Generalized quasi maximum likelihood estimation for generalized autoregressive score models: simulations and real applications

Abstract

In this article, we develop a two-step method for conditional Value at Risk (VaR) estimation in the context of the Generalized Autoregressive Score (GAS) models. The first step consists of estimating the volatility parameter by the generalized Quasi Maximum Likelihood Estimator (gQMLE) and in the second step we estimate the theoretical quantile of the innovations by the empirical quantile of the residuals. When the instrumental density q of the gQMLE is not the Gaussian density used in the standard QMLE, or is not the true distribution of the innovations, both the estimations of the volatility and of the quantile are asymptotically biased. In spite of that the two errors counterbalance each other, and we finally get a consistent estimator of the conditional VaR. We establish the asymptotic properties of the gQMLE for GAS models as well as the conditional VaR two-step estimator. Moreover, we discuss how to apply the gQMLE by giving examples of densities distribution and we get worthwhile results.