Introduction to the Theory and Practice of Econometrics (Probability & Mathematical Statistics) (Wiley series in probability and mathematical statistics)

This Second Edition of the highly acclaimed introduction to econometrics retains its comprehensive nature and strong authorship, while incorporating much new material. New to this edition are a complete treatment of Bayesian inference, sampling theory, an appendix on linear algebra, and a computer handbook. Presentation covers modern statistical models and focuses on the sampling theory process by which the data were generated, and the statistical consequences of alternative decisions under uncertainty. Asymptotics are introduced early on, for use throughout. Includes at least one applied example to illustrate each model, and contains many analytical and numerical exercises. Contents......Page 11 1.1 The Nature of Econometrics......Page 31 1.2.1 Postulation......Page 32 1.2.2 Experimentation......Page 33 1.3 The Nonexperimental Model Building Restriction......Page 34 1.4 Objectives of the Chapters Ahead......Page 35 1.5 Organization of the Book......Page 36 Part 1 Foundations: Statistical Model Specification, Estimation, and Inference......Page 39 2.2 Sampling, the Sample Space, and Random Variables......Page 41 2.2.1 Random Variables and Probability Density Function......Page 42 2.2.2 Mean and Variance......Page 45 2.3 The Linear Statistical Model for Estimating the Mean and Variance of a Random Variable......Page 47 2.4 Estimator for the Mean of a Random Variable......Page 54 2.5 The Sampling Theory Approach to Inference......Page 56 2.6 Estimator for the Variance of a Random Variable......Page 60 2.7 Monte Carlo Experiments For the Linear Statistical Model For the Mean......Page 62 2.8 Large Sample Properties of the Least Squares Estimator of the Mean......Page 64 2.10 Exercises......Page 65 2.10.2 Group Class Exercises......Page 66 2.11 References......Page 67 3.1 The Normal Linear Statistical Model......Page 68 3.2 The Maximum Likelihood Approach to Estimation......Page 70 3.3.2 Empirical......Page 73 3.4.1 Theoretical......Page 75 3.4.2 Empirical......Page 76 3.5 The Cramer-Rao Lower Bound......Page 77 3.6 Summary......Page 80 3.7.2 Group or Class Exercises......Page 81 3.7.3 Algebraic Exercises......Page 82 3.8 References......Page 83 4.2.1 Interval Estimation of B when o2 is Known......Page 84 4.2.2 Interval Estimation of B when o2 is Not Known......Page 86 4.3 An Interval Estimator for o2......Page 89 4.4.1 Likelihood Ratio Tests......Page 90 4.4.2 The Likelihood Ratio Test for the Mean of a Normal Population when o2 is Not Known......Page 95 4.5 The Likelihood Ratio Test for the Variance of a Normal Population......Page 97 4.6 Summary......Page 98 4.7.1 Individual Exercises......Page 99 4.8 References......Page 100 5.1 Bayesian Inference and Bayes’ Theorem......Page 102 5.2.1 A Noninformative Prior......Page 104 5.2.2 An Informative Prior......Page 106 5.2.2a The Gamma Function......Page 108 5.2.3 An Example of an Informative Prior Density......Page 109 5.3 Posterior Distributions......Page 112 5.3.1 Joint Posterior Density from a Noninformative Prior......Page 113 5.4 Marginal Distributions for the Mean and Variance......Page 114 5.4.1 The Example Continued......Page 118 5.5.1 The Bayesian Point Estimator for a Quadratic Loss Function......Page 120 5.5.2 The Bayesian Point Estimator for a Linear Loss Function......Page 121 5.5.3 A Relationship Between Bayesian and Sampling Theory Estimators......Page 122 5.6.1 Interval Estimation......Page 125 5.6.2 Hypothesis Testing......Page 128 5.7 Summary......Page 132 5.8.2 Exercises Using Monte Carlo Data......Page 133 5.9 References......Page 135 Part 2 The General Linear Statistical Model......Page 137 Chapter 6 The General Linear Statistical Model......Page 139 6.1 Specification of the Statistical Model......Page 140 6.1.2 The Sampling Process......Page 142 6.1.3 The Statistical Model......Page 143 6.1.4 An Example......Page 145 6.1.5 A Critique of the Model......Page 146 6.2 Point Estimation......Page 148 6.2.2 The Least Squares Criterion......Page 150 6.2.3 Minimizing the Quadratic Form......Page 152 6.2.4 On Solving a System of Linear Equations......Page 154 6.2.6 An Example......Page 157 6.3.1 The Mean of the Least Squares Estimator......Page 163 6.3.2 The Covariance Matrix......Page 164 6.4 Sampling Performance -- The Gauss—Markov Result......Page 166 6.5 Estimating the Scalar o2......Page 169 6.5.1 Estimating the Covariance Matrix for b......Page 172 6.6.1 Prediction......Page 173 6.6.3 Degree of Explanation......Page 175 6.7.1 The Sampling Experiment......Page 178 6.7.2 The Sampling Results......Page 180 6.8 Some Concluding Remarks......Page 183 6.9 Exercises......Page 184 6.9.3 Other Exercises......Page 185 6.10 References......Page 187 7.1.1 Analytical Representation of the Sample Information......Page 189 7.1.2 The Criterion......Page 190 7.1.3a Maximum Likelihood Estimator for B......Page 191 7.1.3b Maximum Likelihood Estimator for o......Page 193 7.1.3c Distribution of the Quadratic Form e'(I-X(X'X)X')e......Page 194 7.1.4 Sampling Performance of B and o2......Page 198 7.1.5 Summary Statement......Page 199 7.1.6 A Sampling Experiment......Page 200 7.1.6a The Sampling Results......Page 201 7.2.1 The Coefficient Vector for the Orthonormal Case......Page 204 7.2.1a Single Coefficient......Page 205 7.2.1b Two or More Coefficients Jointly......Page 207 7.2.2a Single Linear Combination of the B vector......Page 210 7.2.2b Two or More Linear Combinations of the B Vector Considered Jointly......Page 212 7.2.2c An Example of Joint Confidence Intervals......Page 214 7.2.3 Interval Estimation of o”......Page 215 7.2.4 Prediction Interval Estimates......Page 217 7.3 Hypothesis Testing......Page 219 7.3.1 The Likelihood Ratio Test......Page 220 7.3.1a Distribution of the Ratio of Quadratic Forms......Page 222 7.3.2 General Linear Hypotheses......Page 225 7.3.2a The Restricted Maximum Likelihood Estimator......Page 226 7.3.3 Single Hypothesis......Page 229 7.3.4 Testing a Hypothesis about o?......Page 232 7.4 Summary Statement......Page 233 7.5.1 Individual Exercises......Page 234 7.6 References......Page 236 7.A.2 Matrix Algebra Relevant to Normal Theory......Page 237 7.4.3 Normal Distributions......Page 239 7.A.4 Chi-square, t, and F Distributions......Page 240 7.A.5 Kronecker Products, Partitioned Inverse and Lag Operators......Page 242 8.1 The Bayesian Approach......Page 244 8.2.1 Specification of the Noninformative Prior......Page 245 8.2.2 The Joint Posterior Density Function Derived From a Noninformative Prior......Page 247 8.2.3 Marginal Posterior Density Functions from a Noninformative Prior......Page 248 8.2.3a A Digression on the Multivariate-t Distribution......Page 249 8.2.3b Marginal Posterior Density for a Single Element of B......Page 250 8.3.1 Specification of a Natural Conjugate Prior......Page 251 8.3.2 An Example of a Natural Conjugate Prior......Page 253 8.3.3 Assessment of Informative Priors......Page 257 8.3.4 Posterior Density Functions Derived from an Informative Prior......Page 258 8.3.5 The Example Continued......Page 260 8.4.1 Point Estimation......Page 262 8.4.2 Interval Estimation......Page 264 8.5 Hypothesis Testing......Page 266 8.6 Concluding Remarks......Page 272 8.7.1 General Exercises......Page 273 8.7.2 Exercises Using Monte Carlo Data......Page 274 8.8 References......Page 275 Part 3 The Generalized Linear Statistical Model......Page 277 Chapter 9 Linear Stochastic Regressor Models and Asymptotic Theory......Page 279 9.1.1 Estimation Procedures......Page 280 9.1.2 Sampling Properties of the Least Squares Estimator......Page 288 9.2 The Problem with a Partially Independent Stochastic Regressor Model......Page 290 9.3.1 Consistency and Probability Limits......Page 293 9.3.1a The Probability Limit......Page 295 9.3.1b Probability Limits of Vectors and Matrices......Page 297 9.3.1c Least Squares Consistency......Page 298 9.3.2 Asymptotic Distributions......Page 299 9.3.2a Asymptotic Distribution of a Sample Mean and Least Squares Estimation......Page 301 9.3.2b The Asymptotic Properties of Maximum Likelihood Estimators......Page 304 9.4 Asymptotic Properties of the Least Squares Estimator for the Partially Independent Stochastic Regressor Model......Page 305 9.5 The Problem With General Linear Stochastic Regressor Models......Page 306 9.6 Instrumental Variable Estimation......Page 309 9.7 Summary......Page 311 9.8.2 Individual Numerical Exercises......Page 312 9.9 References......Page 314 9.A Appendix -- Multivariate Normal Regressor Model......Page 315 Chapter 10 General Linear Statistical Model with Non Scalar-Identity Covariance Matrix......Page 319 10.1.1 The Least Squares Estimator of B......Page 320 10.1.2 The Generalized Least Squares Estimator......Page 321 10.1.3 An Unbiased Estimator for o2......Page 325 10.2 The Normal Linear Statistical Model......Page 326 10.4 Interval Estimators......Page 328 10.6 The Consequences of Using Least Squares Procedures......Page 330 10.7 Sampling Experiment......Page 332 10.8 Estimation and Hypothesis Testing when V is Unknown......Page 338 10.9 Summary......Page 340 10.10.1 Algebraic Exercises......Page 341 10.10.2 Individual Numerical Exercises......Page 342 10.11 References......Page 344 11.1 Sets of Equations with Contemporaneously Correlated Disturbances......Page 345 11.1.1 The Kronecker Product......Page 347 11.1.2 Estimation with Known Covariance Matrix......Page 349 11.1.3 Estimation if the Covariance Matrix is Unknown......Page 351 11.1.3a An Example......Page 352 11.1.3b A Monte Carlo Experiment......Page 353 11.1.3d An Iterative Estimation Procedure......Page 354 11.1.4 Hypothesis Testing......Page 356 11.2.2 An Example......Page 358 11.4 Exercises......Page 361 11.4.2 Kronecker Product......Page 362 11.5 References......Page 363 Part 4 Simultaneous Linear Statistical Models......Page 365 12.1 Introduction......Page 367 12.2 Specification of the Sampling Model......Page 369 12.2.1 The Statistical Model......Page 370 12.3 Least Squares Bias......Page 375 12.4 Estimation of the Reduced Form Parameters......Page 378 12.5 The Problem of Going from the Reduced Form Parameters to the Structural Parameters......Page 379 12.5.1 Types of Prior Information for Restrictions......Page 380 12.5.2 Indirect Least Squares; An Example......Page 382 12.6 Identifying an Equation within a System of Equations......Page 386 12.7 Some Examples of Model Formulation, Identification and Estimation......Page 392 12.8 Summary Statement......Page 395 12.9.1 Algebraic Exercises......Page 396 12.9.2 Individual Numerical Exercises......Page 397 12.9.3 Numerical Group Exercises......Page 399 12.10 References......Page 400 Chapter 13 Estimation and Inference for Simultaneous Equation Statistical Models......Page 401 13.1.1 The Indirect Least Squares Approach......Page 402 13.1.2 The Generalized Least Squares Approach......Page 404 13.1.2a A Generalized Least Squares Estimator......Page 405 13.1.2b Sampling Properties......Page 406 13.1.2c A Two-Stage Least Squares Estimator......Page 407 13.2 The Search for an Asymptotically Efficient Estimator......Page 408 13.2.1 The Three-stage Least Squares (3SLS) Estimator......Page 409 13.2.2 Sampling Properties......Page 411 13.2.3 Estimator Comparisons......Page 412 13.2.4 Limited and Full Information Maximum Likelihood Methods......Page 414 13.3 Asymptotic and Finite Sampling Properties of the Alternative Estimators......Page 416 13.4 An Example......Page 418 13.5 On Using the Results of Econometric Models for Forecasting and Decision Processes......Page 425 13.6 Summary Statement......Page 427 13.7.2 Individual Numerical Exercises......Page 429 13.7.3 Group Exercises......Page 430 13.A Appendix -- Asymptotic Sampling Properties of the 2SLS and 3SLS Estimators......Page 431 Part 5 Some Procedures for Handling an Unknown Covariance Matrix......Page 437 14.1 Background......Page 439 14.2 Economic and Statistical Environment......Page 440 14.3 Generalized Least Squares Estimation......Page 442 14.3.1 An Example where Variances are Known......Page 444 14.4 Unknown Variances......Page 445 14.4.1 Estimating the Variances......Page 446 14.4.2 The Estimated Generalized Least Squares Estimator......Page 448 14.5 Testing for Heteroscedasticity......Page 450 14.5.2 The Goldfeld-Quandt Test......Page 451 14.5.3 The Breusch-Pagan Test......Page 452 14.6 General Comments......Page 453 14.7 An Example......Page 454 14.8.1 Algebraic Exercises......Page 459 14.8.2 Individual Numerical Exercises......Page 460 14.8.3 Group Exercises......Page 462 14.9 References......Page 463 15.1 Background and Model......Page 465 15.2.2 Generalized Least Squares Estimation......Page 469 15.2.3 Estimated Generalized Least Squares Estimation......Page 472 15.2.4 Nonlinear Least Squares Estimation......Page 474 15.2.5 Maximum Likelihood Estimation......Page 476 15.3.1 An Asymptotic Test......Page 478 15.3.2 The Durbin-Watson Test......Page 479 15.3.2a Summary......Page 484 15.3.2b An Example......Page 485 15.3.3 Durbin’s h Statistic......Page 486 15.4 Prediction Implications of Autocorrelated Errors......Page 487 15.4.1 Best Linear Unbiased Prediction......Page 489 15.5 An Example......Page 492 15.6 Some Remarks......Page 495 15.7.1 General Exercises......Page 496 15.7.2 Individual Exercises using Monte Carlo Data......Page 499 15.7.3 Group Exercises using Monte Carlo Data......Page 502 15.8 References......Page 503 Part 6 Pooling of Data and Varying Parameter Models......Page 505 16.1 Background and Model......Page 507 16.2 The Dummy Variable Model......Page 508 16.2.1 Parameter Estimation......Page 509 16.2.3 An Alternative Parameterization......Page 512 16.2.4 Testing the Dummy Variable Coefficients......Page 514 16.2.5 An Example......Page 515 16.3 The Error Components Model......Page 518 16.3.1 Generalized Least Squares Estimation......Page 520 16.3.2 Estimation of Variance Components......Page 522 16.3.3 Prediction of Random Components......Page 524 16.3.5 The Example Continued......Page 525 16.4 Fixed or Random Effects?......Page 527 16.5 General Comments......Page 528 16.6.1 Algebraic Exercises......Page 529 16.6.3 Group Exercises......Page 531 16.7 References......Page 532 17.2 Random Coefficient Models......Page 533 17.3 Systematically Varying Parameter Models......Page 535 17.3.1 A Basic Model......Page 536 17.3.2a Seasonality Models......Page 537 17.3.2b Piecewise Regression Models......Page 539 17.4 Summary......Page 540 17.5 Exercises......Page 541 17.6 References......Page 543 Part 7 Unobservable and Qualitative Variables......Page 545 18.2 Binary Choice Models when Repeated Observations are Available......Page 547 18.2.1 The Probit Model......Page 549 18.2.2 The Logit Model......Page 551 18.3 Binary Choice Models when Repeated Observations are not Available......Page 552 18.4 Models with Limited Dependent Variables......Page 556 18.6 Exercises......Page 558 18.7 References......Page 560 Chapter 19 Unobservable Variables......Page 561 19.1 Statistical Consequences of Errors in Variables......Page 562 19.2 A Maximum Likelihood Estimator when both X and Y are Measured Subject to Error......Page 564 19.3 Additional Information in the Form of Additional Equations for Nuisance Parameters......Page 567 19.4 An Example......Page 571 19.5 Summary......Page 573 19.6.1 Algebraic Exercises......Page 574 19.6.2 Individual Exercises......Page 575 19.7 References......Page 577 Part 8 Nonsample Information, Biased Estimation, and Choosing the Dimension and Form of the Design Matrix......Page 579 Chapter 20 The Use of Nonsample Information......Page 581 20.1 Exact Prior Information......Page 582 20.1.1 Mean and Covariance......Page 584 20.1.2 Consequences of Incorrect Restrictions......Page 586 20.1.3 On Gauging Estimator Performance......Page 587 20.1.4 General Linear Hypotheses......Page 590 20.2 Stochastic Linear Restrictions......Page 592 20.2.1 The Statistical Model......Page 593 20.2.2 The Estimator......Page 594 20.2.3 Sampling Comparisons......Page 595 20.2.4 Stochastic Linear Hypotheses......Page 596 20.3.1 The Inequality Restricted Estimator......Page 598 20.3.2 The Sampling Properties......Page 600 20.3.2a The Mean......Page 601 20.3.2b The Risk......Page 603 20.3.3 Hypothesis Testing......Page 604 20.4 Summary Statement......Page 605 20.5.1 Individual Exercises for Section 20.1......Page 606 20.5.4 Joint or Class Exercises for Section 20.2......Page 607 20.6 References......Page 608 21.1 Pretest Estimators......Page 609 21.1.1b Risk......Page 611 21.1.1c The Optimal Level of Significance......Page 614 21.2.1 The James and Stein Rule......Page 615 21.2.2 A Reformulated Rule......Page 617 21.3 A Stein-like Pretest Estimator......Page 618 21.4 Some Remarks......Page 620 21.5.1 Individual Exercises......Page 621 21.6 References......Page 622 22.1 Introduction......Page 623 22.2 Statistical Consequences of an Incorrect Design Matrix......Page 624 22.2.1 Mean Square Error Norms......Page 628 22.4 Ad Hoc Selection Rules......Page 630 22.4.1 The R2 and R2 Criteria......Page 631 22.4.2 The Cp Criterion......Page 632 22.4.3 Amemiya Prediction Criterion (PC)......Page 633 22.5.1 A Stein-Rule Formulation......Page 634 22.5.2 The Positive Stein Rule......Page 636 22.7 Exercises......Page 637 22.7.1 Individual Exercises......Page 638 22.8 References......Page 639 23.1 Introduction......Page 640 23.2.1 The Effects of Pairwise Linear Associations......Page 642 23.2.2 The Effects of General Linear Associations......Page 644 23.2.4 The Effects of Multicollinearity......Page 647 23.3.1 Methods for Detecting Multicollinearity......Page 650 23.3.2 An Example -- The Klein-Goldberger Consumption......Page 652 23.4.1 Additional Sample Information......Page 654 23.4.2 Exact Linear Constraints......Page 656 23.4.3 Stochastic Linear Restrictions......Page 657 23.6 Exercises......Page 658 23.7 References......Page 660 Part 9 The Nonlinear Statistical Model......Page 661 Chapter 24 Nonlinear Regression Models......Page 663 24.1 Introduction......Page 664 24.2 Parameter Estimation in Nonlinear Statistical Models......Page 665 24.2.1 Least Squares Estimation......Page 666 24.2.2 Maximum Likelihood Estimation......Page 667 24.2.3 The Asymptotic Variance Covariance Matrix of Buy......Page 669 24.2.4 The Asymptotic Distribution of B_LS......Page 670 24.3 Computing the Estimates......Page 673 24.3.1 The Gauss Method......Page 675 24.3.2 Gradient Methods......Page 679 24.4 Using Nonsample Information......Page 682 24.4.1 Equality Constraints......Page 683 24.4.2 Inequality Constraints......Page 685 24.5 Confidence Regions and Hypothesis Testing......Page 687 24.5.1 Confidence Intervals and Confidence Regions......Page 688 24.5.2 Hypothesis Testing......Page 689 24.6 Some Extensions......Page 691 24.7 Problems......Page 692 24.8 References......Page 693 Part 10 Time Series and Distributed Lag Models......Page 695 25.1 Introduction......Page 697 25.2.1 Stochastic Processes......Page 698 25.2.2 The Autocovariance and Autocorrelation Functions......Page 699 25.2.4 The Lag Operator......Page 701 25.3 Autoregressive Processes......Page 703 25.3.1 Estimation of Autoregressive Processes......Page 704 25.3.2 Partial Autocorrelations......Page 706 25.4 Moving Average Processes......Page 711 25.4.1 Determining the Order of a Moving Average......Page 712 25.4.2 Estimating the Parameters of a Moving Average......Page 714 25.5 ARIMA Models......Page 716 25.6.1 Identification......Page 720 25.6.2 Estimation......Page 722 25.6.3 Diagnostic Checking......Page 724 25.7.1 Forecasting MA Processes......Page 725 25.7.2 Forecasting ARIMA Processes......Page 727 25.8 Limitations of ARIMA Models and their Relationship to Econometric Models......Page 729 25.9 A Guide to Future Study Topics......Page 732 25.10 Problems......Page 733 25.11 References......Page 736 26.1 Introduction......Page 737 26.2 Bivariate Stochastic Processes......Page 738 26.3 Bivariate Autoregressive Processes......Page 739 26.3.1 Estimating the Parameters of a Bivariate Autoregressive Process......Page 740 26.3.2 Specification of a Bivariate Autoregressive Process......Page 743 26.3.3 An Example......Page 744 26.4 Forecasting Bivariate Autoregressive Processes......Page 746 26.5 Bivariate Time Series and Distributed Lag Models......Page 750 26.7 Problems......Page 752 26.8 References......Page 755 27.1 A Dynamic Economic Model......Page 756 27.2 Finite Distributed Lags......Page 758 27.2.1 The Almon Lag......Page 759 27.2.2 Determining the Polynomial Degree of the Almon Lag......Page 762 27.2.3 Some Problems Related to the Use of Polynomial Lags......Page 763 27.3.1 Geometric Lag Models......Page 766 27.3.2a Using Instrumental Variables......Page 767 27.3.2b Maximum Likelihood Estimation......Page 769 27.4 Seasonality and Distributed Lags......Page 771 27.5 Summary and Concluding Remarks......Page 772 27.6 Problems......Page 774 27.7 References......Page 775 Part 11 Epilogue......Page 777 28.1 Estimating the Mean and Variance of a Normal Population......Page 779 28.2 The General Linear Model......Page 785 28.3 Alternative Design Matrix and Covariance Structures......Page 788 28.4 Simultaneous Equation Linear Statistical Models......Page 793 28.5 Unknown Covariance Matrices......Page 795 28.6 Variable and Varying Parameters, Qualitative Dependent and Nonobservable Variables......Page 798 28.7 Using Nonsample Information and Choosing the Design Matrix......Page 801 28.8 The Nonlinear Statistical Model......Page 803 28.9 Time Series and Distributed Lag Models......Page 804 28.10 A Final Comment......Page 810 Index......Page 855 Introduction -- Analysis Of A Sample Of Data -- Analysis Of A Sample From A Normal Population -- Interval Estimation And Hypothesis Testing In The Normal Linear Model -- The Bayesian Approach To Estimating The Mean And Variance Of A Normal Population -- The General Linear Statistical Model -- The Normal General Linear Statistical Model -- Bayesian Estimation And Interference For The Normal Linear Statistical Model -- Linear Stochastic Regressor Models And Asymptotic Theory -- General Linear Statistical Model With Non Scalar-identity Covariance Matrix -- Disturbance-related Sets Of Regression Equations -- An Introduction To Simultaneous Linear Statistical Models -- Estimation And Interference For Simultaneous Equation Statistical Models -- Heteroscedasticity -- Autocorrelation -- Using Time Series And Cross-sectional Data -- Variable Parameter Models -- Models With Qualitative Or Limited Dependent Variables -- Unobservable Variables -- The Use Of Nonsample Information -- Biased Estimation -- Model Specification--variable Selection -- Multicollinearity -- Nonlinear Regression Models -- Time Series Analysis And Forecasting -- Analysis Of Bivariate Time Series -- Distributed Lag Models -- Summary Of Statistical Models, Estimators And Tests. George G. Judge ... [et Al.]. Includes Bibliographies And Index.