Table of Contents

Summary .XIII

1 Introduction 1

2 Bayesian Statistics and MCMC Methods 9

2.1 Bayesian inference 9

2.2 MCMC methods 10

2.2.1 The Gibbs sampler 11

2.2.2 The Metropolis-Hastings algorithm 12

2.2.3 Dealing with the MCMC output 13

3 Bayesian Estimation of the GARCH(1; 1) Model with

Normal Innovations 17

3.1 The model and the priors 17

3.2 Simulating the joint posterior 18

3.2.1 Generating vector _ 20

3.2.2 Generating parameter _ 20

3.3 Empirical analysis 22

3.3.1 Model estimation 24

3.3.2 Sensitivity analysis 30

3.3.3 Model diagnostics 32

3.4 Illustrative applications 34

3.4.1 Persistence 34

3.4.2 Stationarity 36

4 Bayesian Estimation of the Linear Regression Model with

Normal-GJR(1; 1) Errors 39

4.1 The model and the priors 40

4.2 Simulating the joint posterior 41

4.2.1 Generating vector 41

4.2.2 Generating the GJR parameters 42

Generating vector _ 43

4.3 Empirical analysis 44

4.3.1 Model estimation 46

4.3.2 Sensitivity analysis 52

4.3.3 Model diagnostics 52

4.4 Illustrative applications 53

5 Bayesian Estimation of the Linear Regression Model with

Student-t-GJR(1; 1) Errors 55

5.1 The model and the priors 56

5.2 Simulating the joint posterior 59

5.2.1 Generating vector 59

5.2.2 Generating the GJR parameters 60

Generating vector _ 61

Generating parameter _ 62

5.2.3 Generating vector $ 62

5.2.4 Generating parameter _ 63

5.3 Empirical analysis 64

5.3.1 Model estimation 64

5.3.2 Sensitivity analysis 70

5.3.3 Model diagnostics 70

5.4 Illustrative applications 71

6 Value at Risk and Decision Theory 73

6.1 Introduction 73

6.2 The concept of Value at Risk 76

6.2.1 The one-day ahead VaR under the GARCH(1; 1) dynamics 77

6.2.2 The s-day ahead VaR under the GARCH(1; 1) dynamics 77

6.3 Decision theory 85

6.3.1 Bayes point estimate 85

6.3.2 The Linex loss function 86

6.3.3 The Monomial loss function 90

6.4 Empirical application: the VaR term structure 91

6.4.1 Data set and estimation design 92

6.4.2 Bayesian estimation 94

6.4.3 The term structure of the VaR density 95

6.4.4 VaR point estimates 96

6.4.5 Regulatory capital 100

6.4.6 Forecasting performance analysis 102

6.5 The Expected Shortfall risk measure 104

7 Bayesian Estimation of the Markov-Switching GJR(1; 1)

Model with Student-t Innovations 109

7.1 The model and the priors 111

7.2 Simulating the joint posterior 115

7.2.1 Generating vector s 117

7.2.2 Generating matrix P 118

Generating vector _ 120

Generating vector _ 121

7.2.4 Generating vector $ 122

7.2.5 Generating parameter _ 122

7.3 An application to the Swiss Market Index 122

7.4 In-sample performance analysis 133

7.4.1 Model diagnostics 133

7.4.2 Deviance information criterion 134

7.4.3 Model likelihood 137

7.5 Forecasting performance analysis 144

7.6 One-day ahead VaR density 148

7.7 Maximum Likelihood estimation 152

8 Conclusion 155

A Recursive Transformations 161

A.1 The GARCH(1; 1) model with Normal innovations 161

A.2 The GJR(1; 1) model with Normal innovations 162

A.3 The GJR(1; 1) model with Student-t innovations 163

B Equivalent Speci_cation 165

C Conditional Moments 171

Computational Details 179

Abbreviations and Notations 181

List of Tables 187

List of Figures 189

References 191