Testing for Unit Roots in Time Series Data

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Abstract
Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 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,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.