Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator

Abstract

This paper investigates the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1,1) model. The rescaled variable (the ratio of the disturbance to the conditional standard deviation) is not required to be Gaussian nor independent over time, in contrast to the current literature. The GARCH process may be integrated (α + β = 1), or even mildly explosive (α + β > 1). A bounded conditional fourth moment of the rescaled variable is sufficient for the results. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

Practical Issues in the Analysis of Univariate GARCH Models

“…Under suitable regularity conditions, the ML estimates are consistent and asymptotically normally distributed and an estimate of the asymptotic covariance matrix of the ML estimates is constructed from an estimate of the final Hessian matrix from the optimization algorithm used. Unfortunately, verification of the appropriate regularity conditions has only been done for a limited number of simple GARCH models, see [63], [60], [55], [56] and [81]. In practice, it is generally assumed that the necessary regularity conditions are satisfied.…”

Residual-Based Garch Bootstrap and Second Order Asymptotic Refinement

“…The consistency and asymptotic normality of this QML estimator were first shown in Lee and Hansen (1994) and Lumsdaine (1996) and later refined with much weaker conditions in Berkes et al (2003) and Francq and Zakoïan (2004). The former establishes the asymptotic normality of the estimator under minimal conditions on the innovations {ε t }, and the latter further reduces technical assumptions on the distribution of the innovations at the origin.…”

Joint forecasts of Dow Jones stocks under general multivariate loss function

“…In the constant conditional correlation case (κ = θ = 0), however, the requirements given by Lee and Hansen (1994) for consistent QML estimation of the parameters in a univariate, covariance-stationary GARCH(1,1) model are easily met by our data-generating process. 8 As a direct consequence, the two-step sample correlation estimator ρ ik will be consistent and so too will be the three-step ML estimators γ i and β i .…”

“…All computations in this section were done using the Ox programming language of Doornik (2006). Lee and Hansen (1994) are fulfilled: the process { i k=1 ρ * ik υ k,t }, as a collection of iid random variables, is strictly stationary and ergodic, and, for ρ * ik > 0, all moments of the NIG variable ρ * ik υ k,t do exist and are finite, see Alp (2007). 9 This should not come as a surprise, since, under the Gaussian DCC data gener- that for large sample sizes, available in most financial data sets, the differences are not striking.…”