Conditional scale function estimate in the presence of unknown conditional quantile function

Standard approach for modeling and understanding the variability of statistical data or, generally, dependant data, is often based on the mean variance regression models. However, the assumptions employed on standardized residuals may be too restrictive, in particular, when the data follows heavy-tailed distribution with probably infinite variance. This paper considers the problem of nonparametric estimation of conditional scale function of time series, based on quantile regression methodology of Koenker and Bassett (1978). We use a flexible model introduced in Mwita (2003), that makes no moment assumptions, and discuss an estimate which we get by inverting a kernel estimate of the conditional distribution function. We finally prove the consistency and asymptotic normality for the estimate. Key word and phrases. Conditional quantile, kernel estimate, quantile autoregression, ARCH, QARCH, time series, consistency, asymptotic normality, value-at-risk.