Quasilikelihood and quasi-maximum likelihood for GARCH-type processes: Estimating function approach

When modeling conditionally heteroscedastic processes, it is usually the case that exact likelihood is not known to researchers or it is too complicated for practical purposes. Consequently, a quasi-maximum likelihood (QML) is frequently employed rather than the exact maximum likelihood (cf. Gourieroux (1997, Ch. 4), Straumann (2005, Ch. 5)). When a GARCH process is partially specified only through the first and second order conditional moments, a systematic and unified approach for inference is via the so called quasilikelihood (QL, see Godambe (1985)). This article aims to discriminate between the QL and QML for general GARCH-type processes. It is verified that the QL differs with the QML in terms of conditional skewness and kurtosis. Asymptotics for the QL and QML are obtained and then compared with each other. A class of skewed tt-distributions (cf. Fernandez and Steel (1998)) is considered to illustrate the difference between the QL and QML.