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Table of contents
1 Introduction
1.1 Evolution of the thesis
1.2 Structure and content of the thesis
2 Quantifying pro-cyclicality: An empirical study
2.1 Introduction
2.2 Quantifying the predictive power of risk measures
2.2.1 Sample Quantile Process
2.2.2 Predictive power for the risk using the SQP
2.3 Pro-cyclicality results on real data
2.3.1 Realized volatility as an indicator of market states
2.3.2 SQP predictive power as a function of volatility
2.4 Pro-cyclicality results on different models
2.4.1 IID model
2.4.2 GARCH(1, 1) model
2.4.3 Other influences
2.5 Conclusion
3 Estimators in the IID case: Asymptotic theory
3.1 Introduction
3.2 Sample quantile and r-th absolute centred sample moment
3.2.1 Auxiliary results
3.2.2 Proof of Theorem 3.1
Using Bahadur’s method
Using Taylor’s method
3.2.3 Extensions
3.3 Sample quantile and MedianAD
3.4 Location-scale quantile and measures of dispersion
3.4.1 Location-scale quantile and r-th absolute centred sample moment
3.4.2 Location-scale quantile and MedianAD
3.5 Discussion
3.6 Examples/ finite sample performance
3.6.1 The impact of the choice of the quantile estimator
3.6.2 The effect of sample size in estimation
3.7 Conclusion
4 Estimators in the case of augmented GARCH processes: Asymptotic theory
4.1 Introduction
4.2 The bivariate FCLT
4.2.1 Proof of Theorem 4.3
4.3 Examples
4.4 Conclusion
5 Pro-cyclicality: Connecting empirics and theory
5.1 Introduction
5.2 Pro-cyclicality in IID models
5.2.1 CLT’s between risk and dispersion measure estimators
5.2.2 Results on pro-cyclicality
5.3 Pro-cyclicality in augmented GARCH(p,q) models
5.3.1 FCLT’s between risk and dispersion measure estimators
5.3.2 Results on pro-cyclicality
5.4 Application
5.4.1 Comparing pro-cyclicality in IID models
5.4.2 Pro-cyclicality analysis on real data (reprise)
5.5 Conclusion
6 Conclusion
6.1 Summary
6.2 Further research perspectives
Bibliography
