A Two-step Estimating Approach for Heavy-tailed AR Models with Non-zero Median GARCH-type Noises

This paper develops a novel two-step estimating procedure for heavy-tailed AR models with non-zero median GARCH-type noises, allowing for time-varying volatility. We first establish the self-weighted quantile regression estimator (SQE) across all quantile levels $\tau \in (0,1)$ for the AR parameters ${\theta }_0$. We show that the SQE, less a bias, converges weakly to a Gaussian process at a rate of $n^{-1\mathrm{/}2}$. The bias is zero if and only if $\tau$ equals ${\tau }_0$, the probability that the noise is less than zero. Based on the SQE, we propose an approach to estimate ${\tau }_0$ in the second step and {feed the estimated ${\tau }_0$ back into the SQE to estimate ${\theta }_0$.} Both the estimated ${\tau }_0$ and ${\theta }_0$ are shown to be consistent and asymptotically normal. A random weighting bootstrap method is developed to approximate the complex distribution. The problem we study is non-standard because ${\tau }_0$ may not be identifiable in conventional quantile regression, and the usual methods cannot verify the existence of the SQE bias. Unlike existing procedures for heavy-tailed time series, our method does not require prior information about the symmetry, tail index, or the parametric form of the noise, nor does it require classical identification conditions, such as zero-mean or zero-median.