Tests of

Tests of AR(2)unit roots and cointegration with

the GPH statistic

Jonas Andersson∗and Johan Lyhagen†

Abstract

Time domain tests of unit roots and cointegration are often made after

a model search procedure has been made in order to reduce erroneus in-

ference caused by model misspecification.An alternative to this is to use

a semi parametric method like the GPH(Geweke&Porter-Hudak,1983)

test where the properties of the spectral density close to the zero frequency

are exploited,see Cheung&Lai(1993)who use this to test for cointegra-

tion in a bivariate system.In this paper we further investigate the size and

power properties of this test.The results provide evidence that the test

has not only good power properties against AR(1)and ARFIMA(0,d,0)

alternatives,which was shown by Cheung and Lai(1993)but also against

AR(2)alternatives.We show that the power of the test is not as good

when applied to a raw time series as when applied to residuals from a

cointegrating regression.

The power can,in the AR(1)case,be significantly improved by using

a larger bandwith than is usually done for the GPH estimator without

imposing any size distortions.For the AR(2)alternative,however,severe

size distortions appear when this is done.

1Introduction

Tests for unit roots are usually made in order to investigate whether the effect of a shock in a time series variable persists or eventually vanishes.One of the most common of those tests,the Dickey-Fuller test(Dickey&Fuller,1979),is based on a regression of thefirst difference on thefirst lag of the level.What effectively is tested for is a unit root in the AR(1)polynomial.Since the purpose is to test if the effect of a shock persists an AR(1)alternative is not necessarily the only interesting alternative hypothesis.Aware of this,Dickey and Fuller(1981) proposed a modification of the test,the augmented Dickey-Fuller(ADF)test, where they included lags of thefirst difference in the regression.However,it ∗Department of Information Science,Division of Statistics,Uppsala University,Box513, SE-75120Uppsala.

†Department of Economic Statistics,Stockholm School of Economics,Box6501,SE-113 83Stockholm,Sweden.

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has later been shown that the test has low power against fractional alternatives (see e.g.Sowell,1990).

Since it is the long run properties of the time series we are interested in it seems reasonable to consider the spectral density close to the zero frequency. Cheung and Lai(1993)used this approach in testing for cointegration in a bi-variate system,by basing a test statistic on the semi-parametric GPH estimator. They show that the test in fact is more powerful against both ARFIMA(0,d,0) and,more surprisingly,AR(1)alternatives.In this paper we investigate the properties of this test further.Especially,we study which bandwith to use in order to preserve the attractable size and power properties of the test and also evaluate these properties under AR(2)alternatives.In Section2the GPH es-timator and its properties is reviewed.In Section3pure unit root tests are to be studied while tests of cointegration,based on the GPH estimator,are to be considered in Section4.A conclusion closes the paper.

2The GPH test

We want to test whether the underlying process generating the time series{x t} has a unit root or not.If this is the case the spectral density of thefirst difference G t=(1−L)x t:

f G(ω)=|1−exp(−iω)|2f x(ω),(1) where f x(ω)and f G(ω)are the spectral densities of x t and G t,respectively, should beflat in a neighborhood of the zero frequency.This property is ex-ploited by Cheun

g and Lai(1993)who used the semi parametric GPH estima-tor,originally suggested by Geweke and Porter-Hudak(1983),to test for a unit root in{x t}.The GPH estimator is based on the representation

f G(ω)=|1−exp(−iω)|−2(d−1)f u(ω)

=(2sin(ω/2))−2(d−1)f u(ω),(2) where d is the long memory parameter and f u(ω)is the spectral density of an ARMA process(or a slowly varying function).The estimator is obtained by taking the logaritm of f G(ω),adding and subtracting the log-periodogram of {G t},ln I(ω),and doing the same with ln(f u(0)),i.e.

ln I(ωj)=ln(f u(0))−(d−1)ln¡4sin2(ωj/2)¢+

ln(f u(ωj)/f u(0))+

ln(I(ωj)/f G(ωj))(3)

ωj=2πj

,j=1,2,...m and T=sample size

This is a regression of ln I(ωj)on ln¡4sin2(ωj/2)¢with Gumbel error-term.If f u(ω)isflat,i.e.when the ARMA-part of the process is just a white noise,the

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