Modified CADF and CIPS Panel Unit Root Statistics with Standard Chi‐squared and Normal Limiting Distributions
| Published date | 01 June 2016 |
| DOI | http://doi.org/10.1111/obes.12127 |
| Date | 01 June 2016 |
347
©2016 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 78, 3 (2016) 0305–9049
doi: 10.1111/obes.12127
Modified CADF and CIPS Panel Unit Root Statistics
with Standard Chi-squared and Normal Limiting
Distributions*
Joakim Westerlund†, ‡ and Mehdi Hosseinkouchack§
†Department of Economics, Lund University, P.O. Box 7082, S-220 07 Lund, Sweden
(email: joakim.westerlund@nek.lu.se.)
‡Centre for Financial Econometrics, Deakin Business School, Deakin University, Melbourne
Burwood Campus, 221 Burwood Highway, VIC 3125, Australia
§Goethe University Frankfurt, RuW, P.O. Box 49, Grueneburgplatz 1, 60323 Frankfurt am
Main Germany (email: hosseinkouchack@wiwi.uni-frankfurt.de)
Abstract
In an influential paper Pesaran (‘A simple panel unit root test in presence of cross-section
dependence’, Journal ofApplied Econometrics, Vol. 22, pp. 265–312, 2007) proposes two
unit root tests for panels with a common factor structure. These are the CADF and CIPS test
statistics, which are amongst the most popular test statistics in the literature. One feature
of these statistics is that their limiting distributions are highly non-standard, making for
relatively complicated implementation. In this paper, we take this feature as our starting
point to develop modified CADF and CIPS test statistics that support standard chi-squared
and normal inference.
I. Introduction
Consider the panel data variableyi,t, obser vablefor t=1,…, Ttime periods and i=1,…, N
cross-section units. It is well known that unattended cross-section dependence can lead to
deceptive inference when testing the null hypothesis of a unit root in such variables. This
is certainly true for panel unit root tests devised to test the hypothesis that y1,t,…, yN,tare
jointly unit root non-stationary, but the problem is there also when applying univariate unit
root tests to each cross-section unit (see Westerlund and Breitung, 2013, for an illustration).
This finding has led to the development of factor-based ‘second-generation’test procedures
that are robust to cross-section dependence (see Baltagi, 2008; Breitung and Pesaran, 2008,
chapter 12, for surveys of the literature). Two of the most popular second-generation tests
are the cross-section augmented ADF (CADF) and IPS (CIPS) tests of Pesaran (2007). In
fact, these tests have become two of the workhorses of the industry, with a large number of
JEL Classification numbers: C12; C13; C33.
*The authors thank Anindya Banerjee (Editor) and two anonymous referees for many valuable comments and
suggestions. Our thanks also to the Knut andAlice Wallenberg Foundation for financial support through aWallenberg
Academy Fellowship,and the Jan Wallander andTom Hedelius Foundation for financial support under research grant
number P2014–0112:1.
348 Bulletin
applications (see Frydman and Saks, 2010; Holly, Pesaran andYamagata,2010; Eberhardt,
Helmers and Strauss, 2013; Omri et al., 2015, for a broad range of recent applications)
and also several theoretical extensions (see, e.g., Pesaran, Smith and Yamagata, 2013;
Westerlund, Hosseinkouchack and Solberger, forthcoming).1
As the name suggests, the idea underlying the cross-section augmentation approach
put forth by Pesaran (2007) is to use cross-section averages yt−1and ytof yi,t−1and yi,t,
respectively, as proxies for the common component of the data, which are then included
in the ADF test regression as additional regressors. But if yi,tis unit root non-stationary
then so is yt, suggesting that the asymptotic distributions of the resulting CADF and CIPS
statistics will depend on the Brownian motion generated byyt. They will therefore be highly
non-standard, which in turn makes for complicated implementation. In particular, not only
is it necessary to tabulate critical values for each constellation of (N,T), but there is also
a need to truncate the test statistics in order to ensure finite moments. In the current paper,
we take this feature as our starting point. The purpose is to propose modified CADF and
CIPS test statistics that support standard chi-squared and normal inference.
II. Model and test statistic
The data generating process (DGP) is given by
yi,t=(1 −i)
idt+iyi,t−1+ift+"i,t,(1)
i=exp(T−1ci), (2)
where dt=(1,t,…, tp)for p0 and dt=0 for p=−1, ftis an unobserved common
factor with ibeing the associated factor loading, and "i,tis an idiosyncratic error term.
Most research focus on the case when there is at most a linear trend, and therefore so will
we. Hence, in what follows we assume that p1. The rest of the conditions that we will
be working under, which are similar to those found in Pesaran (2007, section 3), and are
given in Assumption 1.
Assumption 1.
(i) "i,tis independent across both iand twith E("i,t)=0, E("2
i,t)=2
i>0 and E("4
i,t)<
∞;
(ii) ftis independent across twith E(ft)=0, E(f2
t)=1 and E(f4
t)<∞;
(iii) ci0 is independent with j=E(cj
i) and |j|<∞for j0;
(iv) "i,t,ftand ciare mutually independent;
(v) y1,0 =···= yN,0 =0;
(vi) =N−1N
i=1i→= 0asN→∞.
Remark 1. The asymptotic analysis provided in Pesaran (2007) only covers the be-
haviour under the unit root null hypothesis, in which case c1=···=cN=0(1= ···=
N=1). The local-to-unity specification of iin equation (2) therefore represents an ex-
tension of the DGP considered by Pesaran (2007). In fact, the only assumption that is not
1One reason for the popularity of the CADF and CIPS tests is their availability in various statistical computer
programs such as GAUSS and STATA.
©2016 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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