Bootstrapping the coefficients of a polynomial regression with ARFIMA errors

Abstract

The purpose of this paper is to establish the validity of a bootstrap least square estimate of a polynomial regression model exhibiting an autoregressive fractionally integrated moving average ARFIMA (p,d,q) errors. Under standard conditions on the regression parameters and the error components, the bootstrap is shown to be valid. In other words, for a p×1 vector β of unknown parameters, β ̂_m a ’modified’ least square estimate of β, β ̂_ ^*a bootstrap estimate of β, and C ∈ R^k such that C’(β ̂_m −β) has finite variance, it is shown that the distribution of C’(β ̂_ ^* − β ̂_m) converges to that of C’(β ̂_m −β), uniformly in C. This work is an extension of that of Freedman (1981) and Eck (2018) to the case where the error term is a strongly dependent time series.