The residual-based bootstrap is considered one of the most reliable methods for bootstrapping generalized autoregressive conditional heteroscedasticity (GARCH) models. However, in terms of theoretical aspects, only the consistency of the bootstrap has been established, while the higher order asymptotic refinement remains unproven. For example, Corradi and Iglesias (2008) demonstrate the asymptotic refinement of the block bootstrap for GARCH models but leave the results of the residual-based bootstrap as a conjecture. To derive the second order asymptotic refinement of the residual-based GARCH bootstrap, we utilize the analysis in Andrews (2001, 2002) and establish the Edgeworth expansions of the t-statistics, as well as the convergence of their moments. As expected, we show that the bootstrap error in the coverage probabilities of the equal-tailed t-statistic and the corresponding test-inversion confidence intervals are at most of the order of O(n−1), where the exact order depends on the moment condition of the process. This convergence rate is faster than that of the block bootstrap, as well as that of the first order asymptotic test.
Edgeworth Expansions for Centered Random Walks on Covering Graphs of Polynomial Volume Growth
