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Copula Inference for Finance and Insurance
author: ALEXANDRA DA COSTA DIAS
Contents:
Introduction 1
1 Inference for copulae 7
1.1 Definitions, properties and examples . . . . . . . . . . . . . 8
1.2 Tail-dependence . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 The IFM method . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 The pseudo log{likelihood method . . . . . . . . . . . . . . 19
1.5 Pseudo log-likelihood for dependent sequences . . . . . . . 25
1.6 Goodness-of-fit test . . . . . . . . . . . . . . . . . . . . . . 27
1.7 The IFM method for FX weekly returns . . . . . . . . . . . 28
1.7.1 IFM estimates . . . . . . . . . . . . . . . . . . . . . 30
1.7.2 VaR and expected shortfall estimation . . . . . . . . 37
1.7.3 Backtesting VaR and ES . . . . . . . . . . . . . . . . 41
1.7.4 Tail-dependence coefficient estimation . . . . . . . . 44
1.8 A note on jackknife confidence intervals . . . . . . . . . . . 46
2 Stationary copula analysis 51
2.1 Deseasonalisation of the returns . . . . . . . . . . . . . . . . 52
2.2 Dependence structure modelling . . . . . . . . . . . . . . . 60
2.3 Goodness-of-fit tests . . . . . . . . . . . . . . . . . . . . . . 70
2.3.1 Test of elliptical symmetry . . . . . . . . . . . . . . 71
2.3.2 Testing the results of the fittings . . . . . . . . . . . 74
2.4 Tail-dependence . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4.1 Spectral measure estimation . . . . . . . . . . . . . . 76
2.4.2 Multivariate excesses . . . . . . . . . . . . . . . . . . 80
3 Conditional copula models 89
3.1 Time dependence filtering . . . . . . . . . . . . . . . . . . . 90
3.2 Copulae for USD/DEM and USD/JPY residuals . . . . . . 95
3.3 Tail-dependence coefficient . . . . . . . . . . . . . . . . . . 102
3.4 Testing for ellipticity . . . . . . . . . . . . . . . . . . . . . . 104
4 Time-varying copula models 107
4.1 Stochastic dependence structure . . . . . . . . . . . . . . . 108
4.2 The Multivariate GARCH model with time-varying copula 113
4.3 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Fitting the time{varying copula model to the FX returns . 115
5 Change-point analysis for copulae 123
5.1 Statistical change{point analysis . . . . . . . . . . . . . . . 124
5.1.1 The test statistic . . . . . . . . . . . . . . . . . . . . 124
5.1.2 An example: the Gumbel case . . . . . . . . . . . . 129
5.1.3 The power of the test . . . . . . . . . . . . . . . . . 132
5.1.4 The time of the change and corresponding confidence intervals . . . . . . . . . . . . . . . . . . . . . 134
5.1.5 Multiple Changes . . . . . . . . . . . . . . . . . . . . 139
5.2 A comment on pricing . . . . . . . . . . . . . . . . . . . . . 139
5.3 An example with insurance data . . . . . . . . . . . . . . . 141
5.4 Change-point analysis of the FX returns copula . . . . . . . 141
Summary and conclusion 147