Abstract: Conditional quantile estimations are an essential ingredient in modern risk management, and many other applications, where the conditional heteroscedastic structure is usually assumed to capture the volatility in financial time series. This study examines linear quantile regression models with GARCH-X errors. These models include the most popular generalized autoregressive conditional heteroscedasticity (GARCH) as a special case, and incorporate additional covariates into the conditional variance. Three conditional quantile estimators are proposed, and their asymptotic properties are established under mild conditions. A bootstrap procedure is developed to approximate their asymptotic distributions. The finite-sample performance of the proposed estimators is examined using simulation experiments. An empirical application illustrates the usefulness of the proposed methodology.

Key words and phrases: Bootstrap method, GARCH-X errors, joint estimation, quantile regression, two-step procedure, value-at-risk.