GARCH Models, 2nd Edition

7 Estimating GARCH Models by Quasi‐Maximum Likelihood

The quasi‐maximum likelihood (QML) method is particularly relevant for GARCH models because it provides consistent and asymptotically normal estimators for strictly stationary GARCH processes under mild regularity conditions, but with no moment assumptions on the observed process. By contrast, the least‐squares methods of the previous chapter require moments of order 4 at least.

In this chapter, we study in details the conditional QML method (conditional on initial values). We first consider the case when the observed process is pure GARCH. We present an iterative procedure for computing the Gaussian log‐likelihood, conditionally on fixed or random initial values. The likelihood is written as if the law of the variables η _t were Gaussian (we refer to pseudo‐ or quasi‐likelihood), but this assumption is not necessary for the strong consistency of the estimator. In the second part of the chapter, we will study the application of the method to the estimation of ARMA–GARCH models. The asymptotic properties of the quasi‐maximum likelihood estimator (QMLE) are established at the end of the chapter.

7.1 Conditional Quasi‐Likelihood

Assume that the observations ε₁, …, ε _n constitute a realisation (of length n ) of a GARCH(p, q) process, more precisely a non‐anticipative strictly stationary solution of

7.1