A Closed-Form Estimator for the Garch(1,1) Model

Abstract

We propose a closed-form estimator for the linear GARCH~1,1! model+ The estimator has the advantage over the often used quasi-maximum likelihood estimator QMLE! that it can be easily implemented and does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance process+ We derive the asymptotic properties of the estimator, showing T~k Ϫ1!0k-consistency for some k ʦ~1,2! when the fourth moment exists and M T-asymptotic normality when the eighth moment exists+ We demonstrate that a finite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the fourth moment exists+ A simulation study confirms our theoretical results+

Computationally efficient bootstrap prediction intervals for returns and volatilities in ARCH and GARCH processes

“…Although in our studies we have not encountered any stability problems even for a case of α 1 + β 1 being 0.95 and 0.981, in practice that might lead to a failure of convergence and infl ated standard errors, especially for small and moderate samples. Under such circumstances, we can follow the approach of Kristensen and Linton (2006) and censor the LS estimator at 1 − ε for a small positive ε.…”

Simple Estimators for ARCH Models

“…is a TSLS estimator for (32), with the same discussion regarding selection of b in Section 2 remaining applicable.…”

A robust closed-form estimator for the GARCH(1,1) model

“…In this paper, we follow the second alternative and deal with additive outliers by robustifying the closed-form estimator for the GARCH(1,1) proposed by Kristensen and Linton (2006). This estimator has several advantages in comparison with the ML estimator often used to estimate the GARCH(1,1) model that are: it is easy to implement, it does not require the use of any numerical optimization procedure and initial starting values.…”

“…The use of starting values might be a drawback since it can generate different outputs across popular packages (Brooks et al, 2001;McCullough and Renfro, 1999). Our proposal follows that of Kristensen and Linton (2006) and it is based on the autorregressive moving average (ARMA) representation of the squared GARCH model and on the use of the implied autocovariance and autocorrelation functions to obtain closed-form estimators of the parameters. The difference regarding the original estimator is that we replace the sample autocorrelation function by a robust estimator of this function.…”

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ABSTRACTIn this paper we extend the closed-form estimator for the GARCH(1,1) proposed by Kristensen and Linton (2006) to deal with additive outliers. It has the advantage that is per se more robust that the maximum likelihood estimator (ML) often used to estimate this model, it is easy to implement and does not require the use of any numerical optimization procedure.

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