Econometrics 0691010188, 9780691010182

Table of contents :
Cover ... 1
Copyright Page ... 2
Preface ... 3
Prerequisites ... 3
Organization of the Book ... 3
Designing a Course Out of the Book ... 4
Review Questions and Analytical Exercises ... 5
Empirical Exercises ... 5
Mathematical Notation ... 6
Acknowledgments ... 6
Contents ... 8
CHAPTER 1 Finite-Sample Properties of ... 20
ABSTRACT ... 20
1.1 The Classical Linear Regression Model ... 20
The Linearity Assumption ... 21
Iqlatrix Notation ... 23
The Strict Exogeneity Assumption ... 24
Implications of Strict Exogeneity ... 25
Strict Exogeneity in Time-Series Models ... 26
Other Assumptions of the Model ... 27
The Classical Regression Model for Random Samples ... 29
"Fixed" Regressors ... 30
QUESTIONS FOR REVIEW ... 30
1.2 The Algebra of Least Squares ... 32
OLS Minimizes the Sum of Squared Residuals ... 32
Normal Equations ... 33
Two Expressions for the OLS Estimator ... 35
More Concepts and Algebra ... 35
Influential Analysis (optional) ... 38
A Note on the Computation of OLS Estimates ... 40
QUESTIONS FOR REVIEW ... 42
1.3 Finite-Sample Properties of OLS ... 44
Finite-Sample Distribution of b ... 44
Finite-Sample Properties of s^2 ... 47
Estimate of Var(b|X) ... 48
QUESTIONS FOR REVIEW ... 48
1.4 Hypothesis Testing under Normality ... 50
Normally Distributed Error Terms ... 50
Testing Hypotheses about Individual Regression Coefficients ... 52
Decision Rule for the t-Test ... 54
Confidence Interval ... 55
Linear Hypotheses ... 56
The F-Test ... 57
A More Convenient Expression for F ... 59
t versus F ... 60
An Example of a Test Statistic Whose Distribution Depends on X ... 62
QUESTIONS FOR REVIEW ... 63
1.5 Relation to Maximum Likelihood ... 64
The Maximum Likelihood Principle ... 64
Conditional versus Unconditional Likelihood ... 64
The Log Likelihood for the Regression Model ... 65
ML via Concentrated Likelihood ... 65
Cramer-Rao Bound for the Classical Regression Model ... 66
The F-Test as a Likelihood Ratio Test ... 69
Quasi-Maximum Likelihood ... 70
QUESTIONS FOR REVIEW ... 70
1.6 Generalized Least Squares (GLS) ... 71
Consequence of Relaxing Assumption 1.4 ... 72
Efficient Estimation with Known V ... 72
A Special Case: Weighted Least Squares (WLS) ... 75
Limiting Nature of GLS ... 75
QUESTIONS FOR REVIEW ... 76
1.7 Application: Returns to Scale in Electricity Supply ... 77
The Electricity Supply Industry ... 77
The Data ... 77
Why Do We Need Econometrics? 78
The Cobb-Douglas Technology ... 79
How Do We Know Things Are Cobb-Douglas? 80
Are the OLS Assumptions Satisfied? 81
Restricted Least Squares ... 82
Testing the Homogeneity of the Cost Function ... 82
Detour: A Cautionary Note on R^2 ... 84
Testing Constant Returns to Scale ... 84
Importance of Plotting Residuals ... 85
Subsequent Developments ... 85
QUESTIONS FOR REVIEW ... 87
PROBLEM SET FOR CHAPTER 1 ... 88
ANALYTICAL EXERCISES ... 88
MONTE CARLO EXERCISES ... 98
ANSWERS TO SELECTED QUESTIONS ... 101
CHAPTER 2 Large-Sample Theory ... 105
ABSTRACT ... 105
2.1 Review of Limit Theorems for Sequences of Random Variables ... 105
Various Modes of Convergence ... 106
Convergence in Probability ... 106
Almost Sure Convergence ... 106
Convergence in Mean Square ... 107
Convergence in Distribution ... 107
Three Useful Results ... 109
Viewing Estimators as Sequences of Random Variables ... 111
Laws of Large Numbers and Central Limit Theorems ... 112
QUESTIONS FOR REVIEW ... 113
2.2 Fundamental Concepts in Time-Series Analysis ... 114
Need for Ergodic Stationarity ... 114
Various Classes of Stochastic Processes ... 115
Different Formulation of Lack of Serial Dependence ... 123
QUESTIONS FOR REVIEW ... 125
2.3 Large-Sample Distribution of the OLS Estimator ... 126
The Model ... 126
Asymptotic Distribution of the OLS Estimator ... 130
s^2 Is Consistent ... 132
QUESTIONS FOR REVIEW ... 133
2.4 Hypothesis Testing ... 134
Testing Linear Hypotheses ... 134
The Test Is Consistent ... 136
Asymptotic Power ... 137
Testing Nonlinear Hypotheses ... 138
QUESTIONS FOR REVIEW ... 140
2.5 Estimating E(xix) Consistently ... 140
Using Residuals for the Errors ... 140
Data Matrix Representation of S ... 142
Finite-Sample Considerations ... 142
QUESTIONS FOR REVIEW ... 143
2.6 Implications of Conditional Homoskedasticity ... 143
Conditional versus Unconditional Homoskedasticity ... 143
Reduction to Finite-Sample Formulas ... 144
Large-Sample Distribution of t and F Statistics ... 145
Variations of Asymptotic Tests under C. onditional Homoskedasticity ... 145
QUESTIONS FOR REVIEW ... 147
2.7 Testing Conditional Homoskedasticity ... 148
QUESTION FOR REVIEW ... 150
2.8 Estimation with Parameterized Conditional Heteroskedasticity (optional) ... 150
The Functional Form ... 150
WLS with Known α 151
Regression of ei^2 on zi Provides a Consistent Estimate of α 152
WLS with Estimated α 153
OLS versus WLS ... 154
QUESTION FOR REVIEW ... 154
2.9 Least Squares Projection ... 154
Optimally Predicting the Value of the Dependent Variable ... 155
Best Linear Predictor ... 156
OLS Consistently Estimates the Projection Coefficients ... 157
QUESTION FOR REVIEW ... 157
2.10 Testing for Serial Correlation ... 158
Box-Pierce and Ljung-Box ... 159
Sample Autocorrelations Calculated from Residuals ... 161
Testing with Predetermined, but Not Strictly Exogenous, Regressors ... 163
An Auxiliary Regression-Based Test ... 164
QUESTION FOR REVIEW ... 166
2.11 Application: Rational Expectations Econometrics ... 167
The Efficient Market Hypotheses ... 167
Testable Implications ... 169
Testing for Serial Correlation ... 170
Is the Nominal Interest Rate the Optimal Predictor? 173
Rt Is Not Strictly Exogenous ... 175
Subsequent Developments ... 176
QUESTION FOR REVIEW ... 177
2.12 Time Regressions ... 177
The Asymptotic Distribution of the OLS Estimator ... 178
Hypothesis Testing for Time Regressions ... 180
QUESTION FOR REVIEW ... 181
Appendix 2.A: Asymptotics with Fixed Regressors ... 181
Appendix 2.B: Proof of Proposition 2.10 ... 182
PROBLEM SET FOR CHAPTER 2 ... 185
CHAPTER 3 Single-Equation GMM ... 203
ABSTRACT ... 203
3.1 Endogeneity Bias: Working's Example ... 204
A Simultaneous Equations Model of Market Equilibrium ... 204
Endogeneity Bias ... 205
Observable Supply Shifters ... 206
QUESTION FOR REVIEW ... 209
3.2 More Examples ... 210
A Simple Macroeconometric Model ... 210
Errors-in-Variables ... 211
Production Function ... 213
QUESTION FOR REVIEW ... 214
3.3 The General Formulation ... 215
Regressors and Instruments ... 215
Identification ... 217
Order Condition for Identification ... 219
The Assumption for Asymptotic Normality ... 219
QUESTION FOR REVIEW ... 220
3.4 Generalized Method of Moments Defined ... 221
Method of Moments ... 222
Generalized Method of Moments ... 223
Sampling Error ... 224
QUESTION FOR REVIEW ... 224
3.5 Large-Sample Properties of GMM ... 225
Asymptotic Distribution of the GMM Estimator ... 226
Estimation of Error Variance ... 227
Hypothesis Testing ... 228
Estimation of S ... 229
Efficient GMM Estimator ... 229
Asymptotic Power ... 231
Small-Sample Properties ... 232
QUESTION FOR REVIEW ... 232
3.6 Testing Overidentifying Restrictions ... 234
Testing Subsets of Orthogonality Conditions ... 235
QUESTION FOR REVIEW ... 238
3.7 Hypothesis Testing by the Likelihood-Ratio Principle ... 239
The LR Statistic for the Regression Model ... 240
Variable Addition Test (optional) ... 241
QUESTION FOR REVIEW ... 242
3.8 Implications of Conditional Homoskedasticity ... 242
Efficient GMM Becomes 2Sl.S ... 243
J Becomes Sargan's Statistic ... 244
Small-Sample Properties of 2SLS ... 246
Alternative Derivations of 2SLS ... 246
When Regressors Are Predetermined ... 248
Testing a Subset of Orthogonality Conditions ... 249
Testing Conditional Homoskedasticity ... 251
Testing for Serial Correlation ... 251
QUESTION FOR REVIEW ... 252
3.9 Application: Returns from Schooling ... 253
The NLS-Y Data ... 253
The Semi-Log Wage Equation ... 254
Omitted Variable Bias ... 255
IQ as the Measure of Ability ... 256
Errors-in-Variables ... 256
2SLS to Correct for the Bias ... 259
Subsequent Developments ... 260
QUESTION FOR REVIEW ... 260
PROBLEM SET FOR CHAPTER 3
ANALYTICAL EXERCISES ... 261
EMPIRICAL EXERCISES ... 267
CHAPTER 4 Multiple-quation GMM ... 275
ABSTRACT ... 275
4.1 The Multiple-Equation Model ... 276
Linearity ... 276
Stationarity and Ergodicity ... 277
Orthogonality Conditions ... 278
Identification ... 279
The Assumption for Asymptotic Normality ... 281
Connection to the "Complete" System of Simultaneous Equations ... 282
4.2 Multiple-Equation GMM Defined ... 282
4.3 Large-Sample Theory ... 285
4.4 Single-Equation versus Multiple-Equation Estimation ... 288
When Are They "Equivalent"? 289
Joint Estimation Can Be Hazardous ... 290
QUESTION FOR REVIEW ... 291
4.5 Special Cases of Multiple-Equation GMM: FIVE, 3SLS, and SUR ... 291
Conditional Homoskedasticity ... 291
Full-Information Instrumental Variables Efficient (FIVE) ... 292
Three-Stage Least Squares (3SLS) ... 293
Seemingly Unrelated Regressions (SUR) ... 296
SUR versus OLS ... 298
QUESTION FOR REVIEW ... 300
4.6 Common Coefficients ... 303
The Model with Common Coefficients ... 303
The GMM Estimator ... 304
Imposing Conditional Homoskedasticity ... 305
Pooled OLS ... 307
Beautifying the Formulas ... 309
The Restriction That Isn't ... 310
QUESTION FOR REVIEW ... 311
4.7 Application: Interrelated Factor Demands ... 313
The Translog Cost Function ... 313
Factor Shares ... 314
Substitution Elasticities ... 315
Properties of Cost Functions ... 316
Stochastic Specifications ... 317
The Nature of Restrictions ... 318
Multivariate Regression Subject to Cross-Equation Restrictions ... 319
Which Equation to Delete? 321
Results ... 322
QUESTION FOR REVIEW ... 324
PROBLEM SET FOR CHAPTER 4 ... 325
ANALYTICAL EXERCISES ... 325
EMPIRICAL EXERCISES ... 334
CHAPTER 5 Panel Data ... 340
ABSTRACT ... 340
5.1 The Error-Components Model ... 341
Error Components ... 341
Group Means ... 344
A Reparameterization ... 344
QUESTION FOR REVIEW ... 346
5.2 The Fixed-Effects Estimator ... 347
The Formula ... 347
Large-Sample Properties ... 348
Digression: When ηi Is Spherical ... 350
Random Effects versus Fixed Effects ... 351
Relaxing Conditional Homoskedasticity ... 352
QUESTION FOR REVIEW ... 353
5.3 Unbalanced Panels (optional) ... 354
"Zeroing Out" Missing Observations ... 355
Zeroing Out versus Compression ... 356
No Selectivity Bias ... 357
QUESTION FOR REVIEW ... 358
5.4 Application: International Differences in Growth Rates ... 359
Derivation of the Estimation Equation ... 359
Appending the Error Term ... 360
Treatment of αi ... 361
Consistent Estimation of Speed of Convergence ... 362
QUESTION FOR REVIEW ... 363
Appendix 5.A: Distribution of Hausman Statistic ... 363
PROBLEM SET FOR CHAPTER 5 ... 366
ANALYTICAL EXERCISES ... 366
EMPIRICAL EXERCISES ... 375
CHAPTER 6 Serial Correlation ... 382
ABSTRACT ... 382
6.1 Modeling Serial Correlation: Linear Processes ... 382
MA(q) ... 383
MA(∞) as a Mean Square Limit ... 383
Filters ... 386
Inverting Lag Polynomials ... 389
QUESTIONS FOR REVIEW ... 392
6.2 ARMA Processes ... 392
AR(1) and Its MA(∞) Representation ... 393
Autocovariances of AR(1) ... 395
AR(p) and Its MA(∞) Representation ... 395
ARMA(p, q) ... 397
ARMA(p, q) with Common Roots ... 399
Invertibility ... 400
Autocovariance-Generating Function and the Spectrum ... 400
QUESTIONS FOR REVIEW ... 402
6.3 Vector Processes ... 404
QUESTIONS FOR REVIEW ... 408
6.4 Estimating Autoregressions ... 409
Estimation of AR(1) ... 409
Estimation of AR(p) ... 410
Choice of Lag Length ... 411
Estimation of VAR$ 414
Estimation of ARMA(p, q) ... 415
QUESTIONS FOR REVIEW ... 416
6.5 Asymptotics for Sample Means of Serially Correlated Processes ... 417
LLN for Covariance-Stationary Processes ... 418
Two Central Limit Theorems ... 419
Multivariate Extension ... 421
QUESTIONS FOR REVIEW ... 422
6.6 Incorporating Serial Correlation in GMM ... 423
The Model and Asymptotic Results ... 423
Estimating S When Autocovariances Vanish after Finite Lags ... 424
Using Kernels to Estimate S ... 425
VARHAC ... 427
QUESTIONS FOR REVIEW ... 429
6.7 Estimation under Conditional Homoskedasticity (Optional) ... 430
Kernel-Based Estimation of S under Conditional Homoskedasticity ... 430
Data Matrix Representation of Estimated Long-Run Variance ... 431
Relation to GLS ... 432
QUESTIONS FOR REVIEW ... 434
6.8 Application: Forward Exchange Rates as Optimal Predictors ... 435
The Market Efficiency Hypothesis ... 436
Testing Whether the Unconditional Mean Is Zero ... 437
Regression Tests ... 440
QUESTIONS FOR REVIEW ... 444
PROBLEM SET FOR CHAPTER 6 ... 445
ANALYTICAL EXERCISES ... 445
EMPIRICAL EXERCISES ... 455
CHAPTER 7 Extremum Estimators ... 462
ABSTRACT ... 462
7.1 Extremum Estimators ... 463
"Measurability" of θ 463
Two Classes of Extremum Estimators ... 464
Maximum Likelihood (ML) ... 465
Conditional Maximum Likelihood ... 467
Invariance of ML ... 469
Nonlinear Least Squares (NLS) ... 470
Linear and Nonlinear GMM ... 471
QUESTIONS FOR REVIEW ... 472
7.2 Consistency ... 473
Two Consistency Theorems for Extremum Estimators ... 473
Consistency of M-Estimators ... 475
Concavity after Reparameterization ... 478
Identification in NLS and ML ... 479
Consistency of GMM ... 484
QUESTIONS FOR REVIEW ... 485
7.3 Asymptotic Normality ... 486
Asymptotic Normality of M-Estimators ... 487
Consistent Asymptotic Variance Estimation ... 490
Asymptotic Normality of Conditional ML ... 491
Two Examples ... 493
Asymptotic Normality of GMM ... 495
GMM versus ML ... 498
Expressing the Sampling Error in a Common Format ... 500
Consistency of GMM ... 503
7.4 Hypothesis Testing ... 504
The Null Hypothesis ... 504
The Working Assumptions ... 506
The Wald Statistic ... 506
The Lagrange Multiplier (LM) Statistic ... 508
The Likelihood Ratio (LR) Statistic ... 510
Summary of the Trinity ... 511
QUESTIONS FOR REVIEW ... 513
7.5 Numerical Optimization ... 514
Newton-Raphson ... 514
Gauss-Newton ... 515
Writing Newton-Raphson and Gauss-Newton in a Common Format ... 515
Equations Nonlinear in Parameters Only ... 516
QUESTIONS FOR REVIEW ... 517
PROBLEM SET FOR CHAPTER 7 ... 518
ANALYTICAL EXERCISES ... 518
CHAPTER 8 Examples of Maximum Likelihood ... 524
ABSTRACT ... 524
8.1 Qualitative Response (QR) Models ... 524
Score and Hessian for Observation t ... 525
Consistency ... 526
Asymptotic Normality ... 527
QUESTIONS FOR REVIEW ... 527
8.2 Truncated Regression Models ... 528
The Model ... 528
Truncated Distributions ... 529
The Likelihood Function ... 530
Reparameterizing the Likelihood Function ... 531
Verifying Consistency and Asymptotic Normality ... 532
Recovering Original Parameters ... 534
QUESTIONS FOR REVIEW ... 534
8.3 Censored Regression (Tobit) Models ... 535
Tobit Likelihood Function ... 535
Reparameterization ... 536
QUESTIONS FOR REVIEW ... 538
8.4 Multivariate Regressions ... 538
The Multivariate Regression Model Restated ... 539
The Likelihood Function ... 540
Maximum the Likelihood Function ... 541
Consistency and Asymptotic Normality ... 542
QUESTIONS FOR REVIEW ... 542
8.5 FIML ... 543
The Multiple-Equation Model with Common Instruments Restated ... 543
The Complete System of Simultaneous Equations ... 546
Relationship between (F0, !10) and &0 ... 547
The FIML Likelihood Function ... 548
The FIML Concentrated Likelihood Function ... 549
Testing Overidentifying Restrictions ... 550
Properties of the FIML Estimator ... 550
ML Estimation of the SUR Model ... 552
QUESTIONS FOR REVIEW ... 554
8.6 LIML ... 555
LIML Defined ... 555
Computation of LIML ... 557
LIML versus 2SLS ... 559
QUESTIONS FOR REVIEW ... 559
8.7 Serially Correlated Observations ... 560
Two Ouestions ... 560
Unconditional ML for Dependent Observations ... 562
ML Estimation of AR(1) Processes ... 563
Conditional ML Estimation of AR(1) Processes ... 564
Conditional ML Estimation of AR(p) and VAR(p) Processes ... 566
QUESTIONS FOR REVIEW ... 567
PROBLEM SET FOR CHAPTER 8 ... 568
ANALYTICAL EXERCISES ... 568
CHAPTER 9 Unit-Root Econometrics ... 574
ABSTRACT ... 574
9.1 Modeling Trends ... 574
Integrated Processes ... 575
Why Is It Important to Know if the Process Is I(1)? 577
Which Should le Taken as the Null, I(0) or I(1)? 579
Other Approaches to Modeling Trends ... 580
QUESTIONS FOR REVIEW ... 580
9.2 Tools for Unit-Root Econometrics ... 580
Linear I(0) Processes ... 580
Approximating I(1) by a Random Walk ... 581
Relation to ARMA Models ... 583
The Wiener Process ... 584
A Useful Lemma ... 587
QUESTIONS FOR REVIEW ... 589
9.3 Dickey-Fuller Tests ... 590
The AR(1) Model ... 590
Deriving the Limiting Distribution under the I(1) Null ... 591
Incorporating the Intercept ... 594
Incorporating Time Trend ... 598
QUESTIONS FOR REVIEW ... 600
9.4 Augmented Dickey-Fuller Tests ... 602
The Augmented Autoregression ... 602
Limiting Distribution of the OLS Estimator ... 603
Deriving Test Statistics ... 607
Testing Hypotheses about ξ 608
What to Do When p IS Unknown? 609
A Suggestion for the Choice of p_max(T) ... 611
Including the Intercept in the Regression ... 612
Incorporating Time Trend ... 614
Summary of the DF and ADF Tests and Other Unit-Root Tests ... 616
QUESTIONS FOR REVIEW ... 617
9.5 Which Unit-Root Test to Use? 618
Local-to-Unity Asymptotics ... 619
Small-Sample Properties ... 619
9.6 Application: Purchasing Power Parity ... 620
The Embarrassing Resiliency of the Random Walk Model? 621
PROBLEM SET FOR CHAPTER 9 ... 622
ANALYTICAL EXERCISES ... 622
MONTE CARLO EXERCISES ... 628
EMPIRICAL EXERCISES ... 630
CHAPTER 10 Cointegration ... 640
ABSTRACT ... 640
10.1 Cointegrated Systems ... 641
Linear Vector I(0) and I(1) Processes ... 641
The Beveridge-Nelson Decomposition ... 644
Cointegration Defined ... 646
QUESTIONS FOR REVIEW ... 649
10.2 Alternative Representations of Cointegrated Systems ... 650
Phillips's Triangular Representation ... 650
VAR and Cointegration ... 650
The Vector Error-Correction Model (VECM) ... 655
Johansen's ML Procedure ... 657
QUESTIONS FOR REVIEW ... 659
10.3 Testing the Null of No Cointegration ... 660
Spurious Regressions ... 660
The Residual-Based Test for Cointegration ... 661
Testing the Null of Cointegration ... 666
QUESTIONS FOR REVIEW ... 667
10.4 Inference on Cointegrating Vectors ... 667
The Bivariate Example ... 669
Continuing with the Bivariate Example ... 670
Allowing for Serial Correlation ... 671
General Case ... 674
Other Estimators and Finite-Sample Properties ... 675
QUESTIONS FOR REVIEW ... 676
10.5 Application: The Demand for Money in the United States ... 676
The Data ... 677
(m - p, y, R) as a Cointegrated System ... 677
DOLS ... 679
Unstable Money Demand? 680
PROBLEM SET FOR CHAPTER 10 ... 682
EMPIRICAL EXERCISES ... 682