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List of Figures xvii Preface xix 1 Finite-Sample Properties of OLS 31.1 The Classical Linear Regression Model 3 The Linearity Assumption 4Matrix Notation 6 The Strict Exogeneity Assumption 7 Implications of StrictExogeneity 8 Strict Exogeneity in Time-Series Models 9 Other Assumptions ofthe 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 ofSquared Residuals 15 Normal Equations 16 Two Expressions for the OLSEstimator 18 More Concepts and Algebra 18 Influential Analysis (optional)21 A Note on the Computation of OLS Estimates 23 1.3 Finite-SampleProperties of OLS 27 Finite-Sample Distribution of b 27 Finite-SampleProperties of s2 30 Estimate of Var(b | X) 31 1.4 Hypothesis Testing underNormality 33 Normally Distributed Error Terms 33 Testing Hypotheses aboutIndividual Regression Coefficients 35 Decision Rule for the t-Test 37Confidence Interval 38 p-Value 38 Linear Hypotheses 39 The F-Test 40 AMore Convenient Expression for F 42 t versus F 43 An Example of a TestStatistic Whose Distribution Depends on X 45 1.5 Relation to MaximumLikelihood 47 The Maximum Likelihood Principle 47 Conditional versusUnconditional Likelihood 47 The Log Likelihood for the Regression Model 48ML via Concentrated Likelihood 48 Cramer-Rao Bound for the ClassicalRegression Model 49 The F-Test as a Likelihood Ratio Test 52 Quasi-MaximumLikelihood 53 1.6 Generalized Least Squares (GLS) 54 Consequence ofRelaxing Assumption 1.4 55 Efficient Estimation with Known V 55 A SpecialCase: Weighted Least Squares (WLS) 58 Limiting Nature of GLS 58 1.7Application: Returns to Scale in Electricity Supply 60 The ElectricitySupply Industry 60 The Data 60 Why Do We Need Econometrics? 61 TheCobb-Douglas Technology 62 How Do We Know Things Are Cobb-Douglas? 63 Arethe OLS Assumptions Satisfied? 64 Restricted Least Squares 65 Testing theHomogeneity of the Cost Function 65 Detour: A Cautionary Note on R2 67Testing Constant Returns to Scale 67 Importance of Plotting Residuals 68Subsequent Developments 68 Problem Set 71 Answers to Selected Questions 842 Large-Sample Theory 88 2.1 Review of Limit Theorems for Sequences ofRandom Variables 88 Various Modes of Convergence 89 Three Useful Results 92Viewing Estimators as Sequences of Random Variables 94 Laws of LargeNumbers and Central Limit Theorems 95 2.2 Fundamental Concepts inTime-Series Analysis 97 Need for Ergodic Stationarity 97 Various Classes ofStochastic Processes 98 Different Formulation of Lack of Serial Dependence106 The CLT for Ergodic Stationary Martingale Differences Sequences 106 2.3Large-Sample Distribution of the OLS Estimator 109 The Model 109 AsymptoticDistribution of the OLS Estimator 113 s2 Is Consistent 115 2.4 HypothesisTesting 117 Testing Linear Hypotheses 117 The Test Is Consistent 119Asymptotic Power 120 Testing Nonlinear Hypotheses 121 2.5 Estimating E([notdisplayable]) Consistently 123 Using Residuals for the Errors 123 DataMatrix Representation of S 125 Finite-Sample Considerations 125 2.6Implications of Conditional Homoskedasticity 126 Conditional versusUnconditional Homoskedasticity 126 Reduction to Finite-Sample Formulas 127Large-Sample Distribution of t and F Statistics 128 Variations of AsymptoticTests under Conditional Homoskedasticity 129 2.7 Testing ConditionalHomoskedasticity 131 2.8 Estimation with Parameterized ConditionalHeteroskedasticity (optional) 133 The Functional Form 133 WLS with Known[alpha] 134 Regression of e2<sub>i on z<sub>i Provides aConsistent Estimate of [alpha] 135 WLS with Estimated [alpha] 136 OLSversus WLS 137 2.9 Least Squares Projection 137 Optimally Predicting theValue of the Dependent Variable 138 Best Linear Predictor 139 OLSConsistently Estimates the Projection Coefficients 140 2.10 Testing forSerial Correlation 141 Box-Pierce and Ljung-Box 142 Sample AutocorrelationsCalculated from Residuals 144 Testing with Predetermined, but Not StrictlyExogenous, Regressors 146 An Auxiliary Regression-Based Test 147 2.11Application: Rational Expectations Econometrics 150 The Efficient MarketHypotheses 150 Testable Implications 152 Testing for Serial Correlation 153Is the Nominal Interest Rate the Optimal Predictor? 156 R<sub>t IsNot Strictly Exogenous 158 Subsequent Developments 159 2.12 TimeRegressions 160 The Asymptotic Distribution of the OLS Estimates 161Hypothesis Testing for Time Regressions 163 2.A Asymptotics with FixedRegressors 164 2.B Proof of Proposition 2.10 165 Problem Set 168 Answersto Selected Questions 183 3 Single-Equation GMM 186 3.1 Endogeneity Bias:Working's Example 187 A Simultaneous Equations Model of Market Equilibrium187 Endogeneity Bias 188 Observable Supply Shifters 189 3.2 More Examples193 A Simple Macroeconometric Model 193 Errors-in-Variables 194 ProductionFunction 196 3.3 The General Formulation 198 Regressors and Instruments 198Identification 200 Order Condition for Identification 202 The Assumptionfor Asymptotic Normality 202 3.4 Generalized Method of Moments Defined 204Method of Moments 205 Generalized Method of Moments 206 Sampling Error 2073.5 Large-Sample Properties of GMM 208 Asymptotic Distribution of the GMMEstimator 209 Estimation of Error Variance 210 Hypothesis Testing 211Estimation of S 212 Efficient GMM Estimator 212 Asymptotic Power 214Small-Sample Properties 215 3.6 Testing Overidentifying Restrictions 217Testing Subsets of Orthogonality Conditions 218 3.7 Hypothesis Testing bythe Likelihood-Ratio Principle 222 The LR Statistic for the Regression Model223 Variable Addition Test (optional) 224 3.8 Implications of ConditionalHomoskedasticity 225 Efficient GMM Becomes 2SLS 226 J Becomes Sargan'sStatistic 227 Small-Sample Properties of 2SLS 229 Alternative Derivationsof 2SLS 229 When Regressors Are Predetermined 231 Testing a Subset ofOrthogonality Conditions 232 Testing Conditional Homoskedasticity 234Testing for Serial Correlation 234 3.9 Application: Returns from Schooling236 The NLS-Y Data 236 The Semi-Log Wage Equation 237 Omitted VariableBias 238 IQ as the Measure of Ability 239 Errors-in-Variables 239 2SLS toCorrect for the Bias 242 Subsequent Developments 243 Problem Set 244Answers to Selected Questions 254 4 Multiple-Equation GMM 258 4.1 TheMultiple-Equation Model 259 Linearity 259 Stationarity and Ergodicity 260Orthogonality Conditions 261 Identification 262 The Assumption forAsymptotic Normality 264 Connection to the "Complete" System of SimultaneousEquations 265 4.2 Multiple-Equation GMM Defined 265 4.3 Large-Sample Theory268 4.4 Single-Equation versus Multiple-Equation Estimation 271 When AreThey "Equivalent"? 272 Joint Estimation Can Be Hazardous 273 4.5 SpecialCases of Multiple-Equation GMM: FIVE, 3SLS, and SUR 274 ConditionalHomoskedasticity 274 Full-Information Instrumental Variables Efficient(FIVE) 275 Three-Stage Least Squares (3SLS) 276 Seemingly UnrelatedRegressions (SUR) 279 SUR versus OLS 281 4.6 Common Coefficients 286 TheModel with Common Coefficients 286 The GMM Estimator 287 ImposingConditional Homoskedasticity 288 Pooled OLS 290 Beautifying the Formulas292 The Restriction That Isn't 293 4.7 Application: Interrelated FactorDemands 296 The Translog Cost Function 296 Factor Shares 297 SubstitutionElasticities 298 Properties of Cost Functions 299 Stochastic Specifications300 The Nature of Restrictions 301 Multivariate Regression Subject toCross-Equation Restrictions 302 Which Equation to Delete? 304 Results 305Problem Set 308 Answers to Selected Questions 320 5 Panel Data 323 5.1 TheError-Components Model 324 Error Components 324 Group Means 327 AReparameterization 327 5.2 The Fixed-Effects Estimator 330 The Formula 330Large-Sample Properties 331 Digression: When [eta]<sub>i Is Spherical333 Random Effects versus Fixed Effects 334 Relaxing ConditionalHomoskedasticity 335 5.3 Unbalanced Panels (optional) 337 "Zeroing Out"Missing Observations 338 Zeroing Out versus Compression 339 No SelectivityBias 340 5.4 Application: International Differences in Growth Rates 342Derivation of the Estimation Equation 342 Appending the Error Term 343Treatment of [alpha]<sub>i 344 Consistent Estimation of Speed ofConvergence 345 Appendix 5.A: Distribution of Hausman Statistic 346 ProblemSet 349 Answers to Selected Questions 363 6 Serial Correlation 365 6.1Modeling Serial Correlation: Linear Processes 365 MA(q) 366 MA([infinity])as a Mean Square Limit 366 Filters 369 Inverting Lag Polynomials 372 6.2ARMA Processes 375 AR(1) and Its MA([infinity]) Representation 376Autocovariances of AR(1) 378 AR(p) and Its MA([infinity]) Representation 378ARMA(p,q) 380 ARMA(p) with Common Roots 382 Invertibility 383Autocovariance-Generating Function and the Spectrum 383 6.3 Vector Processes387 6.4 Estimating Autoregressions 392 Estimation of AR(1) 392 Estimationof AR(p) 393 Choice of Lag Length 394 Estimation of VARs 397 Estimation ofARMA(p,q) 398 6.5 Asymptotics for Sample Means of Serially CorrelatedProcesses 400 LLN for Covariance-Stationary Processes 401 Two Central LimitTheorems 402 Multivariate Extension 404 6.6 Incorporating SerialCorrelation in GMM 406 The Model and Asymptotic Results 406 Estimating SWhen Autocovariances Vanish after Finite Lags 407 Using Kernels to EstimateS 408 VARHAC 410 6.7 Estimation under Conditional Homoskedasticity(Optional) 413 Kernel-Based Estimation of S under ConditionalHomoskedasticity 413 Data Matrix Representation of Estimated Long-RunVariance 414 Relation to GLS 415 6.8 Application: Forward Exchange Rates asOptimal Predictors 418 The Market Efficiency Hypothesis 419 Testing Whetherthe Unconditional Mean Is Zero 420 Regression Tests 423 Problem Set 428Answers to Selected Questions 441 7 Extremum Estimators 445 7.1 ExtremumEstimators 446 "Measurability" of [theta] 446 Two Classes of ExtremumEstimators 447 Maximum Likelihood (ML) 448 Conditional Maximum Likelihood450 Invariance of ML 452 Nonlinear Least Squares (NLS) 453 Linear andNonlinear GMM 454 7.2 Consistency 456 Two Consistency Theorems for ExtremumEstimators 456 Consistency of M-Estimators 458 Concavity afterReparameterization 461 Identification in NLS and ML 462 Consistency of GMM467 7.3 Asymptotic Normality 469 Asymptotic Normality of M-Estimators 470Consistent Asymptotic Variance Estimation 473 Asymptotic Normality ofConditional ML 474 Two Examples 476 Asymptotic Normality of GMM 478 GMMversus ML 481 Expressing the Sampling Error in a Common Format 483 7.4Hypothesis Testing 487 The Null Hypothesis 487 The Working Assumptions 489The Wald Statistic 489 The Lagrange Multiplier (LM) Statistic 491 TheLikelihood Ratio (LR) Statistic 493 Summary of the Trinity 494 7.5Numerical Optimization 497 Newton-Raphson 497 Gauss-Newton 498 WritingNewton-Raphson and Gauss-Newton in a Common Format 498 Equations Nonlinearin Parameters Only 499 Problem Set 501 Answers to Selected Questions 505 8Examples of Maximum Likelihood 507 8.1 Qualitative Response (QR) Models 507Score and Hessian for Observation t 508 Consistency 509 AsymptoticNormality 510 8.2 Truncated Regression Models 511 The Model 511 TruncatedDistributions 512 The Likelihood Function 513 Reparameterizing theLikelihood Function 514 Verifying Consistency and Asymptotic Normality 515Recovering Original Parameters 517 8.3 Censored Regression (Tobit) Models518 Tobit Likelihood Function 518 Reparameterization 519 8.4 MultivariateRegressions 521 The Multivariate Regression Model Restated 522 TheLikelihood Function 523 Maximizing the Likelihood Function 524 Consistencyand Asymptotic Normality 525 8.5 FIML 526 The Multiple-Equation Model withCommon Instruments Restated 526 The Complete System of SimultaneousEquations 529 Relationship between ([Gamma]<sub>0, [Beta]<sub>0)and [delta]<sub>0 530 The FIML Likelihood Function 531 The FIMLConcentrated Likelihood Function 532 Testing Overidentifying Restrictions533 Properties of the FIML Estimator 533 ML Estimation of the SUR Model 5358.6 LIML 538 LIML Defined 538 Computation of LIML 540 LIML versus 2SLS542 8.7 Serially Correlated Observations 543 Two Questions 543Unconditional ML for Dependent Observations 545 ML Estimation of AR.1/Processes 546 Conditional ML Estimation of AR(1) Processes 547 ConditionalML Estimation of AR(p) and VAR(p) Processes 549 Problem Set 551 9 Unit-RootEconometrics 557 9.1 Modeling Trends 557 Integrated Processes 558 Why IsIt Important to Know if the Process Is I(1)? 560 Which Should Be Taken asthe Null, I(0) or I(1)? 562 Other Approaches to Modeling Trends 563 9.2Tools for Unit-Root Econometrics 563 Linear I(0) Processes 563Approximating I(1) by a Random Walk 564 Relation to ARMA Models 566 TheWiener Process 567 A Useful Lemma 570 9.3 Dickey-Fuller Tests 573 TheAR(1) Model 573 Deriving the Limiting Distribution under the I(1) Null 574Incorporating the Intercept 577 Incorporating Time Trend 581 9.4 AugmentedDickey-Fuller Tests 585 The Augmented Autoregression 585 LimitingDistribution of the OLS Estimator 586 Deriving Test Statistics 590 TestingHypotheses about [zeta] 591 What to Do When p Is Unknown? 592 A Suggestionfor the Choice of p<sub>max(T) 594 Including the Intercept in theRegression 595 Incorporating Time Trend 597 Summary of the DF and ADF Testsand Other Unit-Root Tests 599 9.5 Which Unit-Root Test to Use? 601Local-to-Unity Asymptotics 602 Small-Sample Properties 602 9.6 Application:Purchasing Power Parity 603 The Embarrassing Resiliency of the Random WalkModel? 604 Problem Set 605 Answers to Selected Questions 619 10Cointegration 623 10.1 Cointegrated Systems 624 Linear Vector I(0) and I(1)Processes 624 The Beveridge-Nelson Decomposition 627 Cointegration Defined629 10.2 Alternative Representations of Cointegrated Systems 633 Phillips'sTriangular Representation 633 VAR and Cointegration 636 The VectorError-Correction Model (VECM) 638 Johansen's ML Procedure 640 10.3 Testingthe Null of No Cointegration 643 Spurious Regressions 643 TheResidual-Based Test for Cointegration 644 Testing the Null of Cointegration649 10.4 Inference on Cointegrating Vectors 650 The SOLS Estimator 650 TheBivariate Example 652 Continuing with the Bivariate Example 653 Allowingfor Serial Correlation 654 General Case 657 Other Estimators andFinite-Sample Properties 658 10.5 Application: the Demand for Money in theUnited States 659 The Data 660 (m - p, y, R) as a Cointegrated System 660DOLS 662 Unstable Money Demand? 663 Problem Set 665 Appendix. PartitionedMatrices and Kronecker Products 670 Addition and Multiplication ofPartitioned Matrices 671 Inverting Partitioned Matrices 672