Model-Based Bootstrap

Abstract

In this chapter, we consider bootstrap methods for some popular time series models, such as the autoregressive processes, that are driven by iid random variables through a structural equation. As indicated in Chapter 2, for such models, it is often possible to adapt the basic ideas behind bootstrapping a linear regression model with iid error variables (cf. Freedman (1981)). In Section 8.2, we consider stationary autoregressive processes of a general order and describe a version of the autoregressive bootstrap (ARB) method. Like Efron’s (1979) IID resampling scheme, the ARB also resamples a single value at a time. We describe theoretical and empirical properties of the ARB for the stationary case in Section 8.2. In Section 8.3, we consider the explosive autoregressive processes. In the explosive case, the initial variables defining the model have nontrivial effects on the limit distributions of the least squares estimators of the autoregression (AR) parameters. As a result, the validity of the ARB critically depends on the initial values. In Section 8.3, we describe the relevant issues and provide conditions for the validity of the ARB method in the explosive case.