The Bootstrap of the Mean for Dependent Heterogeneous Arrays

Presently, conditions ensuring the validity of bootstrap methods for the sample mean of (possibly heterogeneous) near epoch dependent (NED) functions of mixing processes are unknown. Here we establish the validity of the bootstrap in this context, extending the applicability of bootstrap methods to a class of processes broadly relevant for applications in economics and Þnance. Our results apply to two block bootstrap methods: the moving blocks bootstrap of Künsch (1989) and Liu and Singh (1992) and the stationary bootstrap of Politis and Romano (1994). In particular, the consistency of the bootstrap variance estimator for the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The Þrst order asymptotic validity of the bootstrap approximation to the actual distribution of the sample mean is also established in this heterogeneous NED context.