美联储-当尾部很重时：GARCH模型的方差目标、非高斯、准最大似然估计的好处（英）

Finance and Economics Discussion SeriesFederal Reserve Board, Washington, D.C.ISSN 1936-2854 (Print)ISSN 2767-3898 (Online)When Tails Are Heavy: The Benefits of Variance-Targeted,Non-Gaussian, Quasi-Maximum Likelihood Estimation ofGARCH ModelsTodd Prono2025-075Please cite this paper as:Prono, Todd (2025). “When Tails Are Heavy: The Benefits of Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models,” Finance and Eco-nomics Discussion Series 2025-075. Washington: Board of Governors of the Federal ReserveSystem, https://doi.org/10.17016/FEDS.2025.075.NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminarymaterials circulated to stimulate discussion and critical comment. The analysis and conclusions set forthare those of the authors and do not indicate concurrence by other members of the research staff or theBoard of Governors. References in publications to the Finance and Economics Discussion Series (other thanacknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.When Tails Are Heavy: The Benefits of Variance-Targeted,Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCHModels1Todd Prono2This Version: July 2025AbstractIn 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 popu-lar 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 thatis highly non-Gaussian, with an ill-defined variance. The rate of convergence to this non-standard limitis slow relative √n and dependent upon unknown parameters. Fortunately, the sub-sample bootstrapis applicable, given a carefully constructed normalization. Surprisingly, both Monte Carlo experimentsand 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 usedto forecast option-implied volatility of volatility (VVIX), thus demonstrating a link between historicalvolatility of VIX and risk-neutral volatility-of-volatility.Keywords: GARCH, VIX, VVIX, heavy tails, robust estimation, variance forecasting, volatility,volatility-of-volatility. JEL codes: C13, C22, C58.1The analysis and conclusions presented herein are those of the author and do not indicate concurrence by either the Federal Reserve Board orthe Federal Reserve System. I owe thanks to seminar participants at the 9th International Workshop on Financial Markets and Nonlinear Dynamicsfor helpful comments and discussions. I additionally owe thanks to Dong Hwan Oh for (many) detailed discussions and reviews.2Federal Reserve Board; (202) 510-2398, todd.a.prono@f