Estimation and tests for TGTACH$\bm{(1, 1)}$ models with heavy-tailed errors: A uniform framework

Abstract

This paper studies the estimation and tests of heavy-tailed TGARCH$(1, 1)$ models based on quasi-maximum likelihood estimation (QMLE) and percentile-$t$ subsample bootstrap method, where heavy-tail means that the distribution of the square errors are in the domain of attraction of a stable law with infinite fourth moment. Under regular conditions, we first establish that the QMLE of ARCH and GARCH coefficients are consistent and their asymptotic distribution is non-Gaussian but is some stable law with exponent $\kappa\in (1, 2)$ despite of whether the heavy tailed TGARCH model is strictly stationary or not. However, the QMLE of the location parameter is consistent only when the model is strictly stationary. Then we propose tests for strict stationarity and symmetry for heavy-tailed TGARCH$(1, 1)$ models based the above asymptotic results and percentile-$t$ subsample bootstrap method to overcome the difficulty that the scale parameter and asymptotic distribution depend on the unknown exponent. These tests can be used in the universal parameter space. Finally, we conduct a simulation study through Monte Carlo methods to illustrate the performance of the proposed methods with finite sample sizes and then give an empirical analysis for 5-year China Treasury Bond Futures to illustrate the application of our methods.