Efficiency Tradeoffs in Estimating the Trend and Error Structure of the Linear Model

Assume that the observed time series has been generated by the model Yt=a + bt + yt, t=l,...,T (1) yt = pyt-i+YiAyt-i+-.--h'p-i^yt-p+i+£t, st~i.i.d.(0,c^) (2) where A denotes the first difference operator and p e (-1,1] is the largest autoregressive root in the autoregressive representation of yt implied by (2). Thus, yt can be an 1(1) or an 1(0) process according to whether p = 1 or p e (-1,1), respectively. If p e (-1,1), the Grenander and Rosenblatt (1957) result implies that the ordinaiy least squares (OLS) estimator of (a,b) in (1) is asymptotically equivalent to the generalized least squares (GLS) estimator of (a,b) using (1) and (2). If p = 1, the parameter a is not identified and although the OLS estimator of b is consistent, it is not asymptotically efficient. In this case, the sample mean of Ayt is an asymptotically efficient estimator of b, being equivalent to the GLS estimator. We will refer to the f sample mean of Ayt as the first-difference estimatorof b. Of course, in practice we do not know a priori whether p is equal to or less than one.