Asymptotic approximations in the near‐integrated model with a non‐zero initial condition

Abstract

This paper considers various asymptotic approximations in the near-integrated first-order autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous-time approximation of Perron (1991a). We assess, via a Monte Carlo simulation study, the extent to which these alternative methods provide adequate approximations to the finite sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991a) continuous-time approximation performs very well while the others only offer improvements when the initial condition is zero.