Abstract

A semiparametric model is proposed in which a parametric filtering of a nonstationary time series, incorporating fractionally differencing with short memory correction, removes correlation but leaves a nonparametric 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. Tests with standard asymptotics for I(1) and other hypotheses are thereby justified. Estimation of the trend function is also considered. We include a Monte Carlo study of finite-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

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Summary

INTRODUCTION

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.

ESTIMATION OF DEPENDENCE PARAMETERS
NONPARAMETRIC REGRESSION ESTIMATION
FINAL REMARKS
XT XT X s 1 XT 0 ktsktq
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