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This monograph is a continuation of my previous research on the influence of the estimation method on risk measures (Ratuszny 2013, 2014, 2015a, 2016) and grew out of several international conferences on analysis of time series and risk management which I have attended since 2016. It is a monograph intended to provide overview of how the models behave during an adverse market movement and to find the best fit in such a period. Multivariate models and simulation techniques based on them are applied to the problems encountered in the risk management to find answers to the questions related to right model specifications and robust methods during the adverse market conditions. Compared to the existing literature on multivariate cases, the originality of this approach is to illustrate how theoretical concepts can be applied in practice, and how they behave during a market downturn. To this end, numerous simulation techniques are applied. Moreover, it enables to better understand the subtleties of the theoretical notions but also to solve real-world problems using the family of multivariate generalised autoregressive conditional heteroskedasticity (GARCH) models and those based on copula. However, the subject of the monograph is not to include the whole family of multivariate models but only the ones most suitable for financial time series and for which estimation of theparameters remains feasible from a practical perspective. Multivariate model selection is always a trade-off between feasibility and accuracy on the one hand and cost on the other. A part of this the monograph is dedicated to techniques related to robust methods to check whether it is a good way to find a better fit, especially in risk modelling where outliers in the tail are of major interest. Generalised autoregressive conditional heteroskedasticity models, developed by Engle (1982), Bollerslev (1986) and Nelson (1991) (exponential GARCH, EGARCH), were a significant achievement in the statistical analysis of univariate financial rates of return and are widely applied to describe and forecast their volatilities. The univariate GARCH models were extended to a multivariate framework by Bollerslev et al. (1988), who proposed the VECH (diagonal VECH–DVECH) model. The VECH model, which is expressed in terms of vectorised conditional variance matrices, is very flexible to represent symmetric responses of conditional variances and covariances to past square rates of return and crossproducts of rates of return (de Almeida et al. 2015). The conditional variances depend on each other and on past conditional covariances. Similarly, the conditional covariances depend not only on past cross-products of rates of return but also on past conditional variances. The main limitation of the VECH model appears in attempts to estimate their parameters for relatively large portfolios (de Almeida et al. 2015).
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Contents
Acronyms
Notation and Conventions
Introduction
Chapter 1 Dynamic Conditional Correlation GARCH Model
1.1 Measures of Multivariate Correlation
1.2 Model Specification
1.3 Estimation
1.4 Outliers
1.4.1 Types
1.4.2 Robust Estimators
1.5 Application
1.5.1 Investment Portfolio and Risk
1.5.2 Vector Autoregressive Model
1.5.3 Co-movements in Volatilities
1.5.4 Marginal Selection
1.5.5 Estimation Results
1.5.6 Model Performance
Chapter 2 Generalised Orthogonal GARCH Model
2.1 Model Specification
2.2 Estimation
2.3 Application
2.3.1 Estimation Results
2.3.2 Model Performance
Chapter 3 Multivariate Copula GARCH Model
3.1 Sklar’s Theorem
3.2 Copula Function and Its Tail Dependence Index
3.3 Model Specification
3.3.1 Time Variation in the Conditional Copula
3.3.2 Copula-MGARCH
3.4 Application
3.4.1 Tail Dependence Index
3.4.2 Estimation Results
3.4.3 Model Performance
Chapter 4 Vine Copula
4.1 Vine Structure
4.2 Simulations
4.3 Estimation
4.4 Application
4.4.1 Estimation Results
4.4.2 Model Performance
Chapter 5 Results
Conclusions
List of Tables
List of Figures
References
Appendix 1
Appendix 2
Opis
Wstęp
This monograph is a continuation of my previous research on the influence of the estimation method on risk measures (Ratuszny 2013, 2014, 2015a, 2016) and grew out of several international conferences on analysis of time series and risk management which I have attended since 2016. It is a monograph intended to provide overview of how the models behave during an adverse market movement and to find the best fit in such a period. Multivariate models and simulation techniques based on them are applied to the problems encountered in the risk management to find answers to the questions related to right model specifications and robust methods during the adverse market conditions. Compared to the existing literature on multivariate cases, the originality of this approach is to illustrate how theoretical concepts can be applied in practice, and how they behave during a market downturn. To this end, numerous simulation techniques are applied. Moreover, it enables to better understand the subtleties of the theoretical notions but also to solve real-world problems using the family of multivariate generalised autoregressive conditional heteroskedasticity (GARCH) models and those based on copula. However, the subject of the monograph is not to include the whole family of multivariate models but only the ones most suitable for financial time series and for which estimation of theparameters remains feasible from a practical perspective. Multivariate model selection is always a trade-off between feasibility and accuracy on the one hand and cost on the other. A part of this the monograph is dedicated to techniques related to robust methods to check whether it is a good way to find a better fit, especially in risk modelling where outliers in the tail are of major interest. Generalised autoregressive conditional heteroskedasticity models, developed by Engle (1982), Bollerslev (1986) and Nelson (1991) (exponential GARCH, EGARCH), were a significant achievement in the statistical analysis of univariate financial rates of return and are widely applied to describe and forecast their volatilities. The univariate GARCH models were extended to a multivariate framework by Bollerslev et al. (1988), who proposed the VECH (diagonal VECH–DVECH) model. The VECH model, which is expressed in terms of vectorised conditional variance matrices, is very flexible to represent symmetric responses of conditional variances and covariances to past square rates of return and crossproducts of rates of return (de Almeida et al. 2015). The conditional variances depend on each other and on past conditional covariances. Similarly, the conditional covariances depend not only on past cross-products of rates of return but also on past conditional variances. The main limitation of the VECH model appears in attempts to estimate their parameters for relatively large portfolios (de Almeida et al. 2015).
Spis treści
Contents
Acronyms
Notation and Conventions
Introduction
Chapter 1 Dynamic Conditional Correlation GARCH Model
1.1 Measures of Multivariate Correlation
1.2 Model Specification
1.3 Estimation
1.4 Outliers
1.4.1 Types
1.4.2 Robust Estimators
1.5 Application
1.5.1 Investment Portfolio and Risk
1.5.2 Vector Autoregressive Model
1.5.3 Co-movements in Volatilities
1.5.4 Marginal Selection
1.5.5 Estimation Results
1.5.6 Model Performance
Chapter 2 Generalised Orthogonal GARCH Model
2.1 Model Specification
2.2 Estimation
2.3 Application
2.3.1 Estimation Results
2.3.2 Model Performance
Chapter 3 Multivariate Copula GARCH Model
3.1 Sklar’s Theorem
3.2 Copula Function and Its Tail Dependence Index
3.3 Model Specification
3.3.1 Time Variation in the Conditional Copula
3.3.2 Copula-MGARCH
3.4 Application
3.4.1 Tail Dependence Index
3.4.2 Estimation Results
3.4.3 Model Performance
Chapter 4 Vine Copula
4.1 Vine Structure
4.2 Simulations
4.3 Estimation
4.4 Application
4.4.1 Estimation Results
4.4.2 Model Performance
Chapter 5 Results
Conclusions
List of Tables
List of Figures
References
Appendix 1
Appendix 2
Opinie
This monograph is a continuation of my previous research on the influence of the estimation method on risk measures (Ratuszny 2013, 2014, 2015a, 2016) and grew out of several international conferences on analysis of time series and risk management which I have attended since 2016. It is a monograph intended to provide overview of how the models behave during an adverse market movement and to find the best fit in such a period. Multivariate models and simulation techniques based on them are applied to the problems encountered in the risk management to find answers to the questions related to right model specifications and robust methods during the adverse market conditions. Compared to the existing literature on multivariate cases, the originality of this approach is to illustrate how theoretical concepts can be applied in practice, and how they behave during a market downturn. To this end, numerous simulation techniques are applied. Moreover, it enables to better understand the subtleties of the theoretical notions but also to solve real-world problems using the family of multivariate generalised autoregressive conditional heteroskedasticity (GARCH) models and those based on copula. However, the subject of the monograph is not to include the whole family of multivariate models but only the ones most suitable for financial time series and for which estimation of theparameters remains feasible from a practical perspective. Multivariate model selection is always a trade-off between feasibility and accuracy on the one hand and cost on the other. A part of this the monograph is dedicated to techniques related to robust methods to check whether it is a good way to find a better fit, especially in risk modelling where outliers in the tail are of major interest. Generalised autoregressive conditional heteroskedasticity models, developed by Engle (1982), Bollerslev (1986) and Nelson (1991) (exponential GARCH, EGARCH), were a significant achievement in the statistical analysis of univariate financial rates of return and are widely applied to describe and forecast their volatilities. The univariate GARCH models were extended to a multivariate framework by Bollerslev et al. (1988), who proposed the VECH (diagonal VECH–DVECH) model. The VECH model, which is expressed in terms of vectorised conditional variance matrices, is very flexible to represent symmetric responses of conditional variances and covariances to past square rates of return and crossproducts of rates of return (de Almeida et al. 2015). The conditional variances depend on each other and on past conditional covariances. Similarly, the conditional covariances depend not only on past cross-products of rates of return but also on past conditional variances. The main limitation of the VECH model appears in attempts to estimate their parameters for relatively large portfolios (de Almeida et al. 2015).
Contents
Acronyms
Notation and Conventions
Introduction
Chapter 1 Dynamic Conditional Correlation GARCH Model
1.1 Measures of Multivariate Correlation
1.2 Model Specification
1.3 Estimation
1.4 Outliers
1.4.1 Types
1.4.2 Robust Estimators
1.5 Application
1.5.1 Investment Portfolio and Risk
1.5.2 Vector Autoregressive Model
1.5.3 Co-movements in Volatilities
1.5.4 Marginal Selection
1.5.5 Estimation Results
1.5.6 Model Performance
Chapter 2 Generalised Orthogonal GARCH Model
2.1 Model Specification
2.2 Estimation
2.3 Application
2.3.1 Estimation Results
2.3.2 Model Performance
Chapter 3 Multivariate Copula GARCH Model
3.1 Sklar’s Theorem
3.2 Copula Function and Its Tail Dependence Index
3.3 Model Specification
3.3.1 Time Variation in the Conditional Copula
3.3.2 Copula-MGARCH
3.4 Application
3.4.1 Tail Dependence Index
3.4.2 Estimation Results
3.4.3 Model Performance
Chapter 4 Vine Copula
4.1 Vine Structure
4.2 Simulations
4.3 Estimation
4.4 Application
4.4.1 Estimation Results
4.4.2 Model Performance
Chapter 5 Results
Conclusions
List of Tables
List of Figures
References
Appendix 1
Appendix 2
