Asymptotics for estimation of quantile regressions with truncated infinite-dimensional processes

Abstract

a b s t r a c tMany processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for least squares estimation of such finite truncations, but for quantile estimation, results are not available at a level of generality that accommodates time series models used as finite approximations to processes of potentially unbounded order. Here we establish consistency and asymptotic normality for conditional quantile estimation of truncations of such infinite-order linear models, with the truncation order increasing in sample size. We focus on estimation of the model at a given quantile. The proofs use the generalized functions approach and allow for a wide range of time series models as well as other forms of regression model. The results are illustrated with both analytical and simulation examples.

Forecasting Financial Volatility with Combined QML and LAD-ARCH Estimators of the GARCH Model

“…Using general results on quantile estimation of truncations of infinite-order processes from Zernov et al 2009, GZZ show that consistent and asymptotically normal quantile (including LAD) estimation of the parameters of this truncated ARCH representation is possible. Correspondingly, consistent and asymptotically normal estimates of the GARCH parameters may be obtained by minimum distance using the parameters of the estimated ARCH representation, and the known relationship between the parameters of the GARCH model and the infinite-order ARCH approximation.…”