An Invariance Principle for Sieve Bootstrap in Time Series

This paper establishes an invariance principle applicable for the asymptotic analysis of sieve bootstrap in time series+ The sieve bootstrap is based on the approximation of a linear process by a finite autoregressive process of order increasing with the sample size, and resampling from the approximated autoregression+ In this context, we prove an invariance principle for the bootstrap samples obtained from the approximated autoregressive process+ It is of the strong form and holds almost surely for all sample realizations+ Our development relies upon the strong approximation and the Beveridge–Nelson representation of linear processes+ For illustrative purposes, we apply our results and show the asymptotic validity of the sieve bootstrap for Dickey–Fuller unit root tests for the model driven by a general linear process with independent and identically distributed innovations+ We thus provide a theoretical justification on the use of the bootstrap Dickey–Fuller tests for general unit root models, in place of the testing procedures by Said and Dickey and by Phillips+