A PANIC ATTACK ON UNIT ROOTS AND COINTEGRATION

Transcription

1 Economerica, Vol. 72, No. 4 (July, 2004, A PANIC AACK ON UNI ROOS AND COINEGRAION BY JUSHAN BAI AND SERENA NG his paper develops a new mehodology ha makes use of he facor srucure of large dimensional panels o undersand he naure of nonsaionariy in he daa. We refer o i as PANIC Panel Analysis of Nonsaionariy in Idiosyncraic and Common componens. PANIC can deec wheher he nonsaionariy in a series is pervasive, or variable-specific, or boh. I can deermine he number of independen sochasic rends driving he common facors. PANIC also permis valid pooling of individual saisics and hus panel ess can be consruced. A disincive feaure of PANIC is ha i ess he unobserved componens of he daa insead of he observed series. he key o PANIC is consisen esimaion of he space spanned by he unobserved common facors and he idiosyncraic errors wihou knowing a priori wheher hese are saionary or inegraed processes. We provide a rigorous heory for esimaion and inference and show ha he ess have good finie sample properies. KEYWORDS: Panel daa, common facors, common rends, principal componens.. INRODUCION KNOWLEDGE OF WHEHER a series is saionary or nonsaionary is imporan for a wide range of economic analysis. As such, uni roo esing is exensively conduced in empirical work. Bu in spie of he developmen of many elegan heories, he powerof univariae unirooess is severely consrained in pracice by he shor span of macroeconomic ime series. Panel uni roo ess have since been developed wih he goal of increasing power hrough pooling informaion across unis. Bu pooling is valid only if he unis are independen, an assumpion ha is perhaps unreasonable given ha many economic models imply, and he daa suppor, he comovemen of economic variables. In his paper, we propose a new approach o undersanding nonsaionariy in he daa, boh on a series by series basis, and from he viewpoin of a panel. Raher han reaing he cross-secion correlaion as a nuisance, we exploi hese comovemens o develop new univariae saisics and valid pooled ess for he null hypohesis of nonsaionariy. Our ess are applied o wo unobserved componens of he daa, one wih he characerisic ha i is srongly correlaed wih many series, and one wih he characerisic ha i is largely uni specific. More precisely, we consider a facor analyic model: X i = D i + λ i F + e i where D i is a polynomial rend funcion, F is an r vecor of common facors, and λ i is a vecor of facor loadings. he series X i is he sum of a his paper was presened a he NSF 200 Summer Symposium on Economerics and Saisics in Berkeley, California, he CEPR/Banca d Ialia Conference in Rome, and a NYU, Princeon, orono, Maryland, Virginia, Michigan, and LSE. We hank hree anonymous referees, he edior, and he seminar paricipans for many helpful commens. he auhors acknowledge financial suppor from he NSF (Grans SBR , SES , and SES

2 28 J. BAI AND S. NG deerminisic componen D i, a common componen λ i F, and an error e i ha is largely idiosyncraic. A facor model wih N variables has N idiosyncraic componens bu a small number of common facors. 2 A series wih a facor srucure is nonsaionary if one or more of he common facors are nonsaionary, or he idiosyncraic error is nonsaionary, or boh. Excep by assumpion, here is nohing ha resrics F o be all I( or all I(0. here is also nohing ha rules ou he possibiliy ha F and e i are inegraed of differen orders. hese are no merely cases of heoreical ineres, bu also of empirical relevance. As an example, le X i be real oupu of counry i. I may consis of a global rend componen F, a global cyclical componen F 2, and an idiosyncraic componen e i ha may or may no be saionary. As anoher example, he inflaion rae of durable goods may consis of a componen ha is common o all prices, and a componen ha is specific o durable goods. Wheher hese componens are saionary or nonsaionary is an empirical maer. I is well known ha he sum of wo ime series can have dynamic properies very differen from he individual series hemselves. If one componen is I( and one is I(0, i could be difficul o esablish ha a uni roo exiss from observaions on X i alone, especially if he saionary componen is large. Uni roo ess on X i can be expeced o be oversized while saionariy ess will have no power. he issue is documened in Schwer (989, and formally analyzed in Panula (99, Ng and Perron (200, and among ohers, in he conex of a negaive moving-average componen in he firs-differenced daa. Insead of esing for he presence of a uni roo in X i, he approach proposed in his paper is o es he common facors and he idiosyncraic componens separaely. We refer o such a Panel Analysis of Nonsaionariy in he Idiosyncraic and Common componens as PANIC. PANIC has wo objecives. he firs is o deermine if nonsaionariy comes from a pervasive or an idiosyncraic source. he second is o consruc valid pooled ess for panel daa when he unis are correlaed. PANIC can also poenially resolve hree economeric problems. he firs is he problem of size disorion jus menioned, namely, exising ess in he lieraure end o over-rejec he nonsaionariy hypohesis when he series being esed is he sum of a weak I( componen and a srong saionary componen. he second is a consequence of he fac ha he idiosyncraic componens in a facor model can only be weakly correlaed across i by design. In conras, X i will be srongly correlaed across unis if he daa obey a facor srucure. hus, pooled ess based upon e i are more likely o saisfy he cross-secion independence assumpion required for pooling. he hird relaes o power, and follows from he fac ha pooled ess exploi cross-secion informaion and are more powerful han univariae uni roo ess. 2 his is a saic facor model, and is o be disinguished from he dynamic facor model being analyzed in Forni, Hallin, Lippi, and Reichlin (2000.

3 A PANIC AACK ON UNI ROOS 29 Since he facors and he idiosyncraic componens are boh unobserved, and our objecive is o es if hey have uni roos, he key o our analysis is consisen esimaion of hese componens when i is no known a priori wheher hey are I( or I(0. o his end, we propose a robus common-idiosyncraic (I-C decomposiion of he daa using large dimensional panels, ha is, daases in which he number of observaions in he ime ( and he cross-secion (N dimensions are boh large. Loosely speaking, he large N permis consisen esimaion of he common variaion wheher or no i is saionary, while a large enables applicaion of he relevan cenral limi heorems so ha limiing disribuions of he ess can be obained. Robusness is achieved by a differencing and recumulaing esimaion procedure so ha I( and I(0 errors can be accommodaed. We provide a rigorous developmen of he heory for his esimaion procedure. Our resuls add o he growing lieraure on large dimensional facor analysis by showing how consisen esimaes of he facors can be obained using he mehod of principal componens even wihou imposing saionariy on he errors. hese resuls can be used o sudy oher dynamic properies of he common facors, such as long memory, ARCH effecs, and srucural change, under very general condiions. Several auhors have also developed panel uni roos o resolve he problem of correlaed errors. In Chang (2002, Moon and Perron (2003, Chang and Song (2002, and Phillips and Sul (2003, for example, cross-secion dependence is reaed as a nuisance. In conras, he naure of he cross-secion dependence is iself an objec of ineres in our analysis. We allow for he possibiliy of muliple facors, and he framework is hus more general han he one-way error componen model of Choi (2002. Furhermore, hese papers are ulimaely ineresed in esing for uni roos in he observed daa. We go beyond his o analyze he source of nonsaionariy. In doing so, we provide a coheren framework for sudying uni roos, common rends, and common cycles in large dimensional panels. Our framework differs from convenional mulivariae ime-series models in which N is small. In small N analysis of coinegraion, common rends, and cycles, he esimaion mehodology being employed ypically depends on wheher he variables considered are all I( or all I(0. 3 Preesing for uni roos is hus necessary. Because N is small, wha is exraced is he rend or he cycle common o jus a small number of variables. No only is he informaion in many poenially relevan series lef unexploied, consisen esimaion of he common facors is in fac no possible when he number of variables is small. In our analysis wih N and large, he common variaion can be exraced wihou appealing o saionariy assumpions and/or coinegraion resricions. his makes i possible o decouple he exracion of common rends and cycles from he issue of esing saionariy. 3 For example, King, Plosser, Sock, and Wason (99, Engle and Kozicki (993, and Gonzalo and Granger (995.

4 30 J. BAI AND S. NG he res of he paper is organized as follows. In Secion 2, we describe he PANIC procedures and presen asympoic resuls for esing single and muliple uni roos. We devoe Secion 3 o he large sample properies of he facor esimaes. As an inermediae resul, we esablish uniform consisency of he facor esimaes wihou assuming he errors are saionary. his resul is of ineres in much broader conexs han uni roo esing. Secion 4 uses simulaions o illusrae he properies of he facor esimaes and he ess in finie samples. 2. PANIC he daa X i (i = N; = are assumed o be generaed by ( (2 (3 X i = c i + β i + λ F i + e i (I LF = C(Lu ( ρ i Le i = D i (Lɛ i where C(L = j=0 C jl j and D i (L = j=0 D ijl j. he idiosyncraic error e i is I( if ρ i =, and is saionary if ρ i <. We allow r 0 saionary facors and r common rends, wih r = r 0 + r. Saed differenly, he rank of C( is r. he objecive is o deermine r and es if ρ i = when neiher F nor e i is observed and will be esimaed by he mehod of principal componens. 2.. Assumpions and Overview Le M< be a generic posiive number, no depending on or N. Le A =race(a A /2. Our analysis is based on he following assumpions: ASSUMPION A: (i For nonrandom λ i, λ i M; for random λ i, E λ i 4 M; (ii N N λ i= iλ p i Σ Λ, an r r posiive definie marix. ASSUMPION B: (i u iid(0 Σ u, E u 4 M; (ii var( F = j=0 C jσ u C j > 0; (iii j=0 j C j <M; and (iv C( has rank r,0 r r. ASSUMPION C: (i For each i, ɛ i iid(0 σ 2 ɛi, E ɛ i 8 M, j=0 j D ij <M, ω 2 ɛi = D i ( 2 σ 2 ɛi > 0; (ii E(ɛ i ɛ j = τ ij wih N i= τ ij M for all j; (iii E N /2 N i= [ɛ isɛ i E(ɛ is ɛ i ] 4 M, for every ( s. ASSUMPION D: he errors ɛ i, u, and he loadings λ i are hree muually independen groups. ASSUMPION E: E F 0 M, and for every i = N, E e i0 M.

5 A PANIC AACK ON UNI ROOS 3 Assumpion A is made on he facor loadings o ensure ha he facor srucure is idenifiable. I is a common assumpion in facor analysis. A se of facors F is deemed o be pervasive if and only if he corresponding loading coefficiens are such ha N N λ i= iλ i converges o a posiive definie marix as N. A variable ha has only a finie number of nonzero loadings does no saisfy his condiion and is no a facor in our large N framework. Insead, is variaion will be considered idiosyncraic, and hus included in ɛ i. Under Assumpion B, he shor run variance of F is required o be posiive definie, bu he long-run covariance of F can be reduced rank o permi linear combinaions of I( facors o be saionary. When r = 0andhere are no sochasic rends, C( is null because F is overdifferenced. On he oher hand, when r 0, one can consider a roaion of F by a marix G such ha he firs r elemens of GF are inegraed, while he final r 0 elemens are saionary. One such roaion is given by G =[β β ],whereβ is r r saisfying β β = I r,andβ β = 0. We define Y = β F o be he r common sochasic rends resuling from such a roaion. Assumpion C(i allows some weak serial correlaion in ( ρ i Le i wih ρ i possibly differen across i, while C(ii and C(iii allow weak cross-secion correlaion. Clearly, C(ii holds if ε i are cross-secionally uncorrelaed. he assumpion obviously holds if here exiss an ordering of he cross secions such ha he ordered ε i (i = 2 Nis a mixing process. 4 Bu he assumpion is more general. I allows weak cross-correlaion in he errors, weak in he sense ha even as N increases, he column sum of he error covariance marix remains bounded. Chamberlain and Rohschild (983 defined an approximae facor model as one in which he larges eigenvalue of Ω is bounded. Bu if e is saionary wih E(e i e j = τ ij, hen from marix heory, he larges eigenvalue of Ω is bounded by max τ j i= ij. Since C(ii requires ha N τ i= ij M N for all j and all N, we have an approximae facor model in he sense of Chamberlain and Rohschild (983. Under Assumpion D, ɛ i, u,andλ i are muually independen across i and. he assumpion is sronger han he one used in Bai and Ng (2002, which permis u and ɛ i o be weakly correlaed. Assumpion E is an iniial condiion assumpion made commonly in uni roo analysis. Our facor esimaes are based on he mehod of principal componens. When e i is I(0, he principal componens esimaors for F and λ i have been shown o be consisen when all he facors are I(0 and when some or all of hem are I(. Bu consisen esimaion of he facors when e i is I( has no been considered in he lieraure. Indeed, when e i has a uni roo, a regression of X i on F is spurious even if F was observed, and he esimaes of λ i and hus of e i will no be consisen. he validiy of PANIC hus hinges on he abiliy o obain esimaes of F and e i ha preserve heir orders of inegraion, 4 Such an assumpion was made in Connor and Korajzcyk (986.

6 32 J. BAI AND S. NG boh when e i is I( and when i is I(0. A conribuion of his paper is o show how his can be accomplished. Essenially, he rick is o apply he mehod of principal componens o he firs-differenced daa. o be precise, suppose we observe X, a daa marix wih ime-series observaions and N cross-secion unis. Suiably ransform X o yield x, a se of ( N saionary variables. Le f = (f 2 f 3 f and Λ = (λ λ N. he principal componen esimaor of f, denoed f ˆ,is imes he r eigenvecors corresponding o he firs r larges eigenvalues of he ( ( marix xx. Under he normalizaion f ˆ f/( ˆ = I r,he esimaed loading marix is ˆΛ = x f/( ˆ. Before urning o he deails, an overview of he inference procedures gives an idea of wha is o follow. If here is one facor, PANIC will es if i is a uni roo process. If here are muliple facors, PANIC will deermine r,he number of independen sochasic rends underlying he r common facors. In addiion, PANIC will es if here is a uni roo in each of he idiosyncraic errors. An imporan aspec of PANIC is ha he idiosyncraic errors can be esed for he presence of a uni roo wihou knowing if he facors are saionary, and vice versa. In fac, he ess on he facors are asympoically independen of he ess on he idiosyncraic errors. In each case, we allow for he possibiliy ha he differenced saionary series are serially correlaed wih (possibly infinie auoregressive represenaions. he wo univariae ess will be denoed by ADFê(i and ADF ˆF respecively, as hey are based on he es of Said and Dickey (984 using an augmened auoregression wih suiably chosen lag lenghs. In he case when r>, we consider wo ess. he firs filers he facors under he assumpion ha hey have finie order VAR represenaions. he second correcs for serial correlaion of arbirary form by nonparamerically esimaing he relevan nuisance parameers. Accordingly, he filered es is denoed by MQ f, and he correced es is denoed by MQ c. hese are modified versions of he Q f and Q c ess developed in Sock and Wason (988. he definiion of x depends on he deerminisic rend funcion. We consider wo specificaions, leading o wha we will call he inercep only model and he linear rend model. he superscrips c and τ will be used o disinguish hese wo cases. he focus ofhis secion is uni roo inference. he properies of he facor esimaes will be deferred o Secion 3. he heory proceeds assuming r is known. We will reurn o he deerminaion of r in pracice in Secion he Inercep Only Case he facor model in he inercep only case is (4 X i = c i + λ i F + e i

7 A PANIC AACK ON UNI ROOS 33 Denoe (5 x i = X i f = F and z i = e i hen he model in firs-differenced form is (6 x i = λ i f + z i Applying he mehod of principal componens o x yields r esimaed facors f ˆ, he associaed loadings ˆλ i, and he esimaed residuals, ẑ i = x i ˆλ ˆ if.define for = 2 : ê i = ˆF = ẑ i fˆ s an r vecor. (i = N. Le ADF c ê (i be he saisic for esing d i0 = 0 in he univariae augmened auoregression (wih no deerminisic erms ê i = d i0 ê i + d i ê i + +d ip ê i p + error 2. If r =, le ADF cˆf be he saisic for esing δ 0 = 0 in he univariae augmened auoregression (wih an inercep: ˆF = c + δ 0 ˆF + δ ˆF + +δ p ˆF p + error 3. If r>, demean ˆF and define ˆF c = ˆF ˆF,where ˆF = ( ˆF. Sar wih m = r: A: Le ˆβ be he m eigenvecors associaed wih he m larges eigenvalues ˆF c.leŷ c of 2 ˆF c = ˆβ ˆF c B.I: Le K(j = j/(j +, j = 0 J: (i Le ˆξ c and le ( J ˆΣ c = K(j j=. wo saisics can be considered: be he residuals from esimaing a firs-order VAR in Ŷ c, ˆξ c c j ˆξ (ii Le ν c c (m be he smalles eigenvalue of [ ]( ˆΦ c (m = 5 c (Ŷ c Ŷ c + Ŷ c Ŷ c (ˆΣ c c + ˆΣ Ŷ Ŷ c c

8 34 J. BAI AND S. NG (iii Define MQ c(m = c [ˆνc c (m ]. B.II: For p fixed ha does no depend on N or : (i Esimae a VAR of order p in Ŷ c o obain ˆΠ(L = I m ˆΠ L ˆΠ p L p.filerŷ c by ˆΠ(L o ge ŷ c = ˆΠ(LŶ c. (ii Le ˆν c f (m be he smalles eigenvalue of [ ˆΦ c (m = 5 f (ŷ c ŷc + ŷc ŷc ]( ŷ c ŷc (iii Define he saisic MQ c (m = f [ˆνc f (m ]. C: If H 0 : r = m is rejeced, se m = m and reurn o sep A. Oherwise, ˆr = m and sop. HEOREM (heinerceponlycase:suppose he daa are generaed by (2, (3, and (4 and Assumpions A E hold. Le W u and W ɛi (i = N be sandard Brownian moions. he following resuls hold as N.. Le p be he order of auoregression chosen such ha p and p 3 / min[n ] 0. Under he null hypohesis ha ρ i =, ADF c ê (i 0 W ɛi(s dw ɛi (s ( 0 W ɛi(s 2 ds /2 (i = N 2. (r =. Le p be he order of auoregression chosen such ha p and p 3 / min[n ] 0. Le W c(s = W u u(s W 0 u(s ds. Under he null hypohesis ha F has a uni roo, ADF cˆf W c(s dw 0 u u(s ( r W c 0 u (s2 ds /2 3. (r >. Le W m be an m-vecor sandard Brownian moion, W c = W m m W 0 m. Le ν c (m be he smalles eigenvalue of Φ c = [ 2 [W c (W c m m ( I m ] W c (sw c m m ds] (s 0 (i Le J be he runcaion lag of he Barle kernel, chosen such ha J and J/min[ N ] 0. hen under he null hypohesis ha F has m sochasic rends, [ˆν c d c (m ] ν c(m. (ii Under he null hypohesis ha F has m sochasic rends wih a finie VAR( p represenaion and a VAR(p is esimaed wih p p, [ˆν c d f (m ] ν c(m.

9 A PANIC AACK ON UNI ROOS 35 Separaely esing F and e i allows us o disenangle he source of nonsaionariy. If F is nonsaionary bu e i is saionary, we say he nonsaionariy of X i is due o a pervasive source. On he oher hand, if F is saionary bu e i is nonsaionary, hen he nonsaionariy of X i is due o a series-specific source.evidenly,ifbohf and e i are nonsaionary, boh common and idiosyncraic variaions conribue o he inegraedness of X i.alhoughdirec esing of F and e i is no feasible, heorem shows ha esing ê i and ˆF are hesameasife i and F were observable. As shown in he Appendix, /2 ê i = /2 e i + o p (,whereheo p ( erm is uniform in. he asympoic disribuion of ADF c ê (i coincides wih he DF es developed by Dickey and Fuller (979 for he case of no consan. he criical value of he es a he 5% significance level is 95. When he firs-differenced daa X i conain no deerminisic erms, /2 ˆF = /2 HF + o p (, whereh is a full rank marix and he o p ( erm is uniform in. his means ha he difference beween he space spanned by esimaed facors and he rue facors is small. esing for a uni roo in demeaned ˆF is asympoically he same as esing for a uni roo in demeaned F. When r =, his is a simple univariae es. he ADF cˆf has he same limiing disribuion as he DF es for he consan only case. he 5% asympoic criical value is Assuming ha he series o be esed is observed, Said and Dickey (984 showed ha he ADF based upon an augmened auoregression has he same limiing disribuion as he DF ifhenumberoflagsischosensuchha p 3 / 0asp. In our analysis, he series o be esed are ê i and ˆF. Since hese are esimaes of e i and F, he allowed rae of increase in p depends on he rae a which he esimaion errors vanish, giving he resul p 3 / min[n ] 0 as saed. If all facors are I(, linear combinaion of he facors will be I(. If all facors are I(0, heir linear combinaions will sill be I(0. However, linear combinaions of I( and I(0 facors canremaini(.sincewecanonlyesimae he space spanned by he facors, individually esing each of he facors for he presence of a uni roo will, in general, oversae he number of common rends. Accordingly, we need o deermine he number of basis funcions spanning he nonsaionary space of F. Sock and Wason (988 proposed wo saisics, denoed Q f and Q c, designed o es if he real par of he smalles eigenvalue of an auoregressive coefficien marix is uniy. While he Q c assumes he nonsaionary componens of F f o be finie order vecorauoregressive processes, he Q c c allows he uni roo process o have more general dynamics, including moving-average errors. Our proposed MQ c c and MQ c f essaremodifiedvariansofsockand Wason s Q c and c Qc f. he basic difference is in he numeraor of he ess. Insead of = ŷc ŷc in Qc,weuseasnumeraor 5[ f = (ŷc ŷc + ŷc ŷc ], which can be hough of as an average of he auocovariance of ŷ c a lead

10 36 J. BAI AND S. NG and lag one. Such a numeraor was also considered by Phillips and Durlauf (986 for esing uni roos in vecor ime series. In he presen conex, he modificaion serves wo purposes. Firs, he modified numeraor is symmeric, and hus he eigenvalues are always real. Second, when r =, he ideniy ŷc ŷc = 5[(ŷ c 2 ( ŷc 2 ] holds under he null hypohesis of a uni roo, leading in he limi o Io s Lemma, W 0 c(s dw (s = 5[W c ( 2 ]. As is well known, he numeraor of he ADF es can be represened eiher way. In he mulivariae case, he analogous ideniy is (ŷc ŷc + ŷ c ŷc = ŷ c ŷc ŷc ŷc, and so, in he limi, he numeraors of our MQ c ess sill have wo equivalen represenaions. A a more echnical level, he modificaions allow us o exploi use of he ideniy, which subsanially simplifies he proofs. he limiing disribuions of [ ˆΦ c (m I f m] and [ ˆΦ c(m I c m] are of he form AΦ c A, which has he same eigenvalues as Φ c.5 he criical values of boh ess can hus be obained by simulaing Φ c, which is based upon a vecor of sandard Brownian moions. hese are repored in able I. Sricly speaking, he MQ c f (m es is valid only when he common rends can be represened as finie order AR(p processes. From a heoreical poin of view, he MQ c c (m is more general, as i only requires he weakly dependen errors o saisfy he momen condiions of Assumpion B. We can hen perform kernel esimaion of he long-run minus he shor-run residual variance of a VAR(. heorem, based upon he Barle kernel as in Newey and Wes (987, shows ha so long as he number of auocovariances, J, does no increase oo fas, serial correlaion of arbirary form can be effecively removed nonparamerically. One can expec he resuls o generalize o oher kernels, wih appropriae resricions o he runcaion poin. ABLE I CRIICAL VALUES FOR MQ c AND MQ f FOR ESING H 0 : r = m A SIGNIFICANCE LEVEL ϕ MQ c c f MQ τ c f m \ ϕ Sock and Wason (988 sugges normalizing he Π(L esimaes so ha ŷ has uni variance.hisisinfacnonecessary.

11 A PANIC AACK ON UNI ROOS 37 In our analysis, r is esimaed by successive applicaion of he MQ ess. If he chosen significance level a each sage is ϕ, henp(ˆr = r ϕ, and he overall asympoic ype I error is also ϕ. his propery is generic of successive esing procedures, including Johansen s race and eigenvalue ess for he number of coinegraing vecors. 6 he resul is a consequence of he fac ha he ess are applied o he same ˆF and hus no independen across sages. When an observed series is esed for a uni roo using he ADF,i is known ha he saisic diverges a rae under he alernaive of saionariy when p is chosen o increase wih such ha p 3 / 0as. HEOREM 2: Suppose ê i is esed for a uni roo using he ADF, he assumpions of heorem hold, and p is chosen such ha p and p 4 / min[n ] 0 as N and. Under he alernaive of saionariy, he saisic diverges a rae min[ N ]. Alhough ê i yields asympoically valid inference abou nonsaionariy of e i,hefachae i is unobserved is no innocuous. As indicaed in heorem, he usual Dickey Fuller limiing disribuion obains only when p 3 / min[n ] 0asp N. heorem 2 shows ha for he es o be consisen, we need p 4 / min[n ] 0. As shown in he Appendix, he divergence rae is N if /N,andis if /N is bounded. he overall rae is hus min[ N ]. Essenially, he power of he es is deermined by how fas he error in esimaing he facors vanishes, and hus depends on boh N and.heaboveraeofdivergencealsoappliesoesing ˆF when r = he Linear rend Case Consider now he facor model in he case of a linear rend: (7 X i = c i + β i + λ i F + e i and hus X i = β i + λ F i + e i Le F = ( F, e i = ( e i, and X i = ( X i. PANIC proceeds as follows for he case of a linear rend. Define X i X i = λ i ( F F + ( e i e i which can be rewrien as (8 x i = λ i f + z i 6 For a deailed discussion, see Johansen (995, pp. 69 and Sock and Wason showed (in an unpublished appendix consisency of he Q ess. Even for observed daa, he proof is quie involved. We conjecure a similar resul o hold for he MQ ess.

12 38 J. BAI AND S. NG where (9 x i = X i X i f = F F z i = e i e i Noe ha he differenced and demeaned daa x i are invarian o c i and β i. As a consequence, here is no loss of generaliy o assume E( F = 0. For example, if F = a + b + η such ha E( η = 0, hen we can rewrie model (7 wih F replaced by η and c i + β i replaced by c i + λ a + (β i i + λ b. i Le f ˆ and ˆλ i be he esimaes obained by applying he mehod of principal componens o he differenced and demeaned daa, x,wihẑ i = x i ˆλ ˆ if,and ê i = ẑi. Alsole ˆF = ˆ f s,anr vecor.. Le ADF τ(i be he saisic for esing d ê i0 = 0 in he univariae augmened auoregression (wih no deerminisic erms ê i = d i0 ê i + d i ê i + +d ip ê i p + error 2. If r =, le ADF τˆf be he saisic for esing δ 0 = 0 in he univariae augmened auoregression (wih an inercep and a ime rend ˆF = c 0 + c + δ 0 ˆF + δ ˆF + +δ p ˆF p + error 3. If r>, le ˆF τ be he residuals from a regression of ˆF on a consan and a ime rend. Repea sep (3 for he inercep only case wih ˆF τ o yield Ŷ τ and ŷ τ.denoeheessbymqτ f (m,andmqτ(m. c replacing ˆF c HEOREM 3 (he Linear rend Case: Suppose he daa are generaed by (2, (3, and (7, and he assumpions of heorem hold. Le W u and W ɛi, i = N be sandard Brownian moions. he following hold as N :. Le p be he order of auoregression chosen such ha p and p 3 / min[n ] 0. Le V ɛi (s = W ɛi (s sw ɛi ( be a Brownian bridge. Under he null hypohesis ha ρ i =, ( /2 ADF τ (i ê V ɛi (s ds 2 (i = N (r =. Le p be he order of auoregression chosen such ha p and p 3 / min[n ] 0. Le W τ( = W u u( (4 7sW 0 u(s ds (2 0 6sW u (s ds. When r = and under he null hypohesis ha F has a uni roo, ADF τˆf W τ(s dw 0 u u(s ( W τ 0 u (s2 ds /2

13 A PANIC AACK ON UNI ROOS (r >. Le W τ m be a vecor of m-dimensional derended Brownian moions. Le ν τ (m be he smalles eigenvalue of Φ τ = [ 2 [W τ (W τ m m ( I m ] W τ (sw τ m m ds] (s 0 (i Le J be he runcaion poin of he Barle kernel, chosen such ha J/min[ N ] 0 as J N. hen under he null hypohesis ha F has m sochasic rends, MQ τ(m d c ν τ(m. (ii Under he null hypohesis ha F has m sochasic rends wih a finie VAR( p represenaion, and a VAR(p is esimaed wih p p, hen MQ τ(m d f ν τ(m. he limiing disribuion of ADF coincides wih he DF for he case wih τˆf a consan and a linear rend. However, as shown in he Appendix, he consequence of having o demean X i is ha /2 ê i converges o a Brownian bridge insead of a Brownian moion. he limiing disribuion of he ADF τ is ê no a DF ype disribuion, bu is proporional o he reciprocal of a Brownian bridge. here are hree imporan feaures of PANIC ha are worhy of highlighing. Firs, he ess on he facors can be performed wihou knowing if he idiosyncraic errors are saionary or nonsaionary. Second, he uni roo es for e i is valid wheher e j j i, is I( or I(0, and in any even, such knowledge is no necessary. hird, he es on he idiosyncraic errors do no depend on wheher F is I( or I(0. In fac, he limiing disribuions of ADF c ê (i and ADF τ ê (i do no depend on he common facors. his propery is useful for consrucing pooled ess Pooled ess A common criicism of univariae uni roo ess is low power, especially when is small. his has generaed subsanial ineres in improving power. A popular mehod is o pool informaion across unis, leading o panel uni roo ess. Recen surveys of panel uni roo ess can be found in Maddala and Wu (999 and Balagi and Kao (200. he early es developed in Quah (994 imposed subsanial homogeneiy in he cross-secion dimension. Subsequen ess such as ha of Levin, Lin, and Chu (2002 and Im, Pesaran, and Shin (2003 allow for heerogeneous inerceps and slopes, while mainaining he assumpion of independence across unis. his assumpion is resricive, and if violaed, can lead o over-rejecions of he null hypohesis. Banerjee, Marcellino, and Osba (200 argued agains use of panel uni roo ess because of his poenial problem. O Connell (998 provides a GLS soluion o his problem, bu he approach is heoreically valid only when N is fixed.

14 40 J. BAI AND S. NG When N also ends o infiniy, as is he case under consideraion, consisen esimaion of he GLS ransformaion marix is no a well defined concep since he sample cross-secion covariance marix will have rank when N>even when he populaion covariance marix is rank N. If cross-secion correlaion can be represened by common facors, hen heorems and 2 show ha univariae ess for ê i do no depend on Brownian moions driven by he common innovaions u asympoically. hus, if e i is independen across i,essbaseduponê i are asympoically independen across i. Consider he following: HEOREM 4: Suppose e i is independen across i and consider esing H 0 : ρ i = i agains H : ρ i < for some i. Le p c(i and ê pτ ê (i be he p-values associaed wih ADF c(i and ADF τ ê ê (i, respecively. hen P c ê = 2 N P τ ê = 2 N 4N i= log pc ê 4N i= log pτ ê (i 2N (i 2N d N(0 d N(0 Under he assumpion ha e i is independen across i, essforê i are independen across i asympoically. he p-values are hus independen U[0,] random variables. his implies ha minus wo imes he logarihm of he p-value is a χ 2 random variable wih wo degrees of freedom. he es 2 N ln p i= X(i was firs proposed in Maddala and Wu (999 for esing a fixed number of observed series. Choi (200 exended he analysis o allow N by sandardizaion. Pooling on he basis of p-values is widely used in mea analysis. I has he advanage of allowing for as much heerogeneiy across unis as possible. For example, i can be used even when he panel is nonbalanced. Alernaively, one can also es if he pooled coefficien esimaed by regressing ê i on ê i is saisically differen from uniy. Such a pooled es would be in he spiri of Levin, Lin, and Chu (2002. A pooled es of he idiosyncraic errors can be seen as a panel es of no coinegraion, as he null hypohesis ha ρ i = for every i holds only if no saionary combinaion of X i can be formed. I differs from oher panel coinegraion ess in he lieraure, such as developed in Pedroni (995, in ha our framework is based on a large N, and he es is applied o ê i insead of X i. While panel uni roo ess for X i are inappropriae if he daa admi a facor srucure, pooling of ess for ê i is asympoically valid under he more plausible assumpion ha e i is independen across i. I should be made clear ha he univariae ess proposed in heorems and 3 permi weak cross-secion correlaion of he idiosyncraic errors. I is only in developing pooled ess ha independence of he idiosyncraic errors is assumed. he independence assumpion can, in principle, be relaxed by allowing he number of cross-correlaed errors

15 A PANIC AACK ON UNI ROOS 4 o be finie so ha as N increases, he p-values are averaged over infiniely many unis ha are no cross-correlaed. 3. CONSISENCY OF ˆF he asympoic resuls saed in he previous secion require consisen esimaion of F and e i when some, none, or all of hese componens are I(. Bai and Ng (2002 considered esimaion of r and showed ha he squared deviaions beween he esimaed facors and he rue facors vanish, while Bai (2003 derived he asympoic disribuions for he esimaed F and λ i.boh sudies assume he errors are all I(0. However, we need consisen esimaes no jus when e i is I(0, bu also when i is I(. he insigh of he presen analysis is ha, by applying he mehod of principal componens o he firs-differenced daa, i is possible o obain consisen esimaes of F and e i, regardless of he dynamic properies of F and e i.o skech he idea why his is he case, assume β i = 0. he facor model in differenced form is X i = λ i F + e i. Clearly, differencing removes he fixed effec c i. his is desirable because a consisen esimae of i canno be obained when e i is I(. Now if e i is I(, e i = z i will be I(0. Under Assumpion C, z i has weak cross-secion and serial correlaion. Consisen esimaes of F can hus be obained. If e i is I(0, e i, alhough over-differenced, is sill saionary and weakly correlaed. hus, consisen esimaion of F can once again be shown. We summarize hese argumens in he following lemma. LEMMA : Le f be defined by (5. Consider esimaion of (6 by he mehod of principal componens and suppose Assumpions A E hold. hen here exiss an H wih rank r such ha as N, (a min{n } ˆ f Hf 2 = O p ( (b min{ N }( f ˆ Hf = O p ( for each given, (c min{ N}(ˆλ i H λ i = O p (, for each given i. he resuls also hold when f is defined by (9 and (8 is esimaed. Asiswellknowninfacoranalysis,λ i and f are no direcly idenifiable. herefore, when assessing he properies of he esimaes, we can only consider he difference in he space spanned by f ˆ and f, and likewise beween ˆλ i and λ i. he marix H is defined (in he Appendix such ha Hf is he projecion of fˆ on he space spanned by he facors, f. Resul (a is proved in Bai and Ng (2002, while (b and (c are proved in Bai (2003. I should be remarked ha when e i is I(0, esimaion using he daa in level form will give a direc and consisen esimae on F. Alhough hese esimaes could be more efficien han he ones based upon firs differencing, hey are no consisen when e i is I(. In Pesaran and Smih (995, i was shown ha spurious correlaions beween wo I( variables do no arise in cross-secion regressions esimaed

16 42 J. BAI AND S. NG wih ime averaged daa under he assumpion of sricly exogenous regressors, i.i.d. errors, and fixed. Phillips and Moon (999 showed ha an average long-run relaion, defined from long-run covariance marices of a panel of I( variables, can be idenified when N and are boh large. Lemma shows ha he individual relaions (no jus he average can be consisenly esimaed under a much wider range of condiions: he regressors are unobserved, hey can be I( or I(0, and he individual regressions may or may no be spurious. Alhough λ i and f can be consisenly esimaed, he series we are ineresed in esing are ˆF = ˆ f s and ê i = ẑi. hus, we need o show ha given esimaes of f and z i, ˆF and ê i are consisen for F and e i, respecively. LEMMA 2: Under he assumpions of Lemma, max ( ˆ f s Hf s = O p(n /2 + O p ( 3/4 he lemma says ha he cumulaive sum of f ˆ is uniformly close o he cumulaive sum of f provided N. 8 Because ˆF = ˆ f s and H f = H F s = HF HF, Lemma 2 can be saed as (0 max ˆF HF + HF =O p (N /2 + O p ( 3/4 Since a locaion shif does no change he nonsaionariy propery of a series, esing he demeaned process ˆF ˆF is asympoically he same as esing H(F F. his resul is insrumenal in obaining he limiing disribuions of uni roo ess for F. I would seem ha for esing ê i, his resul may no be sufficien since ê i also depends on ˆλ i. Bu as shown in he Appendix, we only require (ˆλ i H λ i o be o p ( for uni roo ess on ê i o yield he same inference as esing e i, and by Lemma (c, his holds provided N and end o infiniy. hus, he condiions for valid esing of F and e i using ˆF and ê i are he same. An implicaion of Lemma 2 is ha 2 ˆF ˆF 2 H( F F H p 0. ha is, he sample variaion generaed by ˆF is he same order as F. If 2 F F has r nondegenerae eigenvalues, 2 ˆF ˆF also has r nondegenerae eigenvalues. hus if F has r common rends, ˆF will also have r common rends. his resul is insrumenal in he developmen of he MQ c and MQ f ess. 8 he O p ( 3/4 can be replaced by O p (log /if he momen generaing funcion of f exiss (i.e., if Ee τ f M for all and for some τ>0.

17 A PANIC AACK ON UNI ROOS 43 Uniform convergence of he facor esimaes in large panels was proved in Sock and Wason (2002 under he assumpion ha N 2 and ha F and e i are saionary. However, our analysis provides a more general uniform consisency resul as a by-produc. Upon muliplying (A.2 by,wehave ( max ˆF HF + HF =O p ( /2 N /2 + O p ( /4 As saed in (, ˆF is uniformly consisen for HF (up o a shif facor HF provided /N 0asN. his resul is quie remarkable in ha he common sochasic rends can be consisenly esimaed by he mehod of principal componens, up o a roaion and a shif in level, wihou knowing wheher F or e i is I(0 or I(. his means ha even if each cross-secion equaion is a spurious regression, he common sochasic rends are well defined and can be consisenly esimaed, if hey exis.his is nopossible wihin he framework of radiional ime-series analysis, in which N is fixed. heresulhawhenn and are large, he space spanned by he common facors can be consisenly esimaed under very general condiions is no merely a srong resul of heoreical ineres. I is also of pracical ineres because i opens he possibiliy of esing oher properies of F using ˆF.For example, ARCH and long memory effecs can be assessed, parameer insabiliy ess can be devised, and he relaive imporance of he common and he idiosyncraic componens can be evaluaed even when neiher is observed. Because Lemma 2 is poenially useful in conexs oher han uni roo esing, we saed i as a primary resul. I should be made clear ha uniform consisency is no necessary for PANIC, and hus we do no require /N 0, hough our resuls will hold under hese sronger condiions. For PANIC o be valid, only Lemmas and 2 are necessary. 4. MONE CARLO SIMULAIONS We begin by using a model wih one facor o show ha ˆF consruced as ˆ f is robus o differen saionariy assumpions abou e i,wheref ˆ is esimaed from firs-differenced daa. We generae F as an independen random walk of N(0 errors wih = 00, and λ i is i.i.d. N(. Daa are generaed according o X i = λ i F + e i.wehenconsruc ˆF as discussed in Secion 2 for he inercep only model. In pracice, a comparison of F wih ˆF canno be made because he former is unobservable. Bu F is known in simulaions. hus, for he sake of illusraion, we compare he fied values from he regression F = a + b ˆF + error wih F. An implicaion of Lemma 2 is ha his fied value (which we will coninue o call ˆF should be increasingly close o F as N increases. On he oher hand, esimaion using he daa in levels will no have his consisency propery.

18 44 J. BAI AND S. NG For he case when e i is I(, we simulae a random walk driven by i.i.d. N(0 errors for N=20, 50, and 00, respecively. We hen esimae he facors using (i differenced daa, and (ii he daa in level form. Figures (b, (c, and (d display he rue facor process F along wih ˆF.Evidenly, ˆF ges closer o F as N increases if he daa are differenced. In fac ˆF is close o he rue process even when N = 20. On he oher hand, when he mehod of principal componens is applied o levels of he same daa, all he esimaed series are far from he rue series, showing ha esimaion using he daa in levels is no consisen when e i is I(. We nex assume he idiosyncraic errors are all I(0 by drawing e i from an i.i.d. N(0 disribuion. Figure 2 illusraes ha even hough he daa are over-differenced, he esimaes are very precise. In his case, boh he level and differenced mehods give almos idenical esimaes. We now use simulaions o illusrae he finie sample properies of he proposed ess. hroughou, he number of replicaions is In heory, r is no known. We showed in Bai and Ng (2002 ha he number of facors in saionary daa can be consisenly deermined by informaion crieria (PC p if he penaly on an addiional facor is specified as a funcion of boh N and.in he presen conex, we can consisenly esimae r from he firs-differenced FIGURE. rue and esimaed F when e i is I(.

19 A PANIC AACK ON UNI ROOS 45 FIGURE 2. rue and esimaed F when e i is I(0. daa. In he simulaions, he following is used: ˆr = arg min k=0 k max IC (k where IC (k = log ˆσ 2 (k + k log ( N N + N + N where ˆσ 2 (k = N N i= = ẑ2 i, ẑ i are he esimaed residuals from principal componens esimaion of he firs-differenced daa, and k max = 6. In all he configuraions considered (up o 3 rue facors, he crierion always selecs ˆr = r he Case r = We simulae daa using X i = λ i F + e i,wihe i = ρe i + ɛ i,andf = αf + u,wihλ i N(0, ɛ i N(0, andu N(0 σ 2.Weconsider F hree values of σ 2 F wih he imporance of he common componen increasing in he value of σ 2. In he simulaions, ρ F i is he same across i.wealsoconsider 9 Using IC 2 in Bai and Ng (2002, P(ˆr = r is someimes.98. he choice of a penaly ha saisfies he condiions of Bai and Ng (2002 is imporan.

20 46 J. BAI AND S. NG foureen pairs of (ρ i α.whenρ i = buα<, he errors are nonsaionary bu he facors are saionary. When α = buρ i <, he facors are uni roo processes bu he errors are saionary. We repor resuls for = 00, and N = in able II. he column labeled ˆF is he rejecion rae of he ADF es applied o he esimaed common facor. he remaining hree columns are he average rejecion raes, where he average is aken across N unis over 5000 rials. Resuls for a paricular i are ABLE IIA REJECION RAES FOR HE NULL HYPOHESIS OF A UNI ROO,INERCEP ONLY, r = σ F = 0 σ F = σ F = 5 ρ i α X ˆF ê P c X P c ê X ˆF ê P c X P c ê X ˆF ê P c X P c ê = 00 N = = 00 N = Noe: he daa are generaed as e i = ρ i e i + ɛ i and F = αf + u. Columns under X and ê are rejec raes of he ADF. P c and P τ are rejecion raes of he pooled ess.

21 A PANIC AACK ON UNI ROOS 47 ABLE IIB REJECION RAES FOR HE NULL HYPOHESIS OF A UNI ROO, LINEAR REND MODEL, r = σ F = 0 σ F = σ F = 5 ρ i α X ˆF ê P τ X P τ ê X ˆF ê P τ X P τ ê X ˆF ê P τ X P τ ê = 00 N = = 00 N = Noe: he daa are generaed as e i = ρ i e i + ɛ i and F = αf + u. Columns under X and ê are rejec raes of he ADF. P c and P τ are rejecion raes of he pooled ess. similar. he augmened auoregressions have p = 4[min[N ]/00] /4 lags. Criical values a he 5% level were used. he ADF es applied o X i should have a rejecion rae of.05 when α = or ρ =. In finie samples, his is rue only when ρ = andσ F is small. When σ F =0 and α=.5, for example, he ADF es rejecs a uni roo in X i wih probabiliy around.25 in he inercep model, and.28 in he linear rend model,

22 48 J. BAI AND S. NG even hough ρ =. As noed earlier, esing for a uni roo in X i when i has componens wih differen degrees of inegraion is difficul because of he negaive moving average componen in X i. Because our procedure separaely ess hese componens, our ess are also less sensiive o he choice of runcaion lag compared o convenional esing of X i. urning now o ˆF, he rejecion rae is close o he nominal size of.05 when α is. A oher values of α, he rejecion raes are comparable o he power of oher uni roo ess ha are based on leas squares derending. he ADFê(i has similar properies, wih rejecion raes around 5% when ρ i =. hese resuls sugges ha he error in esimaing F is small even when N=40. Indeed, he resuls for N = 00 are similar excep for small values of α. Resuls for he pooled ess are also repored in ables IIA, B. 0 Boh P c and P τ correcly rejec he null hypohesis when e ê ê i is in fac saionary. When each of he e i is nonsaionary, he rejecion raes roughly equal he nominal size of.05. Consider he sandard pooled ess for X i (see column under P c X Pτ X. When ρ = 5 andα =, all N series are nonsaionary in view of he common sochasic rend. he sandard pooled es should have a rejecion rae close o.05. However, he rejecion rae ranges from.45 o.88 depending on σ F.Consideralso(ρ i α= ( 0. he common facor is i.i.d.; he pooled es has a rejecion rae of.6 when σ F is small and deerioraes o.90 when σ F is large. hese resuls are consisen wih he findings of O Connell (998 ha cross secion correlaion leads he sandard pooled es o over-rejec he null hypohesis r> Incasesofmuliplefacors,wegeneraeheI(facorsassimplerandom walks and he saionary facors as AR( processes wih coefficien α.wecon- inue o assume ha e i is AR( wih parameer ρ i. he facor loadings are aken from an N r marix of N(0 variables. We consider hree cases of σ F as in he previous secion. As he resuls in able III illusrae quie well he consequence of increasing N from 40 o 00, we simply repor resuls for N = 40 o conserve space. Alhough we only presen resuls for r = 3, many addiional configuraions were considered and are available on reques. WebeginwihresulsforesingX i and ê i.wihr = 3, we can vary r from 0 o 3 o assess he case of none, one, wo, and hree common rends. Regardless 0 he p-values required o consruc he pooled ess are obained as follows. We firs simulae he asympoic disribuions repored in heorems and 2 by using parial sums of 500 N(0 errors o approximae he sandard Brownian moion in each of he 0,000 replicaions. A look-up able is hen consruced o map 300 poins on he asympoic disribuions o he corresponding p-values. In paricular, 00 poins are used o approximae he upper ail, 00 o approximae he lower ail, and 00 poins for he middle par of he asympoic disribuions. he p-values mach able IV of MacKinnon (994 very well, whenever hey are available. hese look-up ables are available from he auhors.

23 A PANIC AACK ON UNI ROOS 49 ABLE IIIA REJECION RAES,UNIVARIAE AND POOLED UNI ROO ESS, r = 3, = 00 N = 40, INERCEP MODEL σ F = 0 σ F = σ F = 5 r r ρ i α X ê P c X P c ê X ê P c X P c ê X ê P c X P c ê of he number of common rends, esing X i remains imprecise frequenly. For example, if r = 0andρ i =, here is a uni roo in X i and he ADF es should rejec roughly wih probabiliy.05. Insead, he rejecion raes are.39 for he inercep model, and.48 for he linear rend model. he rejecion raes for ê mirror he resuls for r =, showing ha he behavior of ADFê(i is no sensiive o he rue number of facors in he daa. Wih one random walk facor and wo saionary AR( facors, he ADFê(i has a rejecion rae of.65 when (ρ i α= ( 8 5. his is almos he same rejecion rae as when here was only one saionary facor. he simulaed criical values of he MQ c τ ess are exremely close o hose repored in Sock and Wason for Q c and Q f. We conjecure ha heir ess are also valid in he presen conex. Indeed, because ˆF consisenly esimaes he space spanned by F, we conjecure ha oher ess ha assume F is observed remain valid when F is esimaed using our proposed mehodology. o invesigae his and for he sake of comparison, we also consider he race es of