Maximum likelihood and the bootstrap for nonlinear dynamic models

We provide a unified framework for analyzing bootstrapped extremum estimators of nonlinear dynamic models for heterogeneous dependent stochastic processes. We apply our results to the moving blocks bootstrap of Künsch (Ann. Stat. 17 (1989) 1217) and Liu and Singh (in: R. Lepage, L. Billiard (Eds.), Exploring the Limits of the Bootstrap, Wiley, New York, 1992) and prove the first-order asymptotic validity of the bootstrap approximation to the true distribution of quasi-maximum likelihood estimators. We also consider bootstrap testing. In particular, we prove the first-order asymptotic validity of the bootstrap distribution of suitable bootstrap analogs of Wald and Lagrange Multiplier statistics for testing hypotheses.