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This book offers an up-to-date coverage of the basic principles and of the tools of Bayesian inference in econometrics. Bayesian inference is a branch of statistics that integrates explicitly both data and prior (possibly subjective) information in model building , estimation and evaluation. The book then shows how to use Bayesian methods in a range of models especially suited to the analysis of macroeconomic and financial time series.

This book contains an up-to-date coverage of the last twenty years advances in Bayesian inference in econometrics, with an emphasis on dynamic models. It shows how to treat Bayesian inference in non linear models, by integrating the useful developments of numerical integration techniques based on simulations (such as Markov Chain Monte Carlo methods), and the long available analytical results of Bayesian inference for linear regression models. It thus covers a broad range of rather recent models for economic time series, such as non linear models, autoregressive conditional heteroskedastic regressions, and cointegrated vector autoregressive models. It contains also an extensive chapter on unit root inference from the Bayesian viewpoint. Several examples illustrate the methods.

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CONTENTS

1 Decision Theory and Bayesian Inference = 1

 1.1 Introduction = 1

 1.2 The Baseline Decision Problem = 1

 1.3 The Moral Expectation Theorem = 4

 1.4 The Interpretation of Probabilities = 5

 1.5 Factorizations of Ⅱ : Bayes' Theorem = 8

 1.6 Extensive Form Analysis = 10

 1.7 Normal or Strategic Form Analysis = 12

 1.8 Statistical Inference and Scientific Reporting = 12

 1.9 Estimation = 16

 1.10 Hypothesis Testing = 21

  1.10.1 Introduction = 21

  1.10.2 Classical Hypothesis Testing = 22

  1.10.3 Bayesian Hypothesis Testing = 27

  1.10.4 An Example = 31

2 Bayesian Statistics and Linear Regression = 35

 2.1 Introduction = 35

 2.2 The Likelihood Principle = 35

  2.2.1 Definition = 35

  2.2.2 Nuisance Parameters = 37

  2.2.3 Stopping Rules = 38

  2.2.4 Identification = 40

 2.3 Density and Likelihood Kernels = 43

 2.4 Sufficient Statistics = 46

  2.4.1 Definition = 46

  2.4.2 The Exponential Family = 46

 2.5 Natural Conjugate Inference = 48

  2.5.1 General Principle = 48

  2.5.2 Inference in the Multivariate Normal Process = 49

 2.6 Reductions of Models = 52

  2.6.1 Reduction by Conditioning and Exogeneity = 52

  2.6.2 Conditioning and the Regression Model = 55

 2.7 Inference in the Linear Regression Model = 56

  2.7.1 Model and Likelihood Function = 56

  2.7.2 Natural Conjugate Prior Density = 57

  2.7.3 Posterior Densities = 58

  2.7.4 Predictive Densities = 61

  2.7.5 Tests of Linear Restrictions = 62

3 Methods of Numerical Integration = 65

 3.1 Introduction = 65

 3.2 General Principle for Partially Linear Models = 67

 3.3 Deterministic Integration Methods = 68

  3.3.1 Simpson's Rules = 69

  3.3.2 Other Rules = 71

 3.4 Monte Carlo Methods = 74

  3.4.1 Direct Sampling = 74

  3.4.2 Importance Sampling = 76

  3.4.3 Markov Chain Methods = 83

 3.5 Conclusion = 93

4 Prior Densities for the Regression Model = 94

 4.1 Introduction = 94

 4.2 The Elicitation of a Prior Density = 94

  4.2.1 Distributions Adjusted on Historical Data = 95

  4.2.2 Subjective Prior Information : a Discussion = 97

  4.2.3 The Interval Betting Method for Regression Parameters = 99

  4.2.4 The Predictive Method = 104

  4.2.5 Simplifications for Assigning Prior Covariances = 106

 4.3 The Quantification of Ignorance = 107

  4.3.1 Ancient Justifications for Ignorance Priors = 108

  4.3.2 Modern Justifications for Ignorance Priors = 108

  4.3.3 Stable Inference = 109

  4.3.4 Jeffreys' Invariance Principle = 110

  4.3.5 Non-informative Limit of a Natural Conjugate Prior = 113

  4.3.6 The Reference Prior = 115

 4.4 Restrictive Properties of the NIG Prior = 116

  4.4.1 Diffuse Prior on $$σ^2$$ and Informative Prior on β = 117

  4.4.2 Conflicting Information = 118

 4.5 Student Prior and Poly-t Densities = 118

  4.5.1 Pooling Two Independent Samples = 119

  4.5.2 Student Prior = 122

  4.5.3 A Wage Equation for Belgium = 123

 4.6 Special Topics = 124

  4.6.1 Exact Restrictions = 125

  4.6.2 Exchangeable Priors = 126

5 Dynamic Regression Models = 129

 5.1 Introduction = 129

 5.2 Statistical Issues Specific to Dynamic Models = 129

  5.2.1 Reductions : Exogeneity and Causality = 130

  5.2.2 Reduction of a VAR Model to an ADL Equation = 132

  5.2.3 Treatment of Initial Observations = 134

  5.2.4 Non-stationarity = 136

 5.3 Inference in ADL Models = 136

  5.3.1 Model Specification and Posterior Analysis = 136

  5.3.2 Truncation to the Stationarity Region = 137

  5.3.3 Predictive Analysis = 137

  5.3.4 Inference on Long-run Multipliers = 140

 5.4 Models with AR Errors = 143

  5.4.1 Common Factor Restrictions in ADL Models = 144

  5.4.2 Bayesian Inference = 144

  5.4.3 Testing for Common Factors and Autocorrelation = 146

 5.5 Models with ARMA Errors = 148

  5.5.1 Identification Problems = 148

  5.5.2 The Likelihood Function = 150

  5.5.3 Bayesian Inference = 153

 5.6 Money Demand in Belgium = 154

6 Unit Root Inference = 158

 6.1 Introduction = 158

 6.2 Controversies in the Literature = 159

  6.2.1 The Helicopter Tour = 160

  6.2.2 Bayesian Routes to Unit Root Testing = 162

  6.2.3 What Is Important? = 164

 6.3 Dynamic Properties of the AR(1) Model = 164

  6.3.1 Initial Condition = 164

  6.3.2 Introducing a Constant and a Trend = 166

  6.3.3 Trend and Cycle Decomposition = 168

 6.4 Pathologies in the Likelihood Functions = 169

  6.4.1 Definitions = 169

  6.4.2 The Simple AR(1) Model = 169

  6.4.3 The Non-linear AR(1) Model with Constant = 170

  6.4.4 The Linear AR(1) Model with Constant = 173

  6.4.5 Summary = 174

 6.5 The Exact Role of Jeffreys' Prior = 174

  6.5.1 Jeffreys' Prior Without Deterministic Terms = 175

  6.5.2 Choosing a Prior for the Simple AR(1) Model = 178

  6.5.3 Jeffreys' prior with Deterministic Terms = 179

  6.5.4 Playing with Singularities = 180

  6.5.5 Bayesian Unit Root Testing = 182

  6.5.6 Can We Test for a Unit Root Using a Linear Model? = 184

 6.6 Analysing the Extended Nelson-Plosser Data = 185

  6.6.1 The AR(p) Model with a Deterministic Trend = 185

  6.6.2 The Empirical Results = 188

 6.7 Conclusion = 192

 6.8 Appendix : Jeffreys' Prior with the Exact Likelihood = 193

7 Heteroscedasticity and ARCH = 197

 7.1 Introduction = 197

 7.2 Functional Heteroscedasticity = 199

  7.2.1 Prior Density and Likelihood Function = 199

  7.2.2 Posterior Analysis = 201

  7.2.3 A Test of Homoscedasticity = 202

  7.2.4 Application to Electricity Consumption = 202

 7.3 ARCH Models = 204

  7.3.1 Introduction = 204

  7.3.2 Properties of ARCH Processes = 205

  7.3.3 Likelihood Function and Posterior Density = 208

  7.3.4 Predictive Densities = 209

  7.3.5 Application to the USD／DM Exchange Rate = 211

  7.3.6 Regression Models with ARCH Errors = 211

 7.4 GARCH Models = 215

  7.4.1 Properties of GARCH Processes = 216

  7.4.2 Extensions of GARCH Processes = 217

  7.4.3 Inference in GARCH Processes = 219

  7.4.4 Application to the USD／DM Exchange Rate = 220

 7.5 Stationarity and Persistence = 221

  7.5.1 Stationarity = 221

  7.5.2 Measures of Persistence = 223

  7.5.3 Application to the USD／DM Exchange Rate = 224

 7.6 Bayesian Heteroscedasticity Diagnostic = 225

  7.6.1 Properties of Bayesian Residuals = 226

  7.6.2 A Diagnostic Procedure = 227

  7.6.3 Applications to Electricity and Exchange Rate Data Sets = 229

 7.7 Conclusion = 229

8 Non-Linear Time Series Models = 231

 8.1 Introduction = 231

 8.2 Inference in Threshold Regression Models = 232

  8.2.1 A Typology of Threshold Models = 232

  8.2.2 Notation = 234

  8.2.3 Posterior Analysis in the Homoscedastic Case = 235

  8.2.4 Posterior Analysis for the Heteroscedastic Case = 236

  8.2.5 Predictive Density for the SETAR Model = 237

 8.3 Pathological Aspects of Threshold Models = 238

  8.3.1 The Nature of the Threshold = 239

  8.3.2 Identification in Abrupt Transition Models = 239

  8.3.3 Identification in Smooth Transition Models = 241

 8.4 Testing for Linearity and Model Selection = 244

  8.4.1 Model Selection = 244

  8.4.2 A Lnearity Test Based on the Posterior Density = 245

  8.4.3 A Numerical Example = 247

 8.5 Empirical Applications = 247

  8.5.1 A Consumption Function for France = 248

  8.5.2 United States Business Cycle Asymmetries = 253

 8.6 Disequilibrium Models = 256

  8.6.1 Maximum Likelihood Estimation = 257

  8.6.2 The Structure of the Posterior Density = 258

  8.6.3 Elicitation of Prior Information on β = 260

  8.6.4 Numerical Evaluation of the Posterior Density = 261

  8.6.5 Endogenous Prices and Other Regime Indicators = 262

 8.7 Conclusion = 263

9 Systems of Equations = 265

 9.1 Introduction = 265

 9.2 VAR Models = 265

  9.2.1 Unrestricted VAR Models and Multivariate Regression = 265

  9.2.2 Restricted VAR Models and SURE Models = 267

  9.2.3 The Minnesota Prior for VAR Models = 269

 9.3 Cointegration and VAR Models = 272

  9.3.1 Model Formulation = 272

  9.3.2 Identification Issues = 273

  9.3.3 Likelihood Function and Prior Density = 274

  9.3.4 Posterior Results = 275

  9.3.5 Examples = 278

  9.3.6 Selecting the Cointegration Rank = 283

 9.4 Simultaneous Equation Models = 285

  9.4.1 Limited Information Analysis = 285

  9.4.2 Full Information Analysis = 287

A Probability Distributions = 289

 A.1 Univariate Distributions = 289

  A.1.1 The Uniform Distribution = 289

  A.1.2 The Gamma, Chi-squared, and Beta Distributions = 290

  A.1.3 The Univariate Normal Distribution = 293

  A.1.4 Distributions Related to the Univariate Normal Distribution = 294

 A.2 Multivariate Distributions = 297

  A.2.1 Preliminary : Choleski Decomposition = 297

  A.2.2 The Multivariate Normal Distribution = 298

  A.2.3 The Matricvariate Normal Distribution = 301

  A.2.4 The Normal-Inverted Gamma-2 Distribution = 302

  A.2.5 The Multivariate Student Distribution = 303

  A.2.6 The Inverted Wishart Distribution = 305

  A.2.7 The Matricvariate Student Distribution = 307

  A.2.8 Poly-t Distributions = 309

B Generating random numbers = 312

 B.1 General Methods for Univariate Distributions = 312

  B.1.1 Inverse Transform Method = 313

  B.1.2 Acceptance-Rejection Method = 313

  B.1.3 Compound or Data Augmentation Method = 315

 B.2 Univariate Distributions = 315

  B.2.1 Exponential Distribution = 315

  B.2.2 Gamma Distribution = 316

  B.2.3 Chi-squared Distribution = 316

  B.2.4 Inverted Gamma-2 Distribution = 317

  B.2.5 Beta Distribution = 317

  B.2.6 Normal Distribution = 317

  B.2.7 Student Distribution = 317

  B.2.8 Cauchy Distribution = 318

 B.3 General Methods for Multivariate Distributions = 318

  B.3.1 Multivariate Transformations = 318

  B.3.2 Factorization into Marginals and Conditionals = 319

  B.3.3 Markov Chains = 319

 B.4 Multivariate Distributions = 319

  B.4.1 Multivariate Normal = 319

  B.4.2 Multivariate Student = 320

  B.4.3 Matricvariate Normal = 320

  B.4.4 Inverted Wishart = 320

  B.4.5 Matricvariate Student = 321

  B.4.6 Poly-t 2-0 = 321

  B.4.7 Poly-t 1-1 = 322

References = 323

Subject Index = 340

Author Index = 347