Efficient Estimation in Semiparametric GARCH Models

It is well-knownthat financial data sets exhibit conditional heteroskedasticity. GARCH type models are often used to model this phenomenon. Since the distribution of the rescaled innovations is generally far froma normal distribution, a semiparametric approach is advisable. Several publications observed that adaptive estimation of the Euclidean parameters is not possible in the usual parametrization when the distribution of the rescaled innovations is the unknown nuisance parameter. However, there exists a reparametrization such that the efficient score functions in the parametric model of the autoregression parameters are orthogonal to the tangent space generated by the nuisance parameter, thus suggesting that adaptive estimation of the autoregression parameters is possible. Indeed, we construct adaptive and hence efficient estimators in a general GARCH in mean type context including integrated GARCH models. Our analysis is based on a general LAN Theorem for time-series models, published elsewhere. In contrast to recent literature about ARCH models we do not need any moment condition