A CUSUM TEST OF COMMON TRENDS IN LARGE HETEROGENEOUS PANELS

Transcription

1 A CUSUM TEST OF COMMON TRENDS IN LARGE HETEROGENEOUS PANELS JAVIER HIDALGO AND JUNGYOON LEE A. This paper examies a oparametric CUSUM-type test for commo treds i large pael data sets with idividual fixed effects. We cosider, as i Zhag, Su ad Phillips 202, a partial liear regressio model with ukow fuctioal form for the tred compoet, although our test does ot ivolve local smoothigs. This coveietly forgoes the eed to choose a badwidth parameter, which due to a lack of a clear ad sesible iformatio criteria it is diffi cult for testig purposes. We are able to do so after makig use that the umber of idividuals icreases with o limit. After removig the parametric compoet of the model, whe the errors are homoscedastic, our test statistic coverges to a Gaussia process whose critical values are easily tabulated. We also examie the cosequeces of havig heteroscedasticity as well as discussig the problem of how to compute valid critical values due to the very complicated covariace structure of the limitig process. Fially, we preset a small Mote-Carlo experimet to shed some light o the fiite sample performace of the test.. INTRODUCTION Oe of the oldest issues ad mai cocers i time series is perhaps the study ad modellig of their tred behaviour. The literature is large ad vast, see for istace a overview by Phillips 200, or more recetly by White ad Grager 20 o some practicalities regardig treds for the purpose of, say, predictio. A huge amout of the literature has focused o the topic of stochastic vs. determiistic treds. O the other had, more recetly there has bee a iterest o how aalysis of macroecoomic data may beefit from usig cross-sectioal iformatio. Oe of the earliest works is give by Phillips ad Moo 999 o testig for uit roots i pael data, see also the recet work o macroecoomic covergece by Phillips ad Sul 2007, ad Breitug ad Pesara 2008 for a survey o stochastic treds i pael data models. Recetly there has also bee a surge of iterest o examiig determiistic tredig regressio models. Amog others ad Date: 26 Jue 204. Key words ad phrases. Commo Treds, large data set, Partial liear models, Bootstrap algorithms. JEL Classificatio: C2, C3, C23. We thaks the commets of a referee which have led to a clearer versio of the mauscript. Of course, ay remaiig errors are our sole resposibility.

2 2 JAVIER HIDALGO AND JUNGYOON LEE i a cotext with spatial depedece, see Robiso 20a who cosiders oparametric tredig models or Robiso 20b for a geeral parametric model where the tredig regressors are of a o polyomial type t γ with γ > /2. O the other had, semiparametric models have bee cosidered, see Gao ad Hawthore 2006 ad i pael data models by Atak, Lito ad Xiao 20 or Zhag, Su, Xu ad Phillips 202 or Degras, Xu, Zhag ad Wu 202. The work by Atak et al. 20 assumes that the umber of cross-sectio uits is fiite although the tredig regressio fuctios are commo amog the differet uits. O the cotrary, the latter two refereces allow for the umber of cross-sectio uits to icrease without limit ad they cosider a test for commo treds, or, i the laguage of pael data aalysis for homogeeity/poolability. This paper belogs to the latter framework. More specifically, we are cocered with testig for commo treds i the partial pael liear regressio model y it = β x it + m i t/t + υ it, i =,...,, t =,..., T, υ it = µ i + ε it,. where {x it } t, i N +, are k vector of exogeous variables, m i t/t is left to be of ukow fuctioal form, µ i are the idividual fixed effects ad the error terms {ε it } t, i N +, are sequeces of zero mea radom variables, possibly heteroscedastic. We shall remark that the reaso ot to itroduce fixed time effects i., is because the tests are ivariat to its presece, see 2.5 or 2.6 below. I additio, model. is a geeralizatio of that discussed i Degras, et al. 202, where we have ow some covariates x it i the model. We are iterested, to be more precise the ext sectio, o testig the hypothesis H 0 : m i τ = m τ for all i ad τ [0, ]. This hypothesis testig ca be put ito a much broader cotext of testig for equality amog a set of curves, such as the coditioal expectatio ad/or the dyamic structure i time series. More geerally, our hypothesis ca arise whe we wish to decide how similar two or more groups of idividuals are. Whe the umber of curves is fiite, there is a substatial literature. Relevat examples are the classical poolability test i pael data, see Hsiao 986 ad refereces therei, for parametric models, ad Baltagi, Hidalgo ad Li 996 i a oparametric sceario. I a time series cotext, we ca metio the work by Detter ad Paparoditis 2008 ad refereces there, where the iterest is to decide whether the covariace structure is the same across the sequeces. Nowadays due to the amout of available data, it is ot urealistic to cosider the umber of series large or that they icrease without limit. Amog the latter framework, we ca cite Ji ad Su 200 ad Degras et al. 202 ad i a partially liear regressio model the work by Zhag et al. 202.

3 TESTING FOR COMMON TRENDS 3 Also it worth metioig the recet work by Hidalgo ad Souza 203 who tested if the dyamic structure across sequeces is the same. It is worth otig that we ca geeralize our model. as y it = β x it + g i d t + υ it, where, as i Pesara ad Tosetti 20 or more recetly Brya ad Jekis 203, d t has the iterpretatio of beig a sequece of observed commo effects/factors across idividuals. See also Galvao ad Kato 203, where d t has the meaig of beig fixed effects. With this iterpretatio, if g i d t = g i d t, say, we ca regard the latest display model as a pael data with iteractive fixed effects, so that our hypothesis becomes g i = g, i.e. we are testig the existece of iteractive effects versus a more stadard additive fixed effects type of model. Bearig this i mid, we focus evertheless o the case where d t = t/t merely for historical reasos, although our mai results seem to hold true for the last displayed model. We fiish this sectio relatig the results of the paper with the problem of classificatio with fuctioal data sets, which is a topic of active research. The reaso beig that this paper tackles the problem of whether a set of curves, that is the tredig fuctios, are the same or ot amog a set of idividual uits. Withi the fuctioal data aalysis framework, this questio traslates ito whether there is some commo structure or if we ca split the set of curves ito several classes or groups. See classical examples i Ferraty ad Vieu 2006, although their approach uses oparametric techiques which we try to avoid so that the issues of how to choose a badwidth parameter ad/or the metric to decide closeess are avoided. With this i mid, we believe that our approach ca be used for a classificatio scheme. For istace, i ecoomics are the tred compoets across differet idustries the same? I the laguage of fuctioal data aalysis, this is a problem of supervised classificatio which is othig more tha a moder ame to oe of the oldest statistical problems: amely to decide if a idividual belogs to a particular populatio. The term supervised refers to the case where we have a traiig sample which has bee classified without error. Moreover, we ca evisage that our methodology ca be exteded to problems dealig with fuctioal data i a framework similar to those examied by Chag ad Ogde The remaider of the paper is orgaized as follows. I the ext sectio, we itroduce the test statistics ad their limitig distributios i a model without the regressors x it. Sectio 3 cosiders the model., ad we explicitly examie the cosequeces whe we allow for heteroscedasticity, that is E ε it = σ 2 i for i N+. We show that the limitig process has quite a messy covariace structure to be of ay use with real data sets. Because of this, we the examie a trasformatio to overcome the latter cocered. Sectio 4 describes the local alteratives for which the tests have o trivial power ad their cosistecy. We also describe ad show the validity of a bootstrap versio of the test, as our simulatio results illustrate that tests

4 4 JAVIER HIDALGO AND JUNGYOON LEE based o a subset of proposed statistics yield a very poor fiite sample performace, see Table 2. Sectio 5 presets a Mote Carlo experimet to get some idea of the fiite sample performace of our tests. Sectio 6 cocludes ad fially we cofie the proofs to the Appedix. 2. TESTING FOR HOMOGENEITY As a startig poit we shall first provide a test for commo treds i the absece of exogeous regressors x it. Our motivatio comes from the fact that the results of Sectio 3 are the much easier to follow after those give i Theorems ad 2 below. So, we begi with the model y it = m i t/t + υ it, i =,...,, t =,..., T, υ it = µ i + ε it 2. with both ad T icreasig to ifiity. The ull hypothesis H 0 is H 0 : m i τ = m τ for all i ; τ [0, ] 2.2 ad the alterative hypothesis H becomes H : 0 < ι = Λ / <, 2.3 where Λ= {i : µ Υ i > 0} with Υ i = {τ [0, ] : m i τ m τ} ad A deotes the cardiality of the set A ad λ Υ deotes the Lebesgue measure of the set Υ. Λ deotes the set of idividuals i for which m i τ is differet from the commo tred m τ, ad thus ι =: ι represets the proportio of sequeces i for which m i τ m τ. Oe feature of 2.3 is that ι ca be asymptotically egligible. More specifically, as we shall show i Sectio 4, the test has otrivial power uder local alteratives such that ι = O /2. This situatio whe ι 0 ca be of iterest if we wat to decide whether a ew set of sequeces share the same tredig behaviour as the existig oes, or for supervised classificatio purposes. Before we state our results, let s itroduce some regularity coditios. Coditio C: {ε it } t Z, i N +, are sequeces of zero mea idepedet radom variables with Eε 2 it = σ2 ad fiite fourth momets. I additio, the sequeces {ε it } t ad {ε jt } t are mutually idepedet for all i j. Coditio C2: The fuctios m i τ are bouded for all i N +. Coditio C is stadard, although stroger tha we really eed for our results to follow. A ispectio of our proofs idicate that the results would follow provided some type of weak depedece i both time ad/or crosssectioal dimesios. The oly mai differece, beig that the asymptotic covariace structure of our statistics will be affected by its depedece. So, for simplicity ad to ease argumets, we have decided to keep C as it stads. It is worth emphasizig that we do ot assume that the sequeces {ε it } t Z, i N + are idetically distributed, although their first two momets

5 TESTING FOR COMMON TRENDS 5 are costat, although we shall discuss the cosequeces of allowig for heteroscedasticity i Sectio 3 below. Regardig Coditio C2, we otice that we do ot eed ay type of smoothess coditio o the tred fuctios m i τ. We require oly that they are bouded, although strictly speakig this is oly eeded whe examiig the cosistecy ad local power of the tests. Let s itroduce ow some otatio used i the sequel of the paper. For arbitrary sequeces {ς it } T, i, we defie ς t = ς jt, j= ς i = T T s= ς is, ς = j= s= T ς js. 2.4 For presetatio reasos we shall cosider separately the cases µ i = 0 ad µ i 0. The argumets eeded i the latter case are legthier, but they are based o those give whe µ i = The case without fixed idividual effects µ i. Defie m i t/t := y it y t, i =,, t =,..., T 2.5 m i t/t := y it i y jt, i =,, t =,..., T. 2.6 i j= The motivatio to itroduce the quatities 2.5 ad 2.6 is that, uder the ull hypothesis H 0, it holds that y t = m t/t + ε t. So, t t m i := ε it ε t ; m i := ε it i ε jt, T T i which do ot deped o the tred fuctio m ad they have zero mea. However, uder the alterative hypothesis, m i t/t ad m i t/t will develop a mea differet tha zero, suggestig that tests based o 2.5 ad 2.6 will have the usual desirable statistical properties. Hece, followig ideas i Brow, Durbi ad Evas 975, we base our test for homogeeity o Q T,,r,u = Q T,,r,u = y it y t, y it i i y jt j= =2, r, u [0, ], where [c] deotes the iteger part of a umber c. Let s itroduce the followig otatio. We deote D[0,]2 weak covergece i the space D [0, ] 2, the space of cotiuous fuctios from the right ad equipped with the Skorohod metric, give i p. 662 of Bickel ad

6 6 JAVIER HIDALGO AND JUNGYOON LEE Wichura 97. Recall that a two-parameter process W r, u is said to be a Browia sheet if it is a Gaussia process, satisfyig W 0, 0 = 0 ad Cov W r, u, W r 2, u 2 = r r 2 u u 2, where a a 2 = mi a, a 2. We also itroduce two-parameter process B r, u = W r, u uw r,. Theorem. Assumig C, uder H 0, we have that as T,, a QT,,r,u D[0,] 2 σb r, u, r, u [0, ], 2.7 b Q T,,r,u D[0,] 2 σw r, u, r, u [0, ] The case with idividual fixed effects µ i. Whe the idividual fixed effects µ i s are preset i 2., we eed to modify both Q T,,r,u ad Q T,,r,u, as we eed to remove them from the model. To that ed, for all r, u [0, ], cosider U T,,r,u = U T,,r,u = {y it y t y i + y } t=2 y it i i j= y jt t t Let s itroduce the two-parameter Gaussia process s= y is + B r, u = W r, u uw r, rw, u + ruw,, which satisfies B 0, 0 = 0, B r, = 0, B, u = 0 i t i r, u [0, ] ad Cov B r, u, B r 2, u 2 = r r r r 2 u u 2 u u 2, where a a 2 = max a, a 2. The process B r, u is kow as the Browia pillow. Theorem 2. Assumig C, uder H 0, we have that as T, a UT,,r,u D[0,] 2 σb r, u, r, u [0, ], 2.9 b U T,,r,u D[0,] 2 σw r, u, r, u [0, ]. 2.0 We ow commet briefly o the results of Theorems ad 2. The first poit to otice is that, as we metioed i the abstract, we are able to avoid local smoothig i the estimatio of the fuctios m i, beig the key that both ad T icrease to ifiity. We otice that both U T,,r,u ad Q T,,r,u have the same asymptotic behaviour, which is ot the case for U T,,r,u ad Q,r,u. This is somehow expected after we otice that both y it i i j= y jt ad the term iside the braces i the defiitio of U T,,r,u have the same structure, more specifically they behave as martigale differece sequeces. We just remark that y it i i j= y jt is othig but the martigale trasformatio of y it y t. t y js j= s=.

7 TESTING FOR COMMON TRENDS Tests for homogeeity. We put together results of Sectios to provide test statistics for homogeeity. To that ed we deote by g a cotiuous fuctioal: [0, ] 2 R +. To implemet the tests we eed to estimate the variace σ 2. For that purpose, deote residuals by e it := y it ȳ t ; e 2 it := y it ȳ t ȳ i + ȳ, depedig o whether the model does ot have or has idividual fixed effects. As we metioed earlier, uder the H 0, we have that I additio we estimate σ 2 by e it := ε it ε t ; e 2 it := ε it ε t ε i + ε. σ 2 l := T e l 2 it, for l =, 2. Propositio. Assumig C, uder H 0, as, T, we have that σ 2 l p σ 2, for l =, 2. Proof. The proof is stadard ad so it is omitted. Corollary. Assumig C C2, uder H 0, for ay cotiuous fuctioal g, as T,, a T Q T,,r,u d T Q T,,r,u d g g Br, u ; g g W r, u σ σ b T U T,,r,u d g g B r, u ; σ 2 g T U T,,r,u σ 2 d g W r, u. Proof. The proof follows by stadard argumets usig the cotiuous mappig theorem, Theorems ad 2 ad Propositio, so it is omitted. Oe fuctioal ofte employed to implemet the test is the sup, that is U T,,r,u d σ 2 sup r,u [0,] sup r,u [0,] U T,,r,u σ 2 d sup B r, u, r,u [0,] sup W r, u, r,u [0,] which correspods to a Kolmogorov-Smirov type of test. Critical values for the supremum of absolute value of the three limitig processes are provided i Table for various sigificace levels, which have bee simulated with iteratios.

8 8 JAVIER HIDALGO AND JUNGYOON LEE T. Critical Values for α-sigificace level,, T = 0000 sup r,u [0,] sup r,u [0,] sup r,u [0,] α W r, u Br, u B r, u TEST OF COMMON TRENDS We ow revisit our model of iterest.. Recall that uder the ull hypothesis we ca write. as y it = β x it + m t/t + υ it, υ it = µ i + ε it. 3. It is obvious that if β were kow, the problem of testig for homogeeity would be o differet from that examied i Sectio 2. That is, we would test homogeeity i the model y it x itβ = : υ it = m i t/t + µ i + ε it replacig y it there by υ it. However β is ukow, so give a estimator β, we might the cosider istead υ it =: y it x it β = x it β β + mt/t + µ i + ε it. Thus, it is sesible to cosider U,r,u ad U,r,u but with υ it replacig y it there. More specifically, for r, u [0, ], we defie Û T,,r,u = υ it υ t υ i + υ 3.2 Û T,,r,u = i j= υ it υ t js s= υ i t is j= s= + υ js. i t i t t=2 So, we ow look at how to estimate the slope parameters β i 3.. For that purpose, give a geeric sequece {ς it } i,t ad deotig by ˇς it := ς it ς t ς i + ς, we have that uder H 0, 3. becomes ˇy it = β ˇx it + ˇε it ; i =,..., ; t =,..., T, so that we estimate β by least squares methods, that is T β = ˇx itˇx it ˇx itˇy it. Observe that ˇς it removes the commo tred ad the fixed effects.

9 TESTING FOR COMMON TRENDS 9 Coditio C3: For all i N +, {x it } t Z are k-dimesioal sequeces of idepedet radom variables mutually ucorrelated with the error sequeces {ε it } t Z, where E x it = κ i, Σ = E { x it κ i x it κ i } ad fiite fourth momets. Coditio C4: T, such that T/ 2 0. We have the followig two results. Theorem 3. Assumig C C4, uder H 0, as T,, /2 β β d N 0, σ 2 Σ. Theorem 4. Assumig C to C4, uder H 0, as T, we have that for ay cotiuous fuctioal g, T Û T,,r,u d a g g B r, u T Û T,,r,u d ; b g g W r, u, σ σ where σ 2 := T υ it υ t υ i + υ 2. A typical cotiuous fuctioal g employed is the supremum, that is /2 sup σ Û T,,r,u d sup W r, u, r,u [0,] r,u [0,] which correspods to a Kolmogorov-Smirov type of statistic. Remark. The results of the previous theorem idicates that kowledge of β does ot affect the asymptotic behaviour of our test. 3.. Uequal variace σ 2 i. We ow look at the problem of testig for homogeeity with possible heterogeeous secod momets i the error term. More specifically, we are cocered with testig for homogeeity whe Coditio C5: For all i N +, {ε it } t Z are sequeces of radom variables as i C but such that Eε 2 it = σ2 i satisfyig σ 2 i C <. i= We ca be tempted to write Coditio C5 i a more traditioal format as sayig that E ε 2 it x it = σ 2 x it. However, Coditio C3 implies that the latter coditioal variace would ot play ay relevat role o the asymptotic behaviour of ay cotiuous fuctioal of either ÛT,,r,u or Û T,,r,u, i the sese that the asymptotic behaviour is that give i Theorem 4 but with a differet scalig σ. O the other had, this is ot the case though, if the ucoditioal variace of {ε it } t Z is as stated i Coditio C5 as Theorem 5 below shows. It is also worth metioig that it is obvious how to hadle the situatio where we allow for tredig variaces, say σ 2 i = α + α 2 i γ, as

10 0 JAVIER HIDALGO AND JUNGYOON LEE log as γ <. This relaxatio of C5 is trivial ad it oly complicates the otatio ad argumets. Fially, similar caveats appear if C5 we modified to say that Eε 2 it = σ2 t. However, to save space we decided to keep C5 as it stads. Let s itroduce the followig otatio [u] σ 2 u = lim σ 2 i. Theorem 5. Assumig C3 ad C5, uder H 0 we have that as, T, i= a /2 Û T,,r,u = G r, u b /2 Û T,,r,u = G r, u, where G r, u ad G r, u are Gaussia processes with covariace structures Φ r, u ; r 2, u 2 give i 7.6 below ad Cov G r, u, G r 2, u 2 = r r 2 σ 2 u u 2 u 2 σ 2 u respectively. r r 2 u 2 σ 2 u r r 2 u 2 σ 2 u u 2 +u σ 2 u 2 r r 2 r r 2, The results of Theorem 5 idicate that to obtai asymptotic critical values o fuctioals based o /2 Û T,,r,u ad /2 Û T,,r,u appear to be a dautig task. This is i view of the very complicated ad complex covariace structure of the latter two processes. To overcome this type of problem, we evisage a procedure based o a trasformatio of Û T,,r,u or Û T,,r,u i such a way that the trasformed statistics will have a kow or at least simpler covariace structure. Deote the estimator of σ 2 i by σ 2 i = T T υ it υ t υ i υ 2, i =,...,. 3.3 It is quite stadard i view of C3 ad C5 to show that σ 2 i is a poitwise cosistet estimator for σ 2 i for all i =,...,, ad uder C3 ad C5 ad uder suffi ciet momet coditios, it is easily show that E σ 2 i σ 2 i 2p = O T p, i =,...,. 3.4 So, for istace we modify Û T,,r,u to Û T,,r,u = σ i υ it i j= σ j υ t js s= σ i t i υ is j= + i t s= σ j υ js i t 3.5.

11 TESTING FOR COMMON TRENDS A similar trasformatio is obtaied for ÛT,,r,u. Now, Theorem 3 implies that the right side of 3.5 is i j= σ ε it j ε t js s= σ i t i ε is j= s= σ + j ε js σ i i t i t +o p /2 uiformly i r, u [0, ]. Now, if we would have σ i istead of σ i, the right had side of the last displayed equatio would be the what we had i Theorems 2 or 4, but with σ 2 replaced by there, as we had that ε it /σ i satisfies C but with uit variace. So, if σ i σ i ε it = o p /2 3.6 uiformly i r, u [0, ], we the coclude that the modificatio give by Û,r,u leads to a statistic with a kow covariace structure. To that ed, we first observe that by 3.4 ad that E sup l=,...,q a l q l= E a l k /k, we have that sup σ 2 i σ 2 i = O p /2p T /2 = o. i=,..., The latter uiform covergece, as opposed to poitwise cosistecy. But 3.6 is the case, because by Taylor s expasio, + { 2p l= σi σ i } l σi σ i ε it/σ i σ i σ i 2p+ σ i ε it. 3.7 From here, suitable regularity coditios o how ad T icrease to ifiity ad somewhat legthy but routie algebra lead us to the claim i 3.6 ad coclude that the trasformatio of Û T,τ,ι give i Û,r,u will overcome the problematic issue of how to tabulate valid asymptotic critical values for our test. 4. ASYMPTOTIC PROPERTIES UNDER LOCAL ALTERNATIVE HYPOTHESIS. BOOTSTRAP As usual, it is importat to examie the behaviour of ay test uder alterative hypothesis ad ultimately uder local alteratives. We begi with the discussio uder local alteratives, the cosistecy of the tests

12 2 JAVIER HIDALGO AND JUNGYOON LEE beig a corollary. To that ed, cosider { mi τ = m τ + ϱ H l : T /ξ i τ for all i I, m i τ = m τ for all I < i, 4. where ϱ i τ is differet tha zero i Υ i = Υ, a set of positive Lebesgue measure, ξ /2 ad I C /2, where C is a fiite positive costat. It is worth remarkig that we have assumed that Υ is the same for all i for otatioal simplicity. However i the Mote Carlo simulatios we shall examie what happes with the power of the test whe, give a particular alterative hypothesis, we permute the order of the cross-sectio uits, see Table 8. We begi with the test based o the statistic U T,,r,u leavig the case with regressors x it for later. Propositio 2. Uder H l with ξ = /2 ad I = C /2, assumig C ad C2, we have that /2 U T,,r,u D[0,]2 B r, u+c /2 ϱ τ dτ r ϱ τ dτ, [0,r] Υ where ϱ τ = lim /2 /2 i= ϱ i τ. Proof. By defiitio we have that U T,,r,u = + ε it ε t ε i + ε [T r] C /2 i= ϱ i t/t C /2 j= Υ ϱ j t/t. The secod term o the right of the last displayed equality kow as the drift fuctio, is i= T 3/2 [T r] C /2 i= ϱ i t/t. So, we have that after we ormalize U T,,r,u by /2 the latter displayed expressio becomes [T r] C /2 /2 ϱ i t/t C /2 τ dτ r ϱ τ dτ. [0,r] Υ ϱ Υ Observe that we ca iterchage the limit i ad the itegral usig Lebesgue domiated covergece because ϱ i u C for all i. Recall that C2 implies that the fuctios ϱ i τ are bouded. This cocludes the proof of the propositio.

13 TESTING FOR COMMON TRENDS 3 Oe immediate coclusio that we draw from Propositio 2 is that our tests detect local alteratives shrikig to the ull hypothesis at a parametric rate O T /2 /2. A questio of iterest ca be if test based o U,r,u ca detect local alteratives whe ξ < /2 ad/or I is a fiite iteger. This type of alteratives ca be useful for the purpose of, say, classificatio or whether we wish to decide if a ew set of idividuals, i I, shares the commo tred fuctio m τ. To that ed, we otice that the drift fuctio i /2 U,r,u becomes [T r] T ξ+/2 /2 I i= ϱ i t/t [T τ] T ξ+/2 /2 T /2 ξ /2 I ϱ i t/t i= [0,r] Υ i= I ϱ i u du. So, if T 2ξ, the drift fuctio is ozero ad hece the test based o U,r,u, ad similarly for U T,,r,u, have o trivial asymptotic power. Aother easy coclusio from Propositio 2 is that uder fixed alteratives, sice [0,r] Υ ϱ τ dτ r Υ ϱ τ dτ is ozero, /2 U T,,r,u P 0, so that tests based o U T,,r,u are cosistet, so is U T,,r,u. We ow discuss the local power for the model 3.. We begi by discussig the cosequeces o the asymptotic properties of the least squares estimator of β. To that ed, recall that 3. uder local alteratives with ξ = /2 ad I = C /2 becomes so deotig y it = β x it + m i t/t + υ it, m i t/t = m i t/t T υ it = µ i + ε it T s= m i s/t, we have that stadard maipulatios yields T β β = ˇx itˇx it ˇx it m i t/t + T T ˇx itˇx it m j t/t j= ˇx it ε it. 4.2 But as we ca take β Ex it = 0, the oly cosequece o the asymptotic behaviour of /2 β is that the asymptotic variace will have a additioal term comig from T /2 ˇx it m i t/t m j t/t, i= j=

14 4 JAVIER HIDALGO AND JUNGYOON LEE which is differet tha zero uder fixed alterative hypothesis, whereas uder the local β alteratives it is clear to be o p implyig that the behaviour of /2 β is uchaged. The oly coditio for the latter is that ϱ i τ is square itegrable. See our commets after Corollary 2 below. We ow discuss the behaviour of ÛT,,r,u uder the local alteratives i 4.. Corollary 2. Uder H l with ξ = /2 ad I = C /2 ad assumig C C4, we have that /2 Û T,,r,u D[0,]2 B r, u+c /2 ϱ τ dτ r ϱ τ dτ. [0,r] Υ Proof. The proof follows easily i view of 4.2, commets that follow ad Propositio 2, so it is omitted. Corollary 2 the idicates that, as it was the case uder the ull hypothesis, the behaviour of the test is ualtered by the estimatio of the slope parameters β uder the local alteratives. Fially, we eed to discuss the cosistecy of the tests. But this is a atural cosequece of Propositio 2 ad Corollary 2. The oly differece, but this does ot affect the cosistecy of the test, is that the distributio of β is differet as it will have a additioal term i its asymptotic variace comig from the asymptotic variace of first term o the right of 4.2, which by stadard argumets yield to the expressio Σ V ar = Σ lim,t /2 /2 x it m i t/t m j t/t Σ j= 2 m i t/t m j t/t. Agai regardig the tests based o /2 Û T,,r,u, we obtai the same type of coclusio proceedig as with /2 Û T,,r,u. 4.. Bootstrap of the test. I this subsectio, we provide a valid bootstrap algorithm for our tests. I particular our mai cocer is to do so for the tests based o ÛT,,r,u ad Q T,,r,u as our Mote-Carlo experimet suggests a poor fiite sample performace, especially whe compared to those of Û T,,r,u ad Q T,,r,u. To that ed, because by C the errors ε it are i.i.d., Efro s aive bootstrap would be appropriate. The bootstrap will have the followig 3 STEPS: STEP : For each i =,...,, deote by {ε it }T a radom sample of size T from the empirical distributio of { ε it } t,i, where ε it = υ it υ i υ t + υ. j= Υ

15 TESTING FOR COMMON TRENDS 5 Observe that by defiitio i= T ε it = 0. STEP 2 : We compute our bootstrap pael model as y it = x it β + ε it, i =,..., ; t =,..., T, ad the bootstrap least squares estimator as T β = ˇx itˇx it ˇx itˇy it. STEP 3 : Obtai the least squares residuals υ it = ˇy it ˇx it β, i =,..., ; t =,..., T ad the compute the bootstrap aalogues of ÛT,,r,u ad Û T,,r,u with υ it replacig υ it there. That is, for r, u [0, ], we defie Û T,,r,u = Û T,,r,u = υ it υ it υ t υ i υ i j= υ t i t js s= υ is j= s= + υ js. i t i t Theorem 6. Assumig C to C4, uder the maitaied hypothesis, as T,, /2 β β d N 0, σ 2 Σ, i probability. We the have the followig result. Theorem 7. Assumig C to C4, uder the maitaied hypothesis as T, we have that for ay cotiuous fuctioal g, a g σ Û T,,r,u d g B r, u, i probability b g σ Û T,,r,u d g W r, u i probability, where σ 2 := T υ it υ t υ i υ MONTE CARLO STUDY We ow preset results of a small Mote-Carlo study to shed some light o the fiite sample performace of our tests based o the statistics Q T,,r,u, Q T,,r,u, U T,,r,u ad U T,,r,u. All throughout the study, the results reported are based o 000 iteratios. The sample sizes used were T, = 20, 30, 50.

16 6 JAVIER HIDALGO AND JUNGYOON LEE We will oly preset the results for model., as those for 2. are qualitatively the same. So, we cosider. give by y it = β x it + m i t/t + υ it, i =,...,, t =,..., T, υ it = µ i + ε it, where µ i, N0,, i =,, with µ = j= µ j ad mt/t = 2 log+t/t 2. Recall that uder H 0 m i t/t = m t/t. For ε it, we cosider i.i.d. stadard ormal radom variable as is the case for the regressors x it ad β = 2. I all the tables Q ad Q stads the test i Sectio 3 with µ i = 0 ad the sup orm. Similarly, Û ad Û are those with µ i as defied above. Table 2 reports the Mote Carlo size at the 0%, 5% ad % omial size, whereas Tables 3 to 8 reports the power of the differet tests. T 2. Mote Carlo Size of 0%, 5% ad % tests. T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û Table 2 suggests that the sizes of the test based o the asymptotic distributio are pretty good eve for small samples for those tests based o the statistics Q ad Û. However, the fiite sample performace of the tests based o Q ad U are very poor, i particular whe we compare it with the performace obtaied with Q ad Û. Due to its poor small sample performace, we decided to employ bootstrap methods to see if we ca improve it. The table suggests that bootstrap test defiitely improves the fiite sample performace, beig ow tests based o, say, Q or Q very similar o their behaviour. Our ext experimet deals with the power of the tests. We have cosidered four differet settigs of heterogeeity i the tred fuctio m i.

17 TESTING FOR COMMON TRENDS 7 Settig A sigle break poit. Let b deote the break poit, for which values 0.5 ad 0.75 were tried. For i =,, b, we set mt/t = 0 for t =,, T/2 ad mt/t = for t = +T/2,, T. For i = b+,,, we set mt/t = 2 log+t/t 2, which rages from 0 to aroud.4. Tables 3-4 report the Mote Carlo power of the 0%, 5% ad % tests uder this settig. T 3. Mote Carlo Power of 0%, 5% ad % tests, settig A, b = 0.5 T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û T 4. Mote Carlo Power of 0%, 5% ad % tests, settig A, b = 0.75 T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û

18 8 JAVIER HIDALGO AND JUNGYOON LEE Settig B three break poits. Let b = 0.25, b 2 = 0.5, b 3 = 0.75 be three break poits. For i =,, b ad i = b 2 +,, b 3, we set mt/t = 0 for t =,, T/2 ad mt/t = for t = + T/2,, T. For i = b +,, b 2 ad i = b 3 +,,, we let mt/t = 2 log+t/t 2. Table 5 report the Mote Carlo power of the 0%, 5% ad % tests uder this settig. T 5. Mote Carlo Power of 0%, 5% ad % tests, settig B T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û Settig C sigle break poit with liear tred fuctios. Let b deote the break poit, for which values 0.5 ad 0.75 were tried. For i =,, b, we set mt/t = + t/t, while for i = b +,,, we set mt/t = 2 + 2t/T. Tables 6-7 report the Mote Carlo power of the 0%, 5% ad % tests uder this settig. I Settigs A ad B above, the orderig of the i idex was such that the break poits eatly divide the sample ito subgroups of the same tred fuctios. However, it is plausible i the real data that the orderig of i idex is ot so iformative. This motivated us to cosider the followig settig. Settig D radom permutatio. For i =,, /2, we set mt/t = 0 for t =,, T/2 ad mt/t = for t = + T/2,, T, while for i = + /2,,, we let mt/t = 2 log + t/t 2. After data has bee geerated, the i idex was shuffl ed usig radom permutatio. Table 8 report the Mote Carlo power of the 0%, 5% ad % tests uder this settig. Across all settigs, the Mote Carlo power results seem broadly similar betwee tests based o asymptotic ad bootstrap critical values. Also, as

19 TESTING FOR COMMON TRENDS 9 T 6. Mote Carlo Power of 0%, 5% ad % tests, settig C, b = 0.5 T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û T 7. Mote Carlo Power of 0%, 5% ad % tests, settig C, b = 0.75 T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û with the size results, the presece of regressor ad the resultig eed for the first stage estimatio of the parameter β has ot much altered the Mote Carlo power results betwee the simple ad partial liear models. The results improved with icreasig sample sizes ad the reported Mote Carlo powers were close to at T, = 50 i all settigs. For both settigs A ad C which use a sigle break poit, power performace was better whe the break was at 0.5 compared to whe it was This is to be expected, sice the former break poit represets a

20 20 JAVIER HIDALGO AND JUNGYOON LEE T 8. Mote Carlo Power of 0%, 5% ad % tests, settig D T test stat Asymptotic test Bootstrap test 400 Q Q Û Û Q Q Û Û Q Q Û Û further deviatio away from the ull of commo treds/homogeeity tha the latter. For settigs A, B ad C i which the orderig of the i idex is preserved, the worst power results were obtaied by the Û statistic uder the presece of idividual fixed effects. I Settig D, where the orderig of the i idex was disturbed, the Mote Carlo results differ betwee particular realisatios of the radom permutatio of i idex ad the oes reported i the tables appear to be quite stadard. Mote Carlo power results are poorer for this settig whe compared to others, highlightig the effect of the orderig of i idex o our test statistics. It is ecouragig that the power results improves cosiderably as the sample size icreases eve i this settig. 6. CONCLUSIONS I this paper we have described ad examied a simple test for commo treds of uspecified fuctioal form as the umber of idividuals icreases. We were able to perform the test without resortig to oparametric estimatio techiques so that we avoid the ueasy issue of how to choose a badwidth parameter i a testig procedure like ours. We have allowed the pael regressio model to have covariates eterig i a liear form. We discuss the cosequeces that the estimatio of the slope parameters may have i the asymptotic behaviour of the test, fidig that there were o cosequeces, so that the asymptotic behaviour of the test is ot altered by the estimatio of the parameters.

21 TESTING FOR COMMON TRENDS 2 There are several iterestig issues worth examiig as those already metioed i the itroductio. Oe of them is how we ca exted this methodology to the situatio where the sample sizes T i for the differet sequeces of idividual uits i =,...,, are ot ecessarily the same. We believe that the methodology i the paper ca be implemeted after some smoothig has bee put i place, for istace, via splies. A secod relevat extesio is what happes whe the errors exhibit cross-sectioal ad/or time depedece. We cojecture that, after ispectio of our proofs, the mai coclusios of our results will still hold true if they are weakly depedet. However, the techical details to accomplish this ad i particular those to obtai a cosistet estimator of the asymptotic variace might be cumbersome ad legthy. We evisage, though, that it is possible to obtai a simple computatioal estimator of the log ru variace of the test via bootstrap methods usig results give i Sectio 3, together with those obtaied by Chag ad Ogde However the details are beyod the scope of this paper. 7.. Proof of Theorem. 7. APPENDIX Proof. We first otice that uder H 0, we have that a Q T,,r,u = b Q T,,r,u = ε it ε t, ε it i i ε jt j=, r, u [0, ], We begi with part b. To show weak covergece i 2.8 we eed to establish covergece of fiite dimesioal distributios ad tightess. To that ed, we first we establish that as T,, Deotig we have E Q T,,r,u Q T,,r 2,u 2 σ 2 r r 2 u u z it =: ε it i ε jt, 7.2 i j= Q T,,r,u = z it. Now, C ad stadard argumets imply that E {z i tz i2 s} = 0 7.3

22 22 JAVIER HIDALGO AND JUNGYOON LEE if i i 2 ad t, s. So, we have that [T r ] E [u ] [T r 2 ] z it [u 2 ] z it = [T r r 2 ] [u u 2 ] σ 2 r r 2 u u 2, E zit 2 which completes the proof of 7.. Next, we establish that the fiite dimesioal distributios fidi of Q T,,r,u coverge to those of σw r, u. As usual because the Cramer-Wold device, we oly eed to show the covergece i distributio of Q T,,r,u d σw r, u, r, u [0, ], for ay fixed r ad u. We have already show i 7. that V ar Q T,,r,u σ 2 ru. O the other had, it is well kow that for ay fixed t, Z t = [u] d z it N 0, σ 2 u, ad as Z t ad Z s are idepedet by C, we coclude that [T r] d Z t N 0, σ 2 ur. T So, we have established the CLT whe we take sequetially the limits i ad the i T. However, we eed to show that this is the case whe they are take joitly. To that ed, it suffi ces to verify that Z t satisfy the geeralized Lideberg-Feller coditio: for ay η > 0, as T,, [T r] q T r := E ZtI Z 2 t η 0, see Theorem 2 i Phillips ad Moo 999, beig a suffi ciet coditio E Zt 4 = E [u] z it 4 C. 7.4

23 Proceedig as we did to show 7., E Zt 4 3 = 2 TESTING FOR COMMON TRENDS 23 [u] Ez 2 it 2 + [u] 2 = 3σ 4 + o + [u] 2 + [u] 2 i,...,i 4 =2 l= log = 3σ 4 u O. i,...,i 4 =2 i,...,i 4 =2 cum z i t, z i2 t, z i3 t, z i4 t cum ε i t,..., ε i4 t i i 4 cum ε jt,..., 4 i l j= j= ε jt So the suffi ciet coditio 7.4 holds true ad hece Q T,,r,u = d σw r, u. To complete the proof of part b, we eed to show tightess. For that purpose, defie a icremet of Q T,,r,u as follows. For 0 r r 2 ad 0 u u 2, r,u,r 2,u 2 := [T r 2 ] [u 2 ] t=[t r ]+ i=[u ]+ z it. A suffi ciet coditio for tightess, see Bickel ad Wichura 97, is [T E 2 r,u,r 2,u 2 4 r2 ] [T r ] 2 [u2 ] [u ] 2 C, 7.5 T where we ca assume that u 2 u ad T r 2 r sice otherwise the iequality i 7.5 would hold trivially. Now, proceedig as we did with 7., we have that the left side of 7.5 is 3 T T 2 2 [T r 2 ] [u 2 ] t=[t r ]+ i=[u ]+ [T r 2 ] Ez 2 it [u 2 ] t,...,t 4 =[T r ]+ i,...,i 4 =[u ]+ 2 cum z i t, z i2 t 2, z i3 t 3, z i4 t

24 24 JAVIER HIDALGO AND JUNGYOON LEE By C, we have that cum z i t, z i2 t, z i3 t, z i4 t 0 oly if t =... = t 4, so that the secod term of 7.6 is T 2 2 [T r 2 ] [u 2 ] t=[t r ]+ i,...,i 4 =[u ]+ [T r2 ] [T r ] 2 C T + [u] 2 i,...,i 4 =2 l= 2 i,...,i 4 =2 cum z i t, z i2 t, z i3 t, z i4 t [u] cum ε i t,..., ε i4 t i i 4 cum ε jt,..., 4 i l j= j= [T r2 ] [T r ] 2 [u2 ] [u ] 2 C. T ε jt O the other had, the first term of 7.6 is also bouded by the far right side of the last displayed iequality. So this cocludes the proof of the tightess coditio ad so part b of the theorem. We ow look at part a. The proof of part a proceeds similarly, if it is ot easier, to that give i part b oce we observe that [T r] [u] QT,,r,u = ε it ε jt so that it is omitted. = 7.2. Proof of Theorem 2. [u] j= [T r] ε it u [T r] ε it, 7.7 Proof. As i the proof of Theorem, we shall oly hadled explicitly part b, the proof of part a beig similarly hadled. Usig 7.2 ad the that w it = z it t z is, 7.8 t s= the otatio give i the proof of Theorem, we have that U T,,r,u = w it. t=2 Before we look at the structure of the secod momets of U T,,r,u, it is worth emphasizig that if t s for all i, j E w it w js = 0. Bearig this i mid

25 together with 7.3, we the have that [T r ] [u ] [T r 2 ] [u 2 ] E w it = TESTING FOR COMMON TRENDS 25 w it [T r r 2 ] [u u 2 ] σ 2 r r 2 u u 2, because by defiitio of w it ad C, E w 2 it = Ez 2 it t/ t. Next, we examie the covergece i distributio of [u] U T,,r,u = [rt ] w it. t=2 E wit 2 As we argued i the proof of Theorem, for ay fixed t, we have that [u] d w it N 0, σ 2 + u. t From here we obtai that [T r] T t=2 [u] d w it N 0, σ 2 ru 7.9 sice the sequece [u] w it are idepedet i t by C. However, we eed to show that 7.9 also holds whe the limit i ad T is take joitly. To that ed it suffi ces to show that 4 E [u] C but the left side of the last displayed expressio is 2 [u] 3 2 Ewit 2 + [u 2 ] 2 cum w i t, w i2 t, w i3 t, w i4 t = 3σ 4 + o + [u] 2 + [u] 2 i,...,i 4 =2 l= log = 3σ 4 u O. w it i,...,i 4 =2 i,...,i 4 =2 cum z i t,..., z i4 t i i 4 cum z jt,..., 4 i l j= j= So, we coclude that 7.9 also holds whe the limits are takig joitly. z jt

26 26 JAVIER HIDALGO AND JUNGYOON LEE To fiish the proof it remais to verify the tightess coditio of [u] U T,,r,u = [rt ] w it. t=2 To that ed, ad proceedig as i the proof of Theorem, it is eough to show that for 0 r r 2 ad 0 u u 2, r,u,r 2,u 2 := [r 2 T ] [u 2 ] t=[r T ]+ i=[u ]+ satisfies 7.5. The proof is essetially the same as that give for 7.5, i.e. 7.6, but with the obvious chages ad so is omitted. This completes the proof of part b. The proof of part a is essetially the same as, if ot easier tha, that of part b, oce we otice that UT,,r,u = z it T z jt T ad hece is omitted. = 7.3. Proof of Theorem 3. [u] s= w it [T r] ε it u [T r] [u] T r ε it + ru T ε it ε it, Proof. The proof is quite stadard after usig Moo ad Phillips 999 results. First stadard argumets idicate that T i= T ˇx itˇx P it Σ. se- Ideed this is the case because by C3, x it κ i x it κ i are i.i.d. queces of radom variables with fiite secod momets. Next, T /2 i= T ˇx it ε it = T /2 T /2 i= T x it ε it T i= T T x is ε it {x i + x t x }, where we have assumed that κ i = 0 as the left side of the last displayed equality is ivariat to the value of κ i. It is clear that the asymptotic s=

27 TESTING FOR COMMON TRENDS 27 behaviour of the left side of the last displayed equality is give by that of T /2 T x it ε it. 7.0 i= To show that 7.0 coverges i distributio to a ormal radom variable we employ Theorem 2 of Moo ad Phillips 999. But, 4 T E T /2 x it ε is C, by C, C3 ad stadard results. This cocludes the proof of the theorem Proof of Theorem 4. Proof. First by defiitio, we have that Û T,,r,u = ˇε it β β ˇx it. So compared to what we had i the case of o regressors x it, we have ow the extra term β β [T r] [u] ˇx it. But, as we ca cosider the situatio where E x it = κ i 0 without loss of geerality as Eˇx it = 0, we ca easily coclude that sup ˇx it r,u [0,] = o p. 7. Ideed this is the case because for each r, u [0, ] we have ˇX T r, u =: ˇx it = o p, by ergodicity. O the other had, the tightess coditio of ˇX T r, u proceeds as that i Theorem 3. Thus, we coclude that Û T,,r,u = ˇε it + o p /2 uiformly i r, u [0, ], because by Theorem 4, β β = O p /2. From here the proof of the theorem follows i a straightforward fashio.

28 28 JAVIER HIDALGO AND JUNGYOON LEE 7.5. Proof of Theorem 5. Proof. As we have doe with the results i Theorem ad 3, we shall explicitly deal with the proof of part b, part a beig similarly hadled. To that ed, we first otice that we still have that /2 β β d N 0, σ 2 Σ. Next, proceedig as i the proof of Theorem 4, ad usig the otatio i 7.2 ad 7.8, we have that Û T,,r,u = w it + o p /2, uiformly i r, u [0, ]. However Coditio C5 will yield to the fact that, say E z it z jt ca be differet tha zero eve whe i j. Ideed for, stadard algebra gives that E z it z jt = ij + σ 2 i + σ 2 ij ij i 2 i l= σ2 l i = j ij l= σ 2 l i j 7.2 usig the otatio ij = i j ad ij + = i j. O the other had, E w it w is = 0 if s t. 7.3 So, the latter two displayed expressios idicate that the covariace structure of /2 Û T,,r,u differs from whe the secod momets of ε it are the same. Due to this, we ow proceed to characterize the covariace structure of /2 Û T,,r,u, that is [T r ] E by 7.3. Now, [u ] E w it w jt = E [T r 2 ] w it { [u 2 ] w it z it t = E z it z jt + = [T r r 2 ] t z is s= [u ] t t 2 E z is z js s= [u 2 ] j=2 z jt t t s= E w it w jt z js } 7.4

29 TESTING FOR COMMON TRENDS 29 because by C, E z it z js = 0 if t s. So, the right side of 7.4 is [T r r 2 ] [u ] [u 2 { } ] t E z it z jt + t 2 E z is z js = + [T r r 2 ] [T r r 2 ] j= [u u 2 ] [u ] { [u 2 ] Ez 2 it + j i t t s= Ez2 is t t 2 s= s= Ez 2 is { } t E z it z jt + t 2 E z is z js. } s= 7.5 Now because 7.2, C ad stadard algebra gives that the first term o the right of 7.5 is r r 2 [u u 2 ] σ 2 i log + O + log T T by 7.2. O the other had, 7.2 implies that the cotributio due to t t 2 E z is z js s= ito the secod term o the right of 7.5 is O log log T T, so that the behaviour of secod term o the right of 7.5 is give by that of [T r r 2 ] = r r 2 [u ] [u ] [u 2 ] j i [u 2 ] j i E z it z jt σ 2 ij ij + ij ij Hece we coclude that the right side of 7.4 is govered by r r 2 [u ] [u E 2 ] z it z jt = r r 2 [u ] i= [u 2 ] j=2 j= which implies that as, T, it is [u u 2 ] r r 2 lim l= σ 2 l. [u ] [u 2 ] E ε i ε l i ε j ε l j σ 2 i l=i + log i u + log i u 2 + log i u log i u l=j

30 30 JAVIER HIDALGO AND JUNGYOON LEE Next that the fidi of /2 Û T,,r,u coverge i distributio to a Gaussia radom variable with mea zero ad variace [u] i i Φ r, u; r, u = r lim σ 2 i + 2 log + log 2 u u proceeds as stadard ad it is othig differet to the proof uder C. Fially the tightess coditio, for which a suffi ciet coditio is 2 E [T r 2 ] [u 2 ] t=[t r ]+ i=[u ]+ w it 4 C r 2 r +ψ u 2 u +ψ, for some ψ > 0. Now the proof of 7.6 suggests that it suffi ces all we eed to show that [T r 2 ] [u 2 ] 4 2 E i ε it log ε 2 u i t t=[t r ]+ i=[u ]+ C r 2 r +ψ u 2 u +ψ. However the proof of the last displayed iequality is almost routie by C3. This cocludes the proof of part b of the theorem. The proof of part a follows similarly after we otice that [T r ] E [u ] i= [T r 2 ] ε it [u 2 ] i= ε it = This cocludes the proof of the theorem Proof of Theorem 6. = [T r r 2 ] [T r r 2 ] Proof. By defiitio of β we have that /2 β β = ˇx itˇx it /2 So it suffi ces to show that /2 ˇx itˇε it d N 0, σ 2 Σ [u u 2 ] i= [u u 2 ] i= T E ε it ε jt σ 2 i. i probability. ˇx itˇε it. Arguig as we did i the proof of Theorem 4, say, it is clear that we oly eed to show that /2 x it κ i ε d it N 0, σ 2 Σ i probability.

31 TESTING FOR COMMON TRENDS 3 But this is stadard see Shao ad Tu Proof of Theorem 7. Proof. We would just show part a, with part b beig similarly hadled. From the cotiuous mappig theorem it suffi ces to show that ÛT,,r,u = i υ j= it υ t i t js s= υ is j= s= + υ js i t i t coverges to i probability to W r, u. Deotig we write zit =: ε it i ε i jt, wit =: zit t t j= ÛT,,r,u = [T r] [u] wit. Now, because {ε it }T, i, are i.i.d. sequeces of radom variables, E { zi t z i 2 s} = 0; E { wi t w i 2 s} = 0, which implies that [T r ] [u ] [T r 2 ] [u 2 ] E wit wit = [T r r 2 ] [u u 2 ] E w 2 it 7.7 s= z is = r r 2 u u 2 T But the right side of 7.7 coverges i probability to r r 2 u u 2 σ 2 by stadard argumets. Next, we establish that the fiite dimesioal distributios fidi of Û T,,r,u coverge i probability to those of σw r, u. As usual because the Cramer- Wold device, we oly show the covergece i distributio of ÛT,,r,u d σw r, u, r, u [0, ], i probability 7.8 for ay fixed r ad u. To that ed, it is well kow that for ay fixed t Wt = [u] w it d N 0, σ 2 u, i probability ad as W t is idepedet of W s by costructio, we coclude that [T r] T W t d N 0, σ 2 ur, i probability. So, we have established that whe takig limits i ad the i T sequetially, it yields that Û T,,r,u coverges i bootstrap distributio to σw r, u. However, we eed to show that this is the case whe the limits i w 2 it.

32 32 JAVIER HIDALGO AND JUNGYOON LEE ad T are takig joitly. To that ed, it suffi ces to verify that W t satisfy the geeralized Lideberg-Feller coditio: for ay η > 0, as T,, [T r] E W 2 t I W t η P 0, for which a suffi ciet coditio is E Wt 4 = E [u] w it 4 = C T, 7.9 with C T beig a O p sequece of radom variables. By assumptio, E 2 [u] Wt 4 3 = 2 E wit 2 + [u] 2 cum wi t, wi 2 t, wi 3 t, w i 4 t i,...,i 4 =2 = 3 wit 2 + cum ε it,..., ε it + O p + [u] 2 C T. i,...,i 4 =2 4 i l l= i j= cum ε jt,..., ε jt So, 7.9 holds true so is 7.8. To complete the proof of part a, we eed to show tightess. For that purpose, defie a icremet of Q T,,r,u as follows. For 0 r r 2 ad 0 u u 2, r,u,r 2,u 2 := [T r 2 ] [u 2 ] t=[t r ]+ i=[u ]+ A suffi ciet coditio for tightess give by [T E 2 r,u,r 2,u 2 4 r2 ] [T r ] 2 [u2 ] [u ] 2 C T, T 7.20 ad where u 2 u ad T r 2 r sice otherwise 7.20 holds trivially. Now, proceedig as with 7.7, the left side of 7.20 is 3 T T 2 2 [T r 2 ] [u 2 ] Ewit 2 t=[t r ]+ i=[u ]+ [T r 2 ] [u 2 ] t,...,t 4 =[T r ]+ i,...,i 4 =[u ]+ 2 w it. 7.2 cum w i t, w i 2 t 2, w i 3 t 3, w i 4 t 4.

33 TESTING FOR COMMON TRENDS 33 By costructio, cum wi t, wi 2 t 2, wi 3 t 3, wi 4 t 4 0 oly if t =... = t 4, so that the secod term of 7.2 is bouded by [T r2 ] [T r ] 2 [u 2 ] C T 2 { cum ε i t,..., ε i4 t i,...,i 4 =[u ]+ C T [T r2 ] [T r ] T 4 + i l l= 2 [u2 ] [u ] 2. i cum ε jt,..., ε jt O the other had the first term of 7.2 is also bouded by the right side of 7.2. So this cocludes the proof of the tightess coditio ad part a of the theorem. j=

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