Testing for Unit Roots in Time Series Data

Abstract

Let Y tsatisfy the stochastic difference equation for t= 1,2,…, where e tare independent and identically distributed random variables with mean zero and variance σ 2 and the initial conditions (Y−p+1,…,Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,m pof satisfy the hypothesis H d:m1= … =m d= 1 and |m j| < 1 for j=d+ 1,…,p. We present a reparameterization of the model for Y tthat is convenient for testing the hypothesis H d. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis H d, (ii) a likelihood ratio type t-statistic for testing the hypothesis H dagainst the alternative H d−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.