Bootstrap Unit Root Tests in Panels with Cross-Sectional Dependency

B ootstrap U nitR ootTests inP anels

withCross-SectionalD ependency

A bstract

W e apply bootstrap methodology to unitroottests fordependentpanels withN cross-sectionalunits andT time series observations. M ore

speci…-cally, we leteach panelbe driven byagenerallinearprocess which maybe di¤erentacross cross-sectionalunits, and approximate itby a …nite order autoregressive integrated process oforderincreasingwithT. A s we allow

thedependencyamongtheinnovations generatingtheindividualpanels, we constructourunitroottests from the estimation ofthe system ofthe en-tireN panels. T he limitdistributions ofthe tests are derived by passing T to in…nity, withN …xed. W e then apply the bootstrap method to the

approximatedautoregressions toobtainthecriticalvalues forthepanelunit roottests, andestablishtheasymptoticvalidityofsuchbootstrap panelunit roottestsundergeneralconditions. T heproposedbootstrap testsareindeed quitegeneralcoveringawideclass ofpanelmodels. T heyinparticularallow forverygeneraldynamicstructures whichmayvaryacross individualunits, and more importantly forthe presence ofarbitrary cross-sectionaldepen-dency. T he…nitesampleperformanceofthebootstrap tests is examinedvia simulations, andcomparedtothatofthet-barstatistics byIm, P esaranand Shin (19 9 7 ), which is one ofthe commonly used unitroottests forpanel data. W e…ndthatourbootstrap panelunitroottests perform wellrelative tothe t-barstatistics.

T his version: January, 2000

Keywords andphrases: D ependentpanels, unitroottests, sievebootstrap, A R approxima-tion.

1_{T he paperwas written while I was visitingthe Cowles Foundation forR esearch in Economics atY ale}

U niversityduringthe fallof19 9 9 . I am gratefultoD onA ndrews, B illB rown, Joon P arkandP eterP hillips forhelpfuldiscussions andcomments. M ythanks alsogototheseminarparticipants atY ale. T his research is supported in partbythe CSIV fund from R ice U niversity.

1. Introduction

R ecently, nonstationarypanels have drawn much attention in both theoreticaland empir-icalresearch, as a numberofpaneldata sets covering relatively long time periods have become available. V arious statistics fortestingthe unitroots and cointegration forpanel models have been proposed, and frequently used fortestinggrowth convergence theories, purchasingpowerparityhypothesis and forestimatinglong-runrelationships amongmany macroeconomic and international…nancialseries including exchange rates and spot and future interestrates. Such paneldatabased tests appeared attractive tomany empirical researchers, since they o¤eralternatives tothe tests based only on individualtime series observations thatareknowntohavelowdiscriminatorypower. A numberofunitroots and cointegrationtests havebeen developed forpanelmodels bymanyauthors. See L evin and L in(19 9 2,19 9 3), Q uah(19 9 4), Im, P esaranandShin(19 9 7 ) andM addalaandW u(19 9 6) for someofthe panelunitroottests, andP edroni (19 9 6,19 9 7 ) and M cCoskeyand Kao(19 9 8) forthe panelcointegration tests available in the currentliterature. B anerjee (19 9 9 ) gives agoodsurveyontherecentdevelopments intheeconometricanalysis ofpaneldatawhose time series componentis nonstationary.2

SincetheworkbyL evinandL in(19 9 2), anumberofunitroottests forpaneldatahave beenproposed. L evinandL in(19 9 2,19 9 3) considerunitroottests forhomogeneous panels, which are simply the usualt-statistics constructed from the pooled estimatorwith some appropriate modi…cations. Such unitroottests forhomogeneous panels can therefore be represented as a simple sum taken overi = 1 ;:::;N and t = 1 ;:::;T. T hey showunder

cross-sectionalindependencythatthesequentiallimitofthestandardt-statistics fortesting theunitrootis thestandardnormaldistribution.3 _{Forheterogeneous panels, theunitroot} testcannolongerberepresentedas asimplesum sincethepooledestimatoris inconsistent forsuch heterogeneous panels as shown in P esaran and Smith (19 9 5). Consequently the secondstageN-asymptoticsintheabovesequentialasymptoticsdoesnotworkhere. Im, P

e-saranandShin(19 9 7 ) looks attheheterogeneous panels andproposes unitroottests which arebasedontheaverageoftheindependentindividualunitroottests, t-statistics andL M statistics, computed from each individualpanel. T hey showthattheirtests alsoconverge tothe standard normaldistribution upon takingsequentiallimits. T hough they allowfor theheterogeneity, theirlimittheoryis stillbasedonthecross-sectionalindependecy, which canbequitearestrictiveassumption formostofthe paneldataweencounter.

T he tests suggested by L evin and L in (19 9 3) and Im, P esaran and Shin (19 9 7 ) are notvalid in the presence ofcross-correlations amongthe cross-sectionalunits. T he limit

2_{Stationarypanels haveamuchlongerhistoryandhavebeenintenselyinvestigatedbymanyresearchers.}

T he readers are referred tothe books by H siao(19 86), M atyas and Sevestre (19 9 6) and B altagi (19 9 5) for the literature on the econometricanalysis ofpaneldata.

3_{T he sequentiallimitis taken by …rstpassingT toin…nity withN …xed and subsequently letN tend}

to in…nity. R egression limit theory fornonstationary paneldata is developed rigorously by P hillips and M oon (19 9 9 ). T hey show thatthe limitofthe double indexed processes may depend on the wayN and

T tend toin…nity. T heyformallydevelops the asymptotics ofsequentiallimit, diagonalpath limit(N and

T tend toin…nity ata controled rate ofthe typeT = T(N)) and jointlimits (N andT tend to in…nity simultaneously without any restrictions imposed on the divergence rate). T heir limit thoery, however, assumes cross-sectionalindependence.

limitdistributions oftheirtests are nolongervalid and unknown when the independency assumptionisviolated. Indeed, M addalaandW u(19 9 6)showthroughsimulationsthattheir tests havesubstantialsizedistortions whenusedforcross-sectionallydependentpanels. A s awaytodealwith suchinferentialdi¢cultyinpanels withcross-correlations, theysuggest tobootstrap thepanelunitroottests, suchas thoseproposedbyL evinandL in(19 9 3), Im, P esaran and Shin (19 9 7 ) and Fisher(19 33), forcross-sectinally dependentpanels. T hey showthroughsimulations thatthebootstrap versionofsuchtests perform muchbetter, but donotprovide thevalidityofusingbootstrap methodology.

In this paper, we apply bootstrap methodology tounitroottests forcross-sectionally dependentpanels. M orespeci…cally, weleteachpanelbedrivenbyagenerallinearprocess which may be di¤ erentacross cross-sectionalunits, and approximate itby a …nite order autoregressive integrated process oforderincreasingwithT. A s we allowthe dependency

amongthe innovations generatingthe individualpanels, we constructourunitroottests from theestimationofthesystem oftheentireN panels. T helimitdistributions ofthetests

arederivedbypassingT toin…nity, withN …xed. W ethenapplythebootstrap methodto

theapproximatedautoregressions toobtainthe criticalvalues forthe panelunitroottests basedontheoriginalsample, andestablishtheasymptoticvalidityofsuchbootstrap panel unitroottests undergeneralconditions.

T he restofthe paperis organized as follows. Section 2 introduces the panelunitroot tests forcross-sectionally dependentpanels based on the originalsample and derives the limittheory forthe sample tests. Section 3 applies the sieve bootstrap methodology to thesamplepanelunitroottests consideredinSection2 andestablishes asymptoticvalidity ofthe sieve bootstrap unitroottests. A lsodiscussed in Section 3 are the practicalissues arising from the implementation of the sieve bootstrap methodology. In Section 4, we conductsimulations to investigate …nite sample performance ofthe bootstrap unit root tests. Section5 concludes, and mathematicalproofs are provided in an A ppendix.

2. U nitR ootTests forD ependentP anels

W econsiderapanelmodelgenerated as the following…rstorderautoregressiveregression: 4yit= ®iyi;t¡1+ uit; i = 1 ;:::;N ;j = 1 ;:::;T: (1) A s usual, the index i denotes individualcross-sectionalunits, such as individuals, house-holds, industries orcountries, and the index t denotes time periods. W e are interested in testingtheunitrootnullhypothesis, ®i= 0 forallyitgivenasin(1), againstthealternative, ®i< 0 forsomeyit, i = 1 ;:::;N . T hus, thenullimplies thatallyit’s haveunitroots, andis rejectifanyoneofyit’sisstationarywith®i< 0 . T herejectionofthenullthereforedoesnot imply thatthe entire panelis stationary. T he initialvalues (y10;:::;yN0)of(y1t;:::;yNt) donota¤ectoursubsequentasymptoticanalysisaslongas theyarestochasticallybounded, andthereforewesetthem atzeroforexpositionalbrevity.

Itis assumed thatthe errorterm (uit)in the model(1) is given by a generallinear process speci…ed as

whereL is theusuallagoperatorand ¼i(z)= 1 X k= 0 ¼i;kzk

for i = 1 ;:::;N. N ote that we let ¼ivary across i, thereby allowing heterogeneity in individualserialcorrelation structures. W e alsoallowforthe cross-sectionaldependency through the cross-correlation ofthe innovations "it; i = 1 ;:::;N thatgenerate the error uit’s. T o de…ne the cross-sectionaldependecy more explicitly, we de…ne the time series innovation("t)Tt= 1 by

"t= ("1t;:::;"Nt)0 (3) anddenotebyj¢jtheEuclideannorm: foravectorx= (xi),jxj2 =Pix2i, andforamatrix A = (aij);jA j=Pi;ja2ij. W eassume the following:

A ssumption A 1 L et("t;Ft)be a martingale di¤ erence sequence, with some …ltration (Ft), such thatE("t"0tjFt¡1)= § a.s., andEj"tjr < 1 forsomer¸4.

A ssumption A 2 L et¼i(z)6= 0 foralljzj·1 , andP1k= 0 jkjsj¼i;kj< 1 forsome s ¸1 , foralli = 1 ;:::;N .

T heconditionsinA ssumptions A 1 andA 2 areroutinelyimposedonthelinearprocesses given by(2). Itis wellknown thatan invariance principle holds forapartialsum process of("t)de…ned in (3) underA ssumption A 1. T hatis,

1 p T [T¢] X t= 1 "t! d B = 0 B @ B1 ... BN 1 C A = B M (0 ;§) (4)

asT! 1 , where[x]denotes themaximum integerwhichdoes notexceed x. W e maywrite (uit)as uit= ¼i(1 )"it+ (¹ui;t¡1¡¹uit) (5) where ¹ uit= 1 X k= 0 ¹

¼i;k"i;t¡k; ¹¼i;k= 1 X j= k+ 1

¼i;j

U nderourcondition in A ssumption A 2, we haveP1

k= 0j¹¼i;kj< 1 [see P hillips and Solo (19 9 2)]and therefore (¹uit)is wellde…ned both in a.s. and L r sense [see B rockwelland D avis (19 9 1, P roposition 3.1.1)].

U nderthe unitroothypothesis ®1=¢¢¢=®N = 0 , wemaynowwrite

yit= ¼i(1 )wit+ (¹ui0 ¡¹uit) (6) where wit=Ptk= 1"ik. Consequently, (yit)behaves asymptotically as the constant¼i(1 ) multipleof(wit). N otethat(¹uit)is stochasticallyofsmallerorderofmagnitudethan (wit), andthereforewillnotcontributetoourlimittheory.

U nder A ssumptions A 1 and A 2, we may write the linearprocess given in (2) as an in…niteorderautoregressive (A R ) process ®i(L )uit= "it with ®i(z)= 1 ¡ 1 X k= 1 ®i;kzk

andapproximate(uit)by a…nite orderA R process

uit= ®i;1ui;t¡1+ ¢¢¢+ ®i;piui;t¡pi+ "

pi it (7 ) with "pi it= "it+ 1 X k= pi+ 1 ®i;kui;t¡k

U nderA ssumptions A 1 and A 2, wehaveforeach i = 1 ;:::;N

Ej"pi it¡"itjr ·Ejuitjr 0 @ X1 k= pi+ 1 j®i;kj 1 A r = o(p¡rs_i )

N ote thatwehaveunderA ssumptions A 1 and A 2 Ejuitjr ·c Ã₁ X k= 0 ¼_i;k2 !r=2 Ej"itjr < 1

forsome constantc, due tothe M arcinkiewicz-Z ygmund inequality[see, e.g., Stout(19 7 4, T heorem 3.3.6)]. T heerrorinapproximating(uit)bya…niteorderA R processthusbecomes smallas pigets large.

U singthe A R approximationof(uit)given in (7 ), wewritethemodel(1) as 4yit= ®iyi;t¡1+

pi

X k= 1

®i;kui;t¡k+ "piti

which, since4yit= uitunderthenullhypothesis, canbeseenas anautoregressionof4yit augmentedbyyi;t¡1, viz.

4yit= ®iyi;t¡1+ pi

X k= 1

®i;k4yi;t¡k+ "piti (8) O urunitroottests willbebased on theabove approximated autoregression.

Fortheorderpiinthe regression(8), we assume

T he A R orderpishould, in particular, be increasingwithT.4 W e may choose pi’s using the usualorderselection criteria such as Schwartz information criterion (B IC) orA kaike informationcriterion(A IC).5 _{T heorderselection can be basedeitheron theregression(8)} with no restriction on ®i’s, oron the approximated A R regression in (7 ) where ®i’s are restrictedtobezero. T his willnota¤ectoursubsequentlimittheory.

2.1 U nitR ootTests forH eterogeneous P anels

T he augmented autoregression (8) can be written in the followingmatrix form by taking theindividualunits, with alltheirT observations, one aftertheother, viz.

0 B @ 4y1 ... 4yN 1 C A = 0 B @ y`;1 0 ... 0 y`;N 1 C A 0 B @ ®1 ... ®N 1 C A+ 0 B @ X p1 1 0 ... 0 X pN N 1 C A 0 B @ ¯p1 1 ... ¯pN N 1 C A+ 0 B @ "p1 1 ... "pN N 1 C A ormorecompactly 4y = Y`® + Xp¯p+ "p (9 ) whereweuse thefollowingnotation y`;i= 0 B @ yi;0 ... yi;T¡1 1 C A; X pi i = 0 B @ xpi0 i1 ... xpi0 iT 1 C A and ¯pi i = 0 B @ ®i;1 ... ®i;pi 1 C A with xpi0

it = (4yi;t¡1;:::;4yi;t¡pi), foralli = 1 ;:::;N.

W e constructthe tests forthe nullhypothesis ofthe unitroots in yt= (y1t;:::;yNt)0 generated by(1) and (2) based on the system G L S and O L S estimation ofthe augmented A R (9 ). T hefeasible G L S estimatorof® in (9 ) is givenby

^

®G T = BG T¡1AG T

where AG T and BG T are de…ned below. Forthe testofthe null® = 0 , we considerthe

followingF -type testbasedonthefeasibleG L S estimator^®G T:

FG T = ^®0G T(var(^®G T))¡1®^G T = A0G TBG T¡1AG T (10) where AG T = Y<sub>`</sub>0(~§¡1-IT)"p ¡Y`0(~§¡1-IT)Xp ³ X _p0(~§¡1-IT)Xp ´¡1 X _p0(~§¡1-IT)"p BG T = Y<sub>`</sub>0(~§¡1-IT)Y`¡Y`0(~§¡1-IT)Xp ³ X <sub>p</sub>0(~§¡1-IT)Xp ´¡1 X<sub>p</sub>0(~§¡1-IT)Y`

4_{O urregression (8) here may be viewed as the extension ofthe unitrootregression considered in Said}

andD ickey(19 84) tothepanelmodels. H owever, ourassumptionontheA R orderpiis substantiallyweaker

than thatused bySaid and D ickey(19 8 4), due tothe resultin Changand P ark(19 9 9 ).

5_{A s forthechoiceamongtheselectioncriteria, B IC mightbepreferredif(u}

it) is believedtobegenerated

by a …nite autoregression, since it yields a consistentestimator forpi. See, e.g., A n, Chen and H annan

(19 8 2). Ifnot, A IC maybeabetterchoice, sinceitleads toanasymptoticallye¢cientchoicefortheoptimal orderofsome projected in…nite orderautoregressive process as shown by Shibata(19 80). See Choi (19 9 2) formore discussions on the modelselection issue forA R M A models.

and ~§ is aconsistentestimatorofthe covariance matrix §. T he limitdistribution forthe testFG T is easily drived from the asymptoticbehaviors ofAG T and BG T, and is given in

T heorem 2.1 below.

O nthe otherhand, theO L S estimatorof® in (9 ) is givenby ^

®O T = BO T¡1AO T

andusethefollowingO L S-based F -type testfortesting® = 0

FO T = ^®0O T(var(^®O T))¡1®^O T = A0O TM F O T¡1 AO T (11) where AO T = Y<sub>`</sub>0"p ¡Y`0X p(Xp0X p)¡1X p0"p BO T = Y`0Y`¡Y`0Xp(Xp0Xp)¡1Xp0Y` M F O T = Y<sub>`</sub>0(~§-IT)Y`¡Y`0X p(X p0X p)¡1X p0(~§-IT)Y`¡Y`0(~§-IT)X p(Xp0X p)¡1X p0Y` + Y<sub>`</sub>0X p(X p0Xp)¡1X p0(~§-IT)Xp(Xp0X p)¡1X p0Y`

T heO L S estimator^®O T is less e¢cientthattheG L S estimator^®G T inourcontext. T he

O L S-based testFO T in (11) is thus expected tobe less powerfulthan the G L S-based test

FG T in (10). H owever, we observe in oursimulations thatFO T often performs betterthan

FG T in…nite samples, especiallywhenN is large.

T oconstructaconsistentestimatorforthe covariance matrix §, we may estimate the regression

uit= ~®pi;1iui;t¡1+ ¢¢¢+ ~®pi;piiui;t¡pi+ ~"

pi

it (12)

bysingle-equationO L S fori = 1 ;:::;N, withtheunitrootrestriction®i=0 imposed. T he …ttedresiduals (~"pi

it)areconsistentfor("it), since ~®p_i;ki areconsistentfor®i;kfor1 ·k·pi, and the autoregressive coe¢cients (®i;k)fork> pibecome negligible in the limitas long as we letpi! 1 . T his is shown in Park(19 9 9 , L emma3.1). O fcourse, one may obtain consistent…tted residuals byestimatingtheunrestrictedregession(8). T his againwillnot a¤ectourlimittheory. From (~"pi

it), form thetime series residualvectors ~ "pt= (~"p1t1;:::;~" pN Nt)0 (13) fort= 1 ;:::;T. W ethenestimate§ by ~ §= 1 T T X t= 1 ~ "p_t"~p_t0 N otice that ~ §=1 T T X t= 1 "p_t"p0_t+ op(1 )=1 T T X t= 1 "t"0t+ op(1 )=E"t"0t+ op(1 )

wherethesecondequalityfollows from L emmaA 1 (c) inA ppendix. W euse(~§-IT)as an

L et¾ijand ¾ijdenote, respectively, the (i;j)-elements ofthe covariance matrix§ and its inverse§¡1_{. T helimittheories forthetests F}

G T andFO T aregiven in

T heorem 2.1 U nderA ssumptions A 1, A 2 and A 3, wehave (a) FG T ! d Q0AGQ ¡1 BGQAG (b) FO T ! d Q0AOQ ¡1 M F OQAO asT ! 1 , where QAG = 0 BB B B B B B @ ¼1(1 ) N X j= 1 ¾1j Z1 0 B1dBj ... ¼N(1 ) N X j= 1 ¾NjZ1 0 BNdBj 1 CC C C C C C A ; QAO = 0 B B B B B @ ¼1(1 ) Z1 0 B1dB1 ... ¼N(1 ) Z1 0 BNdBN 1 C C C C C A QBG = 0 B B B B B @ ¾11¼1(1 )2 Z1 0 B 2 1 ::: ¾1N ¼1(1 )¼N (1 ) Z1 0 B1BN ... ... ... ¾N1_¼ N(1 )¼1(1 ) Z1 0 BNB1 ::: ¾ N N ¼N(1 )2 Z1 0 B 2 N 1 C C C C C A and QM F O = 0 B B BB B B @ ¾11¼1(1 )2 Z1 0 B 2 1 ::: ¾1N¼1(1 )¼N (1 ) Z1 0 B1BN ... ... ... ¾N1¼N (1 )¼1(1 ) Z1 0 BNB1 ::: ¾N N¼N(1 ) 2 Z1 0 B 2 N 1 C C CC C C A R emarks

(a) T he limitdistributions ofthe FG T and FO T are nonstandard and depend heavily on

the nuisanceparameters thatde…ne thecross-sectionaldependencyand the heterogeneous serialdependence. T herefore, itis impossible tobase inference on the tests FG T and FO T.

In the nextsection, we propose bootstrap version ofthese tests todealwith the nuisance parameterdependencyproblem andtoovercometheinferentialdi¢culty.

(b) T heF -typetests FG T andFO T consideredherearetwo-tailedtests whichrejectthenull

®i= 0 foralli when ®i6= 0 forsome i. H ence, they rejectthe nullofthe unitroots not only againstthe stationarity ®i< 0 butalso againstthe explosive cases with ®i> 0 for somei. T his willhave anegative e¤ ecton thepowers ofthe tests.

T he frameworkwithin which we may e¤ectively dealwith the problem in R emark(b) abovehas beenrecentlydevelopedbyA ndrews (19 9 9 ).6_{T odealwiththeproblem, wemay}

6_{H ere we consider testing ®}

i= 0 against ®i< 0 , and the parametersetis given by ®i· 0 foreach

cross-sectionaluniti= 1;:::;N. T he value of®iunderthe nullhypothesis is thereforeon theboundaryof

replacezerosforthemembersof^®G T and ^®O T whichhavepositivevalues. T hiscanbeeasily

carriedoutbymultiplyingelementbyelementtheestimators ^®G T = (^®G T;1;:::;^®G T;N )0and

^

®O T = (^®O T;1;:::;^®O T;N)0respectivelytotheN -dimensionalindicatorfunctions1 f^®G T ·0 g

and1 f^®O T ·0 g. D enoteby:¤theelementbyelementmultiplication, andusethistomodify

theestimators ^®G T and ^®O T as follows

^ ®G T:¤1 f^®G T ·0 g = 0 B @ ^ ®G T;11f^®G T;1·0 g ... ^ ®G T;N 1f^®G T;N ·0 g 1 C A (14) ^ ®O T:¤1 f^®O T ·0 g = 0 B @ ^ ®O T;11f^®O T;1·0 g ... ^ ®O T;N 1f^®O T;N ·0 g 1 C A

W enowde…nenewstatistics, whichwecallK-statistics. From themodi…edG L S estimator above, wede…ne the G L S-based K-statistics KG T as follows

KG T = (^®G T:¤1 f^®G T ·0 g)0(var(^®G T))¡1(^®G T:¤1 f^®G T ·0 g)

= (AG T:¤1 f^®G T ·0 g)0B¡1G T(AG T:¤1 f^®G T ·0 g) (15)

andthe O L S-basedK-statistics KO T as

KO T = (^®O T:¤1 f^®O T ·0 g)0(var(^®O T))¡1(^®O T:¤1 f^®O T ·0 g)

= (AO T:¤1 f^®O T ·0 g)0M F O T¡1 (AO T:¤1 f^®O T ·0 g) (16)

T he K-statistics constructed as above are essentiallyone-sided tests, since theye¤ ectively elliminate the probabilityofrejectingthe nullagainstthe explosive alternative. T herefore theyare expected toimprove the powerproperties ofthe correspondingtwo-tailed F -type tests fortestingoftheunitrootnullagainsttheone-waystationaryalternative.

T helimitdistributions oftheK-statistics canbeeasilyobtainedinamannersimilarto thatusedtoderive thelimittheories fortheF -type tests, andaregiven in

Corollary2.1 U nderA ssumptions A 1, A 2 andA 3, wehave

(a) KG T ! d (QAG :¤1 fQ¡1BGQAG ·0 g)0Q¡1BG(QAG :¤1 fQ¡1BGQAG ·0 g) (b) KO T ! d (QAO :¤1 fQ¡1BOQAO ·0 g)0Q¡1M F O(QAO :¤1 fQ¡1BOQAO ·0 g) asT _{! 1 , where} QBO = 0 B B B B B @ ¼1(1 )2 Z1 0 B 2 1 ::: ¼1(1 )¼N(1 ) Z1 0 B1BN ... ... ... ¼N(1 )¼1(1 ) Z1 0 BNB1 ::: ¼N (1 ) 2 Z1 0 B 2 N 1 C C C C C A

andthe terms QAG; QBG; QAO and QM F O arede…ned in T heroem 2.1.

A s can be seen clearlyfrom the above Corollary, the limitdistributions ofthe K-tests are alsononstandard and depend heavilyon the nuisance parameters. In the nextsection, we willalsoconsiderbootstrappingthe K-type tests.

2.2 U nitR ootTests forH omogeneous P anels

Forthe testofthe unitroot, we are testing®i= 0 foralli. T herefore, we are essentially lookingatahomogeneous panel, as faras testingofthenullhypothesis is concerned. IfA R coe¢cients ®i’s in ouroriginalmodel(1) are homogeneous, i.e., ®1 =¢¢¢= ®N = ®, then

thecorrespondingaugmentedA R inmatrixform is given by

4y = y`® + Xp¯p + "p (17 )

whichisthesameastheaugmentedA R inmatrixform fortheoriginalheterogeneousmodel (9 ), exceptthathere we have an (N T £1 )-vectory`= (y`;10 ;:::;y`;N0 )0in the place ofthe (N T <sub>£</sub>N)-matrixY`andtheparameter® is nowascalarinsteadofan(N £1 )-vector.

Itis naturaltoconsiderthe t-statistics fortestingthenullhypothesis ofthe unitroots inthehomogeneous model(17 ), sincetheparameter® tobetestedis nowascalar. H erewe donotallowforthe heterogeneity ofthe A R coe¢cient, as in L evin and L in (19 9 2,19 9 3). O bviously, theunitroottestbasedonthehomogeneous panel(17 ) is valid, sincethemodel is correctlyspeci…ed underthe nullhypothesis ofthe unitroots. T he homogeneous panel, however, maynotprovideappropriatemodellingsunderthealternativehypothesis, andthis may have an adverse e¤ ecton the powerofthe tests. H owever, we mayuse the one-sided t-typetests, ifbasedonthehomogeneous panels, whichhaveaclearaclearadvantageover thetwo-tailedF -typetests constructedfrom the heterogeneous panels.

T heO L S andG L S basedt-statistics areconstructedfrom theG L S andO L S estimators ofthescalarparameter® inthehomogeneous augmentedA R (17 ) andaregiven by

tG T = aG Tb¡1=2G T and tO T = aO TM tO T¡1=2 (18) where aG T = y`0(~§¡1-IT)"p ¡y`0(~§¡1-IT)Xp(X p0(~§¡1-IT)Xp)¡1X p0(~§¡1-IT)"p bG T = y<sub>`</sub>0(~§¡1-IT)y`¡y0`(~§¡1-IT)X p(Xp0(~§¡1-IT)X p)¡1X p0(~§¡1-IT)y` aO T = y<sub>`</sub>0"p ¡y0`Xp(Xp0Xp)¡1Xp0"p M tO T = y`0(~§-IT)y`¡2y0`Xp(Xp0X p)¡1Xp0(~§-IT)y` + y`0X p(Xp0X p)¡1X p0(~§-IT)Xp(Xp0X p)¡1X p0y`

Inthe followingtheorem wepresentthe limittheories forthe tG T and TO T tests.

T heorem 2.2 U nderA ssumptions A 1, A 2 and A 3, wehave (a) tG T ! d QaGQ ¡1=2 bG (b) tO T ! d QaOQ ¡1=2 M tO asT ! 1 , where QaG = N X i= 1 N X j= 1 ¾ij Z1 0 BidBj; QbG = N X i= 1 N X j= 1 ¾ij Z1 0 BiBj

and QaO = N X i= 1 ¼i Z1 0 BidBi; QM tO = N X i= 1 N X j= 1 ¾ij¼i¼j Z1 0 BiBj

T he limitprocesses QM tO appearingin the limitdistributions oftG T and tO T are the

sums ofthe individualelements in the correspondinglimitprocesses QAG, QBG, QAO and

QM F O de…ned in T heorem 2.1, which constitute the statistics KG T and KO T developed for

the heterogenous panels.7 _{T he limitdistributions ofthe t-statistics t}

G T and tO T are also

nonstandard and su¤erfrom nuisance parameterdependency, as in the cases with the F -tests and K-statistics. H ence itis notpossible touse these statistics forinference as they stand. In the nextsection, we considerboostrappingthe panelunitroottests proposed in this section toresolve the nuisance prameterdependency problem and toprovide avalid basis forinferenceinnonstationarypanels withcross-sectionaldependency.

3. B ootstrap U nitR ootTests forD ependentP anels

In this section, we considerthe sieve bootstraps forthe various panelunitroottests, FG T,

FO T, KG T, KO T, tG T andtO T consideredintheprevious section. Inparticular, weestablish

the asymptoticvalidityofthe bootstrapped tests byshowingbootstrap consistencyofthe tests. W eusetheconventionalnotation¤tosignifythebootstrap samples, anduseP¤_and E¤_{todenote, respectively, theprobabilityandexpectationconditionalupontherealization} oftheoriginalsample. W hiledevelopingtheasymptotictheoriesforthebootstrappedtests, wealsodiscuss various issues andproblems arisinginpracticalimplementationofthesieve bootstrap methodologyin this section.

T oconstructthe bootstrapped tests, we …rstgenerate the bootstrap samples for("¤ it), (u¤

it)and (yit¤). Forthe generation of("¤it), we need to make sure that the dependence structure among cross-sectionalunits, i = 1 ;:::;N , is preserved. T o do so, we generate

theN-dimensionalvector("¤_t)= ("¤_1t;:::;"¤Nt)0by resampling from the centered residual vectors (~"p

t)de…nedin(13) from theregression(12). T hatis, obtain("¤t)from theempirical

distribution of _Ã ~ "p_t¡_T1 T X t= 1 ~ "p_t !T t= 1 T he bootstrap samples ("¤

t)constructed as such will, in particular, satisfyE¤"¤t= 0 and E¤_"¤

t"¤t= ~§.8

7 _{L evin and L in (19 9 2,19 9 3) considers t-statistics forhomogeneous panels}

undercross-sectionalindepen-dency. Consequently, theycanapplyN-asymptoticsafterthelimitasT tends toin…nityis taken, andderive the limitdistribution thatis thestandard normal. T heirtheory, however, does notextend toourstatistics, since we allowfordependency across cross-sectionalunits.

8_{O fcourse, we may resample "}¤

it’s individually from the ~"piti’s for i= 1;:::;N and t= 1;:::;T. In

this case, preservingtheoriginalcorrelation structureamongthe cross-sectionalunits needs more care. W e basicallyneedtopre-whiten~"pi

it’sbeforeresampling, andthenre-colortheresamplestorecoverthecorrelation

structure. M ore speci…cally, we …rstpre-whiten ~"pi

it’s by pre-multiplying ~§¡1=2 to ~" p t= (~" p1 1t;:::; ~"pNN t)0, for t= 1;:::;T. N ext, generate "¤

it’s by resampling from the pre-whitened ~"piti’s, and then re-colorthem by

pre-multiplying ~§1=2 _to"¤

N ext, wegenerate(u¤

it)recursivelyfrom ("¤it)as u¤it= ~®pi;1iu¤i;t¡1+ ¢¢¢+ ~®pi;piiu

¤ i;t¡pi+ " ¤ it (19 ) where(~®pi i;1;:::;~® pi

i;pi)arethecoe¢cientestimates from the…ttedregression(12).

Initializa-tionof(u¤

it)is unimportantforoursubsequenttheoreticaldevelopment, thoughitmayplay an importantrole in …nitesamples.9 _{T he coe¢cientestimates (~®}pi

i;1;:::;~®pi;pii)used in (19 )

maybeobtainedfrom estimating(12) bytheY ule-W alkermethodinsteadoftheO L S. T he two methods are asymptotically equivalent. H owever, in smallsamples the Y ule-W alker method may be preferred tothe O L S, since italways yields an invertible autoregression, therebyensuringthe stationarityofthe process (u¤

it). See B rockwelland D avis (19 9 1, Sec-tions 8.1 and 8.2). H owever, the probability ofhavingthe noninvertibility problem in the O L S estimation becomes negligibleas thesamplesizeincreases.

Finally, obtain(y¤

it)bytakingpartialsums of(u¤it), viz.

yit¤= y¤i0 + t X k= 1

u¤ik

with some initialinitialvalue y¤

i0. N otice thatthe bootstrap samples (y¤it)are generated withtheunitrootimposed. T hesamples generatedaccordingtotheunrestrictedregression (1) willnot necessarily have the unit root property, and this willmake the subsequent bootstrap procedureinconsistentas showninB asawaetal(19 9 1). T hechoiceoftheinitial valuey¤

i0 does nota¤ecttheasymptotics as longas itis stochasticallybounded. T herefore, wesimplysetitequaltozeroforthesubsequentanalysis in this section.

W e may obtain the B everidge-N elson representations forthe bootstrapped series (u¤ it) and(y¤

it)similartothosefor(uit)and(yit)givenin(5) and(6) intheprevious section. L et ~

®i(1 )= 1 ¡Ppk= 1i ®~ pi

i;k. T hen itis indeed easytoget u¤_it = 1 ~ ®i(1 ) "¤_it+ pi X k= 1 Ppi j= k®~ pi i;j ~ ®i(1 )

(u¤_i;t¡k¡u¤i;t¡k+ 1) = ~¼i(1 )"¤it+ (¹u¤i;t¡1¡¹u¤it)

where ~¼i(1 )= 1 =~®i(1 )and ¹u¤t= ~¼i(1 )Ppk= 1i ( Ppi

j= k®~ pi

i;j)u¤i;t¡k+ 1, andtherefore, yit¤= t X k= 1 u¤ik= ~¼i(1 )w¤it+ (¹u¤i0 ¡¹u¤it) wherew¤ it= Pt k= 1"¤ik.

Forthedevelopmentofthelimittheories forthebootstrappedteststatistics, weassume

9_{W emayusethe…rstp}

i-valuesof(uit) astheinitialvaluesof(u¤it). T hebootstrap samples(u¤it) generated

as such may notbe stationary processes. A lternatively, we may generate a largernumber, sayT + M , of (u¤

it) and discard …rstM -values of(u¤it). T his willensure that(u¤it) become more stationary. In this case

A ssumption B 1 L et ("t)be a sequence of iid random variables such thatE"t= 0 , E"t"0t= § andEj"tjr < 1 forsomer¸4.

A ssumption B 2 L et¼i(z)6= 0 foralljzj· 1 , andP1k= 0 jkjsj¼i;kj< 1 forsome s ¸ 1 , foralli = 1 ;:::;N .

A ssumption B 3a L etpi! 1 and pi= o(T·)with · < 1 =2 asT ! 1 , foralli = 1 ;:::;N.

A ssumption B 3b L etpi= cn· forsome constantc and 1 =rs < · < 1 =2, foralli = 1 ;:::;N.

T he iid assumption in A ssumption B 1, instead ofthe martingale di¤erence condition in A ssumption A 1, is made tomake the usualbootstrap procedure meaningful. A ssumption B 2 is identicaltoA ssumption A 2. In the place ofA ssumption A 3 forthe expansion rate ofA R orderpi’s, we impose eitherA ssumption B 3aorB 3b. B oth A ssumptions B 3a and B 3b are strongerthan A ssumption A 3. W e willimpose the condition in A ssumption B 3a toprove the consistency ofthe bootstrap tests in the weak form, i.e., the convergence of conditionalbootstrap distributions in probability. T o establish the strongconsistency or thea.s. convergenceofconditionalbootstrap distributions, weneedastrongerconditionin A ssumption B 3b. N otice thatwe only require 0 < · < 1 =2, forthe G aussian modelwith r=1 orthe …nite orderA R M A modelwith s = 1 . T he condition is therefore notvery stringent.

Conventions

(a) A ssumptions B 1, B 2 andB 3atogetherwillberefered toas A ssumption(W ), with‘W ’ standingforweak, andthesetofA ssumptionsB 1, B 2 andB 3b willbecalledasA ssumption (S), with ‘S’forstrong.

(b) W e willuse the symbolo¤

p(1 )tosignify the bootstrap convergence in probability. For a sequence ofbootstrapped random variables Z ¤

n, forinstance, Z n¤= o¤p(1 )a.s. and inP implyrespectivelythat

P¤_fjZ ¤

nj> ±g ! 0 a.s. orinP forany± > 0 . Similarly, we willuse the symbolO¤

p(1 )todenote the bootstrap version of theboundedness inprobability. N eedlesstosay, thede…nitionsofo¤

p(1 )andO¤p(1 )naturally extend too¤

p(cn)and Op¤(cn)forsome nonconstantnumericalsequence(cn).

W e need following lemmas for the derivation ofthe limit distributions for the sieve bootstrap panelunitroottests.

L emma3.1 U nderA ssumptions (W ), wehave (a) 1 T T X t= 1 y_i;t¤_¡1"¤_jt= ~¼i(1 )1 T T X t= 1 w_i;t¤_¡1"¤_jt+ o¤_p(1 ) (b) 1 T2 T X t= 1 y_i;t¤_¡1y_j;t¤_¡1= ~¼i(1 )~¼j(1 )1 T2 T X t= 1 w¤_i;t¡1w¤_j;t¡1+ o¤_p(1 )

then weletkC k= maxxjC xj=jxj. L emma3.2 L etx¤pi

it = (4yi;t¤¡1;:::;4yi;t¤¡pi)0. T henwehave

(a) E¤ ° ° ° ° ° ° Ã 1 T T X t= 1 x¤pi itx¤piti0 !_¡1°_° ° ° °

° = Op(1 ) or O (1 )a:s: underA ssumptions (W ) and (S),

respectively, foralli = 1 ;:::;N . (b) E¤ ¯ ¯ ¯ ¯ ¯ T X t= 1 x¤pi ityj;t¤¡1 ¯ ¯ ¯ ¯ ¯= O (Tp 1=2

i )a:s:underA ssumption (W ), foralli;j = 1 ;:::;N. (c) E¤ ¯ ¯ ¯ ¯ ¯ T X t= 1 x¤pi it"¤jt ¯ ¯ ¯ ¯ ¯= O (T 1=2_p1=2

i )a:s:underA ssumptions (W ), foralli;j = 1 ;:::;N.

3.1 B ootstrap U nitR ootTests forH eterogeneous P anels

T o constructthe bootstrapped tests, we considerthe following bootstrap version ofthe augmentedautoregression (8) whichwas usedtoconstructthe sampleteststatistics

4yit¤= ®iyi;t¤¡1+ pi

X k= 1

®i;k4yi;t¤¡k+ "¤it (20) andwrite this inmatrixform

4y¤= Y_`¤® + X_p¤¯p + "¤ (21) wherethe variables arede…ned in the same manneras in theregression(9 ) with

y`;i¤ = 0 B @ y¤_i;0 ... y¤_i;T_¡1 1 C A; X ¤pi i = 0 B @ x¤pi0 i1 ... x¤pi0 iT 1 C A and "¤i= 0 B @ "¤_i;1 ... "¤_i;T 1 C A fori = 1 ;:::;N .

W e test for the unit root hypothesis ® = 0 in (21), using the bootstrap versions of F -type tests that are de…ned analogously as the sample F -type tests considered in the previous section. T hebootstrap F -tests areconstructedfrom thebootstrap G L S andO L S estimators of® in thebootstrap augmented A R regression (21). M oreexplicitly, we de…ne theboostrap G L S-basedF -testas

FG T¤ = A¤0G TBG T¤¡1A¤G T (22)

analogouslyas thesampleG L S-basedF -testFG T given in (10), where

A¤G T = Y`¤0(~§¡1-IT)"¤¡Y<sub>`</sub>¤0(~§¡1-IT)X_p¤ ³ X_p¤0(~§¡1-IT)X _p¤ ´_¡1 X_p¤0(~§¡1-IT)"¤ BG T¤ = Y`¤0(~§¡1-IT)Y<sub>`</sub>¤¡Y_{`</sub>¤0(~§¡1-IT)X<sub>p</sub>¤ ³ X <sub>p</sub>¤0(~§¡1-IT)X<sub>p</sub>¤ ´<sub>¡1</sub> X <sub>p</sub>¤0(~§¡1-IT)Y<sub>`}¤

T hebootstrap O L S-basedF -testisalsode…nedanalogouslyasthesampleO L S-basedF -test FO T de…nedin(11), viz. FO T¤ = A¤0O TM F O T¤¡1A¤O T (23) where A¤O T = Y`¤0"¤¡Y`¤0X p¤(X p¤0X p¤)¡1X p¤0"¤p M F O T¤ = Y`¤0(~§-IT)Y<sub>`</sub>¤¡Y_{`</sub>¤0X<sub>p</sub>¤(X<sub>p</sub>¤0X <sub>p</sub>¤)¡1X<sub>p</sub>¤0(~§-IT)Y<sub>`}¤ ¡Y`¤0(~§-IT)Xp¤(X p¤0Xp¤)¡1Xp¤0Y`¤ + Y`¤0X p¤(X p¤0Xp¤)¡1X p¤0(~§-IT)Xp¤(Xp¤0Xp¤)¡1X p¤0Y`¤ T he bootstrap F -statistics F ¤

G T and FO T¤ given in (22) and (23) involve the covariance

matrix estimator ~§ de…ned below (13). T he estimate ~§ is the population parameterfor the bootstrap samples ("¤

t), correspondingto § forthe originalsamples ("t). W e may of course use the bootstrap estimate ~§¤, say, forthe construction ofthe statistics F ¤

G T and

F ¤

O T foreachbootstrap iteration. T hetwoversionsofthebootstrap testsareasymptotically

equivalentatleastforthe …rstorderasymptotics, and we use ~§ in the construction ofthe bootsrap tests forconvenience.10

W e nowpresentthelimittheoryforthebootstrap F -typetests F ¤ G T and FO T¤ in T heorem 3.1 W ehaveasT ! 1 , (a) F ¤ G T ! d¤ Q0AGQ ¡1 BGQAG inP ora.s. (b) F ¤ O T ! d¤ Q0AOQ ¡1 M F OQAO inP ora.s.

respectivelyunderA ssumption (W ) or(S), where QAG, QBG, QAO and QM F O are de…ned

inT heorem 2.1.

T he results in P art(a) and (b) above showthatthe bootstrap F -statistics F ¤

G T and FO T¤

have the same limitdistributions as the corresponding sample F -statistics FG T and FO T

given in T heorem 2.1. T his establishes the asymptoticvalidity ofthe boostrap tests F ¤

G T

andF ¤

O T.

T hebootstrap K-statistics areconstructedfrom thebootstrap samplesintheanalogous mannerinwhichthe sampleK-statistics arede…ned in (15) and(16).

KG T¤ = (A¤G T:¤1 f^®¤G T ·0 g)0BG T¤¡1(A¤G T:¤1 f^®G T¤ ·0 g) KO T¤ = (A¤O T:¤1 f^®¤O T ·0 g) 0_M ¤¡1 F O T(A¤O T :¤1 f^®¤O T ·0 g) (24) andtheirlimittheories aregiven in Corollary3.1 W ehaveasT ! 1 , (a) K¤ G T ! d¤ (QAG :¤1 fQ¡1BGQAG ·0 g) 0_Q¡1 BG(QAG :¤1 fQ ¡1 BGQAG ·0 g) inP ora.s. 10_{T hebootstrap testsbasedonthebootstrap estimate ~§}¤_{maybebetterforhigherorderasymptotics, since}

theymorecloselymimicthesamplestatistics thanthebootstrap testsbasedonthepopulationparameter~§ . T he statistics considered in the paperare, however, non-pivotaland thereforethe higherorderasymptotics are irrelevanthere.

(b) K¤ O T ! d¤ (QAO :¤1 fQ¡1BOQAO ·0 g) 0_Q¡1 M F O(QAO :¤1 fQ ¡1 BOQAO ·0 g) in P ora.s.

respectively underA ssumption (W ) or(S), where QAG, QBG, QAO, QM F O and QBO are

de…ned in T heorem 2.1 andCorollary2.1.

Corollary 3.1 shows thatthe bootstrap K-statistics K¤

G T and KO T¤ have the same limiting

distributions as thecorrespondingsampleK-statistics KG T andKG T giveninCorollary2.1,

therebyprovingthe asymptoticvalidityofthebootstrap K-statistics.

3.2 B ootstrap U nitR ootTests forH omogeneous P anels

T hebootstrap t-statisticsarealsoconstructedinananalogousmannerasweconstructedthe sample t-statistics, tG T and tO T, in Section 2.2. T hus, we considerthe homogeneous panel

ofthe bootstrap samples, with ®1 =¢¢¢= ®N = ® imposed, and compute the t-statistics

from the correspondingaugementedA R , whichis writteninmatrixform as

4y¤= y_{`</sub>¤® + X<sub>p</sub>¤¯p + "¤ (25) T he variables appearing in the above regression are de…ned in the same way as in the augmented A R in matrix form forthe bootstrap heterogeneous model(21), exceptthat herewehavean(N T <sub>£1 )-vectory</sub>¤<sub>`}= (y¤0_{`;1</sub>;:::;y¤0<sub>`;N} )0intheplaceofthe(N T _£N )-matrix Y¤

` and theparameter® is nowascalarinsteadofan (N £1 )-vector.

T he bootstrapped G L S and O L S based t-statistics are based on the G L S and O L S estimatorof® in the homogeneous augmented A R (25), andaregiven by

t¤G T = a¤G Tb ¤¡1=2 G T and t¤O T = a¤O TM ¤¡1=2 tO T (26) where a¤G T = y`¤0(~§¡1-IT)"¤¡y¤0`(~§¡1-IT)Xp¤(Xp¤0(~§¡1-IT)X p¤)¡1Xp¤0(~§¡1-IT)"¤ b¤G T = y`¤0(~§¡1-IT)y¤`¡y¤0`(~§¡1-IT)Xp¤(X p¤0(~§¡1-IT)Xp¤)¡1Xp¤0(~§¡1-IT)y`¤ a¤O T = y`¤0"¤¡y`¤Xp¤(Xp¤0X p¤)¡1Xp¤0"¤ M tO T¤ = y`¤0(~§-IT)y¤<sub>`</sub>¡2y¤0_{`</sub>X<sub>p</sub>¤(X<sub>p</sub>¤0X <sub>p</sub>¤)¡1X<sub>p</sub>¤0(~§-IT)y¤<sub>`} + y`¤0X p¤(Xp¤0X p¤)¡1X p¤0(~§-IT)Xp¤(X p¤0X p¤)¡1X p¤0y¤` T helimitdistributions oft¤ G T andt¤O T aregivenin T heorem 3.2 W ehaveasT _{! 1 ,} (a) t¤ G T ! d¤ QaGQ ¡1=2 bG inP ora.s. (b) t¤ O T ! d¤ QaOQ ¡1=2 M tO inP ora.s.

respectivelyunderA ssumption(W ) or(S), whereQaG, QbG, QaO and QM tO are de…nedin

T heorem 2.2.

T heresults inT heorem 3.2 showthatthebootstrap t-statistics t¤

G T andt¤O T havethe limit

distributions thatare equivalentto those ofthe sample t-statistics tG T and tO T given in

4. Simulations

W econductasetofsimulationstoinvestigatethe…nitesampleperformanceofthebootstrap panelunitroottests, F ¤

G T, FO T¤, KG T¤ , KO T¤ , t¤G T and t¤O T, proposed in the paper. Forthe

simulation, weconsiderthe(yt)givenbythemodel(1) with(ut)generatedas eitherA R (1) orM A (1) processes, viz.,

(A ) uit= ½iui;t¡1+ "it (B ) uit= "it+ µi"i;t¡1

T he innovations "t= ("1t;:::;"Nt)0thatgenerate ut= (u1t;:::;uNt)0are drawn from an N -dimensionalmultivariatenormaldistributionwithmeanzeroandcovariancematrix§.11

T he A R and M A coe¢cients, ½i’s and µi’s, used in the generation ofthe errors (uit)are drawnrandomlyfrom theuniform distribution. M orespeci…cally, ½i»U niform[0.2,0.4]and µi»U niform[¡0 :4;0 :4].12

T he parametervalues forthe (N _£N )covariance matrix § = (¾ij)are also randomly drawn, butwith particularattention. T o ensure that§ is a symmetric positive de…nite matrixandtoavoid thenearsingularityproblem, wegenerate§ viafollowingsteps: (1) G eneratean(N _£N)matrixU from U niform[0,1].

(2) Constructfrom U anorthogonalmatrixH = U (U0_{U )}¡1=2_.

(3) G enerate a set ofN eigenvalues, ¸1;:::;¸N . L et ¸1=r> 0 and ¸N = 1 and draw

¸2;:::;¸N_¡1 from U niform[r,1].

(4) Form adiagonalmatrix¤with(¸1;:::;¸N)on the diagonal.

(5) Constructthecovariance matrix§as aspectralrepresentation§= H¤H0_.

T hecovariancematrixconstructedthis waywillsurelybesymmetricandnonsingularwith eigenvalues taking values from r to 1. W e set the maximum eigenvalue at 1 since the scale does notmatter. T he ratioofthe minimum eigenvalue tothe maximum is therefore determinedbythesameparameterr. T hecovariancematrixbecomessingularasrtends to zero, andbecomes sphericalasrapproachesto1. Forthesimulations, wesetratr= 0 :1 .13

Forthetestoftheunitroothypothesis, weset®i=0 foralli=1 ;:::;N, andinvestigate the …nite sample sizes in relation tothe correspondingnominaltestsizes. T oexamine the rejection probabilities ofthe tests underthe alternative ofstationarity, we generate ®i’s randomly from U niform[¡0 :8;0 ]. T he modelis thus heterogenous underthe alternative. T he …nite sample performance ofthe bootstrap tests are compared with thatofthe t-bar statistics by Im, P esaran and Shin (19 9 7 ), which is based on the average ofthe individ-ualt-statistics computed from the sample A D F regressions (8) with mean and variance

11_{T hesimulationmodelforthecase(B ) isgeneratedfrom anM A (1) process(u}

it), whichcanberepresented

as an in…niteorderA R process. U singthelagorderpiselected byA IC, weapproximate(uit) byan A R (pi)

process as in (12). T he approximated autoregression is then estimated bythe Y ule-W alkermethod.

12_{M addalaandW u(19 9 6) andIm, P esaranandShin(19 9 7 ) alsogenerateparameters fortheirsimulation}

models radomlyfrom uniform distributions.

13_{O urbootstrap tests donotseem todepend on the the value ofr, butthet-barstatistics does. T hough}

wedonotreportthedetails, weobservefrom asetofsimulationsthatthet-bartendstohavehigherrejection probabilities whenr is closeto0, andthatitseemstohavesubstantialsizedistortions evenwhen§ isnearly sphericalwith r = 0 :99 .

modi…cations. M oreexplicitly, thet-barstatistics is de…nedas t-bar= p N (¹tN ¡N¡1 PN i= 1E(ti)) q N ¡1PN i= 1var(ti)

where tiis the t-statistics fortesting®i=0 forthe i-th sample A D F regression (8), and ¹

tN =N¡1

PN

i= 1ti. T he values ofthe expectation and variance, E(ti)and var(ti), foreach individualtidepend on T and the lagorderpi, and computed viasimulations from inde-pendentnomalsamples. T able 2 in Im, P esaran and Shin (19 9 7 ) tabulates the values of E(ti)and var(ti)forT= 5;1 0 ;1 5;20 ;25;30 ;40 ;50 ;60 ;70 ;1 0 0 and forpi= 1 ;:::;8.

T hepanels withthecross-sectionaldimensionsN = 5;20 andthetimeseries dimensions

T= 50 ;1 0 0 are considered forthe 1% , 5% and 10% size tests. Since we are usingrandom parametervalues, we simulate 20 times and reportthe ranges ofthe …nite sample perfor-mances ofthe bootstrap tests. Each simulation run is carried outwith 1,000 simulation iterations, eachofwhichuses bootstrap criticalvalues computedfrom 500 bootstrap repeti-tions. T hesimulationresults forthet-barstatistics andourbootstrap testsF ¤

G T, FO T¤, KG T¤ ,

KO T¤ , t¤G T andt¤O T arereportedinT ablesA 1-B 2. T ablesA 1 andA 2 reports, respectively, the

…nite sample sizes and powers ofthe tests forCase A with A R errors, and T ables B 1 and B 2 reports those forCase B with M A errors. Foreach statistics, we reportthe minimum, mean, median and maximum ofthe rejection probabilities underthe nulland underthe alternativehypothesis.

A scanbeseenfrom T ablesA 1 andB 1, thet-bartestsu¤ersfrom serioussizedistortions. T hedirectionofthesizedistortions is, however, notinoneway. Forthe1% tests, thet-bar statistics su¤ers from upwardsizedistortions exceptfortheM A casewithN= 5, wherethe

t-baris slightlydownwardbiased. T hedegreeoftheupwarddistortions seems tobehigher forthe A R case and increases asN gets large. Forthe 5% and 10% tests, the t-bartest

is mostly downward biased exceptforthe 5% testwithN = 20, where the testis upward

biased.14 _{T hedownwarddistortionis moreserious fortheM A casewithsmallerN = 5. O n} theotherhand, the…nitesamplesizes ofthebootstrap tests arequiteclosetothenominal testsizes forbothA R and M A cases and forallN= 5,20 andT= 50,100.

T hebootstrap tests are morepowerfulthan thet-barstatistics formostcases withthe smallerN = 5, ascanbeseenfrom T ablesA 2 andB 2. Indeed, forthe5% and10% testsallof

ourbootstrap tests havehigherrejectionprobabilities thanthet-barforbothA R andM A cases. For1% tests, onlytheG L S basedbootstrap tests F ¤

G T andKG T¤ perform betterthan

the t-bar. A s the numberofthe cross-sectionalunits increases toN = 20, the performance

ofthe t-barstatistics improves. W ith thesmallernumberofobservations overtimeT= 50,

itactuallyperforms betterthan the bootstrap tests excepttheO L S based t-statistics t¤

O T,

butthe di¤ erencebecomes negligible asT increases.

A mongthebootstrap tests, theG L S basedtests, F ¤

G T andKG T¤ , aremorepowerfulthan

the O L S based tests, F ¤

O T and KO T¤ , forthe smallerN = 5, butforthe largerN = 20, the

advantage from the G L S e¢ciency vanishes. T his is perhaps due tothe errorinvolved in

14_{T he downward size distortions ofthe t-barstatistics have been wellnoted in severalsimulation works.}

M addalaandW u(19 9 6), forexample, reportthatthet-barstatistics su¤ ers from substantialdownwardsize distortions in the presence ofcross-correlations amongthe cross-sectionalunits.

estimatinglargedimensionalcovariancematrix. Fort-typetests, theO L S basedt-statistics t¤O T is indeednoticeablymorepowerfulthanits G L S couterpartt¤G T whenthelargerN= 20

is used. T hey are alsomore powerfulthan the F -type tests and K-statistics in this case. T he advantage ofthe one-tailtests based on the homogeneous panels appears tobe quite importantin …nitesamples.

T heK-statistics was proposedas alternatives tothetwo-sidedF -typetests tocomeup with more powerfultests forthe unitroots againstthe one-wayalternative ofthe station-arity. T he simulation results in T ables A 2 and B 2, however, showthatthe improvement theK-statistics makeovertheF -typetests arenotnoticeable. O nepossiblereasonis that the …nite sample distributions ofthe ^®G T and ^®O T, upon which the modi…cations forthe

K-statistics aremade, areskewtotheleftsomuchthatthemodi…cationdoes nothaveac-tuale¤ect. T hus, onemaycorrectforthebiases inthedistributions of^®G T and ^®O T before

applyingthe modi…cations in (14). T his can be done by carryingoutanested bootstrap. W edonotpursuethis inthispaperduetothecomputationtime, butwillreportinafuture work.

A llbootstrap tests aremorepowerfulforthecasewiththesmallerN = 5 andthelarger T= 100 than the cases with the largerN= 20 and the smallerT= 50. T his is because our

bootstrap tests areT-asymptotictests, whichwillworkbetterforalargeT. T het-bartest

is, however, noticeablymorepowerfulforthecaseswithN= 20 andT= 50 thanforthecases

withN = 5 andT= 100. T his indicates thatthet-bartestworksmuchbetterforpanels with

largernumberofN , which is expected since the testis based on the average ofindividual

tests.

4. Conclusion

T here has been much recentempiricaland theoreticaleconometric work on models with nonstationarypaneldata. In particular, muchattentionhas beenpaidtothedevelopment and implementation ofthe panelunitroottests which have been used frequently to test forvarious covergence theories, such as growth covergence theories and purchasingpower parity hypothesis. A variety oftests have been proposed, includingthe tests proposed by L evin and L in (19 9 3) and Im, P esaran and Shin (19 9 7 ) thatappeartobe mostcommonly used. A lltheexistingtests, however, assumetheindependenceacross cross-sectionalunits, whichis quiterestrictive. Cross-sectionaldependencyseems indeedquiteapparentformost ofinterestingpaneldata.

In the paper, we investigate various unitroottests forpanelmodels which explicitly allow forthe cross-correlation across cross-sectionalunits as wellas heterogeneous serial dependence. T he limittheories forthe panelunitroottests are derived by passing the numberoftime series observationsT to in…nity with the numberofcross-sectionalunits N …xed. A s expected the limit distributions of the tests are nonstandard and depend

heavily on the nuisance parameters, renderingthe standard inferentialprocedure invalid. T oovercome the inferentialdi¢culty ofthe panelunitroottests in the presence ofcross-sectional dependency, we propose to use the bootstrap method. L imit theories for the bootstrap tests are developed, andin particulartheirasymptoticvalidityis established by

proving the consistency ofthe boostrap tests. T he simulations showthatthe bootstrap panelunitroottests perform wellin …nite samples relative tothe t-barstatistics by Im, P esaranand Shin(19 9 7 ).

5. A ppendix: M athematicalP roofs

T hefollowinglemmas provideasymptoticresults forthesamplemoments appearinginthe sampleteststatistics FG T, FO T, KG T, KO T, tG T andtO T de…nedin(10), (11), (15), (16) and

(18).

L emmaA 1 U nderA ssumptions A 1, A 2 andA 3, wehave (a) 1 T N X t= 1 yi;t¡1"pjtj= ¼i(1 )1 T T X t= 1 wi;t¡1"jt+ op(1 ), foralli;j = 1 ;:::;N (b) 1 T2 T X t= 1 yi;t¡1yj;t¡1= ¼i(1 )¼j(1 )1 T2 T X t= 1 wi;t¡1wj;t¡1+ op(1 ), foralli;j = 1 ;:::;N (c) 1 T T X t= 1 "p_t"p_t0= 1 T T X t= 1 "t"0t+ op(1 ) P roofofL emmaA 1

P art(a) T he stated results followimmediatelyifweapplytheresults in L emmaA 1 (a) ofChangandP ark(19 9 9 ) toeach (i;j)pair, fori;j = 1 ;:::;N .

P art(b) T hestatedresultfollows directlyfrom P hillips and Solo(19 9 2). P art(c) L et QT = 1 T T X t= 1 "p_t"p_t0¡1 T T X t= 1 "t"0t

T hen foreach(i;j)-elementofQ , the followingholds QT;ij = 1 T T X t= 1 "pi it" pj jt¡ 1 T T X t= 1 "it"jt = 1 T T X t= 1 ("pi it¡"it)"pjtj+ 1 T T X t= 1 "it("pjtj¡"j;t) = op(p¡si )+ op(p¡sj )

foralli;j = 1 ;:::;N, due to L emma A 1 (c) in Chang(19 9 9 ). N owthe stated resultis

immediate.

L emmaA 2 U nderA ssumptions A 1, A 2 andA 3, wehave (a) ° °° ° ° ° Ã 1 T T X t= 1 xpi itx pi0 it !_¡1°_°° ° ° ° = Op(1 ), forallpiandi = 1 ;:::;N (b) ¯ ¯ ¯ ¯ ¯ T X t= 1 xpi ityj;t¡1 ¯ ¯ ¯ ¯ ¯= Op(Tp 1=2 i ), foralli;j = 1 ;:::;N (c) ¯ ¯ ¯ ¯ ¯ T X t= 1 xpi it" pj jt ¯ ¯ ¯ ¯ ¯= Op(T 1=2_p1=2 i )+ op(Tp1=2i p¡sj ), foralli;j = 1 ;:::;N.

P roofofL emmaA 2 T hestatedresultinP art(a) follows directlyfrom theapplication ofthe resultin L emma A 2 (a) foreach i = 1 ;:::;N, and those in P arts (b) and (c) are

easily obtained usingthe results in L emmaA 2 (b) and (c) ofChangand P ark(19 9 9 ) for each(i;j)pairfori;j = 1 ;:::;N , withsomeobviousmodi…cationwithrespecttotheorders

pi’s ofthe A R approximations involved. P roofofT heorem 2.1

P art(a) W e begin by writingoutexplicitly the componentsample moments appearing inAG T and BG T de…ned below(11).

Y`0(~§¡1-IT)Y` = 0 B @ y_{`;1</sub>0 0 ... 0 y0<sub>`;}N 1 C A 0 B @ ~ ¾11_I T ¢¢¢ ~¾1NIT ... ... ... ~ ¾11IT ¢¢¢ ~¾1NIT 1 C A 0 B @ y`;1 0 ... 0 y`;N 1 C A = 0 B B B B B B B B @ ~ ¾11 T X t= 1 y1;t2 ¡1 ¢¢¢ ~¾1N T X t= 1 y1;t¡1yN;t¡1 ... ... ... ~ ¾N1 T X t= 1 yN;t¡1y1;t¡1 ¢¢¢ ¾~N N T X t= 1 yN2;t¡1 1 C C C C C C C C A (27 ) and Xp0(~§¡1-IT)Y` = 0 B @ X p10 1 0 ... 0 X pN0 N 1 C A 0 B @ ~ ¾11IT ¢¢¢ ~¾1NIT ... ... ... ~ ¾11<sub>I</sub> T ¢¢¢ ~¾1NIT 1 C A 0 B @ y`;1 0 ... 0 y`;N 1 C A = 0 B B B B B B B B @ ~ ¾11 T X t= 1 xp1 1ty1;t¡1 ¢¢¢ ~¾1N T X t= 1 xp1 1tyN;t¡1 ... ... ... ~ ¾N1 T X t= 1 xpN Nty1;t¡1 ¢¢¢ ~¾N N T X t= 1 xpN NtyN;t¡1 1 C C C C C C C C A (28)

where ~¾ij_{denotes (i;j)-elementofthe inverse covariance matrix estimate ~§}¡1_{. Similarly,} wehave X_p0(~§¡1-IT)"p = 0 B B B B B B BB @ ~ ¾11 T X t= 1 xp1 1t"p1t1 + ¢¢¢ + ~¾1N T X t= 1 xp1 1t"pNNt ... ... ... ~ ¾N1 T X t= 1 xpN Nt"p1t1 + ¢¢¢ + ~¾N N T X t= 1 xpN Nt"pNNt 1 C C C C C C CC A

= 0 B B B B B B B B @ N X j= 1 ~ ¾1j T X t= 1 xp1 1t" pj jt ... N X j= 1 ~ ¾Nj T X t= 1 xpN Nt" pj jt 1 C C C C C C C C A (29 ) Y_`0(~§¡1-IT)"p = 0 B B B B B B B B @ ~ ¾11 T X t= 1 y1;t¡1"p1t1 + ¢¢¢ + ~¾1N T X t= 1 y1;t¡1"pNNt ... ... ... ~ ¾N1 T X t= 1 yN;t¡1"p1t1 + ¢¢¢ + ~¾N N T X t= 1 yN;t¡1"pNNt 1 C C C C C C C C A = 0 B B B B B B B B @ N X j= 1 ~ ¾1j T X t= 1 y1;t¡1"pjtj ... N X j= 1 ~ ¾Nj T X t= 1 yN;t¡1"pjtj 1 C C C C C C C C A

W e nowexamine the stochasticorders ofthe components included in AG T and BG T. L et

¸(¢)denoteeigenvalues ofamatrix. W ehave

¸min(~§¡1-IT)X p0X p ·Xp0(~§¡1-IT)Xp

N otice that¸min(~§¡1-IT)= ¸min(~§¡1)and ¸min(~§¡1)= 1 =¸max(~§). T hen wehave

à X p0(~§¡1-IT)Xp T !¡1 · ¸max(~§) à Xp0X p T !_¡1 = Op(1 ) (30)

since¸max(~§)! p ¸max(§)< 1 , and à Xp0X p T !_¡1 = 0 B BB B B B B @ à 1 T T X t= 1 xp1 1tx p10 1t !¡1 0 ... 0 à 1 T T X t= 1 xpN NtxpNNt0 !_¡1 1 C CC C C C C A = Op(1 ) (31)

due toL emmaA 2 (a). M oreoveritfollows from L emmaA 2 (b) and(28) that

Xp0(~§¡1-IT)Y`= Op(Tp¹1=2) (32) where ¹p = max

1·i·N pi, and from L emmaA 2 (c) and(29 ) that

where p = min

1·i·N pi, as de…ned earlier. N otice that ¹p = p = o(T

1=2_{)asT ! 1 under} A ssumption 3.

Itfollows from (30), (32) and(33) that

¯ ¯ ¯ ¯Y`0(~§¡1-IT)Xp ³ X p0(~§¡1-IT)Xp ´<sub>¡1</sub> Xp0(~§¡1-IT)"p ¯ ¯ ¯ ¯ · ¯¯¯Y`0(~§¡1-IT)Xp ¯ ¯ ¯°°°_°³X p0(~§¡1-IT)Xp ´_¡1°° ° ° ¯ ¯ ¯X p0(~§¡1-IT)"p ¯ ¯ ¯ = op(Tpp¹¡s)+ Op(T1=2p)¹ whichimplies AG T T = Y0 `(~§¡1-IT)"p T + op(1 )= QAG T + op(1 ) (34)

due toL emmaA 1 (a), where

QAG T = 0 B B B B B B B B @ N X j= 1 ~ ¾1j¼1(1 )1 T T X t= 1 w1;t¡1"jt ... N X j= 1 ~ ¾Nj_¼ N (1 ) 1 T T X t= 1 wN;t¡1"jt 1 C C C C C C C C A + op(1 )

M oreover, we have from (30) and(32) that

¯ ¯ ¯ ¯Y`0(~§¡1-IT)Xp ³ X p0(~§¡1-IT)Xp ´<sub>¡1</sub> Xp0(~§¡1-IT)Y` ¯ ¯ ¯ ¯ · ¯¯¯Y<sub>`</sub>0(~§¡1-IT)Xp ¯ ¯ ¯ ° ° ° ° ³ X <sub>p</sub>0(~§¡1-IT)Xp ´<sub>¡1</sub>°<sub>°</sub> ° ° ¯ ¯ ¯X<sub>p</sub>0(~§¡1-IT)Y` ¯ ¯ ¯ = Op(Tp)¹

which, togetherwith L emmaA 1 (b) and (27 ), gives BG T T2 = Y<sub>`</sub>0(~§¡1-IT)Y` T2 + op(1 )= QBG T + op(1 ) (35) where QBG T = 0 B B B B B B B B @ ~ ¾11¼1(1 )2 1 T2 T X t= 1 w1;t2 ¡1 ¢¢¢ ~¾1N ¼1(1 )¼N (1 ) 1 T2 T X t= 1 w1;t¡1wN;t¡1 ... ... ... ~ ¾N1_¼ N (1 )¼1(1 )1 T2 T X t= 1 wN;t¡1w1;t¡1 ¢¢¢ ¾~N N ¼N (1 )2 1 T2 T X t= 1 w2N;t¡1 1 C C C C C C C C A

U singthe asymptoticresults in(34) and(35, wewrite FG T = µ_A G T T ¶₀µ_B G T T2 ¶_¡1µ_A G T T ¶ = Q0AG TQ ¡1 BG TQAG T + op(1 )

T hen the limitdistribution ofFG T follows immediatelyfrom theinvarianceprinciplegiven

in(4).

P art(b) W ehavefrom L emmaA 2 (b) and(c) that

X_p0Y` = 0 B B B B B B @ T X t= 1 xp1 1ty1;t¡1 0 ... 0 T X t= 1 xpN NtyN t;t¡1 1 C C C C C C A = Op(Tp¹1=2) (36) X p0"p = 0 B B B B B B @ T X t= 1 xp1 1t"p1t1 ... T X t= 1 xpN Nt" pN N tt 1 C C C C C C A = Op(T1=2p¹1=2)+ op(Tp¹1=2p¡s) (37 )

T hese togetherwith(31) give

¯ ¯ ¯Y`0Xp(X p0X p)¡1X p0"p ¯ ¯ ¯·¯¯Y`0Xp ¯ ¯°°_°(X 0 pXp)¡1 ° ° ° ¯¯¯Xp0"p ¯ ¯ ¯= op(Tpp¹¡s)+ Op(T1=2p)¹ whichinturngives AO T T = Y_`0"p T + op(1 )= QAO T + op(1 ) (38)

due toL emmaA 1 (a), where

QAO T = 0 B B B B B BB B @ ¼1(1 )1 T T X t= 1 w1;t¡1"1t ... ¼N (1 ) 1 T T X t= 1 wN;t¡1"Nt 1 C C C C C CC C A

W e have from (30) that

Xp0(~§-IT)Xp · ¸_{max(~§ )}(Xp0X p)= Op(T ) (39 ) W ealsohavefrom L emmaA 2 (b) that

X _p0(~§-IT)Y`= 0 B B B B B B BB @ ~ ¾11 T X t= 1 xp1 1ty1;t¡1 ¢¢¢ ~¾1N T X t= 1 xp1 1tyN;t¡1 ... ... ... ~ ¾N1 T X t= 1 xpN Nty1;t¡1 ¢¢¢ ~¾N N T X t= 1 xpN NtyN;t¡1 1 C C C C C C CC A = Op(Tp¹1=2) (40)

where ~¾ijdenotes (i;j)-elementofthe covariance matrixestimate ~§. T hen we have ¯ ¯ ¯Y<sub>`</sub>0Xp(X p0X p)¡1Xp0(~§-IT)Y` ¯ ¯ ¯= Op(Tp)¹ and _¯ ¯ ¯Y`0X p(Xp0X p)¡1X p0(~§-IT)Xp(Xp0X p)¡1X p0Y` ¯ ¯ ¯= Op(Tp)¹ whichthen give M F O T T2 = Y0 `(~§-IT)Y` T2 + op(1 )= QM F O T + op(1 ) (41)

due toL emmaA 1 (b), where

QM F O T = 0 B B B BB B B B @ ~ ¾11¼1(1 )2 1 T2 T X t= 1 w21;t¡1 ¢¢¢ ~¾1N ¼1(1 )¼N (1 ) 1 T2 T X t= 1 w1;t¡1wN;t¡1 ... ... ... ~ ¾N1¼N (1 )¼1(1 )1 T2 T X t= 1 wN;t¡1w1;t¡1 ¢¢¢ ¾~N N¼N (1 )2 1 T2 T X t= 1 w2N;t¡1 1 C C C CC C C C A

W e nowhavefrom theresults in (38) and (41) that FO T = µ_A O T T ¶0µ_M F O T T2 ¶¡1µ_A O T T ¶ = Q0AO TQ ¡1 M F O TQAO T + op(1 )

from whichthe statedresultfollows immediately. P roofofCorollary2.1

P art(a) Itfollows from (34) and (35) that

T®^G T = µ_B G T T2 ¶¡1µ_A G T T ¶ = Q¡1_{BG T}QAG T + op(1 ) whichimplies 1 T ³ AG T:¤1 f^®G T ·0 g ´ = µ_A G T T :¤1 ½_®^ G T T ·0 ¾¶ = µ_A G T T :¤1 fT®^G T ·0 g ¶ = ³QAG T:¤1 n Q¡1_{BG T}QAG T ·0 o´ + op(1 ) D uetotheaboveresultand (35), wemaywritetheKG T statistics given in (15) as

KG T = µ₁ T ³ AG T:¤1 f^®G T ·0 g ´¶0µ_B G T T2 ¶_¡1µ₁ T ³ AG T:¤1 f^®G T ·0 g ´¶ = ³QAG T :¤1 n Q¡1_{BG T}QAG T ·0 o´₀ Q¡1_{BG T} ³QAG T:¤1 n Q¡1_{BG T}QAG T ·0 o´ + op(1 )

N owthe statedresultfollows immediatelyfrom (4). P art(b) From (31) and(36), wehave

¯ ¯ ¯Y<sub>`</sub>0X p(Xp0Xp)¡1Xp0Y` ¯ ¯ ¯= Op(Tp)¹ whichtogetherwithL emmaA 1 (b) gives

BO T T2 = Y<sub>`</sub>0Y` T2 + op(1 )= QBO T + op(1 ) where QBO T = 0 B B B B BB B B @ ¼1(1 )2 1 T2 T X t= 1 w2_1;t¡1 ¢¢¢ ¼1(1 )¼N(1 ) 1 T2 T X t= 1 w1;t¡1wN;t¡1 ... ... ... ¼N (1 )¼1(1 ) 1 T2 T X t= 1 wN;t¡1w1;t¡1 ¢¢¢ ¼N(1 )2 1 T2 T X t= 1 w2N;t¡1 1 C C C C CC C C A

Itfollows from (38) andtheaboveresultthat

T®^O T = µ_B O T T2 ¶_¡1µ_A O T T ¶ = Q¡1_{BO T}QAO T + op(1 ) and 1 T ³ AO T:¤1 f^®O T ·0 g ´ =³QAO T :¤1 n Q¡1_{BO T}QAO T ·0 o´ + op(1 ) From this andthe resultin(41), we mayexpress thestatistics KO T given in (16) as

KO T = µ₁ T ³ AO T:¤1 f^®O T ·0 g ´¶0µ_M F O T T2 ¶¡1µ₁ T ³ AO T:¤1 f^®O T ·0 g ´¶ = ³QAO T :¤1 n Q¡1_{BO T}QAO T ·0 o´0 Q¡1_{M F O T} ³QAO T :¤1 n Q¡1_{BO T}QAO T ·0 o´ + op(1 ) whichis requiredforthe statedresult.

P roofofT heorem 2.2 T he limittheories forthe G L S and O L S based t-statistics tG T

andtO T de…nedin(18) canbederivedinthesimilarmanneras wedidfortheF -typetests

FG T andFO T intheproofofT heorem 2.1. W ejusthavetotakeintoaccountthatthelagged

levelvariables comeina(N T <sub>£1 )-vectory`</sub>instead ofthe(N T <sub>£</sub>N )-matrix Y`. P art(a) Since X _p0(~§¡1-IT)y` = 0 B B B B BB B B @ N X j= 1 ~ ¾1j T X t= 1 xp1 1t" pj jt ... N X j= 1 ~ ¾Nj T X t= 1 xpN Nt" pj jt 1 C C C C CC C C A = Op(Tp¹1=2)

due toL emmaA 2 (b), itfollows from (30) and(33) that ¯ ¯ ¯ ¯y0`(~§¡1-IT)Xp ³ X <sub>p</sub>0(~§¡1-IT)Xp ´¡1 X<sub>p</sub>0(~§¡1-IT)"p ¯ ¯ ¯ ¯= op(Tpp¹¡s)+ Op(T1=2p)¹ and <sub>¯</sub> ¯ ¯ ¯y0`(~§¡1-IT)Xp ³ X_p0(~§¡1-IT)X p ´¡1 X _p0(~§¡1-IT)y` ¯ ¯ ¯ ¯= Op(Tp)¹

N ext, wewriteoutthefollowingsamplemoments appearinginaG T andbG T, de…nedbelow

(18): y<sub>`</sub>0(~§¡1-IT)y` = N X i= 1 N X j= 1 ~ ¾ij T X t= 1 yi;t¡1yj;t¡1 y_`0(~§¡1-IT)"p = N X i= 1 N X j= 1 ~ ¾ij T X t= 1 yi;t¡1"pjtj

T hen from theaboveresults andL emmaA 1 (a) and (b), itfollows that aG T T = y0_{`</sub>(~§¡1-IT)"p T + op(1 ) = N X i= 1 N X j= 1 ~ ¾ij1 T T X t= 1 yi;t¡1"pjtj+ op(1 ) = QaG T + op(1 ) bG T T2 = y0<sub>`}(~§¡1-IT)y` T2 + op(1 ) = N X i= 1 N X j= 1 ~ ¾ij1 T2 T X t= 1 yi;t¡1yj;t¡1+ op(1 ) = QbG T + op(1 ) where QaG T = N X i= 1 N X j= 1 ~ ¾ij¼i(1 )1 T T X t= 1 wi;t¡1"jt QbG T = N X i= 1 N X j= 1 ~ ¾ij¼i(1 )¼j(1 )1 T2 T X t= 1 wi;t¡1wj;t¡1

W emaynowwrite tG T de…ned in (18) as follows

tG T = aG T T µ_b G T T2 ¶¡1=2 = QaG TQ ¡1=2 bG T + op(1 )

and the limittheory fortG T is directly obtained from applyingthe invariance principle in

(4) toQaG T andQbG T.

P art(b) A gain, we…rstanalyzethecomponents aO T andM tO T, de…nedbelow(18), that

constitutethe O L S basedt-statistics tO T givenin (18). Since

X_p0y` = 0 B B B BB B @ T X t= 1 xp1 1ty1;t¡1 ... T X t= 1 xpN NtyN;t¡1 1 C C C CC C A = Op(Tp¹1=2)

X _p0(~§-IT)y` = 0 B B B B B B B B @ N X j= 1 ~ ¾1j T X t= 1 xp1 1tyj;t¡1 ... N X j= 1 ~ ¾Nj T X t= 1 xpN Ntyj;t¡1 1 C C C C C C C C A = Op(Tp¹1=2)

byL emmaA 2 (b), we havefrom (39 ) that

¯ ¯ ¯Y_{`</sub>0X p(X p0X p)¡1X p0"p ¯ ¯ ¯ = op(Tpp¹¡s)+ Op(T1=2p)¹ ¯ ¯ ¯Y<sub>`}0X p(Xp0Xp)¡1Xp0(~§-IT)Y` ¯ ¯ ¯ = Op(Tp)¹ ¯ ¯ ¯Y<sub>`</sub>0Xp(Xp0X p)¡1X p0(~§-IT)X p(Xp0X p)¡1X p0Y` ¯ ¯ ¯ = Op(Tp)¹ W enowdeducefrom L emmaA 1 (a) and (b) that

aO T T = y_{`</sub>0"p T + op(1 )= N X i= 1 1 T T X t= 1 yi;t¡1"piti+ op(1 )= QaO T + op(1 ) M tO T T2 = y<sub>`}0(~§-IT)y` T2 + op(1 )= N X i= 1 N X j= 1 ~ ¾ij1 T2 T X t= 1 yi;t¡1yj;t¡1+ op(1 )= QM tO T + op(1 ) where QaO T = N X i= 1 ¼i(1 )1 T T X t= 1 wi;t¡1"it QM tO T = N X i= 1 N X j= 1 ~ ¾ij¼i(1 )¼j(1 )1 T2 T X t= 1 wi;t¡1wj;t¡1 T hen we have tO T = aO T T µ_M tO T T2 ¶_¡1=2 = QaO TQ ¡1=2 M tO T + op(1 )

from whichthe statedresultfollows immediately. P roofs forthe B ootstrap A symptotics

P roofofL emma3.1 T hestatedresults inparts (a)–(c) followfrom L emma1 ofChang andP ark(19 9 ).

P roofofL emma3.2 SeeP roofofL emma2 inChangandP ark(19 9 9 ). P roofofT heorem 3.1 P art(a) From à Xp¤0(~§¡1-IT)Xp¤ T !¡1 · ¸max(~§) à X p¤0Xp¤ T !_¡1 = Op¤(1 ) (42)

andthe results inL emma2 (a)–(c), wehave ¯ ¯ ¯ ¯Y`¤0(~§¡1-IT)Xp¤ ³ X p¤0(~§¡1-IT)Xp¤ ´<sub>¡1</sub> X p¤0(~§¡1-IT)"¤ ¯ ¯ ¯ ¯ · ¯¯¯Y`¤0(~§¡1-IT)X p¤ ¯ ¯ ¯°°°_°³Xp¤0(~§¡1-IT)Xp¤ ´_¡1°° ° ° ¯ ¯ ¯X p¤0(~§¡1-IT)"¤ ¯ ¯ ¯ = O_p¤(T1=2p)¹

T his togetherwith L emma1(b) implies that A¤G T T = Y ¤0 `(~§¡1-IT)"¤+ o¤p(1 )= QA¤ G T + o ¤ p(1 ) (43)

inP ora.s. underA ssumption(W ) or(S), where

QA¤ G T = 0 B B B B BB B B @ N X j= 1 ~ ¾1j¼~1(1 )1 T T X t= 1 w_1;t¤_¡1"¤_jt ... N X j= 1 ~ ¾Nj_¼~ N (1 ) 1 T T X t= 1 w¤N;t¡1"¤jt 1 C C C C CC C C A

Similarly, we havefrom (42), L emma2 (a) and(b) that

¯ ¯ ¯ ¯Y`¤0(~§¡1-IT)Xp¤ ³ X p¤0(~§¡1-IT)Xp¤ ´<sub>¡1</sub> X p¤0(~§¡1-IT)Y`¤ ¯ ¯ ¯ ¯ · ¯¯¯Y_{`</sub>¤0(~§¡1-IT)X <sub>p</sub>¤ ¯ ¯ ¯ ° ° ° ° ³ X <sub>p</sub>¤0(~§¡1-IT)X <sub>p</sub>¤ ´<sub>¡1</sub>°<sub>°</sub> ° ° ¯ ¯ ¯X<sub>p</sub>¤0(~§¡1-IT)Y<sub>`}¤ ¯ ¯ ¯ = Op¤(Tp)¹

andthis alongwith L emma1 (a) gives B¤G T

T2 = Y ¤0

`(~§¡1-IT)Y<sub>`</sub>¤+ o¤_p(1 )= QB¤_{G T} + o¤p(1 ) (44) inP ora.s. underA ssumption(W ) or(S), where

QB¤ G T = 0 B B B B B B B B @ ~ ¾11~¼1(1 )2 1 T2 T X t= 1 w_1;t¤2_¡1 ¢¢¢ ~¾1N _¼~ 1(1 )~¼N (1 ) 1 T2 T X t= 1 w¤_1;t¡1wN¤;t¡1 ... ... ... ~ ¾N1_¼~ N (1 )~¼1(1 )1 T2 T X t= 1 wN¤;t¡1w¤1;t¡1 ¢¢¢ ¾~N N ¼~N (1 )2 1 T2 T X t= 1 w¤2N;t¡1 1 C C C C C C C C A

inP ora.s. underA ssumption(W ) or(S), analogouslyas before. W e nowwrite thebootstrappedstatisticF ¤ G T as FG T¤ = µ_A¤ G T T ¶0µ_B¤ G T T2 ¶¡1µ_A¤ G T T ¶ = Q0A¤ G TQ ¡1 B¤ G TQA¤G T + o ¤ p(1 )

due to(43) and (44). Itis showninP ark(19 9 9 ) that ~ ¼i(1 )! a:s:¼i(1 ) (45) and 1 T T X t= 1 w_i;t¤_¡1"¤_jt! d¤ Z1 0 BidBj a:s: and 1 T2 T X t= 1 w_i;t¤_¡1w_j;t¤_¡1! d¤ Z1 0 BiBj a:s: (46) underA ssumption (W ). N ow, the limitingdistribution ofthe F ¤

G T follows immediately.

P art(b) Itfollows from P arts (b) and(c) ofL emma2 that

X_p¤0Y<sub>`</sub>¤= O¤<sub>p</sub>(Tp¹1=2); X<sub>p</sub>¤0"¤= O<sub>p</sub>¤(T1=2p¹1=2) (47 ) whichtogetherwith(42) gives ¯ ¯ ¯Y`¤0X p¤(Xp¤0Xp¤)¡1X p¤0"¤ ¯ ¯ ¯·¯¯¯Y`¤0X p¤ ¯ ¯ ¯°°°(Xp¤0X p)¡1 ° ° ° ¯¯¯X p¤0"¤ ¯ ¯ ¯= Op¤(T1=2p)¹ T hen we havefrom L emma1(a) that

A¤O T T = Y_`¤0"¤ T + o ¤ p(1 )= QA¤_{O T} + o¤p(1 ) (48) where QA¤_{O T} = 0 B B BB B B B B @ ~ ¼1(1 )1 T T X t= 1 w¤_1;t¡1"¤_1t ... ~ ¼N (1 ) 1 T T X t= 1 wN¤;t¡1"¤Nt 1 C C CC C C C C A

N ext, wededuce from (42) andL emma2(b) that

X_p¤0(~§-IT)Xp¤= O¤p(T¡1); Xp¤0(~§-IT)Y`¤= Op¤(Tp¹1=2) (49 ) andthis togetherwith (47 ) gives

¯ ¯ ¯Y`¤0Xp¤(Xp¤0X p¤)¡1Xp¤0(~§-IT)Y`¤ ¯ ¯ ¯= O¤p(Tp)¹ and _¯ ¯ ¯Y`¤0X p¤(Xp¤0X p¤)¡1X p¤0(~§-IT)Xp¤(Xp¤0Xp¤)¡1X p¤0Y`¤ ¯ ¯ ¯= Op¤(Tp)¹ T hen we have M F O T¤ T2 = Y_{`</sub>¤0(~§-IT)Y<sub>`}¤ T2 + o ¤ p(1 )= QM _{F O T}¤ + o¤p(1 ) (50)

due toL emma1(b), where QM ¤ F O T = 0 B B B B B B B B @ ~ ¾11¼~1(1 )2 1 T2 T X t= 1 w¤2_1;t¡1 ¢¢¢ ~¾1N ¼~1(1 )~¼N (1 ) 1 T2 T X t= 1 w_1;t¤_¡1wN¤;t¡1 ... ... ... ~ ¾N1¼~N (1 )~¼1(1 )1 T2 T X t= 1 w¤N;t¡1w1;t¤¡1 ¢¢¢ ¾~N N¼~N (1 )2 1 T2 T X t= 1 w¤2N;t¡1 1 C C C C C C C C A

Finally, wehave from theresults in(49 ) and(50) FO T¤ = µ_A¤ O T T ¶0µ_M ¤ F O T T2 ¶¡1µ_A¤ O T T ¶ = Q0_A¤ O TQ ¡1 M ¤ F O TQA ¤ O T + o ¤ p(1 ) andthe statedresultnowfollows immediatelyfrom (45) and(46).

P roofofCorollary3.1 T heproofis analogous totheproofofCorollary2.1. P art(a) Itfollows from (43) and (44) that

T®^¤G T = µ_{B ¤} G T T2 ¶¡1µ_A¤ G T T ¶ = Q¡1_B¤ G TQA¤G T + o ¤ p(1 ) giving 1 T ³ A¤G T:¤1 f^®¤G T ·0 g ´ = µ_A¤ G T T :¤1 fT®^ ¤ G T ·0 g ¶ = ³QA¤ G T:¤1 n Q¡1_B¤ G TQA¤G T ·0 o´ + o¤_p(1 ) From the aboveresultand(44), wemaywrite theK¤

G T statistics givenin(24) as KG T¤ = µ₁ T ³ A¤G T:¤1 f^®¤G T ·0 g ´¶0µ_B ¤ G T T2 ¶¡1µ₁ T ³ A¤G T:¤1 f^®¤G T ·0 g ´¶ = ³QA¤ G T :¤1 n Q¡1_B¤ G TQA¤G T ·0 o´₀ Q¡1_B¤ G T ³ QA¤ G T:¤1 n Q¡1_B¤ G TQA¤G T ·0 o´ + o¤_p(1 )

N owthe statedresultfollows immediatelyfrom (45) and (46). P art(b) Itfollows from (42) and (47 ) that

¯ ¯ ¯Y_{`</sub>¤0X <sub>p</sub>¤(X<sub>p</sub>¤0X <sub>p</sub>¤)¡1X <sub>p</sub>¤0Y<sub>`}¤¯¯¯·¯¯¯Y_{`</sub>¤0X <sub>p</sub>¤¯¯¯°°°(X<sub>p</sub>¤0X p)¡1 ° ° ° ¯¯¯X <sub>p</sub>¤0Y<sub>`}¤¯¯¯= O¤_p(Tp)¹ whichtogetherwithL emma3.1 (b) gives

BO T¤

T2 =

Y_{`</sub>¤0Y<sub>`}¤

where QB¤_{O T} = 0 BB B B B B B B @ ~ ¼1(1 )2 1 T2 T X t= 1 w¤2_1;t¡1 ¢¢¢ ~¼1(1 )~¼N(1 ) 1 T2 T X t= 1 w_1;t¤_¡1w¤N;t¡1 ... ... ... ~ ¼N (1 )~¼1(1 )1 T2 T X t= 1 wN¤;t¡1w¤1;t¡1 ¢¢¢ ¼~N(1 )2 1 T2 T X t= 1 w¤2N;t¡1 1 CC C C C C C C A

Itfollows from (48) andtheaboveresultthat

T®^¤O T = µ_B¤ O T T2 ¶¡1µ_A¤ O T T ¶ = Q¡1_B¤ O TQA¤O T + o ¤ p(1 ) and 1 T ³ A¤O T:¤1 f^®¤O T ·0 g ´ =³QA¤ O T :¤1 n Q¡1_B¤ O TQA¤O T ·0 o´ + o¤p(1 ) From this andthe resultin(50), we mayexpress thetestK¤

O T de…nedin(24) as KO T¤ = µ₁ T ³ A¤O T:¤1 f^®¤O T ·0 g ´¶0µ_M ¤ F O T T2 ¶¡1µ₁ T ³ A¤O T:¤1 f^®¤O T ·0 g ´¶ = ³QA¤ O T :¤1 n Q¡1_B¤ O TQA¤O T ·0 o´₀ Q¡1_M ¤ F O T ³ QA¤ O T :¤1 n Q¡1_B¤ O TQA¤O T ·0 o´ + o¤_p(1 ) whichtogetherwith(45) and(46) gives the statedresult.

P roofofT heorem 3.2 T he limitdistributions ofthe bootstrap G L S and O L S based t-statistics, t¤

G T and t¤O T, de…ned in (26) are derived analogously as we did forthe sample

t-statistics tG T and tO T in theproofofT heorem 2.2.

P art(a) Itfollows from Parts (b) and (c) ofL emma2 that

X p¤0(~§¡1-IT)y`¤= Op¤(Tp¹1=2); Xp¤0(~§¡1-IT)"¤= O¤p(Tp¹1=2) whichalongwith (42) gives ¯ ¯ ¯ ¯y¤0`(~§¡1-IT)X _p¤ ³ Xp¤0(~§¡1-IT)X _p¤ ´_¡1 X p¤0(~§¡1-IT)"¤ ¯ ¯ ¯ ¯= Op¤(T1=2p)¹ and _¯ ¯ ¯ ¯y`¤0(~§¡1-IT)X<sub>p</sub>¤ ³ X <sub>p</sub>¤0(~§¡1-IT)X<sub>p</sub>¤ ´¡1 X<sub>p</sub>¤0(~§¡1-IT)y¤<sub>`</sub> ¯ ¯ ¯ ¯= Op¤(Tp)¹ T hen we haveduetotheresults in P arts (a) and(b) ofL emma1 that

a¤G T T = y_{`</sub>¤0(~§¡1-IT)"¤ T + o ¤ p(1 ) = Qa¤ G T + o ¤ p(1 ) b¤G T T2 = y<sub>`}¤0(~§¡1-IT)y_`¤ T2 + o ¤ p(1 ) = Qb¤_{G T} + o¤p(1 )

where Qa¤_{G T} = N X i= 1 N X j= 1 ~ ¾ij¼~i(1 )1 T T X t= 1 w¤_i;t¡1"¤_jt Qb¤_{G T} = N X i= 1 N X j= 1 ~ ¾ij¼~i(1 )~¼j(1 )1 T2 T X t= 1 w_i;t¤_¡1w¤_j;t¡1 W emaynowwrite t¤ G T as t¤G T = a¤ G T T µ_b¤ G T T2 ¶¡1=2 = Qa¤_{G T}Q¡1=2b¤ G T + o ¤ p(1 ) andthe limittheoryfort¤ G T is directlyobtainedfrom (45) and (46). P art(b) Since, X p¤0y¤`= O¤p(Tp¹1=2); Xp¤0(~§-IT)y¤`= O¤p(Tp¹1=2) byL emmaA 2 (b), we havefrom (49 ) that

¯ ¯ ¯Y_{`</sub>¤0X <sub>p</sub>¤(X<sub>p</sub>¤0X <sub>p</sub>¤)¡1X <sub>p</sub>¤0"¤¯¯¯ = O¤<sub>p</sub>(T1=2p)¹ ¯ ¯ ¯Y<sub>`}¤0X _p¤(X_p¤0X _p¤)¡1X _p¤0(~§-IT)Y_{`</sub>¤ ¯ ¯ ¯ = O¤<sub>p</sub>(Tp)¹ ¯ ¯ ¯Y<sub>`}¤0X_p¤(X _p¤0X_p¤)¡1X_p¤0(~§-IT)X_p¤(X_p¤0X _p¤)¡1X_p¤0Y_`¤ ¯ ¯ ¯ = O¤_p(Tp)¹ W enowdeducefrom L emma1 that

a¤O T T = y<sub>`</sub>¤0"¤ T + o ¤ p(1 ) = Qa¤ O T + o ¤ p(1 ) M ¤ tO T T2 = y¤0 `(~§-IT)y¤_` T2 + o ¤ p(1 ) = QM _{tO T}¤ + o¤p(1 ) where Qa¤_{O T} = N X i= 1 ~ ¼i(1 )1 T T X t= 1 w_i;t¤_¡1"¤_it QM ¤ tO T = N X i= 1 N X j= 1 ~ ¾ij¼~i(1 )~¼j(1 )1 T2 T X t= 1 w_i;t¤_¡1w¤_j;t¡1 T hen we have t¤O T = a¤ O T T µ_{M ¤} tO T T2 ¶¡1=2 = Qa¤_{O T}Q¡1=2M ¤ tO T+ o ¤ p(1 ) from whichthe statedresultfollows immediatelyfrom (45) and (46).