Properties of the nonparametric autoregressive bootstrap

Abstract

For nonparametric autoregression, we investigate a model based bootstrap procedure ('autoregressive bootstrap') that mimics the complete dependence structure of the original time series. We give consistency results for uniform bootstrap confidence bands of the autoregression function based on kernel estimates of the autoregression function. This result is achieved by global strong approximations of the kernel estimates for the resample and for the original sample. Furthermore, it is obtained that the autoregressive bootstrap also yields asymptotically correct approximations for distributions of parametric statistics, for which regression-type bootstrap-techniques like the wild bootstrap do not work. For this purpose, we prove geometric ergodicity and absolute regularity of the nonparametric autoregressive bootstrap process. We propose some particular estimators of the autoregression function and of the density of the innovations such that the mixing coefficients of the autoregressive bootstrap process can be bounded uniformly by some exponentially decaying sequence. This is achieved by using wellestablished coupling techniques. Moreover, by using some 'decoupling' argument, we show that the stationary density of the bootstrap process converges to that of the original process. The paper may serve as a template for proving similar consistency results for other bootstrap techniques such as the Markov bootstrap.

Bootstrap Methods for Time Series

“…Franke, Kreiss, Mammen, and Neumann (2000) give conditions under which the nonparametric autoregressive bootstrap can be used to obtain uniform confidence bands for and to carry out inference about the parameters of a misspecified (finite-dimensional) parametric model. Neumann and Kreiss (1998) use a wildbootstrap version of the nonparametric autoregressive bootstrap to test parametric models of the conditional mean and variance functions.…”

Bandwidth Selection in Nonparametric Kernel Testing

“…In addition, we may also use a wild bootstrap procedure to generate a sequence of resamples for {e * i }. Note also that the above simulation is based on the so-called regression bootstrap simulation procedure discussed in the literature, such as Li and Wang (1998), Franke, Kreiss and Mammen (2002), and Li and Racine (2007). When X i = Y i−1 , we may also use a recursive simulation procedure, commonly-used in the literature.…”

“…When X i = Y i−1 , we may also use a recursive simulation procedure, commonly-used in the literature. See for example, Hjellvik and Tjøstheim (1995), and Franke, Kreiss and Mammen (2002).…”

Estimation of nonlinear autoregressive models using design-adapted wavelets

“…Neumann and Kreiss (1998), Franke et al (2002c), some conditions on the regression function re(x) are given in order to generate a time series {Xt} that fulfills Assumption (A1).…”