An Introduction to Bayesian Inference in

This is a classical reprint edition of the original 1971 edition ofAn Introduction to Bayesian Inference in Economics. This historical volume is an early introduction to Bayesian inference and methodology which still has lasting value for today's statistician and student. The coverage ranges from the fundamental concepts and operations of Bayesian inference to analysis of applications in specific econometric problems and the testing of hypotheses and models.

Arnold Zellner was a leading economist at the University of Chicago Booth School of Business who pioneered the field of Bayesian econometrics. Zellner was known for the breadth of his contributions to many different areas of econometrics. His pioneering work in systems of equations, Bayesian statistics and econometrics, or time series analysis would each have earned him worldwide recognition. An award-winning teacher, Zellner published more than 200 scholarly articles and 22 books and monographs, including An Introduction to Bayesian Inference in Econometrics, J. Wiley and Sons, Inc., 1971 and Basic Issues in Econometrics, University of Chicago Press, 1984.

I Remarks on Inference in Economics

1

(12)

1.1 The Unity of Science

1

(1)

1.2 Deductive Inference

2

(2)

1.3 Inductive Inference

4

(1)

1.4 Reductive Inference

5

(2)

1.5 Jeffreys' Rules for a Theory of Inductive Inference

7

(1)

1.6 Implications of the Rules

8

(4)

Questions and Problems

12

(1)

II Principles of Bayesian Analysis with Selected Applications

13

(45)

2.1 Bayes' Theorem

13

(4)

2.2 Bayes' Theorem and Several Sets of Data

17

(1)

2.3 Prior Probability Density Functions

18

(3)

2.4 Marginal and Conditional Posterior Distributions for Parameters

21

(3)

2.5 Point Estimates for Parameters

24

(3)

2.6 Bayesian Intervals and Regions for Parameters

27

(1)

2.7 Marginal Distribution of the Observations

28

(1)

2.8 Predictive Probability Density Functions

29

(1)

2.9 Point Prediction

30

(1)

2.10 Prediction Regions and Intervals

31

(1)

2.11 Some Large Sample Properties of Bayesian Posterior Pdf's

31

(3)

2.12 Application of Principles to Analysis of the Pareto Distribution

34

(4)

2.13 Application of Principles to Analysis of the Binomial Distribution

38

(2)

2.14 Reporting the Results of Bayesian Analyses

40

(1)

Appendix

41

(13)

Questions and Problems

54

(4)

III The Univariate Normal Linear Regression Model

58

(28)

3.1 The Simple Univariate Normal Linear Regression Model

58

(7)

3.1.1 Model and Likelihood Function

58

(2)

3.1.2 Posterior Pdf's for Parameters with a Diffuse Prior Pdf

60

(3)

3.1.3 Application to Analysis of the Investment Multiplier

63

(2)

3.2 The Normal Multiple Regression Model

65

(17)

3.2.1 Model and Likelihood Function

65

(1)

3.2.2 Posterior Pdf's for Parameters with a Diffuse Prior Pdf

66

(4)

3.2.3 Posterior Pdf Based on an Informative Prior Pdf

70

(2)

3.2.4 Predictive Pdf

72

(3)

3.2.5 Analysis of Model when X'X is Singular

75

(7)

Questions and Problems

82

(4)

IV Special Problems in Regression Analysis

86

(28)

4.1 The Regression Model with Autocorrelated Errors

86

(12)

4.2 Regressions with Unequal Variances

98

(10)

4.3 Two Regressions with Some Common Coefficients

108

(2)

Appendix 1

110

(1)

Appendix 2

110

(2)

Questions and Problems

112

(2)

V On Errors in the Variables

114

(48)

5.1 The Classical EVM: Preliminary Problems

114

(9)

5.2 Classical EVM: ML Analysis of the Functional Form

123

(4)

5.3 ML Analysis of Structural Form of the EVM

127

(5)

5.4 Bayesian Analysis of the Functional Form of the EVM

132

(13)

5.5 Bayesian Analysis of the Structural Form of the EVM

145

(1)

5.6 Alternative Assumption about the Incidental Parameters

145

(9)

Appendix

154

(3)

Questions and Problems

157

(5)

VI Analysis of Single Equation Nonlinear Models

162

(24)

6.1 The Box-Cox Analysis of Transformations

162

(7)

6.2 Constant Elasticity of Substitution (CES) Production Function

169

(7)

6.3 Generalized Production Functions

176

(7)

Questions and Problems

183

(3)

VII Time Series Models: Some Selected Examples

186

(38)

7.1 First Order Normal Autoregressive Process

186

(5)

7.2 First Order Autoregressive Model with Incomplete Data

191

(3)

7.3 Analysis of a Second Order Autoregressive Process

194

(6)

7.4 "Distributed Lag" Models

200

(7)

7.5 Applications to Consumption Function Estimation

207

(6)

7.6 Some Generalizations of the Distributed Lag Model

213

(3)

Appendix

216

(4)

Questions and Problems

220

(4)

VIII Multivariate Regression Models

224

(24)

8.1 The Traditional Multivariate Regression Model

224

(9)

8.2 Predictive Pdf for the Traditional Multivariate Regression Model

233

(3)

8.3 The Traditional Multivariate Model with Exact Restrictions

236

(2)

8.4 Traditional Model with an Informative Prior Pdf

238

(2)

8.5 The "Seemingly Unrelated" Regression Model

240

(6)

Questions and Problems

246

(2)

IX Simultaneous Equation Econometric Models

248

(43)

9.1 Fully Recursive Models

250

(2)

9.2 General Triangular Systems

252

(1)

9.3 The Concept of Identification in Bayesian Analysis

253

(5)

9.4 Analysis of Particular Simultaneous Equation Models

258

(7)

9.5 "Limited Information" Bayesian Analysis

265

(5)

9.6 Full System Analysis

270

(6)

9.7 Results of Some Monte Carlo Experiments

276

(11)

9.7.1 The Model and Its Specifications

277

(1)

9.7.2 Sampling-Theory Analysis of the Model

278

(1)

9.7.3 Bayesian Analysis of the Model

278

(2)

9.7.4 Experimental Results: Point Estimates

280

(6)

9.7.5 Experimental Results: Confidence Intervals

286

(1)

9.7.6 Concluding Remarks on the Monte Carlo Experiments

286

(1)

Questions and Problems

287

(4)

X On Comparing and Testing Hypotheses

291

(28)

10.1 Posterior Probabilities Associated with Hypotheses

292

(6)

10.2 Analyzing Hypotheses with Diffuse Prior Pdf's for Parameters

298

(4)

10.3 Comparing and Testing Hypotheses with Nondiffuse Prior Information

302

(4)

10.4 Comparing Regression Models

306

(6)

10.5 Comparing Distributed Lag Models

312

(5)

Questions and Problems

317

(2)

XI Analysis of Some Control Problems

319

(41)

11.1 Some Simple One Period Control Problems

320

(7)

11.2 Single-Period Control of Multiple Regression Processes

327

(4)

11.3 Control of Multivariate Normal Regression Processes

331

(2)

11.4 Sensitivity of Control to Form of Loss Function

333

(3)

11.5 Two-Period Control of the Multiple Regression Model

336

(8)

11.6 Some Multiperiod Control Problems

344

(10)

Appendix 1

354

(2)

Appendix 2

356

(2)

Questions and Problems

358

(2)

XII Conclusion

360

(3)

Appendix A

363

(16)

Appendix B

379

(21)

Appendix C

400

(15)

Bibliography

415

(8)

Author Index

423

(4)

Subject Index

427

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An Introduction to Bayesian Inference in Econometrics

Summary

Author Biography

Table of Contents

Supplemental Materials