Journal of Econometrics

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1 Journal of Economerics 76 (203) 8 29 Conens liss available a SciVerse ScienceDirec Journal of Economerics journal homepage: Principal componens esimaion and idenificaion of saic facors Jushan Bai a,, Serena g b a Deparmen of Economics, Columbia Universiy, IAB 09, 420 Wes 8h Sree, ew York, Y 0027, Unied Saes b Deparmen of Economics, Columbia Universiy, IAB 7, 420 Wes 8h Sree, ew York, Y 0027, Unied Saes a r i c l e i n f o a b s r a c Aricle hisory: Received 3 December 202 Received in revised form 3 December 202 Acceped 25 March 203 Available online April 203 JEL classificaion: C30 C33 I is known ha he principal componen esimaes of he facors and he loadings are roaions of he underlying laen facors and loadings. We sudy condiions under which he laen facors can be esimaed asympoically wihou roaion. We derive he limiing disribuions for he esimaed facors and facor loadings when and are large and make precise how idenificaion of he facors affecs inference based on facor augmened regressions. We also consider facor models wih addiive individual and ime effecs. he asympoic analysis can be modified o analyze idenificaion schemes no considered in his analysis. 203 Elsevier B.V. All righs reserved. Keywords: Diffusion indices AVAR Roaion acor space Skew-symmeric marices. Inroducion Large dimensional facor analysis has been found o be useful in an increasingly large number of applicaions, and he heoreical properies of he principal componens esimaes are quie well undersood. he mehod of principal componens esimaes he space spanned by he laen facors insead of he facors hemselves. hus, if is he r vecor of laen facors, and is he vecor of facor esimaes, here exiss an r r inverible marix H such ha esimaes H. Asympoic resuls are saed in erms of H. Similarly, if λ i is he vecor of facor loadings and λ i is he corresponding esimae, asympoic resuls are known for λ i H λ i. In some insances, he objec of ineres is he condiional mean, and inerpreaion of he parameers ha deermine he condiional mean is no necessary. or example, in diffusion index forecasing analysis of Sock and Wason (2002), he primary ineres is he prediced value of he dependen variable. In facor augmened regressions, he facors are merely presen o conrol for laen common effecs. In problems wih errors-in-variables or endogeneiy such as considered in Bai and g (200), one only Corresponding auhor. addresses: jb3064@columbia.edu, jushan.bai@columbia.edu (J. Bai), Serena.g@columbia.edu (S. g). needs he facors o be srongly correlaed wih he endogenous regressor o validae he facors as insrumens. In all hese cases, we are no ineresed in he coefficiens on he facors per se and being able o esimae a roaion of suffices. here are, however, cases when he parameers of ineres are he coefficiens associaed wih he facors, or even he facors hemselves. or example, in arbirage pricing heory, resricions on he facor loadings would be necessary o pin down he sensiiviy o risk facors such as inflaion, real aciviy, and financial markes. In facor augmened regressions of he form y = α + W β + ε, one migh be ineresed in esing a hypohesis concerning α. Since he asympoic heory is only available for ( ˆα H α), he es is uninformaive excep when α is zero. If H is known, ˆα can be given economic inerpreaion. We sudy hree ses of resricions such ha and Λ are exacly idenified. If he underlying and Λ ha generae he daa saisfy hose resricions hen H is asympoically an ideniy marix. his is useful because can be reaed as hough hey were he laen and α can be learn from ˆα. We derive he asympoic disribuions for he esimaed facors and he loadings based on hese resricions. In case here exis no and Λ ha saisfy any of he idenificaion condiions considered here, he roaion marix H will no be an ideniy marix asympoically and we will be esimaing roaions of he underlying and Λ. Oher idenificaion condiions may be considered; he mehod developed in his paper should be useful o derive he corresponding limiing disribuions /$ see fron maer 203 Elsevier B.V. All righs reserved. hp://dx.doi.org/0.06/j.jeconom

2 J. Bai, S. g / Journal of Economerics 76 (203) Our analysis is exended o allow for (i) addiive individual effecs, (ii) common ime effecs, and (iii) heerogeneous ime rends in he panel of daa. 2. acor models and idenificaion Le and denoe he sample size in he ime series and cross-secion dimensions, respecively. or i =,..., and =,...,, he observaion X i has a facor srucure represened as X i = λ i + e i. As wrien, here are no deerminisic erms. Individual fixed effecs and ime rends will be considered subsequenly. Le X and e be marices. he facor model in marix form is X = Λ + e where = (, 2,..., ) is he r marix of facors and Λ = (λ, λ 2,..., λ ) is he r marix of facor loadings. Our objecive is o esimae boh and Λ. We make he following assumpions: Assumpion A. here exiss an M <, no depending on and, such ha (a) E 4 M and = p Σ > 0 is a r r nonrandom marix. (b) λ i is eiher deerminisic such ha λ i M, or i is sochasic such ha E λ i 4 M. In eiher case, Λ Λ p Σ Λ > 0 is a r r non-random marix as. (c.i) E(e i ) = 0 and E e i 8 M. (c.ii) E(e i e js ) = σ ij,s, σ ij,s σ ij for all (, s) and σ ij,s τ s for all (i, j). urhermore, σ i= ij M for each j, τ = s M for each s, and σ i,j,,s= ij,s M. (c.iii) or every (, s), E /2 i= [e ise i E(e is e i )] 4 M. (d) {λ i }, { }, and {e i }, are hree muually independen groups. (e) (i) /2 λ d i= ie i (0, Γ ); (ii) /2 = d e i (0, Φ i ). Assumpions A(a) and (b) imply he exisence of r facors. he idiosyncraic errors e i are allowed o be cross-secionally and serially correlaed, bu only weakly as saed under condiion A(c). If e i are iid, hen A(c.ii) and A(c.iii) are saisfied. Assumpion A(d) allows wihin group dependence, meaning ha can be serially correlaed, λ i can be correlaed over i, and e i can have serial and cross-secional correlaions ha are no oo srong so ha A(a) (c) hold. We assume no dependence beween he facor loadings and he facors, or beween he facors and he idiosyncraic errors, which is he meaning of muual independence beween groups. Par (e) of Assumpion A defines he limiing covariance of he facors. he mehod of principal componens minimizes he objecive funcion r[(x Λ ) (X Λ )] by choosing he normalizaions ha / = I r and Λ Λ is diagonal. he esimaor for, denoed = (,..., ), is a r marix consising of r uniary eigenvecors (muliplied by ) associaed wih he r larges eigenvalues of he marix XX /() in decreasing order. hen Λ = ( λ,..., λ ) = X / is a r marix of esimaed facor loadings. he esimaors and Λ saisfy he normalizaion resricions since / = Ir holds by consrucion and Λ Λ/ = Ṽ where Ṽ is a r r diagonal marix consising of he r larges eigenvalues of XX /(). While he resricions used by he principal componens esimaor idenify he space spanned by he columns of and he space spanned by he columns of Λ, hey do no necessarily idenify he individual columns of or of Λ. o be precise, le H be an r r marix whose ranspose is H = Ṽ ( /)(Λ Λ/). () Under Assumpion A, Sock and Wason (2002) and Bai and g (2002) showed ha H is inverible, esimaes H (a roaion of ), and Λ esimaes ΛH (a roaion of Λ), hough he produc Λ esimaes Λ. We are specifically ineresed in condiions under which we can ideniy he columns of and he columns of Λ from he produc Λ. oice ha Λ = RR Λ for any r r inverible marix R, and R has r 2 free parameers. hus we need a leas r 2 resricions in order o ideniy and Λ, see Lawley and Maxwell (97). While more han r 2 resricions can be imposed as in Heaon and Solo (2004) and Reis and Wason (200), he mehod of principal componens is no suiable for imposing over-idenifying resricions. We consider hree ses of resricions ha will lead o exac idenificaion. We hen show ha if he rue and rue Λ saisfy hese resricions, hen he corresponding roaion marix is asympoically an ideniy marix. Idenifying resricions: Resricions Resricions on Λ on (2.): PC = I r Λ Λ is a diagonal marix wih disinc enries (2.2): PC2 Λ = I r Λ =, Λ Λ = 2 λ 0 0 λ 2 λ , λ ii λ r λ r2 λ rr 0, i =,..., r Ir (2.3): PC3 Unresriced Λ = 2.. PC PC requires ha he diagonal elemens of Λ Λ are disinc and posiive and are arranged in decreasing order. he sandard mehod of principal componens implicily invokes he firs resricion in PC bu does no require he diagonal marix Λ Λ o have disinc elemens. Wihou his resricion, he principal componens esimaor canno ideniy he individual columns of and hose of Λ, and here will be roaional indeerminacy. Under PC, he normalizaion on gives r(r + )/2 resricions since a symmeric marix conains r(r + )/2 free parameers. he diagonaliy of Λ Λ gives r(r )/2 resricions. ogeher, he wo normalizaions lead o exacly r 2 resricions. We show in he Appendix ha if he resricions defined by PC also hold for he underlying and Λ ha generae he daa, hen H = I r + O p (δ 2 where δ denoes min Λ 2 ), (2), hroughou his paper. PC is a saisical resricion and is ofen used in he maximum likelihood esimaion, see Lawley and Maxwell (97). his idenificaion condiion is less resricive han i appears. A block diagonal marix of facor loadings also saisfies PC. 2 or example, wih By symmery, hree differen ses of idenificaion resricions can be obained by swiching and Λ. or example, is diagonal and Λ Λ = I r. Since he asympoic resuls sill hold by swiching he role of and Λ, we only consider he hree ses of resricions given above. 2 An exension of his model is he inclusion of a global facor, see for example, Moench and g (20), Hallin and Liska (2008) and Wang (2008). However, he facor loading marix does no necessarily saisfy PC; i will saisfy PC2 if here is a cross-secion uni which is affeced by he global facor only.

3 20 J. Bai, S. g / Journal of Economerics 76 (203) 8 29 r = 3, he following loading marix will saisfy PC: π 0 0 Λ = 0 π π 3 where π i is a vecor of i wih =. he loading marix implies ha he firs facor affecs he firs individuals, he second facor affecs he nex 2 individuals, and so on. his case is poenially useful for economic applicaions. PC sill holds under an arbirary permuaion of he cross secions. hus in he block diagonal case, i is no required o know which cross secion unis belong o he firs group (affeced by he firs facor) and which belong o he second group, and so forh. he nex wo ses of resricions, PC2 and PC3, involve ordering he daa. Boh of which are frequenly used in empirical work PC2 PC2 resrics Λ o be an inverible lower riangular marix. I hus requires knowledge of which variable is affeced by he firs facor only, which variable is affeced by he firs wo facors only, and so on. 3 PC2 is analogous o a riangular sysem of simulaneous equaions. he choice of he firs r variables of X and heir ordering provide he auxiliary informaion for idenificaion. Given he unresriced esimaes and Λ, i is easy o obain esimaors saisfying PC2. Le Λ be he firs r r block of Λ and le ˆ and ˆΛ denoe he esimaors ha saisfy PC2, i.e., ˆ ˆ/ = Ir and ˆΛ is lower riangular. hen ˆ and ˆΛ can be obained as follows. Sep : obain a QR decomposiion of Λ o yield Λ = Q R, where R is an upper riangular marix wih posiive diagonal elemens, and Q is an r r orhogonal marix such ha Q Q = I r. his decomposiion is unique for any inverible Λ. Sep 2: define R ˆ = Q, ˆΛ = Λ Q =. ˆΛ 2 By consrucion, ˆ ˆ/ = Q ( /)Q = Q Q = I r. he new roaion marix is H = HQ. Since ˆ and ˆΛ are roaions of he principal componen esimaes and Λ, hey are equivalen in a cerain sense. However, heir asympoic disribuions will be differen. We show in he Appendix ha H is asympoically an ideniy marix, bu (H I r ) is asympoically non-negligible unless r =. More specifically, if he rue and Λ saisfy PC2, hen H I r = O p (δ 2 ), r = H I r = O p ( /2 ), r >. his implies ha Ξ = (H I r ) = o p () for r =. In fac, when r =, PC and PC2 are idenical and (2) is in agreemen wih Ξ = o p (). However, Ξ = (H I r ) = O p () when r >. In fac, he limi of Ξ is a skew-symmeric random marix. 4 In consequence, he limiing disribuions of ˆ and ˆλ i will be affeced by Ξ PC3 he hird se of idenificaion resricions specify he firs r r block of Λ (denoed Λ ) o be an ideniy marix and leaves he facor process compleely unresriced oher han requiring / o be inverible so ha r facors exis. Unlike PC and PC2, 3 he srucure of Λ is similar o Sock and Wason (2005), hough hey are ineresed in idenificaion of shocks o he facors raher han he facors. A variaion o PC2 is o normalize he diagonal elemens λ ii (i =, 2,..., r) o be, wih / being diagonal (insead of an ideniy marix). 4 A marix C is skew-symmeric (also known as ani-symmeric) if C + C = 0. So he diagonal elemens of a skew-symmeric marix are zero, and C ij = C ji. all r 2 resricions are imposed on Λ under PC3. he resricions imply ha he firs variable X is affeced by he firs facor only, he second variable X 2 is affeced by he second facor only, ec. he resuling srucure resembles he classical errors-in-variables model in which X i = i + e i for i =,..., r, as in Panula and uller (986), and Wansbeek and Meijer (2000, pp ). While PC3 requires he choice of he firs r variables, he esimaors for Λ and are easy o obain. Given he principal componens esimaes Λ and, le ˆΛ = Λ Λ, ˆ = Λ. he roaion marix in his case is H Ď = H Λ because ˆ = Λ = H Λ + o p(). If he and Λ underlying he daa saisfy PC3, hen H Ď will converge in probabiliy o I r. I follows ha ˆ esimaes and ˆΛ esimaes Λ wihou roaion. We show in he Appendix ha (H Ď I r ) = ξ + o p () (3) where ξ is defined in () below. he fac ha (H Ď I r ) is no negligible for all r will affec he limiing disribuions of ˆλ i and ˆ. Remark (Local vs. Global Idenificaion). Condiions for global and local idenificaion of facor models are discussed, for example, in Bekker (986) and Algina (980). Boh PC and PC2 ideniy and Λ up o a column sign change. Changing he sign of any column of and he sign of he corresponding column of Λ will leave he produc Λ unchanged. he resuling new and new Λ sill saisfy PC, and hence observaionally equivalen o he original and Λ. hus PC and PC2 are local idenificaion condiions. However, once we fix he column signs of Λ (or ), PC and PC2 become global idenificaion condiions. here will be no oher and Λ wih he given column signs and he given produc Λ. o undersand how global idenificaion is achieved, consider PC2. Once Λ is given hen Λ( /)Λ = ΛΛ is known, since / = I r. Le C = ΛΛ. rom Λ = (Λ, Λ 2 ), we have ΛΛ Λ Λ = Λ Λ 2 C C Λ 2 Λ Λ 2 Λ, C = 2 2 C 2 C 22 where we also pariion marix C correspondingly. Suppose for concreeness ha r = 3. Knowing C is equivalen o knowing he elemens of Λ Λ = λ2 λ λ 2 λ λ 3 λ λ2 22 λ 2 λ 3 + λ 22 λ 32. λ λ λ2 33 If he sign of λ is known, hen λ is idenified from λ 2. Since λ 0, λ 2 and λ 3 can be idenified, which furher implies he idenificaion of λ If he sign of λ 22 is known, hen λ 22 is also idenified. Since λ 22 0, his implies he idenificaion of λ 32. he same reasoning implies he idenificaion of λ 33, given is sign. In summary, we can idenify Λ provided ha Λ is inverible and he signs of λ ii (i =, 2, 3) are known. 5 ex, from C 2 = Λ 2 Λ, we idenify Λ 2 from Λ 2 = C 2 (Λ ). hus PC2 ogeher wih he column signs of Λ (or ) imply global idenificaion in he resriced parameer space ha ensures inveribiliy of Λ. 5 Idenificaion of Λ alone does no require λ 33 0, bu furher idenificaion of Λ 2 does need λ 33 0 so ha Λ is inverible.

4 J. Bai, S. g / Journal of Economerics 76 (203) PC3 also implies global idenificaion, bu sign resricions are no necessary. o see his, le C = Λ( /)Λ be given. Under PC3, Λ( /)Λ ( /) ( /)Λ 2 = Λ 2 ( /) Λ 2 Λ. 2 Knowing C is equivalen o knowing /. hus we ideniy Λ 2 from Λ 2 = C 2 ( /) = C 2 C. 3. Asympoic heory We are ineresed in he implicaions of using he facor esimaes idenified using PC, PC2, or PC3 for inference. o his end, le Z i = ( /) /2 e i. = d By Assumpion A(e), Z i Z i where Z i is a zero mean normal vecor as. o derive he limiing disribuion for ˆ and ˆλ i, we use he asympoic represenaions for and λ i, given in heorems and 2 of Bai (2003). Specifically, if / 0, hen ( H ) = Ṽ and if / 0, ( λ i H λ i ) = H λ i e i + o p () (4) i= e i + o p (). (5) = A useful and alernaive expression for (4) is ( H ) = H Λ Λ because () implies Ṽ = H (Λ Λ/). 3.. PC λ i e i + o p () (6) i= = Ṽ (Λ Λ/)(Λ Λ/) Under PC, H = I r + O p (δ 2 ). I follows ha ( ) = ( H )+ ( H I r ) = ( H )+o p (), provided ha /δ 2 = o(), or equivalenly, / 0. hus under PC, we can rewrie (6) as ( ) = Λ Λ λ i e i + o p (). (7) i= his resul says ha is asympoically equivalen o he leas squares esimaor for in a cross-secion regression wih Λ as he regressor, as if Λ were observable. Similarly, if / 0 and H = I r + O p (δ 2 ), hen ( λ i λ i ) = e i + o p () (8) = because / = I r and (H I r ) = o p () if / 0. In view of (8), we can now inerpre λ i as he leas squares esimaor for λ i in a ime series regression wih as regressor, as hough i were observed. hese represenaions and he required relaive rae beween and are he same as in (4) and (5), excep ha we replace H by an ideniy marix in view of he idenificaion resricions. he fac ha H is an r dimensional ideniy marix asympoically simplifies he limiing disribuions for and λ i because he righ hand sides of (7) and (8) do no depend on any esimaed quaniies. heorem. Suppose ha Assumpion A and PC hold. Le and λ i be obained by he mehod of principal componens. hen as, wih / 0, we have ( ) d (0, Σ Λ Γ Σ Λ ). (9) urhermore, if / 0, ( λ i λ i ) d (0, Φ i ). (0) A formal proof is given in he Appendix. In essence, / = I r + O p (δ ), and Ṽ = Λ Λ/ + O p (δ 2 ) under PC. hus he limi of / is I r and he limi of Ṽ is Σ Λ. Since Λ Λ/ Σ Λ by Assumpion A(b), and (9) follows from (7). urhermore, (8) ogeher wih / = I r implies (0). heorem sheds ligh on he role of idenificaion assumpions on he principal componens esimaor. As H and Q are now ideniy marices, he idenificaion assumpions affec no jus where we cener he limiing disribuion of he facor esimaes, bu also heir asympoic variances. Using he limiing resul in (0) we can es if λ i or some componens of λ i are zero. Consider esing he null hypohesis ha Rλ i = λ i, where R is a (q r) known resricion marix (q r) and λ i is q, a known vecor. Under he null hypohesis, (R λ i λ i ) (R ˆΦ i R ) (R λ i λ i ) d χ 2 q. We can also es resricions beween λ i and λ j (i j). Pu δ = (λ i, λ j ) and ˆδ = (ˆλ i, ˆλ j ). Consider he hypohesis Rδ = δ, where R is q 2r and δ is q. By he asympoic represenaion of (8), if E(e i e j ) = 0 for i j, hen ˆλ i and ˆλ j are asympoically independen. So le ˆΦ = diag( ˆΦ i, ˆΦ j ) (a block-diagonal marix), hen (Rˆδ δ) (R ˆΦR ) (Rˆδ δ) d χ 2 q. If E(e i e j ) 0, hen Φ will no be a block diagonal marix, bu i is sraighforward o esimae he join asympoic covariance marix. Saisics for esing hypoheses concerning he facors can be similarly consruced PC2 o derive he asympoic disribuions of ˆ and ˆλ i for PC2, and PC3, we need he following: Assumpion B. (Z i, Z,..., Z r ) d (Z i, Z..., Z r ). he random variables Z i are defined earlier. Assumpion B srenghens A(e) o require he join convergence of Z i and (Z,..., Z r ) o he join limi of Z i and (Z,..., Z r ). Hereafer, we le ξ be an r r marix defined by ξ = ( e,..., e r ) = = (Z,..., Z r ). () he limiing disribuions of he facor esimaes under PC2 depend on wheher r = or r >. If r =, PC and PC2 are idenical, so he limiing disribuions ˆ and ˆλ i are given in heorem. When

5 22 J. Bai, S. g / Journal of Economerics 76 (203) 8 29 r >, he represenaions for ˆ and ˆλ i each has an exra erm because (H I r ) is non-negligible. More specifically, for i > r, (ˆλ i λ i ) = e i and for each, (ˆ ) = = (H I r )λ i + o p () (2) Λ Λ λ i e i i= (/) /2 (H I r ) + o p (). (3) Le Ξ = (H I r ) and le Ξ kh denoe he (k, h)h elemen of Ξ ( k, h r). We show in he Appendix ha (ξ Λ ) kh + o p (), k > h Ξ kh = o p () k = h (4) Ξ hk + o p (), k < h where o p () holds if / 0. he limi of he off-diagonal elemens of Ξ are deermined by he limi of he off-diagonal elemens of ξ (Λ ), where ξ is defined in (). I urns ou ha (2) also holds for i =, 2,..., r, no jus for i > r. or i r, he las r i componens of ˆλ i and of λ i are zero. Using he asympoic represenaion of (H I r ) in (4), i can be shown ha he las r i componens of he righ hand side of (2) indeed have zero limis. = I r under PC2, and Z i is he limiing disri- = e i. Le veck(a) denoe he column Recall ha Σ buion of vecor ha sacks he lower riangular elemens of A (excluding he diagonal elemens). oe ha veck( ) is differen from vech( ). or any skew-symmeric marix A, here is a duplicaion marix D such ha vec(a) = D veck(a). Eq. (4) implies veck(ξ) = veck(ξ Λ () ) + o p (). rom ξ (Λ d ) (Z, Z 2,..., Z r )(Λ ) d we have veck(ξ) η, where η is defined as η = veck((z, Z 2,..., Z r )(Λ ) ). hen (H I r )λ i = Ξλ i = (λ i I r)vec(ξ) = (λ i I r)d veck(ξ) d (λ i I r)dη. Le Z i be he limi of he firs erm on he righ hand side of (2). We have heorem 2. Suppose ha Assumpions A, B, and PC2 hold. Le ˆ and ˆλ i denoe he esimaes wih he resricions of PC2. (i) Le Z i = d (0, Φ i ). hen for each i and as, / 0, (ˆλ i λ i ) d Z i (λ i I r)dη wih where η = veck[(z, Z 2,..., Z r )Λ ] and D is a duplicaion marix linking vec( ) and veck( ). (ii) Le G = d (0, Σ Λ Γ Σ Λ ) and is independen of η. If / c wih 0 c <, (ˆ ) d G + c( I r)dη, In par (i) of heorem 2, (λ i I r)dη is he limi of (H I r )λ i, which is also normal since η is normal. Similarly, for par (ii) of he heorem, G is he limi of he firs erm on he righ hand side of (3), and c( I r)dη is he limi of he second erm of (3). Hypohesis esing can be performed as in Secion PC3 Similar o PC2, he represenaions for ˆ and ˆλ i each has an exra erm because (H Ď I r ) is non-negligible. As λ i is known for i r, we only need o consider i r +. We show in he Appendix ha (ˆλ i λ i ) = and for each, (ˆ ) = = e i (H Ď I r )λ i + o p () (5) Λ Λ λ i e i i= + (/) /2 (H Ď I r ) + o p () (6) where (H Ď I r ) is given in (3). heorem 3. Suppose ha Assumpions A, B, and PC3 hold. Le ˆ and ˆλ i denoe he esimaes wih he resricions of PC3. (i) Le Z i = d (0, Σ Φ i Σ ). hen for i r +, as, wih / 0, (ˆλ i λ i ) d Z i (Z,..., Z r )λ i. (ii) Le G = d (0, Σ Λ Γ Σ Λ ) and is independen of (Z,..., Z r ). If / c wih 0 c <, (ˆ ) d G + c(z,..., Z r ). o undersand par (i) of heorem 3, noe ha Z i is he limi of he firs erm on he righ hand side of (5). Under Assumpions A, B, and PC3, he second erm in (5) saisfies (H Ď I r ) d (Z, Z 2,..., Z r ), which is an r r marix of random variables. 6 Alhough / (whose limi is Σ ) is no required o be an ideniy marix under PC3, Z i is normally disribued. As a consequence, (Z,..., Z r )λ i is also normally disribued if λ i is non-random. I follows ha ˆλ i is sill normally disribued. Similarly, par (ii) of heorem 3 comes from he fac ha G is he limiing random variable for he firs erm on he righ hand side of (6). Again, hypohesis esing can be performed similarly as in Secion Implicaions for facor-augmened regressions Consider he infeasible regression model y = α + W β + ε where is no observable and is replaced by ˆ esimaed under one of he hree idenificaion assumpions. Le ˆδ = ( ˆα, ˆβ ) denoe he leas squares esimaor of he facor augmened regression y = ˆ α + W β + v = ẑ δ + v (7) where v = ε + ( ˆ ) α, ẑ = (ˆ, W ), and δ = (α, β ). o sae he asympoic behavior of ˆδ, we also need he following: 6 he marix convergence in disribuion implicily refers o he convergence wih vecorizaion. In any even, (H Ď I r )λ i is already a vecor, so is convergence o he vecor (Z,..., Z r )λ i is well defined.

6 J. Bai, S. g / Journal of Economerics 76 (203) Assumpion C. or z = (, W ), E z 4 M < ; E(ε z, z 2,...) = 0; z and ε are independen of he idiosyncraic errors e is for all i and s. urhermore, = z z p Σ zz > 0 and /2 = z d ε (0, Σ zz,ε ), where Σ zz,ε = plim = ε2 z z > 0. If was observed, hen under Assumpion C, he asympoic variance of ˆδ would be given by Σ Σ zz zz,εσ zz. As shown in Bai and g (2006), ˆα is an esimae of H α (and no α) when is used in place of. he following heorem sudies he properies of ˆδ when ˆ is used in place of. heorem 4. Suppose / 0 and Assumpions A C hold. Define Σ δ = Σ Σ zz zz,εσ zz. Le δ = (α, β ) and le ˆδ be obained by he leas squares esimaion of facor augmened regression (7), where ˆ is obained under he resricions defined by PC, PC2, or PC3. hen (ˆδ δ) d (0, Avar(ˆδ)) where Avar(ˆδ) = Σ δ under PC, Avar(ˆδ) = Σ δ + diag[(α I r ) Dvar(η)D (α I r ), 0] under PC2, and Avar(ˆδ) = Σ δ + diag(var [(Z,..., Z r )α], 0) under PC3. urhermore, η and D are defined in Secion 3.2, and (Z,..., Z r ) is defined in Secion 3.3; diag(a, B) refers o he block diagonal marix wih blocks A and B. heorem 4 saes ha under PC, ˆδ has properies as hough he laen facors were available as regressors. Alhough he disribuion of ˆβ is invarian o idenificaion assumpions used, he disribuion of ˆα does depend on wheher PC, PC2, or PC3 is used. o undersand heorem 4, noe ha under PC, ( ˆα α) = ( ˆα H α) (H I)H α. he firs erm on he righ is analyzed by Bai and g (2006). Under PC, (H I r ) = o p () provided / 0 since H I r = O p (δ 2 ). As H is asympoically an ideniy marix, ˆα now direcly esimaes α. hus, he limiing disribuion for ( ˆα H α) saed in Bai and g (2006) simplifies o he case of sandard leas squares as if were observed. Under PC, he asympoic variance of Σˆδ can be consisenly esimaed by Σˆδ = ẑ ẑ ẑ ẑ ˆv2 ẑ ẑ = = which is Whie s heeroskedasiciy robus covariance esimaor using ẑ as regressors. Under PC2 and PC3, (H I r ) and (H Ď I r ) are no asympoically negligible when r >. he asympoic variance of ˆα under PC2 has an exra erm given by he variance of (α I r )Dη. Under PC3, he exra erm in he asympoic variance of ˆα is due o var[(z,..., Z r )α]. Deails on esimaion of he asympoic variances are given in Appendix A. I is however useful o noe ha if e j are independen for j =, 2,..., r, hen he normal vecors Z j are also independen. In such a case, var[(z,..., Z r )α] = r k= Φ kα k can be consisenly esimaed by r k= ˆΦ k ˆα k. I is useful o remark ha when ˆ esimaes insead of a roaion of, we can give economic inerpreaion o he coefficiens on he regressors ˆ. or example, in facor augmened auoregressions (AVAR) or for he facor models considered in his paper we can obain he impulse responses of each observable X i in he panel o he common shocks ha drive. 7 Suppose ha = = A + + A p p + A 0 u, where u is a vecor of srucural shocks, and A 0 is a r r marix linking he srucural shocks u o he reduced form shocks v such ha v = A 0 u. 8 Observing (wih economic inerpreaions for each componen) allows us o use sandard srucural VAR analysis o ideniy A 0 and compue he impulse responses +k u impulse responses for he observable variables X i,+k u for each i and for all k acor models wih deerminisic erms. I follows ha we can compue he = λ +k i u In pracice, he daa are demeaned and rends are removed before he facors are esimaed. acor models wih deerminisic erms are of he form X i = µ i + δ i () + λ i + e i where µ i is an individual fixed effec and δ i () is a ime effec. When δ i () = δ, he ime effecs are common. When δ i () = δ i, we have individual specific linear rends. hese reamens of deerminisic erms will be analyzed in he nex hree subsecions. 5.. Individual fixed effecs We firs assume ha he ime effec is absen. he model in vecor form is wrien as X = µ + Λ + e. he model is observaionally equivalen o he following model X = µ + Λ + e where µ = µ + Λ, and =. We impose he resricion = = = 0. Equivalenly, wih ι = (,,..., ), a vecor, he resricion is ι = = 0. = (E) In he absence of fixed effecs, he principal componens esimaor is based on he daa marix X X, where X = [X, X 2,..., X ]. o accoun for he fixed effecs, we need o demean he daa. Equivalenly, we can esimae µ by X = = X and use he residuals o esimae Λ and. he demeaned daa marix is Z = [X X,..., X X] = X Xι. he principalcomponens of, denoed, corresponds o he eigenvecors muliplied by of he r larges eigenvalues of he daa marix Z Z. ha is, () Z Z = Ṽ (8) where Ṽ is r r diagonal marix consising of he firs r larges eigenvalues, arranged in decreasing order. he facor loading esimaor is Λ = Z /. By consrucion, and Λ already saisfy PC, namely, ha / = Ir and Λ Λ = diagonal. We now wan o show ha (i) hese esimaes also saisfy he consrain (E) and (ii) ha λ i has he same expression wih or wihou demeaning. o see (i), firs noe ha ι Z = ι X (ι ι ) X = ι X X which equals zero by he definiion of X. Muliply ι on each side of (8), we have 0 = ι Z Z = ι Ṽ. 7 Similar issues have been considered by Sock and Wason (2005) and orni e al. (2009). 8 he model is sill saic even hough is dynamic.

7 24 J. Bai, S. g / Journal of Economerics 76 (203) 8 29 Since Ṽ is an inverible (diagonal) marix of eigenvalues, i follows ha ι = = = 0, which is (E). he principal componens esimaor for Λ can now be rewrien as Λ = Z / = (X Xι ) / = X / where he las equaliy makes use of he resul ι = 0. herefore, he expression for λ i has he same form wih or wihou demeaning he daa. o show (ii) ha he limiing disribuion for λ is of he same form wih or wihou fixed effecs noe ha since = and = 0 by assumpion, he model in demeaned daa is X i Xi = λ i + e i ē i. Replacing e i wih e i ē i in (8) and since = ē i = = ēi = 0, ( λ i λ i ) = = (e i ē i ) + o p () = e i + o p (). = his represenaion coincides wih (8). hus under Assumpions A, B, PC and (E), he limi is again ( λ i λ i ) (0, Φ i ), which is (0). he limiing disribuion for also has he same form wih or wihou demeaning. Replacing e i wih e i ē i in (7), we have Λ Λ ( ) = Λ Λ = i= = Λ Λ = i= i= λ i (e i ē i ) + o p () Λ Λ λ i e i /2 λ i e i + o p () λ i e i + o p (). i= he second erm on he righ hand side is O p ( /2 ) because () /2 i= = λ ie i = O p (). he asympoic represenaion for is hus he same as when fixed effecs are absen. his implies ha he limiing disribuion has he same form. he esimaors under idenificaion resricions PC2 and PC3 are consruced exacly he same way as when fixed effecs are absen, bu using he newly defined principal componens esimaors and Λ. hus when (E) holds, he expressions for λ i and are he same wih or wihou demeaning. o esimae he model, we firs remove he cross-secion mean and ime series mean from he daa. Le Ẋ i = X i Xi X. + X, where Xi is ime series mean for each i, X is he cross-secion mean for period, and X is he overall mean of X i. he variable Ẋ i is he usual wihin group ransformaion of X i. By similarly defining ė i, he demeaned model is Ẋ i = λ i + ė i. his is now in he form of a pure facor model wihou individual and ime effecs. We can again esimae he model using he daa Ẋ i, wih any of he hree ses of idenificaion resricions, PC, PC2, and PC3. here is no need o direcly impose he fixed effecs resricions (E) and (E2). When (wihin-group) ransformed daa are used, hese resricions are auomaically saisfied. he limiing disribuions can again be derived using represenaion (8) wih e i replaced by ė i = e i ē i ē + ē. Specifically, /2 (e i ē i ē + ē ) = = /2 e i /2 = = /2 = = e i /2 = /2 e i O p ( /2 ) = ē i= = where he firs equaliy follows from = e i = ē i = 0 and ē = 0 since = = 0. hus he limiing disribuion is sill deermined by he limi of ( /) /2 = e i. Similarly, /2 λ i ė i = /2 λ i e i + O p ( /2 ). i= i= I follows ha he limiing disribuion for he facor loadings is of he same form as when fixed effecs are absen. he values of he limiing variances will, however, be general differen. If here are no fixed effecs in he rue model bu demeaned daa are used in esimaion, he resuling esimaes for he facors and heir loadings will, in general, have larger variances han hose wihou demeaning he daa. o see his, recall ha under PC or PC2, he esimaed facor loadings in he fixed effecs model are represened by (ˆλ i λ i ) = e i + o p () = 5.2. Common ime effecs We now allow for common ime effecs. X i = µ i + δ + λ i + e i. or idenificaion, we now need he addiional resricion 9 λ i = 0. i= (E2) wheher or no he fixed effecs are esimaed. If e i (0, σ 2 ), is a saionary vecor, hen he limiing disribuion is (ˆλ i λ i ) d (0, σ 2 [E( )] ). ow esimaion of he fixed effecs will also remove he mean from. 0 Alhough he represenaion looks he same, he limiing variance of ˆλ i is hen σ 2 [var( )]. As can have non-zero mean, he second momen E( ) is in general larger han he variance of. As E( ) var( ) implies [E( )] [var( )], he limiing variance of ˆλ i is smaller when fixed effecs are known o be absen and are no esimaed. 9 he resricion may be replaced by E(λi ) = 0 if each λ i is considered o be a vecor of random variables. 0 Our assumpion ha = 0 is asympoically equivalen o E( ) = 0.

8 J. Bai, S. g / Journal of Economerics 76 (203) Heerogeneous rends Insead of common ime effecs, consider a model wih heerogeneous coefficiens on he linear rends: X i = µ i + δ i + λ i + e i. We now assume ha is a zero mean process ha does no conain a linear rend because in he presence of µ i + δ i, we canno separaely idenify he heerogeneous rends and he facor process. or example, suppose ha = c + d + η, where η is a zero mean process, we can rewrie he model as X i = µ i + δ i + λ η i + e i wih µ i = µ i + λ i c and δ i = δ i + λ i d. We can only idenify η. We focus on he idenificaion resricion PC, i.e., / = I r and Λ Λ is diagonal. Le X τ i denoe he residuals from he leas squares derending for each series i. We have X τ i = λ i τ + e τ i, where τ and e τ i are also he residuals from he leas squares derending (no acual derending is performed on hem since hey are unobservable). Le a and b be he OLS coefficiens when is regressed on [, ], and a i,e and b i,e are similarly defined, we have τ = a b e τ = i e i a i,e b i,e. While τ is no equal o, one can easily show ha τ = + O p ( /2 ). oe ha / = I r implies ha τ τ / = I r + O p (/) because is a zero mean sequence by assumpion in his secion. ogeher wih diagonaliy of Λ Λ under PC, we can use earlier argumens o show ha ( λ i λ i ) = τ τ τ τ = τ eτ + i o p() = τ e i + o p (). oe ha we can replace e τ i by e i because { τ } is orhogonal o he sequence {, }. Similarly, Λ ( τ ) = Λ λ i e τ + i o p() i= Λ Λ Λ Λ = λ i e i = = i= λ i (a i,e + b i,e ) + o p () i= Λ Λ λ i e i + o p (). he las equaliy follows from he fac ha a i,e + b i,e is a linear combinaion of s= e s is and s= e is, each of which is O p ( /2 ). Using he assumpion ha e i has weak cross-secional correlaion, we can show ha /2 λ i= i(a i,e + b i,e ) = O p ( /2 ). Asympoic normaliy for ( λ i λ i ) and for ( τ ) follows from he fac ha /2 = τ e i and /2 λ i= ie i are asympoically normal. Once he daa are demeaned and derended, he esimaion procedure is idenical o he case wih or wihou linear rends. In addiion, he asympoic variances for ˆλ i and ˆ are esimaed as if here were no deerminisic erms. Analogous argumens can be used o esablish ha he limiing disribuions under PC2 and PC3 also have he same form as he case wihou deerminisic inerceps or rends. Deails are omied. i= able Marginal R 2 : ˆ roaed under PC2. Series acor ces ips sfyg puxhs fyg hsbr fmrra fspcom An applicaion Sock and Wason (2005) analyzed 32 series over he sample 959: o 2003:2. he predicors include series in 4 caegories: real oupu and income; employmen and hours; real reail, manufacuring and rade sales; consumpion; housing sars and sales; real invenories; orders; sock prices; exchange raes; ineres raes and spreads; money and credi quaniy aggregaes; price indexes; average hourly earnings; and miscellaneous. he series are ransformed by aking logarihms and/or differencing so ha he ransformed series are approximaely saionary. he IC and IC 2 crieria developed in Bai and g (2002) find 7 saic facors explaining over 40 percen of he variaion in he daa. Sock and Wason (2005) performed variance decomposiions and repored ha he firs facor explains much of he variaion in producion and employmen relaed series, while he second facor explains movemens in ineres raes, consumpion, and sock prices. Variaion in inflaion is mainly explained by he second and hird facor. acor four is highly correlaed wih ineres rae movemens, facor five wih employmen, facor six wih exchange raes, sock reurns, and hourly earnings. We use he Sock Wason daa exended o 2007:2 and used in Ludvigson and g (20). Afer deleing a series ha is no longer published, he new daase has 3 series. We firs ransform he daa o be saionary. he demeaned and sandardized daa are hen used o esimae he facors. he firs 7 facors sill explain 45% of he variaion in he daa, hough he IC 2 crierion now finds he opimal number of facors o be 8. An imporan aspec of PC2 is ha i uses he ordering of he variables o idenify he facors. We reorder he daa such ha he firs eigh series are () ces002, oal employees on non-far payroll; (2) ips0, indusrial producion oal index; (3) sfyg, spread beween one-year -bill rae (fyg) and fed funds rae; (4) puxhs, CPI excluding sheler; (5) fyg, one year -bill rae; (6) hsbr, housing unis auhorized; (7) fmrra, oal reserves; (8) fspcom, S&P 500 index. Under PC2, employmen responds o he firs facor only while indusrial producion responds o he firs wo facors. he ineres rae spread responds o facors one o hree, while inflaion responds o facors one o four, and so on. his in urn implies ha shocks o ˆ are shocks o employmen, while shocks o ˆ2 are indusrial producion shocks orhogonal o employmen, and so forh. able repors he marginal explanaory power of he j-h facor. he (i, j)h enry of he able is compued as follows. Le R 2 (j) be he R 2 in a regression of he series in quesion on he firs j roaed facors. We firs regress he ih series on he firs j roaed facors o ge R 2 (j), and hen regress he same series on he firs j roaed facors o ge R 2 (j ). he (i, j)h enry equals he difference beween R 2 (j) and R 2 (j ). he resuls conform ha under PC2, he firs wo facors are real aciviy facors while facor four is inflaion. acors hree and five are relaed o ineres raes, while facor seven is a moneary facor. acor six is a housing facor, and facor 8 is ha of he sock marke. I is useful o compare he marginal R 2 s obained by regressing hese same series on he sandard principal componen esimaes,

9 26 J. Bai, S. g / Journal of Economerics 76 (203) 8 29 able 2 Marginal R 2 :. Series acor ces ips sfyg puxhs fyg hsbr fmrra fspcom his is repored in able 2. he resuls are in line wih wha was repored in Sock and Wason (2005) ha he firs wo facors highly correlaed wih oupu and employmen daa. However, he remaining facors load on a variey of oher variables. Using he PC2 roaion, he eigh facors are much more concenraed on he variaions in eigh series which faciliaes he inerpreaion of hese facors. his is useful in subsequen facor augmened regressions in which economic inerpreaion of he coefficiens on ˆ is warraned. 7. Conclusion his paper considers principal-componens-based esimaion of facors and facor loadings. In general, he mehod does no separaely idenify he facors and facor loadings bu only heir roaions. his paper considers idenificaion resricions under which he laen facors and he loadings are idenified so ha he esimaes are no roaed. hree ses of resricions are considered. We show ha if he underlying facors and facor loadings saisfy he resricions used in he esimaion, hen he roaion marix is asympoically an ideniy marix. Limiing disribuions are derived, and he asympoic covariance marices are obained for each case separaely. Oher resricions may also be considered and he asympoic properies of he corresponding esimaors may be derived based on similar argumens. Acknowledgmens We hank Silvia Goncalves, Benoi Perron, and Dalibor Sevanovic for helpful commens. he auhors acknowledge financial suppor from he S (SES , SES ). Appendix A his appendix shows how o consisenly esimae he asympoic covariances under PC PC3. PC. his is sraighforward. We esimae Σ Λ by ˆΣ Λ = Λ Λ/. o esimae Φ i and Γ, we can use one of he hree mehods given in Bai and g (2006). Le ˆΦ i and ˆΓ denoe hese esimaes. hen Σ Λ Γ Σ Λ is esimaed by ˆΣ ˆΓ Λ ˆΣ Λ. PC2. o esimae he asympoic variance of ˆλ i, firs consider he case when e i are cross-secionally independen, so ha Z i are independen over i. his implies ha Z i (i > r) is independen of η (he laer depends on (Z,..., Z r )). oing ha ( /) = I r under PC2, Avar(ˆλ i ) = Φ i + (λ i I r)d var(η) D (λ i I r ) which is he sum of he variances of Z i and of (λ i I r)dη. o esimae he variance of η, we le ζ = veck[ (e,..., e r )Λ ]. hen η is he limi of /2 ζ =. In he absence of serial correlaion in e j (j =, 2,..., r), he variance of η is equal o he probabiliy limi of ζ = ζ, and is esimaed by var(η) = = ˆζ ˆζ wih ˆζ = veck[ˆ (ê,..., ê r ) ˆΛ ]. Wih serial correlaion in e j, he variance of η is he limi of = s= E(ζ ζ s ), and i is esimaed by he ewey Wes mehod using he series ˆζ ( =, 2,..., ). Given var(η), we esimae Avar(ˆλ i ) by Avar(ˆλ i ) = ˆΦ i + (ˆλ i I r)d var(η) D (ˆλ i I r ) where ˆΦ i = ˆ = ˆ ê2 i in he absence of serial correlaion in e i, and ˆΦ i is consruced by he ewey Wes mehod based on he series ˆ ê i in he presence of serial correlaion. If he e i s are cross-secionally correlaed, Z i can be correlaed wih η. Especially for he case of i r, Z i is correlaed wih η. o accoun for his correlaion, we le τ be he vecor ha sacks e i and ζ so τ is an r + r(r )/2 dimensional vecor. hen (ˆλ i λ i ) = [I r, (λ i I r )D] /2 τ = + o p (). In he absence of serial correlaion in e i, we esimae he variance of /2 = τ by ˆVτ = ˆτ = ˆτ ; in he presence of serial correlaion, ˆVτ is he ewey Wes esimaor using he series ˆτ. inally, Avar(ˆλ i ) = [I r, (ˆλ i I r)d] ˆVτ [I r, (ˆλ i I r)d]. Consider now esimaing he asympoic variance of ˆ. Wheher or no e i are cross secionally correlaed, G is independen of η since G is obained by he CL wih he enire cross secions, and η only depends on e i for i r. hus Avar(ˆ ) = Σ Λ Γ Σ Λ + c( I r)d var(η) D ( I r ). I is esimaed by Avar(ˆ ) = ˆΣ Λ ˆΓ ˆΣ Λ + (/)(ˆ I r)d var(η) D (ˆ I r ) where ˆΣ Λ = ( ˆΛ ˆΛ/), and ˆΓ is given by any one of he hree mehods in Bai and g (2006) using he series ˆλ i ê i (i =, 2,..., ). urhermore, Our earlier discussion on esimaing var(η) does no assume e,..., e r o be uncorrelaed, so var(η) given earlier is valid wheher or no e i are cross-secionally correlaed. PC3. We separaely discuss wheher or no e i is cross-secionally independen. Case i: If e i are cross-secionally independen, hen Z i are independen over i and r Avar(ˆλ i ) = Σ Φ i + Φ k λ 2 ik Σ k= which is he sum of variance of Z i and ha of (Z,..., Z r )λ i. urhermore, as G is he limi from he cenral limi heorem applied o all he cross secion unis, G is independen of Z,..., Z r. hus Avar(ˆ ) = Σ Λ Γ Σ Λ + c2 k Φ k 2. k (9) r= An esimae of Avar(ˆ ) is given by ˆΣ Λ 2 ˆΦ k ˆ k, and an esimae of Avar(ˆλ i ) is ˆΣ ˆΓ ˆΣ Λ ( ˆΦ i + r + (/) r ˆΦ k= l= k ˆλ 2 ) ik ˆΣ, where ˆΣ = ˆ ˆ/, ˆΣ Λ = ( ˆΛ ˆΛ/), and ˆΓ and ˆΦ i have he same form as under PC and PC2 bu using he new ˆ and ˆΛ. Case ii: If e i is cross-secionally correlaed, hen combining (5) and (3), we have (ˆλ i λ i ) = (I r, I r ) b i + o p () where b i is a 2 by vecor wih e i as he firs elemen and (e,..., e r )λ i = r k= e kλ ik as he second elemen. hus he limiing covariance is given by Avar(ˆλ i ) = Σ (I r, I r )Ψ i (I r, I r ) Σ =

10 J. Bai, S. g / Journal of Economerics 76 (203) where Ψ i = lim = s= E( s b ib is ), which specializes o Ψ i = plim = ( b ib ) i in he absence of ime series correlaion. o esimae Ψ i, apply he ewey Wes esimaor o he sequence ˆ ˆbi. he asympoic variance is esimaed by Avar(ˆλ i ) = ˆΣ (I r, I r ) ˆΨ i (I r, I r ) ˆΣ. Alhough G is sill independen of Z,..., Z r (because G is obained from averaging he enire cross secions), Z,..., Z r are dependen among hemselves. Under PC3 and cross secion dependence, (H Ď I r ) = ( I r)vec s a s + o p () where a = (e,..., e r ). Le s= d ( I r)vec[(z,..., Z r ) ] Υ = Avar(vec[(Z,..., Z r ) ]) = E[vec( s a s)vec( a )] s= = which simplifies o Υ = s= E[vec( s a s)vec( s a s )] in he absence of ime series correlaions. Le c be he limi of /. he limiing variance of (ˆ ) becomes Avar(ˆ ) = Σ Λ Γ Σ Λ + c2 ( I r)υ ( I r ). o esimae Υ, apply he ewey Wes esimaor o he sequence ˆ s â s. he asympoic variance of ˆ is esimaed by Avar(ˆ ) = ˆΣ ˆΓ Λ ˆΣ Λ + (/)2 (ˆ I r) ˆΥ (ˆ I r ). Appendix B Proof of (2). Rewrie / = ( H) / + H / = H / + O p (δ 2 ) (20) because ( H) / = O p (δ 2 ), see Lemma B.2 of Bai (2003). Righ muliply H o boh sides, H/ = H ( /)H + O p (δ 2 ). Rewrie he lef hand side of above as H/ = (H + )/ = Op (δ 2 ) + I r because (H )/ = Op (δ 2 ) and / = Ir, see Lemma B.3 of Bai (2003). Equaing he above wo equaions we obain I r = H ( /)H + O p (δ 2 ). (2) hus if ( /) = I r, we have I r = H H + O p (δ 2 ). (22) Ignore he O p (δ 2 ) erm, he above shows ha H is an orhogonal marix so ha is eigenvalues are eiher or. We need o show ha H is a diagonal marix. rom he definiion of H H = Ṽ ( /)(Λ Λ/) = Ṽ H (Λ Λ/) + O p (δ 2 ) where we use he fac ha / = H +O p (δ 2 ) under / = I r, see (20). Muliplying Ṽ on boh sides and aking he ranspose (Λ Λ/)H = HṼ + O p (δ 2 ). (23) his equaion implies ha H (up o a negligible erm) is a marix consising of eigenvecors of (Λ Λ/). he laer marix is diagonal and has disinc eigenvalues by assumpion. hus, each eigenvalue is associaed wih a unique uniary eigenvecor (up o a sign change) and each eigenvecor has a single non-zero elemen. his implies ha H is a diagonal marix up o an O p (δ 2 ) order. I is already known ha he eigenvalues of H are or, H is a diagonal marix wih elemens of or as is elemens. Wihou loss of generaliy, we can assume all elemens are (oherwise muliply he corresponding columns of and Λ by ). his implies H = I r + O p (δ 2 ). Moreover, from (23) we obain (Λ Λ/) = Ṽ + O p (δ 2 ). Proof of heorem. Resul (2) leads o represenaions (7) and (8). he heorem is a direc consequence of hese represenaions and Assumpion A. Proof of (4). oe H = HQ is he roaion marix under PC2. Under PC2, / = I r, hus (22) holds. his implies ha H is an orhogonal marix, up o a negligible erm, and so is HQ since Q is also orhogonal. urhermore, lef muliply (22) by Q and righ muliply i by Q, and use Q Q = I r, we have I r = Q H HQ + O p (δ 2 ). (24) We nex show HQ is a diagonal marix, up o an O p ( /2 ) erm. By (5), for each i, λ i H λ i = O p ( /2 ), we have Λ = ( λ,..., λ r ) = H (λ,..., λ r ) + O p ( /2 ). ha is, Λ = H Λ + O p( /2 ). By he QR decomposiion, we have QR = Λ = H Λ + O p( /2 ). Since Λ is also an upper riangular marix (an assumpion of PC2) and H is an orhogonal marix up o a negligible erm, by he uniqueness of he QR decomposiion, we have Q = H + O p ( /2 ). Righ muliply H on each side we have HQ = I r + O p ( /2 ). When r =, HQ is a scalar, and combined wih (24), we srenghen he rae o HQ = I r +O p (δ 2 ). or general r >, he rae canno be improved. Le = HQ I r = O p ( /2 ). Eq. (24) implies ( + I r ) ( + I r ) = O p (δ 2 ). ha is, + + = O p (δ 2 ). Bu = O p (/), so + = O p (δ 2 ). his implies ha he diagonal elemens of are all O p (δ 2 ) and is skew-symmeric up o an O p(δ 2 ) erm (and especially for r =, = O p (δ 2 )). We nex derive he asympoic represenaion for. Using (5), we can wrie Λ H Λ = H (e,..., e r ) + o p ( /2 ). = Lef muliplying H and using HH = I r + O p (δ 2 ) = ( /) + O p (δ 2 ) [see (22), which sill holds under PC2], we have H Λ Λ = (e,..., e r ) + o p ( /2 ). = he firs erm on he righ hand side is /2 ξ, where ξ given in (), so ha H Λ Λ = /2 ξ + o p ( /2 ). By he QR decomposiion of Λ, H Λ = HQR = (HQ I)R + R = R + R. hus H Λ Λ = R + (R Λ ). I follows ha = (R Λ )R + /2 ξ R + o p ( /2 ). Since boh R and Λ are upper riangular marices, he below diagonal elemens of are equal o he corresponding elemens of /2 ξ R + o p ( /2 ). Since is skew-symmeric up o an O p (δ 2 ) order, he elemens of above he diagonal are also given.

11 28 J. Bai, S. g / Journal of Economerics 76 (203) 8 29 ha is, ij = /2 (ξ R ) ij + o p ( /2 ) for i > j, and ij = ji + O p (δ 2 ) for i < j, and ii = O p (δ 2 ) (i, j =, 2,..., r). urhermore, we can replace R by Λ. o see his, by he uniqueness of QR decomposiion, R = Λ + o p(). So /2 ξ R = /2 ξ (Λ ) + /2 ξ o p () = /2 ξ (Λ ) + o p ( /2 ). inally, (4) is obained by noing Ξ =. Proof of (2). Using ˆλ i = Q λ i, ˆλ i λ i = Q λ i λ i = Q ( λ i H λ i ) + Q H (I HQ )λ i. Muliplying, (ˆλ i λ i ) = Q ( λ i H λ i ) Q H (HQ I r )λ i. Since Q H = I r + o p (), he second erm on he righ hand side is (H I r )λ i + o p (). Using (5), he firs erm on he righ hand side is Q H /2 = e i + o p (). Bu Q H = I r + o p () = ( /) +o p () under PC2. Combining he resuls yield (2). his argumen holds for all i =, 2,...,. Proof of (3). Using ˆ = Q, ˆ = Q = Q ( H ) + (Q H I r ). Muliplying, (ˆ ) = Q ( H ) + (/) /2 (Q H I r ). rom (6), he firs erm on he righ is Q H (Λ Λ/) λ i= ie i + o p (); bu Q H = I r + o p (). or he second erm on he righ, (Q H I r ) = (HQ I r ) + o p () because (HQ I r ) is skew-symmeric up o an o p () erm when / 0. Combining resuls we obain (3). Proof of heorem 2. his is a direc consequence of (4), (2), (3), Assumpions A and B. Proof of (3). oe H Ď = H Λ is he roaion marix under PC3. Since he principal componens esimaor saisfies λ i H λ i = O p ( /2 ), we have Λ = ( λ,..., λ r ) = H (λ,..., λ r ) + O p ( /2 ). Lef muliply H o obain H Λ = I r + O p ( /2 ) because (λ,..., λ r ) = I r under PC3. ha is, H Ď = I r + O p ( /2 ) so H Ď p I r. Using represenaion (5), we have (H Ď I r ) = HH ( e,..., e r ) + o p (). = However, (2) implies HH (3). Proof of (5). Recall ha ˆλ i λ i = Λ λ i λ i = Λ ( λ i H λ i ) + ( Λ H I r )λ i. Muliply on each side (ˆλ i λ i ) = Λ ( λ i H λ i ) = ( /) + O p (δ 2 ). his proves + Λ H (I r H Ď )λ i. or he firs erm on he righ hand side, using (5), Λ ( λ i H λ i ) = ( Λ H )(HH ) e i + o p (). Since H Λ = I r + o p () is inverse is also I r + o p (). urhermore, as argued earlier, HH = ( /) + O p (δ 2 ). hus Λ ( λ i H λ i ) = = = e i + o p (). he second erm on he righ hand side equals (I r H Ď )λ i + o p (). his proves (5). Proof of (6). irs noe ha ˆ = Λ = Λ ( H )+ Λ H = Λ ( H ) + (H Ď I r ). I follows ha (ˆ ) = Λ ( H ) + (/) /2 (H Ď I r ). rom (6), and using Λ H = I r + o p (), he firs erm on he righ hand side is Λ Λ ( H Λ ) = λ i e i + o p (). i= Combining he wo equaions leads o (6). Proof of heorem 3. his follows from (3), (5), (6), and Assumpions A and B. Proof of heorem 4. We firs consider he case of idenificaion under PC so ha we use in place of in he regression model. We can rewrie he model as in Bai and g (2006) y = W H α + ε β + ( H )H α = ẑ δ + ε + a where ẑ = (, W ), δ = (α H, β ), and a represens he las erm on he righ hand side. When / 0, Bai and g (2006) shows ha he error a is negligible, and he leas squares esimaor ˆδ has he sandard limiing disribuion as if conains no esimaion error (as if H were observable). More specifically, (ˆδ δ ) d (0, Φ Σ zz,εσ zz Φ 0 Σ zz 0 ) where Φ 0 = diag(v Q Σ Λ, I) and V Q Σ Λ is he probabiliy limi of H, where Q represens he probabiliy limi of /. In our case, he limi of H is an ideniy marix (also follows from Q = I r and V = Σ Λ in he presen case) so ha Φ 0 is an ideniy marix. his implies ha (ˆδ δ ) d (0, Σ δ ) Σ zz,εσ where Σ δ = Σ zz zz. urhermore, (ˆδ δ) = (ˆδ δ ) + [(α H α), 0 ]. Bu (α H α) = (H I r )H α = o p () provided ha / 0 because H Ir = O p (δ 2 ). I follows ha under / 0, (ˆδ δ) d (0, Σ δ ). We nex consider PC3. We use ˆ in place of, where ˆ is defined in he main ex. Since ˆ is an esimae of H Ď, we define δ Ď = [(H Ď α), β ]. hen y = ẑ δď + ε + a Ď, here aď = ( HĎ ˆ )H Ď α. he same argumen in Bai and g (2006) leads o