## Bernoulli

• Bernoulli
• Volume 21, Number 3 (2015), 1629-1669.

### Robust estimation and inference for heavy tailed GARCH

Jonathan B. Hill

#### 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.

#### Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1629-1669.

Dates
Revised: February 2014
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.bj/1432732032

Digital Object Identifier
doi:10.3150/14-BEJ616

Mathematical Reviews number (MathSciNet)
MR3352056

Zentralblatt MATH identifier
1319.62192

#### Citation

Hill, Jonathan B. Robust estimation and inference for heavy tailed GARCH. Bernoulli 21 (2015), no. 3, 1629--1669. doi:10.3150/14-BEJ616. https://projecteuclid.org/euclid.bj/1432732032

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#### Supplemental materials

• Supplement to “Robust estimation and inference for heavy tailed GARCH”. We prove Lemmas A.1, A.3, A.4 and A.6, and Lemmas B.1 and B.2. Assume all functions satisfy Pollard’s [54] permissibility criteria, the measure space that governs all random variables in this paper is complete, and therefore all majorants are measurable. Cf. Dudley [19]. Probability statements are therefore with respect to outer probability, and expectations over majorants are outer expectations.