Econometrics

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([not displayable]) 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 [alpha] 134 Regression of e2i on zi Provides a Consistent Estimate of [alpha] 135 WLS with Estimated [alpha] 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 Estimates 161 Hypothesis Testing for Time Regressions 163 2.A Asymptotics with Fixed Regressors 164 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 [eta]i 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 [alpha]i 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([infinity]) as a Mean Square Limit 366 Filters 369 Inverting Lag Polynomials 372 6.2 ARMA Processes 375 AR(1) and Its MA([infinity]) Representation 376 Autocovariances of AR(1) 378 AR(p) and Its MA([infinity]) Representation 378 ARMA(p,q) 380 ARMA(p) 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 [theta] 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 ([Gamma]0, [Beta]0) and [delta]0 530 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 about [zeta] 591 What to Do When p Is Unknown? 592 A Suggestion for the Choice of pmax(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. Partitioned Matrices and Kronecker Products 670 Addition and Multiplication of Partitioned Matrices 671 Inverting Partitioned Matrices 672