Bootstrapping M-estimators in GARCH models

We consider the weighted bootstrap approximation of the distribution of a class of M-estimators of the GARCH (p, q) parameters. We prove that the bootstrap distribution, given the data, is a consistent estimate in probability of the distribution of the M-estimator which is asymptotically normal. We propose an algorithm for the computation of M-estimates which at the same time is software-friendly to compute the bootstrap replicates from the given data. Our simulation study indicates superior coverage rates for various weighted bootstrap schemes compared with the rates based on the normal approximation and the existing bootstrap methods in the literature such as percentile t-subsampling schemes for the GARCH model. Since some familiar bootstrap schemes are special cases of the weighted bootstrap, this paper thus provides a unified theory and algorithm for bootstrapping in GARCH models