Econometrics

Bruce E. Hansen is the Mary Claire Aschenbrener Phipps Distinguished Chair of Economics at the University of Wisconsin–Madison and one of the most cited econometricians in the world.

• PrefaceAcknowledgmentsNotation1 Introduction1.1 What Is Econometrics?1.2 The Probability Approach to Econometrics1.3 Econometric Terms1.4 Observational Data1.5 Standard Data Structures1.6 Econometric Software1.7 Replication1.8 Data Files for Textbook1.9 Reading the BookI Regression2 Conditional Expectation and Projection2.1 Introduction2.2 The Distribution of Wages2.3 Conditional Expectation2.4 Logs and Percentages2.5 Conditional Expectation Function2.6 Continuous Variables2.7 Law of Iterated Expectations2.8 CEF Error2.9 Intercept-Only Model2.10 Regression Variance2.11 Best Predictor2.12 Conditional Variance2.13 Homoskedasticity and Heteroskedasticity2.14 Regression Derivative2.15 Linear CEF2.16 Linear CEF with Nonlinear Effects2.17 Linear CEF with Dummy Variables2.18 Best Linear Predictor2.19 Illustrations of Best Linear Predictor2.20 Linear Predictor Error Variance2.21 Regression Coefficients2.22 Regression Subvectors2.23 Coefficient Decomposition2.24 Omitted Variable Bias2.25 Best Linear Approximation2.26 Regression to the Mean2.27 Reverse Regression2.28 Limitations of the Best Linear Projection2.29 Random Coefficient Model2.30 Causal Effects2.31 Existence and Uniqueness of the Conditional Expectation*2.32 Identification*2.33 Technical Proofs*2.34 Exercises3 The Algebra of Least Squares3.1 Introduction3.2 Samples3.3 Moment Estimators3.4 Least Squares Estimator3.5 Solving for Least Squares with One Regressor3.6 Solving for Least Squares with Multiple Regressors3.7 Illustration3.8 Least Squares Residuals3.9 Demeaned Regressors3.10 Model in Matrix Notation3.11 Projection Matrix3.12 Annihilator Matrix3.13 Estimation of Error Variance3.14 Analysis of Variance3.15 Projections3.16 Regression Components3.17 Regression Components (Alternative Derivation)*3.18 Residual Regression3.19 Leverage Values3.20 Leave-One-Out Regression3.21 Influential Observations3.22 CPS Dataset3.23 Numerical Computation3.24 Collinearity Errors3.25 Programming3.26 Exercises4 Least Squares Regression4.1 Introduction4.2 Random Sampling4.3 Sample Mean4.4 Linear Regression Model4.5 Expectation of Least Squares Estimator4.6 Variance of Least Squares Estimator4.7 Unconditional Moments4.8 Gauss-Markov Theorem4.9 Generalized Least Squares4.10 Residuals4.11 Estimation of Error Variance4.12 Mean-Squared Forecast Error4.13 Covariance Matrix Estimation under Homoskedasticity4.14 Covariance Matrix Estimation under Heteroskedasticity4.15 Standard Errors4.16 Estimation with Sparse Dummy Variables4.17 Computation4.18 Measures of Fit4.19 Empirical Example4.20 Multicollinearity4.21 Clustered Sampling4.22 Inference with Clustered Samples4.23 At What Level to Cluster?4.24 Technical Proofs*4.25 Exercises5 Normal Regression5.1 Introduction5.2 The Normal Distribution5.3 Multivariate Normal Distribution5.4 Joint Normality and Linear Regression5.5 Normal Regression Model5.6 Distribution of OLS Coefficient Vector5.7 Distribution of OLS Residual Vector5.8 Distribution of Variance Estimator5.9 t-Statistic5.10 Confidence Intervals for Regression Coefficients5.11 Confidence Intervals for Error Variance5.12 t-Test5.13 Likelihood Ratio Test5.14 Information Bound for Normal Regression5.15 ExercisesII Large Sample Methods6 A Review of Large Sample Asymptotics6.1 Introduction6.2 Modes of Convergence6.3 Weak Law of Large Numbers6.4 Central Limit Theorem6.5 Continuous Mapping Theorem and Delta Method6.6 Smooth Function Model6.7 Stochastic Order Symbols6.8 Convergence of Moments7 Asymptotic Theory for Least Squares7.1 Introduction7.2 Consistency of Least Squares Estimator7.3 Asymptotic Normality7.4 Joint Distribution7.5 Consistency of Error Variance Estimators7.6 Homoskedastic Covariance Matrix Estimation7.7 Heteroskedastic Covariance Matrix Estimation7.8 Summary of Covariance Matrix Notation7.9 Alternative Covariance Matrix Estimators*7.10 Functions of Parameters7.11 Asymptotic Standard Errors7.12 t-Statistic7.13 Confidence Intervals7.14 Regression Intervals7.15 Forecast Intervals7.16 Wald Statistic7.17 Homoskedastic Wald Statistic7.18 Confidence Regions7.19 Edgeworth Expansion*7.20 Uniformly Consistent Residuals*7.21 Asymptotic Leverage*7.22 Exercises8 Restricted Estimation8.1 Introduction8.2 Constrained Least Squares8.3 Exclusion Restriction8.4 Finite Sample Properties8.5 Minimum Distance8.6 Asymptotic Distribution8.7 Variance Estimation and Standard Errors8.8 Efficient Minimum Distance Estimator8.9 Exclusion Restriction Revisited8.10 Variance and Standard Error Estimation8.11 Hausman Equality8.12 Example: Mankiw, Romer, and Weil (1992)8.13 Misspecification8.14 Nonlinear Constraints8.15 Inequality Restrictions8.16 Technical Proofs*8.17 Exercises9 Hypothesis Testing9.1 Introduction9.2 Hypotheses9.3 Acceptance and Rejection9.4 Type I Error9.5 T-Tests9.6 Type II Error and Power9.7 Statistical Significance9.8 p-Values9.9 t-Ratios and the Abuse of Testing9.10 Wald Tests9.11 Homoskedastic Wald Tests9.12 Criterion-Based Tests9.13 Minimum Distance Tests9.14 Minimum Distance Tests under Homoskedasticity9.15 F Tests9.16 Hausman Tests9.17 Score Tests9.18 Problems with Tests of Nonlinear Hypotheses9.19 Monte Carlo Simulation9.20 Confidence Intervals by Test Inversion9.21 Multiple Tests and Bonferroni Corrections9.22 Power and Test Consistency9.23 Asymptotic Local Power9.24 Asymptotic Local Power, Vector Case9.25 Exercises10 Resampling Methods10.1 Introduction10.2 Example10.3 Jackknife Estimation of Variance10.4 Example10.5 Jackknife for Clustered Observations10.6 The Bootstrap Algorithm10.7 Bootstrap Variance and Standard Errors10.8 Percentile Interval10.9 The Bootstrap Distribution10.10 The Distribution of the Bootstrap Observations10.11 The Distribution of the Bootstrap Sample Mean10.12 Bootstrap Asymptotics10.13 Consistency of the Bootstrap Estimate of Variance10.14 Trimmed Estimator of Bootstrap Variance10.15 Unreliability of Untrimmed Bootstrap Standard Errors10.16 Consistency of the Percentile Interval10.17 Bias-Corrected Percentile Interval10.18 BCa Percentile Interval10.19 Percentile-t Interval10.20 Percentile-t Asymptotic Refinement10.21 Bootstrap Hypothesis Tests10.22 Wald-Type Bootstrap Tests10.23 Criterion-Based Bootstrap Tests10.24 Parametric Bootstrap10.25 How Many Bootstrap Replications?10.26 Setting the Bootstrap Seed10.27 Bootstrap Regression10.28 Bootstrap Regression Asymptotic Theory10.29 Wild Bootstrap10.30 Bootstrap for Clustered Observations10.31 Technical Proofs*10.32 ExercisesIII Multiple Equation Models11 Multivariate Regression11.1 Introduction11.2 Regression Systems11.3 Least Squares Estimator11.4 Expectation and Variance of Systems Least Squares11.5 Asymptotic Distribution11.6 Covariance Matrix Estimation11.7 Seemingly Unrelated Regression11.8 Equivalence of SUR and Least Squares11.9 Maximum Likelihood Estimator11.10 Restricted Estimation11.11 Reduced Rank Regression11.12 Principal Component Analysis11.13 Factor Models11.14 Approximate Factor Models11.15 Factor Models with Additional Regressors11.16 Factor-Augmented Regression11.17 Multivariate Normal*11.18 Exercises12 Instrumental Variables12.1 Introduction12.2 Overview12.3 Examples12.4 Endogenous Regressors12.5 Instruments12.6 Example: College Proximity12.7 Reduced Form12.8 Identification12.9 Instrumental Variables Estimator12.10 Demeaned Representation12.11 Wald Estimator12.12 Two-Stage Least Squares12.13 Limited Information Maximum Likelihood12.14 Split-Sample IV and JIVE12.15 Consistency of 2SLS12.16 Asymptotic Distribution of 2SLS12.17 Determinants of 2SLS Variance12.18 Covariance Matrix Estimation12.19 LIML Asymptotic Distribution12.20 Functions of Parameters12.21 Hypothesis Tests12.22 Finite Sample Theory12.23 Bootstrap for 2SLS12.24 The Peril of Bootstrap 2SLS Standard Errors12.25 Clustered Dependence12.26 Generated Regressors12.27 Regression with Expectation Errors12.28 Control Function Regression12.29 Endogeneity Tests12.30 Subset Endogeneity Tests12.31 Overidentification Tests12.32 Subset Overidentification Tests12.33 Bootstrap Overidentification Tests12.34 Local Average Treatment Effects12.35 Identification Failure12.36 Weak Instruments12.37 Many Instruments12.38 Testing for Weak Instruments12.39 Weak Instruments with k2 > 112.40 Example: Acemoglu, Johnson, and Robinson (2001)12.41 Example: Angrist and Krueger (1991)12.42 Programming12.43 Exercises13 Generalized Method of Moments13.1 Introduction13.2 Moment Equation Models13.3 Method of Moments Estimators13.4 Overidentified Moment Equations13.5 Linear Moment Models13.6 GMM Estimator13.7 Distribution of GMM Estimator13.8 Efficient GMM13.9 Efficient GMM versus 2SLS13.10 Estimation of the Efficient Weight Matrix13.11 Iterated GMM13.12 Covariance Matrix Estimation13.13 Clustered Dependence13.14 Wald Test13.15 Restricted GMM13.16 Nonlinear Restricted GMM13.17 Constrained Regression13.18 Multivariate Regression13.19 Distance Test13.20 Continuously Updated GMM13.21 Overidentification Test13.22 Subset Overidentification Tests13.23 Endogeneity Test13.24 Subset Endogeneity Test13.25 Nonlinear GMM13.26 Bootstrap for GMM13.27 Conditional Moment Equation Models13.28 Technical Proofs*13.29 ExercisesIV Dependent and Panel Data14 Time Series14.1 Introduction14.2 Examples14.3 Differences and Growth Rates14.4 Stationarity14.5 Transformations of Stationary Processes14.6 Convergent Series14.7 Ergodicity14.8 Ergodic Theorem14.9 Conditioning on Information Sets14.10 Martingale Difference Sequences14.11 CLT for Martingale Differences14.12 Mixing14.13 CLT for Correlated Observations14.14 Linear Projection14.15 White Noise14.16 The Wold Decomposition14.17 Lag Operator14.18 Autoregressive Wold Representation14.19 Linear Models14.20 Moving Average Process14.21 Infinite-Order Moving Average Process14.22 First-Order Autoregressive Process14.23 Unit Root and Explosive AR(1) Processes14.24 Second-Order Autoregressive Process14.25 AR(p) Process14.26 Impulse Response Function14.27 ARMA and ARIMA Processes14.28 Mixing Properties of Linear Processes14.29 Identification14.30 Estimation of Autoregressive Models14.31 Asymptotic Distribution of Least Squares Estimator14.32 Distribution under Homoskedasticity14.33 Asymptotic Distribution under General Dependence14.34 Covariance Matrix Estimation14.35 Covariance Matrix Estimation under General Dependence14.36 Testing the Hypothesis of No Serial Correlation14.37 Testing for Omitted Serial Correlation14.38 Model Selection14.39 Illustrations14.40 Time Series Regression Models14.41 Static, Distributed Lag, and Autoregressive Distributed Lag Models14.42 Time Trends14.43 Illustration14.44 Granger Causality14.45 Testing for Serial Correlation in Regression Models14.46 Bootstrap for Time Series14.47 Technical Proofs*14.48 Exercises15 Multivariate Time Series15.1 Introduction15.2 Multiple Equation Time Series Models15.3 Linear Projection15.4 Multivariate Wold Decomposition15.5 Impulse Response15.6 VAR(1) Model15.7 VAR(p) Model15.8 Regression Notation15.9 Estimation15.10 Asymptotic Distribution15.11 Covariance Matrix Estimation15.12 Selection of Lag Length in a VAR15.13 Illustration15.14 Predictive Regressions15.15 Impulse Response Estimation15.16 Local Projection Estimator15.17 Regression on Residuals15.18 Orthogonalized Shocks15.19 Orthogonalized Impulse Response Function15.20 Orthogonalized Impulse Response Estimation15.21 Illustration15.22 Forecast Error Decomposition15.23 Identification of Recursive VARs15.24 Oil Price Shocks15.25 Structural VARs15.26 Identification of Structural VARs15.27 Long-Run Restrictions15.28 Blanchard and Quah (1989) Illustration15.29 External Instruments15.30 Dynamic Factor Models15.31 Technical Proofs*15.32 Exercises16 Nonstationary Time Series16.1 Introduction16.2 Partial Sum Process and Functional Convergence16.3 Beveridge-Nelson Decomposition16.4 Functional CLT16.5 Orders of Integration16.6 Means, Local Means, and Trends16.7 Demeaning and Detrending16.8 Stochastic Integrals16.9 Estimation of an AR(1)16.10 AR(1) Estimation with an Intercept16.11 Sample Covariances of Integrated and Stationary Processes16.12 AR(p) Models with a Unit Root16.13 Testing for a Unit Root16.14 KPSS Stationarity Test16.15 Spurious Regression16.16 Nonstationary VARs16.17 Cointegration16.18 Role of Intercept and Trend16.19 Cointegrating Regression16.20 VECM Estimation16.21 Testing for Cointegration in a VECM16.22 Technical Proofs*16.23 Exercises17 Panel Data17.1 Introduction17.2 Time Indexing and Unbalanced Panels17.3 Notation17.4 Pooled Regression17.5 One-Way Error Component Model17.6 Random Effects17.7 Fixed Effects Model17.8 Within Transformation17.9 Fixed Effects Estimator17.10 Differenced Estimator17.11 Dummy Variables Regression17.12 Fixed Effects Covariance Matrix Estimation17.13 Fixed Effects Estimation in Stata17.14 Between Estimator17.15 Feasible GLS17.16 Intercept in Fixed Effects Regression17.17 Estimation of Fixed Effects17.18 GMM Interpretation of Fixed Effects17.19 Identification in the Fixed Effects Model17.20 Asymptotic Distribution of Fixed Effects Estimator17.21 Asymptotic Distribution for Unbalanced Panels17.22 Heteroskedasticity-Robust Covariance Matrix Estimation17.23 Heteroskedasticity-Robust Estimation—Unbalanced Case17.24 Hausman Test for Random vs. Fixed Effects17.25 Random Effects or Fixed Effects?17.26 Time Trends17.27 Two-Way Error Components17.28 Instrumental Variables17.29 Identification with Instrumental Variables17.30 Asymptotic Distribution of Fixed Effects 2SLS Estimator17.31 Linear GMM17.32 Estimation with Time-Invariant Regressors17.33 Hausman-Taylor Model17.34 Jackknife Covariance Matrix Estimation17.35 Panel Bootstrap17.36 Dynamic Panel Models17.37 The Bias of Fixed Effects Estimation17.38 Anderson-Hsiao Estimator17.39 Arellano-Bond Estimator17.40 Weak Instruments17.41 Dynamic Panels with Predetermined Regressors17.42 Blundell-Bond Estimator17.43 Forward Orthogonal Transformation17.44 Empirical Illustration17.45 Exercises18 Difference in Differences18.1 Introduction18.2 Minimum Wage in New Jersey18.3 Identification18.4 Multiple Units18.5 Do Police Reduce Crime?18.6 Trend Specification18.7 Do Blue Laws Affect Liquor Sales?18.8 Check Your Code: Does Abortion Impact Crime?18.9 Inference18.10 ExercisesV Nonparametric Methods19 Nonparametric Regression19.1 Introduction19.2 Binned Means Estimator19.3 Kernel Regression19.4 Local Linear Estimator19.5 Local Polynomial Estimator19.6 Asymptotic Bias19.7 Asymptotic Variance19.8 AIMSE19.9 Reference Bandwidth19.10 Estimation at a Boundary19.11 Nonparametric Residuals and Prediction Errors19.12 Cross-Validation Bandwidth Selection19.13 Asymptotic Distribution19.14 Undersmoothing19.15 Conditional Variance Estimation19.16 Variance Estimation and Standard Errors19.17 Confidence Bands19.18 The Local Nature of Kernel Regression19.19 Application to Wage Regression19.20 Clustered Observations19.21 Application to Test Scores19.22 Multiple Regressors19.23 Curse of Dimensionality19.24 Partially Linear Regression19.25 Computation19.26 Technical Proofs*19.27 Exercises20 Series Regression20.1 Introduction20.2 Polynomial Regression20.3 Illustrating Polynomial Regression20.4 Orthogonal Polynomials20.5 Splines20.6 Illustrating Spline Regression20.7 The Global/Local Nature of Series Regression20.8 Stone-Weierstrass and Jackson Approximation Theory20.9 Regressor Bounds20.10 Matrix Convergence20.11 Consistent Estimation20.12 Convergence Rate20.13 Asymptotic Normality20.14 Regression Estimation20.15 Undersmoothing20.16 Residuals and Regression Fit20.17 Cross-Validation Model Selection20.18 Variance and Standard Error Estimation20.19 Clustered Observations20.20 Confidence Bands20.21 Uniform Approximations20.22 Partially Linear Model20.23 Panel Fixed Effects20.24 Multiple Regressors20.25 Additively Separable Models20.26 Nonparametric Instrumental Variables Regression20.27 NPIV Identification20.28 NPIV Convergence Rate20.29 Nonparametric vs. Parametric Identification20.30 Example: Angrist and Lavy (1999)20.31 Technical Proofs*20.32 Exercises21 Regression Discontinuity21.1 Introduction21.2 Sharp Regression Discontinuity21.3 Identification21.4 Estimation21.5 Inference21.6 Bandwidth Selection21.7 RDD with Covariates21.8 A Simple RDD Estimator21.9 Density Discontinuity Test21.10 Fuzzy Regression Discontinuity21.11 Estimation of FRD21.12 ExercisesVI Nonlinear Methods22 M-Estimators22.1 Introduction22.2 Examples22.3 Identification and Estimation22.4 Consistency22.5 Uniform Law of Large Numbers22.6 Asymptotic Distribution22.7 Asymptotic Distribution under Broader Conditions*22.8 Covariance Matrix Estimation22.9 Technical Proofs*22.10 Exercises23 Nonlinear Least Squares23.1 Introduction23.2 Identification23.3 Estimation23.4 Asymptotic Distribution23.5 Covariance Matrix Estimation23.6 Panel Data23.7 Threshold Models23.8 Testing for Nonlinear Components23.9 Computation23.10 Technical Proofs*23.11 Exercises24 Quantile Regression24.1 Introduction24.2 Median Regression24.3 Least Absolute Deviations24.4 Quantile Regression24.5 Example Quantile Shapes24.6 Estimation24.7 Asymptotic Distribution24.8 Covariance Matrix Estimation24.9 Clustered Dependence24.10 Quantile Crossings24.11 Quantile Causal Effects24.12 Random Coefficient Representation24.13 Nonparametric Quantile Regression24.14 Panel Data24.15 IV Quantile Regression24.16 Technical Proofs*24.17 Exercises25 Binary Choice25.1 Introduction25.2 Binary Choice Models25.3 Models for the Response Probability25.4 Latent Variable Interpretation25.5 Likelihood25.6 Pseudo-True Values25.7 Asymptotic Distribution25.8 Covariance Matrix Estimation25.9 Marginal Effects25.10 Application25.11 Semiparametric Binary Choice25.12 IV Probit25.13 Binary Panel Data25.14 Technical Proofs*25.15 Exercises26 Multiple Choice26.1 Introduction26.2 Multinomial Response26.3 Multinomial Logit26.4 Conditional Logit26.5 Independence of Irrelevant Alternatives26.6 Nested Logit26.7 Mixed Logit26.8 Simple Multinomial Probit26.9 General Multinomial Probit26.10 Ordered Response26.11 Count Data26.12 BLP Demand Model26.13 Technical Proofs*26.14 Exercises27 Censoring and Selection27.1 Introduction27.2 Censoring27.3 Censored Regression Functions27.4 The Bias of Least Squares Estimation27.5 Tobit Estimator27.6 Identification in Tobit Regression27.7 CLAD and CQR Estimators27.8 Illustrating Censored Regression27.9 Sample Selection Bias27.10 Heckman’s Model27.11 Nonparametric Selection27.12 Panel Data27.13 Exercises28 Model Selection, Stein Shrinkage, and Model Averaging28.1 Introduction28.2 Model Selection28.3 Bayesian Information Criterion28.4 Akaike Information Criterion for Regression28.5 Akaike Information Criterion for Likelihood28.6 Mallows Criterion28.7 Hold-Out Criterion28.8 Cross-Validation Criterion28.9 K-Fold Cross-Validation28.10 Many Selection Criteria Are Similar28.11 Relation with Likelihood Ratio Testing28.12 Consistent Selection28.13 Asymptotic Selection Optimality28.14 Focused Information Criterion28.15 Best Subset and Stepwise Regression28.16 The MSE of Model Selection Estimators28.17 Inference after Model Selection28.18 Empirical Illustration28.19 Shrinkage Methods28.20 James-Stein Shrinkage Estimator28.21 Interpretation of the Stein Effect28.22 Positive Part Estimator28.23 Shrinkage Toward Restrictions28.24 Group James-Stein28.25 Empirical Illustrations28.26 Model Averaging28.27 Smoothed BIC and AIC28.28 Mallows Model Averaging28.29 Jackknife (CV) Model Averaging28.30 Granger-Ramanathan Averaging28.31 Empirical Illustration28.32 Technical Proofs*28.33 Exercises29 Machine Learning29.1 Introduction29.2 Big Data, High Dimensionality, and Machine Learning29.3 High-Dimensional Regression29.4 p-norms29.5 Ridge Regression29.6 Statistical Properties of Ridge Regression29.7 Illustrating Ridge Regression29.8 Lasso29.9 Lasso Penalty Selection29.10 Lasso Computation29.11 Asymptotic Theory for the Lasso29.12 Approximate Sparsity29.13 Elastic Net29.14 Post-Lasso29.15 Regression Trees29.16 Bagging29.17 Random Forests29.18 Ensembling29.19 Lasso IV29.20 Double Selection Lasso29.21 Post-Regularization Lasso29.22 Double/Debiased Machine Learning29.23 Technical Proofs*29.24 ExercisesAppendixesA Matrix AlgebraA.1 NotationA.2 Complex MatricesA.3 Matrix AdditionA.4 Matrix MultiplicationA.5 TraceA.6 Rank and InverseA.7 Orthogonal and Orthonormal MatricesA.8 DeterminantA.9 EigenvaluesA.10 Positive Definite MatricesA.11 Idempotent MatricesA.12 Singular ValuesA.13 Matrix DecompositionsA.14 Generalized EigenvaluesA.15 Extrema of Quadratic FormsA.16 Cholesky DecompositionA.17 QR DecompositionA.18 Solving Linear SystemsA.19 Algorithmic Matrix InversionA.20 Matrix CalculusA.21 Kronecker Products and the Vec OperatorA.22 Vector NormsA.23 Matrix NormsB Useful-InequalitiesB.1-Inequalities for Real NumbersB.2-Inequalities for VectorsB.3-Inequalities for MatricesB.4-Probability InequalitiesB.5-Proofs*ReferencesIndex