A Simple Panel‐CADF Test for Unit Roots*

AuthorClaudio Lupi,Mauro Costantini
Published date01 April 2013
Date01 April 2013
DOIhttp://doi.org/10.1111/j.1468-0084.2012.00690.x
276
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2012. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 75, 2 (2013) 0305-9049
doi: 10.1111/j.1468-0084.2012.00690.x
A Simple Panel-CADF Test for Unit RootsÅ
Mauro Costantini† and Claudio Lupi
Department of Economics and Finance, Brunel University, Kingston Lane, Uxbridge, Middlesex
UB8 3PH, UK (e-mail: Mauro.Costantini@brunel.ac.uk)
Department of Economics, Management, and Social Sciences, University of Molise, Via De
Sanctis, I-86100 Campobasso, Italy (e-mail: lupi@unimol.it)
Abstract
In this paper, we propose a simple extension to the panel case of the covariate-augmented
Dickey–Fuller (CADF) test for unit roots developed in Hansen (1995). The panel test we
propose is based on a Pvalues combination approach that takes into account cross-section
dependence. We show that the test has good size properties and gives power gains with
respect to other popular panel approaches. An empirical application is carried out for illus-
tration purposes on international data to test the purchasing power parity (PPP) hypothesis.
I. Introduction
Toobtain more powerful unit root tests, Hansen (1995) suggests using covariate-augmented
Dickey–Fuller (CADF) tests, that is unit root tests that exploit stationary covariates in an
otherwise standard Dickey–Fuller framework. In this paper, we extend Hansen’s CADF
test to small panels. Although the CADF test is not the covariate-augmented point optimal
test in general, we decided to use it for three main reasons. First, simulations reported in
Elliott and Jansson (2003) show that the feasible point optimal test can give power gains at
the cost of inferior size performances: this is important in our framework, because Hanck
(2008) shows that size distortions tend to cumulate in panel tests of the kind proposed here.
Second, Hansen’s CADF test is based on the familiar ADF framework, so that it can be
more appealing to practitioners once the computational burden related to the computation
of the test Pvalues is eased. Finally, we show that under conditions considered especially
relevant for the panel unit root hypothesis, the CADF test is based on the correct conditional
model.
ÅWethank S. Popp for his contributions in the early stages of this work. Comments from audiences at the Univer-
sity of Oxford, at the Institute for Advanced Studies at Viennaand at the Third Italian Congress of Econometrics and
Empirical Economics atAncona are gratefully acknowledged. We are especially grateful to R. Cerqueti, J. Dolado, L.
Gutierrez, R. Kunst, P.K.Narayan, P. Paruolo, M. Wagner,J. Westerlund and D. Zaykin for comments and discussion
on the paper or specic parts of it. None of them is responsible for any remaining error. We are pleased to thank Y.
Chang and W.Song for having provided their data and GAUSS code. Comments from an Editor and three anonymous
referees greatly helped us in improving on previous versions of the paper. The implementation of the panel unit root
test described in this paper is part of the ongoing R(RDevelopment Core Team, 2011) project punitroots and is
freely available from the R-Forge website (see Kleiber and Lupi, 2011).
JEL Classication numbers: C22, C23.
Simple panel-CADF test 277
The panel CADF (pCADF) test we propose is specically designed for macro-panels,
where the time dimension Tis large and the number of panel units Nis typically fairly
small. The test is based on the inverse-normal Pvalue combination advocated in Choi
(2001) and extended in Demetrescu, Hassler and Tarcolea (2006) to cope with dependence
across the panel units. The advantages of this approach are fourfold. First, provided that
we can compute the Pvalues of the CADF test, the extension to the panel case is straight-
forward. Second, the asymptotics carries through for T→∞, without requiring N→∞.
Third, we do not need balanced panels, so that individual time series may come in different
lengths and span different sample periods. Fourth, the test allows for the stochastic as well
as the non-stochastic components to be different across individual time series. The null
hypothesis of the test is that all the series have a unit root, while the alternative is that at
least one time series is stationary. Some authors consider this as a disadvantage, but we
believe that the extent to which this is a real limitation depends on the specic goal of
the analysis. In fact, from the economist’s point of view, there are instances in which it is
especially interesting to test for the presence of a unit root collectively over a whole panel
of time series precisely because the presence of a unit root in all the series can be inter-
preted as a stylized fact that can give stronger support in favour (or against) a particular
economic interpretation.
Although developed independently, the results reported in the present paper are related
to other recent research on covariate augmented panel tests. Despite some similarities,
even in the name, the pCADF test presented here should not be confused with the cross-
sectionally augmented ADF (CADF) test advocated in Pesaran (2007). Chang and Song
(2009) also start from the observation that using stationary covariates can greatly improve
the power of unit root tests. However, their approach is completely different from ours:
while we use a simple Pvalue combination approach, Chang and Song (2009) propose a
method based on nonlinear IV estimation of the autoregressive coefcient.
The rest of the paper is organized as follows. Section II is devoted to a brief discussion
of the test proposed in Hansen (1995) and illustrates the method we use to obtain the neces-
sary Pvalues. Section III offers a brief account of the inverse-normal combination method
and its modications to deal with cross-dependent time series. In section IV an extensive
Monte Carlo analysis of the pCADF test is carried out. For the purpose of illustration, in
section V, we apply our pCADF test to the purchasing power parity (PPP) hypothesis. The
last section concludes.
II. The CADF test and the Pvalues approximation
Hansen (1995) assumed that the series ytto be tested for a unit root can be written as
yt=dt+st,(1)
a(L)st=st1+vt,(2)
vt=b(L)(xtx)+et,(3)
where dtis a deterministic term (usually a constant or a constant and a linear trend),
a(L):
=(1 a1La2L2−···−apLp) is a polynomial in the lag operator L,xtI(1)isan
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2012

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