Testing for Unit Roots in Time Series Data

Let Y t satisfy the stochastic difference equation for t = 1,2,…, where e t are independent and identically distributed random variables with mean zero and variance σ 2 and the initial conditions ( Y −p+1 ,…, Y 0 ) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m 1 , m 2 ,…, m p of satisfy the hypothesis H d : m 1 = … = m d = 1 and | m j | < 1 for j = d + 1,…, p . We present a reparameterization of the model for Y t that 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 d against the alternative H d−1 . Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.