dynamic factors in the presence of blocks

Journal of Econometrics 163 (2011) 29–41

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Journal of Econometrics

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Dynamic factors in the presence of blocksMarc Hallin a,b,c,d,e,∗, Roman Liška a,1

a ECARES (European Centre for Advanced Research in Economics and Statistics), Université libre de Bruxelles, CP 114, B-1050 Bruxelles, Belgiumb ORFE, Princeton University, United Statesc CentER, Tilburg University, Netherlandsd ECORE, Bruxelles and Louvain-la-Neuve, Belgiume Académie Royale de Belgique

a r t i c l e i n f o

Article history:Available online 11 November 2010

JEL classification:C13C33C43

Keywords:Panel dataHigh dimensional time series dataDynamic factor modelDynamic principal components

a b s t r a c t

Macroeconometric data often comeunder the formof large panels of time series, themselves decomposinginto smaller but still quite large subpanels or blocks. We show how the dynamic factor analysis methodproposed in Forni et al. (2000), combinedwith the identificationmethod of Hallin and Liška (2007), allowsfor identifying and estimating joint and block-specific common factors. This leads to amore sophisticatedanalysis of the structures of dynamic interrelations within and between the blocks in such datasets, alongwith an informative decomposition of explained variances. The method is illustrated with an analysis ofa dataset of Industrial Production Indices for France, Germany, and Italy.

© 2011 Published by Elsevier B.V.

1. Introduction

1.1. Panel data and dynamic factor models

In many fields – macroeconometrics, finance, environmentalsciences, chemometrics, . . . – information comes under the form ofa large number of observed time series or panel data. Panel dataconsist of series of observations (length T ) made on n individualsor ‘‘cross-sectional items’’ that have been put together on purpose,because,mainly, they carry some information about some commonfeature or unobservable process of interest, or are expected todo so. This ‘‘commonness’’ is a distinctive feature of panel data:mutually independent cross-sectional items, in that respect, donot constitute a panel (or then, a degenerate one). Cross-sectionalheterogeneity is another distinctive feature of panels: n (possiblynon-independent) replications of the same time series would beanother form of degeneracy. Moreover, the impact of item-specificor idiosyncratic effects, which have the role of a nuisance, very

∗ Corresponding author at: ECARES (European Centre for Advanced Research inEconomics and Statistics), Université libre de Bruxelles, CP 114, B-1050 Bruxelles,Belgium.

E-mail address:[email protected] (M. Hallin).1 Present address: Institute for Health and Consumer Protection (IHCP), European

Commission Joint Research Centre, I-21027 Ispra (VA), Italy.

0304-4076/$ – see front matter© 2011 Published by Elsevier B.V.doi:10.1016/j.jeconom.2010.11.004

often dominate, quantitatively, that of the common features oneis interested in.

Finally, all individuals in a panel are exposed to the influence ofunobservable or unrecorded covariates, which create complex in-terdependencies, both in the cross-sectional as in the time dimen-sion,which cannot bemodelled, as thiswould require questionablemodelling assumptions and a prohibitive number of nuisance pa-rameters. These interdependencies may affect all (or almost all)items in the panel, in which case they are ‘‘common’’; they alsomay be specific to a small number of items, hence ‘‘idiosyncratic’’.

The idea of separating ‘‘common’’ and ‘‘idiosyncratic’’ effectsis thus at the core of panel data analysis. The same idea is thecornerstone of factor analysis. There is little surprise, thus, to seea time series version of factor analysis emerging as a powerfultool in the context of panel data. This dynamic version of factormodels, however, requires adequate definitions of ‘‘common’’ and‘‘idiosyncratic’’. This definition should not simply allow foridentifying the decomposition of the observation into a ‘‘common’’component and an ‘‘idiosyncratic’’ one, but also should provide anadequate translation of the intuitive meanings of ‘‘common’’ and‘‘idiosyncratic’’.

Denote by Xit the observation of item i (i = 1, . . . , n) at timet (t = 1, . . . , T ); the factor model decomposition of this observa-tion takes the form

Xit = χit + ξit , i = 1, . . . , n, t = 1, . . . , T ,

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30 M. Hallin, R. Liška / Journal of Econometrics 163 (2011) 29–41

where the common component χit and an the idiosyncratic one ξitare mutually orthogonal (at all leads and lags) but unobservable.Some authors identify this decomposition by requiring the idiosyn-cratic components to be ‘‘small’’ or ‘‘negligible’’, as in dimensionreduction techniques. Some others require that the n idiosyncraticprocesses be mutually orthogonal white noises. Such characteri-zations do not reflect the fundamental nature of factor models:idiosyncratic components indeed can be ‘‘large’’ and strongly au-tocorrelated, while white noise can be common. For instance, ina model of the form Xit = χt + ξit , where χt is white noise andorthogonal to ξit = εit + aiεi,t−1, with i.i.d. εit ’s, the white noisecomponent χt , which is present in all cross-sectional items, verymuch qualifies as being ‘‘common’’, while the cross-sectionally in-dependent autocorrelated ξit ’s, being item-specific, exhibit all theattributes one would like to see in an ‘‘idiosyncratic’’ component.

A possible characterization of commonness/idiosyncrasy is ob-tained by requiring the common component to account for allcross-sectional correlations, leading to possibly autocorrelated butcross-sectionally orthogonal idiosyncratic components. This yieldsthe so-called ‘‘exact factor models’’ considered, for instance, bySargent and Sims (1977) and Geweke (1977). These exact mod-els, however, are too restrictive in most real life applications,where it often happens that two (or a small number of) cross-sectional items, being neighbours in some broad sense, exhibitcross-sectional correlation also in variables that are orthogonal, atall leads and lags, to all other observations throughout the panel. A‘‘weak’’ or ‘‘approximate factor model’’, allowing for mildly cross-sectionally correlated idiosyncratic components, therefore alsohas been proposed (Chamberlain, 1983; Chamberlain and Roth-schild, 1983), in which, however, the common and idiosyncraticcomponents are only asymptotically (as n → ∞) identified. Un-der its most general form, the characterization of idiosyncrasy, inthis weak factor model, can be based on the behavior, as n → ∞,of the eigenvalues of the spectral densitymatrices of the unobserv-able idiosyncratic components, but also (Forni and Lippi, 2001) onthe asymptotic behavior of the eigenvalues of the spectral densitymatrices of the observations themselves: see Section 2 for details.This general characterization is the one we are adopting here.

Finally, once the common and idiosyncratic components areidentified, two types of factormodels can be found in the literature,depending on theway factors are driving the common components.In static factor models, it is assumed that common components areof the form

χit =

q−l=1

bilflt , i = 1, . . . , n, t = 1, . . . , T , (1)

that is, the χit ’s are driven by q factors f1t , . . . , fqt which areloaded instantaneously. This static approach is the one adoptedby Chamberlain (1983), Chamberlain and Rothschild (1983), Stockand Watson (1989, 2002a,b, 2005), Bai and Ng (2002, 2007), anda large number of applied studies. The so-called general dynamicmodel decomposes common components into

χit =

q−l=1

bil(L)ult , i = 1, . . . , n, t = 1, . . . , T , (2)

where u1t , . . . , uqt , the unobservable common shocks, are loadedvia one-sided linear filters bil(L). That ‘‘fully dynamic’’ approach(the terminology is not unified and the adjective ‘‘dynamic’’ is oftenused in an ambiguous way) goes back, under exact factor form, toChamberlain (1983) and Chamberlain and Rothschild (1983), butwas developed, mainly, by Forni et al. (2000, 2003, 2004, 2005,2009); Forni and Lippi (2001) and Hallin and Liška (2007).

The static model (1) clearly is a particular case of the generaldynamic one (2). Its main advantage is simplicity. On the otherhand, both models share the same assumption on the asympotic

behavior of spectral eigenvalues—the plausibility of which isconfirmed by empirical evidence. But the static model (1) placesan additional and rather severe restriction on the data-generatingprocess, while the dynamic one (2), as shown by Forni and Lippi(2001), does not—we refer to Section 2 for details. Moreover,the synchronization of clocks and calendars across the panel isoften quite approximative, so that the concept of ‘‘instantaneousloading’’ itself may be questionable.

Both the static and the general dynamic models are receivingincreasing attention in finance andmacroeconometric applicationswhere information usually is scattered through a (very) largenumber n of interrelated time series (n values of the order ofseveral hundreds, or even one thousand, are not uncommon).Classical multivariate time series techniques are totally helplessin the presence of such values of n, and factor model methods, tothe best of our knowledge, are the only ones that can handle suchdatasets. In macroeconomics, factor models are used in businesscycle analysis (Forni and Reichlin, 1998; Giannone et al., 2006),in the identification of economy-wide and global shocks, in theconstruction of indices and forecasts exploiting the informationscattered in a huge number of interrelated series (Altissimo et al.,2001), in themonitoring of economic policy (Giannone et al., 2005),and in monetary policy applications (Bernanke and Boivin, 2003;Favero et al., 2005). In finance, factor models are at the heart ofthe extensions proposed by Chamberlain and Rothschild (1983)and Ingersol (1984) of the classical arbitrage pricing theory; theyalso have been considered in performance evaluation and riskmeasurement (Chapters 5 and 6 of Campbell et al., 1997), and inthe statistical analysis of the structure of stock returns (Yao, 2008).

Factormodels in the recent years also generated a huge amountof applied work: see d’Agostino and Giannone (2005), Artis et al.(2005), Bruneau et al. (2007), Den Reijer (2005), Dreger andSchumacher (2004), Schumacher (2007), Nieuwenhuyzen (2004),Schneider and Spitzner (2004), Giannone and Matheson (2007),and Stock and Watson (2002b) for applications to data from UK,France, the Netherlands, Germany, Belgium, Austria, New Zealand,and the US, respectively; Altissimo et al. (2001), Angelini et al.(2001), Forni et al. (2003), and Marcellino et al. (2003) for the Euroarea and Aiolfi et al. (2006) for South American data—to quote onlya few. Dynamic factor models also have entered the practice of anumber of economic and financial institutions, including severalcentral banks and national statistical offices, who are using themin their current analysis and prediction of economic activity. Areal time coincident indicator of the EURO area business cycle(EuroCOIN), based on Forni et al. (2000), is published monthlyby the London-based Center for Economic Policy Research andthe Banca d’Italia: see (http://www.cepr.org/data/EuroCOIN/). Asimilar index, based on the same methods, is established for theUS economy by the Federal Reserve Bank of Chicago.

1.2. Dynamic factor models in the presence of blocks: outline of thepaper

Although heterogeneous, panel data very often are obtainedby pooling together several ‘‘blocks’’ which themselves constitute‘‘large’’ subpanels. In macroeconometrics, for instance, datatypically are organized either by country or sectoral origin: thedatabase which is used in the construction of EuroCOIN, themonthly indicator of the euro area business cycle published byCEPR, includes almost 1000 time series that cover six Europeancountries and are organized into eleven subpanels includingindustrial production, producer prices, monetary aggregates, etc.Depending on the objectives of the analysis, such a panel could bedivided into six blocks (one for each country), or into eleven blocks(one for each subpanel). When these blocks are large enough,several dynamic factor models can be considered and analyzed,

http://www.cepr.org/data/EuroCOIN/

M. Hallin, R. Liška / Journal of Econometrics 163 (2011) 29–41 31

allowing for a refined analysis of interblock relations. In the simpletwo-block case, ‘‘marginal common factors’’ can be defined foreach block, and need not coincide with the ‘‘joint common factors’’resulting from pooling the two blocks.

The objective of this paper is to provide a theoretical basisfor that type of analysis. For simplicity, we start with the simplecase of two blocks. We show (Section 2) how a factorization ofthe Hilbert space spanned by the n observed series leads to adecomposition of each of them into fourmutually orthogonal (at allleads and lags) components: a strongly idiosyncratic one, a stronglycommon one, a weakly common, and a weakly idiosyncratic one.In Sections 3 and 4, we show how projections onto appropriatesubspaces provide consistent data-driven reconstructions of thosevarious components. Section 5 is devoted to the general case ofK ≥

2 blocks, allowing for various decompositions of each observationinto mutually orthogonal (at all leads and lags) components. Thetools we are using throughout are Brillinger’s theory of dynamicprincipal components, a key result (Proposition 2) by Forni et al.(2000), and the identification method developed by Hallin andLiška (2007). Proofs are concentrated in Appendix.

The potential of the method is briefly illustrated, in Section 6,with a panel of Industrial Production Index data for France andGer-many (K = 2blocks, q = 3 factors), then France, Germany and Italy(K = 3 blocks, q = 4 factors). Simple as it is, the analysis of thatdataset reveals some striking facts. For instance, both Germanyand Italy exhibit a ‘‘national common factor’’ which is idiosyncraticto the other two countries, while France’s common factors areincluded in the space spanned by Germany’s. The (estimated) per-centages of explained variation associatedwith the various compo-nents also are quite illuminating: Germany, with 25% of commonvariation, is the ‘‘most common’’ out of the three countries. But italso is, with only 6.4% of its total variation, the ‘‘least strongly com-mon’’ one. France has the highest proportion (82.4%) of marginalidiosyncratic variation but also the highest proportions of stronglyand weakly idiosyncratic variations (72.6% and 9.8%, respectively).We do not attempt here to provide an economic interpretation forsuch facts. Nor do we apply the method to a more sophisticateddataset.2 But we feel that the simple application we are proposingprovides sufficient evidence of the potential power of the method,both from a structural as from a quantitative point of view.3

2. The dynamic factor model in the presence of blocks

We throughout assume that all stochastic variables in thispaper belong to the Hilbert space L2(Ω,F , P), where (Ω,F , P) issome given probability space. We will study two double-indexedsequences Y := Yit; i ∈ N, t ∈ Z and Z := Zjt; j ∈ N, t ∈

Z, where t stands for time and i, j are cross-sectional indices, ofobserved random variables. Let Yny := Yny,t; t ∈ Z and Znz :=

Znz ,t; t ∈ Z be the ny- and nz-dimensional subprocesses of Y andZ, respectively, where Yny,t :=

Y1t , . . . , Ynyt

′ and Znz ,t :=Z1t , . . . , Znz t

′, and write

Xn,t := (Y1t , . . . , Ynyt , Z1t , . . . , Znz t)′:= (Y′

ny,tZ′

nz ,t)′

with n := (ny, nz) and n := ny+nz . The Hilbert subspaces spannedby the processes Y, Z and X are denoted by Hy,Hz and H ,respectively.

2 Another application, of an entirely different nature and scope, is considered inHallin et al. (2011).3 A similar problem is considered in two recent working papers (Ng et al., 2008;

Ng and Moench, in press), where the authors adopt a hierarchical model with‘‘national shocks’’ that are pervasive throughout all blocks, while ‘‘regional shocks’’are region-specific; their approach, however, resorts to the static factor modelliterature.

The following assumption is made throughout the paper.

Assumption A1. For all n, the vector process Xn,t; t ∈ Z is azero-mean second-order stationary process.

Denoting by 6y;ny(θ) and 6z;nz (θ) the (ny × ny) and (nz × nz)spectral density matrices of Yny,t and Znz ,t , and by 6yz;n(θ) =

6′

zy;n(θ) their (ny × nz) cross-spectrum matrix, write

6n(θ) =:

6y;ny(θ) 6yz;n(θ)

6zy;n(θ) 6z;nz (θ)

, θ ∈ [−π, π]

for the (n × n) spectral density matrix of Xn,t , with elementsσi1i2(θ), σj1j2(θ) or σkk(θ), k = 1, . . . , n, i1, i2 = 1, . . . , ny,j1, j2 = 1, . . . , nz . On these matrices, we make the followingassumption.

Assumption A2. For any k ∈ N, there exists a real ck > 0 such thatσkk(θ) ≤ ck for any θ ∈ [−π, π].

For any θ ∈ [−π, π], let λy;ny,i(θ) be 6y;ny(θ)’s i-th eigenvalue(in decreasing order of magnitude). The function θ → λy;ny,i(θ)is called 6y;ny(θ)’s i-th dynamic eigenvalue. The notation θ →

λz;nz ,j(θ) and θ → λn,k(θ) is used in an obvious way for thedynamic eigenvalues of 6z;nz (θ) and 6n(θ), respectively. Thecorresponding dynamic eigenvectors, of dimensions (ny × 1), (nz ×

1), and (n × 1), are denoted by py;ny,i(θ), pz;nz ,j(θ), and pn,k(θ),respectively.

Throughout, we repeatedly use the classical correspondence

M(L) :=12π

∞−s=−∞

[∫ π

−π

M(θ)eisθ dθ]Ls

between a matrix-values function M(θ) and the filter M(L): forinstance,

λ−11;k(L) :=

12π

∞−s=−∞

[∫ π

−π

λ−11;k(θ)e

isθdθ]Ls,

6−1/2(L) :=12π

∞−s=−∞

[∫ π

−π

6−1/2(θ)eisθ dθ]Ls, etc.

Dynamic eigenvectors, in particular, can be expanded in Fourierseries, e.g.

pn,k(θ) =12π

∞−s=−∞

[∫ π

−π

pn,k(ω)eisωdω]e−isθ

where the series on the right-hand side converge in quadraticmean, which in turn defines square summable filters of the form

pn,k(L) =

12π

∞−s=−∞

[∫ π

−π

pn,k(ω)eisωdω]Ls.

Similarly definepy;ny,i

(L) andpz;nz ,j

(L) frompy;ny,i(θ) andpz;nz ,j(θ),respectively.

On these dynamic eigenvalues, wemake the following assump-tions.

Assumption A3. For some qy, qz ∈ N,

(i) the qy-th dynamic eigenvalue of 6y;ny(θ), λy;ny,qy(θ), divergesas ny → ∞, a.e. in [−π, π], while the (qy+1)-th one,λy;ny,qy+1(θ), is θ-a.e. bounded;

(ii) the qz-th dynamic eigenvalue of 6z;nz (θ), λz;nz ,qz (θ), divergesas nz → ∞, a.e. in [−π, π], while the (qz+1)-th one,λz;nz ,qz+1(θ), is θ-a.e. bounded.


The following lemma shows that this behavior of the dynamiceigenvalues of the subpanel spectralmatrices6y;ny(θ) and6z;nz (θ)entails a similar behavior for the dynamic eigenvalues λn,k(θ) of6n(θ).

Lemma 1. Let Assumptions A1–A3 hold. Then, there exists q ∈ N,with max(qy, qz) ≤ q ≤ qy + qz , such that 6n(θ)’s q-th dynamiceigenvalue λn,q(θ) diverges as min(ny, nz) → ∞, a.e. in [−π, π],while the (q + 1)-th one, λn,q+1(θ), is θ-a.e. bounded.

Proof. See the Appendix.

Theorem 2 in Forni and Lippi (2001) establishes that thebehavior of dynamic eigenvalues described in Assumption A3and Lemma 1 characterizes the existence of a dynamic factorrepresentation. We say that a process X := Xkt , k ∈ N, t ∈

Z admits a dynamic factor representation with q factors if Xktdecomposes into a sum

Xkt = χkt + ξkt , with χkt :=

q−l=1

bkl(L)ult and

bkl(L) :=

∞−m=1

bklmLm, k ∈ N, t ∈ Z,

such that

(i) the q-dimensional vector process ut := (u1tu2t . . . uqt)′; t ∈

Z is orthonormal white noise;(ii) the (unobservable) n-dimensional processes ξn := (ξ1tξ2t · · ·

ξnt)′; t ∈ Z are zero-mean stationary for any n, with (idiosyn-

crasy) θ-a.e. bounded (as n → ∞) dynamic eigenvalues;(iii) ξk,t1 and ul,t2 are mutually orthogonal for any k, l, t1 and t2;(iv) the filters bkl(L) are square summable:

∑∞

m=1 b2klm < ∞ for all

k ∈ N and l = 1, . . . , q, and(v) q is minimal with respect to (i)–(iv).

The processes ult; t ∈ Z, l = 1, . . . , q, are called the commonshocks or factors, the random variables ξkt and χkt the idiosyncraticand common components of Xkt , respectively. Actually, Forni andLippi define idiosyncrasy via the behavior of dynamic aggregates,then show (their Theorem 1) that this definition is equivalent tothe condition on dynamic eigenvalues given here.

That result of Forni and Lippi (2001), alongwith Lemma 1, leadsto the following proposition.

Proposition 1. Let Assumptions A1 and A2 hold. Then,

(a) Assumption A3(i) is satisfied iff the process Y has dynamic factorrepresentation (qy factors; call them the (common) y-factors,spanning the y-common space H

χy , with orthogonal complement

Hξy )

Yit = χy;it + ξy;it =

qy−l=1

by;il(L)uy;lt + ξy;it , i ∈ N, t ∈ Z; (3)

(b) Assumption A3(ii) is satisfied iff the process Z has dynamic factorrepresentation (qz factors; call them the (common) z-factors,spanning the z-common space H

χz , with orthogonal complement

Hξz )

Zjt = χz;jt + ξz;jt =

qz−l=1

bz;jl(L)uz;lt + ξz;jt , j ∈ N, t ∈ Z; (4)

(c) Assumption A3 is satisfied iff the process X has dynamic factorrepresentation (q factors, with q as in Lemma 1; call them thejoint common factors, spanning the joint common spaceH

χx , with

orthogonal complement Hξx ).

Xkt =

Yit = χxy;it + ξxy;it =

q−l=1

bxy;il(L)ult + ξxy;it ,

k ∈ N, t ∈ Z in case Xkt = Yit

Zjt = χxz;jt + ξxz;jt =

q−l=1

bxz;jl(L)ult + ξxz;jt ,

k ∈ N, t ∈ Z in case Xkt = Zjt .

(5)

All filters involved have square-summable coefficients.

Proof. The proof follows directly from the characterizationtheorem of Forni and Lippi (2001).

It follows that, under Assumption A3, the processes Y and Zadmit twodistinct decompositions each: themarginal factormodels(a) and (b), with marginal common shocks uy;lt (l = 1, . . . , qy) anduz;lt (l = 1, . . . , qz), respectively, and the joint factormodel (c),withjoint common shocks ult (l = 1, . . . , q). This double representationallows for refining the factor decomposition. Call x-, y-, or z-idiosyncratic a process which is orthogonal (at all leads and lags) tothe x-, y-, or z-factors (to H

χx ,H

χy , or H

χz ), respectively. Similarly,

call x-, y-, or z-common any process belonging to Hχx ,H

χy , or H

χz .

The x-common componentsχxy;it andχxz;jt further decompose into

χxy;it = χy;it + νy;it and χxz;jt = χz;jt + νz;jt ,

where νy;it := ξy;it − ξxy;it is y-idiosyncratic, hence orthogonal toχy;it , and νz;jt := ξz;jt − ξxz;jt is z-idiosyncratic, hence orthogonalto χz;jt ; since we also have νy;it := χxy;it − χy;it where both χxy;itand χy;it are orthogonal to ξxy;it , it follows that νy;it and ξxy;it alsoare mutually orthogonal.

Define Hχy∩z := H

χy

Hχz and qy∩z := qy + qz − q: H

χy∩z is

spanned by a (qy∩z)-tuple of white noises, which are both y- andz-common (in case qy +qz = q,Hχ

y∩z = 0). Denoting by φy;it andφz;jt the projections of χy;it and χz;jt onto H

χy∩z , and by ψy;it and

ψz;jt the corresponding residuals, we obtain the decompositions

Yit =

χxy;it φy;it + ψy;it + νy;it +ξxy;it

χy;it

ξy;it

and

Zjt =

χxz;jt φz;jt + ψz;jt + νz;jt +ξxz;jt ,

χz;jt

ξz;jt

i, j ∈ N, t ∈ Z (6)

of the original observations into four mutually orthogonalcomponents. The φy;it and φz;jt components will be calledstrongly common, ξxy;it and ξxz;jt strongly idiosyncratic, ψy;it andψz;jt weakly common, νy;it and νz;jt weakly idiosyncratic. Thesedecompositions induce additive decompositions of the variancesof the observations into a sum of four terms indicating the relativecontributions of each component.

In the following sections, we propose a procedure that providesconsistent estimates of φy;it , ψy;it , νy;it , ξxy;it and φz;jt , ψz;jt , νz;jt ,

ξxz;jt , hence ξy;it , ξxy;it , ξz;jt , and ξxz;jt .

3. Identifying the factor structure; population results

Based on the n-dimensional vector processXn,t =

Y′ny,t , Z

′nz ,t

′

,we first asymptotically identify φy;it , ψy;it , νy;it , φz;jt , ψz;jt and νz;jtas min(ny, nz) → ∞. More precisely, we show that, under speci-fied spectral structure, all those quantities can be consistentlyrecovered from the finite-n subpanels Xn,t as min(ny, nz) → ∞.


3.1. Recovering the joint common and strongly idiosyncratic compo-nents

Under the joint factor model, Proposition 2 in Forni et al. (2000)provides Xn,t-measurable reconstructions – denoted by χn

xy;it andχnxz;jt , respectively – of the joint common components χxy;it andχxz;jt , which converge in quadratic mean for any i, j and t , asmin(ny, nz) → ∞; we are using the terminology ‘‘reconstruction’’rather than ‘‘estimation’’ to emphasize that spectral densities here,unlike in Section 4, are assumed to be known.

Write M∗ for the adjoint (transposed, complex conjugate) of amatrixM. The scalar process Vn,kt := p∗

n,k(L)Xn,t; t ∈ Z, the spec-

tral density of which is λn,k(θ), will be called Xn,t ’s (equivalently,6n(θ)’s) k-th dynamic principal component, k = 1, . . . , n. The ba-sic properties of dynamic principal components imply that Vn,k1tand Vn,k2t, for k1 = k2, are mutually orthogonal at all leads andlags. Forni et al. (2000) show that the projections of Yit and Zjt ontothe closed space spanned by the present, past and future valuesof Vn,kt , k = 1, . . . , q yield the desired reconstructions of χxy;itand χxz;jt . They also provide (up to minor changes due to the factthat they consider row- rather than column-eigenvectors, as we dohere) the explicit forms

χnxy;it = K∗

y;n,i(L)Xn,t and

χnxz;jt = K∗

z;n,j(L)Xn,t i = 1, . . . , ny, j = 1, . . . , nz, (7)

with

Ky;n,i(L) :=

q−k=1

p∗

n,k,i(L)p

n,k(L) and

Kz,n,j(L) =

q−k=1

p∗

n,k,j(L)p

n,k(L),

where pn,k,i(L) denotes the i-th component of p

n,k(L), i such that

Xit belongs to the y-subpanel and pn,k,j(L) the j-th component of

pn,k(L), j such that Xjt belongs to the z-subpanel. We then can state

a first consistency result.

Proposition 2. Let Assumptions A1–A3 hold. Then,

limmin(ny,nz )→∞

χnxy;it = χxy;it and lim

min(ny,nz )→∞

χnxz;jt = χxz;jt

in quadratic mean, for any i, j, and t.

Proof. The proof consists in applying Proposition 2 in Forni et al.(2000) to the joint panel.

It follows from (7) that χnxy;it has variance

Var(χnxy;it) =

q−k=1

∫ π

−π

|pn,k,i(θ)|2λn,k(θ)dθ.

Averaging this variance over the subpanel produces a measure

1ny

ny−i=1

Var(χnxy;it) =

1ny

ny−i=1

q−k=1

∫ π

−π

|pn,k,i(θ)|2λn,k(θ)dθ

of the contribution of joint common factors in the variability of they-subpanel. Dividing it by the averaged variance

1ny

ny−i=1

Var(Yit) =1ny

ny−i=1

∫ π

−π

λy;ny,i(θ)dθ

of the y-subpanel yields an evaluationny−i=1

q−k=1

∫ π

−π

|pn,k,i(θ)|2λn,k(θ)dθ/ny−i=1

∫ π

−π

λy;ny,i(θ)dθ (8)

of its ‘‘degree of commonness’’ within the joint panel. For the z-subpanel, this measure takes the formnz−j=1

q−k=1

∫ π

−π

|pn,k,j(θ)|2λn,k(θ)dθ/nz−j=1

∫ π

−π

λz;nz ,j(θ)dθ.

As for the strongly idiosyncratic components ξxy;it , and ξxz;jt ,they are consistently recovered, as min(ny, nz) → ∞, by ξnxy;it :=

Yit − χnxy;it and ξnxz;jt := Zjt − χn

xz;jt , respectively. In view of themutual orthogonality of common and idiosyncatic components,the variance of ξnxy;it writes

Var(ξnxy;it) = Var(Yit)−

q−k=1

∫ π

−π

|pn,k,i(θ)|2λn,k(θ)dθ;

the complement to one of (8) therefore constitutes ameasure of the‘‘degree of idiosyncrasy’’ of the y-subpanel within the joint panel.Similar formulas hold for the strongly idiosyncratic componentξnxz;jt .

3.2. Recovering the marginal common, marginal idiosyncratic, andweakly idiosyncratic components

If qy = q, the marginal common and idiosyncratic componentsχy;it and ξy;it coincide with their joint counterparts χxy;it and ξxy;it ,whichwere taken care of in the previous section. Assume thereforethat q > qy; the marginal and joint y-common spaces then do notcoincide anymore.

Applying to the y-subpanels the same type of technique asin Section 3.1, consistent reconstructions of the χy;it ’s could beobtained from the spectral submatrices 6y;ny(θ). Now, χy;it is alsothe common component of χxy;it , so that the same result canbe obtained from factorizing the joint common spectral densitymatrices. As a reconstruction of χy;it we therefore consider theprojection χn

y;it of χnxy;it onto the space spanned by the first qy

dynamic principal components

Vny;t :=

V ny;1t , . . . , V

ny;qyt

′

, with V ny;kt := p∗

χxy;n,k(L)χn

xy;t , (9)

of the spectral density matrix 6χxy;n(θ) of χnxy;t = (χn

xy;1t , . . . ,

χnxy;nyt)

′; pχxy;n,k(θ) here denotes the dynamic eigenvector associ-atedwith6χxy;n(θ)’s k-th dynamic eigenvalue λχxy;n,k(θ). This pro-jection takes the form

χny;it = K∗

χxy;n,i(L)χnxy;t = K∗

χxy;n,i(L)K∗

y;n,i(L)Xn,t , (10)

with Kχxy;n,i(L) :=∑qy

k=1 pχxy;n,k,i(L)pχxy;n,k(L), where pχxy;n,k,i

(L)stands for p

χxy;n,k(L)’s i-th component.

Similarly, the reconstruction χnz;jt of χz;jt is, with obvious

notation,

χnz;jt = K∗

χxz ;n,i(L)χnxz;t = K∗

χxz ;n,j(L)K∗

y;n,j(L)Xn,t .

We then have a second consistency result.

Proposition 3. Let Assumptions A1–A3 hold. Then


χny;it = χy;it and lim

min(ny,nz )→∞

χnz;jt = χz;jt


Proof. The proof again is a direct application of Proposition 2 inForni et al. (2000) to the y- and z-subpanels, respectively.

The variance of the reconstructed marginal y-common compo-nent χn

y;it is

Var(χny;it) =

qy−k=1

∫ π

−π

|pχxy;n,k,i(θ)|2λχxy;n,k(θ)dθ.


The averaged variance explained by the y-common factors in they-subpanel is thus

1ny

ny−i=1

Var(χny;it) =

1ny

ny−i=1

qy−k=1

∫ π

−π

|pχxy;n,k,i(θ)|2λχxy;n,k(θ)dθ

=1ny

qy−k=1

∫ π

−π

λχxy;n,k(θ)dθ. (11)

Similarly, the averaged variance explained by the z-commonfactors in the z-subpanel is

1nz

∫ π

−π

qz−k=1

λχxz ;n,k(θ)dθ. (12)

Consistent reconstructions of the marginal idiosyncratic com-ponents ξy;it and ξz;jt are straightforwardly obtained as ξny;it :=

Yit − χny;it and ξ

nz;jt := Zjt − χn

z;jt , whereas the weakly idiosyn-cratic components νy;it and νz;it can be recovered as νny;it := χn

xy;it −

χny;it = ξny;it − ξ

nxy;it and ν

nz;jt := χn

xz;jt −χnz;jt = ξnz;jt − ξ

nxz;jt , respec-

tively.The averaged variance of weakly idiosyncratic components

(or its ratio to∑ny

i=1 Var(Yit)) can be interpreted as measuringthe extent to which the z-common factors contribute to y-idiosyncratic variation. Clearly, since

Var(Yit) = Var(χxy;it)+ Var(ξxy;it) andVar(χxy;it) = Var(χy;it)+ Var(νy;it),

we have

Var(νy;it) = Var(χxy;it)− Var(χy;it) = Var(ξy;it)− Var(ξxy;it).

In the finite-n panel, Var(νy;it) is estimated by

Var(νny;it) = Var(ξny;it)− Var(ξnxy;it), (13)

so that the evaluation of the z-common factors contribution to y-idiosyncratic variation is

1ny

ny−i=1

Var(νny;it) =1ny

ny−i=1

q−k=1

∫ π

−π

|pn,k,i(θ)|2λn,k(θ)dθ

−

qy−k=1

∫ π

−π

λχxy;n,k(θ)dθ.

Similar formulas hold for the z-subpanel.

3.3. Disentangling the strongly and weakly common components

By definition, φy;it is obtained as the projection of χy;it ontoHχy∩z , and ψy;it follows as the residual χy;it − φy;it . Unlike H

χy

and Hχz , however, H

χy∩z is not characterized via an explicit

sequence of orthonormal bases. The methods developed in theprevious sections, thus, do not apply unless such a sequencecan be computed first. This however requires some preparation:Proposition 4 is adapted from Theorem 8.3.1 in Brillinger (1981);Proposition 5, to the best of our knowledge, is new.

Proposition 4. Assume that the (r + s)-dimensional second-ordermean-zero stationary process (ζ′

t , η′t)

′; t ∈ Z is such that the spec-

tral density matrix 6ηη(θ) of ηt is nonsingular. Then, the projectionof ζt onto the closed space Hη spanned by ηt; t ∈ Z—that is, the r-tuple A∗(L)ηt of square-summable linear combinations of the present,past and future of ηt minimizing E[(ζt −A∗(L)ηt)(ζt −A∗(L)ηt)

′] is

6ζη(L)6−1ηη (L)ηt , where6ζη(θ) denotes the cross-spectrum of ζt and

ηt .

Actually, Brillinger also requires (ζ′t , η

′t)

′ to have absolutelysummable autocovariances, so that the filter 6ζη(L)6

−1ηη (L) also is

absolutely summable. We, however, do not need this here.Next, let H1,H2 and H12 be the Hilbert spaces spanned by

V1;t; t ∈ Z, V2;t; t ∈ Z, and (V′

1;t ,V′

1;t)′; t ∈ Z, respectively,

where V1;t := (V1;1,t , . . . , V1;q1,t)′ is a q1-tuple (resp. V2;t :=

(V2;1,t , . . . , V2;q2,t)′ a q2-tuple) of mutually orthogonal (at all

leads and lags) nondegenerate stochastic processes: the dynamicdimensions ofH1 andH2 are thus q1 and q2, respectively. Denotingby

6(θ) =:

611(θ) 612(θ)621(θ) 622(θ)

, θ ∈ [−π, π],

the spectral density matrix of (V′

1;t ,V′

2;t)′, with 611(θ) = diag

(λ1;1(θ), . . . , λ1;q1(θ)) and 622(θ) = diag(λ2;1(θ), . . . , λ2;q2(θ)), assume that 6(θ) has rank q12θ-a.e., so that H12 has dynamicdimension q12, and the intersection H1∩2 := H1 ∩ H2 dynamicdimension q1∩2 = q1+q2−q12. We then have the following result.

Proposition 5. (i) The spectral density

6−1/222 (θ)621(θ)diag(λ−1

1;1(θ), . . . , λ−11;q1(θ))

× 612(θ)6−1/222 (θ), θ ∈ [−π, π] (14)

of the q2-dimensional process 6−1/222 (L)621(L)6

−111 (L)V1;t has a

maximal eigenvalue equal to one, with multiplicity q1∩2.(ii) Denoting by p1∩2;1(θ), . . . , p1∩2;q1∩2(θ) an arbitrary orthonor-

mal basis of the corresponding q1∩2-dimensional eigenspace, theprocess ϒt := (Υ1,t , . . . ,Υq1∩2,t)

′; t ∈ Z, with

Υk,t := p∗

1∩2;k(L)6−1/2

22 (L)V2;t , k = 1, . . . , q1∩2, (15)

provides an orthonormal basis for H1∩2.

Proof. A random variable Υ ∈ H2, that is, a variable of the formΥ = a∗

Υ (L)V2;t with a∗

Υ (L) := (aΥ ,1(L), . . . , aΥ ,q2(L)) belongs toH1∩2 iff it coincides with its projection onto the space H1 spannedby the V1;t ’s. In view of Proposition 4, that projection is

a∗

Υ (L)621(L)diag(λ−11;1(L), . . . , λ

−11;q1(L))V1;t . (16)

The variance of a projection being less than or equal to the varianceof the projected variable, the variance of (16) is less than or equalto the variance of Υ itself:∫ π

−π

a∗

Υ (θ)621(θ)diag(λ−11;1(θ), . . . , λ

−11;q1(θ))612(θ)aΥ (θ)dθ

=

∫ π

−π

a∗

Υ (θ)61/222 (θ)6

−1/222 (θ)621(θ)diag(λ−1

1;1(θ), . . . , λ−11;q1(θ))

× 612(θ)6−1/222 (θ)6

1/222 (θ)aΥ (θ)dθ

≤

∫ π

−π

a∗

Υ (θ)622(θ)aΥ (θ)dθ,

irrespective of 61/222 (θ)aΥ (θ). It follows that the spectral matrix

(14) has eigenvalues less than or equal to one (θ-a.e.), and thatΥ isinH1∩2 iff6

1/222 (θ)aΥ (θ)belongs to the eigenspacewith eigenvalue

one. The q1∩2 random variables Υk,t defined in (15) clearly satisfythat condition, and it is easy to check that the spectral density ofϒt; t ∈ Z moreover is the q1∩2 × q1∩2 identity matrix. The resultfollows.

The strongly common component φy;it is defined as the projec-tion of χy;it onto H

χy∩z : recovering it as the projection φn

y;it of χny;it

onto the intersection of Hχ

y;n (spanned by Vny;t ) and H

χ

z;n (spannedby Vn

z;t ) seems a natural idea. Proposition 5, with H1 = Hχ

y;n


and H2 = Hχ

z;n, hence q1 = qy, q2 = qz and q1∩2 = qy∩z ,provides an orthonormal basis for that intersection. More pre-cisely, denote by

6nV(θ) :=

6n

VyVy(θ) 6n

VyVz(θ)

6nVzVy

(θ) 6nVzVz

(θ)

the spectrum of (Vn′

y;t ,Vn′

z;t)′. The matrix (14) here takes the form

[6nVzVz

(θ)]−1/26nVzVy

(θ)[6nVyVy

(θ)]−16nVyVz

(θ)[6nVzVz

(θ)]−1/2;

denote by py∩z;n,1(θ), . . . , py∩z;n,qy∩z (θ) its qy∩z first eigenvectors.An orthonormal basis of H

χ

y;n ∩ Hχ

z;n is Vny∩z;t := (V n

y∩z;1,t , . . . ,

V ny∩z;qy∩z ,t)

′ where, in view of (15),

V ny∩z;k,t := p∗

y∩z;n,k(L)6n

VzVz

−1/2(L)Vn

z;t , k = 1, . . . , qy∩z .

Since

χny;it =

qy−k=1

pχxy;n,k,i

(L)V ny:k,t = (p

χxy;n,1,i(L), . . . , p

χxy;n,qy,i(L))Vn

y:t ,

we first compute the projection onto Hχ

y;n ∩ Hχ

z;n of Vny:t . That

projection is obtained by applying Proposition 4 to the (qy +qy∩z)-dimensional random vector (Vn′

y:t ,Vn′

y∩z;t)′, with spectral density

6nVyVy

(θ) 6nVyVy∩z

(θ)

6nVy∩zVy

(θ) 6nVy∩zVy∩z

(θ)

=

diag(λχxy;n,1(θ), . . . , λχxy;n,qy(θ)) 6n

VyVy∩z(θ)

6nVy∩zVy

(θ) Iny∩z×y∩z(θ)

.

Letting

Py∩z;n(θ) := diag(λ−1/2χxz ;n,1(θ), . . . , λ

−1/2χxz ;n,qz (θ))

×py∩z;n,1(θ), . . . , py∩z;n,qy∩z (θ)

,

Pχxy;n(θ) :=pχxy;n,1(θ), . . . , pχxy;n,qy(θ)

,

Pχxz ;n(θ) :=pχxz ;n,1(θ), . . . , pχxy;n,qz (θ)

,

Pn(y)(θ) :=pn,1(y)(θ), . . . , pn,q(y)(θ)

, and

Pn(z)(θ) :=pn,1(z)(θ), . . . , pn,q(z)(θ)

,

with pn,k(y)(θ) collecting the components pn,k,i(θ) of pn,k(θ) suchthat Xit belongs to the y-subpanel (resp. pn,k,(z)(θ) collecting thecomponents pn,k,j(θ) such that Xjt belongs to the z-subpanel), wehave

6nVyVy∩z

(θ) = P∗

χxy;n(θ)Pn(y)(θ)diag(λn,1(θ), . . . , λn,q(θ))

× P∗

n(z)(θ)Pχxz ;n(θ)Py∩z;n(θ).

The desired projection of Vny:t , in view of Proposition 4, is

6nVyVy∩z

(L)Vny∩z;t ; hence, the reconstructions we are proposing are,

for the strongly common component φy;it ,

φny;it := (p

χxy;n,1,i(L), . . . , p

χxy;n,qy,i(L))6n

VyVy∩z(L)Vn

y∩z;t

= (pχxy;n,1,i

(L), . . . , pχxy;n,qy,i

(L))P∗

χxy;n(L)Pn(y)(L)

× diag(λn,1(L), . . . , λn,q(L))

× P∗

n(z)(L)Pχxz ;n(L)Py∩z;n(L)P∗

y∩z;n(L)Vnz;t

=: H∗

y;n,i(L)Vnz;t , (17)

and, for the weakly common one ψy;it , ψny;it := χn

y;it − φny;it . With

obvious changes, we similarly define φnz;jt and ψ

nz;jt .

Parallel with Propositions 2 and 3, we then have the followingconsistency result for φn

y;it and φnz;jt (hence ψ

ny;it and ψ

nz;jt ).

Proposition 6. Let Assumptions A1–A3 hold. Then


φny;it = φy;it and lim

min(ny,nz )→∞

φnz;jt = φz;jt


Proof. The proof still follows from Proposition 2 of Forni et al.(2000), and the fact that all spectral densities involved, for givenn, are locally continuous functions of 6n(θ).

It follows from (17) that the reconstructed strongly commoncomponent φn

y,it has variance

Var(φny,it) =

∫ π

−π

(pχxy;n,1,i(θ), . . . , pχxy;n,qy,i(θ))6nVyVy∩z

(θ)

× 6nVy∩zVy

(θ)(pχxy;n,1,i(θ), . . . , pχxy;n,qy,i(θ))′dθ.

The average 1ny

∑nyi=1 Var(φ

ny;it) measures the contribution of the

strongly common factors in the total variation of the y-subpanel.Similar quantities are easily computed for the z-subpanel.

4. Recovering the factor structure; estimation results

The previous section shows how all components of Yit and Zjtcan be recovered asymptotically as min(ny, nz) → ∞, providedthat the spectral density6n and the numbers q, qy, and qz of factorsare known. The estimates φn

y;it , ψny;it and ν

ny;it all take the form of a

filtered series of the observed process Xn,t . We have indeed

φny;it = H∗

y;n,i(L)Vnz;t =: K∗

φy;n,i(L)Xn,t

ψny;it = χn

y;it − φny;it = [K∗

χxy;n,i(L)K∗

y;n,i(L)− K∗

φy;n,i(L)]Xn,t

=: K∗

ψy;n,i(L)Xn,t , and

νny;it = χnxy;it − χn

y;it = [K∗

y;n,i(L)− K∗

χxy;n,i(L)K∗

y;n,i(L)]Xn,t

=: K∗

νy;n,i(L)Xn,t ,

with

K∗

φy;n,i(L) := H∗

y;n,i(L)P∗

χxz ;n(L)Pn(z)(L)P∗

n(L)Xn,t ,

and Pn(L) := (pn,1(L), . . . , p

n,q(L)).

These three filters all are functions of the spectral densitymatrix 6n(θ) which of course in practice is unknown, as we onlyobserve a finite realization XT

n := (Xn1,Xn2, . . . ,XnT ) of Xn.We therefore need an estimator 6T

n(θ) of 6n(θ), the consistencyof which requires strengthening slightly Assumption A1 into thefollowing Assumption A1′:

Assumption A1′. For all n, the vector process Xn,t; t ∈ Z admitsa linear representationXn,t =

∑∞

k=−∞Ckζt−k, where ζt is full-rank

n-dimensional white noise with finite fourth order moments, andthe n × nmatrices Ck =

Cij,k

are such that

∑∞

k=−∞|k||Cij,k|

1/2 <∞ for all i, j.

Under Assumption A1′, if6Tn(θ), with elements σ T

n,ij(θ), denotesany periodogram-smoothing or lag-window estimator of 6n(θ),

limT→∞

P[

supθ∈[−π,π ]

σ Tn,ij(θ)− σij(θ)

> ε

]= 0

for all n, i, j, and ε > 0, (see Forni et al., 2000).4 In Section 6, weconsider the lag-window estimators

6Tn(θ) :=

MT−k=−MT

0Tnkωke−ikθ (18)

4 Actually, Forni et al. (2000)wrongly borrow the result fromBrockwell andDavis(1987); a more appropriate reference is Robinson (1991), Theorem 2.1.


where 0Tnk is the sample covariance matrix of Xn,t and Xn,t−k and

ωk := 1−|k|/(MT+1) are theweights corresponding to theBartlettlag window of sizeMT . Consistency then is achieved provided thatthe following assumption holds:

Assumption B. MT → ∞, andMTT−1→ 0, as T → ∞.

For simplicity, we consider a single MT for the y- and z-subpanels.One also could base the estimation of 6y;ny ,6z;nz and 6yz;n onthree distinct bandwidth parameter valuesMy

T ,MzT , and Mzy

T .A consistent estimator6T

n(θ) of6n(θ) however is not sufficienthere. Deriving, from this estimator 6T

n(θ), estimated versionsKTφy;n,i(L),K

Tψy;n,i(L) andKT

νy;n,i(L), of the filtersKφy;n,i(L),Kψy;n,i(L)and Kνy;n,i(L) indeed also requires an estimation of the numbersof factors q, qy and qz involved. The only method allowing forsuch estimation is the identification method developed in Hallinand Liška (2007), which we now briefly describe, with a fewadjustments taking into account the notation of this paper. Fora detailed description of the procedure, we refer to the sectionentitled ‘‘A practical guide to the selection of q’’ in Hallin and Liška(2007).

The lag windowmethod described in (18) provides estimations6T

n(θl) of the spectral density at frequencies θl := π l/(MT + 1/2)for l = −MT , . . . ,MT . Based on these estimations, consider theinformation criterion

ICTn;c(k) := log

1n

n−i=k+1

12MT + 1

MT−l=−MT

λTni(θl)

+ kcp(n, T ), 0 ≤ k ≤ qmax, c ∈ R+

0 , (19)

where the penalty function p(n, T ) is o(1) while p−1(n, T ) isomin(n,M2

T ,M−1/2T T 1/2)

as both n and T tend to infinity, and

qmax is some predetermined upper bound; the eigenvalues λTni(θl)are those of 6T

n(θl). Depending on c > 0, the estimated number offactors, for given n and T , is

qTn;c := argmin0≤k≤qmax

ICTn;c(k).

Hallin and Liška (2007) prove that this qTn;c is consistent forany c > 0. ‘‘Optimal’’ tuning c∗ of c is then performed as follows.Consider a J-tuple of the form q

Tjc,nj , j = 1, . . . , J , where nj =

(ny;j, nz;j) with 0 < ny;1 < · · · < ny;J = ny, 0 < nz;1 < · · ·

< nz;J = nz , and 0 < T1 ≤ · · · ≤ TJ = T . This J-tuplecan be interpreted as a ‘‘history’’ of the identification procedure,and characterizes, for each c > 0, a sequence q

Tjc,nj , j = 1, . . . ,

J of estimated factor numbers. In order to keep a balancedrepresentation of the two blocks, we only consider J-tuples alongwhich ny;j/nz;j is as close as possible to ny/nz .

The selection of c∗ is based on the inspection of two mappings:c → qTn;c , and c → Sc , where

S2c := J−1J−

j=1

qTjnj;c

− J−1J−

j=1

qTjnj;c

2

measures the variability of qTjnj;c

over the ‘‘history’’. For n and T largeenough, Sc exhibits ‘‘stability intervals’’, that is, intervals of c valuesover which Sc = 0. The definition of Sc implies that c → qTn;cis constant over such intervals. Starting in the neighbourhood ofc = 0, a first stability interval (0, c+

1 ) corresponds to qTn;c = qmax;choose c∗ as any point in the next one, (c−

2 , c+

2 ). The selectednumber of factors is then qTn = qTn;c∗ . The same method, appliedto the y- and z-subpanels, yields estimators qTny and qTnz of qy andqz ; qTn;yz := qTny + qTnz − qTn provides a consistent estimator of qyz .

The success of this identification method however alsorequires strengthening somewhat the assumptions; from nowon, we reinforce Assumption A1′ into Assumption A1′′ andAssumptions A2 and A3 into Assumptions A2′ and A3′:

Assumption A1′′. Same as Assumption A1′, but (i) the conver-gence condition on theCij,k’s is uniform, supi,j∈N

∑∞

k=−∞|Cij,k||k|1/2

< ∞, and (ii) writing ci1,...,iℓ(k1, . . . , kℓ−1) for the cumulant of or-der ℓ of Xi1(t + k1), . . . , Xiℓ−1(t + kℓ−1), Xiℓ(t), for all 1 ≤ ℓ ≤ 4and 1 ≤ j < ℓ, supi1,...,iℓ [

∑∞

k1=−∞. . .∑

∞

kℓ−1=−∞|ci1,...,iℓ(k1, . . . ,

kℓ−1)|] < ∞.

Assumption A2′. The entries σij(θ) of 6n(θ) (i) are bounded,uniformly in n and θ – that is, there exists a real c > 0 such that|σij(θ)| ≤ c for any i, j ∈ N and θ ∈ [−π, π] – and (ii) they havebounded, uniformly in n and θ , derivatives up to the order two—namely, there exists Q < ∞ such that supi,j∈N supθ

dk

dθkσij(θ)

≤

Q , k = 0, 1, 2.

Assumption A3′. Same as Assumption A3, but moreover (i)λy;ny,qy(θ) and λz;nz ,qz (θ) diverge at least linearly in ny andnz , respectively, that is, lim infny→∞ infθ n−1

y λy;ny,qy(θ) > 0, andlim infnz→∞ infθ n−1

z λz;nz ,qz (θ) > 0, and (ii) both ny/nz and nz/nyare O(1) as min(ny, nz) → ∞.

This ‘‘at least linear’’ divergence assumption is also made inHallin and Liška (2007), and can be considered as a form of cross-sectional stability of the two panels under study.

Once estimated values of the numbers q, qy and qz of factorsare available, the estimated counterparts of of Kφy;n,i(L),Kψy;n,i(L)and Kνy;n,i(L) are obtained by substituting 6T

n(θ), qTn, q

Tny and

qTnz for 6n(θ), q, qy and qz in all definitions of Section 3, thentruncating infinite sums as explained in Section B of Forni et al.(2000) (a truncation which depends on t , which explains thenotation), yielding KTt

φy;n,i(L),KTtψy;n,i(L) and KTt

νy;n,i(L). Parallel withProposition 3 in Forni et al. (2000), we then have the followingresult.

Proposition 7. Let Assumptions A1′′, A2′, A3′ and B hold. Then,for all ϵk > 0 and ηk > 0, k = 1, 2, 3, there exists N0(ϵ1, ϵ2,

ϵ3, η1, η2, η3) such that PKTt ∗

φy;n,i(L)Xn,t − φy;it

> ϵ1

≤ η1,

PKTt∗

ψy;n,i(L)Xn,t − ψy;it

> ϵ2

≤ η2, and P

KTt∗νy;n,i(L)Xn,t − νy;it

> ϵ3

≤ η3, for all t = t(T ) satisfying a ≤ lim infT→∞(t(T )/T ) ≤

lim supT→∞(t(T )/T ) ≤ b, for some a, b such that 0 < a < b < 1,all n ≥ N0 and all T larger than some T0(n, ϵ1, ϵ2, ϵ3, η1, η2, η3).

Proof. The proof consists in reproducing, for each projectioninvolved in the reconstruction of φy;it , ψy;it and νy;it , the proof ofProposition 3 in Forni et al. (2000). Lengthy but obvious details areomitted.

Consistent estimations of the various contributions to the totalvariance of each subpanel can be obtained either by substitutingestimated spectral eigenvalues and eigenvectors for the exactones in the formulas of Section 3, and replacing integrals withthe corresponding finite sums over Fourier frequencies, or bycomputing the empirical variances of the estimated strongly andweakly common, strongly and weakly idiosyncratic components.

5. Dynamic factors in the presence of K blocks (K > 2)

The ideas developed in the previous sections extend to themoregeneral case of K > 2 blocks, with, however, rapidly increasing


complexity. Each subset k1, . . . , kℓ, ℓ = 0, 1, . . . , K of 1, . . . ,K indeed characterizes a decomposition of H into mutuallyorthogonal common and idiosyncratic subspaces, H

χ

k1,...,kℓand

Hξ

k1,...,kℓ, say, leading to 2K distinct implementations of the

Hallin–Liška and Forni et al. procedures.Instead of Yit and Zjt , denote all observations as Xk;it (i =

1, . . . , n; k = 1, . . . , K ), with the additional label k specifyingthat Xk;it belongs to block k. Projecting onto H

χ

k1,...,kℓand

Hξ

k1,...,kℓyields the orthogonal decomposition (for simplicity,

we are dropping the subscripts n and T ) Xk;it = χk;k1,...,kℓ;it +

ξk;k1,...,kℓ;it of Xk;it into a k1, . . . , kℓ-common and a k1, . . . , kℓ-idiosyncratic component; in particular, ξk;1,...,K;it , as in Section 2,can be called strongly idiosyncratic. For l1, . . . , lm ⊂ k1, . . . , kℓand k ∈ k1, . . . , kℓ \ l1, . . . , lm, call weakly idiosyncratic thedifference

νk;l1,...,lm,k1,...,kℓ\l1,...,lm;it

:= ξk;k1,...,kℓ\l1,...,lm;it − ξk;k1,...,kℓ;it

= χk;k1,...,kℓ;it − χk;k1,...,kℓ\l1,...,lm;it .

Since, for A ⊂ B,Hχ

A is a subspace of Hχ

B and Hξ

B a subspaceof H

ξ

A , this weakly idiosyncratic component is orthogonal to bothχk;l1,...,lm;it and ξk;k1,...,kℓ;it .

Further projections can be performed, yielding componentswith various degrees of commonness/idiosyncrasy. We restrict toprojections onto common spaces of the form

Hχ

l1∩···∩lm := Hχ

l1

. . .

Hχ

lm

decomposing χk;k1,...,kℓ;it , k ∈ l1, . . . , lm ⊇ k1, . . . , kℓ, into

φk;l1∩···∩lm;it andψk;k1,...,kℓ,l1∩···∩lm;it := χk;k1,...,kℓ;it − φk;l1∩···∩lm;it

which, in analogy with the two-block case, we respectivelycall strongly and weakly common components. Projections onto(nondegenerate) subspaces of the form H

χ

l1∩···∩lm can be obtainedvia the method described in Section 3.3. Sequences of projectionsonto decreasing sequences of Hχ ’s yield decompositions ofthe original observations into sums of mutually orthogonalcomponents, along with decompositions of their variances; thesedecompositions, however, depend on the sequence of projectionsadopted.

In view of the rapidly increasing notational burden, we will notpursue any further with formal developments; an application forK = 3 is considered in Section 6.2.

6. Real data applications

We applied our method to a dataset of monthly IndustrialProduction Indexes for France, Germany, and Italy, observed fromJanuary 1995 through December 2006. All data were preadjustedby taking a log-difference transformation (T = 143 throughout—one observation is lost due to differencing), then centered andnormalized using their sample means and standard errors.

In practice, some care has to be taken, however, due to the factthat, for finite n and T , the joint and marginal common spacesreconstructed from estimated spectral densities need not beingnested. When defined from population spectral densities, the y-common spaceH

χy associatedwith the Yit ’s and the common space

associated with the χxy;it ’s coincide, and are a subspace of the jointcommon space H

χxy. When based on finite n and T estimations,

those two spaces in general are distinct, and only the second one isa subspace ofHχ

xy. In order to avoid ambiguities and inconsistenciesin sums of squares, it is important, under finite n and T , toproceed with projections and spectral estimation in a sequence.For instance, one first should estimate the global spectrum, and

project the Yit ’s onto the reconstruction ofHχxy based on that global

spectral estimation. From those projections, χn;Txy;it , say, one should

then estimate the spectrum of the χxy;it ’s, the dynamic principalcomponents of which in turn yield the reconstruction of H

χy , etc.

This sequence of projections and estimations is carefully describedin the applications we are treating below.

6.1. A two-block analysis

First consider the data for France and Germany. Using XF;it forthe French observations Yit and XG;jt for the German ones Zjt , wehave ny = nF = 96, nz = nG = 114, hence n = nFG = 210.Spectral densities were estimated from the pooled panel using alag-window estimators of the form (18), with truncation parame-ter MT = 0.5

√T = 5. Based on this estimation, we ran the Hallin

and Liška (2007) identification method on the French and Germansubpanels, with sequences nF ,j = 96 − 2j, j = 1, . . . , 5 and nG,j =

96− 2j, j = 1, . . . , 5, respectively, then on the pooled panel, withsequence nFG,j = 210 − 2j, j = 1, . . . , 8 and an ‘‘almost constant’’proportion 96/210, 114/210 of French and German observations(namely, ⌈96nFG,j/210⌉ French observations, and ⌊114nFG,j/210⌋German ones. In all cases, we put Tj = T = 143, j = 1, . . . , 5.The range for c values, after some preliminary exploration, wastaken as [0, 0.0002, 0.0004, . . . , 0.5], and qmax was set to 10. In allcases, the panels were randomly ordered prior to the analysis. The

penalty function was p(n, T ) =

min

n,M2

T ,M−1/2T T 1/2

−1/2.

The results are shown in Fig. 1, and very clearly concludefor qT(nF ,nG) = 3 (for c ∈ [0.1798, 0.1894]), qTnF ,F = 2 (for c∈ [0.2222, 0.2344]), and qTnG,G = 3 (for c ∈ [0.2032, 0.2138]).Since qT(nF ,nG) = max(qTnF ,F , q

TnG,G

), this identification yieldsa block structure with 3 joint common factors, 3 German-common and 2 French-common factors, the French-commonspace being included in the German-common one. The French-common components thus are strongly common (no weaklycommon French components), whereas one German-commonfactor is French-idiosyncratic (no weakly idiosyncratic Germancomponents).

Taking these facts into account, the three factor analysesdescribed in Sections 2 and 3 yield(a) (global analysis) an analysis based on spectral estimation

in the global panel (three factors) decomposes the Frenchobservation XF;it (resp. the German observation XG;jt ) into astrongly idiosyncratic ξxF;it (resp. ξxG;jt = ξG;jt ) and a jointcommon χxF;it (resp. χxG;jt = χG;jt ) component;

(b) (French block analysis) an analysis based on spectral estima-tion in the subpanel consisting of theχxF;it ’s obtained under (a)(two factors) decomposes the French joint commonχxF;it into aFrench-common component χF;it (coinciding, since H

χ

F ⊂ Hχ

Gwith the strongly common one φF;it ) and a French-weakly-idiosyncratic one νF;it ; the same projection decomposes theGerman joint common componentχxG;jt = χG;jt into a stronglycommon φG;jt and a weakly common ψG;jt (we already knowthat νG;jt = 0).

The results, along with the corresponding percentages ofexplained variance, are provided in Fig. 2. For each of the fourmutually orthogonal subspaces appearing in the decomposition,we provide the percentage of total variation explained in eachcountry. The two strongly common (Franco-German) factorsjointly account for 3.5% only of the German total variability,but 14.6% of the French total variability. Germany’s ‘‘all-German’’(French-idiosyncratic) common factor explains 22.5% of Germany’stotal variance. Although French-idiosyncratic, that German factornevertheless still accounts for 8.5% of the French total variability.These estimated percentages of explained variation were obtainedvia estimated eigenvectors and eigenvalues.


c

0

0.5

1

1.5

2

2.5

qc;

NT

0123456789

1011

0.320.18 0.2 0.22 0.24 0.26 0.28 0.3

Sc

c

qc;

NT Sc

0123456789

1011

0.32 0.34 0.36 0.380.18 0.2 0.22 0.24 0.26 0.28 0.30

0.5

1

1.5

2

2.5

3

(a) France. (b) Germany.

0

0.5

1

1.5

2

2.5

3

3.5

Sc

qc;

NT

0123456789

1011

c0.18 0.2 0.22 0.24 0.26 0.28

(c) France & Germany.

Fig. 1. Identification of the numbers of factors for the France–Germany Industrial Production dataset. The three figures show the simultaneous plots of c → Sc and c → qTc,nneeded for this identification, ((a) and (b)) in the marginal French and German subpanels, and (c) in the complete panel, respectively.

6.2. A three-block analysis

Next, consider the three-block case resulting from adding tothe French and German data the corresponding Italian IndustrialProduction indices, with nI = 91, yielding a panel with K = 3blocks. The series length is still T = 143. Adapting the notationof Section 5, let XF;it , XG;it and XI;it correspond to the French, theGerman, and the Italian subpanels, respectively.

From the global panel (n = nFGI = 301 series), we can extractsix subpanels—the three panels we already analyzed in Section 6.1(the two-block French–German panel, the French and the Germanone-block subpanels), one new one-block subpanel (the marginalItalian one, with nI = 91), two new two-block subpanels (theFrench–Italian one, with nFI = 187 and the German–Italianone, with nGI = 205, respectively). Analyzing these new subpanelsalong the same lines as in the previous section (with, using obviousnotation, nI,j = 91 − 2j, j = 1, . . . , 5, nGI,j = 191 − 2j, j =

1, . . . , 8, nFI,j = 187− 2j, j = 1, . . . , 8, and nFGI,j = 301− 2j, j =

1, . . . , 15), still with MT = 0.5√T = 5, the same penalty function

and the same qmax = 10 as before, we obtain the identificationresults shown in the four graphs of Fig. 3.

These graphs again very clearly identify a total numberof qTn,FGI = 4 joint common factors (for c ∈ [0.1710, 0.1718]),qT(nF ,nI ),FI = 3 (for c ∈ [0.1838, 0.1886]) French–Italian, andqT(nG,nI ),GI = 4 (for c ∈ [0.1786, 0.1800]) German–Italian marginal‘‘binational’’ factors, and qTnI ,I = 2 (for c ∈ [0.2118, 0.22218])marginal Italian factors. Along with the figures obtained inSection 6.1 for France and Germany, this implies that H

χ

F ⊂ Hχ

G ,hence H

χ

F ∩ Hξ

G = 0 = Hχ

GI ∩ Hξ

F . The relations between thosevarious (dynamic) dimensions are easily obtained; for instance,q(nF ,nG),FG = qnF ,F + qnG,G − q(nF ,nG), a relation we already usedin Section 6.1, or

q(nF ,nG),FG = qnF ,F + qnG,G + qnI ,I − q(nF ,nG),FG− q(nF ,nI ),FI − q(nG,nI ),GI + q(nF ,nG,nI ),FGI .

These relations imply that the three countries share one stronglycommon factor. As already noted, France (two factors) has no

Fig. 2. Decomposition of the France–Germany panel data into four mutually or-thogonal components, with the corresponding percentages of explained variation.

specific common factor, but one (the strongly common one) sharedwith Germany and Italy, and one shared with Germany alone. BothItaly (two factors) and Germany (three factors) have a ‘‘national’’factor. Italy’s ‘‘non-national’’ factor is the strongly common one;Germany’s ‘‘non-national’’ factors are those shared with France,and include the strongly common one. The Italian and German‘‘national’’ factors need not be mutually orthogonal.

Proceeding with the various projections described in Section 5,we successively obtain

(a) (global analysis) a four-factor analysis based on spec-tral estimation in the global panel: projecting onto theresulting reconstruction of H

ξ

FGI and Hχ

FGI decomposesXF;it , XG;it , and XI;it into their strongly idiosyncratic compo-nents ξF;FGI;it , ξG;FGI;it , and ξI;FGI;it , with respective orthog-onal complements χF;FGI;it , χG;FGI;it , and χI;FGI;it ;

(b1) (French–German block analysis) a three-factor analysis basedon spectral estimation in the subpanel consisting of theχF;FGI;it ’s and χG;FGI;it ’s obtained under (a): projection ontothe resulting reconstructions of H

χ

FG = Hχ

G decomposes


0

0.5

1

1.5

2

2.5

3

3.5

4

qc;

NT

0123456789

1011

c0.32 0.34 0.34 0.380.18 0.2 0.22 0.24 0.26 0.28 0.3

Sc

c

qc;

NT

0123456789

1011

0.260.18 0.2 0.22 0.24 0.28

Sc

00.20.40.60.811.21.41.61.82

(a) Italy. (b) France & Italy.

0

0.5

1

1.5

2

2.5

3

3.5

4

qc;

NT

0123456789

1011

c0.16 0.18 0.2 0.22 0.24 0.26

Sc

0

0.5

1

1.5

2

2.5

3

3.5

qc;

NT

c

0123456789

1011

0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24

Sc

(c) Germany & Italy. (d) France & Germany & Italy.

Fig. 3. Identification of the numbers of factors for the French–German–Italian Industrial Production dataset. The four figures show the simultaneous plots of c → Sc andc → qTc,n needed for this identification: (a) for the marginal Italian subpanel, ((b) and (c)) for the France–Italy and Germany–Italy subpanels, and (d) for the completethree-country panel, respectively.

χF;FGI;it into χF;FG;it and νF;I,FG;it , and χG;FGI;it intoχG;FG;it and νG;I,FG;it , respectively;

(b2) (French–Italian block analysis) a three-factor analysis, similarto that in (b1), based on spectral estimation in the subpanelconsisting of the χF;FGI;it ’s and χI;FGI;it ’s obtained under(a): projection onto the resulting reconstruction of H

χ

FIdecomposes χI;FGI;it into χI;FI;it and νI;G,FI;it ;

(c1) (French block analysis) a two-factor analysis, similar to step(b) in Section 6.1: projecting χF;FG;it obtained in (b1) ontoHχ

F yields χF;F;it = χF;F∩G;it and νF;G,FG;it ; for χG;FG;it ,the same projection actually coincides with a projection ontoHχ

F∩G, yielding φG;F∩G;it and ψG;FG,F∩G;it = ψG;FG,F;it ;(c2) (Italian block analysis) a two-factor analysis based on

spectral estimation in the subpanel consisting of the χI;FI;it ’sobtained in step (b2): projecting them onto the resultingreconstruction of H

χ

I yields χI;I;it and νI;FG,I;it ;(d) (French-and-Italian block analysis) a final projection of

χF;F;it and φG;F∩G;it obtained in (c1), and χI;I;it obtainedin (c2) onto H

χ

F∩G∩I = Hχ

F∩I yields the strongly commoncomponents φF;F∩G∩I;it , φG;F∩G∩I;it and φI;F∩G∩I;it , alongwith theweakly commononesψF;F,F∩G∩I;it , ψG;F,F∩G∩I;it ,and ψI;I,F∩G∩I;it .

Conclusions are summarized in the diagram of Fig. 4, alongwith the various percentages of explained variances. Inspectionof that diagram reveals that the three countries all exhibit a highpercentage of about 71% of strongly idiosyncratic variation. Franceand Italy, with about 9% of strongly common variation, are the‘‘most strongly common’’ in the group of three.

Acknowledgements

Marc Hallin gratefully acknowledges the support of the Son-derforschungsbereich ‘‘Statistical modelling of nonlinear dynamic

processes’’ (SFB 823) of the Deutsche Forschungsgemeinschaft anda Discovery Grant of the Australian Research Council. Part of thiswork was completed while visiting the Economics Departmentat the European University Institute in Florence under a FernandBraudel Fellowship. The authors are grateful to Christine De Mol,Richard Spady, Marco Lippi and Mario Forni for many stimulatingdiscussions andhelpful suggestions. They also thankAlexeiOnatskifor pointing out a flaw in an earlier stage of themanuscript, as wellas Franz Palm, Jean-Pierre Urbain, and two anonymous referees fortheir comments on the original version of the paper.

Appendix. Proof of Lemma 1

Denote by Θy the set (with Lebesgue measure zero) ofθ values for which divergence in Assumption A2(i) does nothold. Similarly define Θz , and let Θ := Θy ∪ Θz : Θ

also has Lebesgue measure zero. Since 6y;ny(θ) is a principalsubmatrix of 6n(θ), a classical result (see Corollary 1, page293, in Lancaster and Tismenetsky, 1985) implies that, forany n = (ny, nz) and θ ∈ [−π, π], λy;ny,i(θ) ≤ λn,i(θ), i = 1, . . . , ny. Since λy;ny,qy(θ) diverges for all θ ∈ Θ as ny →

∞, so does λn,qy(θ). A similar result also holds for the λz;nz ,j’s, sothat, for all θ ∈ Θ, λn,max(qy,qz )(θ) diverges as min(ny, nz) → ∞.Note that the same result by Lancaster and Tismenetsky (1985) alsoimplies that, for all θ and k, λn,k(θ) is a monotone nondecreasingfunction of both ny and nz and, therefore, either is bounded or goesto infinity as either ny or nz → ∞.

Next, let us show that λn,qy+qz+1(θ) is bounded as min(ny, nz)→ ∞, for all θ ∈ Θ . For all θ ∈ Θ , consider the sequences ofn-dimensional vectors ζn(θ) := (ζ′

y;ny(θ), ξ′

z;nz (θ))′ which are or-

thogonal to the qy + qz vectors (p′

y;ny,1(θ), 0, . . . , 0)′, . . . , (p′

y;ny,qy(θ), 0, . . . , 0)′ and (0, . . . , 0, p′

z;nz ,1(θ))′, . . . , (0, . . . , 0, p′

z;nz ,qz(θ))′. The collection of all such ξn’s is a linear subspace 4n(θ) ofdimension at least n − qy − qz . For any such ξn(θ), in view ofthe orthogonality of ξy;ny(θ) and py;ny,1(θ), . . . , py;ny,qy(θ) (resp.,of ξz;nz (θ) and pz;nz ,1(θ), . . . , pz;nz ,qz (θ)),


Fig. 4. Decomposition of the France–Germany–Italy panel data into eight components, with the corresponding percentages of explained variation.

‖ξn(θ)‖−2ξ∗

n(θ)6n(θ)ξn(θ)

= ‖ξn‖−2ξ∗

y;ny(θ)6y;ny(θ)ξy;ny(θ)

+ ‖ξn(θ)‖−2ξ∗

z;nz (θ)6z;nz (θ)ξz;nz (θ)

+ ‖ξn(θ)‖−2ξ∗

y;ny(θ)6yz;n(θ)ξz;nz (θ)

+ ‖ξn(θ)‖−2ξ∗

z;nz (θ)6zy;n(θ)ξy;ny(θ)

≤ 2(‖ξy;ny(θ)‖−2ξ∗

y;ny(θ)6y;nyξy;ny(θ)

+ ‖ξz;nz(θ)‖−2ξ∗

z;nz (θ)6z;nz ξz;nz (θ))

≤ 2(λ2y;ny,qy+1(θ)+ λ2z;nz ,qz+1(θ))

for all θ ∈ Θ and n = (ny, nz). Since λ2y;ny,qy+1(θ) and λ2z;nz ,qz+1(θ)

are bounded, for any θ ∈ Θ , as min(ny, nz) → ∞, so is ξ∗

n(θ)6n(θ)ξn(θ). Hence, for all θ ∈ Θ and n = (ny, nz),4n (withdimension at least n − qy − qz) is orthogonal to any eigenvectorassociated with a diverging sequence of eigenvalues of 6n(θ). Itfollows that the number of such eigenvalues cannot exceed qy+qz .Summing up, for all θ ∈ Θ , the number of diverging eigenvaluesof 6n(θ) is finite – denote it by q – and comprised betweenmax(qy, qz) and qy + qz , as was to be shown.

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