GARCH Models

9

Optimal Inference and Alternatives to the QMLE*

The most commonly used estimation method for GARCH models is the QML method studied in Chapter 7. One of the attractive features of this method is that the asymptotic properties of the QMLE are valid under mild assumptions. In particular, no moment assumption is required on the observed process in the pure GARCH case. However, the QML method has several drawbacks, motivating the introduction of alternative approaches. These drawbacks are the following: (i) the estimator is not explicit and requires a numerical optimization algorithm; (ii) the asymptotic normality of the estimator requires the existence of a moment of order 4 for the noise η_t; (iii) the QMLE is inefficient in general; (iv) the asymptotic normality requires the existence of moments for _t in the general ARMA-GARCH case; (v) a complete parametric specification is required.

In the ARCH case, the QLS estimator defined in Section 6.2 addresses point (i) satisfactorily, at the cost of additional moment conditions. The maximum likelihood (ML) estimator studied in Section 9.1 of this chapter provides an answer to points (ii) and (iii), but it requires knowledge of the density f of η_t. Indeed, it will be seen that adaptive estimators for the set of all the parameters do not exist in general semi-parametric GARCH models. Concerning point (iii), it will be seen that the QML can ...