Heavy Tails And Copulas: Topics In Dependence Modelling In Economics And Finance

by Rustam Ibragimov, Artem Prokhorov

Overview

'Overall, the book is highly technical, including full mathematical proofs of the results stated. Potential readers are post-graduate students or researchers in Quantitative Risk Management willing to have a manual with the state-of-the-art on portfolio diversification and risk aggregation with heavy tails, including the fundamental theorems as well as collateral (but most useful) results on majorization and copula theory.'
Quantitative Finance This book offers a unified approach to the study of crises, large fluctuations, dependence and contagion effects in economics and finance. It covers important topics in statistical modeling and estimation, which combine the notions of copulas and heavy tails — two particularly valuable tools of today's research in economics, finance, econometrics and other fields — in order to provide a new way of thinking about such vital problems as diversification of risk and propagation of crises through financial markets due to contagion phenomena, among others. The aim is to arm today's economists with a toolbox suited for analyzing multivariate data with many outliers and with arbitrary dependence patterns. The methods and topics discussed and used in the book include, in particular, majorization theory, heavy-tailed distributions and copula functions — all applied to study robustness of economic, financial and statistical models, and estimation methods to heavy tails and dependence.

Product Details

ISBN-13:	9789814689793
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/09/2017
Pages:	304
Product dimensions:	6.00(w) x 9.00(h) x 0.69(d)

Table of Contents

Preface vii

Foreword ix

Acknowledgments xi

1 Introduction and Overview 1

1.1 Crises, contagion and other features of modern economic and financial data 1

1.2 Econometric tools for modern financial and economic: data 3

1.2.1 Multivariate distributions and copulas 3

1.2.2 Heavy tailed stable and power law distributions 9

1.3 Robustness to heavy tails and to copula misspecification 14

1.3.1 Robustness of models to heavy tails 14

1.3.2 Robustness of methods to heavy tails and to copula, misspecification 15

1.4 Plan for the book 16

2 Portfolio Diversification under Independent Fat Tailed Risks 19

2.1 Introduction 19

2.2 Notation and classes of distributions 22

2.3 Value-at-Risk (VaR): Definition and main properties 25

2.4 Majorization, diversification and (non-)coherency of VaR 26

2.4.1 Majorization of random vectors and diversification of portfolio riskiness 26

2.4.2 Subadditivity of VaR 28

2.4.3 Extensions to heterogeneity and skewness 31

2.5 Concluding remarks 38

2.6 Appendix Al: VaR and unimodality properties of log-concave and stable distributions 39

2.7 Appendix A2: Proofs of theorems and propositions 40

3 From Independence to Dependence via Copulas and U-statistics 47

3.1 Introduction 47

3.2 Characterizations of dependence 49

3.2.1 Characterizations of joint distributions 49

3.2.2 Characterizations of copulas 52

3.2.3 Characterizations of expectations 56

3.2.4 Characterizations of certain dependence classes 57

3.2.5 Reduction property for multiplicative systems 60

3.3 Characterizations of Markov processes 60

3.3.1 Copula-based characterizations of Markovness 62

3.3.2 Combining Markovness with other dependence properties 68

3.3.3 Reduction property for Markov processes 74

3.3.4 Fourier copulas 79

3.4 Measures of dependence 80

3.4.1 Problems with correlation 81

3.4.2 Some alternative measures 82

3.4.3 Sharp moment and probability inequalities 87

3.4.4 Vector version of Hoeffiding's φ² 92

3.5 Bounds on options 97

3.5.1 Bounds on European options 97

3.5.2 Bounds for Asian options 102

3.6 Concluding remarks 103

3.7 Appendix: Proofs 105

4 Limits of Diversification under Fat Tails and Dependence 113

4.1 Introduction 113

4.2 Dependence vs independence 116

4.3 Diversification and copulas 118

4.3.1 Power-type copulas 118

4.3.2 Subadditivity of VaR 121

4.4 Diversification and common shocks 125

4.4.1 CT-symmetric and spherical distributions 126

4.4.2 Multiplicative common shocks 127

4.4.3 Additive common shocks 130

4.5 Further results for common shock models 134

4.5.1 Further applications: Portfolio component VaR 135

4.5.2 When heavy-tailedness helps: VaR for financial indices 139

4.5.3 From risk management to econometrics: Efficiency of random effects estimators 146

4.5.4 Extensions: Multiple additive and multiplicative common shocks 150

4.6 Conclusion 155

4.7 Appendix: Proofs 157

5 Robustness of Econometric Methods to Copula Misspecification and Heavy Tails 171

5.1 Introduction 171

5.2 Copula estimation 172

5.2.1 Parametric models: MLE and IFM 173

5.2.2 Nonparametric models: Empirical and Bernstein copulas 176

5.2.3 Semiparametric estimation: Copulas vs marginals 179

5.3 Copula-based estimation of time series models 181

5.3.1 Parametric and semiparametric estimation of Markov processes 181

5.3.2 Nonparametric copula, inference for time series 182

5.3.3 Dependence properties of copula-based time series 183

5.4 Improved and robust parametric estimators 184

5.4.1 QMLE and improved QMLE 185

5.4.2 Full MLE as GMM 188

5.4.3 Efficiency and redundancy of copulas 191

5.4.4 Validity and robustness of copulas 200

5.4.5 Efficiency and redundancy under misspecified but robust copulas 204

5.5 Robustness and efficiency of nonparametric copulas 207

5.5.1 Efficient semiparametric estimation of parameters in marginals 208

5.5.2 Bayesian efficiency and consistency 213

5.6 Robustness of estimators to heavy tails 216

5.6.1 Trimming 216

5.7 Concluding remarks 220

5.8 Appendix: Proofs 222

6 Copula Tests Using Information Matrix 229

6.1 Introduction 229

6.2 Tests of copula robustness 232

6.2.1 Test of overidentifying restrictions 232

6.2.2 Two step test 234

6.3 Tests of copula correctness 235

6.3.1 Copulas and information matrix equivalence 235

6.3.2 Information matrix test 237

6.3.3 Generalized information matrix tests 241

6.3.4 Power study 242

6.4 Concluding remarks 248

6.5 Appendix: Proofs 250

7 Summary and Conclusion 257

7.1 Summary 257

7.2 Future research 259

Bibliography 261

Index 281