Limit Theory for Moderate Deviation from Integrated GARCH Processes

TL;DR

This paper establishes the limit theory for GARCH(1,1) processes that are close to the integrated GARCH regime, extending previous results by allowing slower convergence rates of parameters, with applications to unit root testing and estimator theory.

Contribution

It develops a new asymptotic framework for GARCH processes near the integrated boundary, broadening the scope of limit theory for mildly integrated GARCH models.

Findings

Extended Berkes et al. (2005) results to slower convergence rates.

Applicable to unit root tests for mildly-integrated GARCH innovations.

Provides limit theory for estimators in models with mildly-integrated GARCH processes.

Abstract

This paper develops the limit theory of the GARCH(1,1) process that moderately deviates from IGARCH process towards both stationary and explosive regimes. The GARCH(1,1) process is defined by equations $u_t = \sigma_t \varepsilon_t$, $\sigma_t^2 = \omega + \alpha_n u_{t-1}^2 + \beta_n\sigma_{t-1}^2$ and $\alpha_n + \beta_n$ approaches to unity as sample size goes to infinity. The asymptotic theory developed in this paper extends Berkes et al. (2005) by allowing the parameters to have a slower convergence rate. The results can be applied to unit root test for processes with mildly-integrated GARCH innovations (e.g. Boswijk (2001), Cavaliere and Taylor (2007, 2009)) and deriving limit theory of estimators for models involving mildly-integrated GARCH processes (e.g. Jensen and Rahbek (2004), Francq and Zako\"ian (2012, 2013)).