Abstract
A semi-parametric model is proposed in which a parametric filtering of a non-stationary time series, incorporating fractionally differencing with short memory correction, removes correlation but leaves a non-parametric deterministic trend. Estimates of the memory parameter and other dependence parameters are proposed, and shown to be consistent and asymptotically normally distributed with parametric rate. Unit root tests with standard asymptotic distributions are thereby justified. Estimation of the trend function is also considered. We include a Monte Carlo study of nite-sample performance.
Highlights
A long-established vehicle for smoothing a deterministically-trending time series yt; t = 1; :::; T; is the ...xed-design nonparametric regression model given by t yt = g T + ut; t = 1; :::; T; (1)
Where g(x); x 2 [0; 1] ; is an unknown, smooth, nonparametric function, and ut is an unbservable sequence of random variables with zero mean
A more basic trend function is a polynomial in t of given degree, as still frequently employed in various econometric models
Summary
The autoregressive unit root literature suggests that estimates of in (8) will have a nonstandard limit distribution under = 1 (7); but a normal one in the "stationary" region j j < 1: By contrast we can anticipate, for example from literature concerning (4) with g(x) a priori constant; that estimates of 0 such as ones optimizing an approximate pseudo-Gaussian likelihood, and Wald and other test statistics, will enjoy standard asymptotics, with the usual parametric convergence p rate, T ; whatever the value of 0; due essentially to smoothness properties of the fractional operator; tests are expected to have the classical local e¢ ciency properties.
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