Maximum Likelihood Estimation of Time-Varying Loadings in High-Dimensional Factor Models. Jakob Guldbæk Mikkelsen, Eric Hillebrand and Giovanni Urga

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1 Maximum Likelihood Esimaion of Time-Varying Loadings in High-Dimensional Facor Models Jakob Guldbæk Mikkelsen, Eric Hillebrand and Giovanni Urga CREATES Research Paper Deparmen of Economics and Business Economics Aarhus Universiy Fuglesangs Allé 4 DK-8210 Aarhus V Denmark oekonomi@au.dk Tel:

2 Maximum Likelihood Esimaion of Time-Varying Loadings in High-Dimensional Facor Models Jakob Guldbæk Mikkelsen, Eric Hillebrand, and Giovanni Urga December 15, 2015 Absrac In his paper, we develop a maximum likelihood esimaor of ime-varying loadings in high-dimensional facor models. We specify he loadings o evolve as saionary vecor auoregressions VAR) and show ha consisen esimaes of he loadings parameers can be obained by a wo-sep maximum likelihood esimaion procedure. In he firs sep, principal componens are exraced from he daa o form facor esimaes. In he second sep, he parameers of he loadings VARs are esimaed as a se of univariae regression models wih ime-varying coefficiens. We documen he finie-sample properies of he maximum likelihood esimaor hrough an exensive simulaion sudy and illusrae he empirical relevance of he ime-varying loadings srucure using a large quarerly daase for he US economy. Keywords: High-dimensional facor models, dynamic facor loadings, maximum likelihood, principal componens. JEL classificaion: C33, C55, C13. We acknowledge suppor from The Danish Council for Independen Research DFF ) and CREATES - Cener for Research in Economeric Analysis of Time Series DNRF78), funded by he Danish Naional Research Foundaion. Par of he research was conduced wile he firs named auhor was visiing he Cenre for Economeric Analysis of Cass Business School in he Fall Corresponding auhor. CREATES, Deparmen of Economics and Business Economics, Aarhus Universiy, Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark. jmikkelsen@econ.au.dk. CREATES, Deparmen of Economics and Business Economics, Aarhus Universiy, Fuglesangs Allé 4, DK-8210 Aarhus V, Denmark. ehillebrand@econ.au.dk. Cenre for Economeric Analysis, Faculy of Finance, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ U.K.) and Bergamo Universiy Ialy). g.urga@ciy.ac.uk.

3 1 Inroducion In his paper, we develop a consisen maximum likelihood esimaor of ime-varying loadings in high-dimensional facor models where facors are esimaed wih principal componens. The problem of ime-varying loadings in facor models is imporan because he assumpion of consan loadings has been found o be implausible in a number of sudies considering srucural insabiliy in facor models. In a large macroeconomic daase for he U.S., Sock and Wason 2009) find considerable insabiliy in facor loadings around 1984, and hey improve facor-based forecas regressions of individual variables by allowing facor coefficiens o change afer he break poin. Breiung and Eickmeier 2011) develop Chow-ype ess for srucural breaks in facor loadings and find similar evidence of srucural insabiliy around They also find evidence of srucural breaks in he Euro area around 1992 and Del Negro and Orok 2008), Liu e al. 2011), and Eickmeier e al. 2015) esimae facor models in which he facor loadings are modelled as random walks using large panels of daa, bu heoreical resuls for models wih ime-varying parameers in a high-dimensional seing are scan. The economeric heory on facor models explicily addresses he high dimensionaliy of hese daases by developing resuls in a large N and large T framework. The cenral resuls in he lieraure on consisen esimaion of he facor space by principal componens as N, T have been developed in Sock and Wason 1998, 2002), and Bai and Ng 2002). Forni e al. 2000) consider esimaion in he frequency domain. Principal componens have he advanage of being easy o compue and feasible even when he cross-secional dimension N is larger han he sample size T. Baes e al. 2013) characerize he ypes and magniudes of srucural insabiliy in facor loadings under which he principal componens esimaor of he facor space is consisen. Anoher srand of lieraure is concerned wih esimaion by maximum likelihood. Bai and Li 2012a,b) consider maximum likelihood esimaion of facor loadings and idiosyncraic variances, while Doz e al. 2012) sudy funcions of maximum likelihood esimaors, also in a large N, T seing. Their analyses applies o facor models wih consan loadings. We consider a facor model of he form X i = λ i F +e i for i = 1,...N and = 1,..., T, where he daa X i depend on a small number r N of unobserved common facors F. The r 1 vecor of facor loadings λ i evolves over ime. We model λ i for each i as a saionary vecor auoregression, and our main conribuion is o show ha he parameers of hese ime-varying loadings can be consisenly esimaed by maximum likelihood. Our esimaion procedure consiss of wo seps. In he firs sep, he common facors are esimaed by principal componens, and in he second sep we esimae he loadings parameers by maximum likelihood, reaing he principal componens as observed daa. The principal componens esimaor is robus o saionary variaions in he loadings. By averaging over he cross-secion, he emporal insabiliies in he loadings are smoohed ou and he facor space is consisenly esimaed. Average consisency in of he facor space is shown by Baes e al. 2013), and we exend he resul o uniform consisency in o analyse he maximum likelihood esimaor. 2

4 In he second sep, we esimae a panel of regression models wih ime-varying coefficiens where he principal componens are reaed as he observed regressors, and he loadings are he ime-varying coefficiens. We allow for heeroskedasiciy and serial correlaion in he idiosyncraic errors e i, bu resric our aenion o cross-secionally uncorrelaed errors o avoid proliferaion of parameers. Condiional on he facor esimaes, he variables in he panel of regressions are herefore uncorrelaed, and he loadings parameers can be esimaed as a se of N univariae regression models wih ime-varying coefficiens. Under he condiion ha T 0, he maximum N 2 likelihood esimaor of he ime-varying loadings is consisen as N, T, and esimaion error from he principal componens can be ignored. We poin ou ha he compuaion of he maximum likelihood esimaor is relaively simple. Principal componens are simple o compue, and he se of N univariae regression models wih ime-varying parameers can be readily esimaed by Kalman-filer procedures. The res of he paper is organized as follows. Secion 2 inroduces he model and he wo-sep esimaion procedure. Secion 3.1 saes he assumpions and consisency resuls for he principal componens esimaor, and Secion 3.2 discusses idenificaion of he loadings parameers. Our main resul on consisency of he maximum likelihood esimaor of he ime-varying loadings and he associaed assumpions are saed in Secion 3.3. In Secion 4 we repor he resuls of a Mone Carlo sudy, and in Secion 5 we provide an empirical illusraion. Secion 6 concludes. 2 Model and Esimaion We consider he following model: X = Λ F + e, 1) where X = X 1,..., X N ) is he N-dimensional vecor of observed daa a ime. The observaions are generaed by a small number r N of unobserved common facors F = F 1,..., F r ), ime-varying facor loadings Λ = λ 1,..., λ N ), and idiosyncraic errors e = e 1,..., e N ) wih covariance marix Ee e ) = Ψ 0. The N r loadings marix Λ = λ 1,..., λ N ) is ime-varying and each λ i R r 1 evolves as an r-dimensional vecor auoregression: B 0 i L)λ i λ 0 i ) = η i, 2) where λ 0 i = Eλ i) is he uncondiional mean, and B 0 i L) = I B0 i,1 L... B0 i,p Lp is a p h -order lag polynomial wih roos ouside he uni circle. The auoregressive order p can be allowed o vary over i such ha p i differs over i. We suppress he subscrip for noaional convenience. The innovaions η i have covariance marix Eη i η i ) = Q0 i. Our goal is o esimae he parameers of each of he loadings processes 2) and he idiosyncraic variance marix Ψ 0. To achieve a sufficienly parsimonious paramerizaion of he N N marix Ψ 0, we specify i o be diagonal, Ψ 0 = diagψ1 0,..., ψ0 N ), such ha he idiosyncraic errors are cross-secionally uncorrelaed. Condiional on he facors, X i is herefore uncorrelaed over i, and 3

5 he model can be wrien as: X i = FΛ i + e i, 3) where X i = X i1,..., X it ), e i = e i1,..., e it ), F = diag {F },...,T is a T rt block-diagonal marix, and Λ i = λ i1,..., λ it ). The mean and variance of X i are EX i ) = F 1 λ0 i,..., F T λ0 i ) and Σ i := V arx i ) = FΦ i F + ψ i I T where Φ i = V arλ i ) is of dimension rt rt. We can hus specify a Gaussian likelihood funcion for X i condiional on he facors F = F 1,..., F T ) as: L T X i F ; θ i ) = 1 2 log2π) 1 2T log Σ i 1 2T X i EX i )) Σ 1 i X i EX i )), 4) wih parameer vecor θ i = {B i L), λ i, Q i, ψ i }. Equaions 2) and 3) can be wrien as a linear sae-space model and he likelihood can herefore be calculaed wih he Kalman filer. I is no feasible o esimae θ i wih 4), however, as he likelihood depends on he unobservable facors F. We herefore replace he unobservable facors F in 4) wih an esimae F o form he feasible likelihood funcion L T X i F ; θ i ). This gives us a se of N likelihood funcions o esimae he parameers θ i for each i. funcion as: Define he esimaor θ i which maximizes he feasible likelihood θ i = argmax θ L T X i F ; θ i ). 5) This is our objec of ineres and we show ha he esimaor θ p i θ 0 i for each i, where θi 0 = { B 0 i L), λ 0 i, Q0 i, } ψ0 i is he rue value of he parameers. We use he principal componens esimaor o esimae he facors. The principal componens esimaor reas he loadings as being consan over ime, Λ Λ, and solves he minimizaion problem: F, Λ) = minnt ) 1 F,Λ i=1 X i λ if ) 2, 6) where F is T r and Λ is N r. To uniquely define he minimizers, i is necessary o impose idenifying resricions on he esimaors, as only X i is observed. By concenraing ou Λ and using he normalizaion F F/T = I r, he problem is equivalen o maximizing rf XX )F ), where X = X 1,..., X T ) is he T N marix of observaions. The resuling esimaor F is given by T imes he eigenvecors corresponding o he r larges eigenvalues of he T T marix XX. The soluion is no unique: any orhogonal roaion of F is also a soluion. Bai and Ng 2008b) give an exensive reamen of he principal componens esimaor. We use F o form he feasible likelihood funcion L T X i F ; θ i ). The esimaion procedure hus consiss of wo seps. In he firs sep, we exrac principal componens from he observable daa o esimae he facors F, under he assumpion of consan loadings. In he second sep, we use he facor esimaes ogeher wih he observable daa o maximize he likelihood funcion and esimae he parameers θ i of he ime-varying loadings. Our main resul in Secion 3.3 shows ha his yields a consisen esimaor for he parameers of he 4

6 ime-varying loadings. 3 Asympoic Theory In his secion, we presen he asympoic heory for he wo-sep esimaion mehod discussed in Secion 2. The main resul is Theorem 1 on consisen esimaion of he loadings parameers by maximum likelihood; i is given in Secion 3.3. Our resul builds on he work by Baes e al. 2013), who show average consisency of he principal componens esimaor when loadings are subjec o srucural insabiliy. We use a differen roaion of he principal componens esimaor, and in Secion 3.1 we herefore resae heir resul in Lemma 1. Furhermore, we provide a resul on uniform consisency in of he principal componens esimaor in Proposiion 1. Secion 3.2 discusses idenificaion of he facors and loadings parameers. All resuls are for N, T, and he facor rank r is assumed o be known. We inroduce he following noaion. A = [ra A)] 1/2 denoes he Frobenius norm of he marix A. The subscrips i, j are cross-secional indices,, s are ime indices, and p, q are facor indices. The consan M 0, ) is a consan common o all he assumpions below. Finally, define C NT = min{ N, T }. 3.1 Principal Componens Esimaion Le ξ i := λ i λ 0 i = B0 i L) 1 η i be he loadings innovaions and wrie 1) as: X = Λ 0 F + ξ F + e, where Λ 0 = λ 0 1,..., λ0 N ) and ξ = ξ 1,..., ξ N ) are he N r marices of loadings means and innovaions, respecively. The vecor ξ i is he moving average represenaion of he loadings. Assumpions A-C are sandard for facor models and are he same as Assumpions A-C in Bai and Ng 2002): Assumpion A Facors). E F k M < for some k 4, and T F F p Σ F for some r r posiive definie marix Σ F. Assumpion B Loadings). λ 0 i M <, and Λ0 Λ 0 /N Σ Λ 0 for some posiive definie marix Σ Λ. Assumpion C Idiosyncraic Errors). There exiss a posiive consan M < such ha for all N and T : 1. Ee i ) = 0, E e i 8 M. 5

7 2. Ee se /N) = EN 1 N i=1 e ise i ) = γ N s, ), γ N s, s) M for all s, and T s, γ Ns, ) M. 3. Ee i e j ) = τ ij, wih τ ij, τ ij for some τ ij and for all. In addiion N 1 N i,j=1 τ ij M. 4. Ee i e js ) = τ ij,s, and NT ) 1 N i,j=1 T, τ ij,s M. 5. For every s, ), E N 1/2 N i=1 [e ise i Ee is e i )] 4 M. We leave he momen condiion on he facors in Assumpion A unspecified, as he resul in Proposiion 1 depends on k. Assumpion B requires he columns of Λ 0 o be linearly independen, such ha he marix Σ Λ is non-singular. Assumpions A and B ogeher imply he exisence of r common facors. Assumpion C allows for heeroskedasiciy and limied ime-series and cross-secion dependence in he idiosyncraic errors. I will laer be srenghened o e i being independen over i and. Noe ha if e i is independen for all i and, Assumpions C.2-C.5 follow from C.1. We impose he following assumpion on he facor loadings innovaions and he facors: Assumpion D Facor Loadings Innovaions). The following condiions hold for all N, T and facor indices p 1, q 1, p 2, q 2 = 1,..., r: 1. sup N i,j=1 Eξ isp 1 ξ jq1 F sp1 F q1 ) = ON). s, 2. N s, i,j=1 Eξ isp 1 ξ jsq1 F sp1 F sq1 F p2 F q2 ) = ONT 2 ). 3. sup N i,j=1 Eξ isp 1 ξ jsq1 ξ ip2 ξ jq2 F sp1 F sq1 F p2 F q2 ) = ON 2 ) + ONT ). s, Assumpion D is idenical o Baes e al. 2013) excep for D.3, which is sronger han heir corresponding assumpion. componens and is sill reasonable. Assumpion D.3 is needed for uniform consisency of he principal We do no require independence beween he facors and loadings, as he effec of he facors on he observable variables migh reasonably be expeced o change when he facors differ subsanially from heir mean levels. However, if he facors and loadings are assumed o be independen, and he loadings evolve as saionary vecor auoregressions ha are independen over i, Assumpions D.1-D.3 can easily be shown o hold: For simpliciy, ake r = 1. By Assumpion A and cross-secional independence of he loadings, he supremum in D.1 can be bounded by: sup s, { EF s F ) i,j=1 Eξ is ξ j ) } Msup s, Eξ is ξ j ) = M i,j=1 i=1 sup Eξ is ξ i ). s, The erms Eξ is ξ i ) are he auocovariances of he moving average represenaion of he loadings. As he loadings are saionary, hese auocovariances are bounded, and he rae ON) follows. The rae ONT 2 ) in D.2 follows from D.1 when he facors and he loadings are independen. The sum 6

8 in D.3 can be bounded by: Msup s, M i,j=1 i=1 Eξ is ξ js ξ i ξ j ) = Msup s, sup Eξis) 4 + Msup s, s, i=1 Eξ 2 isξ 2 i) + Msup s, i j Eξ is ξ i )Eξ js ξ j ) T ) 1/2 T ) 1/2 Eξ is ξ i ) 2 E ξ js ξ j ) 2. i j The firs erm is ONT ) if Eξ 4 is ) <, and he second erm is ON 2 ) if he auocovariances Eξ is ξ i ) are square-summable. Assumpion D.3 is herefore saisfied when he loadings and he facors are independen. We assume he same raes o hold wihou imposing independence beween he facors and he loadings. Finally, we impose independence beween he idiosyncraic errors and he facors and loadings innovaions. Assumpion E Independence). For all i, j, s, ), e i is independen of F s, ξ js ). Assumpions A-E are sufficien o consisenly esimae he space spanned by he facors. For his purpose, we use he resul of Lemma 1 below, which is a modified version of Theorem 1 in Baes e al. 2013). 1 We use a rescaled esimaor ha is more convenien for he res of he analysis and herefore resae heir resul: Lemma 1. Under Assumpions A-E here exiss an r r marix H such ha as N, T. Proof. See he Appendix. F H F 2 = O p C 2 NT ) Lemma 1 shows ha he mean-squared deviaion beween he principal componens and he common facors disappears as he sample size T and he cross-secional dimension N end o infiniy. 2 The convergence rae C NT is he same as in Bai and Ng 2002), and he principal componens esimaor is hus robus o saionary deviaions in he loadings around a consan mean. Noe ha he common facors are only idenified up o a roaion, so he principal componens converge o a roaion of he common facors. 1 Baes e al. 2013) use he esimaor ˆF = F V NT, where V NT is he diagonal marix of he r larges eigenvalues of NT ) 1 XX. 2 Lemma 1 also holds when he facor rank is unknown. By seing he number of esimaed facors o any fixed k 1, he Lemma can be saed as T F k H k F 2 = O pc 2 NT ), where F k is k 1 and H k is a r k marix, and F k consisenly esimaes he space spanned by k of he rue facors 7

9 Lemma 1 does no imply uniform convergence in, bu only average consisency of he principal componens. In order o analyse he properies of he feasible likelihood funcion L T X i, F θ i ), we need uniform consisency of he esimaed facors, in addiion o he average consisency of Lemma 1. To esablish uniform convergence, we make addiional assumpions, as in Bai and Ng 2006, 2008a): Assumpion F There exiss a posiive consan M < such ha for all N and T : 1. T γ Ns, ) M for all. 2. E NT ) 1/2 T N k=1 F s[e ks e k Ee ks e k )] 2 M for all. 3. E N 1/2 N i=1 λ0 i e i 8 M for all. Assumpion F.1 is sronger han C.2, bu sill reasonable: If e i is assumed o be saionary wih absoluely summable auocovariances, Assumpion F.1 holds. Assumpions F.2 and F.3 are reasonable as hey involve zero-mean random variables. We can now presen he uniform consisency resul for he esimaed facors. Proposiion 1. Under Assumpions A-F and addiionally if max max F H F = O p T 1/8 N 1/2 ) F = O p α T ), and T/N 2 0, + O p α T N 1/2 ) + O p α T ) + O p C 1 NT ). Proof. See he Appendix. Proposiion 1 shows ha he maximum deviaion beween he facors and he principal componens depends on α T. The convergence rae hus depends on he assumpion imposed on max F. The facors can be modelled as a dynamic process wih arbirary dynamics o deermine α T. However, if he parameers governing hese dynamics are no of direc ineres, nohing is los by assuming he facors o be a sequence of fixed and bounded consans. Thus max F M. 3 We can ake O p α T ) o be O1) in our resuls, and max F H F = o p 1). However, Proposiion 1 is of independen ineres, e.g. for deriving he limiing disribuion for he maximum likelihood esimaor, so we sae Proposiion 1 in is more general form. Bai 2003) and Bai and Ng 2008a) derive a similar resul for facor models wih consan loadings. Uniform convergence when loadings undergo small variaions is also considered by Sock and Wason 1998), who obain a much slower convergence rae and require T = on 1/2 ). Thus, Proposiion 1 exends he uniform consisency resul o he case of ime-varying loadings. 3 Bai and Li 2012a,b) rea he facors as a sequence of fixed consans when providing inferenial heory for maximum likelihood esimaion of facor models wih consan loadings. 8

10 3.2 Idenificaion I is well known ha wihou idenifying resricions, facors and loadings are no separaely idenified in 1). The common componen C = Λ F is idenified, bu normalizaions are needed o separae facor and loadings from he common componen. This has implicaions for he idenificaion of he loadings parameers as well, which we now illusrae. The model defined by 1) and 2) is observaionally equivalen o: X = Λ H 1 H F + e, B i L)H 1 λ i λ i ) = H 1 η i, for i = 1,..., N. Lemma 1 saes ha he principal componens esimaor F is a consisen esimae of a roaion of he rue facors, H F. The wo-sep esimaion procedure fixes he roaional indeerminacy by imposing he normalizaion in he principal componens sep. By replacing he unobserved facors F wih F for maximum likelihood esimaion, we are hus esimaing he parameers of λ i = H 1 λ i. To clarify he issue, consider he following example. Using he same noaion as previously, he elemens of he r 1 vecor λ i = λ i,1..., λ i,r ) refer o he loadings of variable i a ime on each of he r facors, and λ i = Eλ i ) = λ i,1,..., λ i,r ) are he corresponding uncondiional expecaions of he facor loadings. Assume ha he marices Σ F and Σ Λ are diagonal. In his case i is no hard o show ha he roaion marix H converges o Σ 1/2 F. Le he number of facors r = 2 wih variance Σ F = diagσ1 2, σ2 2 ) and le he daa-generaing parameers of he loadings be ) ) ) λ 0 λ i,1 i =, Q 0 q i,1 0 i =, B 0 b i,11 0 i L) = I 2. λ i,2 0 q i,2 0 b i,22 We can now make precise wha θ is esimaing. Wih he normalizaion F F /T = I2, he principal componens will be close o Σ 1/2 F F in large samples. Using he principal componens in place of he unobserved facors means ha we are esimaing he following model: where λ i = Σ 1/2 F λ i = X = Λ F + e, λ i λ i = B i λ i, 1 λ i ) + v i, for i = 1,..., N, σ 1 1 λ i,1 σ2 1 λ i,2 ) and v i = Σ 1/2 F η i. The loadings λ i are scaled by he sandard deviaions of he unobserved facors, and i is he parameers of he roaed loadings λ i ha can be esimaed. In large samples he esimae of he loadings mean λ i will be herefore close o Σ 1/2 F λ 0 i = σ 1 1 λ i,1 σ 1 2 λ i,2 ), 9

11 and he variance esimae V arv i ) = V arσ 1/2 F η i ) will be close o Σ 1/2 F Q 0 i Σ 1/2 F = σ 2 1 q i,1 0 0 σ 2 2 q i,2 The mean and variance parameers are hus scaled by he sandard deviaion of he facors. The marices B i L) and Q 0 i of he daa-generaing model are diagonal in his example, so he diagonal elemens of Bi are he auocorrelaions of λ i,1 and λ i,2. In large samples he firs diagonal elemen of Bi will herefore be close o b i,11 = Covλ i,1, λ i, 1,1 ) V arλ i,1 ) ) = Covσ 1 1 λ i,1, σ 1 1 λ i, 1,1) V arσ 1 1 λ i,1) = σ 2 1 Covλ i,1, λ i, 1,1 ) σ1 2 V arλ = Covλ i,1, λ i, 1,1 ) = b i,11, i,1) V arλ i,1 ) and similarly for b i,22. The esimaes of he auoregressive marix B i he normalizaion imposed on he principal componens, and he esimae of Bi he auoregressive parameers B i of he daa-generaing process λ i.. are herefore unaffeced by is consisen for The argumens of his example apply o he general seing as well. The maximum likelihood esimaor 5) of he loadings parameers is esimaing B i L), H 1 λ i, and H 1 Q i H 1. The mean and variance parameers of 2) are idenified up o he unknown roaion marix H, while he dynamic parameers B i L) are no subjec o any roaion. resricion used o idenify he principal componens. The roaion is deermined by he Using anoher normalizaion in he firs sep will hus change he esimaes of λ i and Q i, while he esimae of B i L) is unaffeced, excep for small numerical differences owing o numerical opimizaion of he likelihood. The dynamic properies of he loadings are herefore uniquely idenified. In he following, we assume for simpliciy ha H = I r. This is jus a normalizaion and can be achieved by imposing furher assumpions on he marices Σ F and Σ Λ. 3.3 Maximum Likelihood Esimaion Our mehod of proof relies on showing ha he likelihood funcion 4) wih principal componens is asympoically equivalen o he likelihood funcion wih unobserved facors. To esablish our resul, we impose disribuional assumpions on he loadings and idiosyncraic errors ha enable maximum likelihood esimaion of he parameers θ i = {B i L), λ i, Q i, ψ i }. We make he following assumpions: Assumpion G Disribuions) For all i = 1,..., N, he following saemens hold: 1. The loadings λ i follow a finie-order Gaussian VAR: B i L)λ i λ i ) = η i, 10

12 wih he r r filer B i L) = I B i,1 L... B i,p L p having roos ouside he uni circle, and η i is an r-dimensional Gaussian whie noise process, η i i.i.d. N 0, Q i ), where Q i is posiive definie wih all elemens bounded. 2. The idiosyncraic errors e are cross-secionally independen Gaussian whie noise, e N 0, Ψ), where Ψ is a diagonal marix wih elemens ψ i > 0 and bounded for all i. G.1 assumes he loadings o evolve as saionary vecor auoregressions. We rule ou he possibiliy of I1) loadings as his would be in violaion of Assumpion D. Wih non-saionary loadings he principal componens esimaor canno consisenly esimae he facor space. 4 G.2 assumes he idiosyncraic errors o be i.i.d. over boh and i. This assumpion can be relaxed o allow for serial correlaion. The key par of Assumpion G.2 is he cross-secional independence. This enables us o analyse he likelihood separaely for each i. This is a se of N independen univariae regressions wih ime-varying parameers. Wih observed regressors, consisency is known o hold, see e.g. Pagan 1980). Raher han proving consisency for he maximum likelihood esimaor wih observed facors we herefore assume consisency in he following assumpion. Assumpion H MLE wih observed facors) For each i, he funcion L T X i F ; θ i ) saisfies: 1. There exiss a funcion L 0 X i F ; θ i ) ha is uniquely maximized a θ θi 0 is in he inerior of a convex se Θ i, and L T X i F ; θ i ) is concave. 3. L T X i F ; θ i ) p L 0 X i F ; θ i ) for all θ i Θ i. Under Assumpions G and H, he maximum likelihood esimaor wih observed facors ˆθ i = p argmax L T X i F ; θ i ) exiss wih probabiliy approaching 1 and is consisen for each i: ˆθi θ 0 i. θ This follows from sandard argumens as in Newey and McFadden 1994). Replacing he unobserved facors wih he principal componens esimaes yields he feasible likelihood funcion L T X i F ; θ i ) and he maximum likelihood esimaor defined in 5). We now sae our main resul. Theorem 1. Le Assumpion A-H hold. For each i, he esimaor θ i defined in 5) exiss wih probabiliy approaching 1 and p θ i θ 0 i. Proof. See he Appendix. 4 Baes e al. 2013) consider random walk loadings of he form λ i = λ i, 1 + T 3/4 ζ i and show ha Assumpion D is saisfied wih his specificaion. However, he scaling of he loadings innovaions by he facor T 3/4 is crucial for Lemma 1 o hold. Wih a pure random walk of he form λ i = λ i, 1 + ζ i, principal componens canno esimae he facor space consisenly. 11

13 Theorem 1 saes ha using he principal componen esimaes insead of he unobserved facors does no affec he consisency of he maximum likelihood esimaor. The main argumen in proving Theorem 1 is ha he feasible likelihood funcion convergences uniformly o he infeasible likelihood funcion. Asympoically, he feasible likelihood funcion herefore has he same properies as he infeasible likelihood funcion, for which consisency is known o hold. Assumpion H hus holds for L T X i F ; θ i ) and consisency follows. In he proof of Theorem 1 we use he following normalizaion ha is convenien for he calculaions: If F F/T = I r and Λ 0 Λ 0 is a diagonal marix wih disinc elemens, we show in he Appendix ha he roaion marix H converges o he ideniy I r. Lemma 1 and Proposiion 1 hen holds wih H replaced by he ideniy marix, and θ i can be esimaed asympoically wihou roaion. Such normalizaions are inconsequenial for he resuls as H is asympoically bounded, and hey are only imposed o avoid unnecessary complicaions. Wihou such normalizaions he feasible likelihood converges o L T X i F H; θ i ) and θ i is consisen for he parameers of he process λ i = H 1 λ i as discussed in Secion 3.2. We have assumed ha he facors are esimaed by he mehod of principal componens. Noe, however, ha he proof of Theorem 1 does no rely on he principal componens esimaor. Theorem 1 holds for all esimaors F ha saisfy he condiions for Lemma 1 and Proposiion 1. Our analysis does no make any formal saemens abou he limiing disribuion of θ i. Simulaion evidence in Secion 4 does, however, sugges ha an asympoic normaliy resul holds for θ i as well. The simulaions furher indicae ha he limiing disribuion of θ i is unaffeced by he esimaion error of F. Two-sep esimaors ypically require adjusing he limiing disribuion o accoun for esimaion error from firs-sep esimaion as in Newey 1984) and Pagan 1986), bu ha does no seem o be he case here. Bai and Ng 2006) show ha he esimaed-regressor problem can be ignored when using principal componens in place of he unobserved facors in facor-augmened VARs. We expec ha a similar resul holds for our model, bu leave a formal proof for fuure research. In Assumpion G.2 and he proof of Theorem 1 we assume ha he model has an exac facor srucure in he sense ha idiosyncraic errors have no cross-secional or emporal dependence. I is sraighforward o relax he assumpion of no emporal dependence. We could model he idiosyncraic errors as cross-secionally uncorrelaed auoregressions and esimae he parameers by including e i in he sae equaion of he sae space represenaion of he model and compue he likelihood wih he Kalman filer. The proof of Theorem 1 applies wih very minor changes. The assumpion of no emporal dependence in e i is hus only for exposiional simpliciy. Relaxing he assumpion of no cross-secional correlaion o allow for an approximae facor srucure requires subsanially more work. The essenial conribuion of Assumpion G.2 is ha i enables he likelihood of he full panel of daa X = X 1,..., X N ) o be analysed separaely for each X i. When he idiosyncraic elemens of he model are cross-secionally correlaed, Ee e ) = Ψ 0 is non-diagonal, and we canno condiion on he facors o make X i independen over i. Analysis of he model wih cross-secionally correlaed errors requires a differen mehod of proof and is be- 12

14 yond he scope of his paper. 5 However, in he nex secion we provide simulaion evidence showing ha our resuls are robus o he assumpion of no cross-secional correlaion in he idiosyncraic elemens. 4 Mone Carlo Simulaions In his secion, we conduc a simulaion sudy o assess he finie-sample performance of he wo-sep esimaor. We provide resuls for boh principal componens and maximum likelihood esimaes. Secion 4.1 describes he simulaion design, and Secion 4.2 repors and discusses he resuls. 4.1 Design The simulaion design broadly follows ha of Sock and Wason 2002): X i = λ if + e i, I r B i L)λ i λ i ) = η i, F p = ρf 1,p + u p, 1 αl)e i = v i, η i i.i.d. N 0, Q i ), u p i.i.d. N 0, 1 ρ 2 ), v i.i.d. N 0, Ω), where i = 1,..., N, = 1,..., T, p = 1,..., r. The processes {η i },{u p }, and {v } are muually independen. The auoregressive marix B i deermines he degree of persisence of he loadings and has eigenvalues inside he uni circle in all simulaions. The uncondiional mean of he loadings is λ i = λ i1,..., λ ir ) and λ ip i.i.d. N 0, 1) in all simulaions. The marix Q i is he covariance marix of he loadings innovaions. The model allows for cross-secional and emporal dependence in he errors e i. The parameer α deermines he degree of serial correlaion in he idiosyncraic errors, and cross-secional ) correlaion is modelled by specifying he variance marix of v as Ω = β i j ψ i ψ j for i, j = 1,..., N. The marix is hus a Toepliz marix and he cross-secional ij correlaion beween he idiosyncraic elemens is herefore limied and deermined by he coefficien β. If β = 0, α = 0, e i is independen across i and, and he model is an exac facor model and correcly specified according o Assumpion G. We allow for facor persisence hrough he coefficien ρ. We generae he model 2000 imes for each of he differen combinaions of T and N. avoid any dependence on iniial values of he simulaed processes we have a burn-in period of 200 observaions for each simulaion. The principal componens are calculaed wih he esimaor F defined in 6). The daa X i are sandardized o have mean zero and variance equal o one prior o 5 Bai and Li 2012a) analyse facor models wih consan loadings in a maximum likelihood seing under weak cross-secional correlaion in e i. They show ha he esimaes of ψi 0 are consisen for he diagonal elemens of T Eee ). However, hey rely on a mehod of proof ha differs subsanially from ours and heir resuls do no readily apply o our model. To 13

15 exracing principal componens. The principal componens are idenified only up o an orhogonal roaion. In order o direcly compare he maximum likelihood esimaes wih daa-generaing parameers, we herefore roae he principal componens o resemble he simulaed facors. More specifically, we solve for he orhogonal r r marix A ha maximizes r[corrf, F A)]. 6 The esimaes are hen rescaled o have he same sandard deviaion as he rue simulaed facors: F p = σf p) σ F F p, p ) p = 1,..., r where F p is he p h column of he roaed principal componens marix F A. Such roaions are innocuous and allow us o direcly compare he esimaed parameer values wih he daageneraing parameers. The principal componens are reaed as daa, and we maximize he likelihood L T X i F ; θ i ) o esimae θ i. The performance of he principal componen esimaor F is measured by he race saisic: R 2 F,F = Ê[rF F F F ) 1 F F )], Ê[rF F )] where Ê denoes he average over he Mone Carlo simulaions. The race saisic R2 F,F is a mulivariae R 2 from a regression of he rue daa-generaing facors on he principal componens. I is smaller han 1 and ends o 1 as he canonical correlaion beween he facors and he principal componens ends o 1. For he maximum likelihood esimaes θ i we compue he mean esimaes over he Mone Carlo repeiions for each parameer. 7 However, for he mean parameer λ i we repor he bias of he esimaes λ i as he rue value of λ i changes for each combinaion of N, T. Furhermore, we calculae he roo-mean-squared error of he esimaes θ i and also of he infeasible esimaes ˆθ i where he rue daa-generaing facors are used in he maximum likelihood esimaion. We repor he relaive roo-mean-squared error beween he esimaes θ i and ˆθ i. This gives us a measure of he esimaion error in θ i ha is due o esimaion error from he principal componens esimaes. The parameers are idenically chosen across he cross-secion. 8 The properies of he esimaed parameers θ i are hus he same for all i and we only repor he resuls for a single cross-secion index. 9 In he baseline case, we se B i = diag{b ip } p=1,...,r, Q i = diag{q ip } p=1,...,r, and choose he 6 The soluion o his is A = V U where V and U are he orhogonal marices of he singular value decomposiion corrf, F ) = USV. When he number of principal componens k is no equal o he rue number of facors r, we only roae he firs l = min{k, r} principal componens. Eickmeier e al. 2015) use he same roaion. 7 Convergence is generally very good, wih all 1-facor calibraions having over 99% convergence rae, and mos calibraions wih 2 and 3 facors have over 98% convergence rae. Excepions are sample sizes of T = 50 for he 2- and 3-facor models where he lowes convergence rae is 92%. However, his is expeced as we are esimaing up o 10 parameers in a highly non-linear model wih 50 observaions. Convergence saisics using he rue facors are similar, bu wih somewha beer convergence raes for calibraions wih 2 and 3 facors and T = The mean parameers λ i are no he same for all i. This is necessary for Assumpion B o be saisfied. Wih λ i idenical over i, he marix Λ 0 does no have full rank and Λ 0 Λ 0 /N will no converge o a posiive definie marix. 9 Simulaions wih loadings parameers calibraed wih heerogeneous values across i show similar resuls as in Table 1. The resuls are available upon reques. 14

16 loadings persisence and variance parameers o be b ip = 0.9 and q ip = 0.2. The idiosyncraic errors are cross-secionally and emporally uncorrelaed, i.e. α = 0, β = 0, and he variance is se a ψ i = 1. Finally, we se ρ = 0 such ha he facors are whie noise. We inroduce serial correlaion and cross-secional dependence separaely in he idiosyncraic errors. We se α = 0.5 and esimae his parameer by including e i in he sae equaion. To consider he effec of cross-secional correlaion, i.e. misspecifying he model, we se β = 0.5. We also repor resuls wih persisen facors wih he facor persisence se a boh 0.9 and 0.5. Finally, we consider he consequences of esimaing he wrong number of facors, i.e. exracing one fewer or one addiional principal componen han he rue number of facors. 4.2 Resuls Table 1 repors he resuls for one facor, r = 1. Panel I shows he resuls for he baseline model wih no serial, no cross-secional dependence in errors, and no facor dependence. The R 2 F,F saisics show ha he facor esimaes are close o he rue facors even for small sample sizes. For he auoregressive parameer b i, he esimaes improve as he sample size T increases. Increasing he cross-secional dimension N only gives minor improvemens for fixed T. This is unsurprising as a larger N can only improve he parameer esimaes hrough beer facor esimaes which are already quie good even for N = 50. The esimae of he loadings innovaion variance q i is closely relaed o he esimae of b i. As b i ges closer o is rue value, so does q i, and vice versa. For T 200 he esimaes are close o he rue values. The small-sample bias of b i is no a consequence of esimaion error from principal componens. Using he rue facors insead of principal componens o esimae he parameers of he laen process λ i also shows ha T 200 is needed for he bias of b i and q i o be less han 10% of he rue value. The loadings mean λ i and he error variance ψ i are very precisely esimaed for all sample sizes. In Panel II, he idiosyncraic errors are serially correlaed, and he auoregressive parameers for he errors are esimaed along wih he oher parameers. The R 2 F,F saisic is hardly affeced by serially correlaed errors. The resuls are very close o he corresponding values in he firs panel. The resuls for he loadings parameers are also very similar and are no markedly affeced. The auoregressive parameer for he errors α and he variance parameer ψ are very close o heir rue value for all sample sizes. The model wih serially correlaed errors can hus be esimaed equally well as he model wih i.i.d. errors. 15

17 Table 1 - Simulaion resuls for 1-facor model T N R 2 F,F b i λ i q i ψ i α Panel True values α = 0, β = 0, ρ = I α = 0.5, β = 0, ρ = II α = 0, β = 0.5, ρ = III α = 0, β = 0, ρ = IV α = 0, β = 0, ρ = V Noes: The columns T and N repor he sample sizes. The column R 2 F,F repors he convergence saisic for he principal componens esimaor. The remaining columns repor he mean of he parameer esimaes over he Mone Carlo simulaions. For he parameer λ i, he bias is repored. 16

18 Table 2 - Relaive roo-mean-squared error for 1-facor model T N R 2 F,F b i λ i q i ψ i α Panel True values α = 0, β = 0, ρ = I α = 0.5, β = 0, ρ = II α = 0, β = 0.5, ρ = III α = 0, β = 0, ρ = IV α = 0, β = 0, ρ = V Noes: The columns T and N repor he sample sizes. The column R 2 F,F repors he convergence saisic for he principal componens esimaor. The remaining columns repor he relaive roo-meansquared error of he parameer esimaes using principal componens and he rue simulaed facors. 17

19 Table 3 - Simulaion resuls for 2- and 3-facor model T N R 2 F,F b i1 λ i1 q i1 b i2 λ i2 q i2 b i3 λ i3 q i3 ψ i Panel True values Two facors α = 0, β = 0, ρ = I Three facors α = 0, β = 0, ρ = II Noes: The columns T and N repor he sample sizes. The column R 2 F,F repors he convergence saisic for he principal componens esimaor. The remaining columns repor he mean of he parameer esimaes over he Mone Carlo simulaions. For he parameer λ i, he bias is repored. Table 4 - Relaive roo mean squared errors for 2- and 3-facor model T N R 2 F,F b i1 λ i1 q i1 b i2 λ i2 q i2 b i3 λ i3 q i3 ψ i Panel True values Two facors α = 0, β = 0, ρ = I Three facors α = 0, β = 0, ρ = II Noes: The columns T and N repor he sample sizes. The column R 2 F,F repors he convergence saisic for he principal componens esimaor. The remaining columns repors he relaive roo-mean-squared error of he parameer esimaes using principal componens and he rue simulaed facors. 18

20 Nex, in Panel III we consider he effec of cross-secional correlaion in he errors. The facor esimaes are again no affeced. The misspecificaion of he variance marix for he idiosyncraic errors deerioraes he esimaes of b i and q i for he smaller sample sizes. Compared o he resuls for i.i.d. errors, esimaes are worse, bu do converge for he larger sample sizes. The mean parameer λ i is no affeced. One can show ha he informaion marix of he likelihood is block diagonal beween λ i and he oher parameers, so misspecifying he variance does no lead o esimaion error in he mean. Cross-secionally correlaed errors inflae he esimae of he idiosyncraic error variance ψ i, however, bu he loadings parameers remain consisen. This indicaes ha our resul in Theorem 1 is robus o he assumpion of no cross-secional correlaion in e i. High facor persisence has a larger impac on he R 2 F,F saisic. Panel IV shows much lower values of hese saisics for all bu he larges sample sizes. However, his esimaion error does no seem o influence he esimae of he loadings parameers. The esimaes for b i, and accordingly q i, are similar o he case of whie noise facors. The mos noable impac of he lower R 2 F,F is in he esimae of ψ i. The increase in facor esimaion error seems o inflae he error variance, which is larger for all sample sizes, bu he resuls do show convergence for he larges sample size. Resuls for more moderae levels of facor persisence are shown in Panel V. The drop in he R 2 F,F is less severe in his case and he esimae of ψ i hus less biased. In Table 2, he relaive roo-mean-squared errors of he esimaes using principal componens and he rue simulaed facors are repored. Values close o 1 indicae ha he asympoic variance of he parameer esimaes is unaffeced by he esimaion error from principal componens esimaion of he facors. In Panels I-III, all he saisics are close o 1 even for he smalles sample sizes. In Panel IV, he saisics for he loadings parameers are somewha higher for he smaller sample sizes, bu close o one for large sample sizes. The saisics for he idiosyncraic variances are much larger han 1. This is parly due o he bias of hese esimaes eviden in Panel IV of Table 1, bu also reflecs higher variabiliy of he esimaes. High facor persisence hus mainly affecs he idiosyncraic variance parameers. Unrepored resuls show ha he esimaes improve for larger sample sizes. In Panel V, he facor persisence is more moderae and he relaive roo-mean-squared errors are much closer o 1. Table 3 displays he simulaions resuls for he model wih 2 and 3 facors wih i.i.d. errors and whie noise facors. Compared o he 1-facor model, he R 2 F,F saisics are lower, reflecing he increasing difficulies in exracing addiional facors. In Panel I, he esimaes for he second se of loadings parameers are worse han for he firs se and he same paern is eviden for he 3-facor model Panel II). The resuls for he hird se of loadings parameers are worse han for he second, which are worse han for he firs. However, all he esimaes are converging o heir rue values. Compared o he 1-facor model, larger sample sizes are generally needed o ge precise esimaes due o he increased number of parameers. Inroducing serial and cross-secional correlaion in he errors or persisence in facors does no reveal any addiional insighs compared o he 1-facor model. The resuls generalize and are herefore omied. Table 4 shows he relaive roo-meansquared errors for he 2- and 3-facor model. The saisics are somewha larger han 1 for he 19

21 smaller sample sizes, bu ge increasingly closer o one as he sample sizes grow. This indicaes ha he esimaion error of he principal componens does no affec he asympoic variance of he esimaes. Table 5 - Simulaion resuls for incorrec number of principal componens T N R 21) R 22) b F,F F,F i1 λ i1 q i1 b i2 λ i2 q i2 ψ i Panel True values principal componen, wo facors α = 0, β = 0, ρ = I principal componens, 1 facor α = 0, β = 0, ρ = II Noes: The columns T and N repor he sample sizes. The columns R 21) and R21) are he F,F F,F wo convergence saisics for he principal componens esimaor. The remaining columns repor he mean of he parameer esimaes over he Mone Carlo simulaions. For he parameer λ i, he bias is repored. Table 5 shows he resuls of esimaing he wrong number of facors. For hese simulaions, we repor wo convergence saisics for he principal componens. The firs is he R 2 from a regression of he principal componens on he rue facors, R 21) F = Ê[ F F F F ) 1 F F ],F Ê[ F F and he second is he ] R 2 from a regression of he rue facors on he principal componens. In Panel I, he simulaed daa have wo facors, bu only 1 principal componen is exraced. The firs saisic R 21) is close F,F o 1 for all sample sizes. Hence, he wo facors explain all he variaion in he single principal componen. The second saisic R 22) does no end o 1, as a single principal componen canno F,F span he wo-dimensional facor space. The resuls show ha he loadings parameers for he firs facor can sill be esimaed consisenly. The consequence of excluding a facor is ha he esimae of he error variance ψ i ges larger, reflecing he variabiliy in he daa from he excluded facor and is loadings. Panel II displays resuls for he 1-facor model wih wo principal componens exraced from he daa. R 22) ends o 1, and he wo principal componens hus explain all he F,F variaion in he single facor. The oher measure R 21) ends o 0.5 as he single facor can only F,F span half of he wo-dimensional space of he principal componens. The loadings on he firs facor 20

22 are esimaed consisenly. For he second facor, he mean and variance of he loadings are being esimaed as zero. 10 The esimaed parameers hus show ha he daa do no load on he second facor and herefore correcly dismiss he second facor. The resuls are hus very encouraging even wih he number of principal componens differen from he rue number of facors. The resuls can be summarized as follows: The loadings and idiosyncraic variance parameers are esimaed consisenly. The sample size T needs o be sufficienly large 200) for he bias in he auoregressive parameers o be less han 10%. The resuls are robus o he assumpion of cross-secionally uncorrelaed errors. The loadings parameers are no affeced by his misspecificaion, only he esimae of he error variance. Loadings parameers are consisenly esimaed even when an incorrec number of principal componens are exraced. Too few principal componens increase he error variance esimae, and loadings means and variances are correcly esimaed as zero for principal componens in excess of he rue number of facors. The relaive roo-mean-squared errors indicae ha he asympoic variance is unaffeced by replacing he facors wih he principal componens esimaes and ha he esimaes have he same limiing disribuion as if he facors are observed. 5 An Empirical Illusraion We provide an empirical illusraion of he model using he daa se of Sock and Wason 2009), who analyse a balanced panel of 144 quarerly ime series for he Unied Saes, focusing on srucural insabiliy in facor loadings and is consequences for forecas regression. The daa se consis of 144 quarerly ime series for he Unied Saes, spanning 1959:I-2006:IV. The daa series are ransformed o be saionary, and he firs wo quarers are hus excluded because of differencing, resuling in T = 190 observaions used for esimaion. We exclude a number of series ha are higher-level aggregaes of he included series, which brings he number of series used for esimaion o N = 109. For a complee daa descripion and deails on daa ransformaions, see he appendix of Sock and Wason 2009). Sock and Wason 2009) argue for 4 facors in he sample, and perform robusness checks of heir resuls using differen numbers of facors. We herefore exrac 4 principal componens from he sandardized daa and esimae he loadings parameers and he idiosyncraic variances for each of he 109 variables. The sysem marices B i and Q i are specified o be diagonal, i.e., he loadings are esimaed as univariae auoregressions uncorrelaed over he facor indices. The lag polynomials B i L) are of order one for all i. 10 The resuls for b i2 are no indicaive of any convergence. Hisograms of he esimaed values show ha he parameer is no idenified as he values are randomly esimaed anywhere beween -1 and 1. 21

23 Sock and Wason 2009) es for breaks in he facor model and find evidence of srucural insabiliy in a large number of he facor loadings. Using Chow saisics o es for breaks in loadings in 1984:I, hey rejec he null of no insabiliy for 41% of he variables. We provide similar evidence of srucural insabiliy in he loadings. For each variable X i, we es he null hypohesis of consan loadings by likelihood raio saisics. In he resriced model, we hus se he diagonal elemens of he marix Q i equal o zero, in which case he maximum likelihood esimaor of he resriced model can be compued by ordinary leas squares. The las column of Table 6 repors he rejecion frequencies grouped by variable caegory as well as he rejecion frequency for he enire panel. For 85% of he series, he likelihood raio saisics rejecs a he 5% significance level, and 76% of he series are rejeced a he 1% level. When comparing he rejecion frequencies across caegories, no obvious paern emerges: The rejecion frequencies are high for all caegories. The caegory wih he lowes rejecion frequency is consumpion variables, where he null of consan loadings canno be rejeced for a hird of he variables. Table 6 - Rejecion frequencies of likelihood raio saisics Caegory F 1 F 2 F 3 F 4 All Facors Oupu Consumpion Labour marke Housing Invesmen Prices & Wages Financial variables Money & Credi Oher All Noes: Column F 1 repors he rejecion frequencies across variable groups of he likelihood raio saisics for esing he null hypohesis of consan loadings on he firs facor and similarly for columns F 2 F 4. The las column repors he rejecion frequencies for he null hypohesis of consan loadings on all facors. All ess are evaluaed a he 5% significance level. We also es he null hypohesis of consan loadings on each individual facor. We reesimae he model for each variable 4 imes and le he loadings on a single facor be consan each ime, while he res of he loadings follows firs-order auoregressions. Columns F 1 F 4 of Table 6 repor he rejecion frequencies for hese likelihood raio saisics. For he firs wo facors, approximaely wo hirds of he series show evidence of ime-varying loadings. For he las wo facors, he rejecion frequencies are smaller, bu ime-variaion in he loadings is sill eviden for a non-rivial number of he series. Comparing he rejecion frequencies across variable caegories does no reveal any general paerns: The rejecion frequencies vary a lo across caegories and facors. Closer inspecion of he individual series indeed reveals ha he way in which he daa load on he facors is very heerogeneous across series. 22

24 Figure 1: Facor loadings for he one-year/hree-monh Treasury erm spread Figure 2: One-year/hree-monh Treasury erm spread and wo esimaes of is common componen CC) 23

25 Comparing our resuls o hose of Sock and Wason 2009), we find ha a larger share of he variables have ime-varying loadings. However, as our model considers saionary variaions around a consan mean, and Sock and Wason 2009) consider a one-ime break a a fixed poin in ime, i is possible for he likelihood raio saisics o rejec he null of consan loadings, even when he Chow ess of Sock and Wason 2009) canno rejec he null of no breaks. Given he large number of series for which we rejec he null of consan loadings, we can obain a beer in-sample fi of he common componen by leing he loadings vary over he ime dimension. Figure 1 displays he esimaed loadings pahs for he one-year/hree-monh Treasury erm spread. This series exhibis very srong evidence of srucural insabiliy in he loadings. The p-values of he likelihood raio saisics for join and single resricions are all zero o a las 4 decimal places. The loadings show a considerable amoun of variaion over he sample period wih quie persisen dynamics. Figure 2 displays he sandardized Treasury erm spread and wo esimaes of is common componen, compued wih ime-varying loadings and consan loadings, respecively. The common componen based on ime-varying loadings racks he daa much closer han he one wih consan loadings. The correlaions of he wo esimaes of he common componen wih he daa are 0.56 and 0.19, respecively. This clearly illusraes he improvemen in he in-sample fi by modelling he loadings as auoregressive processes. 6 Conclusion We proposed a wo-sep maximum likelihood esimaor for ime-varying loadings in high-dimensional facor models. The loadings parameers are esimaed by a se of N univariae regression models wih ime-varying coefficiens, where he unobserved regressors are esimaed by principal componens. Replacing he unobservable facors wih principal componens gives a feasible likelihood funcion ha is asympoically equivalen o he infeasible one wih unobservable facors and herefore gives consisen esimaes of he loadings parameers as N, T. The finie-sample properies of our esimaor were assessed via an exensive simulaion sudy. The resuls showed ha he loadings means and idiosyncraic error variances are esimaed precisely even for small sample sizes. A somewha larger sample size is needed o ge precise esimaes of he loadings variance and dynamic parameers. Furhermore, he simulaions showed very saisfacory resuls when he number of principal componens is differen from he number of facors in he daa. We illusraed he empirical relevance of he ime-varying loadings srucure using he large quarerly daase of Sock and Wason 2009) for he US economy. For he majoriy of he variables we found evidence of ime-varying loadings, and we showed ha a large increase in he in-sample fi of he common componen can be obained by modelling he loadings as ime-varying. 24

26 Appendix Le X = X 1,..., X T ) be he T N marix of observaions, and le V NT be he r r diagonal marix of he r larges eigenvalues of NT ) 1 XX in decreasing order. By he definiion of eigenvalues and eigenvecors, we have NT ) 1 XX F = F VNT or NT ) 1 XX F V 1 NT = F, where F F /T = Ir. Le H = Λ 0 Λ 0 /N)F F /T )V 1 NT be he r r roaion marix. Assumpion A and B ogeher wih Lemma A.1 below implies ha H = O p 1). Le w = ξ F. We can wrie 1) as: X = Λ 0 F + ξ F + e = Λ 0 F + w + e. Define e = e 1,..., e T ) and w = w 1,..., w T ). We use he following expression from Baes e al. 2013): XX = F Λ 0 Λ 0 F + F Λ 0 e + w) + e + w)λ 0 F + e + w)e + w). 7) Le v denoe a conforming uni vecor wih zeros in all enries excep he h. We hen have: XX v = F Λ 0 Λ 0 F + F Λ 0 e + w ) + e + w)λ 0 F + e + w)e + w ). Using he definiion of F and H, we can hen wrie: F H F = V 1 NT NT ) 1 F XX v V 1 NT F F/T )Λ 0 Λ 0 /N)F = V 1 NT NT ) 1 { F F Λ 0 e + F eλ 0 F + F ee + F F Λ 0 w + F wλ 0 F + F ww + F ew + F we }. Denoe each erm on he righ-hand as A 1,..., A 8, respecively. We ge: F H F = V 1 NT 8 A n. 8) The following is a generalizaion of Lemma A.3 in Bai 2003). They consider consan loadings; we generalize he proof o auoregressive loadings. n=1 i) ii) Lemma A.1. Under Assumpions A-E, as N,T : VNT F F Λ 0 Λ 0 T N F F T Λ 0 Λ 0 N F F T p V, F F 2 T = O p C 2 NT ), where V is he diagonal marix consising of he eigenvalues of Σ Λ Σ F. 25

27 Proof. From V NT = F NT ) 1 XX F we ge using 7): V NT F F T Λ 0 Λ 0 N F F T = F NT ) 1 { F Λ 0 e + w) +e + w)λ 0 F + e + w)e + w) ) } F T = F 8 n=1 A n. Hence, 8 F A n 2 n=1 T 8r F 2 ) n=1 8 A n 2, 8 ) 2 A n n=1 where he las inequaliy uses r F F /T ) = rir ) = r and Loève s inequaliy. The righ-hand side is O p C 2 NT ) by Theorem 1 of Baes e al. 2013).11 Saemen i) follows. Saemen ii) is implicily proven by Sock and Wason 1998). I should be noed ha heir paper considers he model X = Λ 0 F + e, i.e. a facor model wih consan loadings. However, + o p 1) and he normal- heir proof only uses he asympoic represenaion V NT = F F F F T T izaion F F /T = Ir. Their proof is hus applicable for our model as well. Λ 0 Λ 0 N Proof of Lemma 1. From 8) we have: F H F 2 V 1 NT 2 8 n=1 8 A n 2. Since V NT converges o a posiive definie marix, i follows ha V 1 NT 2 = O p 1). The righ-hand side is hus O p C 2 NT ) by Theorem 1 in Baes e al. 2013). Proof of Proposiion 1. Using 8) we have: max F H F = max V 1 NT n=1 8 8 A n V 1 NT n=1 max A n. 11 Our Assumpion D.3 differs from he corresponding assumpion in Baes e al. 2013). They assume ha T N s, i,j=1 Eξisp 1ξ jsq1 ξ ip2 ξ jq2 F sp1 F sq1 F p2 F q2 ) is bounded by he envelope funcion Q 3N, T ), wih CNT 2 Q 3N, T ) = ON 2 T 2 ). This is implied by Assumpion D.3, and heir Theorem 1 hus holds wih Assumpion A-E. 26

28 Lemma 1 implies ha V 1 NT = O p1). We can wrie A 1 as: 12 A 1 = NT ) 1 The firs erm is less han: F T s H F s )F sλ 0 e + NT ) 1 H F s F sλ 0 e. F s H F s 2 ) 1/2 N 2 F sλ 0 e 2 ) 1/2. We have: N 2 F sλ 0 e 2 N 1 N 1/2 Λ 0 e 2 F s 2. By Assumpion F.3, he maximum of N 1/2 Λ 0 e i 2 over is O p T 1/4 ), and Assumpion A implies T F s 2 = O p 1). By Lemma 2, we have T F s H F s 2 = O p C 2 NT ). Taking he ) square roo hen gives ha he firs erm is O p C 1 T 1/8 NT Op. For he second erm, we have: NT ) 1 N 1/2 ) H F s F sλ 0 e N 1/2 H N 1/2 Λ 0 e F s 2, where H = O p 1) and T F s 2 = O p 1) by Assumpion A. The maximum of N 1/2 Λ 0 e over is O p T 1/8 ). The second erm is hus equal o O T 1/8 p and dominaes he firs. Consider A 2, which can be wrien as: NT ) 1 The firs erm is bounded by Now, N 2 N 1/2 ) F T s H F s )e sλ 0 F + NT ) 1 H F s e sλ 0 F. F H F s 2 ) 1/2 N 2 e sλ 0 F 2 max F 2 N 1 e sλ 0 F 2 ) 1/2 N 1/2 e sλ 0 2 = O p α 2 T )N 1 by Assumpion F.3. The firs erm is hus equal o O p C 1 NT α T N 1/2 ). The second erm is equal 12 The erms A 1, A 2, A 3 have been shown o be O pα T ) + O pt 1/8 )N 1/2 by Bai and Ng 2008a). They do, however, rely on inermediae resuls, which we have no proved for he model wih ime-varying loadings. We herefore provide an alernaive proof for A 1, A 2, A

29 o: which is bounded by: N 1/2 Mmax F H NT ) 1 i=1 H F s e is λ 0 i F, ) 1/2 F s 2 NT ) 1 i,j=1 e is e js This is equal o O p α T )N 1/2 by Assumpion C.3 and dominaes he firs erm. We can wrie A 3 as: NT ) 1 F T s H F s )[e se Ee se )] + NT ) 1 H F s [e se Ee se )] T + NT ) 1 F T s H F s )Ee se ) + NT ) 1 H F s Ee se ). The firs erm is bounded by: ) 1/2 F s H F s 2 N 1 N 1/2 [e ise i Ee ise i )] i=1 1/2. 2 1/2 By Assumpion C.5, max N 1/2 N i=1 [e is e i Ee is e i)] 2 = O p T ), so he firs erm is equal o O p C 1 NT )O p T 1/4 ). The second erm is bounded by: N 1/2 NT ) 1/2 H NT ) 1/2 i=1 F s [e ise i Ee ise i )]. By Assumpion F.2, he maximum of his expression over is O p N 1/2 ). bounded by: /2 ) 1/2 T ) 1/2 F s H F s 2 γ N s, ) 2.. The hird erm is By Assumpion F.1 and Lemma 2, his is equal o /2 O p C 1 NT ). The fourh erm is bounded by: which is O p α T ). max F H γ N s, ), 28

30 For A 4, we have: NT ) 1 F F Λ 0 w /2 F /2 F N 1 Λ 0 w. The firs wo erms are boh O p 1). The las erm can be bounded in expecaion: E Λ 0 w 2 N N N 2 i,j=1 M 2 r 2 N 2 sup p.q Ew i w j )λ 0 i λ 0 j M 2 N 2 Eξ i F ξ j F ) i,j=1 Eξ ip ξ jq F p F q ) = O p N 1 ) i,j=1 uniformly in by Assumpion D.1, so he maximum of he las erm over is O p N 1/2 ). Consider A 5 : NT ) 1 F s w sλ 0 F max F By Assumpion D.1, his is equal o O p α T N 1/2 ). For A 6, we have: NT ) 1 F s w sw N 1 /2 By Assumpion D.3, we have: Ew sw ) 2 = r 4 sup p 1,p 2,q 1,q 2 i,j Ew is w i w js w j ) i,j F s 2 ) 1/2 N 1 w sλ 0 2 ) 1/2 ) 1/2 T ) 1/2 F s 2 w sw 2. Eξ isp1 ξ jsq1 ξ ip2 ξ jq2 F sp1 F sq1 F p2 F q2 ) = ON 2 ) + ONT ) uniformly in. We herefore have ha max A 6 = N 1 /2 [O p N)+O p N 1/2 T 1/2 )] = O p C 1 The sevenh erm is bounded by: F s 2 ) 1/2 N 2 e sw 2 ) 1/2.. NT ). 29

31 The firs erm is O1). We can bound he second erm in expecaion: N 2 E e sw 2 = N 2 N 2 i,j=1 i,j=1 Mr 2 N 2 sup p,q Ee is e js )Ew i w j ) Ee 2 is) 1/2 Ee 2 js) 1/2 Ew i w j ) i,j=1 Eξ ip ξ jq F p F q ), which is O p N 1 ) uniformly in by Assumpion D.1. Taking he square roo hen gives O p N 1/2 ). Finally, A 8 is bounded by: NT ) 1 F we = F s 2 ) 1/2 N 2 w se 2 ) 1/2 The firs erm is again O1), and he las erm can be bounded in expecaion: N 2 E w se 2 = N 2 N 2 i,j=1 i,j=1 Mr 2 N 2 sup p,q Ee i e j )Ew is w js ) Ee 2 i) 1/2 Ee 2 j) 1/2 Ew is w js ) i,j=1 Eξ ips ξ jqs F ps F sq ), which is O p N 1 ) uniformly in by Assumpion D.1. The las erm is hus O p N 1/2 ). All erms are dominaed by O p T 1/8 ) + O N 1/2 p α T N 1/2 ) + O p α T ) + O p C 1/2 NT ), and Proposiion 1 follows.. Lemma A.2. Le Assumpion A-E hold. If F F/T = I r and Λ 0 Λ 0 is a diagonal marix wih disinc enries, H = I r + O p C 2 NT ). Proof. Firs we need o show ha F F H) F/T and F F H) F /T are boh Op C 2 NT ). We have: F F H) F/T 2 = F H F 2 ) F H F )F 2 F 2 ) = O p C 2 NT ), 30

32 where he las equaliy follows from Lemma 1 and Assumpion A. By similar argumens F F H) F /T = Op C 2 NT ). The res of he proof is idenical o he proof of equaion 2) in Bai and Ng 2013). Lemma A.2 shows ha if he imposed normalizaion holds for he process generaing he daa, he facors can be esimaed wihou roaion. This implies ha θ i can be esimaed wihou roaion as well. In he proof of Theorem 1 below, we assume ha H = I r, and noe ha in general, he feasible likelihood converges o L T X i F H; θ i ), and θ i is consisen for a roaion of θi 0 as discussed in Secion 3.2. Proof of Theorem 1. I suffices o show ha he feasible likelihood funcion L T X i F ; θ i ) converges uniformly o he infeasible one L T X i F ; θ i ). 13 This will imply ha L T X i F ; θ i ) saisfies he condiions of Assumpion H and θ p i θ 0 i. We hus need: sup L T X i F ; θ i ) L T X i F ; θ i ) p 0. θ Θ By he mean value expansion, we can wrie: L T X i F ; θ i ) = L T X i F ; θ i ) + F L T X i F ; θ i ) F F ), where F L T X i F ; θ i ) = L T X i F ;θ i ) F F, and F is beween F and F. For uniform convergence =F he las erm needs o be o p 1) uniformly in Θ, when F is in a neighbourhood of F, such ha max F F = o p 1). Le λ max A) and λ min A) denoe he larges and smalles eigenvalue of a marix A, and le A) s,) denoe enry s, ) of a T T marix A. Furhermore, le φ i be he r r block marix on he diagonal of Φ i, i.e. φ i = V arλ i ). The derivaive of L T X i F ; θ i ) akes he form: 14,15 F L T X i F ; θ i ) = φ i F Σ 1 i,,) + λ i + φ i F Σ 1 i X is F sλ i )Σ 1 i,s,) X i EX i ))X i EX i )) Σ 1 ) where F is o be evaluaed a F. Denoe he hree erms above by B n, for n = 1,..., 3. We can 13 Poinwise convergence would suffice as Assumpion Hiii) requires convergence for all θ i Θ i. However, here are no addiional difficulies in showing uniform convergence, and we herefore prove convergence uniformly in Θ i. 14 The calculaions of he derivaive are omied for breviy. They are available upon reques. 15 Wih auocorrelaed errors, he derivaive akes he same form, bu he variance marix is Σ i = F Φ if + Ψ i, where Ψ i = Ee ie i) is non-diagonal. i,, 31

33 hen wrie: sup L T X i F 3 ; θ i ) L T X i F ; θ i ) = sup θ Θ θ Θ F F ) B n n=1 sup F F ) B 1 + sup F F ) B 2 + sup F F ) B 3. θ Θ θ Θ θ Θ 9) For he erm involving B 1, we have: F F ) φ i F Σ 1 i,,) λ maxσ 1 ) F F φ i F, 10) since each enry in Σ 1 i is bounded by he larges eigenvalue. For he larges eigenvalue of Σi 1, we have λ max Σ 1 i ) = [λ min Σ i )] 1, and i herefore follows from he Weyl inequaliy ha λ max Σ 1 i ) M as: 16 λ minσ i ) λ minfφ i F ) + λ minψ i I T ) ψ i > 0 uniformly in Θ i. The erm φ i is also uniformly bounded, as he parameers of B i L) are in he saionary region, and he elemens of Q i are bounded. We can herefore bound 10) by: O1) F F F F + O1) F F F. Since F is beween F and F, he firs erm is less han F F 2 and is O p C 2 NT ) by Lemma 1. Noe ha F F 2 does no depend on θ i, and he resul is hus uniform in Θ i. For he second erm, we can wrie: F F F F F 2 ) 1/2 which is O p C 1 NT ) by Lemma 1 and Assumpion A, also uniformly in Θ i. For he erm involving B 3 in 9), we can wrie: max F F ) φ i F Σ 1 i F F ) φ i F T X i EX i ))X i EX i )) Σ 1 i Σ 1 i F 2 ) 1/2, ), X i EX i ))X i EX i )) Σ 1 ) i 16 This also holds wih Ψ i = Ee ie i) non-diagonal, as we can bound he smalles eigenvalue of Σ i uniformly i Θ i.,. 32

34 For he erm ouside he sum, we have: max F F ) φ i F φ i max F F F O1)max F F 2 + O1)max F F F. If we ake F o be a sequence of fixed and bounded consans, max F M, and he second erm is hen o p 1) by Proposiion 1, which is uniform in Θ i as he proof of Proposiion 1 does no depend on θ i. The firs erm is bounded by he second. The erm involving he sum can be wrien as which is bounded by Σ 1 i = r Σ 1 i X i EX i ))X i EX i )) Σ 1 i ), X i EX i ))X i EX i )) Σ 1 ), λ max Σ 2 ) rx i EX i ))X i EX i )) M 2 4M 2 i F λ 0 i 2 + F λ i λ 0 i ) 2 + e i 2 + F λ i 2). X i F λ i 2 The firs erm in he sum is bounded by M 2 T F 2 = O p 1). For he second erm in he sum, we can wrie: F λ i, λ 0 i ) 2 F 4 ) 1/2 λ i, λ 0 i 4 ) 1/2 This is O p 1) by Assumpion A and G. By Assumpion C we have T e2 i = O p1), and for he las erm, we can wrie: F λ i 2 M 2 F F 2 + M 2 F 2 = O p C 2 NT ) + O p1),. 11) as λ i is esimaed in a bounded parameer space. F O p 1) = o p 1) uniformly in Θ i. The second erm in 9) is hus max F 33

35 For he erm involving B 2 in 9), we can wrie: F T F ) λ i X is Fs λ i )Σ 1 ) 1/2 F F ) λ i 2 i,s,) X is Fs λ i )Σ 1 s, 2 1/2 The firs erm in parenheses is less han M 2 T F F 2 = O p C 2 NT ) uniformly in Θ i. The second erm in parenheses is equal o r Σ 1 i X i EX i ))X i EX i )) Σ 1 ), which is O p 1) uniformly in Θ i from he argumens above, see 11). By aking he square roo, he second erm is hus O p C 1 NT ) and dominaed by he hird. Collecing he resuls gives: sup L T X i F ; θ i ) L T X i F ; θ i ) = O p max F ) F = o p 1). θ Θ i. References Bai, J. 2003): Inferenial heory for facor models of large dimensions, Economerica, 71, Bai, J. and K. Li 2012a): Maximum likelihood esimaion and inference for approximae facor models of high dimension, forhcoming, Review of Economics and Saisics. 2012b): Saisical analysis of facor models of high dimension, The Annals of Saisics, 40, Bai, J. and S. Ng 2002): Deermining he number of facors in approximae facor models, Economerica, 70, ): Confidence inervals for diffusion index forecass and inference for facoraugmened regressions, Economerica, 74, a): Exremum esimaion when he predicors are esimaed from large panels, Annals of Economics and Finance, 9, b): Large dimensional facor analysis, Foundaions and Trends in Economerics, 3,

36 2013): Principal componens esimaion and idenificaion of saic facors, Journal of Economerics, 176, Baes, B., M. Plagborg-Møller, J. Sock, and M. Wason 2013): Consisen facor esimaion in dynamic facor models wih srucural insabiliy, Journal of Economerics, 177, Breiung, J. and S. Eickmeier 2011): Tesing for srucural breaks in dynamic facor models, Journal of Economerics, 163, Del Negro, M. and C. Orok 2008): Dynamic facor models wih ime-varying parameers: measuring changes in inernaional business cycles, Saff repor 326, Federal Reserve Bank of New York, New York. Doz, C., D. Giannone, and L. Reichlin 2012): A quasi maximum likelihood approach for large, approximae dynamic facor models, Review of Economics and Saisics, 94, Eickmeier, S., W. Lemke, and M. Marcellino 2015): Classical ime varying facoraugmened vecor auo-regressive models esimaion, forecasing and srucural analysis, Journal of he Royal Saisical Sociey: Series A Saisics in Sociey), 178, Forni, M., M. Hallin, M. Lippi, and L. Reichlin 2000): The generalized dynamic-facor model: idenificaion and esimaion, Review of Economics and Saisics, 82, Liu, P., H. Mumaz, and A. Theophilopoulou 2011): Inernaional ransmission of shocks: A ime-varying facor-augmened VAR approach o he open economy, Working paper 425, Bank of England, London. Newey, W. K. 1984): A mehod of momens inerpreaion of sequenial esimaors, Economics Leers, 14, Newey, W. K. and D. McFadden 1994): Large sample esimaion and hypohesis esing, Handbook of Economerics, 4, Pagan, A. 1980): Some idenificaion and esimaion resuls for regression models wih sochasically varying coefficiens, Journal of Economerics, 13, ): Two sage and relaed esimaors and heir applicaions, The Review of Economic Sudies, 53, Sock, J. H. and M. Wason 2009): Forecasing in dynamic facor models subjec o srucural insabiliy, in Hendry, D.F., Casle, J., Shephard, N. Eds.), The Mehodology and Pracice of Economerics. A Fesschrif in Honour of David F. Hendry, Oxford Universiy Press, Oxford,

37 Sock, J. H. and M. W. Wason 1998): Diffusion indexes, Tech. rep., Naional Bureau of Economic Research. 2002): Forecasing using principal componens from a large number of predicors, Journal of he American Saisical Associaion, 97,

38 Research Papers : Jonas Nygaard Eriksen: Expeced Business Condiions and Bond Risk Premia : Kim Chrisensen, Mark Podolskij, Nopporn Thamrongra and Bezirgen Veliyev: Inference from high-frequency daa: A subsampling approach : Asger Lunde, Anne Floor Brix and Wei Wei: A Generalized Schwarz Model for Energy Spo Prices - Esimaion using a Paricle MCMC Mehod : Annasiina Silvennoinen and Timo Teräsvira: Tesing consancy of uncondiional variance in volailiy models by misspecificaion and specificaion ess : Harri Pönkä: The Role of Credi in Predicing US Recessions : Palle Sørensen: Credi policies before and during he financial crisis : Shin Kanaya: Uniform Convergence Raes of Kernel-Based Nonparameric Esimaors for Coninuous Time Diffusion Processes: A Damping Funcion Approach : Tommaso Proiei: Exponenial Smoohing, Long Memory and Volailiy Predicion : Mark Podolskij, Chrisian Schmid and Mahias Veer: On U- and V-saisics for disconinuous Iô semimaringale : Mark Podolskij and Nopporn Thamrongra: A weak limi heorem for numerical approximaion of Brownian semi-saionary processes : Peer Chrisoffersen, Mahieu Fournier, Kris Jacobs and Mehdi Karoui: Opion-Based Esimaion of he Price of Co-Skewness and Co-Kurosis Risk Kadir G. Babaglou, Peer Chrisoffersen, Seven L. Heson and Kris Jacobs: Opion Valuaion wih Volailiy Componens, Fa Tails, and Nonlinear Pricing Kernels : Andreas Basse-O'Connor, Raphaël Lachièze-Rey and Mark Podolskij: Limi heorems for saionary incremens Lévy driven moving averages : Andreas Basse-O'Connor and Mark Podolskij: On criical cases in limi heory for saionary incremens Lévy driven moving averages : Yunus Emre Ergemen, Niels Haldrup and Carlos Vladimir Rodríguez-Caballero: Common long-range dependence in a panel of hourly Nord Pool elecriciy prices and loads : Niels Haldrup and J. Eduardo Vera-Valdés: Long Memory, Fracional Inegraion, and Cross-Secional Aggregaion : Mark Podolskij, Bezirgen Veliyev and Nakahiro Yoshida: Edgeworh expansion for he pre-averaging esimaor : Jakob Guldbæk Mikkelsen, Eric Hillebrand and Giovanni Urga: Maximum Likelihood Esimaion of Time-Varying Loadings in High-Dimensional Facor Models