Introduction to the Theory and Practice of Econometrics

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