When Simplicity Offers a Benefit, Not a Cost: Closed-Form Estimation of the GARCH(1,1) Model that Enhances the Efficiency of Quasi-Maximum Likelihood

Abstract

Simple, multi-step estimators are developed for the popular GARCH(1,1) model, where these estimators are either available entirely in closed form or dependent upon a preliminary estimate from, for example, quasi-maximum likelihood. Identification sources to asymmetry in the model's innovations, casting skewness as an instrument in a linear, two-stage least squares estimator. Properties of regular variation coupled with point process theory establish the distributional limits of these estimators as stable, though highly non-Gaussian, with slow convergence rates relative to the √n-case. Moment existence criteria necessary for these results are consistent with the heavy-tailed features of many financial returns. In light-tailed cases that support asymptotic normality for these simple estimators, conditions are discovered where the simple estimators can enhance the asymptotic efficiency of quasi-maximum likelihood estimation. In small samples, extensive Monte Carlo experiments reveal these efficiency enhancements to be available for (very) heavy tailed cases. Consequently, the proposed simple estimators are members of the class of multi-step estimators aimed at improving the efficiency of the quasi-maximum likelihood estimator.

Keywords: GARCH models, closed form estimation, heavy tails, instrumental variables, regular variation

JEL Classification: C13, C22, C58