Regular Variation and Extremal Dependence of GARCH Residuals with Application to Market Risk Measures

Abstract

Stock returns exhibit heavy tails and volatility clustering. These features, motivating the use of GARCH models, make it difficult to predict times and sizes of losses that might occur. Estimation of losses, like the Value-at-Risk, often assume that returns, normalized by the level of volatility, are Gaussian. Often under ARMA-GARCH modeling, such scaled returns are heavy tailed and show extremal dependence, whose strength reduces when increasing extreme levels. We model heavy tails of scaled returns with generalized Pareto distributions, while extremal dependence can be reduced by declustering data.Declustering, Expected shortfalls, Extremal dependence, Generalized Pareto distribution, Regular variation, Value-at-Risk, C22, C53,

Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review

“…Extremal dependence and leverage are essential to capture the serial dependence observed in the extreme returns. This might also explain why Laurini and Tawn (2008) find that the extremes of the residuals of a simple GARCH(1,1) model are still somewhat correlated, requiring further declustering. Figure 3 depicts the one-day ahead forecasts of the VaR and ES at the α = 0.01 level using the volatility predictions and the formulas in (7) and (8).…”

“…Refinements and improvements of this approach have been proposed over the years. After prewhitening the returns with a GARCH model (Bollerslev 1986); Laurini and Tawn (2008) noted that the scaled residuals still present some dependence in the extremes, and proposed a declustering procedure to account for this effect. Following recent trends in financial econometrics, Bee et al (2016) propose extending the information set F t−1 to include intra-day observations and use volatility models based on high-frequency (HF) data to pre-whiten the returns.…”

Marked point process adjusted tail dependence analysis for high-frequency financial data

“…Variance estimates and confidence intervals either have to be based on rather strong mixing or independence assumptions or non-parametric techniques such as subsampling and bootstrapping can be used. When estimating the tail dependence index for stationary time series, which is closely related to estimating the ordinary tail index, Laurini and Tawn [31] and Ledford and Tawn [32] state that confidence intervals based on iid assumptions will be too small when the extremes are dependent. They propose a block bootstrapping method to obtain proper variance estimates for their estimators.…”

Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets

“…This unpleasant result, which constitutes a serious warning about the capability of such models to capture entirely the variation in volatility, is generally addressed by modeling the residuals by some ad hoc distribution. The most popular ones used to this aim are the (eventually asymmetric) Student's t [18,19,20,21], the generalized Pareto [22,23], the normal inverse Gaussian distribution [24] or the double exponential, as a particular case of the generalized error distribution when the tail thickness parameter equals 1 [25]. In spite of the many efforts devoted to the analysis of this topic, nevertheless the choice of such distributions appears quite arbitrary and -in authors' opinion -makes questionable the capability of the above recalled models to capture the real nature of the volatility dynamics.…”