When Tails Are Heavy: The Benefits of Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models

In heavy-tailed cases, variance targeting the Student's-t estimator proposed in Bollerslev (1987) for the linear GARCH model is shown to be robust to density misspecification, just like the popular Quasi-Maximum Likelihood Estimator (QMLE). The resulting Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimator (VTNGQMLE) is shown to possess a stable limit, albeit one that is highly non-Gaussian, with an ill-defined variance. The rate of convergence to this non-standard limit is slow relative √n and dependent upon unknown parameters. Fortunately, the sub-sample bootstrap is applicable, given a carefully constructed normalization. Surprisingly, both Monte Carlo experiments and empirical applications reveal VTNGQMLE to sizably outperform QMLE and other performance-enhancing (relative to QMLE) alternatives. In an empirical application, VTNGQMLE is applied to VIX (option-implied volatility of the S&P 500 Index). The resulting GARCH variance estimates are then used to forecast option-implied volatility of volatility (VVIX), thus demonstrating a link between historical volatility of VIX and risk-neutral volatility-of-volatility.