Chapter 9
BAYESIAN ANALYSIS OF SIMULTANEOUS EQUATION
SYSTEMS
JACQUES H. DREZE and JEAN-FRANCOIS RICHARD*
Universiti Catholique de Louvain
Contents
1. Introduction and summary 519
1.1. The simultaneous equation model 519
1.2. Bayesian inference and identification 521
1.3. Bayesian treatment of exact restrictions 522
1.4. Bayesian analysis of the reduced form 523
1.5. Bayesian analysis of the structural form 524
1.6. Summary 525
1.7. Bibliographical note 526
2. A special case 526
2.1. Limited information maximum likelihood estimation 526
2.2. A Bayesian analogue 529
2.3. The normalization issue 531
2.4. An application 533
3. Identification 535
3.1. Classical concepts 535
3.2. Posterior densities and identification 536
3.3. Prior densities and identification 537
3.4. .Choice of models and identification 538
4. Reduced-form analytics 539
4.1. Natural-conjugate prior densities 539
4.2. Further results 541
5. Limited information analysis 544
5.1. Introduction 544
5.2. Parametetixation and invariance 544
5.3. Posterior conditional densities and moments 550
5.4. Posterior marginal densities 552
5.5. An application 555
5.6. Normalization and invariance 557
5.1. Two generalizations 559
6. Full information analysis 561
6.1. Introduction 561
6.2. A special case 561
6.3. Extended natural-conjugate prior densities 563
6.4. Seemingly unrelated regression models 567
6.5. Two-equation models 568
6.6. Applications 571
7. Numerics 579
7. I. Introduction 579
7.2. Evaluation of poly-t densities 579
7.3. Numerical integration 581
Appendix A: Elements of multivariate analysis 585
Appendix B: Proofs 589
Reference table 595
References 596