A Simple Panel Unit‐Root Test with Smooth Breaks in the Presence of a Multifactor Error Structure

Published date01 June 2016
Date01 June 2016
DOIhttp://doi.org/10.1111/obes.12109
365
©2015 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.12109
A Simple Panel Unit-Root Test with Smooth Breaks in
the Presence of a Multifactor Error Structure*
Chingnun Lee, Jyh-Lin Wu,†,‡ and Lixiong Yang§
Institute of Economics, National Sun Yat-sen University, Kaohsiung, Taiwan
(e-mail: lee econ@mail.nsysu.edu.tw)
Department of Economics, National Chung Cheng University, Chia-Yi, Taiwan
(e-mail: jlwu2@mail.nsysu.edu.tw; ecdjlw@ccu.edu.tw)
§School of Management, Lanzhou University, Gansu, China (e-mail: ylx@lzu.edu.cn)
Abstract
This paper extends the cross-sectionally augmented panel unit-root test (CIPS) developed
by Pesaran et al. (2013, Journal of Econometrics, Vol. 175, pp. 94–115) to allow for
smoothing structural changes in deterministic terms modelled by a Fourier function. The
proposed statistic is called the break augmented CIPS (BCIPS) statistic. We show that the
non-standard limiting distribution of the (truncated) BCIPS statistic exists and tabulate its
critical values. Monte-Carlo experiments point out that the sizes and powers of the BCIPS
statistic are generally satisfactory as long as the number of time periods, T, is not less than
fifty.The BCIPS test is then applied to examine the validity of long-run purchasing power
parity.
I. Introduction
The development of panel unit-root tests has been a hot research topic during the past
decade. The first generation articles assume that idiosyncratic errors are cross-sectionally
independent (Banerjee, 1999; Levin, Lin and Chu, 2002; Im, Pesaran and Shin, 2003, IPS;
Maddala and Wu, 1999) and the second generation articles focus on the tests that allow
cross-dependent errors (Chang, 2002; Breitung and Das, 2003; Phillips and Sul, 2003; Bai
and Ng, 2004; Moon and Perron, 2004; Smith et al., 2004; Choi and Chue, 2007; Pesaran,
2007; Pesaran, Smith andYamagata,2009, 2012, 2013). Nonetheless, these ar ticles assume
no structural changes in the models.
Two recent papers proposed panel unit-root tests that allow for multiple structural
changes and cross-sectional dependence. Bai and Carrion-i-Silvestre (2009) propose a
modified Sargan–Bhargava (1983, MSB) test in the panel setting. Although this test is
invariant to both mean and trend break parameters, the limiting distribution of the indi-
JEL Classification numbers: C12, C33.
*We are grateful to Anindya Banerjee (editor) and three anonymous referees for very valuable comments and
suggestions on previous versions of the paper.We thank Row-WeiWu for research assistance. Remaining errors are
our own.
366 Bulletin
vidual MSB (MSB*()) test depends on the number of structural breaks. Following the
cross-sectionally augmented procedure of Pesaran (2007), Im, Lee andTieslau (2010, ILT)
develop an LM-type panel unit-root test to account for possible heterogeneity in both the
level and the trend of the series. The ILT test is invariant to nuisance parameters, but its
limiting distribution depends on the number of trend breaks.
Instead of adopting dummy variables to capture discrete breaks, several articles
develop unit-root tests by applying Gallant’s (1981) flexible Fourier form to take into
account smoothing breaks in the deterministic components (Becker, Enders and Hurn,
2004; Becker, Enders and Lee, 2006; Enders and Lee, 2012a,b; Rodrigues and Taylor,
2012). Enders and Lee (2012a,b) point out several advantages of the Fourier form ap-
proximation. First, it works reasonably well for types of breaks often used in economic
analysis. Second, the Fourier function with a single-frequency component () can be a
reasonable approximation for breaks of an unknown form even if the function itself is not
periodic. Third, it involves only the determination of the appropriate component in the
model and hence avoids the complication of selecting break dates, the number of breaks
and the form of breaks. Enders and Lee (2012a,b) find that their proposed tests are robust
to a variety of possible break mechanisms in the deterministic trend function of unknown
forms and numbers. Their Fourier tests complement the unit-root tests using dummy vari-
ables.
This paper extends Pesaran et al.s (2013) multifactor error structure model to allow
for smoothing breaks in deterministic components and then develops a new simple panel
unit-root test that accommodates cross-sectional dependence among variables and smooth-
ing changes in deterministic components. We first develop the breaks and cross-sectional
dependence augmented ADF (BCADF) statistic and its average statistic by generaliz-
ing their cross-sectionally augmented ADF (CADF) regression to incorporate a single-
frequency Fourier function with heterogeneousamplitudes. The breaks and cross-sectional
dependence augmented IPS (BCIPS) statistic is proposed by averaging the BCADF statis-
tics across individuals. An important advantage of the tests is their simplicity in empirical
applications.
To analyse the impact of Fourier terms in the BCADF regression in both finite and
infinite T, new asymptotic results of the BCADF and BCIPS statistics are derived based on
the sequential and joint limit approaches respectively. In the case of serially uncorrelated
errors, Theorems 1 and 2 show that the asymptotic distribution of the BCADF statistic does
not depend on nuisance parameters when the number of individuals, N, tends to infinity
under a fixed Tor when both Nand Tsequentially and jointly tend to infinity. Theorem 3
examines the limiting distribution of the CADF statistic provided by Pesaran et al. (2013)
when Fourier form breaks exist in the data-generating process (DGP) but are ignored in the
regression. We show that, because of the omitted-variable bias, the asymptotic distribution
of the CADF statistic under a fixed Tdepends on nuisance parameters even when Ntends
to infinity, but the dependence vanishes when both Nand Tapproach infinity. Besides, the
limiting distribution of the (truncated) BCIPS statistic is shown to exist. Theorem 4 shows
that the BCADF statistic, under first-order autocorrelated errors, has the same asymptotic
distribution as one that is obtained based on serially uncorrelated errors when both N
and Ttend to infinity. Furthermore, this paper extends the discussion to the case with a
general autoregressive and moving average, ARMA(l,s), specification of errors. In such a
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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