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ECONOMETRICSBRUCE E. HANSEN
©2000, 20201
University of Wisconsin
Department of Economics
This Revision: January, 2020Comments Welcome
1This manuscript may be printed and reproduced for individual or
instructional use, but may not be printed forcommercial
purposes.
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Contents
Preface xv
About the Author xvi
1 Introduction 11.1 What is Econometrics? . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The
Probability Approach to Econometrics . . . . . . . . . . . . . . .
. . . . . . . . . . . . 21.3 Econometric Terms and Notation . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4
Observational Data . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 41.5 Standard Data Structures . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 41.6 Econometric Software . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 61.7 Replication . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 71.8 Data Files for Textbook . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 Reading
the Manuscript . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 91.10 Common Symbols . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
I Regression 13
2 Conditional Expectation and Projection 142.1 Introduction . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 142.2 The Distribution of Wages . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3
Conditional Expectation . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 162.4 Log Differences . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 182.5 Conditional Expectation Function . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 192.6 Continuous Variables
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 212.7 Law of Iterated Expectations . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 222.8 CEF Error
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 232.9 Intercept-Only Model . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
252.10 Regression Variance . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 252.11 Best Predictor . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 262.12 Conditional Variance . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.13
Homoskedasticity and Heteroskedasticity . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 292.14 Regression Derivative . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 302.15 Linear CEF . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 312.16 Linear CEF with
Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 322.17 Linear CEF with Dummy Variables . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 332.18 Best
Linear Predictor . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 352.19 Illustrations of Best Linear
Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 40
i
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CONTENTS ii
2.20 Linear Predictor Error Variance . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 422.21 Regression
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 422.22 Regression Sub-Vectors . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432.23 Coefficient Decomposition . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 442.24 Omitted Variable
Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 452.25 Best Linear Approximation . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.26
Regression to the Mean . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 472.27 Reverse Regression . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 482.28 Limitations of the Best Linear Projection . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 492.29 Random
Coefficient Model . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 502.30 Causal Effects . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
512.31 Existence and Uniqueness of the Conditional Expectation* . .
. . . . . . . . . . . . . . . . 562.32 Identification* . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 562.33 Technical Proofs* . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 58Exercises . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 61
3 The Algebra of Least Squares 643.1 Introduction . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 643.2 Samples . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Moment
Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 653.4 Least Squares Estimator . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
663.5 Solving for Least Squares with One Regressor . . . . . . . .
. . . . . . . . . . . . . . . . . . 673.6 Solving for Least Squares
with Multiple Regressors . . . . . . . . . . . . . . . . . . . . .
. . 693.7 Illustration . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 713.8 Least Squares
Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 733.9 Demeaned Regressors . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.10
Model in Matrix Notation . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 753.11 Projection Matrix . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 763.12 Orthogonal Projection . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 783.13 Estimation
of Error Variance . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 793.14 Analysis of Variance . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
803.15 Projections . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 803.16 Regression
Components . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 813.17 Regression Components (Alternative
Derivation)* . . . . . . . . . . . . . . . . . . . . . . . 833.18
Residual Regression . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 843.19 Leverage Values . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 853.20 Leave-One-Out Regression . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 863.21 Influential
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 883.22 CPS Data Set . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
903.23 Numerical Computation . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 913.24 Collinearity Errors .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 923.25 Programming . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Exercises
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 97
4 Least Squares Regression 1014.1 Introduction . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 1014.2 Random Sampling . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1014.3 Sample Mean . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 102
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CONTENTS iii
4.4 Linear Regression Model . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1034.5 Mean of
Least-Squares Estimator . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 1034.6 Variance of Least Squares Estimator .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7
Unconditional Moments . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 1074.8 Gauss-Markov Theorem . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1084.9 Generalized Least Squares . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1094.10 Modern Gauss Markov
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1104.11 Residuals . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 1114.12
Estimation of Error Variance . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1124.13 Mean-Square Forecast
Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 1134.14 Covariance Matrix Estimation Under
Homoskedasticity . . . . . . . . . . . . . . . . . . . 1154.15
Covariance Matrix Estimation Under Heteroskedasticity . . . . . . .
. . . . . . . . . . . . 1154.16 Standard Errors . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1194.17 Covariance Matrix Estimation with Sparse Dummy Variables .
. . . . . . . . . . . . . . . 1204.18 Computation . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 1214.19 Measures of Fit . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 1234.20 Empirical
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 1244.21 Multicollinearity . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1254.22 Clustered Sampling . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 1284.23 Inference with
Clustered Samples . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1344.24 At What Level to Cluster? . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.25
Technical Proofs* . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 137Exercises . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 140
5 Normal Regression 1445.1 Introduction . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1445.2 The Normal Distribution . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1445.3 Multivariate Normal
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 1465.4 Joint Normality and Linear Regression . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 1485.5 Normal
Regression Model . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 1485.6 Distribution of OLS Coefficient
Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1505.7 Distribution of OLS Residual Vector . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 1515.8 Distribution of
Variance Estimator . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1525.9 t-statistic . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1525.10 Confidence Intervals for Regression Coefficients . . . . .
. . . . . . . . . . . . . . . . . . . 1535.11 Confidence Intervals
for Error Variance . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 1555.12 t Test . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.13
Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1575.14 Information Bound for
Normal Regression . . . . . . . . . . . . . . . . . . . . . . . . .
. . 159Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 161
II Large Sample Methods 162
6 A Review of Large Sample Asymptotics 1636.1 Introduction . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1636.2 Modes of Convergence . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.3 Weak
Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 164
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CONTENTS iv
6.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 1646.5 Continuous Mapping
Theorem and Delta Method . . . . . . . . . . . . . . . . . . . . .
. . 1656.6 Smooth Function Model . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 1666.7 Best Unbiased
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1676.8 Stochastic Order Symbols . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.9
Convergence of Moments . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 1696.10 Uniform Stochastic Bounds . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
7 Asymptotic Theory for Least Squares 1707.1 Introduction . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1707.2 Consistency of Least-Squares Estimator . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1707.3
Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1727.4 Joint Distribution . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 1767.5 Consistency of Error Variance Estimators . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1787.6 Homoskedastic
Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . .
. . . . . . 1797.7 Heteroskedastic Covariance Matrix Estimation . .
. . . . . . . . . . . . . . . . . . . . . . . 1807.8 Summary of
Covariance Matrix Notation . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1817.9 Alternative Covariance Matrix Estimators* .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1827.10
Functions of Parameters . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 1837.11 Best Unbiased Estimation .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 1857.12 Asymptotic Standard Errors . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1867.13 t-statistic . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 1887.14 Confidence Intervals . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1897.15 Regression Intervals . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1907.16 Forecast
Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 1927.17 Wald Statistic . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 1937.18 Homoskedastic Wald Statistic . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 1937.19 Confidence
Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 1947.20 Edgeworth Expansion* . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1957.21 Uniformly Consistent Residuals* . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 1967.22 Asymptotic Leverage*
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 197Exercises . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
8 Restricted Estimation 2068.1 Introduction . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 2068.2 Constrained Least Squares . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 2078.3 Exclusion
Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 2088.4 Finite Sample Properties . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2098.5 Minimum Distance . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 2138.6 Asymptotic
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 2148.7 Variance Estimation and Standard
Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2158.8 Efficient Minimum Distance Estimator . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 2158.9 Exclusion Restriction
Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 2178.10 Variance and Standard Error Estimation . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 2188.11 Hausman
Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 2198.12 Example: Mankiw, Romer and Weil
(1992) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2198.13 Misspecification . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 2248.14 Nonlinear
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 226
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CONTENTS v
8.15 Inequality Restrictions . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 2278.16 Technical Proofs*
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 229Exercises . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
231
9 Hypothesis Testing 2349.1 Hypotheses . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2349.2 Acceptance and Rejection . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 2359.3 Type I Error . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2369.4 t tests . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379.5
Type II Error and Power . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 2389.6 Statistical Significance .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 2399.7 P-Values . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 2409.8 t-ratios
and the Abuse of Testing . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 2429.9 Wald Tests . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2439.10 Homoskedastic Wald Tests . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 2459.11 Criterion-Based
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 2459.12 Minimum Distance Tests . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2469.13
Minimum Distance Tests Under Homoskedasticity . . . . . . . . . . .
. . . . . . . . . . . 2479.14 F Tests . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2489.15 Hausman Tests . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 2499.16 Score Tests . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 2509.17 Problems with Tests of Nonlinear
Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . .
2529.18 Monte Carlo Simulation . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 2559.19 Confidence
Intervals by Test Inversion . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 2579.20 Multiple Tests and Bonferroni
Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
2589.21 Power and Test Consistency . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 2599.22 Asymptotic Local
Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 2619.23 Asymptotic Local Power, Vector Case . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 264Exercises .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 265
10 Resampling Methods 27210.1 Introduction . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27210.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 27210.3 Jackknife
Estimation of Variance . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 27310.4 Example . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27610.5 Jackknife for Clustered Observations . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 27710.6 The Bootstrap
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 27910.7 Bootstrap Variance and Standard Errors .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28010.8
Percentile Interval . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 28210.9 The Bootstrap
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 28310.10 The Distribution of the Bootstrap
Observations . . . . . . . . . . . . . . . . . . . . . . . .
28410.11 The Distribution of the Bootstrap Sample Mean . . . . . .
. . . . . . . . . . . . . . . . . . 28510.12 Bootstrap Asymptotics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 28610.13 Consistency of the Bootstrap Estimate of
Variance . . . . . . . . . . . . . . . . . . . . . . . 28910.14
Trimmed Estimator of Bootstrap Variance . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 29110.15 Unreliability of Untrimmed
Bootstrap Standard Errors . . . . . . . . . . . . . . . . . . . .
29210.16 Consistency of the Percentile Interval . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 29310.17 Bias-Corrected
Percentile Interval . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 295
-
CONTENTS vi
10.18 BCa Percentile Interval . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 29710.19 Percentile-t
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 29810.20 Percentile-t Asymptotic Refinement
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30110.21 Bootstrap Hypothesis Tests . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 30210.22 Wald-Type
Bootstrap Tests . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 30510.23 Criterion-Based Bootstrap Tests . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30510.24 Parametric Bootstrap . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 30710.25 How Many
Bootstrap Replications? . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 30810.26 Setting the Bootstrap Seed . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30810.27 Bootstrap Regression . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 30910.28 Bootstrap
Regression Asymptotic Theory . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 31010.29 Wild Bootstrap . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31210.30 Bootstrap for Clustered Observations . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 31310.31 Technical Proofs* .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 315Exercises . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
319
III Multiple Equation Models 325
11 Multivariate Regression 32611.1 Introduction . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 32611.2 Regression Systems . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 32611.3
Least-Squares Estimator . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 32711.4 Mean and Variance of
Systems Least-Squares . . . . . . . . . . . . . . . . . . . . . . .
. . . 32911.5 Asymptotic Distribution . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 33011.6 Covariance
Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 33211.7 Seemingly Unrelated Regression . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33311.8
Equivalence of SUR and Least-Squares . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 33511.9 Maximum Likelihood Estimator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33511.10 Restricted Estimation . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 33611.11 Reduced Rank
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 33711.12 Principal Component Analysis . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34011.13
PCA with Additional Regressors . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 34211.14 Factor-Augmented
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 343Exercises . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
345
12 Instrumental Variables 34712.1 Introduction . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 34712.2 Overview . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 34712.3 Examples .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 34812.4 Instruments . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35012.5 Example: College Proximity . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 35212.6 Reduced Form . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 35312.7 Reduced Form Estimation . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 35412.8
Identification . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 35512.9 Instrumental
Variables Estimator . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 35612.10 Demeaned Representation . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35912.11
Wald Estimator . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 359
-
CONTENTS vii
12.12 Two-Stage Least Squares . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 36112.13 Limited
Information Maximum Likelihood . . . . . . . . . . . . . . . . . .
. . . . . . . . . 36412.14 JIVE . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36612.15 Consistency of 2SLS . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 36712.16 Asymptotic
Distribution of 2SLS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 36912.17 Determinants of 2SLS Variance . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37012.18 Covariance Matrix Estimation . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 37112.19 LIML Asymptotic
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 37312.20 Functions of Parameters . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37412.21
Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 37512.22 Finite Sample Theory .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 37612.23 Bootstrap for 2SLS . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.24 The
Peril of Bootstrap 2SLS Standard Errors . . . . . . . . . . . . . .
. . . . . . . . . . . . . 38012.25 Clustered Dependence . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38012.26 Generated Regressors . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 38112.27 Regression with
Expectation Errors . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 38512.28 Control Function Regression . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38712.29
Endogeneity Tests . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 39012.30 Subset Endogeneity Tests
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 39312.31 OverIdentification Tests . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 39512.32 Subset
OverIdentification Tests . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 39712.33 Bootstrap Overidentification
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 40012.34 Local Average Treatment Effects . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 40112.35 Identification
Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 40412.36 Weak Instruments . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40612.37 Many Instruments . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 40812.38 Testing for Weak
Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41212.39 Weak Instruments with k2 > 1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41912.40
Example: Acemoglu, Johnson and Robinson (2001) . . . . . . . . . .
. . . . . . . . . . . . 42112.41 Example: Angrist and Krueger
(1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 42312.42 Programming . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 425Exercises . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 428
13 Generalized Method of Moments 43613.1 Introduction . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 43613.2 Moment Equation Models . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 43613.3 Method of
Moments Estimators . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 43713.4 Overidentified Moment Equations . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43813.5
Linear Moment Models . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 43913.6 GMM Estimator . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 43913.7 Distribution of GMM Estimator . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 44013.8 Efficient GMM . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 44113.9 Efficient GMM versus 2SLS . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44213.10
Estimation of the Efficient Weight Matrix . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 44313.11 Iterated GMM . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 44413.12 Covariance Matrix Estimation . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 44413.13 Clustered
Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 44413.14 Wald Test . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
445
-
CONTENTS viii
13.15 Restricted GMM . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 44613.16 Nonlinear
Restricted GMM . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 44813.17 Constrained Regression . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44913.18 Multivariate Regression . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 44913.19 Distance Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45113.20 Continuously-Updated GMM . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45213.21
OverIdentification Test . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 45313.22 Subset
OverIdentification Tests . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 45413.23 Endogeneity Test . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 45513.24 Subset Endogeneity Test . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 45513.25 Nonlinear GMM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 45613.26 Bootstrap for GMM . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45813.27 Conditional Moment Equation Models . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 45913.28 Technical Proofs* . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 461Exercises . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
463
IV Dependent and Panel Data 470
14 Time Series 47114.1 Introduction . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47114.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 47114.3 Differences and
Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 47214.4 Stationarity . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47414.5 Transformations of Stationary Processes . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 47714.6 Convergent Series . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 47714.7 Ergodicity . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47714.8
Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 47914.9 Conditioning on Information
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 48014.10 Martingale Difference Sequences . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 48114.11 CLT for
Martingale Differences . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 48414.12 Mixing . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 48414.13 CLT for Correlated Observations . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 48614.14 Linear
Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 48714.15 White Noise . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 48914.16 The Wold Decomposition . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 48914.17 Linear Models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 49114.18 Moving Average Processes . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49114.19 Infinite-Order Moving Average Process . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 49314.20 Lag Operator . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 49314.21 First-Order Autoregressive Process . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49414.22
Unit Root and Explosive AR(1) Processes . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 49714.23 Second-Order Autoregressive
Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49814.24 AR(p) Processes . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 50014.25 Impulse
Response Function . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 50214.26 ARMA and ARIMA Processes . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50414.27 Mixing Properties of Linear Processes . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 50414.28 Identification . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 505
-
CONTENTS ix
14.29 Estimation of Autoregressive Models . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 50814.30 Asymptotic
Distribution of Least Squares Estimator . . . . . . . . . . . . . .
. . . . . . . . 50914.31 Distribution Under Homoskedasticity . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 50914.32
Asymptotic Distribution Under General Dependence . . . . . . . . .
. . . . . . . . . . . . 51014.33 Covariance Matrix Estimation . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51114.34 Covariance Matrix Estimation Under General Dependence . .
. . . . . . . . . . . . . . . . 51214.35 Testing the Hypothesis of
No Serial Correlation . . . . . . . . . . . . . . . . . . . . . . .
. . 51314.36 Testing for Omitted Serial Correlation . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 51414.37 Model
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 51514.38 Illustrations . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 51614.39 Time Series Regression Models . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 51814.40 Static,
Distributed Lag, and Autoregressive Distributed Lag Models . . . .
. . . . . . . . . 51914.41 Time Trends . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52014.42 Illustration . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 52214.43 Granger
Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 52314.44 Testing for Serial Correlation
in Regression Models . . . . . . . . . . . . . . . . . . . . . .
52514.45 Bootstrap for Time Series . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 52614.46 Technical
Proofs* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 52814.47 Exercises . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 537Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 537
15 Multivariate Time Series 54115.1 Introduction . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 54115.2 Multiple Equation Time Series Models . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 54115.3 Linear
Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 54215.4 Multivariate Wold Decomposition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54315.5 Impulse Response . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 54515.6 VAR(1) Model . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 54615.7 VAR(p) Model . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54715.8 Regression Notation . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 54715.9 Estimation . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 54815.10 Asymptotic Distribution . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54915.11 Covariance Matrix Estimation . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 55015.12 Selection of Lag
Length in an VAR . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 55115.13 Illustration . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55115.14 Predictive Regressions . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 55315.15 Impulse Response
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 55415.16 Local Projection Estimator . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55515.17
Regression on Residuals . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 55515.18 Orthogonalized Shocks . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 55615.19 Orthogonalized Impulse Response Function . . . . . .
. . . . . . . . . . . . . . . . . . . . 55815.20 Orthogonalized
Impulse Response Estimation . . . . . . . . . . . . . . . . . . . .
. . . . . 55815.21 Illustration . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55915.22
Forecast Error Decomposition . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 56015.23 Identification of
Recursive VARs . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 56115.24 Oil Price Shocks . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56215.25 Structural VARs . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 56315.26
Identification of Structural VARs . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 567
-
CONTENTS x
15.27 Long-Run Restrictions . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 56815.28 Blanchard and
Quah (1989) Illustration . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 57015.29 External Instruments . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57215.30
Dynamic Factor Models . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 57315.31 Technical Proofs* . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 57515.32 Exercises . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 577Exercises .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 577
16 Non Stationary Time Series 58116.1 Introduction . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 58116.2 Trend Stationarity . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 58116.3
Autoregressive Unit Roots . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 58116.4 Cointegration . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 58316.5 Cointegrated VARs . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 584
17 Panel Data 58617.1 Introduction . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58617.2
Time Indexing and Unbalanced Panels . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 58717.3 Notation . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 58817.4 Pooled Regression . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 58817.5 One-Way
Error Component Model . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 59017.6 Random Effects . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59017.7
Fixed Effect Model . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 59317.8 Within Transformation .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 59417.9 Fixed Effects Estimator . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 59617.10
Differenced Estimator . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 59717.11 Dummy Variables
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 59917.12 Fixed Effects Covariance Matrix Estimation .
. . . . . . . . . . . . . . . . . . . . . . . . . . 60117.13 Fixed
Effects Estimation in Stata . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 60217.14 Between Estimator . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60317.15 Feasible GLS . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 60417.16 Intercept in
Fixed Effects Regression . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 60517.17 Estimation of Fixed Effects . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60617.18 GMM Interpretation of Fixed Effects . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 60617.19 Identification in
the Fixed Effects Model . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 60817.20 Asymptotic Distribution of Fixed Effects
Estimator . . . . . . . . . . . . . . . . . . . . . . 60817.21
Asymptotic Distribution for Unbalanced Panels . . . . . . . . . . .
. . . . . . . . . . . . . 61017.22 Heteroskedasticity-Robust
Covariance Matrix Estimation . . . . . . . . . . . . . . . . . .
61217.23 Heteroskedasticity-Robust Estimation – Unbalanced Case . .
. . . . . . . . . . . . . . . . 61317.24 Hausman Test for Random vs
Fixed Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
61417.25 Random Effects or Fixed Effects? . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 61517.26 Time Trends . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 61517.27 Two-Way Error Components . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 61617.28
Instrumental Variables . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 61717.29 Identification with
Instrumental Variables . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 61917.30 Asymptotic Distribution of Fixed Effects 2SLS
Estimator . . . . . . . . . . . . . . . . . . . 61917.31 Linear GMM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 62117.32 Estimation with Time-Invariant
Regressors . . . . . . . . . . . . . . . . . . . . . . . . . . .
622
-
CONTENTS xi
17.33 Hausman-Taylor Model . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 62317.34 Jackknife
Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 62617.35 Panel Bootstrap . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62617.36 Dynamic Panel Models . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 62717.37 The Bias of Fixed
Effects Estimation . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 62817.38 Anderson-Hsiao Estimator . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 62917.39
Arellano-Bond Estimator . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 63017.40 Weak Instruments . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 63217.41 Dynamic Panels with Predetermined Regressors . . . .
. . . . . . . . . . . . . . . . . . . . 63317.42 Blundell-Bond
Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 63517.43 Forward Orthogonal Transformation . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63717.44
Empirical Illustration . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 638Exercises . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 640
18 Difference in Differences 64318.1 Introduction . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 64318.2 Minimum Wage in New Jersey . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 64318.3
Identification . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 64618.4 Multiple Units . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 64818.5 Do Police Reduce Crime? . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 64918.6 Trend
Specification . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 65118.7 Do Blue Laws Affect Liquor
Sales? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 65218.8 Check Your Code: Does Abortion Impact Crime? . . . .
. . . . . . . . . . . . . . . . . . . . 65318.9 Inference . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 654Exercises . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
656
V Nonparametric and Nonlinear Methods 658
19 Nonparametric Regression 65919.1 Introduction . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 65919.2 Binned Means Estimator . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 65919.3 Kernel
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 66119.4 Local Linear Estimator . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 66219.5 Local Polynomial Estimator . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 66319.6 Asymptotic Bias .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 66419.7 Asymptotic Variance . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66719.8
AIMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 66719.9 Boundary Bias . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 66919.10 Reference Bandwidth . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 67019.11
Nonparametric Residuals and Prediction Errors . . . . . . . . . . .
. . . . . . . . . . . . . 67119.12 Cross-Validation Bandwidth
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 67219.13 Asymptotic Distribution . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 67419.14 Undersmoothing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 67619.15 Conditional Variance Estimation . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67719.16
Variance Estimation and Standard Errors . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 67819.17 Confidence Bands . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 67819.18 The Local Nature of Kernel Regression . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 679
-
CONTENTS xii
19.19 Application to Wage Regression . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 68019.20 Clustered
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 68219.21 Application to Testscores . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68419.22 Multiple Regressors . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 68519.23 Curse of
Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 68819.24 Computation . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68819.25 Technical Proofs* . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 690Exercises . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 695
20 Series Regression 69720.1 Introduction . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69720.2 Polynomial Regression . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 69820.3 Illustrating
Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 69920.4 Orthogonal Polynomials . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70020.5
Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 70120.6 Illustrating Spline
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 70220.7 The Global/Local Nature of Series Regression
. . . . . . . . . . . . . . . . . . . . . . . . . . 70420.8
Stone-Weierstrass and Jackson Approximation Theory . . . . . . . .
. . . . . . . . . . . . 70620.9 Regressor Bounds . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70820.10 Matrix Convergence . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 70820.11 Consistent
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 71020.12 Convergence Rate . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71120.13 Asymptotic Normality . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 71320.14 Regression
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 71420.15 Undersmoothing . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71420.16 Residuals and Regression Fit . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 71520.17 Cross-Validation
Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 71620.18 Variance and Standard Error Estimation . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 71720.19
Clustered Observations . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 71820.20 Confidence Bands . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 71920.21 Uniform Approximations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 71920.22 Partially
Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 72020.23 Panel Fixed Effects . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 72020.24 Multiple Regressors . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 72120.25 Additively
Separable Models . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 72120.26 Nonparametric Instrumental Variables
Regression . . . . . . . . . . . . . . . . . . . . . . . 72220.27
NPIV Identification . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 72320.28 NPIV Convergence Rate .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 72520.29 Nonparametric vs Parametric Identification . . . . .
. . . . . . . . . . . . . . . . . . . . . . 72520.30 Example:
Angrist and Lavy (1999) . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 72620.31 Technical Proofs* . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
730Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 736
21 Regression Discontinuity 739
22 Nonlinear Econometric Models 74022.1 Introduction . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 74022.2 Nonlinear Least Squares . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 740
-
CONTENTS xiii
22.3 Least Absolute Deviations . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 74322.4 Quantile Regression
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 74622.5 Limited Dependent Variables . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 74722.6 Binary
Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 74822.7 Count Data . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 74922.8 Censored Data . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 75022.9 Sample
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 751Exercises . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 753
23 Machine Learning 75523.1 Introduction . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75523.2 Model Selection . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 75523.3 Bayesian
Information Criterion . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 75823.4 Akaike Information Criterion for
Regression . . . . . . . . . . . . . . . . . . . . . . . . . .
76023.5 Akaike Information Criterion for Likelihood . . . . . . . .
. . . . . . . . . . . . . . . . . . . 76323.6 Mallows Criterion . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 76423.7 Cross-Validation Criterion . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 76523.8
K-Fold Cross-Validation . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 76623.9 Many Selection Criteria are
Similar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 76723.10 Relation with Likelihood Ratio Testing . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 76823.11 Consistent
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 76923.12 Asymptotic Selection Optimality .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77223.13 Focused Information Criterion . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 77423.14 Best Subset and
Stepwise Regression . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 77623.15 The MSE of Model Selection Estimators . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 77723.16
Inference After Model Selection . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 77923.17 Empirical Illustration . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 78123.18 Shrinkage Methods . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 78223.19
James-Stein Shrinkage Estimator . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 78323.20 Interpretation of the Stein
Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 78523.21 Positive Part Estimator . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 78623.22
Shrinkage Towards Restrictions . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 78623.23 Group James-Stein . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 78923.24 Empirical Illustrations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 79023.25 Model
Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 79223.26 Smoothed BIC and AIC . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79523.27 Mallows Model Averaging . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 79723.28 Jackknife (CV)
Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 80023.29 Empirical Illustration . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80123.30 Ridge Regression . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 80123.31 LASSO . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 80623.32 Computation of the LASSO Estimator .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80823.33
Elastic Net . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 80923.34 Regression Sample
Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 80923.35 Regression Trees . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81123.36
Bagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 81323.37 Random Forests . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 81323.38 Ensembling . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 814
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CONTENTS xiv
23.39 Technical Proofs* . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 815Exercises . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 824
Appendices 827
A Matrix Algebra 827A.1 Notation . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
827A.2 Complex Matrices* . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 828A.3 Matrix Addition . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 829A.4 Matrix Multiplication . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 829A.5
Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 830A.6 Rank and Inverse . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 831A.7 Orthogonal and Orthonormal Matrices . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 832A.8 Determinant . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 832A.9 Eigenvalues . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
833A.10 Positive Definite Matrices . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 834A.11 Idempotent
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 835A.12 Singular Values . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
835A.13 Matrix Decompositions . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 836A.14 Generalized
Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 836A.15 Extrema of Quadratic Forms . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838A.16
Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 839A.17 QR Decomposition . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 840A.18 Solving Linear Systems . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 841A.19 Algorithmic
Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 843A.20 Matrix Calculus . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
844A.21 Kronecker Products and the Vec Operator . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 846A.22 Vector Norms . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 847A.23 Matrix Norms . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 847
B Useful Inequalities 850B.1 Inequalities for Real Numbers . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
850B.2 Inequalities for Vectors . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 851B.3 Inequalities for
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 852B.4 Probabability Inequalities . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852B.5
Proofs* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 857
References 872
-
Preface
This textbook is the second in a three-part series covering the
core material typically taught in a one-year Ph.D. course in
econometrics. The sequence is
1. Statistical Theory for Economists (first volume, abbreviated
as STFE)
2. Econometrics (this volume)
The textbooks are written as an integrated series. Each volume
is reasonably self-contained, but eachbuilds on the material
introduced in the previous volume(s).
This volume assumes that students have a background in
multivariate calculus, probability theory,linear algebra, and
mathematical statistics. A prior course in undergraduate
econometrics would behelpful but not required. Two excellent
undergraduate textbooks are Wooldridge (2015) and Stock andWatson
(2014). The relevant background in probability theory and
mathematical statistics is provided inStatistical Theory for
Economists.
For reference, the basic tools of matrix algebra and probability
inequalites are reviewed in the Ap-pendix.
For students wishing to deepen their knowledge of matrix algebra
in relation to their study of econo-metrics, I recommend Matrix
Algebra by Abadir and Magnus (2005).
For further study in econometrics beyond this text, I recommend
White (1984) and Davidson (1994)for asymptotic theory, Hamilton
(1994) and Kilian and Lütkepohl (2017) for time series methods,
Cameronand Trivedi (2005) and Wooldridge (2010) for panel data and
discrete response models, and Li and Racine(2007) for
nonparametrics and semiparametric econometrics. Beyond these texts,
the Handbook ofEconometrics series provides advanced summaries of
contemporary econometric methods and theory.
Alternative PhD-level econometrics textbooks include Theil
(1971), Amemiya (1985), Judge, Griffiths,Hill, Lütkepohl, and Lee
(1985), Goldberger (1991), Davidson and MacKinnon (1993), Johnston
and Di-Nardo (1997), Davidson (2000), Hayashi (2000), Ruud (2000),
Davidson and MacKinnon (2004), Greene(2017) and Magnus (2017). For
a focus on applied methods see Angrist and Pischke (2009).
The end-of-chapter exercises are important parts of the text and
are meant to help teach students ofeconometrics. Answers are not
provided, and this is intentional.
I would like to thank Ying-Ying Lee and Wooyoung Kim for
providing research assistance in preparingsome of the numerical
analysis, graphics, and empirical examples presented in the
text.
This is a manuscript in progress. Parts I-III are near complete.
Parts IV and V are incomplete, inparticular Chapters 16, 21, 22 and
23.
xv
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About the Author
Bruce E. Hansen is the Mary Claire Aschenbrenner Phipps
Distinguished Chair of Economics at theUniversity of
Wisconsin-Madison. Bruce is originally from Los Angeles,
California, has an undergrad-uate degree in economics from
Occidental College, and a Ph.D. in economics from Yale University.
Hepreviously taught at the University of Rochester and Boston
College.
Bruce is a Fellow of the Econometric Society, the Journal of
Econometrics, and the InternationalAssociation of Applied
Econometrics. He has served as Co-Editor of Econometric Theory and
as AssociateEditor of Econometrica. He has published 62 papers in
refereed journals which have received over 30,000citations.
xvi
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Chapter 1
Introduction
1.1 What is Econometrics?
The term “econometrics” is believed to have been crafted by
Ragnar Frisch (1895-1973) of Norway,one of the three principal
founders of the Econometric Society, first editor of the journal
Econometrica,and co-winner of the first Nobel Memorial Prize in
Economic Sciences in 1969. It is therefore fittingthat we turn to
Frisch’s own words in the introduction to the first issue of
Econometrica to describe thediscipline.
A word of explanation regarding the term econometrics may be in
order. Its definitionis implied in the statement of the scope of
the [Econometric] Society, in Section I of theConstitution, which
reads: “The Econometric Society is an international society for the
ad-vancement of economic theory in its relation to statistics and
mathematics.... Its main objectshall be to promote studies that aim
at a unification of the theoretical-quantitative and
theempirical-quantitative approach to economic problems....”
But there are several aspects of the quantitative approach to
economics, and no singleone of these aspects, taken by itself,
should be confounded with econometrics. Thus, econo-metrics is by
no means the same as economic statistics. Nor is it identical with
what we callgeneral economic theory, although a considerable
portion of this theory has a defininitelyquantitative character.
Nor should econometrics be taken as synonomous with the
appli-cation of mathematics to economics. Experience has shown that
each of these three view-points, that of statistics, economic
theory, and mathematics, is a necessary, but not by itselfa
sufficient, condition for a real understanding of the quantitative
relations in modern eco-nomic life. It is the unification of all
three that is powerful. And it is this unification thatconstitutes
econometrics.
Ragnar Frisch, Econometrica, (1933), 1, pp. 1-2.
This definition remains valid today, although some terms have
evolved somewhat in their usage.Today, we would say that
econometrics is the unified study of economic models, mathematical
statistics,and economic data.
Within the field of econometrics there are sub-divisions and
specializations. Econometric theoryconcerns the development of
tools and methods, and the study of the properties of econometric
meth-ods. Applied econometrics is a term describing the development
of quantitative economic models andthe application of econometric
methods to these models using economic data.
1
-
CHAPTER 1. INTRODUCTION 2
1.2 The Probability Approach to Econometrics
The unifying methodology of modern econometrics was articulated
by Trygve Haavelmo (1911-1999)of Norway, winner of the 1989 Nobel
Memorial Prize in Economic Sciences, in his seminal paper
“Theprobability approach in econometrics” (1944). Haavelmo argued
that quantitative economic modelsmust necessarily be probability
models (by which today we would mean stochastic). Deterministic
mod-els are blatently inconsistent with observed economic
quantities, and it is incoherent to apply determin-istic models to
non-deterministic data. Economic models should be explicitly
designed to incorporaterandomness; stochastic errors should not be
simply added to deterministic models to make them ran-dom. Once we
acknowledge that an economic model is a probability model, it
follows naturally that anappropriate tool way to quantify,
estimate, and conduct inferences about the economy is through
thepowerful theory of mathematical statistics. The appropriate
method for a quantitative economic analy-sis follows from the
probabilistic construction of the economic model.
Haavelmo’s probability approach was quickly embraced by the
economics profession. Today noquantitative work in economics shuns
its fundamental vision.
While all economists embrace the probability approach, there has
been some evolution in its imple-mentation.
The structural approach is the closest to Haavelmo’s original
idea. A probabilistic economic modelis specified, and the
quantitative analysis performed under the assumption that the
economic modelis correctly specified. Researchers often describe
this as “taking their model seriously”Ṫhe structuralapproach
typically leads to likelihood-based analysis, including maximum
likelihood and Bayesian esti-mation.
A criticism of the structural approach is that it is misleading
to treat an economic model as correctlyspecified. Rather, it is
more accurate to view a model as a useful abstraction or
approximation. In thiscase, how should we interpret structural
econometric analysis? The quasi-structural approach to infer-ence
views a structural economic model as an approximation rather than
the truth. This theory has ledto the concepts of the pseudo-true
value (the parameter value defined by the estimation problem),
thequasi-likelihood function, quasi-MLE, and quasi-likelihood
inference.
Closely related is the semiparametric approach. A probabilistic
economic model is partially spec-ified but some features are left
unspecified. This approach typically leads to estimation methods
suchas least-squares and the Generalized Method of Moments. The
semiparametric approach dominatescontemporary econometrics, and is
the main focus of this textbook.
Another branch of quantitative structural economics is the
calibration approach. Similar to thequasi-structural approach, the
calibration approach interprets structural models as approximations
andhence inherently false. The difference is that the
calibrationist literature rejects mathematical statistics(deeming
classical theory as inappropriate for approximate models) and
instead selects parameters bymatching model and data moments using
non-statistical ad hoc1 methods.
Trygve Haavelmo
The founding ideas of the field of econometrics are largely due
to the Nor-weigen econometrician Trygve Haavelmo (1911-1999). His
advocacy of proba-bility models revolutionized the field, and his
use of formal mathematical rea-soning laid the foundation for
subsequent generations. He was awarded the No-bel Memorial Prize in
Economic Sciences in 1989.
1Ad hoc means “for this purpose” – a method designed for a
specific problem – and not based on a generalizable principle.
-
CHAPTER 1. INTRODUCTION 3
1.3 Econometric Terms and Notation
In a typical application, an econometrician has a set of
repeated measurements on a set of variables.For example, in a labor
application the variables could include weekly earnings,
educational attainment,age, and other descriptive characteristics.
We call this information the data, dataset, or sample.
We use the term observations to refer to the distinct repeated
measurements on the variables. Anindividual observation often
corresponds to a specific economic unit, such as a person,
household, cor-poration, firm, organization, country, state, city
or other geographical region. An individual observationcould also
be a measurement at a point in time, such as quarterly GDP or a
daily interest rate.
Economists typically denote variables by the italicized roman
characters y , x, and/or z. The conven-tion in econometrics is to
use the character y to denote the variable to be explained, while
the charactersx and z are used to denote the conditioning
(explaining) variables.
Following mathematical convention, real numbers (elements of the
real line R, also called scalars)are written using lower case
italics such as x, and vectors (elements of Rk ) by lower case bold
italics suchas x , e.g.
x =
x1x2...
xk
.Upper case bold italics such as X are used for matrices.
We denote the number of observations by the natural number n,
and subscript the variables by theindex i to denote the individual
observation, e.g. yi , x i and z i . In some contexts we use
indices otherthan i , such as in time series applications where the
index t is common. In panel studies we typically usethe double
index i t to refer to individual i at a time period t .
The i th observation is the set (yi , x i , z i ).The sample is
the set {(yi , x i , z i ) : i = 1, ...,n}.
It is proper mathematical practice to use upper case X for
random variables and lower case x forrealizations or specific
values. Since we use upper case to denote matrices, the distinction
betweenrandom variables and their realizations is not rigorously
followed in econometric notation. Thus thenotation yi will in some
places refer to a random variable, and in other places a specific
realization.This is undesirable but there is little to be done
about it without terrifically complicating the notation.Hopefully
there will be no confusion as the use should be evident from the
context.
We typically use Greek letters such as β, θ and σ2 to denote
unknown parameters of an econometricmodel, and use boldface, e.g. β
or θ, when these are vector-valued. Estimators are typically
denoted byputting a hat “^”, tilde “~” or bar “-” over the
corresponding letter, e.g. β̂ and β̃ are estimators of β.
The covariance matrix of an econometric estimator will typically
be written using the capital bold-face V , often with a subscript
to denote the estimator, e.g. V β̂ = var
[β̂
]as the covariance matrix for β̂.
Hopefully without causing confusion, we will use the notation V
β = avar[β̂
]to denote the asymptotic
covariance matrix ofp
n(β̂−β) (the variance of the asymptotic distribution).
Estimators will be denoted
by appending hats or tildes, e.g. V̂ β is an estimator of V
β.
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CHAPTER 1. INTRODUCTION 4
1.4 Observational Data
A common econometric question is to quantify the causal impact
of one set of variables on anothervariable. For example, a concern
in labor economics is the returns to schooling – the change in
earningsinduced by increasing a worker’s education, holding other
variables constant. Another issue of interestis the earnings gap
between men and women.
Ideally, we would use experimental data to answer these
questions. To measure the returns toschooling, an experiment might
randomly divide children into groups, mandate different levels of
ed-ucation to the different groups, and then follow the children’s
wage path after they mature and enter thelabor force. The
differences between the groups would be direct measurements of the
effects of differ-ent levels of education. However, experiments
such as this would be widely condemned as immoral!Consequently, in
economics non-laboratory experimental data sets are typically
narrow in scope.
Instead, most economic data is observational. To continue the
above example, through data col-lection we can record the level of
a person’s education and their wage. With such data we can
measurethe joint distribution of these variables, and assess the
joint dependence. But from observational data itis difficult to
infer causality as we are not able to manipulate one variable to
see the direct effect on theother. For example, a person’s level of
education is (at least partially) determined by that person’s
choices.These factors are likely to be affected by their personal
abilities and attitudes towards work. The fact thata person is
highly educated suggests a high level of ability, which suggests a
high relative wage. This is analternative explanation for an
observed positive correlation between educational levels and wages.
Highability individuals do better in school, and therefore choose
to attain higher levels of education, and theirhigh ability is the
fundamental reason for their high wages. The point is that multiple
explanations areconsistent with a positive correlation between
schooling levels and education. Knowledge of the jointdistribution
alone may not be able to distinguish between these
explanations.
Most economic data sets are observational, not experimental.
This means thatall variables must be treated as random and possibly
jointly determined.
This discussion means that it is difficult to infer causality
from observational data alone. Causalinference requires
identification, and this is based on strong assumptions. We will
discuss these issueson occasion throughout the text.
1.5 Standard Data Structures
There are five major types of economic data sets:
cross-sectional, time series, panel, clustered, andspatial. They
are distinguished by the dependence structure across
observations.
Cross-sectional data sets have one observation per individual.
Surveys and administrative recordsare a typical source for
cross-sectional data. In typical applications, the individuals
surveyed are per-sons, households, firms or other economic agents.
In many contemporary econometric cross-sectionstudies the sample
size n is quite large. It is conventional to assume that
cross-sectional observationsare mutually independent. Most of this
text is devoted to the study of cross-section data.
Time series data are indexed by time. Typical examples include
macroeconomic aggregates, pricesand interest rates. This type of
data is characterized by serial dependence. Most aggregate economic
datais only available at a low frequency (annual, quarterly or
perhaps monthly) so the sample size is typicallymuch smaller than
in cross-section studies. An exception is financial data where data
are available at ahigh frequency (weekly, daily, hourly, or by
transaction) so sample sizes can be quite large.
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CHAPTER 1. INTRODUCTION 5
Panel data combines elements of cross-section and time series.
These data sets consist of a set ofindividuals (typically persons,
households, or corporations) measured repeatedly over time. The
com-mon modeling assumption is that the individuals are mutually
independent of one another, but a givenindividual’s observations
are mutually dependent. In some panel data contexts, the number of
time se-ries observations T per individual is small while the
number of individuals n is large. In other panel datacontexts (for
example when countries or states are taken as the unit of
measurement) the number ofindividuals n can be small while the
number of time series observations T can be moderately large.
Animportant issue in econometric panel data is the treatment of
error components.
Clustered samples are increasing popular in applied economics
and are related to panel data. In clus-tered sampling, the
observations are grouped into “clusters” which are treated as
mutually independentyet allowed to be dependent within the cluster.
The major difference with panel data is that clusteredsampling
typically does not explicitly model error component structures, nor
the dependence withinclusters, but rather is concerned with
inference which is robust to arbitrary forms of within-cluster
cor-relation.
Spatial dependence is another model of interdependence. The
observations are treated as mutuallydependent according to a
spatial measure (for example, geographic proximity). Unlike
clustering, spatialmodels allow all observations to be mutually
dependent, and typically rely on explicit modeling of thedependence
relationships. Spatial dependence can also be viewed as a
generalization of time seriesdependence.
Data Structures
• Cross-section
• Time-series
• Panel
• Clustered
• Spatial
As we mentioned above, most of this text will be devoted to
cross-sectional data under the assump-tion of mutually independent
observations. By mutual independence we mean that the i th
observation(yi , x i , z i
)is independent of the j th observation
(y j , x j , z j
)for i 6= j . In this case we say that the data
are independently distributed. (Sometimes the label
“independent” is misconstrued. It is a statementabout the
relationship between observations i and j , not a statement about
the relationship between yiand x i and/or z i .)
Furthermore, if the data is randomly gathered, it is reasonable
to model each observation as a drawfrom the same probability
distribution. In this case we say that the data are identically
distributed.If the observations are mutually independent and
identically distributed, we say that the observationsare
independent and identically distributed, i.i.d., or a random
sample. For most of this text we willassume that our observations
come from a random sample.
Definition 1.1 The observations (yi , x i , z i ) are a sample
from the distributionF if they are identically distributed across i
= 1, ...,n with joint distribution F .
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CHAPTER 1. INTRODUCTION 6
Definition 1.2 The observations (yi , x i , z i ) are a random
sample if they aremutually independent and identically distributed
(i.i.d.) across i = 1, ...,n.
In the random sampling framework, we think of an individual
observation(yi , x i , z i
)as a realization
from a joint probability distribution F(y, x , z
)which we can call the population. This “population” is
infinitely large. This abstraction can be a source of confusion
as it does not correspond to a physicalpopulation in the real
world. It is an abstraction since the distribution F is unknown,
and the goal ofstatistical inference is to learn about features of
F from the sample. The assumption of random samplingprovides the
mathematical foundation for treating economic statistics with the
tools of mathematicalstatistics.
The random sampling framework was a major intellectual
breakthrough of the late 19th century,allowing the application of
mathematical statistics to the social sciences. Before this
conceptual devel-opment, methods from mathematical statistics had
not been applied to economic data as the latter wasviewed as
non-random. The random sampling framework enabled economic samples
to be treated asrandom, a necessary precondition for the
application of statistical methods.
1.6 Econometric Software
Economists use a variety of econometric, statistical, and
programming software.Stata (www.stata.com) is a powerful
statistical program with a broad set of pre-programmed econo-
metric and statistical tools. It is quite popular among
economists, and is continuously being updatedwith new methods. It
is an excellent package for most econometric analysis, but is
limited when youwant to use new or less-common econometric methods
which have not yet been programed. At manypoints in this textbook
specific Stata estimation methods and commands are described. These
com-mands are valid for Stata version 15.
MATLAB (www.mathworks.com), GAUSS (www.aptech.com), and
OxMetrics (www.oxmetrics.net)are high-level matrix programming
languages with a wide variety of built-in statistical functions.
Manyeconometric methods have been programed in these languages and
are available on the web. The ad-vantage of these packages is that
you are in complete control of your analysis, and it is easier to
programnew methods than in Stata. Some disadvantages are that you
have to do much of the programming your-self, programming
complicated procedures takes significant time, and programming
errors are hard toprevent and difficult to detect and eliminate. Of
these languages, GAUSS used to be quite popular
amongeconometricians, but currently MATLAB is more popular.
An intermediate choice is R (www.r-project.org). R has the
capabilities of the above high-level matrixprogramming languages,
but also has many built-in statistical environments which can
replicate muchof the functionality of Stata. R is the dominate
programming language in the statistics field, so methodsdeveloped
in that arena are most commonly available in R. Uniquely, R is
open-source, user-contributed,and best of all, completely free! A
smaller but growing group of econometricians are enthusiastic fans
ofR.
For highly-intensive computational tasks, some economists write
their programs in a standard pro-gramming language such as Fortran
or C. This can lead to major gains in computational speed, at
thecost of increased time in programming and debugging.
There are many other packages which are used by econometricians,
include Eviews, Gretl, PcGive,Python, Julia, RATS, and SAS.
As the packages described above have distinct advantages, many
empirical economists end up usingmore than one package. As a
student of econometrics, you will learn at least one of these
packages, and
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CHAPTER 1. INTRODUCTION 7
probably more than one. My advice is that all students of
econometrics should develop a basic level offamiliarity with Stata,
and either Matlab or R (or all three).
1.7 Replication
Scientific research needs to be documented and replicable. For
social science research using obser-vational data, this requires
careful documentation and archiving of the research methods, data
manipu-lations, and coding.
The best practice is as follows. Accompanying each published
paper an author should create a com-plete replication package (set
of data files, documentation, and program code files). This package
shouldcontain the source (raw) data used for analysis, and code
which executes the empirical analysis and othernumerical work
reported in the paper. In most cases this is a set of programs
which may need to be ex-ecuted sequentially. (For example, there
may be an initial program which �