3.2.2 Application: An Investment Equation 283.2.3 Algebraic Aspects of the Least Squares Solution 30 3.5.1 The Adjusted R-Squared and a Measure of Fit 42 3.5.2 R-Squared and the Constant
Trang 4Editorial Director: Sally Yagan Cover Image: Ralf Hiemisch/Getty Images
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Library of Congress Cataloging-in-Publication Data
Trang 5The Economics of Women,
Men and Work
Russian and Soviet Economic
Performance and Structure
Women and the Economy:
Family, Work, and Pay
Macroeconomics: Policy and Practice*
Trang 6Part II Generalized Regression Model and Equation Systems
Part III Estimation Methodology
Part IV Cross Sections, Panel Data, and Microeconometrics
iv
Trang 7Part V Time Series and Macroeconometrics
Part VI Appendices
Trang 82.3.1 Linearity of the Regression Model 15
2.3.4 Spherical Disturbances 21 2.3.5 Data Generating Process for the Regressors 23
CHAPTER 3 Least Squares 26
3.2.1 The Least Squares Coefficient Vector 27
vi
Trang 93.2.2 Application: An Investment Equation 28
3.2.3 Algebraic Aspects of the Least Squares Solution 30
3.5.1 The Adjusted R-Squared and a Measure of Fit 42
3.5.2 R-Squared and the Constant Term in the Model 44
CHAPTER 4 The Least Squares Estimator 51
4.2.1 The Population Orthogonality Conditions 52
4.2.2 Minimum Mean Squared Error Predictor 53
4.2.3 Minimum Variance Linear Unbiased Estimation 54
4.3.1 Unbiased Estimation 55
4.3.2 Bias Caused by Omission of Relevant Variables 56
4.3.3 Inclusion of Irrelevant Variables 58
4.3.4 The Variance of the Least Squares Estimator 58
4.3.6 The Implications of Stochastic Regressors 60
4.3.7 Estimating the Variance of the Least Squares Estimator 61
4.4.1 Consistency of the Least Squares Estimator of β 63
4.4.2 Asymptotic Normality of the Least Squares Estimator 65 4.4.3 Consistency of s2and the Estimator of Asy Var[b] 67
4.4.4 Asymptotic Distribution of a Function of b: The Delta
4.4.5 Asymptotic Efficiency 69
4.4.6 Maximum Likelihood Estimation 73
4.5.1 Forming a Confidence Interval for a Coefficient 76
4.5.2 Confidence Intervals Based on Large Samples 78
4.5.3 Confidence Interval for a Linear Combination of Coefficients:
The Oaxaca Decomposition 79
4.6.1 Prediction Intervals 81
4.6.2 Predicting y When the Regression Model Describes Log y 81
Trang 104.6.3 Prediction Interval for y When the Regression Model Describes
Log y 83
4.7.1 Multicollinearity 89 4.7.2 Pretest Estimation 91
4.7.4 Missing Values and Data Imputation 94
4.7.6 Outliers and Influential Observations 99
CHAPTER 5 Hypothesis Tests and Model Selection 108
5.2.1 Restrictions and Hypotheses 109
5.4.1 Testing a Hypothesis about a Coefficient 115 5.4.2 The F Statistic and the Least Squares Discrepancy 117
5.5.1 The Restricted Least Squares Estimator 121 5.5.2 The Loss of Fit from Restricted Least Squares 122 5.5.3 Testing the Significance of the Regression 126 5.5.4 Solving Out the Restrictions and a Caution about
Using R2 126
5.8.1 Testing Nonnested Hypotheses 134
5.8.3 Comprehensive Approach—The J Test 136
5.10.1 Model Selection Criteria 139 5.10.2 Model Selection 140
5.10.3 Classical Model Selection 140 5.10.4 Bayesian Model Averaging 141
Trang 11CHAPTER 6 Functional Form and Structural Change 149
6.2.1 Binary Variables in Regression 149
6.2.2 Several Categories 152
6.2.3 Several Groupings 152
6.2.4 Threshold Effects and Categorical Variables 154
6.2.5 Treatment Effects and Differences in Differences
Regression 155
6.3.1 Piecewise Linear Regression 158
6.3.2 Functional Forms 160
6.3.3 Interaction Effects 161
6.3.4 Identifying Nonlinearity 162
6.3.5 Intrinsically Linear Models 165
6.4.1 Different Parameter Vectors 168
6.4.2 Insufficient Observations 169
6.4.3 Change in a Subset of Coefficients 170
6.4.4 Tests of Structural Break with Unequal Variances 171
6.4.5 Predictive Test of Model Stability 174
CHAPTER 7 Nonlinear, Semiparametric, and Nonparametric
Regression Models 181
7.2.1 Assumptions of the Nonlinear Regression Model 182
7.2.2 The Nonlinear Least Squares Estimator 184
7.2.3 Large Sample Properties of the Nonlinear Least Squares
Estimator 186 7.2.4 Hypothesis Testing and Parametric Restrictions 189
7.2.6 Computing the Nonlinear Least Squares Estimator 200
7.3.1 Least Absolute Deviations Estimation 203
7.3.2 Quantile Regression Models 207
CHAPTER 8 Endogeneity and Instrumental Variable Estimation 219
Trang 128.3 Estimation 224
8.3.2 The Instrumental Variables Estimator 225 8.3.3 Motivating the Instrumental Variables Estimator 227 8.3.4 Two-Stage Least Squares 230
8.4.1 The Hausman and Wu Specification Tests 234 8.4.2 A Test for Overidentification 238
8.5.1 Least Squares Attenuation 240 8.5.2 Instrumental Variables Estimation 242 8.5.3 Proxy Variables 242
PART II Generalized Regression Model and Equation SystemsCHAPTER 9 The Generalized Regression Model and Heteroscedasticity 257
9.3.1 Generalized Least Squares (GLS) 264 9.3.2 Feasible Generalized Least Squares (FGLS) 266
9.4.1 Ordinary Least Squares Estimation 269 9.4.2 Inefficiency of Ordinary Least Squares 270 9.4.3 The Estimated Covariance Matrix of b 270 9.4.4 Estimating the Appropriate Covariance Matrix for Ordinary
Least Squares 272
9.5.1 White’s General Test 275 9.5.2 The Breusch–Pagan/Godfrey LM Test 276
9.6.1 Weighted Least Squares with Known 278
9.6.2 Estimation When Contains Unknown Parameters 279
Trang 139.7 Applications 280
9.7.1 Multiplicative Heteroscedasticity 280
9.7.2 Groupwise Heteroscedasticity 282
CHAPTER 10 Systems of Equations 290
10.2.1 Generalized Least Squares 293
10.2.2 Seemingly Unrelated Regressions with Identical Regressors 295 10.2.3 Feasible Generalized Least Squares 296
10.2.4 Testing Hypotheses 296
10.2.5 A Specification Test for the SUR Model 297
10.2.6 The Pooled Model 299
10.5.1 Cobb–Douglas Cost Function 307
10.5.2 Flexible Functional Forms: The Translog Cost Function 310
10.6.1 Systems of Equations 315
10.6.2 A General Notation for Linear Simultaneous Equations
Models 318 10.6.3 The Problem of Identification 321
10.6.4 Single Equation Estimation and Inference 326
10.6.5 System Methods of Estimation 329
10.6.6 Testing in the Presence of Weak Instruments 334
CHAPTER 11 Models for Panel Data 343
11.2.1 General Modeling Framework for Analyzing Panel Data 345 11.2.2 Model Structures 346
11.2.3 Extensions 347
11.2.4 Balanced and Unbalanced Panels 348
11.2.5 Well-Behaved Panel Data 348
11.3.1 Least Squares Estimation of the Pooled Model 349
11.3.2 Robust Covariance Matrix Estimation 350
11.3.3 Clustering and Stratification 352
11.3.4 Robust Estimation Using Group Means 354
Trang 1411.3.5 Estimation with First Differences 355 11.3.6 The Within- and Between-Groups Estimators 357
11.4.1 Least Squares Estimation 360 11.4.2 Small T Asymptotics 362 11.4.3 Testing the Significance of the Group Effects 363 11.4.4 Fixed Time and Group Effects 363
11.4.5 Time-Invariant Variables and Fixed Effects Vector
Decomposition 364
11.5.1 Least Squares Estimation 372 11.5.2 Generalized Least Squares 373 11.5.3 Feasible Generalized Least Squares When Is Unknown 374
11.5.4 Testing for Random Effects 376 11.5.5 Hausman’s Specification Test for the Random Effects
11.6.1 Robust Estimation of the Fixed Effects Model 385 11.6.2 Heteroscedasticity in the Random Effects Model 387 11.6.3 Autocorrelation in Panel Data Models 388
11.6.4 Cluster (and Panel) Robust Covariance Matrices for Fixed and
Random Effects Estimators 388
11.9.1 A Robust Covariance Matrix for Nonlinear Least Squares 411 11.9.2 Fixed Effects 412
Models 421
Trang 15PART III Estimation Methodology
CHAPTER 12 Estimation Frameworks in Econometrics 432
12.2.1 Classical Likelihood-Based Estimation 434
12.2.2 Modeling Joint Distributions with Copula Functions 436
12.3.1 GMM Estimation in Econometrics 439
12.3.2 Maximum Empirical Likelihood Estimation 440
12.3.3 Least Absolute Deviations Estimation and Quantile
Regression 441 12.3.4 Kernel Density Methods 442
12.3.5 Comparing Parametric and Semiparametric Analyses 443
12.5.5 Testing Hypotheses 453
CHAPTER 13 Minimum Distance Estimation and the Generalized
Method of Moments 455
13.2.1 Random Sampling and Estimating the Parameters of
Distributions 457 13.2.2 Asymptotic Properties of the Method of Moments
Estimator 461
13.4.1 Estimation Based on Orthogonality Conditions 468
13.4.2 Generalizing the Method of Moments 470
13.4.3 Properties of the GMM Estimator 474
13.5.1 Testing the Validity of the Moment Restrictions 479
13.5.2 GMM Counterparts to the WALD, LM, and LR
Tests 480
Trang 1613.6 GMM Estimation of Econometric Models 482
13.6.1 Single-Equation Linear Models 482 13.6.2 Single-Equation Nonlinear Models 488 13.6.3 Seemingly Unrelated Regression Models 491 13.6.4 Simultaneous Equations Models with Heteroscedasticity 493 13.6.5 GMM Estimation of Dynamic Panel Data Models 496
CHAPTER 14 Maximum Likelihood Estimation 509
14.4.1 Regularity Conditions 514 14.4.2 Properties of Regular Densities 515 14.4.3 The Likelihood Equation 517 14.4.4 The Information Matrix Equality 517 14.4.5 Asymptotic Properties of the Maximum Likelihood
Estimator 517 14.4.5.a Consistency 518 14.4.5.b Asymptotic Normality 519 14.4.5.c Asymptotic Efficiency 520 14.4.5.d Invariance 521
14.4.5.e Conclusion 521 14.4.6 Estimating the Asymptotic Variance of the Maximum
Likelihood Estimator 521
14.6.1 The Likelihood Ratio Test 526
14.6.3 The Lagrange Multiplier Test 529 14.6.4 An Application of the Likelihood-Based Test Procedures 531 14.6.5 Comparing Models and Computing Model Fit 533
14.6.6 Vuong’s Test and the Kullback–Leibler Information
Criterion 534
14.8.1 Maximum Likelihood and GMM Estimation 543 14.8.2 Maximum Likelihood and M Estimation 543 14.8.3 Sandwich Estimators 545
14.8.4 Cluster Estimators 546
Trang 1714.9 Applications of Maximum Likelihood Estimation 548
14.9.1 The Normal Linear Regression Model 548
14.9.2 The Generalized Regression Model 552
14.9.2.a Multiplicative Heteroscedasticity 554 14.9.2.b Autocorrelation 557
14.9.3 Seemingly Unrelated Regression Models 560
14.9.3.a The Pooled Model 560 14.9.3.b The SUR Model 562 14.9.3.c Exclusion Restrictions 562 14.9.4 Simultaneous Equations Models 567
14.9.5 Maximum Likelihood Estimation of Nonlinear Regression
Models 568 14.9.6 Panel Data Applications 573
14.9.6.a ML Estimation of the Linear Random Effects
14.9.6.b Nested Random Effects 576 14.9.6.c Random Effects in Nonlinear Models: MLE Using
Quadrature 580 14.9.6.d Fixed Effects in Nonlinear Models: Full MLE 584
14.10.1 A Finite Mixture Model 589
14.10.2 Measured and Unmeasured Heterogeneity 591
14.10.3 Predicting Class Membership 591
14.10.4 A Conditional Latent Class Model 592
14.10.5 Determining the Number of Classes 594
14.10.6 A Panel Data Application 595
CHAPTER 15 Simulation-Based Estimation and Inference and Random Parameter
Models 603
15.2.1 Generating Pseudo-Random Numbers 605
15.2.2 Sampling from a Standard Uniform Population 606
15.2.3 Sampling from Continuous Distributions 607
15.2.4 Sampling from a Multivariate Normal Population 608
15.2.5 Sampling from Discrete Populations 608
15.5.1 A Monte Carlo Study: Behavior of a Test Statistic 617
15.5.2 A Monte Carlo Study: The Incidental Parameters Problem 619
15.6.1 Random Effects in a Nonlinear Model 621
Trang 1815.6.2 Monte Carlo Integration 623
15.6.2.a Halton Sequences and Random Draws for
Simulation-Based Integration 625 15.6.2.b Computing Multivariate Normal Probabilities Using
the GHK Simulator 627 15.6.3 Simulation-Based Estimation of Random Effects Models 629
CHAPTER 16 Bayesian Estimation and Inference 655
16.3.1 Analysis with a Noninformative Prior 659 16.3.2 Estimation with an Informative Prior Density 661
16.4.1 Point Estimation 664 16.4.2 Interval Estimation 665 16.4.3 Hypothesis Testing 666 16.4.4 Large-Sample Results 668
PART IV Cross Sections, Panel Data, and MicroeconometricsCHAPTER 17 Discrete Choice 681
17.2.1 Random Utility Models for Individual Choice 684 17.2.2 A Latent Regression Model 686
17.2.3 Functional Form and Regression 687
17.3.1 Robust Covariance Matrix Estimation 692 17.3.2 Marginal Effects and Average Partial Effects 693
Trang 1917.3.2.a Average Partial Effects 696 17.3.2.b Interaction Effects 699 17.3.3 Measuring Goodness of Fit 701
17.3.4 Hypothesis Tests 703
17.3.5 Endogenous Right-Hand-Side Variables in Binary Choice
Models 706 17.3.6 Endogenous Choice-Based Sampling 710
17.3.7 Specification Analysis 711
17.3.7.a Omitted Variables 713 17.3.7.b Heteroscedasticity 714
17.4.1 The Pooled Estimator 717
17.4.2 Random Effects Models 718
17.4.3 Fixed Effects Models 721
17.4.4 A Conditional Fixed Effects Estimator 722
17.4.5 Mundlak’s Approach, Variable Addition, and Bias
Reduction 727 17.4.6 Dynamic Binary Choice Models 729
17.4.7 A Semiparametric Model for Individual Heterogeneity 731 17.4.8 Modeling Parameter Heterogeneity 733
17.4.9 Nonresponse, Attrition, and Inverse Probability Weighting 734
17.5.1 Maximum Likelihood Estimation 739
17.5.2 Testing for Zero Correlation 742
17.5.3 Partial Effects 742
17.5.4 A Panel Data Model for Bivariate Binary Response 744 17.5.5 Endogenous Binary Variable in a Recursive Bivariate Probit
17.5.6 Endogenous Sampling in a Binary Choice Model 749
17.5.7 A Multivariate Probit Model 752
CHAPTER 18 Discrete Choices and Event Counts 760
18.2.1 Random Utility Basis of the Multinomial Logit Model 761 18.2.2 The Multinomial Logit Model 763
18.2.3 The Conditional Logit Model 766
18.2.4 The Independence from Irrelevant Alternatives
Assumption 767 18.2.5 Nested Logit Models 768
18.2.6 The Multinomial Probit Model 770
18.2.7 The Mixed Logit Model 771
18.2.8 A Generalized Mixed Logit Model 772
Trang 2018.2.9 Application: Conditional Logit Model for Travel Mode
Choice 773 18.2.10 Estimating Willingness to Pay 779 18.2.11 Panel Data and Stated Choice Experiments 781 18.2.12 Aggregate Market Share Data—The BLP Random Parameters
18.3.1 The Ordered Probit Model 787 18.3.2 A Specification Test for the Ordered Choice Model 791 18.3.3 Bivariate Ordered Probit Models 792
18.3.4 Panel Data Applications 794
18.3.4.a Ordered Probit Models with Fixed Effects 794 18.3.4.b Ordered Probit Models with Random Effects 795 18.3.5 Extensions of the Ordered Probit Model 798
18.3.5.a Threshold Models—Generalized Ordered Choice
Models 799 18.3.5.b Thresholds and Heterogeneity—Anchoring
Vignettes 800
18.4.1 The Poisson Regression Model 803 18.4.2 Measuring Goodness of Fit 804 18.4.3 Testing for Overdispersion 805 18.4.4 Heterogeneity and the Negative Binomial Regression
18.4.5 Functional Forms for Count Data Models 807 18.4.6 Truncation and Censoring in Models for Counts 810 18.4.7 Panel Data Models 815
18.4.7.a Robust Covariance Matrices for Pooled
Estimators 816 18.4.7.b Fixed Effects 817 18.4.7.c Random Effects 818 18.4.8 Two-Part Models: Zero Inflation and Hurdle Models 821 18.4.9 Endogenous Variables and Endogenous Participation 826
CHAPTER 19 Limited Dependent Variables—Truncation, Censoring, and Sample
Selection 833
19.2.1 Truncated Distributions 834 19.2.2 Moments of Truncated Distributions 835 19.2.3 The Truncated Regression Model 837 19.2.4 The Stochastic Frontier Model 839
19.3.1 The Censored Normal Distribution 846
Trang 2119.3.2 The Censored Regression (Tobit) Model 848
19.3.3 Estimation 850
19.3.4 Two-Part Models and Corner Solutions 852
19.3.5 Some Issues in Specification 858
19.3.5.a Heteroscedasticity 858 19.3.5.b Nonnormality 859 19.3.6 Panel Data Applications 860
19.4.1 Models for Duration Data 862
19.4.3 A Regression-Like Approach: Parametric Models of
Duration 863 19.4.3.a Theoretical Background 863 19.4.3.b Models of the Hazard Function 864 19.4.3.c Maximum Likelihood Estimation 866 19.4.3.d Exogenous Variables 867
19.4.3.e Heterogeneity 868 19.4.4 Nonparametric and Semiparametric Approaches 869
19.5.1 Incidental Truncation in a Bivariate Distribution 873
19.5.2 Regression in a Model of Selection 873
19.5.3 Two-Step and Maximum Likelihood Estimation 876
19.5.4 Sample Selection in Nonlinear Models 880
19.5.5 Panel Data Applications of Sample Selection Models 883
19.5.5.a Common Effects in Sample Selection Models 884 19.5.5.b Attrition 886
19.6.1 Regression Analysis of Treatment Effects 890
19.6.1.a The Normality Assumption 892 19.6.1.b Estimating the Effect of Treatment on
the Treated 893 19.6.2 Propensity Score Matching 894
19.6.3 Regression Discontinuity 897
PART V Time Series and Macroeconometrics
CHAPTER 20 Serial Correlation 903
Trang 2220.4 Some Asymptotic Results for Analyzing Time-Series Data 912
20.4.1 Convergence of Moments—The Ergodic Theorem 913 20.4.2 Convergence to Normality—A Central Limit Theorem 915
20.5.1 Asymptotic Properties of Least Squares 918 20.5.2 Estimating the Variance of the Least Squares Estimator 919
20.7.1 Lagrange Multiplier Test 922 20.7.2 Box and Pierce’s Test and Ljung’s Refinement 922 20.7.3 The Durbin–Watson Test 923
20.7.4 Testing in the Presence of a Lagged Dependent
Variable 923 20.7.5 Summary of Testing Procedures 924
20.9.1 AR(1) Disturbances 926 20.9.2 Application: Estimation of a Model with Autocorrelation 927 20.9.3 Estimation with a Lagged Dependent Variable 929
20.10.2 ARCH(q), ARCH-in-Mean, and Generalized ARCH
Models 932 20.10.3 Maximum Likelihood Estimation of the Garch Model 934 20.10.4 Testing for Garch Effects 936
20.10.5 Pseudo–Maximum Likelihood Estimation 937
CHAPTER 21 Nonstationary Data 942
21.2.1 Integrated Processes and Differencing 942 21.2.2 Random Walks, Trends, and Spurious Regressions 944 21.2.3 Tests for Unit Roots in Economic Data 947
21.2.4 The Dickey–Fuller Tests 948 21.2.5 The KPSS Test of Stationarity 958
Trang 2321.4 Nonstationary Panel Data 970
Appendix A Matrix Algebra 973
A.2.1 Equality of Matrices 973
A.2.4 Vector Multiplication 975
A.2.5 A Notation for Rows and Columns of a Matrix 975
A.2.6 Matrix Multiplication and Scalar Multiplication 975
A.2.8 A Useful Idempotent Matrix 978
A.3.2 Linear Combinations of Vectors and Basis Vectors 981
A.3.5 Rank of a Matrix 984
A.3.6 Determinant of a Matrix 986
A.3.7 A Least Squares Problem 987
A.4.1 Systems of Linear Equations 989
A.4.2 Inverse Matrices 990
A.4.3 Nonhomogeneous Systems of Equations 992
A.4.4 Solving the Least Squares Problem 992
A.5.1 Addition and Multiplication of Partitioned Matrices 993 A.5.2 Determinants of Partitioned Matrices 993
A.5.3 Inverses of Partitioned Matrices 993
A.5.4 Deviations from Means 994
A.5.5 Kronecker Products 994
A.6.1 The Characteristic Equation 995
A.6.2 Characteristic Vectors 996
A.6.3 General Results for Characteristic Roots and Vectors 996 A.6.4 Diagonalization and Spectral Decomposition of a Matrix 997 A.6.5 Rank of a Matrix 997
A.6.6 Condition Number of a Matrix 999
A.6.7 Trace of a Matrix 999
A.6.8 Determinant of a Matrix 1000
A.6.9 Powers of a Matrix 1000
A.6.10 Idempotent Matrices 1002
Trang 24A.6.11 Factoring a Matrix 1002 A.6.12 The Generalized Inverse of a Matrix 1003
A.7.1 Nonnegative Definite Matrices 1005 A.7.2 Idempotent Quadratic Forms 1006
A.8.1 Differentiation and the Taylor Series 1007
B.4.1 The Normal Distribution 1019 B.4.2 The Chi-Squared, t, and F Distributions 1021 B.4.3 Distributions with Large Degrees of Freedom 1023 B.4.4 Size Distributions: The Lognormal Distribution 1024 B.4.5 The Gamma and Exponential Distributions 1024 B.4.6 The Beta Distribution 1025
B.4.7 The Logistic Distribution 1025 B.4.8 The Wishart Distribution 1025 B.4.9 Discrete Random Variables 1026
B.7.1 Marginal Distributions 1030 B.7.2 Expectations in a Joint Distribution 1031 B.7.3 Covariance and Correlation 1031 B.7.4 Distribution of a Function of Bivariate Random
Variables 1032
B.8.1 Regression: The Conditional Mean 1034 B.8.2 Conditional Variance 1035
B.8.3 Relationships Among Marginal and Conditional
B.8.4 The Analysis of Variance 1037
Trang 25B.10.2 Sets of Linear Functions 1039
B.10.3 Nonlinear Functions 1040
B.11.1 Marginal and Conditional Normal Distributions 1041
B.11.2 The Classical Normal Linear Regression Model 1042
B.11.3 Linear Functions of a Normal Vector 1043
B.11.4 Quadratic forms in a Standard Normal Vector 1043
B.11.5 The F Distribution 1045
B.11.6 A Full Rank Quadratic Form 1045
B.11.7 Independence of a Linear and a Quadratic Form 1046
Appendix C Estimation and Inference 1047
C.5.1 Estimation in a Finite Sample 1055
C.5.2 Efficient Unbiased Estimation 1058
C.7.1 Classical Testing Procedures 1062
C.7.2 Tests Based on Confidence Intervals 1065
D.2.4 Convergence to a Random Variable 1074
D.2.5 Convergence in Distribution: Limiting Distributions 1076 D.2.6 Central Limit Theorems 1078
D.3.1 Asymptotic Distribution of a Nonlinear Function 1086
D.3.2 Asymptotic Expectations 1087
Appendix E Computation and Optimization 1089
E.2.1 Computing Integrals 1090
Trang 26E.2.2 The Standard Normal Cumulative Distribution Function 1090 E.2.3 The Gamma and Related Functions 1091
E.2.4 Approximating Integrals by Quadrature 1092
E.3.2 Computing Derivatives 1096
E.3.4 Aspects of Maximum Likelihood Estimation 1100 E.3.5 Optimization with Constraints 1101
E.3.6 Some Practical Considerations 1102
E.4.1 Function of One Parameter 1106 E.4.2 Function of Two Parameters: The Gamma Distribution 1107 E.4.3 A Concentrated Log-Likelihood Function 1108
Appendix F Data Sets Used in Applications 1109
References 1115
Combined Author and Subject Index 1161
Trang 27CHAPTER 1 Econometrics 1
CHAPTER 2 The Linear Regression Model 11
CHAPTER 3 Least Squares 26
CHAPTER 4 The Least Squares Estimator 51
xxv
Trang 28CHAPTER 5 Hypothesis Tests and Model Selection 108
CHAPTER 6 Functional Form and Structural Change 149
CHAPTER 7 Nonlinear, Semiparametric, and Nonparametric Regression
Models 181
CHAPTER 8 Endogeneity and Instrumental Variable Estimation 219
Trang 29Example 8.7 Hausman Test for a Consumption Function 237
CHAPTER 9 The Generalized Regression Model and Heteroscedasticity 257
CHAPTER 10 Systems of Equations 290
CHAPTER 11 Models for Panel Data 343
Trang 30Example 11.20 Fannie Mae’s Pass Through 420
CHAPTER 12 Estimation Frameworks in Econometrics 432
CHAPTER 13 Minimum Distance Estimation and the Generalized Method
of Moments 455
CHAPTER 14 Maximum Likelihood Estimation 509
Trang 31Example 14.16 Latent Class Regression Model for Grade Point
CHAPTER 15 Simulation-Based Estimation and Inference and Random Parameter
Models 603
CHAPTER 16 Bayesian Estimation and Inference 655
CHAPTER 17 Discrete Choice 681
Trang 32Example 17.3 Probability Models 694
CHAPTER 18 Discrete Choices and Event Counts 760
CHAPTER 19 Limited Dependent Variables—Truncation, Censoring, and Sample
Selection 833
Trang 33Example 19.9 Incidental Truncation 872
CHAPTER 20 Serial Correlation 903
CHAPTER 21 Nonstationary Data 942
Appendix C Estimation and Inference 1047
Trang 34Appendix D Large-Sample Distribution Theory 1066
Appendix E Computation and Optimization 1089
Trang 35ECONOMETRIC ANALYSIS
Econometric Analysis provides a broad introduction to the field of econometrics This
field grows continually—a list of journals devoted at least in part, if not completely,
to econometrics now includes The Journal of Applied Econometrics, The Journal of Econometrics, The Econometrics Journal, Econometric Theory, Econometric Reviews, Journal of Business and Economic Statistics, Empirical Economics, Foundations and Trends in Econometrics, The Review of Economics and Statistics, and Econometrica.
Constructing a textbook-style survey to introduce the topic at a graduate level has
become increasingly ambitious Nonetheless, I believe that one can successfully seek
that objective in a single textbook This text attempts to present, at an entry level, enough
of the topics in econometrics that a student can comfortably move from here to practice
or more advanced study in one or more specialized areas The book is also intended as
a bridge for students and analysts in the social sciences between an introduction to thefield and the professional literature
NEW TO THIS EDITION
This seventh edition is a major revision of Econometric Analysis Among the most
obvious changes are
including
chapter
Carlo studies
xxxiii
Trang 36THE SEVENTH EDITION OF ECONOMETRIC ANALYSIS
The book has two objectives The first is to introduce students to applied econometrics,
including basic techniques in linear regression analysis and some of the rich variety
of models that are used when the linear model proves inadequate or inappropriate.Modern software has made complicated modeling very easy to do, and an understanding
of the underlying theory is also important The second objective is to present students
with sufficient theoretical background so that they will recognize new variants of the
models learned about here as merely natural extensions that fit within a common body
of principles This book contains a substantial amount of theoretical material, such asthat on GMM, maximum likelihood estimation, and asymptotic results for regressionmodels
This text is intended for a one-year graduate course for social scientists uisites should include calculus, mathematical statistics, and an introduction to econo-
Prereq-metrics at the level of, say, Gujarati’s (2002) Basic EconoPrereq-metrics, Stock and Watson’s (2006) Introduction to Econometrics, Kennedy’s (2008) Guide to Econometrics, or Wooldridge’s (2009) Introductory Econometrics: A Modern Approach I assume, for ex-
ample, that the reader has already learned about the basics of econometric methodologyincluding the fundamental role of economic and statistical assumptions; the distinctionsbetween cross-section, time-series, and panel data sets; and the essential ingredients ofestimation, inference, and prediction with the multiple linear regression model Self-contained (for our purposes) summaries of the matrix algebra, mathematical statistics,and statistical theory used throughout the book are given in Appendices A through
D I rely heavily on matrix algebra throughout This may be a bit daunting to someearly on but matrix algebra is an indispensable tool and I hope the reader will come
to agree that it is a means to an end, not an end in itself With matrices, the unity of avariety of results will emerge without being obscured by a curtain of summation signs.All the matrix algebra needed in the text is presented in Appendix A Appendix E andChapter 15 contain a description of numerical methods that will be useful to practicingeconometricians (and to us in the later chapters of the book)
Contemporary computer software has made estimation of advanced nonlinear els as routine as least squares I have included five chapters on estimation methods used
mod-in current research and five chapters on applications mod-in micro- and macroeconometrics.The nonlinear models used in these fields are now the staples of the applied economet-rics literature As a consequence, this book also contains a fair amount of material thatwill extend beyond many first courses in econometrics Once again, I have included this
in the hope of laying a foundation for study of the professional literature in these areas.One overriding purpose has motivated all seven editions of this book The vastmajority of readers of this book will be users, not developers, of econometrics I be-lieve that it is simply not sufficient to recite the theory of estimation, hypothesis testing,and econometric analysis Although the often-subtle theory is extremely important,the application is equally crucial To that end, I have provided hundreds of numericalexamples My purpose in writing this work, and in my continuing efforts to update it,
is to show readers how to do econometric analysis I unabashedly accept the
unflatter-ing assessment of a correspondent who once likened this book to a “user’s guide toeconometrics.”
Trang 37PLAN OF THE BOOK
The arrangement of the book is as follows:
Part I begins the formal development of econometrics with its fundamental pillar,
the linear multiple regression model Estimation and inference with the linear least squares estimator are analyzed in Chapters 2 through 6 The nonlinear regression model
is introduced in Chapter 7 along with quantile, semi- and nonparametric regression, all
as extensions of the familiar linear model Instrumental variables estimation is developed
The second half of the book is devoted to topics that will extend the linear regressionmodel in many directions Beginning with Chapter 12, we proceed to the more involvedmethods of analysis that contemporary researchers use in analysis of “real-world” data.Chapters 12 to 16 in Part III present different estimation methodologies Chapter 12
presents an overview by making the distinctions between parametric, semiparametric and nonparametric methods The leading application of semiparametric estimation in the current literature is the generalized method of moments (GMM) estimator presented
in Chapter 13 This technique provides the platform for much of modern
economet-rics Maximum likelihood estimation is developed in Chapter 14 Monte Carlo and
simulation-based methods such as bootstrapping that have become a major
compo-nent of current research are developed in Chapter 15 Finally, Bayesian methods are
introduced in Chapter 16
Parts IV and V develop two major subfields of econometric methods, metrics, which is typically based on cross-section and panel data, and macroeconomet- rics, which is usually associated with analysis of time-series data In Part IV, Chapters 17
microecono-to 19 are concerned with models of discrete choice, censoring, truncation, sample tion, duration, treatment effects, and the analysis of counts of events In Part V, Chap-ters 20 and 21, we consider two topics in time-series analysis, models of serial correlationand regression models for nonstationary data—the usual substance of macroeconomicanalysis
selec-REVISIONS
I have substantially rearranged the early part of the book to produce what I hope is amore natural sequence of topics for the graduate econometrics course Chapter 4 is nowdevoted entirely to point and interval estimation, including prediction and forecasting.Finite sample, then asymptotic properties of least squares are developed in detail All
Trang 38of the material on hypothesis testing and specification search is moved into Chapter 5,rather than fragmented over several chapters as in the sixth edition I have also broughtthe material on instrumental variables much farther forward in the text, from after thedevelopment of the generalized regression model in the sixth edition to Chapter 8 in thisone, immediately after full development of the linear regression model This accordswith the greater emphasis on this method in recent applications A very large number
of other rearrangements of the material will also be evident Chapter 7 now contains
a range of advanced extensions of the linear regression model, including nonlinear,quantile, partially linear, and nonparametric regression This is also a point at which thedifferences between parametric, semiparametric, and nonparametric methods can beexamined One conspicuous modification is the excision of the long chapter on linearsimultaneous equations models Some of the material from this chapter appears else-where Two-stage least squares now appears with instrumental variables estimation.Remaining parts of this chapter that are of lesser importance in recent treatments, such
as rank and order conditions for identification of linear models and 3SLS and FIMLestimation, have been deleted or greatly reduced and placed in context elsewhere in thetext The material on discrete choice models has been rearranged to orient the topics
to the behavioral foundations Chapter 17 now broadly introduces discrete choice andrandom utility models, and then builds on variants of the binary choice model Theanalysis is continued in Chapter 18 with unordered, then ordered choice models and,finally, models for counts The last chapter of the section studies models for continu-ous variables in the contexts of particular data-generating mechanisms and behavioralcontexts
I have added new material and some different examples and applications at ous points Topics that have been expanded or given greater emphasis include treat-ment effects, bootstrapping, simulation-based estimation, robust estimation, missingand faulty data, and a variety of different new methods of discrete choice analysis inmicroeconometrics I have also added or expanded material on techniques recently ofinterest, such as quantile regression and stochastic frontier models
numer-I note a few specific highlights of the revision: numer-In general terms, numer-I have increased thefocus on robust methods a bit I have placed discussions of specification tests at severalpoints, consistent with the trend in the literature to examine more closely the fragility
of heavily parametric models A few of the specific new applications are as follows:
now includes substantially more material on bootstrapping standard errors andconfidence intervals The Krinsky and Robb (1986) approach to asymptoticinference has been placed here as well
interaction effects in nonlinear models Chapter 7 contains a lengthy application
of interaction effects in a nonlinear (exponential) regression model The issue isrevisited in Chapter 17
distribution theory, I have added a detailed proof of the Murphy and Topel (2002)result for two-step estimation in Chapter 14
application of inverse probability weighting to deal with attrition in Chapter 17
Trang 39The seventh edition is a major revision of Econometric Analysis both in terms of
organization of the material and in terms of new ideas and treatments I hope thatreaders will find the changes helpful
SOFTWARE AND DATA
There are many computer programs that are widely used for the computations described
in this book All were written by econometricians or statisticians, and in general, allare regularly updated to incorporate new developments in applied econometrics Asampling of the most widely used packages and Internet home pages where you canfind information about them are
Shazam econometrics.com (Northwest Econometrics Ltd., Gibsons, Canada)
A more extensive list of computer software used for econometric analysis can be
With only a few exceptions, the computations described in this book can be carried
out with any of the packages listed NLOGIT was used for the computations in the
ap-plications This text contains no instruction on using any particular program or language.(The author’s Web site for the text does provide some code and data for replication
of the numerical examples.) Many authors have produced RATS, LIMDEP/NLOGIT, EViews, SAS, or Stata code for some of our applications, including, in a few cases,
the documentation for their computer programs There are also quite a few volumesnow specifically devoted to econometrics associated with particular packages, such asCameron and Trivedi’s (2009) companion to their treatise on microeconometrics.The data sets used in the examples are also available on the Web site for thetext, http://pages.stern.nyu.edu/∼wgreene/Text/econometricanalysis.htm Throughoutthe text, these data sets are referred to “Table Fn.m,” for example Table F4.1 The
“F” refers to Appendix F at the back of the text which contains descriptions of the datasets The actual data are posted in generic ASCII and portable formats on the Website with the other supplementary materials for the text There are now thousands ofinteresting Web sites containing software, data sets, papers, and commentary on econo-metrics It would be hopeless to attempt any kind of a survey here One code/data sitethat is particularly agreeably structured and well targeted for readers of this book is
Trang 40the data archive for the Journal of Applied Econometrics They have archived all the
nonconfidential data sets used in their publications since 1988 (with some gaps before1995) This useful site can be found at http://qed.econ.queensu.ca/jae/ Several of the
examples in the text use the JAE data sets Where we have done so, we direct the reader
to the JAE’s Web site, rather than our own, for replication Other journals have begun
to ask their authors to provide code and data to encourage replication Another vast,easy-to-navigate site for aggregate data on the U.S economy is www.economagic.com
ACKNOWLEDGMENTS
It is a pleasure to express my appreciation to those who have influenced this work I main grateful to Arthur Goldberger (dec.), Arnold Zellner (dec.), Dennis Aigner, BillBecker, and Laurits Christensen for their encouragement and guidance After seveneditions of this book, the number of individuals who have significantly improved itthrough their comments, criticisms, and encouragement has become far too large for
re-me to thank each of them individually I am grateful for their help and I hope that all
of them see their contribution to this edition I would like to acknowledge the manyreviewers of my work whose careful reading has vastly improved the book throughthis edition: Scott Atkinson, University of Georgia; Badi Baltagi, Syracuse Univer-sity; Neal Beck, New York University; William E Becker (Ret.), Indiana University;Eric J Belasko, Texas Tech University; Anil Bera, University of Illinois; John Burkett,University of Rhode Island; Leonard Carlson, Emory University; Frank Chaloupka,University of Illinois at Chicago; Chris Cornwell, University of Georgia; Craig Depken
II, University of Texas at Arlington; Frank Diebold, University of Pennsylvania;Edward Dwyer, Clemson University; Michael Ellis, Wesleyan University; Martin Evans,Georgetown University; Vahagn Galstyan, Trinity College Dublin; Paul Glewwe, Uni-versity of Minnesota; Ed Greenberg, Washington University at St Louis; Miguel Herce,University of North Carolina; Joseph Hilbe, Arizona State University; Dr Uwe Jensen,Christian-Albrecht University; K Rao Kadiyala, Purdue University; William Lott, Uni-versity of Connecticut; Thomas L Marsh, Washington State University; Edward Mathis,Villanova University; Mary McGarvey, University of Nebraska–Lincoln; Ed Melnick,New York University; Thad Mirer, State University of New York at Albany; CyrilPasche, University of Geneva; Paul Ruud, University of California at Berkeley; SherrieRhine, Federal Deposit Insurance Corp.; Terry G Seaks (Ret.), University of NorthCarolina at Greensboro; Donald Snyder, California State University at Los Angeles;Steven Stern, University of Virginia; Houston Stokes, University of Illinois at Chicago;Dmitrios Thomakos, Columbia University; Paul Wachtel, New York University; MaryBeth Walker, Georgia State University; Mark Watson, Harvard University; and Ken-neth West, University of Wisconsin My numerous discussions with Bruce McCullough
of Drexel University have improved Appendix E and at the same time increased myappreciation for numerical analysis I am especially grateful to Jan Kiviet of the Uni-versity of Amsterdam, who subjected my third edition to a microscopic examinationand provided literally scores of suggestions, virtually all of which appear herein Pro-fessor Pedro Bacao, University of Coimbra, Portugal, and Mark Strahan of Sand HillEconometrics did likewise with the sixth edition