Econometrics 

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Comment from the Stata technical groupBruce E. Hansen’s Econometrics provides one of the best introductions to this foundational subject in economics. The book presents the core of econometric theory and its methods intuitively and accessibly to readers of many backgrounds without sacrificing statistical or mathematical rigor. It also provides readers with realworld examples, researchlevel datasets (available in a supplementary website), and a brief description on how to fit many different statistical models using Stata. This book is the ideal companion for graduate students, researchers, and practitioners. Some prior background in multivariate calculus, linear algebra, probability, and mathematical statistics is required. However, previous knowledge in econometrics or economics is not. The relevant mathematical background in probability and statistics can be found in the accompanying textbook Probability and Statistics for Economists, by the same author. The broad scope of Econometrics is truly exceptional. It covers the fundamentals of linear regression, hypothesis testing, time series, paneldata methods, nonparametric econometrics, general method of moments (GMM), and modern machine learning. It also contains indepth appendices on matrix algebra and inequalities. For each topic, there is a clear exposition of the underlying theory, a practical guide on how to apply the methods covered, realworld examples, historical notes, and an appendix containing all the mathematical proofs. This book is ideal for both readers who want to better understand the theoretical aspects of econometrics and those more interested in its applications. Econometrics contains the core material typically taught in a firstyear PhD course in econometrics. Some of its content can also be used as part of a secondyear graduate course. The book features hundreds of theoretical and empirical exercises to help students learn by doing. The rigor, clarity, and simplicity of exposition make Econometrics a fundamental reference in the field.  
Table of contentsView table of contents >> Preface
Acknowledgments
Notation
1 Introduction
1.1 What Is Econometrics?
1.2 The Probability Approach to Econometrics 1.3 Econometric Terms 1.4 Observational Data 1.5 Standard Data Structures 1.6 Econometric Software 1.7 Replication 1.8 Data Files for Textbook 1.9 Reading the Book Part I Regression
2 Conditional Expectation and Projection
2.1 Introduction
2.2 The Distribution of Wages 2.3 Conditional Expectation 2.4 Logs and Percentages 2.5 Conditional Expectation Function 2.6 Continuous Variables 2.7 Law of Iterated Expectations 2.8 CEF Error 2.9 InterceptOnly Model 2.10 Regression Variance 2.11 Best Predictor 2.12 Conditional Variance 2.13 Homoskedasticity and Heteroskedasticity 2.14 Regression Derivative 2.15 Linear CEF 2.16 Linear CEF with Nonlinear Effects 2.17 Linear CEF with Dummy Variables 2.18 Best Linear Predictor 2.19 Illustrations of Best Linear Predictor 2.20 Linear Predictor Error Variance 2.21 Regression Coefficients 2.22 Regression Subvectors 2.23 Coefficient Decomposition 2.24 Omitted Variable Bias 2.25 Best Linear Approximation 2.26 Regression to the Mean 2.27 Reverse Regression 2.28 Limitations of the Best Linear Projection 2.29 Random Coefficient Model 2.30 Causal Effects 2.31 Existence and Uniqueness of the Conditional Expectation* 2.32 Identification* 2.33 Technical Proofs* 2.34 Exercises 3 The Algebra of Least Squares
3.1 Introduction
3.2 Samples 3.3 Moment Estimators 3.4 Least Squares Estimator 3.5 Solving for Least Squares with One Regressor 3.6 Solving for Least Squares with Multiple Regressors 3.7 Illustration 3.8 Least Squares Residuals 3.9 Demeaned Regressors 3.10 Model in Matrix Notation 3.11 Projection Matrix 3.12 Annihilator Matrix 3.13 Estimation of Error Variance 3.14 Analysis of Variance 3.15 Projections 3.16 Regression Components 3.17 Regression Components (Alternative Derivation)* 3.18 Residual Regression 3.19 Leverage Values 3.20 LeaveOneOut Regression 3.21 Influential Observations 3.22 CPS Dataset 3.23 Numerical Computation 3.24 Collinearity Errors 3.25 Programming 3.26 Exercises 4 Least Squares Regression
4.1 Introduction
4.2 Random Sampling 4.3 Sample Mean 4.4 Linear Regression Model 4.5 Expectation of Least Squares Estimator 4.6 Variance of Least Squares Estimator 4.7 Unconditional Moments 4.8 GaussMarkov Theorem 4.9 Generalized Least Squares 4.10 Residuals 4.11 Estimation of Error Variance 4.12 MeanSquared Forecast Error 4.13 Covariance Matrix Estimation under Homoskedasticity 4.14 Covariance Matrix Estimation under Heteroskedasticity 4.15 Standard Errors 4.16 Estimation with Sparse Dummy Variables 4.17 Computation 4.18 Measures of Fit 4.19 Empirical Example 4.20 Multicollinearity 4.21 Clustered Sampling 4.22 Inference with Clustered Samples 4.23 At What Level to Cluster? 4.24 Technical Proofs* 4.25 Exercises 5 Normal Regression
5.1 Introduction
5.2 The Normal Distribution 5.3 Multivariate Normal Distribution 5.4 Joint Normality and Linear Regression 5.5 Normal Regression Model 5.6 Distribution of OLS Coefficient Vector 5.7 Distribution of OLS Residual Vector 5.8 Distribution of Variance Estimator 5.9 tStatistic 5.10 Confidence Intervals for Regression Coefficients 5.11 Confidence Intervals for Error Variance 5.12 tTest 5.13 Likelihood Ratio Test 5.14 Information Bound for Normal Regression 5.15 Exercises Part II Large Sample Methods
6 A Review of Large Sample Asymptotics
6.1 Introduction
6.2 Modes of Convergence 6.3 Weak Law of Large Numbers 6.4 Central Limit Theorem 6.5 Continuous Mapping Theorem and Delta Method 6.6 Smooth Function Model 6.7 Stochastic Order Symbols 6.8 Convergence of Moments 7 Asymptotic Theory for Least Squares
7.1 Introduction
7.2 Consistency of Least Squares Estimator 7.3 Asymptotic Normality 7.4 Joint Distribution 7.5 Consistency of Error Variance Estimators 7.6 Homoskedastic Covariance Matrix Estimation 7.7 Heteroskedastic Covariance Matrix Estimation 7.8 Summary of Covariance Matrix Notation 7.9 Alternative Covariance Matrix Estimators* 7.10 Functions of Parameters 7.11 Asymptotic Standard Errors 7.12 tStatistic 7.13 Confidence Intervals 7.14 Regression Intervals 7.15 Forecast Intervals 7.16 Wald Statistic 7.17 Homoskedastic Wald Statistic 7.18 Confidence Regions 7.19 Edgeworth Expansion* 7.20 Uniformly Consistent Residuals* 7.21 Asymptotic Leverage* 7.22 Exercises 8 Restricted Estimation
8.1 Introduction
8.2 Constrained Least Squares 8.3 Exclusion Restriction 8.4 Finite Sample Properties 8.5 Minimum Distance 8.6 Asymptotic Distribution 8.7 Variance Estimation and Standard Errors 8.8 Efficient Minimum Distance Estimator 8.9 Exclusion Restriction Revisited 8.10 Variance and Standard Error Estimation 8.11 Hausman Equality 8.12 Example: Mankiw, Romer, and Weil (1992) 8.13 Misspecification 8.14 Nonlinear Constraints 8.15 Inequality Restrictions 8.16 Technical Proofs* 8.17 Exercises 9 Hypothesis Testing
9.1 Introduction
9.2 Hypotheses 9.3 Acceptance and Rejection 9.4 Type I Error 9.5 tTests 9.6 Type II Error and Power 9.7 Statistical Significance 9.8 pValues 9.9 tRatios and the Abuse of Testing 9.10 Wald Tests 9.11 Homoskedastic Wald Tests 9.12 CriterionBased Tests 9.13 Minimum Distance Tests 9.14 Minimum Distance Tests under Homoskedasticity 9.15 F Tests 9.16 Hausman Tests 9.17 Score Tests 9.18 Problems with Tests of Nonlinear Hypotheses 9.19 Monte Carlo Simulation 9.20 Confidence Intervals by Test Inversion 9.21 Multiple Tests and Bonferroni Corrections 9.22 Power and Test Consistency 9.23 Asymptotic Local Power 9.24 Asymptotic Local Power, Vector Case 9.25 Exercises 10 Resampling Methods
10.1 Introduction
10.2 Example 10.3 Jackknife Estimation of Variance 10.4 Example 10.5 Jackknife for Clustered Observations 10.6 The Bootstrap Algorithm 10.7 Bootstrap Variance and Standard Errors 10.8 Percentile Interval 10.9 The Bootstrap Distribution 10.10 The Distribution of the Bootstrap Observations 10.11 The Distribution of the Bootstrap Sample Mean 10.12 Bootstrap Asymptotics 10.13 Consistency of the Bootstrap Estimate of Variance 10.14 Trimmed Estimator of Bootstrap Variance 10.15 Unreliability of Untrimmed Bootstrap Standard Errors 10.16 Consistency of the Percentile Interval 10.17 BiasCorrected Percentile Interval 10.18 BC_{a} Percentile Interval 10.19 Percentilet Interval 10.20 Percentilet Asymptotic Refinement 10.21 Bootstrap Hypothesis Tests 10.22 WaldType Bootstrap Tests 10.23 CriterionBased Bootstrap Tests 10.24 Parametric Bootstrap 10.25 How Many Bootstrap Replications? 10.26 Setting the Bootstrap Seed 10.27 Bootstrap Regression 10.28 Bootstrap Regression Asymptotic Theory 10.29 Wild Bootstrap 10.30 Bootstrap for Clustered Observations 10.31 Technical Proofs* 10.32 Exercises Part III Multiple Equation Models
11 Multivariate Regression
11.1 Introduction
11.2 Regression Systems 11.3 Least Squares Estimator 11.4 Expectation and Variance of Systems Least Squares 11.5 Asymptotic Distribution 11.6 Covariance Matrix Estimation 11.7 Seemingly Unrelated Regression 11.8 Equivalence of SUR and Least Squares 11.9 Maximum Likelihood Estimator 11.10 Restricted Estimation 11.11 Reduced Rank Regression 11.12 Principal Component Analysis 11.13 Factor Models 11.14 Approximate Factor Models 11.15 Factor Models with Additional Regressors 11.16 FactorAugmented Regression 11.17 Multivariate Normal* 11.18 Exercises 12 Instrumental Variables
12.1 Introduction
12.2 Overview 12.3 Examples 12.4 Endogenous Regressors 12.5 Instruments 12.6 Example: College Proximity 12.7 Reduced Form 12.8 Identification 12.9 Instrumental Variables Estimator 12.10 Demeaned Representation 12.11 Wald Estimator 12.12 TwoStage Least Squares 12.13 Limited Information Maximum Likelihood 12.14 SplitSample IV and JIVE 12.15 Consistency of 2SLS 12.16 Asymptotic Distribution of 2SLS 12.17 Determinants of 2SLS Variance 12.18 Covariance Matrix Estimation 12.19 LIML Asymptotic Distribution 12.20 Functions of Parameters 12.21 Hypothesis Tests 12.22 Finite Sample Theory 12.23 Bootstrap for 2SLS 12.24 The Peril of Bootstrap 2SLS Standard Errors 12.25 Clustered Dependence 12.26 Generated Regressors 12.27 Regression with Expectation Errors 12.28 Control Function Regression 12.29 Endogeneity Tests 12.30 Subset Endogeneity Tests 12.31 Overidentification Tests 12.32 Subset Overidentification Tests 12.33 Bootstrap Overidentification Tests 12.34 Local Average Treatment Effects 12.35 Identification Failure 12.36 Weak Instruments 12.37 Many Instruments 12.38 Testing for Weak Instruments 12.39 Weak Instruments with k_{2} > 1 12.40 Example: Acemoglu, Johnson, and Robinson (2001) 12.41 Example: Angrist and Krueger (1991) 12.42 Programming 12.43 Exercises 13 Generalized Method of Moments
13.1 Introduction
13.2 Moment Equation Models 13.3 Method of Moments Estimators 13.4 Overidentified Moment Equations 13.5 Linear Moment Models 13.6 GMM Estimator 13.7 Distribution of GMM Estimator 13.8 Efficient GMM 13.9 Efficient GMM versus 2SLS 13.10 Estimation of the Efficient Weight Matrix 13.11 Iterated GMM 13.12 Covariance Matrix Estimation 13.13 Clustered Dependence 13.14 Wald Test 13.15 Restricted GMM 13.16 Nonlinear Restricted GMM 13.17 Constrained Regression 13.18 Multivariate Regression 13.19 Distance Test 13.20 Continuously Updated GMM 13.21 Overidentification Test 13.22 Subset Overidentification Tests 13.23 Endogeneity Test 13.24 Subset Endogeneity Test 13.25 Nonlinear GMM 13.26 Bootstrap for GMM 13.27 Conditional Moment Equation Models 13.28 Technical Proofs* 13.29 Exercises Part IV Dependent and Panel Data
14 Time Series
14.1 Introduction
14.2 Examples 14.3 Differences and Growth Rates 14.4 Stationarity 14.5 Transformations of Stationary Processes 14.6 Convergent Series 14.7 Ergodicity 14.8 Ergodic Theorem 14.9 Conditioning on Information Sets 14.10 Martingale Difference Sequences 14.11 CLT for Martingale Differences 14.12 Mixing 14.13 CLT for Correlated Observations 14.14 Linear Projection 14.15 White Noise 14.16 The Wold Decomposition 14.17 Lag Operator 14.18 Autoregressive Wold Representation 14.19 Linear Models 14.20 Moving Average Process 14.21 InfiniteOrder Moving Average Process 14.22 FirstOrder Autoregressive Process 14.23 Unit Root and Explosive AR(1) Processes 14.24 SecondOrder Autoregressive Process 14.25 AR(p) Process 14.26 Impulse Response Function 14.27 ARMA and ARIMA Processes 14.28 Mixing Properties of Linear Processes 14.29 Identification 14.30 Estimation of Autoregressive Models 14.31 Asymptotic Distribution of Least Squares Estimator 14.32 Distribution under Homoskedasticity 14.33 Asymptotic Distribution under General Dependence 14.34 Covariance Matrix Estimation 14.35 Covariance Matrix Estimation under General Dependence 14.36 Testing the Hypothesis of No Serial Correlation 14.37 Testing for Omitted Serial Correlation 14.38 Model Selection 14.39 Illustrations 14.40 Time Series Regression Models 14.41 Static, Distributed Lag, and Autoregressive Distributed Lag Models 14.42 Time Trends 14.43 Illustration 14.44 Granger Causality 14.45 Testing for Serial Correlation in Regression Models 14.46 Bootstrap for Time Series 14.47 Technical Proofs* 14.48 Exercises 15 Multivariate Time Series
15.1 Introduction
15.2 Multiple Equation Time Series Models 15.3 Linear Projection 15.4 Multivariate Wold Decomposition 15.5 Impulse Response 15.6 VAR(1) Model 15.7 VAR(p) Model 15.8 Regression Notation 15.9 Estimation 15.10 Asymptotic Distribution 15.11 Covariance Matrix Estimation 15.12 Selection of Lag Length in a VAR 15.13 Illustration 15.14 Predictive Regressions 15.15 Impulse Response Estimation 15.16 Local Projection Estimator 15.17 Regression on Residuals 15.18 Orthogonalized Shocks 15.19 Orthogonalized Impulse Response Function 15.20 Orthogonalized Impulse Response Estimation 15.21 Illustration 15.22 Forecast Error Decomposition 15.23 Identification of Recursive VARs 15.24 Oil Price Shocks 15.25 Structural VARs 15.26 Identification of Structural VARs 15.27 LongRun Restrictions 15.28 Blanchard and Quah (1989) Illustration 15.29 External Instruments 15.30 Dynamic Factor Models 15.31 Technical Proofs* 15.32 Exercises 16 Nonstationary Time Series
16.1 Introduction
16.2 Partial Sum Process and Functional Convergence 16.3 BeveridgeNelson Decomposition 16.4 Functional CLT 16.5 Orders of Integration 16.6 Means, Local Means, and Trends 16.7 Demeaning and Detrending 16.8 Stochastic Integrals 16.9 Estimation of an AR(1) 16.10 AR(1) Estimation with an Intercept 16.11 Sample Covariances of Integrated and Stationary Processes 16.12 AR(p) Models with a Unit Root 16.13 Testing for a Unit Root 16.14 KPSS Stationarity Test 16.15 Spurious Regression 16.16 NonStationary VARs 16.17 Cointegration 16.18 Role of Intercept and Trend 16.19 Cointegrating Regression 16.20 VECM Estimation 16.21 Testing for Cointegration in a VECM 16.22 Technical Proofs* 16.23 Exercises 17 Panel Data
17.1 Introduction
17.2 Time Indexing and Unbalanced Panels 17.3 Notation 17.4 Pooled Regression 17.5 OneWay Error Component Model 17.6 Random Effects 17.7 Fixed Effects Model 17.8 Within Transformation 17.9 Fixed Effects Estimator 17.10 Differenced Estimator 17.11 Dummy Variables Regression 17.12 Fixed Effects Covariance Matrix Estimation 17.13 Fixed Effects Estimation in Stata 17.14 Between Estimator 17.15 Feasible GLS 17.16 Intercept in Fixed Effects Regression 17.17 Estimation of Fixed Effects 17.18 GMM Interpretation of Fixed Effects 17.19 Identification in the Fixed Effects Model 17.20 Asymptotic Distribution of Fixed Effects Estimator 17.21 Asymptotic Distribution for Unbalanced Panels 17.22 HeteroskedasticityRobust Covariance Matrix Estimation 17.23 HeteroskedasticityRobust Estimation—Unbalanced Case 17.24 Hausman Test for Random vs. Fixed Effects 17.25 Random Effects or Fixed Effects? 17.26 Time Trends 17.27 TwoWay Error Components 17.28 Instrumental Variables 17.29 Identification with Instrumental Variables 17.30 Asymptotic Distribution of Fixed Effects 2SLS Estimator 17.31 Linear GMM 17.32 Estimation with TimeInvariant Regressors 17.33 HausmanTaylor Model 17.34 Jackknife Covariance Matrix Estimation 17.35 Panel Bootstrap 17.36 Dynamic Panel Models 17.37 The Bias of Fixed Effects Estimation 17.38 AndersonHsiao Estimator 17.39 ArellanoBond Estimator 17.40 Weak Instruments 17.41 Dynamic Panels with Predetermined Regressors 17.42 BlundellBond Estimator 17.43 Forward Orthogonal Transformation 17.44 Empirical Illustration 17.45 Exercises 18 Difference in Differences
18.1 Introduction
18.2 Minimum Wage in New Jersey 18.3 Identification 18.4 Multiple Units 18.5 Do Police Reduce Crime? 18.6 Trend Specification 18.7 Do Blue Laws Affect Liquor Sales? 18.8 Check Your Code: Does Abortion Impact Crime? 18.9 Inference 18.10 Exercises Part V Nonparametric Methods
19 Nonparametric Regression
19.1 Introduction
19.2 Binned Means Estimator 19.3 Kernel Regression 19.4 Local Linear Estimator 19.5 Local Polynomial Estimator 19.6 Asymptotic Bias 19.7 Asymptotic Variance 19.8 AIMSE 19.9 Reference Bandwidth 19.10 Estimation at a Boundary 19.11 Nonparametric Residuals and Prediction Errors 19.12 CrossValidation Bandwidth Selection 19.13 Asymptotic Distribution 19.14 Undersmoothing 19.15 Conditional Variance Estimation 19.16 Variance Estimation and Standard Errors 19.17 Confidence Bands 19.18 The Local Nature of Kernel Regression 19.19 Application to Wage Regression 19.20 Clustered Observations 19.21 Application to Test Scores 19.22 Multiple Regressors 19.23 Curse of Dimensionality 19.24 Partially Linear Regression 19.25 Computation 19.26 Technical Proofs* 19.27 Exercises 20 Series Regression
20.1 Introduction
20.2 Polynomial Regression 20.3 Illustrating Polynomial Regression 20.4 Orthogonal Polynomials 20.5 Splines 20.6 Illustrating Spline Regression 20.7 The Global/Local Nature of Series Regression 20.8 StoneWeierstrass and Jackson Approximation Theory 20.9 Regressor Bounds 20.10 Matrix Convergence 20.11 Consistent Estimation 20.12 Convergence Rate 20.13 Asymptotic Normality 20.14 Regression Estimation 20.15 Undersmoothing 20.16 Residuals and Regression Fit 20.17 CrossValidation Model Selection 20.18 Variance and Standard Error Estimation 20.19 Clustered Observations 20.20 Confidence Bands 20.21 Uniform Approximations 20.22 Partially Linear Model 20.23 Panel Fixed Effects 20.24 Multiple Regressors 20.25 Additively Separable Models 20.26 Nonparametric Instrumental Variables Regression 20.27 NPIV Identification 20.28 NPIV Convergence Rate 20.29 Nonparametric vs. Parametric Identification 20.30 Example: Angrist and Lavy (1999) 20.31 Technical Proofs* 20.32 Exercises 21 Regression Discontinuity
21.1 Introduction
21.2 Sharp Regression Discontinuity 21.3 Identification 21.4 Estimation 21.5 Inference 21.6 Bandwidth Selection 21.7 RDD with Covariates 21.8 A Simple RDD Estimator 21.9 Density Discontinuity Test 21.10 Fuzzy Regression Discontinuity 21.11 Estimation of FRD 21.12 Exercises Part VI Nonlinear Methods
22 MEstimators
22.1 Introduction
22.2 Examples 22.3 Identification and Estimation 22.4 Consistency 22.5 Uniform Law of Large Numbers 22.6 Asymptotic Distribution 22.7 Asymptotic Distribution under Broader Conditions* 22.8 Covariance Matrix Estimation 22.9 Technical Proofs* 22.10 Exercises 23 Nonlinear Least Squares
23.1 Introduction
23.2 Identification 23.3 Estimation 23.4 Asymptotic Distribution 23.5 Covariance Matrix Estimation 23.6 Panel Data 23.7 Threshold Models 23.8 Testing for Nonlinear Components 23.9 Computation 23.10 Technical Proofs* 23.11 Exercises 24 Quantile Regression
24.1 Introduction
24.2 Median Regression 24.3 Least Absolute Deviations 24.4 Quantile Regression 24.5 Example Quantile Shapes 24.6 Estimation 24.7 Asymptotic Distribution 24.8 Covariance Matrix Estimation 24.9 Clustered Dependence 24.10 Quantile Crossings 24.11 Quantile Causal Effects 24.12 Random Coefficient Representation 24.13 Nonparametric Quantile Regression 24.14 Panel Data 24.15 IV Quantile Regression 24.16 Technical Proofs* 24.17 Exercises 25 Binary Choice
25.1 Introduction
25.2 Binary Choice Models 25.3 Models for the Response Probability 25.4 Latent Variable Interpretation 25.5 Likelihood 25.6 PseudoTrue Values 25.7 Asymptotic Distribution 25.8 Covariance Matrix Estimation 25.9 Marginal Effects 25.10 Application 25.11 Semiparametric Binary Choice 25.12 IV Probit 25.13 Binary Panel Data 25.14 Technical Proofs* 25.15 Exercises 26 Multiple Choice
26.1 Introduction
26.2 Multinomial Response 26.3 Multinomial Logit 26.4 Conditional Logit 26.5 Independence of Irrelevant Alternatives 26.6 Nested Logit 26.7 Mixed Logit 26.8 Simple Multinomial Probit 26.9 General Multinomial Probit 26.10 Ordered Response 26.11 Count Data 26.12 BLP Demand Model 26.13 Technical Proofs* 26.14 Exercises 27 Censoring and Selection
27.1 Introduction
27.2 Censoring 27.3 Censored Regression Functions 27.4 The Bias of Least Squares Estimation 27.5 Tobit Estimator 27.6 Identification in Tobit Regression 27.7 CLAD and CQR Estimators 27.8 Illustrating Censored Regression 27.9 Sample Selection Bias 27.10 Heckman’s Model 27.11 Nonparametric Selection 27.12 Panel Data 27.13 Exercises 28 Model Selection, Stein Shrinkage, and Model Averaging
28.1 Introduction
28.2 Model Selection 28.3 Bayesian Information Criterion 28.4 Akaike Information Criterion for Regression 28.5 Akaike Information Criterion for Likelihood 28.6 Mallows Criterion 28.7 HoldOut Criterion 28.8 CrossValidation Criterion 28.9 KFold CrossValidation 28.10 Many Selection Criteria Are Similar 28.11 Relation with Likelihood Ratio Testing 28.12 Consistent Selection 28.13 Asymptotic Selection Optimality 28.14 Focused Information Criterion 28.15 Best Subset and Stepwise Regression 28.16 The MSE of Model Selection Estimators 28.17 Inference after Model Selection 28.18 Empirical Illustration 28.19 Shrinkage Methods 28.20 JamesStein Shrinkage Estimator 28.21 Interpretation of the Stein Effect 28.22 Positive Part Estimator 28.23 Shrinkage Toward Restrictions 28.24 Group JamesStein 28.25 Empirical Illustrations 28.26 Model Averaging 28.27 Smoothed BIC and AIC 28.28 Mallows Model Averaging 28.29 Jackknife (CV) Model Averaging 28.30 GrangerRamanathan Averaging 28.31 Empirical Illustration 28.32 Technical Proofs* 28.33 Exercises 29 Machine Learning
29.1 Introduction
29.2 Big Data, High Dimensionality, and Machine Learning 29.3 HighDimensional Regression 29.4 pnorms 29.5 Ridge Regression 29.6 Statistical Properties of Ridge Regression 29.7 Illustrating Ridge Regression 29.8 Lasso 29.9 Lasso Penalty Selection 29.10 Lasso Computation 29.11 Asymptotic Theory for the Lasso 29.12 Approximate Sparsity 29.13 Elastic Net 29.14 PostLasso 29.15 Regression Trees 29.16 Bagging 29.17 Random Forests 29.18 Ensembling 29.19 Lasso IV 29.20 Double Selection Lasso 29.21 PostRegularization Lasso 29.22 Double/Debiased Machine Learning 29.23 Technical Proofs* 29.24 Exercises Appendixes
A Matrix Algebra
A.1 Notation
A.2 Complex Matrices A.3 Matrix Addition A.4 Matrix Multiplication A.5 Trace A.6 Rank and Inverse A.7 Orthogonal and Orthonormal Matrices A.8 Determinant A.9 Eigenvalues A.10 Positive Definite Matrices A.11 Idempotent Matrices A.12 Singular Values A.13 Matrix Decompositions A.14 Generalized Eigenvalues A.15 Extrema of Quadratic Forms A.16 Cholesky Decomposition A.17 QR Decomposition A.18 Solving Linear Systems A.19 Algorithmic Matrix Inversion A.20 Matrix Calculus A.21 Kronecker Products and the Vec Operator A.22 Vector Norms A.23 Matrix Norms B UsefulInequalities
B.1 Inequalities for Real Numbers
B.2 Inequalities for Vectors B.3 Inequalities for Matrices B.4 Probability Inequalities B.5 Proofs* References
Index

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