Introduction

In time-series regression theory, much attention has been given to models with autoregressive roots at unity or in the vicinity of unity. The limit theory has relied on functional laws to Brownian motion and diffusions, and weak convergence to stochastic integrals. The treatment of local to unity roots has relied exclusively on specifications of the form ρ = 1+c/n, where n is the sample size (Phillips, 1987a; Chan and Wei, 1987) or matrix versions of this form (Phillips, 1988). The theory has been particularly useful in defining power functions for unit-root tests (Phillips, 1987a) under alternatives that are immediately local to unity.

To characterize greater deviations from unity Phillips and Magdalinos (2004; hereafter simply PM) have recently investigated time series with an autoregressive root of the form ρn = 1+c/nα, where the exponent α lies in the interval (0, 1). Such roots represent moderate deviations from unity in the sense that they belong to larger neighbourhoods of one than conventional local to unity roots. The parameter α measures the radial width of the neighbourhood with smaller values of α being associated with larger neighbourhoods.