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Linear Trend with Fractionally Integrated Errors
Author(s) -
Deo Rohit S.,
Hurvich Clifford M.
Publication year - 1998
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00099
Subject(s) - mathematics , estimator , autoregressive model , series (stratigraphy) , asymptotic distribution , invertible matrix , ordinary least squares , autoregressive–moving average model , asymptotic analysis , statistics , paleontology , pure mathematics , biology
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.