Performance Evaluation of a Family of GARCH Processes Based on Value at Risk Forecasts: Data Envelopment Analysis Approach

Abstract The aim of this study is to form a detailed comparison of the predicting power of some generalized autoregressive conditional heteroskedasticity processes paired with several parametric distributions in application to Value at Risk evaluation. Selected processes are able to incorporate known volatility characteristics such as the memory or the leverage effect. In the same manner, the distributions have been selected to be able to incorporate the asymmetry or heavy tails. We decide to form the criteria based on the statistical tests and the loss function, both measures popular in value at risk backtesting, as well as various confidence levels as the results can vary with changing the confidence level value estimated on 46 European stocks traded over a 4-year period. As this approach yields a large amount of different criteria about each separate model, the aggregation of data is used in order to create summary metrics and subsequently those metrics are passed to Russel model in order to produce clear and reasonable comparison of the approaches. We found that the data envelopment analysis methods are adequate for tasks regarding performance evaluation of the models. Second, we found that in the long position value at risk forecasting the distribution is more valuable than the volatility specification, which has been already shown to be present for various time series data in the literature. As for the short position value at risk forecasting, we came to a conclusion that the relationship is reversed and the volatility specification should be given the primary attention as it produces better value at risk forecasts.