Statistical Inference of Conditional Quantiles in Nonlinear Time Series Models

This paper studies the statistical properties of a two-step conditional quantile estimator in nonlinear time series models with unspecified error distribution. The asymptotic distribution of the quasi-maximum likelihood estimators and the filtered empirical percentiles is derived. Three applications of the asymptotic result are considered. First, we construct an interval estimator of the conditional quantile without any distributional assumptions. Second, we develop a specification test for the error distribution. Finally, using the specification test, we propose methods for estimating the tail index of the error distribution that supports the construction of a new estimator for the conditional quantile at the extreme tail. The asymptotic results and their applications are illustrated by simulations and real data analyses in which our methods for analyzing daily and intraday financial return series have been adopted.