . . . . . . . . . . . . . . . . . . . . . .5.13 Likelihood Ratio Test . . . . . . . . . . . . . . . .5.14 Information Bound for Normal Regression . . .5.15 Exercises . . . . . . . . . . . . . . . . . . . . . . 147149150151153153II Large Sample Methods1556156156156157A Review of Large Sample Asymptotics6.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2Modes of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3Weak Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS6.46.56.66.76.878vCentral Limit Theorem . . . . . . . . . . . . . . . .Continuous Mapping Theorem and Delta MethodSmooth Function Model . . . . . . . . . . . . . . .Stochastic Order Symbols . . . . . . . . . . . . . .Convergence of Moments . . . . . . . . . . . . . .157158159160160Asymptotic Theory for Least Squares7.1Introduction . . . . . . . . . . . . . . . . . . . .7.2Consistency of Least Squares Estimator . . . .7.3Asymptotic Normality . . . . . . . . . . . . . . .7.4Joint Distribution . . . . . . . . . . . . . . . . .7.5Consistency of Error Variance Estimators . . .7.6Homoskedastic Covariance Matrix Estimation7.7Heteroskedastic Covariance Matrix Estimation7.8Summary of Covariance Matrix Notation . . .7.9Alternative Covariance Matrix Estimators* . .7.10 Functions of Parameters . . . . . . . . . . . . .7.11 Asymptotic Standard Errors . . . . . . . . . . .7.12 t-statistic . . . . . . . . . . . . . . . . . . . . . .7.13 Confidence Intervals . . . . . . . . . . . . . . .7.14 Regression Intervals . . . . . . . . . . . . . . . .7.15 Forecast Intervals . . . . . . . . . . . . . . . . .7.16 Wald Statistic . . . . . . . . . . . . . . . . . . . .7.17 Homoskedastic Wald Statistic . . . . . . . . . .7.18 Confidence Regions . . . . . . . . . . . . . . . .7.19 Edgeworth Expansion* . . . . . . . . . . . . . .7.20 Uniformly Consistent Residuals* . . . . . . . .7.21 Asymptotic Leverage* . . . . . . . . . . . . . . .7.22 Exercises . . . . . . . . . . . . . . . . . . . . . 84185185186187188189Restricted Estimation8.1Introduction . . . . . . . . . . . . . . . . .8.2Constrained Least Squares . . . . . . . . .8.3Exclusion Restriction . . . . . . . . . . . .8.4Finite Sample Properties . . . . . . . . . .8.5Minimum Distance . . . . . . . . . . . . .8.6Asymptotic Distribution . . . . . . . . . .8.7Variance Estimation and Standard Errors8.8Efficient Minimum Distance Estimator .8.9Exclusion Restriction Revisited . . . . . .8.10 Variance and Standard Error Estimation .8.11 Hausman Equality . . . . . . . . . . . . . .8.12 Example: Mankiw, Romer and Weil (1992)8.13 Misspecification . . . . . . . . . . . . . . .8.14 Nonlinear Constraints . . . . . . . . . . .8.15 Inequality Restrictions . . . . . . . . . . .8.16 Technical Proofs* . . . . . . . . . . . . . .8.17 Exercises . . . . . . . . . . . . . . . . . . 17218.
CONTENTS9Hypothesis Testing9.1Hypotheses . . . . . . . . . . . . . . . . . . . . . . . .9.2Acceptance and Rejection . . . . . . . . . . . . . . .9.3Type I Error . . . . . . . . . . . . . . . . . . . . . . . .9.4t tests . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5Type II Error and Power . . . . . . . . . . . . . . . . .9.6Statistical Significance . . . . . . . . . . . . . . . . .9.7P-Values . . . . . . . . . . . . . . . . . . . . . . . . . .9.8t-ratios and the Abuse of Testing . . . . . . . . . . .9.9Wald Tests . . . . . . . . . . . . . . . . . . . . . . . . .9.10 Homoskedastic Wald Tests . . . . . . . . . . . . . . .9.11 Criterion-Based Tests . . . . . . . . . . . . . . . . . .9.12 Minimum Distance Tests . . . . . . . . . . . . . . . .9.13 Minimum Distance Tests Under Homoskedasticity9.14 F Tests . . . . . . . . . . . . . . . . . . . . . . . . . . .9.15 Hausman Tests . . . . . . . . . . . . . . . . . . . . . .9.16 Score Tests . . . . . . . . . . . . . . . . . . . . . . . .9.17 Problems with Tests of Nonlinear Hypotheses . . .9.18 Monte Carlo Simulation . . . . . . . . . . . . . . . .9.19 Confidence Intervals by Test Inversion . . . . . . . .9.20 Multiple Tests and Bonferroni Corrections . . . . .9.21 Power and Test Consistency . . . . . . . . . . . . . .9.22 Asymptotic Local Power . . . . . . . . . . . . . . . .9.23 Asymptotic Local Power, Vector Case . . . . . . . . .9.24 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 3523623824124324424524624925010 Resampling Methods10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .10.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 Jackknife Estimation of Variance . . . . . . . . . . . . .10.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 Jackknife for Clustered Observations . . . . . . . . . . .10.6 The Bootstrap Algorithm . . . . . . . . . . . . . . . . . .10.7 Bootstrap Variance and Standard Errors . . . . . . . . .10.8 Percentile Interval . . . . . . . . . . . . . . . . . . . . . .10.9 The Bootstrap Distribution . . . . . . . . . . . . . . . . .10.10 The Distribution of the Bootstrap Observations . . . .10.11 The Distribution of the Bootstrap Sample Mean . . . .10.12 Bootstrap Asymptotics . . . . . . . . . . . . . . . . . . .10.13 Consistency of the Bootstrap Estimate of Variance . . .10.14 Trimmed Estimator of Bootstrap Variance . . . . . . . .10.15 Unreliability of Untrimmed Bootstrap Standard Errors10.16 Consistency of the Percentile Interval . . . . . . . . . .10.17 Bias-Corrected Percentile Interval . . . . . . . . . . . .10.18 BCa Percentile Interval . . . . . . . . . . . . . . . . . . .10.19 Percentile-t Interval . . . . . . . . . . . . . . . . . . . . .10.20 Percentile-t Asymptotic Refinement . . . . . . . . . . .10.21 Bootstrap Hypothesis Tests . . . . . . . . . . . . . . . . 77279281283285286.
.3010.3110.32viiWald-Type Bootstrap Tests . . . . . . . . .Criterion-Based Bootstrap Tests . . . . . .Parametric Bootstrap . . . . . . . . . . . .How Many Bootstrap Replications? . . . .Setting the Bootstrap Seed . . . . . . . . .Bootstrap Regression . . . . . . . . . . . .Bootstrap Regression Asymptotic TheoryWild Bootstrap . . . . . . . . . . . . . . . .Bootstrap for Clustered Observations . .Technical Proofs* . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . .III Multiple Equation Models28828929029129229329429529729830130611 Multivariate Regression11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .11.2 Regression Systems . . . . . . . . . . . . . . . . . . .11.3 Least Squares Estimator . . . . . . . . . . . . . . . .11.4 Expectation and Variance of Systems Least Squares11.5 Asymptotic Distribution . . . . . . . . . . . . . . . .11.6 Covariance Matrix Estimation . . . . . . . . . . . . .11.7 Seemingly Unrelated Regression . . . . . . . . . . .11.8 Equivalence of SUR and Least Squares . . . . . . . .11.9 Maximum Likelihood Estimator . . . . . . . . . . . .11.10 Restricted Estimation . . . . . . . . . . . . . . . . . .11.11 Reduced Rank Regression . . . . . . . . . . . . . . .11.12 Principal Component Analysis . . . . . . . . . . . .11.13 Factor Models . . . . . . . . . . . . . . . . . . . . . .11.14 Approximate Factor Models . . . . . . . . . . . . . .11.15 Factor Models with Additional Regressors . . . . . .11.16 Factor-Augmented Regression . . . . . . . . . . . . .11.17 Multivariate Normal* . . . . . . . . . . . . . . . . . .11.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 2732833012 Instrumental Variables12.1 Introduction . . . . . . . . . . . .12.2 Overview . . . . . . . . . . . . . .12.3 Examples . . . . . . . . . . . . . .12.4 Endogenous Regressors . . . . . .12.5 Instruments . . . . . . . . . . . .12.6 Example: College Proximity . . .12.7 Reduced Form . . . . . . . . . . .12.8 Identification . . . . . . . . . . . .12.9 Instrumental Variables Estimator12.10 Demeaned Representation . . . .12.11 Wald Estimator . . . . . . . . . . .12.12 Two-Stage Least Squares . . . . .332332332333335336337339340341343343345.
.4112.4212.43viiiLimited Information Maximum Likelihood . . . . .Split-Sample IV and JIVE . . . . . . . . . . . . . . . .Consistency of 2SLS . . . . . . . . . . . . . . . . . . .Asymptotic Distribution of 2SLS . . . . . . . . . . .Determinants of 2SLS Variance . . . . . . . . . . . .Covariance Matrix Estimation . . . . . . . . . . . . .LIML Asymptotic Distribution . . . . . . . . . . . . .Functions of Parameters . . . . . . . . . . . . . . . .Hypothesis Tests . . . . . . . . . . . . . . . . . . . . .Finite Sample Theory . . . . . . . . . . . . . . . . . .Bootstrap for 2SLS . . . . . . . . . . . . . . . . . . . .The Peril of Bootstrap 2SLS Standard Errors . . . . .Clustered Dependence . . . . . . . . . . . . . . . . .Generated Regressors . . . . . . . . . . . . . . . . . .Regression with Expectation Errors . . . . . . . . . .Control Function Regression . . . . . . . . . . . . . .Endogeneity Tests . . . . . . . . . . . . . . . . . . . .Subset Endogeneity Tests . . . . . . . . . . . . . . . .OverIdentification Tests . . . . . . . . . . . . . . . .Subset OverIdentification Tests . . . . . . . . . . . .Bootstrap Overidentification Tests . . . . . . . . . .Local Average Treatment Effects . . . . . . . . . . . .Identification Failure . . . . . . . . . . . . . . . . . .Weak Instruments . . . . . . . . . . . . . . . . . . . .Many Instruments . . . . . . . . . . . . . . . . . . . .Testing for Weak Instruments . . . . . . . . . . . . .Weak Instruments with k 2 1 . . . . . . . . . . . . .Example: Acemoglu, Johnson, and Robinson (2001)Example: Angrist and Krueger (1991) . . . . . . . . .Programming . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . .13 Generalized Method of Moments13.1 Introduction . . . . . . . . . . . . . . . . .13.2 Moment Equation Models . . . . . . . . .13.3 Method of Moments Estimators . . . . . .13.4 Overidentified Moment Equations . . . .13.5 Linear Moment Models . . . . . . . . . . .13.6 GMM Estimator . . . . . . . . . . . . . . .13.7 Distribution of GMM Estimator . . . . . .13.8 Efficient GMM . . . . . . . . . . . . . . . .13.9 Efficient GMM versus 2SLS . . . . . . . . .13.10 Estimation of the Efficient Weight Matrix13.11 Iterated GMM . . . . . . . . . . . . . . . .13.12 Covariance Matrix Estimation . . . . . . .13.13 Clustered Dependence . . . . . . . . . . .13.14 Wald Test . . . . . . . . . . . . . . . . . . .13.15 Restricted GMM . . . . . . . . . . . . . . 4414415416417417418419420420421421422423424
.2413.2513.2613.2713.2813.29Nonlinear Restricted GMM . . . . . . . .Constrained Regression . . . . . . . . . .Multivariate Regression . . . . . . . . . .Distance Test . . . . . . . . . . . . . . . .Continuously-Updated GMM . . . . . .OverIdentification Test . . . . . . . . . .Subset OverIdentification Tests . . . . .Endogeneity Test . . . . . . . . . . . . . .Subset Endogeneity Test . . . . . . . . .Nonlinear GMM . . . . . . . . . . . . . .Bootstrap for GMM . . . . . . . . . . . .Conditional Moment Equation Models .Technical Proofs* . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . .ix.IV Dependent and Panel Data14 Time Series14.1 Introduction . . . . . . . . . . . . . . . . .14.2 Examples . . . . . . . . . . . . . . . . . . .14.3 Differences and Growth Rates . . . . . . .14.4 Stationarity . . . . . . . . . . . . . . . . . .14.5 Transformations of Stationary Processes .14.6 Convergent Series . . . . . . . . . . . . . .14.7 Ergodicity . . . . . . . . . . . . . . . . . . .14.8 Ergodic Theorem . . . . . . . . . . . . . .14.9 Conditioning on Information Sets . . . .14.10 Martingale Difference Sequences . . . . .14.11 CLT for Martingale Differences . . . . . .14.12 Mixing . . . . . . . . . . . . . . . . . . . . .14.13 CLT for Correlated Observations . . . . .14.14 Linear Projection . . . . . . . . . . . . . .14.15 White Noise . . . . . . . . . . . . . . . . . .14.16 The Wold Decomposition . . . . . . . . .14.17 Lag Operator . . . . . . . . . . . . . . . . .14.18 Autoregressive Wold Representation . . .14.19 Linear Models . . . . . . . . . . . . . . . .14.20 Moving Average Processes . . . . . . . . .14.21 Infinite-Order Moving Average Process . .14.22 First-Order Autoregressive Process . . . .14.23 Unit Root and Explosive AR(1) Processes .14.24 Second-Order Autoregressive Process . .14.25 AR(p) Processes . . . . . . . . . . . . . . .14.26 Impulse Response Function . . . . . . . .14.27 ARMA and ARIMA Processes . . . . . . . .14.28 Mixing Properties of Linear Processes . .14.29 Identification . . . . . . . . . . . . . . . . 3463464464465466470471474475476476477
.48xEstimation of Autoregressive Models . . . . . . . . . . . . . . . . . .Asymptotic Distribution of Least Squares Estimator . . . . . . . . .Distribution Under Homoskedasticity . . . . . . . . . . . . . . . . .Asymptotic Distribution Under General Dependence . . . . . . . .Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . .Covariance Matrix Estimation Under General Dependence . . . . .Testing the Hypothesis of No Serial Correlation . . . . . . . . . . . .Testing for Omitted Serial Correlation . . . . . . . . . . . . . . . . .Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Time Series Regression Models . . . . . . . . . . . . . . . . . . . . .Static, Distributed Lag, and Autoregressive Distributed Lag ModelsTime Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Testing for Serial Correlation in Regression Models . . . . . . . . .Bootstrap for Time Series . . . . . . . . . . . . . . . . . . . . . . . . .Technical Proofs* . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Multivariate Time Series15.1 Introduction . . . . . . . . . . . . . . . . . . . .15.2 Multiple Equation Time Series Models . . . . .15.3 Linear Projection . . . . . . . . . . . . . . . . .15.4 Multivariate Wold Decomposition . . . . . . .15.5 Impulse Response . . . . . . . . . . . . . . . . .15.6 VAR(1) Model . . . . . . . . . . . . . . . . . . . .15.7 VAR(p) Model . . . . . . . . . . . . . . . . . . . .15.8 Regression Notation . . . . . . . . . . . . . . . .15.9 Estimation . . . . . . . . . . . . . . . . . . .
CONTENTS vii 10.22 Wald-Type Bootstrap Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
