Linear Trend with Fractionally Integrated Errors

We consider the estimation of linear trend for a time series in the presence of additive long-memory noise with memory parameter d∈[0, 1.5). Although no parametric model is assumed for the noise, our assumptions include as special cases the random walk with drift as well as linear trend with stationary invertible autoregressive moving-average errors. Moreover, our assumptions include a wide variety of trend-stationary and difference-stationary situations. We consider three different trend estimators: the ordinary least squares estimator based on the original series, the sample mean of the first differences and a class of weighted (tapered) means of the first differences. We present expressions for the asymptotic variances of these estimators in the form of one-dimensional integrals. We also establish the asymptotic normality of the tapered means for d∈[0, 1.5) −{0.5} and of the ordinary least squares estimator for d∈ (0.5, 1.5). We point out connections with existing theory and present applications of the methodology.