A quasi-maximum likelihood estimation strategy for value-at-risk forecasting: Application to equity index futures markets

Abstract

We present the first empirical evidence for the validity of the ARMA-GARCH model with tempered stable innovations to estimate 1-day-ahead value at risk in futures markets for the S & P 500, DAX, and Nikkei. We also provide empirical support that GARCH models based on normal innovations appear not to be as well suited as infinitely divisible modelsfor predicting financial crashes. The results are compared with the predictions based on data in the cash market. We also provide the first empirical evidence on how adding trading volume to the GARCH model improves its forecasting ability. In our empirical analysis, we forecast 1 % value at risk in both spot and futures markets using normal and tempered stable GARCH models following a quasi-maximum likelihood estimation strategy. In order todetermine the accuracy of forecasting for each specific model, backtesting using Kupiec’sproportion of failures test is applied. For each market, the model with a lower number ofviolations is preferred. Our empirical result indicates the usefulness of classical tempered stable distributions for market risk management and asset pricing.