Title: Simple estimators for GARCH models
Authors: Todd Prono - Federal Reseve Board (United States) [presenting]
Abstract: The aim is to propose closed-form and variance-targeted two stage least squares (VTTSLS) estimators for the popular GARCH(1,1), threshold GARCH(1,1), and general GARCH($p$,$q$) models, where identification depends either on skewness in the rescaled errors or asymmetry in the conditional variance function. Limit theory for these estimators is established in the empirically relevant case of an ill-defined fourth moment for the GARCH process. The resulting distributional limits, determined using point process theory developed for regularly varying and (weakly) dependent sequences, are highly non-normal though stable, with ill-defined variances. The rate of convergence of these estimators depends on the tail index of the GARCH process, and tend to be quite a bit slower than the usual root $n$ case. Relative to kurtosis-focused, closed-form estimators, the VTTSLS estimators only require a well-defined third moment for the GARCH process and so are better aligned with empirical findings for historical asset returns. In a Monte Carlo study, the VTE benchmarks the VTTSLS, where the VTE has a comparable limit to the VTTSLS in the case of an ill-defined fourth moment.