INCIDENTAL TRENDS AND THE POWER OF PANEL …korora.econ.yale.edu/phillips/pubs/art/p1215.pdfMaddala...

INCIDENTAL TRENDS AND THE POWER OF PANEL UNIT ROOT TESTS

BY

HYUNGSIK ROGER MOON, BENOIT PERRON and PETER C. B. PHILIPS

COWLES FOUNDATION PAPER NO. 1215

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

2007

http://cowles.econ.yale.edu/

Journal of Econometrics 141 (2007) 416–459

Incidental trends and the power ofpanel unit root tests

Hyungsik Roger Moona,�, Benoit Perronb, Peter C.B. Phillipsc,d,e

aDepartment of Economics, University of Southern California, USAbDepartement de sciences economiques, CIREQ, and CIRANO, Universite de Montreal, Canada

cCowles Foundation, Yale University, USAdUniversity of Auckland, New Zealand

eUniversity of York, UK

Available online 28 November 2006

Abstract

The asymptotic local power of various panel unit root tests is investigated. The (Gaussian) power

envelope is obtained under homogeneous and heterogeneous alternatives. The envelope is compared

with the asymptotic power functions for the pooled t-test, the Ploberger and Phillips [2002. Optimal

testing for unit roots in panel data. Mimeo] test, and a point optimal test in neighborhoods of unity

that are of order n�1=4T�1 and n�1=2T�1; depending on whether or not incidental trends are extracted

from the panel data. In the latter case, when the alternative hypothesis is homogeneous across

individuals, it is shown that the point optimal test and the Ploberger–Phillips test both achieve the

power envelope and are uniformly most powerful, in contrast to point optimal unit root tests for time

series. Some simulations examining the finite sample performance of the tests are reported.

r 2006 Elsevier B.V. All rights reserved.

JEL classification: C22; C23

Keywords: Asymptotic power envelope; Common point optimal test; Incidental trends; Local asymptotic power

function; Panel unit root test

ARTICLE IN PRESS

www.elsevier.com/locate/jeconom

0304-4076/$ - see front matter r 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jeconom.2006.10.003

�Corresponding author. Tel.: +1 213 740 2108; fax: +1 213 740 8543.

E-mail address: [email protected] (H.R. Moon).

1. Introduction

In the past decade, much research has been conducted on panels in which both the cross-sectional and time dimensions are large. Testing for a unit root in such panels has been amajor focus of this research. For example, Quah (1994), Levin et al. (2002), Im et al.(2003), Maddala and Wu (1999), and Choi (2001) have all proposed various tests. Thesestudies derived the limit theory for the tests under the null hypothesis of a common panelunit root and power properties were investigated by simulation. On the other hand,Bowman (2002) studies the exact power of panel unit root tests against fixed alternativehypotheses. He characterizes the class of admissible tests for unit roots in panels and showsthat the averaging-up tests of Im et al. (2003) and the test based on Fisher-type statistics inMaddala and Wu (1999) and Choi (2001) are not admissible.

The asymptotic local power properties of some panel unit root tests have become knownrecently. Breitung (2000)1 and Moon and Perron (2004) independently find that without

incidental trends in the panel, their panel unit root test, which is based on a t-ratio typestatistic, has significant asymptotic local power in a neighborhood of unity that shrinks tothe null at the rate of n�1=2T�1 (where n and T denote the size of the cross-section and timedimensions, respectively). However, in the presence of incidental trends, Moon and Perron(2004) show that their t-ratio type test statistic constructed from ordinary least squares(OLS) detrended data has no power (beyond size) in a n�kT�1-neighborhood of unity withk4 1

6. For a panel with incidental trends, Ploberger and Phillips (2002) proposed an

optimal invariant panel unit root test that maximizes average local power. They show thatthe optimal invariant test has asymptotic local power in a neighborhood of unity thatshrinks at the rate n�1=4T�1, thereby dominating the t-ratio test of Moon and Perron(2004) when there are incidental trends.

The present study makes three contributions. First, the local asymptotic power envelopeof the panel unit root testing problem is derived under Gaussian assumptions for fourscenarios: (i) with no fixed effects; (ii) with fixed effects that are parameterized byheterogeneous intercept terms (deemed incidental intercepts); (iii) with fixed effects that areparameterized by heterogeneous linear deterministic trends (deemed incidental trends);and (iv) with incidental intercepts but with a common trend. For cases (ii)–(iv) we restrictthe class of tests to be invariant with respect to the incidental intercepts and trends.We show that in cases (i) and (ii), the power envelope is defined withinn�1=2T�1-neighborhoods of unity and that it depends on the first two moments of thelocal-to-unity parameters. On the other hand, in case (iii), the power envelope is defined withinn�1=4T�1-neighborhoods of unity and it depends on the first four moments of the local-to-unity parameters. Finally, in case (iv), we demonstrate that the power envelope is definedwithin n�1=2T�1-neighborhoods of unity and that it is identical to that of cases (i) and (ii).2

Second, we derive the asymptotic local power of some existing panel unit root tests andcompare these to the power envelope. For case (i), we investigate the t-ratio statisticsstudied by Quah (1994), Levin et al. (2002), and Moon and Perron (2004). For case (ii), we

ARTICLE IN PRESS

1We thank a referee for bringing this paper to our attention. Breitung (2000) derives his results under a

homogeneous local alternative and with cross-sectional independence, while Moon and Perron (2005) consider a

more general model with heterogeneous local alternatives and cross-sectional dependence arising from the

presence of common factors.2This result can also be found in Breitung (1999), the working paper version of Breitung (2000).

H.R. Moon et al. / Journal of Econometrics 141 (2007) 416–459 417

discuss results from Moon and Perron (2005) on a modified t-ratio statistic that isasymptotically equivalent to the test proposed by Levin et al. For case (iii), we compare theoptimal invariant test proposed by Ploberger and Phillips (2002), the LM test proposed byMoon and Phillips (2004), the unbiased test proposed by Breitung (2000), and a new t-testthat is asymptotically equivalent to the Levin et al. (2002) test. First, we show that in allthree cases the existing tests do not achieve maximal power. Next, when the alternativehypothesis is homogeneous across individuals, it is shown that some tests (the t-test in case(i) and the optimal invariant test of Ploberger and Phillips (2002) in cases (ii) and (iii)) doachieve the power envelope and are uniformly most powerful (UMP).Third, we propose a simple point optimal invariant panel unit root test for each case.

These tests are UMP when the alternative hypothesis is homogeneous, in contrast to pointoptimal unit root tests for time series (Elliott et al., 1996) where no UMP test exists.The paper is organized as follows. Section 2 lays out the model, the hypotheses to test,

and the assumptions maintained throughout the paper. Section 3 studies the model wherethere are no fixed effects (or where the fixed effects are known), develops the Gaussianpower envelope, gives a point optimal test and performs some power comparisons.Sections 4 and 5 perform similar analyses for panel models with incidental intercepts andtrends. Section 6 discusses various extensions and generalizations of our framework.Section 7 reports some simulations comparing the finite sample properties of the main testsstudied in Sections 4 and 5. Section 8 concludes, and the Appendix contains the maintechnical derivations and proofs; the remaining proofs can be found in a companion paper,Moon et al. (2006b).

2. Model

The observed panel zit is assumed to be generated by the following component model:

zit ¼ b0igt þ yit,

yit ¼ riyit�1 þ uit; i ¼ 1; . . . ; t ¼ 0; 1 . . . , ð1Þ

where uit is a mean zero error, gt ¼ ð1; tÞ0, and bi ¼ ðb0i; b1iÞ

0.The focus of interest is the problem of testing for the presence of a common unit root in

the panel against local alternatives when both n and T are large. For a local alternativespecification, we assume that

ri ¼ 1�yi

nkTfor some constant k40, (2)

where yi is a sequence of iid random variables.3 The main goal of the paper is to findefficient tests for the null hypothesis

H0 : yi ¼ 0 a.s. ði.e., ri ¼ 1Þ for all i, (3)

against the alternative

H1 : yia0 ði.e., ria1Þ for some i’s. (4)

ARTICLE IN PRESS

3Notice that under the local alternative, ri depends on n and T . Thus, the sequences of panel data zit and yit

should be understood as triangular arrays.

H.R. Moon et al. / Journal of Econometrics 141 (2007) 416–459418

A common special case of interest for the alternative hypothesis H1 is

H2 : yi ¼ y40 for all i, (5)

where the local-to-unity coefficients take on a common value y40 for all i. In this case, theseries are homogeneously locally stationary, that is ri ¼ r ¼ 1� y=nkTo1 for all i.

In (1) the nonstationary panel zit has two different types of trends. The first componentb0igt is a deterministic linear trend that is heterogeneous across individuals i. Thiscomponent characterizes individual effects in the panel. The second component yit is astochastic trend or near unit-root process with ri close to unity.

The following sections look at four different cases. In the first case, there are nofixed effects in the panel that have to be estimated, i.e., bi ¼ ð0; 0Þ

0 (or alternatively bi isknown). The second case arises when the panel data zit contain fixed effects thatare parameterized by heterogeneous intercept terms b0i; which are incidental parametersto be estimated. The third case arises when the panel contains fixed effects thatare parameterized by heterogeneous linear deterministic trends, b0i þ b1it whereboth sets of parameters b0i and b1i need to be estimated. A final caseconsiders panels with heterogeneous intercepts and a common trend of the formb0i þ b1t.

In each case, under the assumptions that the error terms uit are iid normal with zeromean and known variance s2i and that the initial conditions are yi:t�1 ¼ 0 for all i; weconstruct point optimal test statistics. By deriving the limits of the test statistics, weestablish the asymptotic power envelopes of the panel unit root testing problems. Then,we discuss the implementation of these procedures using feasible point optimal teststatistics. To develop these, we relax some of the assumptions made in deriving the powerenvelopes.

We maintain the following assumptions in deriving the limits of the feasible pointoptimal tests and some other tests available in the literature.

Assumption 1. For i ¼ 1; 2; . . . and over t ¼ 0; 1; . . . ; uit�iidð0;s2i Þ with supi E½u8it�oM and

inf i s2i XM 40 for some finite constants M and M.

Assumption 2. The initial observations yi0 are iid with Ejyi0j8oM for some constant M

and are independent of uit, tX1 for all i.

Assumption 3. 1=T þ 1=nþ n=T ! 0.

Before proceeding, we introduce the following notation. Define

zt ¼ ðz1t; . . . ; zntÞ0; yt ¼ ðy1t; . . . ; yntÞ

0; ut ¼ ðu1t; . . . ; untÞ0,

Z ¼ ðz1; . . . ; zT Þ; Y ¼ ðy1; . . . ; yT Þ; Y�1 ¼ ðy0; y1; . . . ; yT�1Þ; U ¼ ðu1; . . . ; uT Þ,

so the ði; tÞth elements of Z;Y ;Y�1, and U are zit, yit; yit�1, and uit, respectively. Define theT-vectors G0 ¼ ð1; . . . ; 1Þ

0, G1 ¼ ð1; 2; . . . ;TÞ0, set G ¼ ðG0;G1Þ ¼ ðg1; . . . ; gT Þ

0, and define

b0 ¼ ðb01; . . . ; b0nÞ0; b1 ¼ ðb11; . . . ; b1nÞ

0,

b ¼ ðb0; b1Þ ¼ ðb1; . . . ; bnÞ0.

Let Zi, Y i, Y�1;i, and Ui denote the transpose of the ith row of Z;Y ;Y�1, and U ,respectively, and write the model in matrix form as

ARTICLE IN PRESSH.R. Moon et al. / Journal of Econometrics 141 (2007) 416–459 419

Z ¼ bG0 þ Y ,

Y ¼ rY�1 þU ,

where r ¼ diagðr1; . . . ;rnÞ. Define S ¼ diagðs21; . . . ; s2nÞ.

3. No fixed effects

This section investigates the model in which b0igt is observable or equivalently gt ¼ 0. Inthis case, the model becomes

Z ¼ Y ,

Y ¼ rY�1 þU .

We consider local neighborhoods of unity that shrink at the rate of 1=n1=2T and one-sidedalternatives, as indicated in the following assumptions.

Assumption 4. k ¼ 1=2 in (2).

Assumption 5. yi is a sequence of iid random variables whose support is a subset of abounded interval ½0;My� for some MyX0.

Let my;k ¼ Eðyki Þ. The assumption of a bounded support for yi is made for convenience,

and could be relaxed at the cost of stronger moment conditions. It is also convenientto assume that the yi are identically distributed, and this assumption could be relaxedas long as cross-sectional averages of the moments ð1=nÞ

Pni¼1 Eðy

ki Þ have limits such

as my;k.According to Assumption 5, yiX0 for all i, so that rip1. In this case, the null hypothesis

of a unit root in (3) is equivalent to my;1 ¼ 0 or My ¼ 0 (i.e., yi ¼ 0 a.s. and the variance ofy, s2y, is 0), and the alternative hypothesis in (4) implies my;140. Hence, in this section weset the hypotheses in terms of the first moment of yi as follows:

H0 : my;1 ¼ 0 (6)

and

H1 : my;140. (7)

To test these hypotheses, Moon and Perron (2004) proposed t-ratio tests based on a modifiedpooled OLS estimator of the autoregressive coefficient and show that they have significantasymptotic local power in neighborhoods of unity shrinking at the rate 1=

ffiffiffinp

T : This sectionfirst derives the (asymptotic) power envelope and shows that the power function of a feasiblepoint optimal test forH0 achieves the envelope for the hypotheses above. We then compare theasymptotic local power of this point-optimal test with that of the Moon–Perron test.

3.1. Power envelope

The power envelope is found by computing the upper bound of power of all pointoptimal tests for each local alternative. To proceed, we define

rci¼ 1�

ci

n1=2T,

ARTICLE IN PRESSH.R. Moon et al. / Journal of Econometrics 141 (2007) 416–459420

where ci is an iid sequence of random variables on ½0;Mc� for some Mc40. Denote by mc;k

the kth raw moment of ci, i.e., mc;k ¼ Eðcki Þ.

Define

DciððTþ1Þ�ðTþ1ÞÞ

¼

1 0 . . . 0 0

�rci1 . .

. ... ..

.

0 . .. . .

.0 0

..

.�rci

1 0

0 . . . 0 �rci1

26666666664

37777777775,

C ¼ diagðc1; . . . ; cnÞ, and DC ¼ diagðDc1 ; . . . ;DcnÞ.When uit are iid Nð0;s2i Þ with s2i known and the initial conditions yi;�1 are all zeros, so

that yi0 ¼ ui0 for all i, the log-likelihood function is

LnT ðCÞ ¼ �12ðvecðY 0ÞÞ0D0CðS

�1 � ITþ1ÞDCðvecðY 0ÞÞ.

Denote by LnT ð0Þ the log-likelihood function when ci ¼ 0 for all i.Define

VnT ðCÞ ¼ �2LnT ðCÞ þ 2LnT ð0Þ �12mc;2.

The statistic V nT ðCÞ is the (Gaussian) likelihood ratio statistic of the null hypothesis ri ¼ 1against an alternative hypothesis ri ¼ rci

for i ¼ 1; . . . ; n. According to the Neyman–Pearson lemma, rejecting the null hypothesis for small values of V nT ðCÞ is themost powerful test of the null hypothesis H0 against the alternative hypothesisri ¼ rci

. When the alternative hypothesis is given by H1, the test is a point optimaltest (see, e.g., King, 1988). Let CnT ðCÞ be the test that rejects H0 for small values ofV nT ðCÞ.

Theorem 6. Assume that bi ¼ 0 for all i or gt ¼ 0 for all t in (1). Suppose that Assumptions

1–5 hold. Then,

VnT ðCÞ ) Nð�EðciyiÞ; 2mc;2Þ.

The asymptotic critical values of the test CnT ðCÞ can be readily computed. In a notationwe will use throughout the paper, let za denote the ð1� aÞ-quantile of the standard normaldistribution, i.e., PðZp� zaÞ ¼ a, where Z�Nð0; 1Þ. Then, the size a asymptotic critical

value cðC; aÞ of the test CnT ðCÞ is cðC; aÞ ¼ �ffiffiffiffiffiffiffiffiffiffi2mc;2

pza, and its asymptotic local power is

FEðciyiÞffiffiffiffiffiffiffiffiffiffi2mc;2

p � za

!, (8)

where FðxÞ is the cumulative distribution function of Z.Using (8), it is easy to find the power envelope, i.e., the values of ci for which power is

maximized. By the Cauchy–Schwarz inequality

FEðciyiÞffiffiffiffiffiffiffiffiffiffi2mc;2

p � za

!pF

ffiffiffiffiffiffiffiffimy;22

r� za

� �,


and the upper bound of Fðffiffiffiffiffiffiffiffiffiffiffiffimy;2=2

p� zaÞ is achieved with ci ¼ yi: Then, by the

Neyman–Pearson lemma, Fðffiffiffiffiffiffiffiffiffiffiffiffimy;2=2

p� zaÞ traces out a power envelope and we have the

following theorem.


1–5 hold. Then, the power envelope for testing H0 in (3) against H1 in (4) is Fðffiffiffiffiffiffiffiffiffiffiffiffimy;2=2

p� zaÞ,

where my;2 ¼ Eðy2i Þ and za is the ð1� aÞ-quantile of the standard normal distribution.

3.2. Implementation of the test

In order to implement a test that achieves the power envelope, estimates of the variances, s2i ;are necessary. The estimator we propose computes the variances under the null hypothesis. Tosimplify notation, let the first difference matrix D0 be simply denoted by D: Our estimator justtakes the sample average of the squared first differences for each cross-section:

s21;iT ¼1

TðDZiÞ

0DZi ¼1

Ty2

i0 þXT

t¼1

ðDyitÞ2

!.

Denote by S1 ¼ diagðs21;1T ; . . . ; s21;nT Þ the estimated covariance matrix and by LnT ðCÞ and

LnT ð0Þ the log-likelihood functions where the unknown S has been replaced by S1.The feasible point-optimal statistic is

VnT ðCÞ ¼ � 2LnT ðCÞ þ 2LnT ð0Þ �12mc;2

¼Xn

i¼1

1

s21;iTz2i0 þ

XT

t¼1

ðDcizitÞ

2

" #�Xn

i¼1

1

s21;iTz2i0 þ

XT

t¼1

ðDzitÞ2

" #�

1

2mc;2.

The following theorem establishes asymptotic equivalence between the feasible andinfeasible versions of the test:


1–5 hold. Then, V nT ðCÞ ¼ VnT ðCÞ þ opð1Þ.

3.3. Power comparison

3.3.1. The t-ratio test

We start by investigating the t-ratio test of Quah (1994), Levin et al. (2002), and Moonand Perron (2004), which is based on the pooled OLS estimator.4 For simplicity we assumethat the error variances s2i are known. Let

r ¼

Pni¼1

1s2

i

PTt¼1 yityit�1Pn

i¼11s2

i

PTt¼1 y2

it�1

ARTICLE IN PRESS

4When the error term uit is serially correlated, one can use a modified version of the pooled OLS estimator.

Details of this modification can be found in Moon and Perron (2005). A more detailed discussion of the case

where the errors are serially correlated can be found in Section 6.4.


be the pooled OLS estimator with corresponding t-statistic

t ¼r� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1Pni¼1

1

s2i

PTt¼1 y2

it�1

vuuut.

Under the conditions assumed above, we have t) Nð�my;1=ffiffiffi2p

; 1Þ (see Moon and Perron,2004). The power of the t-test with size a is then

Fmy;1ffiffiffi2p � za

� �. (9)

Remarks.

(a) By the Cauchy–Schwarz inequality, it is straightforward to show that


� �pF

ffiffiffiffiffiffiffiffimy;22

r� za

� �. (10)

In view of (10), the t-ratio test achieves optimal power only when the alternative is

homogeneous as in H2, that is when yi ¼ y a.s., so that EðyiÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiEðy2i Þ

q. Otherwise, the

power of the t-ratio test is strictly sub-optimal. This implies that the t-ratio test is theUMP test for testing H0 against H2 but not against H1. The result is not surprisingsince the t-ratio test is constructed based on the pooled OLS estimator and pooling isefficient only under the homogeneous alternative.

(b) Notice from (9) that the asymptotic local power of the t-test is determined by my;1, themean of the local-to-unity parameters yi. In the given formulation, the local alternativeis restricted to be one sided in Assumption 5. If we allow two-sided alternatives, thisopens the possibility that my;1 ¼ 0 even under the alternative hypothesis, in which casethe power of the pooled t-test is equivalent to size.

(c) The pooled OLS estimator defined above can be interpreted as a GLS estimator since itgives weights that are inversely related to the variance of each observation. Moon andPerron (2004) do not make this adjustment and use a conventional OLS estimator.However, Levin et al. (2002) first correct for heteroskedasticity by dividing through bythe estimated standard deviation before using pooled OLS on this transformed data.Their procedure can thus also be interpreted as a GLS estimator although it iscommonly called pooled OLS. To avoid confusion with the previous literature, we willkeep referring to estimators with weights that are the reciprocal of the standarddeviation as pooled OLS estimators.

3.3.2. A common-point optimal test with ci ¼ c

As shown earlier, to achieve the power envelope, one needs to choose ci ¼ yi a.s. forCnT ðCÞ. Denote this test CnT ðYÞ. Of course, the test CnT ðYÞ is infeasible because it is notpossible to identify the distribution of yi in the panel and generate a sequence from itsdistribution. Indeed, if the yi were known, there would be no need to test the null of a panel unitroot.


One way of implementing the test CnT ðCÞ is to use randomly generated ci’s from somedomain that is considered relevant. The variates ci are independent of yi and the power ofthe test CnT ðCÞ is

Fmc;1my;1ffiffiffiffiffiffiffiffiffiffi2mc;2

p � za

!. (11)

Since mc;1pffiffiffiffiffiffiffimc;2p

, the power (11) is bounded by


� �, (12)

which is achieved when we choose ci ¼ c; where c is any positive constant. We denote thistest CnT ðcÞ.

Remarks.

(a) Not surprisingly, the power (12) of the testCnT ðcÞ is identical to that of the t-ratio test in theprevious section. Of course, both tests are based on the homogeneous alternative hypothesis.

(b) Note that the power of the testCnT ðcÞ does not depend on c. The test is optimal againstthe special homogeneous alternative hypothesis H2 for any choice of c. This result is incontrast to the power of the point optimal test for unit root time series in Elliott et al.(1996), where power does depend on the value of c. The reason is that the localalternative in the panel unit root case, rci

¼ 1� ci=n1=2T ; is closer to the nullhypothesis than the alternative rci

¼ 1� c=T that applies in the case where there isonly time series data. In effect, when we are this close to the null hypothesis with ahomogeneous local alternative, it suffices to use any common local alternative insetting up the panel point optimal test.

4. Fixed effects I: incidental intercepts case

We extend the analysis in the previous section by allowing for fixed effects, i.e., b0igt ¼

b0i; so that gt ¼ 1. In this case, the model has the matrix form Z ¼ b0G00 þ Y .

4.1. Power envelope

This section derives the power envelope of panel unit root tests for H0 that are invariantto the transformation Z! Z þ bn

0G00 for arbitrary bn

0. When uit are iid Nð0;s2i Þ with s2iknown and the initial conditions yi;�1 are zeros, i.e., yi0 ¼ ui0, the log-likelihood function is

LnT ðC;b0Þ ¼ �12½vecðZ0 � G0b

00Þ�0D0CðS

�1 � ITþ1ÞDC½vecðZ0 � G0b00Þ�.

We denote by LnT ð0; b0Þ the log-likelihood function when ci ¼ 0 for all i.A (Gaussian) point optimal invariant test statistic for this case can be constructed as

follows (see, for example, Lehmann, 1959; Dufour and King, 1991; Elliott et al., 1996):

V fe1;nT ðCÞ ¼ �2 minb0

LnT ðC; b0Þ �minb0

LnT ð0;b0Þ� �

�1

2mc;2.


For given ci’s; the point optimal invariant test, say Cfe1;nT ðCÞ, rejects the null hypothesisfor small values of V fe1;nT ðCÞ.

Theorem 9. Suppose Assumptions 1–5 hold and that b1i ¼ 0 or are known. Then, as

ðn;TÞ ! 1(a) Vfe1;nT ðCÞ ) Nð�EðciyiÞ; 2mc;2Þ.(b) The power envelope for invariant testing of H0 in (3) against H1 in (4) is Fð

ffiffiffiffiffiffiffiffiffiffiffiffimy;2=2

p� zaÞ,

where my;2 ¼ Eðy2i Þ and za is the ð1� aÞ-quantile of the standard normal distribution.

Remarks.

(a) As in the case of CnT ðcÞ, we define the test Cfe1;nT ðcÞ with a common constant pointci ¼ c. Then, the power of the test Cfe1;nT ðcÞ is


� �, (13)

which is the same as for the CnT ðcÞ test in the previous section without fixedeffects.

(b) Note that the asymptotic power envelope is the same as in the case without incidentalintercepts, so estimation of intercepts does not affect maximal achievable power. Theresult is analogous to the time series case in Elliott et al. (1996, p. 816).

(c) With incidental intercepts in the model, Levin et al. (2002) proposed a panel unit roottest based on the pooled OLS estimator. Let ~zit ¼ zit � ð1=TÞ

PTt¼1 zit and

~zit�1 ¼ zit�1 � ð1=TÞPT

t¼1 zit�1. When the error variances s2i are known, the t-statistic

proposed by Levin et al. is asymptotically equivalent to the following t-statistic:

tþ ¼

ffiffiffiffiffi30

51

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

1

s2i

XT

t¼1

~z2it�1

vuut ðrþpool � 1Þ,

where

rþpool ¼Xn

i¼1

1

s2i

XT

t¼1

~z2it�1

" #�1 Xn

i¼1

1

s2i

XT

t¼1

~zit�1 ~zit

" #þ

3

T.

As shown by Moon and Perron (2005), the tþ-test also has significant asymptotic localpower within n�1=2T�1 neighborhoods of unity, and its power is given by

F 32

ffiffiffiffi551

qmy;1 � za

� �,

which is below that of the Cfe1;nT ðcÞ test.


As in the case without fixed effects, we need to estimate the unknown quantities to makethe point-optimal test feasible. In this case, the unknown quantities are the intercepts, bi0,


and variances, s2i . The fixed effects will be estimated by generalized least squares (GLS)under the null hypothesis, or

b0ið0Þ ¼ ðDG00DG0Þ�1DG00DZi,

where DG0 ¼ ð1; 0; . . . ; 0Þ0, and the resulting estimate is simply the first observation, zi0:

The variance estimator for each cross-section is then:

s22;iT ¼1

TDZi � DG0b0ið0Þh i0

DZi � DG0b0ið0Þh i

¼1

T

XT

t¼1

ðDzitÞ2.

Define S2 ¼ diagðs22;1T ; . . . ; s22;nT Þ as before, and let LnT ðC;b0Þ and LnT ð0; b0Þ be the log-

likelihood function values with the unknown S replaced by S2. The feasible statistic is then

V fe1;nT ðCÞ ¼ �2 minb0

LnT ðC; b0Þ �minb0

LnT ð0;b0Þ� �

�1

2mc;2,

leading to an asymptotically equivalent test.

Theorem 10. Suppose that Assumptions 1–5 hold and that b1i ¼ 0 or are known. Then,V fe1;nT ðCÞ ¼ VnT ðCÞ þ opð1Þ.

5. Fixed effects II: incidental trends case

This section considers the important practical case where heterogeneous linear trendsneed to be estimated. Set gt ¼ ð1; tÞ

0 and for this case, we consider local neighborhoods ofunity that shrink at the slower rate of 1=n1=4T .

Assumption 11. k ¼ 14in (2).

We relax Assumption 5 to allow for two-sided alternatives, so that the time seriesbehavior of yit can be either stationary or explosive under the alternative hypothesis.

Assumption 12. yi�iid with mean my and variance s2y with a support that is a subset of abounded interval ½�Mly, Muy�, where Mly, MuyX0.

Under Assumption 12, we can re-express hypotheses (3) and (4) using the second rawmoment of yi as follows:

H0 : my;2 ¼ 0 (14)

and

H1 : my;240. (15)

The usual one-sided version where the series has a unit root or is stationary is the specialcase with Mly ¼ 0:We proceed as above by first deriving the power envelope, developing afeasible implementation of the resulting statistic, and then investigating the asymptoticlocal power of different panel unit root tests.

5.1. Power envelope

This section derives the Gaussian power envelope of panel unit root tests for H0 that areinvariant to the transformation Z! Z þ bnG0 for arbitrary bn. When uit are iid Nð0; s2i Þ


with s2i known and the initial conditions yi;�1 are zeros, that is, yi0 ¼ ui0, the log-likelihoodfunction is

LnT ðC; bÞ ¼ �12½vecðZ0 � Gb0Þ�0D0CðS

�1 � ITþ1ÞDC½vecðZ0 � Gb0Þ�.

We denote by LnT ð0;bÞ the log-likelihood function when ci ¼ 0 for all i. As above, a(Gaussian) point optimal invariant test statistic can be constructed as

V fe2;nT ðCÞ ¼ � 2 minb

LnT ðC;bÞ �minb

LnT ð0; bÞ� �

þ1

n1=4

Xn

i¼1

ci

!þ

1

n1=2

Xn

i¼1

c2i

!op2T þ

1

n

Xn

i¼1

c4i

!op4T ,

where

op2T ¼ �1

T

XT

t¼1

t� 1

Tþ

2

T

XT

t¼1

t

T

t� 1

T

� ��

1

3,

op4T ¼1

T2

XT

t¼1

XT

s¼1

t� 1

T

s� 1

Tmin

t� 1

T;s� 1

T

� ��

2

3

1

T

XT

t¼1

t� 1

T

� �2

þ1

9.

For given ci’s, the point optimal invariant test, say Cfe2;nT ðCÞ, rejects the null hypothesisfor small values of V fe2;nT ðCÞ.

The asymptotic behavior of V fe2;nT ðCÞ is given in the following result.

Theorem 13. Suppose that Assumptions 1–3, 11, and 12. Then, V fe2;nT ðCÞ )

Nð� 190Eðc2i y

2i Þ;

145Eðc4i ÞÞ.

From Theorem 13, the size a asymptotic critical value is

cfe2ðC; aÞ ¼ �

ffiffiffiffiffiffiffimc;4

45

rza,

and the asymptotic power of the test is given by

F1

6ffiffiffi5p

Eðc2i y2i Þffiffiffiffiffiffiffiffiffiffiffi

Eðc4i Þp � za

!. (16)

By the Cauchy–Schwarz inequality, we have

F1

6ffiffiffi5p

Eðc2i y2i Þffiffiffiffiffiffiffiffiffiffiffi

Eðc4i Þp � za

!pF

1

6ffiffiffi5p

ffiffiffiffiffiffiffiffimy;4p

� za

� �. (17)

Again, the maximal power, Fð 16ffiffi5p


� zaÞ, is achieved by choosing ci ¼ yi. According to

the Neyman–Pearson lemma, Fð 16ffiffi5p


� zaÞ traces out the power envelope. Summariz-

ing, we have the following theorem.

Theorem 14. Suppose that the trends b0igt in (1) are unknown and need to be estimated and

Assumptions 1–3, 11, and 12 hold. Then, the power envelope for testing the null hypothesis H0

in (3) against the alternative hypothesis H1 in (4) is Fð 16ffiffi5p


� zaÞ; where my;4 ¼ Eðy4i Þ and

zais the ð1� aÞ-quantile of the standard normal distribution.


Remarks.

(a) An important finding of Theorem 14 is that in the panel unit root modelwith incidental trends,the POI test has significant asymptotic local power in localneighborhoods of unity that shrink at the rate 1=n1=4T : By contrast, in the panelunit root model either without fixed effects or only with incidental intercepts,the POI test has significant asymptotic power in local neighborhoods of unitythat shrink at the faster rate 1=n1=2T : This difference in the neighborhood radiusof nonnegligible power is a manifestation of the difficulty in detecting unit rootsin panels in the presence of heterogeneous trends, a problem that wasoriginally discovered in Moon and Phillips (1999) and called the ‘incidental trend’problem.

(b) The power envelope of invariant tests of H0 in (3) against H1 depends on the fourthmoment of the local-to-unity parameters yi’s. This dependence suggests that panelswith more dispersed autoregressive coefficients will tend to more easily reject the nullhypothesis.

(c) When the alternative hypothesis is the homogeneous alternative H2 (i.e., yi ¼ yÞ, thepower envelope is

Fð 16ffiffi5p y2 � zaÞ (18)

and, in this case, the power envelope is attained by using ci ¼ c for any choice of c.(d) If the yi are symmetrically distributed about my;1 and k4 is the fourth cumulant, thenffiffiffiffiffiffiffiffimy;4p

¼ m2y;1f1þ 6s2y=m2y;1 þ ð3s

4y þ k4Þ=m4y;1g

1=2 and this will be close to m2y;1 when the

ratios 6s2y=m2y;1 and ð3s4y þ k4Þ=m4y;1 are both small. In such cases, it is clear from (17)

that the test with ci ¼ c for any choice of c will be close to the power envelope.


Again, the covariance matrix S is generally unknown and needs to be estimated: To doso, we use the GLS estimator of bi under the null hypothesis,

bið0Þ ¼ ðDG0DGÞ�1DG0DZi ¼

zi0

1

T

PTt¼1

Dzit

0B@

1CA,

where

DG ¼1 0 � � � 0

0 1 � � � 1

� �0,

and define the estimator of the error variance for cross-section i as

s23;iT ¼1

T½DZi � DGbið0Þ�

0½DZi � DGbið0Þ� ¼1

T

XT

t¼1

Dzit �1

T

XT

t¼1

Dzit

!2

.


Denote S3 ¼ diagðs23;1T ; . . . ; s23;nT Þ. Let LnT ðCÞ and LnT ð0Þ be the log-likelihood function

with the unknown S replaced with S3: The feasible statistic is then:

V fe2;nT ðCÞ ¼ � 2 minb

LnT ðC;bÞ �minb

LnT ð0; bÞ� �

þ1

n1=4

Xn

i¼1

ci

!þ

1

n1=2

Xn

i¼1

c2i

!op2T þ

1

n

Xn

i¼1

c4i

!op4T .

Again, we have an asymptotically equivalent test.

Theorem 15. Suppose that Assumptions 1–5 hold. Then, V fe2;nT ðCÞ ¼ Vfe2;nT ðCÞ þ opð1Þ.

5.3. Power comparison

We compare the power of five tests, and for simplicity assume that the error variances s2iare known.

5.3.1. The optimal invariant test of Ploberger and Phillips (2002)

We start with the optimal invariant panel unit root test proposed by Plobergerand Phillips (2002). To construct the test statistic, we first estimate the trend coefficients bby GLS b ¼ ðDZDGÞðDG0DGÞ�1; and detrend the panel data Z giving E ¼ Z � bG0:Define

Vg;nT ¼ffiffiffinp 1

nT2trðS�1=2EE0S�1=2Þ � o1T

� �, (19)

where o1T ¼ ð1=TÞPT

t¼1 ðt=TÞð1� t=TÞ. In summation notation, we have

Vg;nT ¼1ffiffiffinp

Xn

i¼1

1

Ts2i

XT

t¼1

Z2it;T � o1T

" #, (20)

where

Zit;T ¼1ffiffiffiffiTp ðzit � zi0Þ �

t

TðziT � zi0Þ

h i,

a maximal invariant statistic. In view of (19) and (20), we may interpret V g;nT as thestandardized information of the GLS detrended panel data. The test Cg;nT proposed byPloberger and Phillips (2002) rejects the null hypothesis H0 for small values of Vg;nT :

To investigate the asymptotic power of Cg;nT ; we first derive the asymptotic distributionof V g;nT :

Lemma 1. Suppose Assumptions 1–3, 11, and 12 hold. Then, Vg;nT ) Nð� 190my;2;

145Þ.

Using Lemma 1, it is straightforward to find the size a asymptotic critical values fgðaÞ ofthe test Cg;nT . For za; the ð1� aÞ-quantile of Z; the critical value is fgðaÞ ¼ �

13ffiffi5p za; and

the asymptotic local power is given by


� �, (21)

showing that the test Cg;nT has significant asymptotic power against the local alternative H1:


Remarks.

(a) Notice that the asymptotic power of the test Cg;nT is determined by the second momentof yi; my;2, so that it relies on the variance of yi as well as the mean of yi:

(b) According to Ploberger and Phillips (2002), the test Cg;nT is an optimal invariant test.Let Qy;nT ðyÞ be the joint probability measure of the data for the given yi’s and let v bethe probability measure on the space of yi. Ploberger and Phillips (2002) show that thetestCg;nT is asymptotically the optimal invariant test that maximizes the average powerRðRCg;nT dQy;nT ðyÞÞdv, a quantity which also represents the power of Cg;nT against the

Bayesian mixtureR

Qy;nT ðyÞdv.(c) Comparing the power (21) of the test Cg;nT to the power envelope is straightforward.

By the Cauchy–Schwarz inequality we have


� �pF


6ffiffiffi5p � za

� �.

The test Cg;nT achieves the power envelope if the yi are constant a.s. That is, the powerenvelope is achieved against the special alternative hypothesis H2.

5.3.2. The LM test in Moon and Phillips (2004)

The second test we investigate is the LM test proposed by Moon and Phillips (2004),which is constructed in a fashion similar to V g;nT : The main difference is that Moon andPhillips (2004) use ordinary least squares (OLS) to detrend the data. To fix ideas, defineQG ¼ IT � PG with PG ¼ GðG0GÞ�1G0. Let DT ¼ diagð1;TÞ and

V o;nT ¼ffiffiffinp 1

nT2trðS�1=2ZQGZ0S�1=2Þ � o2T

� �,

where

o2T ¼1

T

XT

t¼1

t

T�

1

T2

XT

t¼1

XT

s¼1

minðt; sÞ

ThT ðt; sÞ,

hT ðt; sÞ ¼ g0tD�1T

1

T

XT

p¼1

D�1T gpg0pD�1T

!�1D�1T gs.

Define

~Zit;T ¼1ffiffiffiffiTp zit � g0t

XT

t¼1

gtg0t

!�1 XT

t¼1

gtzit

!24

35,

a scaled version of the OLS detrended panel. Then, we can write

V o;nT ¼1ffiffiffinp

Xn

i¼1

1

Ts2i

XT

t¼1

~Z2

it;T � o2T

" #,

which can also be interpreted as the standardized information of the detrended panel data.The LM test, say Co;nT ; of Moon and Phillips (2004) is to reject the null hypothesis H0 forsmall values of V o;nT ðcÞ.


The following theorem gives the limit distribution of V o;nT ðcÞ.

Lemma 2. Suppose Assumptions 1–3, 11, and 12 hold. Then, Vo;nT ) Nð� 1420

my;2;11

6300Þ.

The size a asymptotic critical value of Co;nT ; say foðaÞ, is given by foðaÞ ¼ �ffiffiffiffiffiffiffi11

6300

qza;

and the asymptotic power is Fð my;22ffiffiffiffi77p � zaÞ.

Remarks.

(a) Similar to the test Cg;nT ; the test Co;nT has significant asymptotic power against thelocal alternative H1; and its power depends on the second moment of yi; my;2:

(b) The asymptotic power of the optimal invariant test Cg;nT dominates that of the testCo;nT because ðm2y þ s2yÞ=2

ffiffiffiffiffi77p

oðm2y þ s2yÞ=2ffiffiffiffiffi45p

: This is not so surprising since theoptimal invariant test Cg;nT is based on GLS-detrended data, while the test Co;nT isbased on OLS-detrended data.

(c) As remarked earlier, the test Vfe2;nT ðcÞ will achieve power close to the power envelopewhen the ratios 6s2y=m

2y;1 and ð3s4y þ k4Þ=m4y;1 are both small.

5.3.3. The unbiased test of Breitung (2000)

Breitung (2000) has proposed an alternative test to the Levin et al. (2002) test that doesnot require bias adjustment. The idea is to transform the data as

yn

it ¼ st Dzit �1

T � tðDzitþ1 þ � � � þ DziT Þ

� �,

xn

it ¼ zit�1 � zi0 �t� 1

TðziT � zi0Þ,

and note that ynit and xn

it are orthogonal to each other. The pooled estimator proposed byBreitung is then

rn ¼ 1þ

Pni¼1

PT�1t¼2 s�2i yn

itxnitPn

i¼1

PT�1t¼2 s�2i xn2

it

,

and is correctly centered and does not require bias adjustment in contrast to the Levin etal. (2002) pooled OLS estimator. Breitung suggests testing the panel unit root nullhypothesis by looking at the corresponding t-statistic:

UBnT ¼

Pni¼1

PT�1t¼2 s�2i yn

itxnitffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1

PT�1t¼2 s�2i xn2

it

q .

Under a homogeneous local alternative, Breitung claims (Theorem 5, p. 172) that thisstatistic has power in a local neighborhood defined with k ¼ 1

2; and that the expectation in

the asymptotic normal distribution under the alternative is

yffiffiffi6p

limT!1

qEðT�1P

xnity

nitÞ

qðy=ffiffiffinpÞ

��y¼0

� �.

In a separate paper (Moon et al., 2006a), we show analytically that the limit above is 0,and therefore that Breitung’s test does not have power in a neighborhood that shrinks atthe faster rate 1=n1=2T towards the null. Instead, we show that the necessary rate is the


same slower 1=n1=4T rate that applies to the other tests with incidental trends. Indeed, weshow that under the assumptions in this section, the UB statistic has the followingdistribution.

Lemma 3. Suppose Assumptions 1–3, 11, and 12 hold. Then, UBnT ) Nðmy;2=6ffiffiffi6p

; 1Þ.

Remark. The above lemma shows that the asymptotic power of Breitung’s test isFðmy;2=6

ffiffiffi6p� zaÞ, which is obviously below the power envelope.

5.3.4. A common-point optimal invariant test

The test Vfe2;nT ðYÞ that achieves the power envelope is infeasible. If we use randomlygenerated c0is that are independent of yi and the panel data zit in constructing the test,according to (16), the power of the test V fe2;nT ðCÞ is

F1

6ffiffiffi5p

mc;2my;2ffiffiffiffiffiffiffimc;4p � za

!. (22)

Since mc;2pffiffiffiffiffiffiffimc;4p

, the power (22) is bounded by

Fð 16ffiffi5p my;2 � zaÞ, (23)

which is achieved when we choose ci ¼ c for V fe2;nT ðCÞ, where c is any positive constant.We denote this test V fe2;nT ðcÞ.

Remarks.

(a) The power (23) of the test V fe2;nT ðcÞ is identical to that of the Ploberger–Phillipsoptimal invariant test V g;nT :

(b) The power of the test Vfe2;nT ðcÞ also does not depend on c. It is optimal against thespecial homogeneous alternative hypothesis H2 for any choice of c.

5.3.5. A t-test

In a manner similar to Moon and Perron (2005), we can define statistics that areasymptotically equivalent to the Levin et al. (2002) statistic based on the pooled OLSestimator for this case. When there are incidental trends, the Levin et al. statistic isasymptotically equivalent to the following t-statistic:

tþ ¼

ffiffiffiffiffiffi112193

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þq

ðrþpool � 1Þ,

where the bias-corrected pooled OLS estimator is

rþpool ¼ ½trðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þ��1½trðS�1=2 ~Z�1 ~Z0S�1=2Þ� þ

7:5

T.

On the other hand, Moon and Perron (2004) consider the following t-ratio test based on adifferent bias-corrected pooled estimator

t# ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þq

ðr#pool � 1Þ,


where

r#pool ¼ ½trðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þ��1 trðS�1=2 ~Z�1 ~Z0S�1=2Þ þ

nT

2

� �.

By definition,

rþpool � r#pool ¼15

2T

trðS�1=2 ~Z�1 ~Z0

�1S�1=2Þ �nT2

15

trðS�1=2 ~Z�1 ~Z0

�1S�1=2Þ

0BB@

1CCA

and

tþ ¼

ffiffiffiffiffiffiffiffi112

193

rt# þ

15

2

ffiffiffiffiffiffiffiffi112

193

r ffiffiffinp 1

nT2trðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þ �1

15

� 1

nT2trðS�1=2 ~Z�1 ~Z

0

�1S�1=2Þ1=2

.

Using Theorem 4 of Moon and Perron (2004) and Lemma 2, it is possible to show thefollowing.

Lemma 4. Suppose Assumptions 1–3, 11, and 12 hold. Then, tþ )

Nð� 15ffiffiffiffi15p

2

ffiffiffiffiffiffi112193

qmy;2=420; 1Þ.

6. Discussion

6.1. Case with incidental intercepts but a common trend

This section investigates the panel model for zit in (1) where there are incidentalintercepts but a common trend, viz.,

zit ¼ b0i þ b1tþ yit,

yit ¼ riyit�1 þ uit; i ¼ 1; . . . ; t ¼ 0; 1 . . . .

This model is relevant because there is a tradition of imposing such a common trend inempirical work in microeconometrics. In addition, the analysis of asymptotic local powerfor this model provides further evidence that it is the presence of incidental trends, b1it;rather than incidental intercepts b0i that makes the detection of unit roots morechallenging.

To proceed, we make the same assumptions as in Sections 2–4, so that

ri ¼ 1�yi

n1=2T.

Let ln ¼ ð1; . . . ; 1Þ0, n-vector of ones. Using notation defined in Section 2, we write the

model as

Z ¼ b0G00 þ b1lnG01 þ Y ,

Y ¼ rY�1 þU .


In the following theorem we show that the power envelope of panel unit root tests for H0

that are invariant to the transformation Z! Z þ bn

0G00 þ bn

1lnG01 for arbitrary bn

0 and bn

1 isthe same as the one we found in Sections 3 and 4.When uit are iid Nð0;s2i Þ with s2i known and the initial conditions yi;�1 are zeros, that is,

yi0 ¼ ui0; the log-likelihood function is

LnT ðC; b0; b1Þ ¼ �12½vecðZ0 � G0b

00 � G1l0nb1Þ�

0D0CðS�1 � ITþ1Þ

�DC½vecðZ0 � G0b00 � G1l0nb1Þ�.

As before, a (Gaussian) point optimal invariant test statistic for this case can beconstructed as follows:

V fe3;nT ðCÞ ¼ �2 minb0;b1

LnT ðC; b0; b1Þ �minb0;b1

LnT ð0;b0; b1Þ

� ��

1

2mc;2.

For given ci’s, the point optimal invariant test, say Cfe3;nT ðCÞ, rejects the null hypothesisfor small values of Vfe3;nT ðCÞ.

Theorem 16. Suppose Assumptions 1–5 hold. Then,

V fe3;nT ðCÞ ¼ V fe1;nT ðCÞ þ opð1Þ.

6.2. Initial conditions

In the derivations above, we have assumed that all series in the panel were initialized atthe origin (yi;�1 ¼ 0Þ. It is well-known in the time series case that the initial condition canplay an important role in the performance of unit root tests (Evans and Savin, 1984;Phillips, 1987; Elliott, 1999; Elliott and Muller, 2003). A common assumption made inthe time series context is that the initial condition is drawn from the unconditionaldistribution under the stationary alternative, i.e., y0�Nð0; 1=ð1� r2ÞÞ. In the local-to-unitycase, r ¼ 1� y=T ; this formulation of the initial condition gives y0 ¼ Opð

ffiffiffiffiTpÞ, which has

some appeal because the order of magnitude of the initial condition is the same as that ofthe sample data yt.This commonly used set up for the time series case does not extend naturally to the panel

model. Indeed, under the assumption yi;�1�Nð0; 1=ð1� r2i ÞÞ, and with local alternativesri ¼ 1� yi=n1=2T or ri ¼ 1� yi=n1=4T (depending on whether trends are present or not),we have yi;�1 ¼ Opðn

1=4ffiffiffiffiTpÞ or yi;�1 ¼ Opðn

1=8ffiffiffiffiTpÞ, respectively, in which case yi;�1

diverges with n. The sample data yi;t for this series is then dominated by the initialcondition yi;�1: There is, of course, no reason in empirical panels why the order ofmagnitude of the initial condition for an individual series should depend on the totalnumber of individuals (nÞ observed in the panel and such a formulation would be hard tojustify. In this sense, the situation is quite different from the time series case, where thereare good reasons for expecting initial observations for nonstationary or nearlynonstationary time series to have stochastic orders comparable to those of the sample.Moreover, under the initialization yi;�1�Nð0; 1=ð1� r2i ÞÞ, the likelihood ratio statisticdiverges to negative infinity under the local alternative, as we show below.To illustrate, consider the case with no fixed effect and with uit�iid Nð0; 1Þ across i over

t: Here we assume that ri ¼ 1� yi=n1=2T ; as in Sections 3 and 4. Assume that if yia0; yi;�1


are iid Nð0; 1=ð1� r2i ÞÞ and independent of ujt, and if yi ¼ 0; yi;�1 are iid Nð0; 1Þ andindependent of ujt: Denote deviations from the initial condition as

~yit ¼ yit � yi;�1

¼ uit þ riuit�1 þ r2i uit�2 þ � � � þ rtiui0 þ ðrtþ1

i � 1Þyi;�1.

All quantities based on ~yit will behave as in the case of a fixed initial condition. Define thenotation

DyiY i ¼ ðð1� r2i Þ

1=2yi;�1;Dyiyi0; . . . ;Dyi

yiT Þ0

¼ ð1� r2i Þ1=2yi;�1;Dyi0 �

yi

n1=2Tyi;�1; . . . ;DyiT �

yi

n1=2TyiT�1

� �0,

D0Y i ¼ ðyi;�1;Dyi0; . . . ;DyiT Þ0.

Then, the likelihood ratio is

�1

2LnT ;A þ

1

2LnT ;0

¼Xn

i¼1

ðDyiY iÞ0Dyi

Y i �Xn

i¼1

ðD0Y iÞ0D0Y i

¼Xn

i¼1

ð1� r2i Þy2i0 þ

XT

t¼0

D ~yit �yi

n1=2T~yit�1 �

yi

n1=2Tyi;�1

� �2

� y2i;�1 �

XT

t¼0

D ~y2it

" #

¼Xn

i¼1

y2inT� r2i

� �y2

i;�1 � 21

n1=2T

Xn

i¼1

yiyi;�1

XT

t¼0

ðD ~yitÞ þ 21

nT2

Xn

i¼1

y2i yi;�1

XT

t¼0

~yit�1

� 21

n1=2T

Xn

i¼1

yi

XT

t¼0

D ~yitð ~yit�1Þ þ1

nT2

Xn

i¼1

y2iXT

t¼0

~y2it�1.

The last two terms behave as in the case of fixed initial conditions in the limit since they aredeviations from the initial condition. As for the other three terms, we concentrate on thehomogeneous case, yi ¼ y for simplicity. We can show that the first term is

Xn

i¼1

y2

nT� r2i

� �y2

i;�1 ¼ Opðn3=2TÞ,

while the second term is

1

n1=2T

Xn

i¼1

yyi;�1ðyiT � yi;�1Þ ¼ Opðn1=4Þ,

and the third term is

1

nT2

Xn

i¼1

y2yi;�1

XT

t¼0

~yit�1 ¼ Op1

n1=4

� �.

Thus, the behavior of the likelihood ratio statistic is dominated by the first term. This firstterm has a negative mean and thus the likelihood ratio statistic diverges to negative infinityunder the local alternative.


This example makes it clear that mechanical extensions of time series formulations thatare commonly used for initial conditions can lead to quite unrealistic and unjustifiablefeatures in a panel context. It is therefore necessary to consider initializations that aresensible for panel models, while at the same time having realistic time series properties.Given the more limited focus of the present study, we will not pursue this discussion ofinitial conditions further here but retain the (simplistic) assumption of zero initialconditions. Clearly, it is an important matter for future research to extend the theory andrelax this condition.

6.3. Cross-sectional dependence

As with most of the early panel unit root tests that have been proposed in the literature,the above analysis supposes that the observational units that make up the panel areindependent of each other. This assumption is not realistic in many applications, such asthe analysis of cross-country macroeconomic series, where individual series are likely to beaffected by common, worldwide shocks. Accordingly, more recent panel tests such as thosein Bai and Ng (2004), Moon and Perron (2004), Phillips and Sul (2003), Chang (2002), andPesaran (2005) allow for the presence of cross-sectional dependence among the units,typically through the presence of dynamic factors.In order to handle such cross-sectional dependence, we can combine the defactoring

method of Bai and Ng (2004), Moon and Perron (2004) or Phillips and Sul (2003) to theanalysis of this paper. The idea is to apply the optimal tests developed here to the data afterthe common factors have been extracted. Once the extraction process has been completed,there is, of course, no claim of optimality of the resulting tests, and we do not prove herethat this approach has any optimality property. However, intuition suggests that thisapproach should perform well in practice, and simulation evidence provided in Moon andPerron (2006) confirms this.For illustration, we will use the model of Moon and Perron (2004). Thus, the

assumption is that the disturbance in (1) has a factor structure

uit ¼ g0if t þ eit. (24)

The proposed procedure is as follows:

1. Estimate the deterministic components (bi) by GLS to obtain yit.2. Use the pooled OLS estimate to compute residuals uit.3. Use principal components on the covariance matrix of these estimated residuals to

estimate the common factor(s), f t and factor loadings, gi: Post-multiply the data matrixZ by Qg ¼ I � gðg0gÞ�1g0 so that ZQg is no longer affected by the common factors.

4. Use the common point-optimal (CPO) test proposed earlier in the paper on ZQg.

6.4. Serial correlation

Serial correlation can be accounted for in the construction of the test statistics byreplacing variances with long-run variances, o2

i ¼P1

j¼�1 gij ; where gij ¼ Eðui;tui;t�jÞ. Sinceserial correlation is not accommodated in the above derivation of the power envelope, thisprocedure will not in general be optimal, but should result in tests with correct asymptoticsize under quite general short memory autocorrelation (as in Elliott et al., 1996). Standard


kernel-based estimators of the long-run variance as in Andrews (1991) and Newey andWest (1994) can be used to estimate the long-run variances. The development of optimalprocedures that accommodate serial correlation is of interest but beyond the scope of thepresent contribution.

7. Simulations

This section reports the results of a small Monte Carlo experiment designed to assessand compare the finite-sample properties of the tests presented earlier in the paper. For thispurpose, we use the following data generating process:

zit ¼ b0i þ b1itþ yit,

yit ¼ riyit�1 þ uit,

yi;�1 ¼ 0; uit�iid Nð0;s2i Þ,

s2i�U½0:5; 1:5�.

We consider both the incidental intercepts case ðb1i ¼ 0Þ of Section 4 and the incidentaltrends case ðb1ia0Þ of Section 5. In each case, the heterogeneous intercepts and/or trendsare iid Nð0; 1Þ. We assume that the error term is independent in both the time and cross-sectional dimensions with a Gaussian distribution and heteroskedastic variances. Initialconditions are set to zero and, as discussed earlier, this is a limitation of the experimentsand may lead to more favorable results for many of the tests than under randominitializations where there is some dependence on the localization parameters.

We focus the study on three main questions. The first is the sensitivity of the point-optimal invariant test to the choice of ci: The second is how far the feasible and infeasiblepoint-optimal tests are from the theoretical power envelope in finite samples. Finally, welook at the impact of the distribution of the local-to-unity parameters under the alternativehypothesis.

We consider the following eight distributions for the local-to-unity parameters: yi ¼ 0 8ifor size, and for local power, (1) yi�iid U½0; 2�, (2) yi�iid U½0; 4�, (3) yi�iid U½0; 8�, (4)yi�iid w2ð1Þ, (5) yi�iid w2ð2Þ, (6) yi�iid w2ð4Þ, (7) yi ¼ 1 8i; and (8) yi ¼ 2 8i. Thesedistributions enable us to examine performance of the tests as the mass of the distributionof the localizing parameters moves away from the null hypothesis. We can also look at theeffect of homogeneous versus heterogeneous alternatives (cases (1) and (4) versus (7), andcases (2) and (5) versus (8)) together with the role of the higher-order moments of thedistribution. For instance, case (1) has the same mean as case (4) but smaller higher-ordermoments. The same situation arises for cases (2) and (5), and cases (3) and (6). Note thatthe alternatives with w2 distributions do not fit our asymptotic framework since they haveunbounded support.

We take three values for each of n (10, 25, and 100) and T (50, 100, and 250). All tests areconducted at the 5% significance level, and the number of replications is set at 10,000.

Table 1 presents the results for the incidental intercepts case. The tests we consider arethe infeasible point-optimal test with ci ¼ yi (the finite-sample analog of the powerenvelope which uses the local-to-unity parameters generated in the simulation), our CPOinvariant test for three values of c (1, 2, and 0.5), the t-ratio type test as in Moon and


Perron (2005), and the t-bar statistic of Im, Pesaran, and Shin for which no analyticalpower result is available.5 The first panel of the table provides the size and power predictedby the asymptotic theory in Section 4 using the moments of yi and ci. The other panels inthe table report the size and size-adjusted power of the tests for the various combinationsof n and T. Thus, if asymptotic theory were a reliable guide to finite-sample behavior,subsequent panels in the table would mirror the first panel.The main outcomes from the first panel of the table can be summarized as follows:

� The power envelope is higher for the w2 alternatives than for the uniform alternativeswith the same mean. This is because the power envelope depends on the seconduncentered moment of yi.� The power of the feasible CPO test is the same for the uniform and w2 alternatives sincepower in this case depends only on the mean of yi.� The test based on the tþ-statistic is less powerful than the CPO test.� The power envelope is higher for the heterogeneous alternatives than the homogeneousalternatives with the same mean.

For the other panels of the table, the second column gives the expected value of theautoregressive parameter implied by the distribution of the local-to-unity parameter andthe values of n and T. As can be seen, the alternatives considered are very close to 1, and ata qualitative level, the results match closely the asymptotic predictions. The mainconclusions are:

� The size properties of the CPO test appear to be mildly sensitive to the choice of c. Thesize of the test tends to increase with c.� In terms of power, the choice of c is much less important, as predicted by asymptotictheory. In fact, most of the variation is within two simulation standard deviations, andmuch of the difference is probably due to experimental randomness.� In all cases, power is far below what is predicted by theory and below the powerenvelope defined by ci ¼ yi: The differences are reduced as both n and T are increased.� In all cases, the tþ-test is less powerful than the CPO tests, but it does dominate the t-bar statistic.� In the homogeneous cases, there is less power difference between the CPO tests and theoptimal test. This is expected since the CPO test is most powerful against thesealternatives.� Finally, despite the theoretical predictions that they should be equal, the actual powerfor the w2 alternatives is slightly below that for the corresponding uniform alternatives.

Table 2 reports the same information as Table 1 for the incidental trends case. Inaddition to the above tests, in this case we also consider the optimal test of Ploberger andPhillips (2002), the LM test of Moon and Phillips (2004), and the unbiased test of Breitung(2000). Once again, the first panel of the table gives the predictions for size and powerbased on our asymptotic theory.

ARTICLE IN PRESS

5We have also considered tests with randomly generated values for the ci’s. Since the results were inferior to

those with fixed choices of c, we do not report them here, but they are available from the authors upon request.


ARTICLE IN PRESS

Table 1

Size and size-adjusted power of tests—incidental intercepts case

DGP: zit ¼ b0i þ z0it,

z0it ¼ 1�yi

n12T

� �z0it�1 þ sieit,

b0i; eit�iid Nð0; 1Þ,

si�iid U½0:5; 1:5�

(A) Theoretical values

ci ¼ yi ci ¼ 1 ci ¼ 2 ci ¼ 0:5 tþ IPS

yi ¼ 0 (size) 5.0 5.0 5.0 5.0 5.0 5.0

yi�U½0; 2� 20.4 17.4 17.4 17.4 12.0 –

yi�U½0; 4� 49.5 40.9 40.9 40.9 24.0 –

yi�U½0; 8� 94.7 88.2 88.2 88.2 59.2 –

yi�w2ð1Þ 33.7 17.4 17.4 17.4 12.0 –

yi�w2ð2Þ 63.9 40.9 40.9 40.9 24.0 –

yi�w2ð4Þ 96.6 88.2 88.2 88.2 59.2 –

yi ¼ 1 17.4 17.4 17.4 17.4 12.0 –

yi ¼ 2 40.9 40.9 40.9 40.9 24.0 –

EðriÞ ci ¼ yi ci ¼ 1 ci ¼ 2 ci ¼ 0:5 tþ IPS

(B) n ¼ 10;T ¼ 50

yi ¼ 0 (size) 1 – 2.8 5.2 1.9 7.1 5.4

yi�U½0; 2� 0.9684 14.0 11.9 11.9 12.0 9.1 8.0

yi�U½0; 4� 0.9368 41.0 23.1 23.5 22.9 14.4 9.8

yi�U½0; 8� 0.8735 88.9 46.4 48.2 45.6 25.9 14.7

yi�w2ð1Þ 0.9684 15.9 11.2 11.2 11.2 9.1 7.4

yi�w2ð2Þ 0.9368 46.5 20.6 20.9 20.7 13.2 9.5

yi�w2ð4Þ 0.8735 87.8 43.9 45.5 43.1 24.6 15.1

yi ¼ 1 0.9684 7.9 12.9 12.9 13.1 9.2 7.1

yi ¼ 2 0.9368 28.5 27.5 27.6 27.5 15.5 10.5

(C) n ¼ 25;T ¼ 50

yi ¼ 0 (size) 1 – 3.8 5.5 3.2 8.4 6.4

yi�U½0; 2� 0.9817 20.1 13.5 13.7 13.4 10.2 7.9

yi�U½0; 4� 0.9635 47.0 27.2 27.8 26.9 16.8 10.7

yi�U½0; 8� 0.9270 90.8 58.8 59.7 57.9 32.3 16.9

yi�w2ð1Þ 0.9817 23.4 12.6 12.7 12.4 9.4 7.5

yi�w2ð2Þ 0.9635 55.1 24.6 25.2 24.3 15.1 9.9

yi�w2ð4Þ 0.9270 91.7 56.8 57.8 55.9 32.0 16.8

yi ¼ 1 0.9817 12.2 15.4 15.2 15.3 10.9 7.8

yi ¼ 2 0.9635 34.2 32.6 32.6 32.4 18.9 11.3

(D) n ¼ 100;T ¼ 50

yi ¼ 0 (size) 1 – 4.4 5.3 4.1 12.9 8.3

yi�U½0; 2� 0.99 23.7 14.1 14.2 14.1 10.5 8.0

yi�U½0; 4� 0.98 49.4 29.4 29.6 29.4 19.1 11.6

yi�U½0; 8� 0.96 91.6 67.7 68.2 67.6 40.1 21.1

yi�w2ð1Þ 0.99 31.7 13.2 13.4 13.2 9.6 7.9

yi�w2ð2Þ 0.98 60.0 27.8 28.1 27.8 17.8 12.0


ARTICLE IN PRESS

Table 1 (continued )


yi�w2ð4Þ 0.96 93.9 66.8 67.2 66.8 40.9 21.1

yi ¼ 1 0.99 14.4 15.9 15.8 15.8 10.7 7.8

yi ¼ 2 0.98 38.6 37.3 37.2 37.4 21.2 12.0

(E) n ¼ 10;T ¼ 100

yi ¼ 0 (size) 1 – 2.6 5.1 1.7 6.7 5.1

yi�U½0; 2� 0.9968 13.8 13.5 13.8 13.4 9.1 7.6

yi�U½0; 4� 0.9937 39.3 23.2 23.9 23.2 13.7 9.3

yi�U½0; 8� 0.9874 89.3 48.1 50.4 46.9 24.3 14.1

yi�w2ð1Þ 0.9968 14.4 11.0 11.2 11.0 8.2 7.2

yi�w2ð2Þ 0.9937 44.3 21.1 21.6 20.7 11.8 8.6

yi�w2ð4Þ 0.9874 88.0 47.7 49.7 46.7 24.1 14.9

yi ¼ 1 0.9968 8.6 14.2 14.3 14.0 9.9 7.9

yi ¼ 2 0.9937 28.4 27.7 28.0 27.1 16.0 10.4

(F) n ¼ 25;T ¼ 100

yi ¼ 0 (size) 1 – 4.0 5.6 3.4 6.7 5.4

yi�U½0; 2� 0.9982 19.5 13.7 13.6 13.6 9.9 7.7

yi�U½0; 4� 0.9963 45.8 28.5 28.5 28.3 16.6 11.3

yi�U½0; 8� 0.9927 91.0 58.9 59.3 58.4 31.0 16.7

yi�w2ð1Þ 0.9982 21.9 12.8 12.9 12.7 9.2 7.7

yi�w2ð2Þ 0.9963 53.5 25.8 25.9 25.5 15.1 10.6

yi�w2ð4Þ 0.9927 91.6 57.9 58.8 57.2 29.8 16.4

yi ¼ 1 0.9982 12.1 14.7 14.7 14.6 9.6 7.6

yi ¼ 2 0.9963 33.7 31.9 32.1 31.7 17.4 11.5

(G) n ¼ 100;T ¼ 100

yi ¼ 0 (size) 1 – 4.6 5.3 4.3 8.4 6.3

yi�U½0; 2� 0.999 22.9 14.7 14.6 14.8 10.8 8.2

yi�U½0; 4� 0.998 48.9 31.5 31.4 31.6 19.7 11.4

yi�U½0; 8� 0.996 92.8 71.4 71.7 71.5 42.1 20.8

yi�w2ð1Þ 0.999 30.3 13.3 13.2 13.3 10.2 8.2

yi�w2ð2Þ 0.998 59.9 28.7 28.8 28.8 18.6 11.9

yi�w2ð4Þ 0.996 94.1 68.6 68.6 68.5 40.4 20.8

yi ¼ 1 0.999 14.5 15.5 15.8 15.4 10.2 7.6

yi ¼ 2 0.998 37.8 36.4 36.7 36.4 19.3 12.0

(H) n ¼ 10;T ¼ 250

yi ¼ 0 (size) 1 – 3.2 5.5 2.1 6.2 4.3

yi�U½0; 2� 0.9989 12.6 11.7 12.0 11.6 9.4 7.4

yi�U½0; 4� 0.9979 37.5 21.8 22.6 21.9 13.3 9.3

yi�U½0; 8� 0.9958 88.8 46.4 48.7 45.2 23.5 13.9

yi�w2ð1Þ 0.9989 13.7 10.3 10.6 10.2 8.0 6.8

yi�w2ð2Þ 0.9979 43.9 19.2 20.0 18.8 12.7 8.9

yi�w2ð4Þ 0.9958 87.4 44.3 46.7 43.3 23.7 14.3

yi ¼ 1 0.9989 8.6 12.9 12.9 12.8 9.5 7.9

yi ¼ 2 0.9979 28.1 26.0 26.3 25.7 14.4 9.9

(I) n ¼ 25;T ¼ 250

yi ¼ 0 (size) 1 – 3.8 5.2 3.2 6.1 4.8

yi�U½0; 2� 0.9994 18.7 13.9 14.1 13.9 9.8 7.9

yi�U½0; 4� 0.9988 45.1 28.1 28.4 28.0 16.0 10.2

yi�U½0; 8� 0.9976 91.1 60.0 60.9 59.6 30.9 16.7


Just as in unit root testing with time series models, power is much lower when trends arepresent or fitted. In fact, power is much lower than it may first appear in the table since theactual local alternative approaches the null hypothesis at the slower rate Oðn�1=4T�1Þ thanin the incidental intercepts case. Thus, for the same distribution of the local-to-unityparameters, the alternative hypothesis is actually further from unity than in Table 1.

The main predictions contained in the first panel of the table for the incidental trendscase are as follows:

� In contrast to the incidental intercepts case, power of the CPO test is higher for w2

alternatives than for uniform alternatives since it depends on higher-order moments inthis case.� The Moon and Phillips test, although dominated, is expected to perform well.� The tþ-test has lowest power as is expected.� Breitung’s unbiased test has power that lies between the CPO test and the Moon andPhillips test.� The power envelope is lower for homogeneous alternatives.

The simulation findings reported in the remaining panels of Table 2 conform wellto these predictions. We have not reported the finite-sample analog of the powerenvelope because of numerical problems encountered in the computation. In a finitesample, the terms involving high powers of ci dominate for distant alternatives, and thispushes the distribution of the statistic to the right, leading to negligible rejectionprobabilities.

Our other findings for this case are:

� The size properties of the point-optimal test are much more sensitive to the choice of c

and values of n and T than for the incidental intercepts case. It is therefore difficult to

ARTICLE IN PRESS



yi�w2ð1Þ 0.9994 21.1 12.2 12.4 12.2 8.5 7.0

yi�w2ð2Þ 0.9988 52.4 25.0 25.6 24.9 15.0 10.7

yi�w2ð4Þ 0.9976 91.5 58.3 59.1 57.9 30.6 17.1

yi ¼ 1 0.9994 12.0 14.8 14.6 14.9 10.6 8.2

yi ¼ 2 0.9988 33.6 32.7 32.8 32.9 18.0 11.1

(J) n ¼ 100;T ¼ 250

yi ¼ 0 (size) 1 – 4.8 5.6 4.5 6.4 5.3

yi�U½0; 2� 0.9997 21.6 14.9 14.9 14.8 10.9 7.9

yi�U½0; 4� 0.9993 49.7 33.3 33.2 32.9 20.4 12.2

yi�U½0; 8� 0.9987 92.3 73.3 73.4 73.1 41.9 20.7

yi�w2ð1Þ 0.9997 30.4 14.5 14.3 14.4 10.5 7.6

yi�w2ð2Þ 0.9993 59.9 30.4 30.5 30.1 18.4 11.3

yi�w2ð4Þ 0.9987 94.4 71.3 71.3 71.0 40.9 20.6

yi ¼ 1 0.9997 15.2 15.7 15.8 15.7 12.4 8.7

yi ¼ 2 0.9993 36.8 34.9 35.2 34.9 21.6 12.0


ARTICLE IN PRESS

Table 2

Size and size-adjusted power of tests—incidental trends case

DGP: zit ¼ b0i þ b1i tþ z0it,

z0it ¼ 1�yi

n14T

� �z0it�1 þ sieit,

b0i; b1i; eit�iid Nð0; 1Þ,

si�iid U½0:5; 1:5�


ci ¼ yi ci ¼ 1 ci ¼ 2 ci ¼ 0:5 Ploberger–Phillips Moon–Phillips tþ IPS UB

yi ¼ 0 (size) 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

yi�U½0; 2� 6.5 6.1 6.1 6.1 6.1 5.8 5.8 – 6.0

yi�U½0; 4� 13.3 10.6 10.6 10.6 10.6 9.0 8.6 – 10.0

yi�U½0; 8� 68.7 47.8 47.8 47.8 47.8 33.4 30.1 – 42.3

yi�w2ð1Þ 18.9 7.8 7.8 7.8 7.8 7.0 6.9 – 7.5

yi�w2ð2Þ 42.7 14.7 14.7 14.7 14.7 11.7 11.1 – 13.6

yi�w2ð4Þ 94.7 55.7 55.7 55.7 55.7 39.1 35.2 – 49.5

yi ¼ 1 5.8 5.8 5.8 5.8 5.8 5.6 5.6 – 5.7

yi ¼ 2 8.9 8.9 8.9 8.9 8.9 7.8 7.6 – 8.5

EðriÞ ci ¼ 1 ci ¼ 2 ci ¼ 0:5 Ploberger–Phillips Moon–Phillips tþ IPS UB

(B) n ¼ 10;T ¼ 50

yi ¼ 0 (size) 1 2.2 0.1 3.2 1.3 1.0 6.1 7.1 6.0

yi�U½0; 2� 0.944 5.9 6.0 5.8 5.8 5.8 5.4 5.2 5.9

yi�U½0; 4� 0.888 8.3 8.4 8.3 8.3 8.1 7.3 6.2 8.3

yi�U½0; 8� 0.775 18.3 18.4 18.2 18.1 15.3 13.3 10.6 16.0

yi�w2ð1Þ 0.944 6.4 6.6 6.4 6.4 6.1 6.2 5.8 6.3

yi�w2ð2Þ 0.888 9.4 9.5 9.3 9.3 8.7 7.7 7.0 8.1

yi�w2ð4Þ 0.775 18.1 18.3 18.0 18.1 15.5 13.5 10.8 15.2

yi ¼ 1 0.944 5.7 5.6 5.7 5.7 6.0 5.8 5.9 5.9

yi ¼ 2 0.888 8.3 8.2 8.2 8.2 7.8 7.4 6.9 7.4

(C) n ¼ 25;T ¼ 50

yi ¼ 0 (size) 1 5.6 1.8 6.7 2.5 1.3 7.8 9.0 5.0

yi�U½0; 2� 0.957 5.3 5.3 5.3 5.3 4.8 4.5 4.8 5.6

yi�U½0; 4� 0.915 8.7 8.6 8.7 8.7 7.3 6.0 6.2 7.9

yi�U½0; 8� 0.829 22.6 22.6 22.5 22.5 17.7 14.2 11.7 18.8

yi�w2ð1Þ 0.957 6.2 6.1 6.2 6.3 5.7 4.8 5.2 6.7

yi�w2ð2Þ 0.915 9.1 9.0 9.1 9.1 7.9 6.6 6.4 9.2

yi�w2ð4Þ 0.829 22.2 22.3 22.1 22.2 17.4 13.9 11.5 18.5

yi ¼ 1 0.957 5.6 5.5 5.6 5.6 5.1 5.5 5.0 6.0

yi ¼ 2 0.915 8.1 8.1 8.1 8.1 6.9 6.9 6.1 7.5

(D) n ¼ 100;T ¼ 50

yi ¼ 0 (size) 1 12.9 7.9 14.0 3.2 0.1 10.6 12.8 4.2

yi�U½0; 2� 0.968 5.4 5.4 5.4 5.4 6.0 6.1 5.3 5.1

yi�U½0; 4� 0.937 9.2 9.4 9.3 9.3 8.9 8.7 7.0 7.9

yi�U½0; 8� 0.874 29.0 29.3 29.0 29.0 23.6 20.4 13.5 21.7

yi�w2ð1Þ 0.968 7.0 7.2 7.0 7.0 7.2 7.2 5.8 5.6

yi�w2ð2Þ 0.937 10.5 10.6 10.5 10.5 10.1 10.0 8.0 8.8


ARTICLE IN PRESS




yi�w2ð4Þ 0.874 27.9 28.3 27.9 27.9 22.6 20.4 13.9 21.4

yi ¼ 1 0.968 6.4 6.3 6.4 6.4 5.6 5.7 5.4 4.8

yi ¼ 2 0.937 8.4 8.4 8.4 8.4 7.0 6.8 6.4 7.4

(E) n ¼ 10;T ¼ 100

yi ¼ 0 (size) 1 1.2 0.1 1.8 1.3 1.5 5.5 5.7 6.2

yi�U½0; 2� 0.994 5.8 5.6 5.7 5.8 5.7 5.4 5.7 6.0

yi�U½0; 4� 0.989 8.6 8.5 8.6 8.6 8.7 7.4 6.6 7.6

yi�U½0; 8� 0.978 19.3 19.3 19.3 19.4 16.6 14.1 11.2 16.1

yi�w2ð1Þ 0.994 6.7 6.7 6.8 6.8 6.7 6.5 6.4 6.5

yi�w2ð2Þ 0.989 9.6 9.6 9.6 9.6 8.4 8.1 7.4 8.5

yi�w2ð4Þ 0.978 18.2 18.1 18.1 18.2 15.7 13.6 11.2 16.0

yi ¼ 1 0.994 5.3 5.4 5.3 5.4 5.1 5.3 5.3 5.6

yi ¼ 2 0.989 6.9 6.8 6.9 6.9 6.5 6.4 6.9 7.0

(F) n ¼ 25;T ¼ 100

yi ¼ 0 (size) 1 3.6 1.0 4.6 2.7 2.1 6.0 6.2 5.7

yi�U½0; 2� 0.996 5.7 5.7 5.7 5.7 5.7 5.6 5.6 5.8

yi�U½0; 4� 0.992 8.8 8.8 8.7 8.7 8.5 7.7 7.1 7.9

yi�U½0; 8� 0.983 22.7 22.7 22.7 22.6 18.4 16.4 12.9 18.4

yi�w2ð1Þ 0.996 6.8 6.8 6.7 6.7 6.7 6.7 6.4 6.0

yi�w2ð2Þ 0.992 9.5 9.4 9.4 9.3 8.3 8.4 7.6 8.1

yi�w2ð4Þ 0.983 21.2 21.2 21.1 21.1 17.5 15.7 12.6 18.0

yi ¼ 1 0.996 5.9 6.0 5.8 5.8 5.9 5.5 5.7 5.4

yi ¼ 2 0.992 7.4 7.5 7.3 7.3 7.2 6.8 6.2 7.8

(G) n ¼ 100;T ¼ 100

yi ¼ 0 (size) 1 7.1 3.5 7.9 3.4 1.6 8.0 8.6 4.7

yi�U½0; 2� 0.997 6.2 6.3 6.2 6.2 5.9 6.0 5.2 5.6

yi�U½0; 4� 0.994 10.2 10.4 10.3 10.3 8.7 8.7 7.4 8.4

yi�U½0; 8� 0.988 28.8 29.1 28.8 28.8 21.6 19.6 13.7 23.2

yi�w2ð1Þ 0.997 7.1 7.1 7.1 7.1 6.3 6.4 6.1 6.5

yi�w2ð2Þ 0.994 10.7 10.8 10.7 10.7 9.3 9.0 7.4 9.6

yi�w2ð4Þ 0.987 30.0 30.4 30.0 30.1 22.2 20.1 14.3 23.1

yi ¼ 1 0.997 5.9 6.0 5.9 5.9 6.2 6.0 5.2 5.6

yi ¼ 2 0.994 9.2 9.3 9.2 9.3 8.0 7.2 6.2 7.7

(H) n ¼ 10;T ¼ 250

yi ¼ 0 (size) 1 1.2 0.0 2.0 1.8 2.5 6.0 5.2 6.2

yi�U½0; 2� 0.998 5.1 5.1 5.0 5.0 5.1 5.2 5.4 6.1

yi�U½0; 4� 0.996 7.8 7.8 7.9 7.9 6.6 6.2 6.0 7.5

yi�U½0; 8� 0.993 18.1 18.4 18.2 18.2 14.4 12.6 9.8 16.6

yi�w2ð1Þ 0.998 6.3 6.4 6.3 6.3 6.0 5.8 5.8 6.3

yi�w2ð2Þ 0.996 9.1 9.1 9.2 9.2 7.4 7.0 6.7 8.3

yi�w2ð4Þ 0.993 17.2 17.2 17.2 17.2 13.9 12.1 10.2 15.9

yi ¼ 1 0.998 5.8 5.7 5.8 5.8 5.7 5.7 5.2 5.2

yi ¼ 2 0.996 7.2 7.2 7.2 7.3 7.0 6.6 5.9 7.8

(I) n ¼ 25;T ¼ 250

yi ¼ 0 (size) 1 2.6 0.6 3.2 2.8 2.7 5.4 5.2 5.8

yi�U½0; 2� 0.999 6.6 6.5 6.5 6.5 6.2 6.1 5.8 5.4

yi�U½0; 4� 0.997 8.9 8.8 8.8 8.8 8.2 7.8 7.1 7.4


come up with a good choice of c based on these results, although values between 1 and 2seem to provide a good balance for all values of n and T .� Both the Ploberger–Phillips and Moon–Phillips tests tend to underreject, sometimesquite severely.� The t-type test tends to overreject, and its power is close to that of Moon and Phillips.� As in the incidental intercepts case, the power properties of the CPO test do not appearsensitive to the choice of c. There is a slight tendency for c ¼ 2 to achieve highest power.� The fatter-tailed distributions have higher power than the corresponding uniformdistributions for the two closest alternatives. For the alternatives that are furthest away(cases (3) and (6)), the reverse is true.� The Ploberger–Phillips test behaves in a similar way to the CPO test, as predicted by theasymptotics.� The LM test of Moon and Phillips has good power but appears to be slightly dominatedby the other two tests, as again predicted by our theory.� Power of the unbiased test of Breitung is generally between that of the Ploberger–Phillips and Moon–Phillips test, again as predicted.� When the alternative hypothesis is homogeneous (cases (7) and (8)), the tests based on acommon value of ci have higher power than for the corresponding heterogeneousalternative case. This phenomenon is more pronounced for the w2 alternativehypothesis.

These results suggest that the asymptotic theory generally provides a useful guide to thefinite sample performance of the tests statistics in the vicinity of the panel unit root null.However, the presence of more complex deterministic components and increasing distancefrom the null hypothesis reduces the accuracy of the analytic results from asymptotic

ARTICLE IN PRESS




yi�U½0; 8� 0.994 23.1 23.2 22.9 22.9 19.1 16.1 12.5 19.0

yi�w2ð1Þ 0.999 6.6 6.5 6.5 6.5 6.3 6.0 5.9 6.2

yi�w2ð2Þ 0.997 9.4 9.5 9.3 9.3 9.0 8.5 7.3 8.6

yi�w2ð4Þ 0.994 21.5 21.4 21.4 21.3 17.4 15.0 12.4 18.9

yi ¼ 1 0.999 5.4 5.4 5.3 5.3 5.4 5.3 5.7 5.6

yi ¼ 2 0.997 7.4 7.3 7.4 7.4 6.8 6.5 6.5 7.4

(J) n ¼ 100;T ¼ 250

yi ¼ 0 (size) 1 4.7 2.2 5.4 3.9 3.3 6.6 6.2 5.2

yi�U½0; 2� 0.999 5.9 5.9 5.8 5.8 5.2 5.2 5.5 5.7

yi�U½0; 4� 0.998 9.2 9.2 9.1 9.2 8.3 7.5 7.3 8.5

yi�U½0; 8� 0.996 29.6 29.8 29.6 29.6 21.9 18.6 14.3 23.5

yi�w2ð1Þ 0.999 6.6 6.6 6.5 6.5 6.2 5.8 5.4 6.6

yi�w2ð2Þ 0.998 10.4 10.5 10.4 10.4 9.2 8.3 7.7 9.5

yi�w2ð4Þ 0.996 27.4 27.5 27.4 27.4 20.8 17.8 14.5 24.1

yi ¼ 1 0.999 6.0 5.9 5.9 5.9 5.8 5.9 5.5 5.4

yi ¼ 2 0.998 9.1 9.0 9.1 9.0 8.1 8.3 7.2 7.7


theory. Overall, the simulation findings strongly suggest that use of the CPO test (and thePloberger–Phillips test in the trends case) improves power over the commonly used t-ratiotype statistics.

8. Conclusion

In terms of their asymptotic power functions, the Ploberger–Phillips (2002) test and theCPO test have good discriminatory power against a unit root null in shrinkingneighborhoods of unity. When the alternative is homogeneous it is possible to attain theGaussian asymptotic power envelope and both the Ploberger–Phillips test and the CPOtest are UMP in this case. Interestingly, the CPO test has this property irrespective of thepoint chosen to set up the test. This is in contrast to point optimal tests of a unit root thatare based solely on time series data (Elliott et al., 1996), where no test is UMP, and anarbitrary selection of a common point is needed in the construction of the test.

An important empirical consequence of the present investigation is that increasing thecomplexity of the fixed effects in a panel model inevitably reduces the potential power ofunit root tests. This reduction in power has a quantitative manifestation in the radial orderof the shrinking neighborhoods around unity for which asymptotic power is nonnegligible.When there are no fixed effects or constant fixed effects, tests have power in aneighborhood of unity of order n�1=2T�1: When incidental trends are fitted, the tests onlyhave power in a larger neighborhood of order n�1=4T�1: A continuing reduction in poweris to be expected as higher-order incidental trends are fitted in a panel model. The situationis analogous to what happens in time series models where unit root nonstationary data arefitted by a lagged variable and deterministic trends. In such cases, both the lagged variableand the deterministic trends compete to model the nonstationarity in the data with theupshot that the rate of convergence is affected. In particular, Phillips (2001) showed thatthe rate of convergence to a unit root is slowed by the presence of increasing numbers ofdeterministic regressors. In the panel model context, the present paper shows thatdiscriminatory power against a unit root is generally weakened as more complexdeterministic regressors are included.

Acknowledgments

We thank Peter Robinson, an associate editor, and a referee for their helpful commentsand suggestions. Moon thanks the Faculty Development Awards of USC for researchsupport. Perron thanks SSHRCC, FQRSC, and MITACS for financial support. Phillipsthanks the NSF for support under Grant No. SES 04-142254.

Appendix A. Technical results and proofs

Let zitð0Þ and yitð0Þ, respectively, denote the panel observations zit and yit that aregenerated by model (1) with ri ¼ 1, that is, yi ¼ 0: Also define Zð0Þ; Y ð0Þ; Y�1ð0Þ,respectively, in a similar fashion to Z; Y ; and Y�1. For notational simplicity, set ui0 ¼ yi0.


Throughout the proofs, we will use the notation

~s2iT ¼1

T

XT

t¼1

u2it,

and

hðr; sÞ ¼ ð1; rÞ1

R 10 rdrR 1

0 rdrR 10 r2 dr

0@

1A�1

1

s

� �¼ 4� 6r� 6sþ 12rs.

Readers are referred to the preprint of this paper, Moon et al. (2006b), for detailedderivations of the results presented here, and to Phillips and Moon (1999) for themultidimensional limit theory.

A.1. Preliminary results

Lemma 5. Suppose that Assumption 1 is satisfied. Then, as n;T !1 with n=T ! 0; the

following hold:

(a)Pn

i¼1ð ~s2iT � s2i Þ

2¼ opð1Þ.

(b) sup1pipn j ~s2iT � s2i j ¼ opð1Þ.

(c) With probability approaching one, there exists a constant M40 such that inf i ~s2iTXM.

Proof. See Moon et al. (2006b). &

Suppose that ci is a sequence of iid random variables, independent of uit for all i and t,with a bounded support.

Lemma 6. Suppose that Assumptions 1–3, 11, and 12 hold. Then, the following hold as

ðn;T !1Þ with n=T ! 0:

ðaÞ1ffiffiffinp

Xn

i¼1

c2i1

T2s2i

XT

t¼1

ðyit � yi0Þ �t

TðyiT � yi0Þ

n o2

� o1T

" #

) N �Eðc2i y

2i Þ

90;Eðc4i Þ

45

� �.

ðbÞ1ffiffiffinp

Xn

i¼1

1

T2s2i

XT

t¼1

y2it �

1

T3s2i

XT

t¼1

XT

s¼1

yityishT ðt; sÞ � o2T

" #

) N �Eðy2i Þ420

;11

6300

� �.



A.2. Proofs and derivations for Section 3

Proof of Theorem 6. Since Dyit ¼ �ðyi=n1=2TÞyit�1 þ uit under Assumption 4, we can write

V nT ðCÞ

¼Xn

i¼1

1

s2iy2

i0 þXT

t¼1

ðDciyitÞ

2

" #�

1

s2i

Xn

i¼1

y2i0 þ

XT

t¼1

ðDyitÞ2

" #�

1

2mc;2

¼2

n1=2T

Xn

i¼1

ci

s2i

XT

t¼1

Dyityit�1 þ1

nT2

Xn

i¼1

c2is2i

XT

t¼1

y2it�1 �

1

2mc;2

¼ �2

nT2

Xn

i¼1

ciyi

s2i

XT

t¼1

y2it�1 þ

2

n1=2T

Xn

i¼1

ci

s2i

XT

t¼1

uityit�1

þ1

nT2

Xn

i¼1

c2is2i

XT

t¼1

y2it�1 �

1

2mc;2.

Direct calculation shows that under the assumptions of the theorem, we have

�2

nT2

Xn

i¼1

XT

t¼1

ciyi

s2iy2

it�1!p � EðciyiÞ,

1

nT2

Xn

i¼1

c2is2i

XT

t¼1

y2it�1!p

1

2mc;2,

and

2

n1=2T

Xn

i¼1

ci

s2i

XT

t¼1

uityit�1 ) Nð0; 2mc;2Þ,

thereby giving the required result. &

Lemma 7. Let M be a finite constant. Under Assumptions 1 and 2, the following hold:

(a) supi E½ðð1=TÞPT

t¼1 uityit�1Þ2�oM.

(b) supi E½ðð1=T2ÞPT

t¼1 y2it�1Þ

2�oM.

(c) supi E½y2i0�oM.

Proof. The lemma follows by direct calculation and we omit the proof. &

Lemma 8. Suppose that Assumptions 1–4 hold. Then, the following hold:

(a)Pn

i¼1ðs21;iT � s2i Þ

2¼ opð1Þ.

(b) sup1pipn js21;iT � s2i j ¼ opð1Þ.

(c) With probability approaching one, there exists a constant M40 such that inf i s21;iTXM.



Proof of Theorem 8. By definition,

VnT ðCÞ ¼ �2

nT2

Xn

i¼1

ciyi

s21;iT

XT

t¼1

y2it�1 þ

2

n1=2T

Xn

i¼1

ci

s21;iT

XT

t¼1

uityit�1

þ1

nT2

Xn

i¼1

c2i

s21;iT

XT

t¼1

y2it�1 �

1

2mc;2.

First, by the Cauchy–Schwarz inequality,

1

n

Xn

i¼1

1

s21;iT�

1

s2i

!ciyi

T2

XT

t¼1

y2it�1

��

p1

n

Xn

i¼1

s21;iT � s2is21;iT

!20@

1A

1=2

1

n

Xn

i¼1

ciyi

s2i

1

T2

XT

t¼1

y2it�1

!20@

1A

1=2

psupi js

21;iT � s2i j

inf i s21;iT

M

inf i s2i

1

n

Xn

i¼1

1

T2

XT

t¼1

y2it�1

!20@

1A

1=2

¼ opð1ÞOpð1Þ ¼ opð1Þ,

where the last line holds by Lemmas 7 and 8, the assumption that ci and yi have uniformlybounded supports, and inf i s2i 40. Similarly, by Lemmas 7 and 8, the assumption that ci

has a bounded support, and inf i s2i 40, we have

1

n1=2

Xn

i¼1


!ci

Ts2i

XT

t¼1

uityit�1

��

pXn

i¼1


!20@

1A

1=2

1

n

Xn

i¼1

ci

Ts2i

XT

t¼1

uityit�1

!20@

1A

1=2

pðPn

i¼1 ðs21;iT � s2i Þ

2Þ1=2

inf i s21;iT

M

inf i s2i

1

n

Xn

i¼1

1

T

XT

t¼1

uityit�1

!20@

1A

1=2

¼ opð1ÞOpð1Þ,

and

1

nT2

Xn

i¼1

c2i

s21;iT

XT

t¼1

y2it�1 ¼

1

nT2

Xn

i¼1

c2is2i

XT

t¼1

y2it�1 þ opð1Þ.

Combining these, we complete the proof that VnT ðCÞ ¼ V nT ðCÞ þ opð1Þ. &



Proof of Theorem 9. For the theorem, it is enough to show that

Vfe1;nT ðCÞ ¼ VnT ðCÞ þ opð1Þ.

Let b0iðciÞ ¼ ðDciG00Dci

G0Þ�1ðDci

G00DciZiÞ. Then Zi � G0b0iðciÞ ¼ Y i � G0ðb0iðciÞ � b0iÞ, and

we can rewrite V fe1;nT ðCÞ as

V fe1;nT ðCÞ

¼Xn

i¼1

1

s2i

ðDciY i � Dci

G0ðb0iðciÞ � b0iÞÞ0ðDci

Y i � DciG0ðb0iðciÞ � b0iÞÞ

�ðDY i � DG0ðb0iðciÞ � b0iÞÞ0ðDY i � DG0ðb0iðciÞ � b0iÞÞ

24

35� 1

2mc;2

¼ V nT ðCÞ þ V fe11;nT ðCÞ,

where

Vfe11;nT ðCÞ ¼Xn

i¼1

1

s2i

ðDY 0iDG0ÞðDG00DG0Þ�1ðDG00DY iÞ

�ðDciY 0iDci

G0ÞðDciG00Dci

G0Þ�1ðDci

G00DciY iÞ

" #.

For the required result, it is enough to show that

Vfe11;nT ðCÞ ¼ opð1Þ

as n;T !1 with n=T ! 0; which follows by Lemmas 7(c) and 9 and the assumption thatinf i s2i 40; since

V fe11;nT ðCÞ

¼Xn

i¼1

1

s2iy2

i0 �1

1þc2in

1

T

yi0 þci

n1=2

1

TðyiT � yi0Þ þ

c2in

1

T2

XT

t¼1

yit�1

!2

2664

3775

¼ I1 � I2 � I3 � 2I4 � 2I5 � 2I6,

and

I1 ¼1

nT

Xn

i¼1

y2i0

s2i

c2i

1þc2inT

0BB@

1CCA ¼ Op

1

T

� �¼ opð1Þ,

I2 ¼1

nT

Xn

i¼1

c2i

s2i 1þc2inT

� � yiT � yi0ffiffiffiffiTp

� �2

¼ Op1

T

� �¼ opð1Þ,

I3 ¼1

n2T

Xn

i¼1

c4i

s2i 1þc2inT

� � 1

TffiffiffiffiTp

XT

t¼1

yit�1

!2

¼ Op1

nT

� �¼ opð1Þ,


jI4j ¼1ffiffiffiffiffiffiffinTp

Xn

i¼1

ci

s2i 1þc2inT

� � yi0

yiT � yi0ffiffiffiffiTp

� ��

��

pffiffiffiffin

T

r1

n

Xn

i¼1

ci

s2i 1þc2inT

� � y2i0

0BBB@

1CCCA

1=2

1

n

Xn

i¼1

ci

s2i 1þc2inT


� �2

0BBB@

1CCCA

1=2

¼

ffiffiffiffin

T

rOpð1ÞOpð1Þ ¼ opð1Þ,

and, similarly,

I5 ¼1

n3=2T

Xn

i¼1

c3i

s2i 1þc2inT


� �1

TffiffiffiffiTp

XT

t¼1

yit�1

!¼ opð1Þ,

I6 ¼1

nffiffiffiffiTp

Xn

i¼1

c2i

s2i 1þc2inT

� � yi0

1

TffiffiffiffiTp

XT

t¼1

yit�1

!¼ opð1Þ,

as required. &

Lemma 9. Let M be a finite constant. Under Assumptions 1 and 2, the following hold:

(a) supi E½ððyiT � yi0Þ=ffiffiffiffiTpÞ2�oM.

(b) supi E½ðð1=TffiffiffiffiTpÞPT

t¼1 yit�1Þ2�oM.

Proof. The lemma follows by direct calculation, and its proof is omitted. &

Lemma 10. Suppose that Assumptions 1–4 hold. Then, the following hold:

(a) sup1pipnðs22;iT � s2i Þ ¼ opð1Þ.

(b) With probability approaching one, there exists a constant M40 such that inf i s22;iTXM.


Proof of Theorem 10. Using Lemmas 7(c), 9, and 10 and the assumptions that the supportsof yi and ci are bounded and inf i s2i 40, we can show using arguments similar to those used


in the proof of Theorem 8 that

V fe11;nT ðCÞ

¼Xn

i¼1

1

s2iy2

i0 �1

1þc2in

1

T

yi0 þci

n1=2

1

TðyiT � yi0Þ þ

c2in

1

T2

XT

t¼1

yit�1

!2

2664

3775

¼ V fe11;nT ðCÞ þ opð1Þ.

The required result now follows. &


Lemma 11. Under Assumptions 1–3, 11, and 12,

V fe2;nT ðCÞ

¼1

n1=4

Xn

i¼1

ci

s2i

2

T

XT

t¼1

Dyityit�1 �yiTffiffiffiffi

Tp

� �2

þyi0ffiffiffiffi

Tp

� �2

þ s2i

" #

þ1

n1=2

Xn

i¼1

c2is2i

1

T2

PTt¼1

y2it�1 � 2

yiTffiffiffiffiTp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �

þ1

3

yiTffiffiffiffiTp

� �2

þ s2i op2T

266664

377775

þ1

n

Xn

i¼1

c4is2i

�1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �2

þ2

3

yiTffiffiffiffiTp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �

�1

9

yiTffiffiffiffiTp

� �2

þ s2i op4T

2666664

3777775

þ1

n1=4T

Xn

i¼1

S1iT

s2iþ

1

n1=2T1=2

Xn

i¼1

S2iT

s2iþ

1

n5=4

Xn

i¼1

S3iT

s2i,

with ð1=nÞPn

i¼1E½S2kiT � ¼ Oð1Þ, for k ¼ 1; 2; 3 when ðn;T !1Þ with n=T ! 0:


Lemma 12. Under Assumptions 1–3, 11, and 12, the following hold:

ðaÞ1

n1=4

Xn

i¼1

ci

s2i

2

T

XT

t¼1


Tp

� �2

þyi0ffiffiffiffi

Tp

� �2

þ s2i

" #¼ opð1Þ;

ðbÞ1

n1=2

Xn

i¼1

c2is2i

1

T2

PTt¼1

y2it�1 � s2i

1

T

PTt¼1

t� 1

T

� �þ

1

3

yiTffiffiffiffiTp

� �2

� s2i

( )

� 2yiTffiffiffiffi

Tp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �� s2i

2

T

PTt¼1

t

T

� t� 1

T

� �� 2666664

3777775


) N �1

90E c2i y

2i

� ;1

45Eðc4i Þ

� �;

ðcÞ1

n

Xn

i¼1

c4is2i

�1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �2

þ2

3

yiTffiffiffiffiTp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �

�1

9

yiTffiffiffiffiTp

� �2

þ s2i op4T

266664

377775 ¼ opð1Þ.


Lemma 13. Let M be a finite constant. Under Assumptions 1–3, 11, and 12, the following

hold:

(a) supi E½y4i0�oM.

(b) supi E½ðyiT=ffiffiffiffiTpÞ4�oM.

(c) supi E½ðð1=TÞPT

t¼1 yit�1uitÞ2�oM.

(d) supi E½ðð1=T2ÞPT

t¼1 y2it�1Þ

2�oM.

(e) supi E½ðð1=TffiffiffiffiTpÞPT

t¼1 yit�1Þ4�oM.

(f) supi E½ðð1=TffiffiffiffiTpÞPT

t¼1 ½ðt� 1Þ=T �yit�1Þ4�oM.

Proof. The lemma follows by direct calculations and we omit the proof. &

Lemma 14. Suppose that Assumptions 1–3, and 11 hold. Then, the following hold:

(a)Pn


2¼ opð1Þ.

(b) sup1pipnðs21;iT � s2i Þ ¼ opð1Þ.

(c)Pn


2¼ opð1Þ.

(d) sup1pipn ðs23;iT � s2i Þ ¼ opð1Þ.

(e) With probability approaching one, there exists a constant M40 such that inf i s23;iTXM.


Proof of Theorem 15. For the required result, it is enough to show that

ðaÞ1

n1=4

Xn

i¼1

1

s23;iT�

1

s2i

!ci

2

T

XT

t¼1


Tp

� �2

þyi0ffiffiffiffi

Tp

� �2

þ s2i

" #¼ opð1Þ,

ðbÞ1

n1=4

Xn

i¼1


!¼ opð1Þ,

ðcÞ1

n1=2

Xn

i¼1

1

s23;iT�

1

s2i

!c2i

1

T2

PTt¼1

y2it�1 � 2

yiTffiffiffiffiTp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �

þ1

3

yiTffiffiffiffiTp

� �2

þ s2i op2T

266664

377775 ¼ opð1Þ,


ðdÞ1

n

Xn

i¼1

1

s23;iT�

1

s2i

!c4i

�1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �2

þ2

3

yiTffiffiffiffiTp

� �1

TffiffiffiffiTp

PTt¼1

t

Tyit�1

� �

�1

9

yiTffiffiffiffiTp

� �2

þ s2i op4T

2666664

3777775

¼ opð1Þ,

ðeÞ1

n1=4T

Xn

i¼1

1

s23;iT�

1

s2i

!S1iT ¼ opð1Þ,

ðfÞ1

n1=2T1=2

Xn

i¼1

1

s23;iT�

1

s2i

!S2iT ¼ opð1Þ,

ðgÞ1

n5=4

Xn

i¼1

1

s23;iT�

1

s2i

!S3iT ¼ opð1Þ.

Parts (c)–(g) hold by arguments similar to those used in the proof of Theorem 8, that is, usethe Cauchy–Schwarz inequality, Lemmas 13, 14 and the assumptions that the supports ofyi and ci are uniformly bounded and inf i s2i 40. For Part (a), notice by definition that

1

n1=4

Xn

i¼1

1

s23;iT�

1

s2i

!ci

2

T

XT

t¼1


Tp

� �2

þyi0ffiffiffiffi

Tp

� �2

þ s2i

" #

¼1

n1=4

Xn

i¼1

1

s23;iT�

1

s2i

!ci �ðri � 1Þ2

1

T

XT

t¼1

y2it�1 þ 2ð1� riÞ

1

T

XT

t¼1

yit�1uit

"

�1

T

XT

t¼1

u2it � s2i

!#

¼1

n3=4T

Xn

i¼1

s23;iT � s2is23;iTs

2i

!ciy

2i

T2

XT

t¼1

y2it�1 �

2

n1=2T

Xn

i¼1


2i

!ciyi

T

XT

t¼1

yit�1uit

þ1

n1=4T1=2

Xn

i¼1


2i

!1

T1=2

XT

t¼1

ðu2it � s2i Þ

!.

Using similar arguments to those in the proofs of Parts (c)–(g), we can show that the firstand the second terms are of Opð1=n1=4TÞ and Opð1=TÞ, respectively. Also the third term isopð1Þ since by the Cauchy–Schwarz inequality, Lemma 14, and the assumption inf i s2i 40;


it follows that

1

n1=4T1=2

Xn

i¼1


2i

!1

T1=2

XT

t¼1

ðu2it � s2i Þ

!��

pn1=4

T1=2

Xn

i¼1


2i

!20@

1A

1=2

1

n

Xn

i¼1

T1=2XT

t¼1

ðu2it � s2i Þ

!20@

1A

1=2

pn1=4

T1=2

1

inf i s23;iT

1

inf i s2i

Xn

i¼1

ðs23;iT � s2i Þ2

!1=21

n

Xn

i¼1

T1=2XT

t¼1

ðu2it � s2i Þ

!20@

1A

1=2

¼n1=4

T1=2Opð1Þopð1ÞOpð1Þ ¼ opð1Þ,

which yields Part (a).For Part (b), notice that

1

n1=4

Xn

i¼1


!¼

1

n1=4

Xn

i¼1

s23;iT � s2is2i

!þ

1

n1=4

Xn

i¼1


s23;iT�

1

s2i

!.

The second term is opð1Þ by Lemma 14 and the assumption inf is2i 40; since

1

n1=4

Xn

i¼1


s23;iT�

1

s2i

!�� ¼ 1

n1=4

Xn

i¼1


s23;iTs2i

��

p1

n1=4

1

inf i s23;iT

1

inf i s2i

Xn

i¼1


!

¼ opð1Þ.

To complete the proof of Part (b), it is enough to show that the first term is opð1Þ. Write thefirst term as

1

n1=4

Xn

i¼1

s23;iT � s2is2i

!

¼1

n1=4

Xn

i¼1

s23;iT � s21;iTs2i

!þ

1

n1=4

Xn

i¼1

s21;iT � ~s2iTs2i

!þ

1

n1=4

Xn

i¼1

~s2iT � s2is2i

� �.

By definition and by Lemma 13, we have

1

n1=4

Xn

i¼1

s23;iT � s21;iTs2i

!¼

1

n1=4T

Xn

i¼1

1

s2iy2

i0 þyiT � yi0ffiffiffiffi

Tp

� �2 !

¼ Opn3=4

T

� �¼ opð1Þ,


1

n1=4

Xn

i¼1

s21;iT � ~s2iTs2i

!¼

1

n1=4

Xn

i¼1

1

s2i

y2i0

Tþ

y2in1=2T

1

T2

XT

t¼1

y2it�1

!

� 2yi

n1=4T

1

T

XT

t¼1

uityit�1

!!

¼ Opn3=4

T

� �þOp

n1=4

T

� �þOp

1

T

� �¼ opð1Þ,

and

1

n1=4

Xn

i¼1

~s2iT � s2is2i

� �¼

1

n1=4 T1=2

Xn

i¼1

1

s2i

1

T1=2

XT

t¼1

ðu2it � s2i Þ

!

¼ Opn1=4

T1=2

� �¼ opð1Þ,

the last line holding because

E1

n1=2

Xn

i¼1

1

s2i

1

T1=2

XT

t¼1

ðu2it � s2i Þ

!" #2¼ Oð1Þ.

Combining these, we have

1

n1=4

Xn

i¼1

s23;iT � s2is2i

!¼ opð1Þ,

as required. &

Proof of Lemma 1. The lemma holds by Lemma 6(a) with ci ¼ 1. &

Proof of Lemma 2. The lemma holds by Lemma 6(b). &


Proof of Theorem 16. Denote ZnC ¼ ðS

�1=2 � ITþ1ÞDCvecðZ0Þ, Gn

0;C ¼ ðS�1=2 � ITþ1Þ

DCðIn � G0Þ, Gn

1;C ¼ ðS�1=2 � ITþ1ÞDCðIn � G0Þln, Yn

C ¼ ðS�1=2 � ITþ1Þ DCvecðY 0Þ, and

Mn0;C ¼ InðTþ1Þ � Gn

0;CðGn00;CGn

0;CÞ�1Gn0

0;C. Under the null, when C ¼ 0; we denote thesequantities by Zn

0 ; Gn

0;0; Gn

1;0 Y n0 ; and Mn

0, respectively. Then, by definition

Zn

C ¼ Gn

0;Cb0 þ Gn

1;Cb1 þ Yn

C.

Using this notation, we may express

V fe3;nT ðCÞ ¼ � 2 minb0;b1

LnT ðC;b0; b1Þ �minb0;b1

LnT ð0; b0; b1Þ

� ��

1

2mc;2

¼ Yn0CMn

0;CYn

C � Yn0CMn

0;CGn

1;CðGn01;CMn

0;CGn

1;CÞ�1Gn0

1;CMn

0;CYn

C

� Yn00 Mn

1;0Yn

0 þ Yn00 Mn

1;0Gn

1;0ðG01;0Mn

1;0Gn

1;0Þ�1G01;0Mn

1;0Yn

0 �12mc;2.


In what follows we show that

Y 0CMn

0;CGn

1;CðG01;CMn

0;CGn

1;CÞ�1G01;CMn

0;CYn

C

� Y 00 Mn

1;0Gn

1;0ðG01;0M

n

1;0Gn

1;0Þ�1G01;0M

n

1;0Yn

0

¼ opð1Þ. ð25Þ

Then, by definition, it follows that

Vfe3;nT ðCÞ ¼ Y 0CMn

0;CYn

C � Y 00 Mn

1;0Yn

0 �12mc;2 þ opð1Þ

¼ � 2 minb0

LnT ðC;b0Þ �minb0

LnT ð0; b0Þ� �

�1

2mc;2 þ opð1Þ

¼ Vfe1;nT ðCÞ þ opð1Þ,

as required for the theorem. &

Proof of (25). By definition

Y 0CMn

0;CGn

1;CðG01;CMn

0;CGn

1;CÞ�1G01;CMn

0;CY n

C � Y 00 Mn

1;0Gn

1;0ðG01;0Mn

1;0Gn

1;0Þ�1G01;0Mn

1;0Yn

0

¼

1ffiffiffiffiffiffiffinTp Y 0CMn

0;CGn

1;C

� �2

�1ffiffiffiffiffiffiffinTp Y 00 Mn

1;0Gn

1;0

� �2

1

nTG01;CMn

0;CGn

1;C

þ1ffiffiffiffiffiffiffinTp Yn0

0 Mn

1;0Gn

1;0

� �21

1

nTG01;CMn

0;CGn

1;C

�1

1

nTG01;0M

n1;0Gn

1;0

0B@

1CA

¼ I þ II ; say.

For term I ; with probability approaching one,

G01;CMn0;CGn

1;C

nT40,

since

G01;CMn0;CGn

1;C

nT¼

1

nT

Xn

i¼1

1

s2iðDci

G1Þ0ðDci

G1Þ �1

nT

Xn

i¼1

1

s2i

½ðDciG1Þ0ðDci

G0Þ�2

ðDciG0Þ0ðDci

G0Þ

¼1

nT

Xn

i¼1

1

s2i

XT

t¼1

1þci

n1=2

t

T

� �2

�1

n2T

Xn

i¼1

c2is2i

1

T

PTt¼1 1þ

ci

n1=2

t

T

� �� 2

1þc2inT

X1þ oð1Þ

inf i s2i.

Next,


0;CGn

1;C ¼1ffiffiffiffiffiffiffinTp Y 0CGn

1;C �1ffiffiffiffiffiffiffinTp Y 0CGn

0;CðG00;CGn

0;CÞ�1G00;CGn

1;C


¼1ffiffiffinp

Xn

i¼1

1

s2i

yiT � yi0ffiffiffiffiTp þ

ciffiffiffinp

yiTffiffiffiffiTp þ

c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!þ opð1Þ

because

1ffiffiffiffiffiffiffinTp Y 0CGn

0;CðG00;CGn

0;CÞ�1G00;CGn

1;C

¼1

nffiffiffiffiTp

Xn

i¼1

ci

s2i

yi0 þci

n1=2

yiT � yi0

� T

þc2in

1

T2

PTt¼1 yit�1

� �1þ

ci

n1=2

1

T

PTt¼1

t

T

� �

1þc2inT

¼ Op1ffiffiffiffiTp

� �.

Similarly, we have

1ffiffiffiffiffiffiffinTp Y 00 Mn

1;0Gn

1;0 ¼1ffiffiffinp

Xn

i¼1

1

s2i


� �þ opð1Þ.

Then, since

1ffiffiffinp

Xn

i¼1

1

s2i


� �¼ Opð1Þ

and

1ffiffiffinp

Xn

i¼1

1

s2i


ciffiffiffinp


c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!¼ Opð1Þ,

the numerator of term I is


0;CGn

1;C

� �2

�1ffiffiffiffiffiffiffinTp Y 00 Mn

1;0Gn

1;0

� �2

¼1ffiffiffinp

Xn

i¼1

1

s2i


ciffiffiffinp


c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!( )2

�1ffiffiffinp

Xn

i¼1

1

s2i


� �( )2

þ opð1Þ

¼ 21ffiffiffinp

Xn

i¼1

1

s2i


� �( )1ffiffiffinp

Xn

i¼1

1

s2i

ciffiffiffinp


c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!( )

þ1ffiffiffinp

Xn

i¼1

1

s2i

ciffiffiffinp


c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!( )2

þ opð1Þ

¼ opð1Þ,


where the last line holds since

1ffiffiffinp

Xn

i¼1

1

s2i

ciffiffiffinp


c2in

1

TffiffiffiffiTp

XT

t¼1

t

Tyit�1

!¼ Op

1

n1=2

� �¼ opð1Þ.

Therefore, we have

I ¼ opð1Þ.

Next, we show that II ¼ opð1Þ. Since

1ffiffiffiffiffiffiffinTp Y 00 Mn

1;0Gn

1;0 ¼1ffiffiffinp

Xn

i¼1

1

s2i


� �þ opð1Þ ¼ Opð1Þ,

the required result II ¼ opð1Þ follows if we show that

1

1

nTG01;CMn

0;CGn

1;C

�1

1

nTG01;0Mn

1;0Gn

1;0

0B@

1CA ¼ opð1Þ,

which follows because with probability approaching one,

G01;0Mn1;0Gn

1;0

nT¼

1

n

Xn

i¼1

1

s2iX

1

inf i s2i,

and

G01;CMn0;CGn

1;C

nT�

G01;0Mn1;0G

n

1;0

nT

¼1

n

Xn

i¼1

1

s2i

1

T

XT

t¼1

1þci

n1=2

t

T

� �2

� 1

( )�

1

n2T

Xn

i¼1

c2is2i

1

T

PTt¼1 1þ

ci

n1=2

t

T

� �� 2

1þc2inT

¼1

n3=2

Xn

i¼1

ci

s2i

1

T

XT

t¼1

2t

Tþ

ci

n1=2

t

T

�2� ��

1

n2T

Xn

i¼1

c2is2i

1

T

PTt¼1 1þ

ci

n1=2

t

T

� �� 2

1þc2inT

¼ O1

n1=2

� �þO

1

n

� �¼ oð1Þ.

Therefore,

II ¼ opð1Þ: &

References

Andrews, D., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econome-

trica 59, 817–858.

Bai, J., Ng, S., 2004. A PANIC attack on unit roots and cointegration. Econometrica 72, 1127–1177.

Bowman, D., 2002. Efficient tests for autoregressive unit roots in panel data. Mimeo.


Breitung, J., 1999. The local power of some unit root tests for Panel data. SFB 373 Discussion paper, No. 69-1999,

Humboldt University, Berlin.

Breitung, J., 2000. The local power of some unit root tests for panel data. In: Baltagi, B. (Ed.), Nonstationary

Panels, Panel Cointegration, and Dynamic Panels, Advances in Econometrics, vol. 15. JAI, Amsterdam,

pp. 161–178.

Choi, I., 2001. Unit root tests for panel data. Journal of International Money and Finance 20, 249–272.

Dufour, J., King, M., 1991. Optimal invariant tests for the autocorrelation coefficient in linear regressions with

stationary or nonstationary AR(1) errors. Journal of Econometrics 47, 115–143.

Elliott, G., 1999. Efficient tests for a unit root when the initial observation is drawn from its unconditional

distribution. International Economic Review, 40, 767–783.

Elliott, G., U. Muller, 2003. Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.

Elliott, G.T., Rothenberg, T.J., Stock, J., 1996. Efficient tests for an autoregressive unit root. Econometrica 64,

813–836.

Evans, G.B.A., Savin, N.E., 1984. Testing for unit roots: 2. Econometrica 52, 1241–1269.

Im, K., Pesaran, H., Shin, Y., 2003. Testing for unit roots in heterogeneous panels. Journal of Econometrics 115,

53–74.

King, M., 1988. Towards a theory of point optimal testing. Econometric Reviews 6, 169–218.

Lehmann, E., 1959. Testing Statistical Hypotheses. Wiley, New York.

Levin, A., Lin, F., Chu, C., 2002. Unit root tests in panel data: asymptotic and finite-sample properties. Journal of

Econometrics 108, 1–24.

Maddala, G.S., Wu, S., 1999. A comparative study of unit root tests with panel data and a new simple test.

Oxford Bulletin of Economics and Statistics 61, 631–651.

Moon, H.R., Perron, B., 2004. Testing for a unit root in panels with dynamic factors. Journal of Econometrics

122, 81–126.

Moon, H.R., Perron, B., 2005. Asymptotic local power of pooled t-ratio tests for unit roots in panels with fixed

effects. Mimeo.

Moon, H.R., Perron, B., 2006, An empirical analysis of nonstationarity in a panel of interest rates with factors.

Journal of Applied Econometrics, forthcoming.

Moon, H.R., Phillips, P.C.B., 1999. Maximum likelihood estimation in panels with incidental trends. Oxford

Bulletin of Economics and Statistics 61, 771–748.

Moon, H.R., Phillips, P.C.B., 2004. GMM estimation of autoregressive roots near unity with panel data.

Econometrica 72, 467–522.

Moon, H.R., Perron, B., Phillips, P.C.B., 2006a. A note on ‘‘The local power of some unit root tests for panel

data’’ by J. Breitung. Econometric Theory 22, 1177–1188.

Moon, H.R., Perron, B., Phillips, P.C.B., 2006b. Unpublished appendix for ‘‘Incidental trends and the power of

panel unit root tests’’. Mimeo.

Newey, W.K., West, K.D., 1994. Automatic lag selection in covariance matrix estimation. Review of Economic

Studies 61, 631–653.

Phillips, P.C.B., 1987. Time series regression with a unit root. Econometrica 55, 277–302.

Phillips, P.C.B., 2001. New unit root asymptotics in the presence of deterministic trends. Journal of Econometrics

11, 323–353.

Phillips, P.C.B., Moon, H.R., 1999. Linear regression limit theory for nonstationary panel data. Econometrica 67,

1057–1111.

Phillips, P.C.B., Sul, D., 2003. Dynamic panel estimation and homogeneity testing under cross-section

dependence. Econometrics Journal 6, 217–239.

Ploberger, W., Phillips, P.C.B., 2002. Optimal testing for unit roots in panel data. Mimeo.

Quah, D., 1994. Exploiting cross-section variations for unit root inference in dynamic panels. Economics Letters

44, 9–19.


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