Non-Gaussian Quasi Maximum Likelihood Estimation of GARCH Models

TL;DR

This paper introduces a two-step non-Gaussian QMLE method for GARCH models that corrects bias, achieves consistency, and improves efficiency, especially with heavy-tailed innovations, supported by simulations and empirical analysis.

Contribution

It proposes a novel two-step non-Gaussian QMLE (2SNG-QMLE) that corrects bias and enhances efficiency in GARCH parameter estimation.

Findings

2SNG-QMLE is consistent and asymptotically normal.

The method outperforms Gaussian QMLE with heavy-tailed errors.

Extensions further improve estimation efficiency.

Abstract

The non-Gaussian quasi maximum likelihood estimator is frequently used in GARCH models with intension to improve the efficiency of the GARCH parameters. However, unless the quasi-likelihood happens to be the true one, non-Gaussian QMLE methods suffers inconsistency even if shape parameters in the quasi-likelihood are estimated. To correct this bias, we identify an unknown scale parameter that is critical to the consistent estimation of non-Gaussian QMLE, and propose a two-step non-Gaussian QMLE (2SNG-QMLE) for estimation of the scale parameter and GARCH parameters. This novel approach is consistent and asymptotically normal. Moreover, it has higher efficiency than the Gaussian QMLE, particularly when the innovation error has heavy tails. Two extensions are proposed to further improve the efficiency of 2SNG-QMLE. The impact of relative heaviness of tails of the innovation and quasi-likelihood distributions on the asymptotic efficiency has been thoroughly investigated. Monte Carlo simulations and an empirical study confirm the advantages of the proposed approach.