Sample quantile analysis for long-memory stochastic volatility models

Abstract

This study investigates asymptotic properties of sample quantile estimates in the context of long-memory stochastic volatility models in which the latent volatility component is an exponential transformation of a linear long-memory time series. We focus on the least absolute deviation quantile estimator and show that while the underlying process is a sequence of stationary martingale differences, the estimation errors are asymptotically normal with the convergence rate which is slower than n and determined by the dependence parameter of the volatility sequence. A non-parametric resampling method is employed to estimate the normalizing constants by which the confidence intervals are constructed. To demonstrate the methodology, we conduct a simulation study as well as an empirical analysis of the Value-at-Risk estimate of the S&P 500 daily returns. Both are consistent with the theoretical findings and provide clear evidence that the coverage probabilities of confidence intervals for the quantile estimate are severely biased if the strong dependence of the unobserved volatility sequence is ignored.