• Adaptive estimation;
  • Efficiency;
  • GARCH;
  • Kernel;
  • Semi-parametric

Summary  Financial time series exhibit time-varying volatilities and non-Gaussian distributions. There has been considerable research on the GARCH models for dealing with these issues related to financial data. Since in practice the true error distribution is unknown, various quasi maximum likelihood methods based on different assumptions on the error distribution have been studied in the literature. However, the specification of the distribution family or in particular the shape parameter of the density function is often incorrect. This leads to an efficiency loss quite common in such estimation procedures. To avoid the inaccuracy, semi-parametric maximum likelihood approaches were introduced, where the estimators of the GARCH parameters are derived from the likelihood based on a non-parametrically estimated density function. In general, the semi-parametric likelihood function is trimmed for the extreme observations in order to derive a inline image-consistent estimator. In this paper we consider the situation in which the untrimmed likelihood function is maximized to develop a new semi-parametric estimator. The resulting estimator is consistent, asymptotically Gaussian with a vanishing bias term, and a limiting variance–covariance matrix that attains the information lower bound. This work also provides insight into the efficiencies (bias–variability trade-off) of a general class of semi-parametric estimators of GARCH models.