Greene W.H. Econometric Analysis
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Contents
xix
19.3.2 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 Models for Duration 861
19.4.1 Models for Duration Data 862
19.4.2 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 Incidental Truncation and Sample Selection 872
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 Evaluating Treatment Effects 888
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
19.7 Summary and Conclusions 898
PART V Time Series and Macroeconometrics
CHAPTER 20 Serial Correlation 903
20.1 Introduction 903
20.2 The Analysis of Time-Series Data 906
20.3 Disturbance Processes 909
20.3.1 Characteristics of Disturbance Processes 909
20.3.2 AR(1) Disturbances 910
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Contents
20.4 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 Least Squares Estimation 918
20.5.1 Asymptotic Properties of Least Squares 918
20.5.2 Estimating the Variance of the Least Squares Estimator 919
20.6 GMM Estimation 921
20.7 Testing for Autocorrelation 922
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
20.8 Efficient Estimation When Is Known 924
20.9 Estimation When Is Unknown 926
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 Autoregressive Conditional Heteroscedasticity 930
20.10.1 The ARCH(1) Model 931
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
20.11 Summary and Conclusions 939
CHAPTER 21 Nonstationary Data 942
21.1 Introduction 942
21.2 Nonstationary Processes and Unit Roots 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
21.3 Cointegration 959
21.3.1 Common Trends 962
21.3.2 Error Correction and VAR Representations 963
21.3.3 Testing for Cointegration 965
21.3.4 Estimating Cointegration Relationships 967
21.3.5 Application: German Money Demand 967
21.3.5.a Cointegration Analysis and a Long-Run Theoretical
Model 968
21.3.5.b Testing for Model Instability 969
Contents
xxi
21.4 Nonstationary Panel Data 970
21.5 Summary and Conclusions 971
PART VI Appendices
Appendix A Matrix Algebra 973
A.1 Terminology 973
A.2 Algebraic Manipulation of Matrices 973
A.2.1 Equality of Matrices 973
A.2.2 Transposition 974
A.2.3 Matrix Addition 974
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.2.7 Sums of Values 977
A.2.8 A Useful Idempotent Matrix 978
A.3 Geometry of Matrices 979
A.3.1 Vector Spaces 979
A.3.2 Linear Combinations of Vectors and Basis Vectors 981
A.3.3 Linear Dependence 982
A.3.4 Subspaces 983
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 Solution of a System of Linear Equations 989
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 Partitioned Matrices 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 Characteristic Roots and Vectors 995
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
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Contents
A.6.11 Factoring a Matrix 1002
A.6.12 The Generalized Inverse of a Matrix 1003
A.7 Quadratic Forms and Definite Matrices 1004
A.7.1 Nonnegative Definite Matrices 1005
A.7.2 Idempotent Quadratic Forms 1006
A.7.3 Comparing Matrices 1006
A.8 Calculus and Matrix Algebra 1007
A.8.1 Differentiation and the Taylor Series 1007
A.8.2 Optimization 1010
A.8.3 Constrained Optimization 1012
A.8.4 Transformations 1014
Appendix B Probability and Distribution Theory 1015
B.1 Introduction 1015
B.2 Random Variables 1015
B.2.1 Probability Distributions 1015
B.2.2 Cumulative Distribution Function 1016
B.3 Expectations of a Random Variable 1017
B.4 Some Specific Probability Distributions 1019
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.5 The Distribution of a Function of a Random Variable 1026
B.6 Representations of a Probability Distribution 1028
B.7 Joint Distributions 1030
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 Conditioning in a Bivariate Distribution 1034
B.8.1 Regression: The Conditional Mean 1034
B.8.2 Conditional Variance 1035
B.8.3 Relationships Among Marginal and Conditional
Moments 1035
B.8.4 The Analysis of Variance 1037
B.9 The Bivariate Normal Distribution 1037
B.10 Multivariate Distributions 1038
B.10.1 Moments 1038
Contents
xxiii
B.10.2 Sets of Linear Functions 1039
B.10.3 Nonlinear Functions 1040
B.11 The Multivariate Normal Distribution 1041
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.1 Introduction 1047
C.2 Samples and Random Sampling 1048
C.3 Descriptive Statistics 1048
C.4 Statistics as Estimators—Sampling Distributions 1051
C.5 Point Estimation of Parameters 1055
C.5.1 Estimation in a Finite Sample 1055
C.5.2 Efficient Unbiased Estimation 1058
C.6 Interval Estimation 1060
C.7 Hypothesis Testing 1062
C.7.1 Classical Testing Procedures 1062
C.7.2 Tests Based on Confidence Intervals 1065
C.7.3 Specification Tests 1066
Appendix D Large-Sample Distribution Theory 1066
D.1 Introduction 1066
D.2 Large-Sample Distribution Theory 1067
D.2.1 Convergence in Probability 1067
D.2.2 Other forms of Convergence and Laws of Large
Numbers 1070
D.2.3 Convergence of Functions 1073
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.2.7 The Delta Method 1083
D.3 Asymptotic Distributions 1084
D.3.1 Asymptotic Distribution of a Nonlinear Function 1086
D.3.2 Asymptotic Expectations 1087
D.4 Sequences and the Order of a Sequence 1088
Appendix E Computation and Optimization 1089
E.1 Introduction 1089
E.2 Computation in Econometrics 1090
E.2.1 Computing Integrals 1090
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Contents
E.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 Optimization 1093
E.3.1 Algorithms 1095
E.3.2 Computing Derivatives 1096
E.3.3 Gradient Methods 1097
E.3.4 Aspects of Maximum Likelihood Estimation 1100
E.3.5 Optimization with Constraints 1101
E.3.6 Some Practical Considerations 1102
E.3.7 The EM Algorithm 1104
E.4 Examples 1106
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
EXAMPLES AND APPLICATIONS
Q
CHAPTER 1 Econometrics 1
Example 1.1 Behavioral Models and the Nobel Laureates 2
Example 1.2 Keynes’s Consumption Function 4
CHAPTER 2 The Linear Regression Model 11
Example 2.1 Keynes’s Consumption Function 13
Example 2.2 Earnings and Education 14
Example 2.3 The U.S. Gasoline Market 17
Example 2.4 The Translog Model 18
Example 2.5 Short Rank 19
Example 2.6 An Inestimable Model 19
Example 2.7 Nonzero Conditional Mean of the Disturbances 20
CHAPTER 3 Least Squares 26
Section 3.2.2 Application: An Investment Equation 28
Example 3.1 Partial Correlations 38
Example 3.2 Fit of a Consumption Function 41
Example 3.3 Analysis of Variance for an Investment Equation 42
Example 3.4 Art Appreciation 46
CHAPTER 4 The Least Squares Estimator 51
Example 4.1 The Sampling Distribution of a Least Squares
Estimator 54
Example 4.2 Omitted Variable 57
Example 4.3 Sampling Variance in the Two-Variable Regression
Model 59
Example 4.4 Nonlinear Functions of Parameters: The Delta Method 69
Example 4.5 Least Squares vs. Least Absolute Deviations—A Monte
Carlo Study 71
Example 4.6 MLE with Normally Distributed Disturbances 74
Example 4.7 The Gamma Regression Model 74
Example 4.8 Confidence Interval for the Income Elasticity of Demand for
Gasoline 77
Example 4.9 Confidence Interval Based on the Asymptotic
Distribution 78
Example 4.10 Pricing Art 85
Example 4.11 Multicollinearity in the Longley Data 90
Example 4.12 Predicting Movie Success 93
xxv
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Examples and Applications
CHAPTER 5 Hypothesis Tests and Model Selection 108
Example 5.1 Art Appreciation 116
Example 5.2 Earnings Equation 116
Example 5.3 Restricted Investment Equation 120
Example 5.4 Production Function 124
Example 5.5 F Test for the Earnings Equation 126
Example 5.6 A Long-Run Marginal Propensity to Consume 132
Example 5.7 J Test for a Consumption Function 136
Example 5.8 Size of a RESET Test 138
Example 5.9 Bayesian Averaging of Classical Estimates 142
CHAPTER 6 Functional Form and Structural Change 149
Example 6.1 Dummy Variable in an Earnings Equation 150
Example 6.2 Value of a Signature 150
Example 6.3 Genre Effects on Movie Box Office Receipts 152
Example 6.4 Analysis of Covariance 153
Example 6.5 A Natural Experiment: The Mariel Boatlift 157
Example 6.6 Functional Form for a Nonlinear Cost Function 162
Example 6.7 Intrinsically Linear Regression 166
Example 6.8 CES Production Function 167
Example 6.9 Structural Break in the Gasoline Market 172
Example 6.10 The World Health Report 173
CHAPTER 7 Nonlinear, Semiparametric, and Nonparametric Regression
Models 181
Example 7.1 CES Production Function 182
Example 7.2 Identification in a Translog Demand System 183
Example 7.3 First-Order Conditions for a Nonlinear Model 185
Example 7.4 Analysis of a Nonlinear Consumption Function 191
Example 7.5 The Box–Cox Transformation 193
Example 7.6 Interaction Effects in a Loglinear Model for Income 195
Example 7.7 Linearized Regression 201
Example 7.8 Nonlinear Least Squares 202
Example 7.9 LAD Estimation of a Cobb–Douglas Production
Function 205
Example 7.10 Income Elasticity of Credit Card Expenditure 208
Example 7.11 Partially Linear Translog Cost Function 211
Example 7.12 A Nonparametric Average Cost Function 214
CHAPTER 8 Endogeneity and Instrumental Variable Estimation 219
Example 8.1 Models with Endogenous Right-Hand-Side Variables 219
Example 8.2 Instrumental Variable Analysis 228
Example 8.3 Streams as Instruments 228
Example 8.4 Instrumental Variable in Regression 229
Example 8.5 Instrumental Variable Estimation of a Labor Supply
Equation 232
Example 8.6 Labor Supply Model (Continued) 236
Examples and Applications
xxvii
Example 8.7 Hausman Test for a Consumption Function 237
Example 8.8 Overidentification of the Labor Supply Equation 239
Example 8.9 Income and Education in a Study of Twins 244
Example 8.10 Instrumental Variables Estimates of the Consumption
Function 248
Example 8.11 Does Television Cause Autism? 252
Example 8.12 Is Season of Birth a Valid Instrument? 254
CHAPTER 9 The Generalized Regression Model and Heteroscedasticity 257
Example 9.1 Heteroscedastic Regression 269
Example 9.2 The White Estimator 274
Example 9.3 Testing for Heteroscedasticity 277
Section 9.7.1 Multiplicative Heteroscedasticity 280
Example 9.4 Multiplicative Heteroscedasticity 281
Section 9.7.2 Groupwise Heteroscedasticity 282
Example 9.5 Groupwise Heteroscedasticity 284
CHAPTER 10 Systems of Equations 290
Example 10.1 A Regional Production Model for Public Capital 300
Example 10.2 Stone’s Expenditure System 307
Example 10.3 A Cost Function for U.S. Manufacturing 313
Example 10.4 Structure and Reduced Form in a Small Macroeconomic
Model 319
Example 10.5 Identification 324
Example 10.6 Klein’s Model I 332
CHAPTER 11 Models for Panel Data 343
Example 11.1 Wage Equation 351
Example 11.2 Repeat Sales of Monet Paintings 354
Example 11.3 Robust Estimators of the Wage Equation 355
Example 11.4 Analysis of Covariance and the World Health Organization
Data 358
Example 11.5 Fixed Effects Wage Equation 368
Example 11.6 Testing for Random Effects 377
Example 11.7 Estimates of the Random Effects Model 378
Example 11.8 Hausman Test for Fixed versus Random Effects 380
Example 11.9 Variable Addition Test for Fixed versus Random Effects 381
Example 11.10 Hospital Costs 384
Example 11.11 Robust Standard Errors for Fixed and Random Effects
Estimators 389
Example 11.12 Spatial Autocorrelation in Real Estate Sales 392
Example 11.13 Spatial Lags in Health Expenditures 393
Example 11.14 The Returns to Schooling 397
Example 11.15 Dynamic Labor Supply Equation 408
Example 11.16 Health Care Utilization 411
Example 11.17 Exponential Model with Fixed Effects 413
Example 11.18 Demand for Electricity and Gas 416
Example 11.19 Random Coefficients Model 418
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Examples and Applications
Example 11.20 Fannie Mae’s Pass Through 420
Example 11.21 Dynamic Panel Data Models 421
Example 11.22 A Mixed Fixed Growth Model for Developing
Countries 426
CHAPTER 12 Estimation Frameworks in Econometrics 432
Example 12.1 The Linear Regression Model 435
Example 12.2 The Stochastic Frontier Model 435
Example 12.3 Joint Modeling of a Pair of Event Counts 438
Example 12.4 Semiparametric Estimator for Binary Choice Models 442
Example 12.5 A Model of Vacation Expenditures 443
CHAPTER 13 Minimum Distance Estimation and the Generalized Method
of Moments 455
Example 13.1 Euler Equations and Life Cycle Consumption 455
Example 13.2 Method of Moments Estimator for N[μ, σ
2
] 457
Example 13.3 Inverse Gaussian (Wald) Distribution 458
Example 13.4 Mixtures of Normal Distributions 459
Example 13.5 Gamma Distribution 460
Example 13.5 (Continued) 462
Example 13.6 Minimum Distance Estimation of a Hospital Cost
Function 466
Example 13.7 GMM Estimation of a Nonlinear Regression Model 472
Example 13.8 Empirical Moment Equation for Instrumental
Variables 475
Example 13.9 Overidentifying Restrictions 479
Example 13.10 GMM Estimation of a Dynamic Panel Data Model of Local
Government Expenditures 503
CHAPTER 14 Maximum Likelihood Estimation 509
Example 14.1 Identification of Parameters 510
Example 14.2 Log-Likelihood Function and Likelihood Equations for the
Normal Distribution 513
Example 14.3 Information Matrix for the Normal Distribution 520
Example 14.4 Variance Estimators for an MLE 522
Example 14.5 Two-Step ML Estimation 540
Example 14.6 Multiplicative Heteroscedasticity 557
Example 14.7 Autocorrelation in a Money Demand Equation 559
Example 14.8 ML Estimates of a Seemingly Unrelated Regressions
Model 565
Example 14.9 Identification in a Loglinear Regression Model 568
Example 14.10 Geometric Regression Model for Doctor Visits 571
Example 14.11 Maximum Likelihood and FGLS Estimates of a Wage
Equation 576
Example 14.12 Statewide Productivity 579
Example 14.13 Random Effects Geometric Regression Model 584
Example 14.14 Fixed and Random Effects Geometric Regression 588
Example 14.15 Latent Class Model for Grade Point Averages 590