CONTENTS
List of Figures = xvii
Preface = xix
1 Finite-Sample Properties of OLS = 3
1.1 The Classical Linear Regression Model = 3
The Linearity Assumption = 4
Matrix Notation = 6
The Strict Exogeneity Assumption = 7
Implications of Strict Exogeneity = 8
Strict Exogeneity in Time-Series Models = 9
Other Assumptions of the Model = 10
The Classical Regression Model for Random Samples = 12
"Fixed" Regressors = 13
1.2 The Algebra of Least Squares = 15
OLS Minimizes the Sum of Squared Residuals = 15
Normal Equations = 16
Two Expressions for the OLS Estimator = 18
More Concepts and Algebra = 18
Influential Analysis (optional) = 21
A Note on the Computation of OLS Estimates = 23
1.3 Finite-Sample Properties of OLS = 27
Finite-Sample Distribution of b = 27
Finite-Sample Properties of s2 = 30
Estimate of Var(b｜X) = 31
1.4 Hypothesis Testing under Normality = 33
Normally Distributed Error Terms = 33
Testing Hypotheses about Individual Regression Coefficients = 35
Decision Rule for the t-Test = 37
Confidence Interval = 38
p-Value = 38
Linear Hypotheses = 39
The F-Test = 40
A More Convenient Expression for F = 42
t versus F = 43
An Example of a Test Statistic Whose Distribution Depends on Ⅹ = 45
1.5 Relation to Maximum Likelihood = 47
The Maximum Likelihood Principle = 47
Conditional versus Unconditional Likelihood = 47
The Log Likelihood for the Regression Model = 48
ML via Concentrated Likelihood = 48
Cramer-Rao Bound for the Classical Regression Model = 49
The F-Test as a Likelihood Ratio Test = 52
Quasi-Maximum Likelihood = 53
1.6 Generalized Least Squares (GLS) = 54
Consequence of Relaxing Assumption 1.4 = 55
Efficient Estimation with Known Ⅴ = 55
A Special Case : Weighted Least Squares (WLS) = 58
Limiting Nature of GLS = 58
1.7 Application : Returns to Scale in Electricity Supply = 60
The Electricity Supply Industry = 60
The Data = 60
Why Do We Need Econometrics? = 61
The Cobb-Douglas Technology = 62
How Do We Know Things Are Cobb-Douglas? 63
Are the OLS Assumptions Satisfied? = 64
Restricted Least Squares = 65
Testing the Homogeneity of the Cost Function = 65
Detour : A Cautionary Note on R2 = 67
Testing Constant Returns to Scale = 67
Importance of Plotting Residuals = 68
Subsequent Developments = 68
Problem Set = 71
Answers to Selected Questions = 84
2 Large-Sample Theory = 88
2.1 Review of Limit Theorems for Sequences of Random Variables = 88
Various Modes of Convergence = 89
Three Useful Results = 92
Viewing Estimators as Sequences of Random Variables = 94
Laws of Large Numbers and Central Limit Theorems = 95
2.2 Fundamental Concepts in Time-Series Analysis = 97
Need for Ergodic Stationarity = 97
Various Classes of Stochastic Processes = 98
Different Formulation of Lack of Serial Dependence = 106
The CLT for Ergodic Stationary Martingale Differences Sequences = 106
2.3 Large-Sample Distribution of the OLS Estimator = 109
The Model = 109
Asymptotic Distribution of the OLS Estimator = 113
s2 Is Consistent = 115
2.4 Hypothesis Testing = 117
Testing Linear Hypotheses = 117
The Test Is Consistent = 119
Asymptotic Power = 120
Testing Nonlinear Hypotheses = 121
2.5 Estimating E( ?i 2 xi xi ) Consistently = 123
Using Residuals for the Errors = 123
Data Matrix Representation of S = 125
Finite-Sample Considerations = 125
2.6 Implications of Conditional Homoskedasticity = 126
Conditional versus Unconditional Homoskedasticity = 126
Reduction to Finite-Sample Formulas = 127
Large-Sample Distribution of t and F Statistics = 128
Variations of Asymptotic Tests under Conditional Homoskedasticity = 129
2.7 Testing Conditional Homoskedasticity = 131
2.8 Estimation with Parameterized Conditional Heteroskedasticity(optional) = 133
The Functional Form = 133
WLS with Known α = 134
Regression of ei 2 on zi Provides a Consistent Estimate of α = 135
WLS with Estimated α = 136
OLS versus WLS = 137
2.9 Least Squares Projection = 137
Optimally Predicting the Value of the Dependent Variable = 138
Best Linear Predictor = 139
OLS Consistently Estimates the Projection Coefficients = 140
2.10 Testing for Serial Correlation = 141
Box-Pierce and Ljung-Box = 142
Sample Autocorrelations Calculated from Residuals = 144
Testing with Predetermined, but Not Strictly Exogenous, Regressors = 146
An Auxiliary Regression-Based Test = 147
2.11 Application : Rational Expectations Econometrics = 150
The Efficient Market Hypotheses = 150
Testable Implications = 152
Testing for Serial Correlation = 153
Is the Nominal Interest Rate the Optimal Predictor? = 156
Rt Is Not Strictly Exogenous = 158
Subsequent Developments = 159
2.12 Time Regressions = 160
The Asymptotic Distribution of the OLS Estimator = 161
Hypothesis Testing for Time Regressions = 163
Appendix 2.A : Asymptotics with Fixed Regressors = 164
Appendix 2.B : Proof of Proposition 2.10 = 165
Problem Set = 168
Answers to Selected Questions = 183
3 Single-Equation GMM = 186
3.1 Endogeneity Bias : Working's Example = 187
A Simultaneous Equations Model of Market Equilibrium = 187
Endogeneity Bias = 188
Observable Supply Shifters = 189
3.2 More Examples = 193
A Simple Macroeconometric Model = 193
Errors-in-Variables = 194
Production Function = 196
3.3 The General Formulation = 198
Regressors and Instruments = 198
Identification = 200
Order Condition for Identification = 202
The Assumption for Asymptotic Normality = 202
3.4 Generalized Method of Moments Defined = 204
Method of Moments = 205
Generalized Method of Moments = 206
Sampling Error = 207
3.5 Large-Sample Properties of GMM = 208
Asymptotic Distribution of the GMM Estimator = 209
Estimation of Error Variance = 210
Hypothesis Testing = 211
Estimation of S = 212
Efficient GMM Estimator = 212
Asymptotic Power = 214
Small-Sample Properties = 215
3.6 Testing Overidentifying Restrictions = 217
Testing Subsets of Orthogonality Conditions = 218
3.7 Hypothesis Testing by the Likelihood-Ratio Principle = 222
The LR Statistic for the Regression Model = 223
Variable Addition Test(optional) = 224
3.8 Implications of Conditional Homoskedasticity = 225
Efficient GMM Becomes 2SLS = 226
J Becomes Sargan's Statistic = 227
Small-Sample Properties of 2SLS = 229
Alternative Derivations of 2SLS = 229
When Regressors Are Predetermined = 231
Testing a Subset of Orthogonality Conditions = 232
Testing Conditional Homoskedasticity = 234
Testing for Serial Correlation = 234
3.9 Application : Returns from Schooling = 236
The NLS-Y Data = 236
The Semi-Log Wage Equation = 237
Omitted Variable Bias = 238
IQ as the Measure of Ability = 239
Errors-in-Variables = 239
2SLS to Correct for the Bias = 242
Subsequent Developments = 243
Problem Set = 244
Answers to Selected Questions = 254
4 Multiple-Equation GMM = 258
4.1 The Multiple-Equation Model = 259
Linearity = 259
Stationarity and Ergodicity = 260
Orthogonality Conditions = 261
Identification = 262
The Assumption for Asymptotic Normality = 264
Connection to the "Complete" System of Simultaneous Equations = 265
4.2 Multiple-Equation GMM Defined = 265
4.3 Large-Sample Theory = 268
4.4 Single-Equation versus Multiple-Equation Estimation = 271
When Are They "Equivalent"? = 272
Joint Estimation Can Be Hazardous = 273
4.5 Special Cases of Multiple-Equation GMM : FIVE, 3SLS, and SUR = 274
Conditional Homoskedasticity = 274
Full-Information Instrumental Variables Efficient(FIVE) = 275
Three-Stage Least Squares(3SLS) = 276
Seemingly Unrelated Regressions(SUR) = 279
SUR versus OLS = 281
4.6 Common Coefficients = 286
The Model with Common Coefficients = 286
The GMM Estimator = 287
Imposing Conditional Homoskedasticity = 288
Pooled OLS = 290
Beautifying the Formulas = 292
The Restriction That Isn't = 293
4.7 Application : Interrelated Factor Demands = 296
The Translog Cost Function = 296
Factor Shares = 297
Substitution Elasticities = 298
Properties of Cost Functions = 299
Stochastic Specifications = 300
The Nature of Restrictions = 301
Multivariate Regression Subject to Cross-Equation Restrictions = 302
Which Equation to Delete? = 304
Results = 305
Problem Set = 308
Answers to Selected Questions = 320
5 Panel Data = 323
5.1 The Error-Components Model = 324
Error Components = 324
Group Means = 327
A Reparameterization = 327
5.2 The Fixed-Effects Estimator = 330
The Formula = 330
Large-Sample Properties = 331
Digression : When ηi Is Spherical = 333
Random Effects versus Fixed Effects = 334
Relaxing Conditional Homoskedasticity = 335
5.3 Unbalanced Panels(optional) = 337
"Zeroing Out" Missing Observations = 338
Zeroing Out versus Compression = 339
No Selectivity Bias = 340
5.4 Application : International Differences in Growth Rates = 342
Derivation of the Estimation Equation = 342
Appending the Error Term = 343
Treatment of αi = 344
Consistent Estimation of Speed of Convergence = 345
Appendix 5.A : Distribution of Hausman Statistic = 346
Problem Set = 349
Answers to Selected Questions = 363
6 Serial Correlation = 365
6.1 Modeling Serial Correlation : Linear Processes = 365
MA(q) = 366
MA(∞) as a Mean Square Limit = 366
Filters = 369
Inverting Lag Polynomials = 372
6.2 ARMA Processes = 375
AR(1) and Its MA(∞) Representation = 376
Autocovariances of AR(1) = 378
AR(p) and Its MA(∞) Representation = 378
ARMA(p, q) = 380
ARMA(p, q) with Common Roots = 382
Invertibility = 383
Autocovariance-Generating Function and the Spectrum = 383
6.3 Vector Processes = 387
6.4 Estimating Autoregressions = 392
Estimation of AR(1) = 392
Estimation of AR(p) = 393
Choice of Lag Length = 394
Estimation of VARs = 397
Estimation of ARMA(p, q) = 398
6.5 Asymptotics for Sample Means of Serially Correlated Processes = 400
LLN for Covariance-Stationary Processes = 401
Two Central Limit Theorems = 402
Multivariate Extension = 404
6.6 Incorporating Serial Correlation in GMM = 406
The Model and Asymptotic Results = 406
Estimating S When Autocovariances Vanish after Finite Lags = 407
Using Kernels to Estimate S = 408
VARHAC = 410
6.7 Estimation under Conditional Homoskedasticity(Optional) = 413
Kernel-Based Estimation of S under Conditional Homoskedasticity = 413
Data Matrix Representation of Estimated Long-Run Variance = 414
Relation to GLS = 415
6.8 Application : Forward Exchange Rates as Optimal Predictors = 418
The Market Efficiency Hypothesis = 419
Testing Whether the Unconditional Mean Is Zero = 420
Regression Tests = 423
Problem Set = 428
Answers to Selected Questions = 441
7 Extremum Estimators = 445
7.1 Extremum Estimators = 446
"Measurability" of = 446
Two Classes of Extremum Estimators = 447
Maximum Likelihood(ML) = 448
Conditional Maximum Likelihood = 450
Invariance of ML = 452
Nonlinear Least Squares(NLS) = 453
Linear and Nonlinear GMM = 454
7.2 Consistency = 456
Two Consistency Theorems for Extremum Estimators = 456
Consistency of M-Estimators = 458
Concavity after Reparameterization = 461
Identification in NLS and ML = 462
Consistency of GMM = 467
7.3 Asymptotic Normality = 469
Asymptotic Normality of M-Estimators = 470
Consistent Asymptotic Variance Estimation = 473
Asymptotic Normality of Conditional ML = 474
Two Examples = 476
Asymptotic Normality of GMM = 478
GMM versus ML = 481
Expressing the Sampling Error in a Common Format = 483
7.4 Hypothesis Testing = 487
The Null Hypothesis = 487
The Working Assumptions = 489
The Wald Statistic = 489
The Lagrange Multiplier(LM) Statistic = 491
The Likelihood Ratio(LR) Statistic = 493
Summary of the Trinity = 494
7.5 Numerical Optimization = 497
Newton-Raphson = 497
Gauss-Newton = 498
Writing Newton-Raphson and Gauss-Newton in a Common Format = 498
Equations Nonlinear in Parameters Only = 499
Problem Set = 501
Answers to Selected Questions = 505
8 Examples of Maximum Likelihood = 507
8.1 Qualitative Response (QR) Models = 507
Score and Hessian for Observation t = 508
Consistency = 509
Asymptotic Normality = 510
8.2 Truncated Regression Models = 511
The Model = 511
Truncated Distributions = 512
The Likelihood Function = 513
Reparameterizing the Likelihood Function = 514
Verifying Consistency and Asymptotic Normality = 515
Recovering Original Parameters = 517
8.3 Censored Regression(Tobit) Models = 518
Tobit Likelihood Function = 518
Reparameterization = 519
8.4 Multivariate Regressions = 521
The Multivariate Regression Model Restated = 522
The Likelihood Function = 523
Maximizing the Likelihood Function = 524
Consistency and Asymptotic Normality = 525
8.5 FIML = 526
The Multiple-Equation Model with Common Instruments Restated = 526
The Complete System of Simultaneous Equations = 529
Relationship between ( Γ0 , B0 ) and δ0 = 530
The FIML Likelihood Function = 531
The FIML Concentrated Likelihood Function = 532
Testing Overidentifying Restrictions = 533
Properties of the FIML Estimator = 533
ML Estimation of the SUR Model = 535
8.6 LIML = 538
LIML Defined = 538
Computation of LIML = 540
LIML versus 2SLS = 542
8.7 Serially Correlated Observations = 543
Two Questions = 543
Unconditional ML for Dependent Observations = 545
ML Estimation of AR(1) Processes = 546
Conditional ML Estimation of AR(1) Processes = 547
Conditional ML Estimation of AR(p) and VAR(p) Processes = 549
Problem Set = 551
9 Unit-Root Econometrics = 557
9.1 Modeling Trends = 557
Integrated Processes = 558
Why Is It Important to Know if the Process Is Ⅰ(1)? = 560
Which Should Be Taken as the Null, Ⅰ(0) or Ⅰ(1)? = 562
Other Approaches to Modeling Trends = 563
9.2 Tools for Unit-Root Econometrics = 563
Linear Ⅰ(0) Processes = 563
Approximating Ⅰ(1) by a Random Walk = 564
Relation to ARMA Models = 566
The Wiener Process = 567
A Useful Lemma = 570
9.3 Dickey-Fuller Tests = 573
The AR(1) Model = 573
Deriving the Limiting Distribution under the Ⅰ(1) Null = 574
Incorporating the Intercept = 577
Incorporating Time Trend = 581
9.4 Augmented Dickey-Fuller Tests = 585
The Augmented Autoregression = 585
Limiting Distribution of the OLS Estimator = 586
Deriving Test Statistics = 590
Testing Hypotheses aboutζ = 591
What to Do When p Is Unknown? = 592
A Suggestion for the Choice of Pm ax (T) = 594
Including the Intercept in the Regression = 595
Incorporating Time Trend = 597
Summary of the DF and ADF Tests and Other Unit-Root Tests = 599
9.5 Which Unit-Root Test to Use? = 601
Local-to-Unity Asymptotics = 602
Small-Sample Properties = 602
9.6 Application : Purchasing Power Parity = 603
The Embarrassing Resiliency of the Random Walk Model? = 604
Problem Set = 605
Answers to Selected Questions = 619
10 Cointegration = 623
10.1 Cointegrated Systems = 624
Linear Vector Ⅰ(0) and Ⅰ(1) Processes = 624
The Beveridge-Nelson Decomposition = 627
Cointegration Defined = 629
10.2 Alternative Representations of Cointegrated Systems = 633
Phillips's Triangular Representation = 633
VAR and Cointegration = 636
The Vector Error-Correction Model(VECM) = 638
Johansen's ML Procedure = 640
10.3 Testing the Null of No Cointegration = 643
Spurious Regressions = 643
The Residual-Based Test for Cointegration = 644
Testing the Null of Cointegration = 649
10.4 Inference on Cointegrating Vectors = 650
The SOLS Estimator = 650
The Bivariate Example = 652
Continuing with the Bivariate Example = 653
Allowing for Serial Correlation = 654
General Case = 657
Other Estimators and Finite-Sample Properties = 658
10.5 Application : The Demand for Money in the United States = 659
The Data = 660
(m-p, y, R) as a Cointegrated System = 660
DOLS = 662
Unstable Money Demand? = 663
Problem Set = 665
Appendix A : Partitioned Matrices and Kronecker Products = 670
Addition and Multiplication of Partitioned Matrices = 671
Inverting Partitioned Matrices = 672
Index = 675
