Limit theory and bootstrap for explosive and partially explosive autoregression

Abstract

Consistency of the least squares estimator \\ ̂ gb of the autoregressive parameter vector is established in a pth order autoregression model Y t = β 1 Y t−1 + … + β p Y t− p + ε t , when all the roots of the characteristic polynomial Φ( ξ) = ξ p − β 1 ξ p−1 − … − β p lie outside the unit circle and { ε t } is an arbitrary collection of independent random variables satisfying a uniform integrability of log + (| ε t |) and a condition in terms of the concentration functions. For i.i.d. errors, a limiting distribution result for \\ ̂ gb is obtained under the finiteness of Elog + (| ε t |). The asymptotics for bootstrapping the sampling distribution of \\ ̂ gb is also considered under the same moment condition and is shown to match (in probability) the limiting distribution of \\ ̂ gb. Thus, for the explosive case, the bootstrap principle works with the usual choice of the resample size even if the error distribution is heavy tailed. Furthermore, we show that the error in the bootstrap approximation (as measured by the Kolmogorov distance) goes to zero, almost surely, if E| ε t | < ∞. Partially explosive models, where the characteristic polynomial Φ has some roots inside and some roots outside the unit circle are also considered. For such models, the Kolmogorov distance between the true sampling distribution of \\ ̂ gb and its bootstrap approximation is shown to converge to zero when Eε 2 t < ∞.