Robust Estimation and Inference for Heavy Tailed GARCH

Abstract

We develop two new estimators for a general class of stationary GARCH models with possibly heavy tailed asymmetrically distributed errors, covering processes with symmetric and asymmetric feedback like GARCH, Asymmetric GARCH, VGARCH and Quadratic GARCH. The first estimator arises from negligibly trimming QML criterion equations according to error extremes. The second imbeds negligibly transformed errors into QML score equations for a Method of Moments estimator. In this case, we exploit a sub-class of redescending transforms that includes tail-trimming and functions popular in the robust estimation literature, and we re-center the transformed errors to minimize small sample bias. The negligible transforms allow both identification of the true parameter and asymptotic normality. We present a consistent estimator of the covariance matrix that permits classic inference without knowledge of the rate of convergence. A simulation study shows both of our estimators trump existing ones for sharpness and approximate normality including QML, Log-LAD, and two types of non-Gaussian QML (Laplace and Power-Law). Finally, we apply the tail-trimmed QML estimator to financial data.

Non-standard inference for augmented double autoregressive models with null volatility coefficients

“…Third, our entire statistical inference procedure aforementioned is valid as long as y t is stationary, and hence it can have a wide applicable scope in dealing with the heavy-tailed data. Heavy-tailedness is often observed in many empirical data (see, e.g., Rachev (2003), Hill (2015) and Zhu and Ling (2015)).…”

Fat tails and spurious estimation of consumption‐based asset pricing models

“…Specifications that average across Euler equations introduce nonexistent moments, which cause the GMM criterion to be asymptotically random. Complementary to the tail-trimming explored in Hill (2015) and Hill and Prokhorov (2016), our Monte Carlo experiments show that age cohort GMM, and overidentifying restrictions in general, yield improvements in mean squared error and test size/power.…”

“…In the generalized autoregressive conditional heteroskedasticity (GARCH) setting, when error moments become infinite between 2 and 4, the convergence rate of quasi-maximum likelihood (QML) estimation falls below n 1/2 (Berkes and Horváth, 2003), and the asymptotic distribution may be non-Gaussian and difficult to estimate (Hall and Yao, 2003). To address fat-tail issues in the GARCH context, Hill (2015) and Hill and Prokhorov (2016) introduce, respectively, tail-trimmed QML and tail-trimmed GEL, each of which yields asymptotic normality and better finite-sample properties relative to a variety of standard methods.…”

Inference for Structural Impulse Responses in Svar-Garch Models

“…) in the first step instead of the QMTTL estimator of Hill (2015). We replaced the standard QML estimator with the QMTTL estimator of Hill (2015) because the aforementioned estimator enjoys improved convergence properties in small samples compared to the standard QML estimator.…”