Working PaPer SerieS. Weak and Strong cross Section dependence and estimation of Large PaneLS. no 1100 / october 2009

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1 Working PaPer SerieS no / october 29 Weak and Strong cross Section dependence and estimation of Large PaneLS b y Alexander Chudik, M. Hashem Pesaran and Elisa osetti

2 WORKIG PAPER SERIES O / OCOBER 29 WEAK AD SROG CROSS SECIO DEPEDECE AD ESIMAIO OF LARGE PAELS by Alexander Chudik 2, M. Hashem Pesaran 3 and Elisa osetti 4 In 29 all ublications feature a motif taken from the 2 banknote. his aer can be downloaded without charge from htt:// or from the Social Science Research etwork electronic library at htt://ssrn.com/abstract_id= A reliminary version of this aer was resented at he Econometrics Journal Secial Session, Royal Economic Society Annual Conference, Surrey, Aril 29, and at the Institute for Advanced Studies, Vienna, May 29. We have benefited from comments by articiants, and articularly from our discussions with Manfred Deistler. 2 Euroean Central Bank, Kaiserstrasse 29, D-63 Frankfurt am Main, Germany; alexander.chudik@ecb.euroa.eu 3 Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, UK; mh@econ.cam.ac.uk; htt:// 4 Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, UK; et268@cam.ac.uk. Elisa osetti acknowledges financial suort from ESRC (Ref.no. RES ).

3 Euroean Central Bank, 29 Address Kaiserstrasse Frankfurt am Main, Germany Postal address Postfach Frankfurt am Main, Germany elehone Website htt:// Fax All rights reserved. Any reroduction, ublication and rerint in the form of a different ublication, whether rinted or roduced electronically, in whole or in art, is ermitted only with the exlicit written authorisation of the or the author(s). he views exressed in this aer do not necessarily reflect those of the Euroean Central Bank. he statement of urose for the Working Paer Series is available from the website, htt:// eu/ub/scientific/ws/date/html/index. en.html ISS (online)

4 COES Abstract 4 on-technical summary 5 Introduction 6 2 Cross section deendence in large anels 8 2. Weak and strong deendence Weak and strong cross section deendence Proerties of weakly and strongly cross sectionally deendent rocesses 3 Dynamic anels 4 4 Common factor models 9 5 CCE estimation of anel data models with infinite factors 23 6 Monte Carlo exeriments Results 3 7 Concluding remarks 32 ables and figures 33 References 44 Aendix 46 Euroean Central Bank Working Paer Series 55 Working Paer Series o 3

5 Abstract his aer introduces the concets of time-secific weak and strong cross section deendence. A double-indexed rocess is said to be cross sectionally weakly deendent at a given oint in time, t, if its weighted average along the cross section dimension () converges to its exectation in quadratic mean, as is increased without bounds for all weights that satisfy certain granularity conditions. Relationshi with the notions of weak and strong common factors is investigated and an alication to the estimation of anel data models with an infinite number of weak factors and a finite number of strong factors is also considered. he aer concludes with a set of Monte Carlo exeriments where the small samle roerties of estimators based on rincial comonents and CCE estimators are investigated and comared under various assumtions on the nature of the unobserved common effects. Keywords: Panels, Strong and Weak Cross Section Deendence, Weak and Strong Factors. JEL Classification: C, C3, C33. 4 Working Paer Series o

6 ontechnical Summary Cross section deendence is a raidly growing eld of study in anel data analysis. he aim of this aer is to characterize the correlation attern over the cross sectional dimension for a general class of rocesses, regardless of whether they are reresented by factor or satial models or any other rocess featuring cross section dimension roosed in the literature. Unlike in the case of time series, data along the cross sectional dimension do not tyically have a natural ordering. his aer rooses a new characterization of cross section deendence into weak and strong, which are more widely alicable than the existing de nitions characterizing the attern of cross section correlation in the factor literature. We consider the asymtotic behaviour of weighted averages at each oint in time, which does not require any stationarity assumtions to be imosed on the underlying time series rocesses. We de ne a rocess to be cross sectionally weakly deendent at a given oint in time if its weighted average at that time converges to its exectation in quadratic mean, as the cross section dimension is increased without bounds for all weights that satisfy certain granularity conditions. If this requirement does not hold, then the rocess is said to be cross sectionally strongly deendent. Convergence roerties of weighted averages is of great imortance for the asymtotic theory of various estimators and tests commonly used in anel data econometrics, as well as for arbitrage ricing theory and ortfolio otimization with a large number of assets. In this aer we also investigate relationshi with the notions of weak and strong common factors and consider the roblem of estimating the sloe coe cients of large anels, where cross section units are subject to a number of unobserved common factors that may rise with the number of cross section units. It is established that Common Correlated E ects (CCE) estimator introduced by Pesaran (26) remains asymtotically normal under certain conditions on the loadings of the in nite factor structure, including cases where methods relying on rincial comonents fail. A Monte Carlo study documents these theoretical ndings by investigating the small samle erformance of estimators based on rincial comonents and the CCE estimators under alternative assumtions on the nature of unobserved common e ects. In articular, we examine and comare the erformance of these estimator when the errors are subject to a nite number of unobserved strong factors and an in nite number of weak and/or semi-weak unobserved common factors. Working Paer Series o 5

7 Introduction here exists a growing literature on econometric methods for reresenting and measuring cross section deendence in anel data regression models. Conditioning on variables seci c to the cross section units alone tyically does not deliver cross section error indeendence and it is well known that neglecting cross section deendence can lead to biased estimates and surious inference. How to account for contemoraneous error correlations deends on the number of cross section units,, relative to the time series dimension,, and in most cases on the nature and the degree of cross section deendencies observed. When is small relative to, the nature of cross section deendence is unimortant as long as the errors are not correlated with the regressors, in which case the Seemingly Unrelated Regression Equations (SURE) aroach can be used (Zellner (962)). But when is large relative to, the SURE rocedure is not alicable and the nature of cross section deendence needs to be taken into account. In such cases there are two main aroaches to modelling cross section deendence in anels : (i) satial rocesses ioneered by Whittle (954) and develoed further by Anselin (988), Kelejian and Prucha (999), and Lee (24); and (ii) factor models introduced by Hotelling (933) and rst alied in economics by Stone (947). Factor models have been used extensively in nance (Chamberlain and Rothschild (983), Connor and Korajzcyk (993); Stock and Watson (998); Kaetanios and Pesaran (27)), and in macroeconomics (Forni and Reichlin (998); Stock and Watson (22)). While in rincile, as we shall see, cross sectionally deendent rocesses, including satial and network rocesses, can be set u as an unobserved factor structure with ossibly in nite number of factors, the original idea for using latent factors is to characterize co-movements of individual cross section units by a small number of latent factors lus a white noise, in order to overcome the curse of dimensionality. he aim of this aer is to characterize the correlation attern over the cross sectional dimension for a general class of rocesses, regardless of whether they are reresented by factor or satial models or any other rocess featuring cross section dimension roosed in the literature. Unlike in the case of time series, data along the cross sectional dimension do not tyically have a natural ordering. One way to characterize the correlation structure of double index rocesses has been roosed in the factor literature. he idiosyncratic (or weak deendence) roerty, advanced by Forni and Lii (2), alies to both dimensions and requires that the weighted average of a stationary rocess, comuted both over time and across sections, converges to zero in quadratic mean for all sets of weights satisfying a certain condition. his notion is used by the authors to characterize dynamic factor models. heir framework is a generalization of the static model for asset markets by Chamberlain (983) and Chamberlain and Rothschild (983), and extends some of the results resented by Forni and Reichlin (998). Forni and Lii (2) show that a necessary and su cient condition for a rocess to be idiosyncratic (or weakly deendent over time and across the units) is the boundedness of the largest eigenvalue of its sectral density matrix at all frequencies. Using this result, Anderson et al. (29) (see their De nition 4) formally de ne a double index stochastic rocess as weakly deendent if the largest eigenvalues of its sectral density is bounded in (at all frequencies), as 6 Working Paer Series o

8 oosed to a strongly deendent rocess, for which a nite, nonzero number of eigenvalues diverge to in nity as goes to in nity. We remark that these assumtions on the asymtotic behaviour of eigenvalues of the sectral density are needed for identi cation of common factors and their loadings, and their estimation by rincial comonents analysis. Further, to ensure the existence of the sectral density, this literature assumes that the underlying time series rocesses are stationary with absolutely summable autocovariances. his aer rooses a new characterization of cross section deendence into weak and strong, which are more widely alicable than the de nitions introduced by Anderson et al. (29). We consider the asymtotic behaviour of weighted averages at each oint in time, which does not require any stationarity assumtions to be imosed on the underlying time series rocesses. We de ne a rocess to be cross sectionally weakly deendent at a given oint in time if its weighted average at that time converges to its exectation in quadratic mean, as the cross section dimension is increased without bounds for all weights that satisfy certain granularity conditions. If this requirement does not hold, then the rocess is said to be cross sectionally strongly deendent. Convergence roerties of weighted averages is of great imortance for the asymtotic theory of various estimators and tests commonly used in anel data econometrics, as well as for arbitrage ricing theory and ortfolio otimization with a large number of assets. It is clear that the underlying time series rocesses in either of the two literature need not be stationary, and concets of weak and strong deendence that are more generally alicable are needed. In this aer we focus on the econometric literature and consider the roblem of estimating the sloe coe cients of large anels, where cross section units are subject to a number of unobserved common factors that may rise with. It is established that Common Correlated E ects (CCE) estimator introduced by Pesaran (26) remains asymtotically normal under certain conditions on the loadings of the in nite factor structure, including cases where methods relying on rincial comonents fail. A Monte Carlo study documents these theoretical ndings by investigating the small samle erformance of estimators based on rincial comonents and the CCE estimators under alternative assumtions on the nature of unobserved common e ects. In articular, we examine and comare the erformance of these estimator when the errors are subject to a nite number of unobserved strong factors and an in nite number of weak and/or semi-weak unobserved common factors. he lan of the remainder of the aer is as follows. Section 2 introduces the concets of strong and weak cross section deendence, and exlores the relationshi between the deendence structure of rocesses. Section 3 focuses on cross section deendence in dynamic anels. Section 4 resents common factor models and discusses the notions of weak, semi-strong and strong factors. Section 5 introduces the CCE estimators in the context of anels with an in nite number of common factors. Section 6 describes the Monte Carlo design and discusses the results. Finally, Section 7 rovides some concluding remarks. otation: j (A)j j 2 (A)j ::: j n (A)j are the eigenvalues of a matrix A 2 M nn, where Working Paer Series o 7

9 M nn is the sace of nn comlex valued matrices. A denotes a generalized inverse of A. he sectral radius of A 2 M nn is (A) = max [j P i(a)j], and its column norm is kak jn = max n jn i= ja ijj. P he row norm of A is kak = max n in j= ja ijj. he sectral norm of A is kak = [(AA )] =2, and kak 2 = [ r (AA )] =2. K is used for a xed ositive constant that does not deend on. 2 Cross section deendence in large anels In this section, we study the structure of correlation of the double index rocess fz it ; i 2 ; t 2 Zg where z it are random variables de ned on a robability sace (; F; P ); the index t refers to an ordered set, the time, while the index i indicates the units of an unordered oulation. Our rimary focus is on characterizing the correlation structure of the double index rocess fz it g over the cross sectional dimension. We start by reviewing de nitions rovided in the existing literature to characterize the correlation attern of fz it g; and next we introduce our general notions of weakly and strongly cross sectionally deendent rocesses. 2. Weak and strong deendence Forni and Lii (2) introduce the notion of idiosyncratic rocess to characterize a weak form of deendence that involves both time series and cross sectional dimensions under the following assumtion: Assumtion (Forni and Lii, 2, Assumtion ) For each 2, the rocess z t = (z t ; :::; z t ) is covariance stationary and the sectral measure of z t is absolutely continuous. otice that Assumtion guarantees the sectral density for the vector z t to exist. Consider any sequence of weights vectors w = (w ; w 2 ; :::; w ) such that lim kw k =. ()! Let F z (!) denote the sectral density matrix for z t and de ne the norm kw k Fz as kw k Fz = 2 Z w F z () w d: Forni and Lii (2) de ne the rocess fz it g as idiosyncratic if, for all weights w satisfying condition (), we have lim kw k Fz =.! he idiosyncratic roerty imlies that the variance of the weighted average of fz it g, comuted both over time and across sections, vanishes to zero as tends to in nity. he authors show that the 8 Working Paer Series o

10 sequence fz it g is idiosyncratic if and only if the largest eigenvalue of F z (!), z ; (!), is bounded in! and. Further, a rocess fz it g for which the (m + )th eigenvalue of F z (!) is bounded in! and, and the mth eigenvalue diverges in for all frequencies!, can be reresented by the so-called generalized factor structure, namely a linear combination of m dynamic factors, lus an idiosyncratic rocess (see their heorems and 2). his is an extension to the dynamic case of the static factor model used in arbitrage ricing theory as advanced by Ross (976) and further develoed by Chamberlain (983), Chamberlain and Rothschild (983), and Ingersoll (984). Based on the above results, Anderson et al. (29) de ne the concets of weak and strong deendence for rocesses fz it g satisfying Assumtion, on the basis of the asymtotic behaviour of the eigenvalues of F z (!). De nition (Weak and strong deendence) he double index rocesses fz it ; i 2 ; t 2 Zg is weakly deendent if z ; (!) is uniformly bounded in! and. he rocess fz itg is strongly deendent if the rst m (m < K) eigenvalues ( z ; (!) ; :::; z ;m (!)) diverge to in nity as!, for all frequencies. For further details on the above de nitions we refer to Forni and Lii (2) (see their Assumtion, De nitions, 6 and 9; heorems and 2), and Anderson et al. (29) (see their Assumtions 4 and 5). We note that the stationarity of the time series rocesses in z t set in Assumtion is needed for estimation by (dynamic) rincial comonents analysis of common factors and their loadings in the generalized factor structure. However, this assumtion is likely to be quite restrictive and is unlikely to hold in many alications, esecially in nance where time series often exhibit time-varying volatility. 2.2 Weak and strong cross section deendence We now resent our de nitions of weak and strong cross section deendence at a given oint in time. For ease of exosition, in the following we omit the subscrit where not necessary. We make the following assumtions: Assumtion 2 Let w t = (w t ; :::; w t ), for t 2 Z and 2, be a vector of non-stochastic weights. For any t 2, the sequence of weights vectors fw t g of growing dimension (! ) satis es the following granularity conditions: kw t k = O 2 ; (2) and w jt kw t k = O 2 for any j 2 : (3) Working Paer Series o 9

11 Assumtion 3 Let I t be the information set available at time t. For each t 2, z t = (z t ; :::; z t ) has conditional mean and variance E (z t ji t ) = ; (4) V ar (z t ji t ) = t ; (5) where t is a symmetric, nonnegative de nite matrix, with generic (i; j) th element ij;t, and such that < ii;t K, for i = ; :::;, where K is a nite constant indeendent of. Assumtion 2, known in nance as the granularity condition, ensures that the weights fw it g are not dominated by a few of the cross section units. Although we have assumed the weights to be nonstochastic, this is done for exositional convenience and can be relaxed by requiring that conditional on the information set the weights, w t, are distributed indeendently of z t. In Assumtion 3 we imose some regularity conditions on the time series roerties of fz it g. Assumtion 3 is also standard in nance and seci es that z t has conditional means and variances. he rst art, (4), can be relaxed to E (z t ji t ) = ;t, with ;t being a re-determined function of the elements of I t. But to kee the exosition simle and without loss of generality we have set ;t =. o simlify the notations we suress the exlicit deendence of z t, w t and other vectors and matrices on, unless this is needed to avoid ossible confusions. Consider now the weighted averages, z wt = P i= w itz it = w tz t, for t 2, where z t and w t satisfy Assumtions 2 and 3. We are interested in the limiting behavior of z wt at a given oint in time t 2 as!. De nition 2 (Weak and strong cross section deendence) he rocess fz it g is said to be cross sectionally weakly deendent (CWD) at a given oint in time t 2 conditional on information set I t, if for any sequence of weight vectors fw t g satisfying the granularity conditions (2)-(3) we have lim V! ar(w tz t ji t ) = : (6) fz it g is said to be cross sectionally strongly deendent (CSD) at a given oint in time t 2 conditional on information set I t, if there exists a sequence of weights vectors fw t g satisfying (2)-(3) and a constant K indeendent of such that for any su ciently large V ar(w tz t ji t ) K > : (7) he concets of weak and strong cross section deendence roosed here are de ned conditional on an information set, namely the set I t in the de nition above. In this way we are able to consider cross section deendence roerties of fz it g without having to limit the time series features of the rocess. Various information sets could be considered in ractise, deending on alications. One examle is the set containing lagged realizations of the rocess fz it g, that is I t = fz t ; z t 2 ; ::::g. Working Paer Series o

12 In the context of dynamic models, it is useful to condition on the initialization of the dynamic rocess (i.e. starting values) only. In stationary anels, unconditional variances of cross section averages could be considered. In the remainder of the aer, if not stated exlicitly, the concets of CWD and CSD are always de ned on the information set I t. Remark In contrast to the notions of weak and strong deendence advanced by Forni and Lii (2) and Anderson et al. (29), our concets of CWD and CSD do no require the underlying rocesses to be covariance stationary and have sectral density at all frequencies. Remark 2 A articular form of a CWD rocess arises when airwise correlations take non-zero values only across nite subsets of units that do not sread widely as samle size increases. A similar case occurs in satial rocesses, where for examle local deendency exists only among adjacent observations. However, we observe that the notion of weak deendence does not necessarily involve an ordering of the observations or the seci cation of a distance metric. 2.3 Proerties of weakly and strongly cross sectionally deendent rocesses he following roosition establishes the relationshi between weak cross section deendence and the asymtotic behaviour of the sectral radius of t (namely, ( t )). Proosition he following statements hold: (i) he rocess fz it g is CWD at a oint in time t 2 if ( t ) is bounded in. (ii) he rocess fz it g is CSD at a oint in time t 2 if and only if lim! ( t ) = K >. Proof. First, suose ( t ) is bounded in. We have V ar(w tz t ji t ) = w t t w t w tw t ( t ) ; (8) and under the granularity conditions (2)-(3) it follows that lim V! ar(w tz t ji t ) = ; namely that fz it g is CWD, which roves (i). ow suose that fz it g is CSD at time t. hen, from (8), it follows that ( t ) tends to in nity at least at the rate. oting that ( t ) where, under Assumtion 3, ii;t are nite, ( t ) cannot diverge to in nity at a rate larger than, and hence it follows that under CSD lim! ( t ) = K >. o rove the reverse relation, rst note that, from the Rayleigh-Ritz theorem, See Horn and Johnson (985),.76. i= ii;t ( t ) = max v t vt=v t t v t = v t t v t : (9) Working Paer Series o

13 Let w t = v t and notice that w t satis es (2)-(3). Hence, we can rewrite ( t ) as ( t ) = V ar(w t z t ji t ): () It follows that if lim! ( t ) = K >, then lim! V ar(wt z t ji t CSD, which roves (ii). ) >, i.e. the rocess is Since 2 ( t ) k t k ; it follows from (8) that if lim! ( t ) > then also lim! k tk >. Hence, both the sectral radius and the column norm of the covariance matrix of a CSD rocess are unbounded in. his result for a CSD rocess is similar to the condition of not absolutely summable autocorrelations that characterizes time series rocesses with strong temoral deendence (Robinson (23)). A number of remarks concerning the above concets of CWD and CSD are in order. Remark 3 he de nition of idiosyncratic rocess by Forni and Lii (2) and our de nition of CWD di er in the way weights used to build weighted averages are de ned. While Forni and Lii assume lim! kwk =, our granularity conditions (2)-(3) imly that, for any t 2, lim! 2 kw t k = for any >. his di erence in the de nition of weights has some imlications on the roerties of our rocesses. In articular, under (), it is ossible to show that the idiosyncratic rocess (and hence also the de nition of weak deendence à la Anderson et al. (29)) imly bounded eigenvalues of the sectral density matrix. Conversely, under (2)-(3), it is clear that if ( t ) = O( ) for any >, then, using (8), lim! w tw t ( t ) = ; and the underlying rocess will be CWD. Hence, the bounded eigenvalue condition is su cient but not necessary for CWD. According to our de nition a rocess could be CWD even if its maximum eigenvalue is rising with, so long as its rate of increase is bounded aroriately. In Section 3, we investigate the relation between bounded eigenvalues of the sectral density matrix, and bounded eigenvalues of the covariance matrix, t, in the case of dynamic anels. One rationale for characterizing rocesses with increasing largest eigenvalues at the slower ace than as weakly deendent is that bounded eigenvalues is not a necessary condition for consistent estimation in general, although in some cases, such as the method of rincial comonents, this condition is necessary. More on this below in Section 5, where we consider estimation of sloe coe cients in anels with an in nite factor structure. We conclude this section with two results concerning the relationshi between strongly and weakly cross sectionally correlated variables. Following De nition 2, we say that two rocesses fz it;a g and 2 See Horn and Johnson (985), Working Paer Series o

14 fz it;b g are weakly correlated at time t if lim! E(z wt;az wt;b ji t ) =, for all sets of weights that satisfy the granularity conditions. he next roosition considers correlation of two rocesses with di erent cross deendence structures. We then investigate the correlation structure of linear combinations of strongly correlated and weakly correlated variables. Proosition 2 Suose that fz it;a g and fz it;b g are CSD and CWD rocesses, resectively. hen for all sets of weights fw a it g and w b it satisfying conditions (2)-(3), we have Proof. Let have lim E(z wt;az wt;b ji t ) = :! n o n o wi;t a and wi;t b be two sets of weights satisfying conditions (2)-(3). For t 2, we [E(z wt;a z wt;b ji t )] 2 E(z 2 wt;a ji t )E(z 2 wt;b ji t ): Further, under Assumtion 3 the rocess z it;a satis es E(z 2 wt;a ji t ) < K; where K is a nite constant. Also from (6), and considering that z it;b is a CWD rocess we have lim! E(z2 wt;b ji t ) = : herefore, for all sets of weights satisfying (2)-(3), we obtain lim E(z wt;az wt;b ji t ) = :! Proosition 3 Consider two indeendent rocesses fz it;a g and fz it;b g ; and their linear combinations de ned by z it;c = a z it;a + b z it;b ; () where a and b are non-zero xed coe cients. hen the following statements hold: (i) Suose fz it;a g and fz it;b g are CSD, then fz it;c g is CSD, (ii) Suose fz it;a g and fz it;b g are CWD, then fz it;c g is CWD, (iii) Suose fz it;a g is CSD and fz it;b g is CWD, then fz it;c g is CSD. Proof. Let t;a and t;b be the covariance matrices of z t;a = (z t;a ; :::; z t;a ) and z t;b = (z t;b ; :::; z t;b ), and t;c the covariance of their linear combination that is, given the assumtion of indeendence between z t;a and z t;b t;c = 2 a t;a + 2 b t;b: Working Paer Series o 3

15 he variance of the weighted average w tz t;c satis es V ar(w tz t;c ji t ) 2 jv ar(w tz t;j ji t ); j = a; b; which imlies that, if there exists a weights vector w t satisfying the granularity conditions such that either V ar(w tz t;a ji t ) or V ar(w tz t;b ji t ) or both are bounded away from zero, then also V ar(wtz t;c ji t ) is bounded away from zero and fz it;c g is cross sectionally strongly deendent (this roves (i) and (iii)). Also, we know that V ar(w tz t;c ji t ) = V ar(w tz t;a ji t ) + V ar(w tz t;b ji t ): oting that V ar(wtz t;a ji t ) and V ar(wtz t;b ji t ) satisfy (6), then lim! V ar(wtz t;c ji t ) =, and hence fz it;c g is cross sectionally weakly correlated (this roves (ii)). he above result can be generalized to linear functions of more than two rocesses. In general, linear combinations of indeendent rocesses that are strongly (weakly) correlated is strongly (weakly) deendent, while linear combinations of a nite number of weakly and strongly correlated rocesses is strongly correlated, since on aggregation only terms involving the strong comonent will be of any relevance. his result will be emloyed in Section 4. 3 Dynamic anels Suose that for each 2, cross section units collected into the vector z t = (z t ; z 2t ; :::; z t ) are generated from the following VAR model, z t = t z t + u t, (2) where t is a dimensional matrix of unknown coe cients, which could be time-varying, the vector u t of reduced-form errors has mean and variance E (u t ) =, E u t u t = t ; (3) where t, t = ; :::;, are symmetric, nonnegative de nite matrix, and u t is indeendently distributed of u t for any t 6= t. he initialization of the dynamic rocess could be from a nite ast, t 2 f M + ; ::; ; ::g Z, M being a xed ositive integer; or we can let M!, as in Chudik and Pesaran (29). he in nite-dimensional satio-temoral model (2) can also be viewed more generally as a dynamic network, with t and t caturing the static and dynamic forms of inter-connections that might exist in the network. All linear dynamic anel data models existing in the literature could be written as secial cases of (2). Sequence of models (2) of growing dimension (! ) is non-nested since the deendence between unit i and j could change with the inclusion of new unit(s). For this reason, the rocess fz it ; 2 ; i 2 f; ::; g ; t 2 g given by (2) is a trile 4 Working Paer Series o

16 index rocess, but we continue to omit subscrit (were not necessary) to simlify the exosition. Object of this section is to investigate the correlation attern of fz it g across the cross sectional units in the dynamic setting given by (2). In our analysis, we set I t to contain only the starting values, z M, i.e. I t = I = fz M g. Consider the following assumtions on the coe cient matrices, t, and the error vector, u t : Assumtion 4 here exist a constant K < and an arbitrarily small ositive constant > such that for any xed t 2 and any 2, we have k t k < K, (4) and k t k < K. (5) Remark 4 Equation (5) of Assumtion 4 imlies that fu it g is CWD. he initialization of a dynamic rocess could be from a non-stochastic oint or could have been from a stochastic oint, ossibly generated from a rocess di erent from the DGP of fu it g. Proosition 4 Consider model (2) and suose Assumtion 4 holds. hen for any sequence of weight vectors fw t g satisfying condition (2), and for a xed M and a xed t 2, lim V ar! w tz t j z M =. (6) Proof. he vector di erence equation (2) can be solved backwards, taking z M as given: z t = t+m Y s=! t+m t s z M + `=! `Y t s u t `: s= he variance of z t (conditional on initial values) is t; M = V ar (z t j z M ) = t+m `=! `Y t s t ` ` s= s=! Y t s : For any t 2, k t; M k is under Assumtion 4 bounded by k t; M k t+m `=! `Y k t s k 2 k t `k = O : s= It follows that for any arbitrary vector of weights satisfying (2), V ar w tz t j z M = w t t; M w t ( t; M ) w tw t = o () ; (7) Working Paer Series o 5

17 where ( t; M ) k t; M k = O, and wtw t = kw t k 2 = O. Hence, the dynamic rocess fz it g given by (2) under Assumtion 4 is CWD at any oint in time t 2, conditional on starting values z M. he result of the above roosition can be readily extended to situations where M and/or t!. In such cases we need the stronger requirement that k t k <, for all t 2. It is then easily seen that the VAR() model, (2), yields a cross sectionally weakly deendent rocess if for all t and, k t k < K, and k t k <, irresective of the values of t and M. 3 here are several interesting imlications of this nding. Consider the following additional assumtion on the coe cients matrix t, which states that for some units the o -diagonal elements of the matrix t are small. Assumtion 5 Let K be a non-emty index set. De ne vector t; i = ti ; :::; t;i;i ; ; t;i;i+ ; :::; t;i where tij for i; j 2 f; 2; :::; g is the (i; j) element of matrix t. For any i 2 K and any t 2, vector t; i satis es t; i j=;j6=i 2 tij A =2 = O 2. (8) Remark 5 Assumtion 5 imlies that for i 2 K, P i=;i6=j tij t; i = O (). 4 herefore, it is ossible for the deendence of each individual unit on the rest of the units in the system to be large. However, as we shall see below, in the case where fz it g is a CWD rocess, the model for the i th cross section unit de-coules from the rest of the system as!. Corollary Consider model (2) and suose Assumtions 4 and 5 hold. hen, a xed M, a xed t 2, and any i 2 K, If, in addition to Assumtions 4 and 5, k t k < lim V ar (z it tii z i;t u it j z M ) =. (9)! and M!, we have lim V ar (z it tii z i;t u it ) = for any i 2 K and any t 2. (2)! 3 Under these assumtions the unconditional variance of z t is bounded by kv ar (z t)k = k tk `= < su k tk t2! `Y k t sk 2 k t `k s= ( ) 2` = O. `= 4 ote that t; i t; i. See Horn and Johnson (985,. 34). An examle of vector t; i for which P lim! i=;i6=j tij 6= is when tij = k= for i 6= j and any xed non-zero constant k. 6 Working Paer Series o

18 Proof. Assumtion 5 imlies that for i 2 K, vector t; i satis es condition (2). It follows from Proosition 4 that lim V ar! t; iz t j z M = for any i 2 K and any t 2 : (2) Similarly, under the assumtion k t k < and M!, we have kv ar (z t )k = O (see Footnote 3), which imlies System (2) imlies lim V ar! t; iz t = for any i 2 K and any t 2 : (22) z it tii z i;t u it = t; iz t ; for any i 2 f; ::; g and any t 2 : (23) aking conditional variance of (23) and using (2)-(22) now yields (9)-(2). Strong deendence in in nite-dimensional VAR models could arise as a result of CSD errors fu it g, or could be due to dominant atterns in the coe cients of t, or both. An examle of the former is the residual common factor model where the weighted averages of factor loadings do not converge to zero. Further examles of CSD IVAR models, featuring also dominant unit, are rovided in Chudik and Pesaran (29). he following roosition resents su cient conditions for the VAR() rocess to be weakly deendent in the sense of Anderson et al. (29). Since the concet of weak deendence by Anderson et al. (29) is de ned only for stationary rocesses, we have to assume that t and t are time invariant. Proosition 5 Consider model (2) with time invariant coe cient matrix t =, and suose that for each t 2, u t satis es E (u t ) =, E (u t u t) = ;where is a time invariant symmetric, nonnegative de nite matrix, u t is indeendently distributed of u t for any t 6= t, and () <, so that z t is a covariance stationary rocess. hen z t is weakly deendent, in the sense of Anderson et al. (29), if () K < and kk <. Proof. he sectral density of z t is given by (i = ) F z (!) = 2 I e i! I e i! : For each 2, we have [F z (!)] = kf z (!)k ; and kf z (!)k I e i! kk I e i! : 2 Working Paer Series o 7

19 Under the assumtion that () <, I e i! = I + e i! +e 2i! 2 + :::: ow we assume kk <, and since e ij! =, it follows Similarly If, in addition, () K < we have I e i! + kk + kk 2 + ::::: = kk : I e i! k k = kk ; [F z (!)] I e i! kk I e i! 2 = 2 () I e i! I e i! 2 () kk which is bounded in since both () and kk 2 = O () ; are bounded. his comletes the roof. Remark 6 otice that under the assumtion that kk < and if, for at least one frequency!, the matrix (I e i! ) (I e i! ) is non-singular, it is ossible to show that weak deendence in the sense of Anderson et al. (29) imlies () K <. o rove this, rst notice that if A; B are two n n comlex valued matrices then 5 Alying (24)-(25) to [F z (! )], we obtain kabk kak min BB =2 ; (24) kabk kbk min AA =2 : (25) [F z (! )] = kf z (! )k = (I e i! ) (I e i! ) 2 h (I e i! ) min (I e i! ) (I e i! ) i =2 2 h(i e i! ) (I e i! ) i =2 min h(i e i! ) (I e i! ) i =2 2 kk min = 2 () min h (I e i! ) (I e i! ) i > : Given that [F z (!)] K < at all frequencies!, it must follow that () K <. 5 See Bernstein (25), age Working Paer Series o

20 4 Common factor models Consider the following in nite factor model for fz it g: z it = i f t + i2 f 2t + ::: + i f t + " it ; i = ; :::; ; (26) where the common factors, f`t, and the idiosyncratic errors, " it, satisfy the following assumtions: Assumtion 6 he vector f t is a covariance stationary rocess, with absolute summable autocovariances, distributed indeendently of " it for all i; t; t, and such that E(f 2`t ji t ) = and E(f`t f t ji t ) = ; for ` 6= = ; 2; :::; : Assumtion 7 V ar (" it ji t ) = 2 i K <, and " it, " jt are indeendently distributed for all i 6= j and for all t. he rocess z it in (26) has conditional variance V ar(z it ji t ) = V ar (u it ji t ) + V ar (" it ji t ) = 2 i` + 2 i : Finiteness of the conditional variance of z it as stated in Assumtion 3 imlies that his could arise if, for examle, `= 2 i` `= K < ; for i = ; :::; : (27) i` = O(), for ` = ; :::; m; i = ; :::; ; (28) i` = O, for ` = m + ; :::; ; i = ; :::; ; (29) where m < does not deend on. We now introduce the de nition of weak and strong factors. De nition 3 (Weak and strong factors) he factor f`t is said to be strong if he factor f`t is said to be weak if lim! lim! E j i`j = K > : (3) i= E j i`j = K < : (3) i= Working Paer Series o 9

21 In the case where the loadings attached to f`t do not satisfy either of the above conditions (3)- (3), we refer to the corresonding common factor f`t as semi-weak (or semi-strong). For examle, a factor is semi-weak when the the absolute sum of its loadings, P i= E j i`j, increases at a rate slower than. here exists a relationshi between the notions of CSD and CWD and the de nitions of weak and strong factors. his is rovided in the following theorem. heorem Consider the factor model (26), and suose that Assumtions 3-7 hold and factor loadings are non-stochastic. hen under the condition that lim! P Ǹ = j i`j = K < (for any i 2 ), the following statements hold: (i) he rocess fz it g is cross sectionally weakly deendent at a given oint in time t 2 if f`t is weak for ` = ; :::;. (ii) he rocess fz it g is cross sectionally strongly deendent at a given oint in time t 2 if and only if there exists at least one strong factor. Proof. In matrix form, the covariance of z t = (z t ; :::; z t ) is t = + " : where " is a diagonal matrix with elements 2 i. If f`t is weak for ` = ; :::; then k k is bounded in, and ( t ) + " k k + 2 max K; (32) and, from Proosition, fz it g is CWD, which roves oint (i). ow suose that fz it g is CSD. hen < lim! ( t ) lim! k k + lim! 2 max Given that, by assumtion, k k is bounded in, it follows that lim! k k = K >, and there exists at least one strong factor in (26). o rove the reverse relation, assume that there exists at least one strong factor in (26) (i.e., lim! k k = K > ). oting that6 =2 ( t ) =2 k k : (33) it follows that lim! ( t ) = K > and the rocess is CSD, which roves oint (ii). Under (3)-(3), z it can rewritten as 6 See Bernstein (25),.368, eq. xiv. z it = u it + e it ; (34) 2 Working Paer Series o

22 where u it = m i`f`t ; e it = `= `=m+ i`f`t + " it ; (35) and i` satisfy conditions (3) for ` = ; :::; m, and (3) for ` = m + ; :::;. In the light of heorem, it follows that u it is CSD and e it is CWD. Also, notice that when m =, we have a model with an in nite number of weak factors. Remark 7 Consider the following general satial rocess z t = Rv t ; (36) where R is an matrix and v t is an vector of indeendently distributed random variables. Pesaran and osetti (29) have shown that satial rocesses commonly used in the emirical literature, such as the Satial Autoregressive (SAR) rocess, or the Satial Moving Average (SMA), can be written as secial cases of (36). Seci cally, for a SMA rocess R = I + S, where is a scalar arameter (jj < K) and S is nonnegative matrix that exresses the ordering or network linkages among errors, while in the case of an invertible SAR rocess, we have R = (I S). Standard satial literature assumes that R has bounded column and row norms. It is easy to see that under these conditions the above rocess can be reresented by a factor rocess with in nite weak factors (i.e., with m = ), and no idiosyncratic error (i.e., " it = ). For examle by setting z it = i`f`t ; `= where i` = r i`, and f`t = v`t for i; ` = ; :::;. Clearly, under the bounded column and row norms of R, the loadings of the above factor structure satisfy (3) and hence carry weak cross section deendence. Remark 8 Consistent estimation of factor models with weak or semi-weak factors may be roblematic. o see this, consider the following single factor model where suose that loadings are known z it = i f t + " it ; " it IID ; 2 : he least squares estimator of f t, which is the best linear unbiased estimator, is given by P i= ^f t = iz it P ; V ar ^ft = i= 2 i 2 P : i= 2 i If for examle P i= 2 i is bounded, as in the case of weak factors, then V ar ^ft! ; for each t. does not vanish as In the literature on factor models, it is quite common to imose conditions on the loadings or on the eigenvalues of the conditional covariance matrix, ut, of u t = (u t ; :::; u t ) that constrain the Working Paer Series o 2

23 form of cross section deendence carried by the factor structure. For examle, Bai (29) imoses P that factor loadings satisfy lim! i= 2 i` >, for ` = ; :::; m. Onatski (26) and Paul (27) consider the case where the idyosyncratic errors are indeendent with a homogeneous variance, 2, and consider the `th factor as strong if P i= 2 i` > c 2, and weak if P i= 2 i` c 2 ; where c is such that c = o =2. In the literature on asset ricing models, one common assumtion is that m ( ut ) is bounded away from zero at rate (Chamberlain (983); Forni and Lii (2)). Consider now factor model (34)-(35). Since rank ( ut ) = m, and i ( ut ) >, for i = ; 2; ::; m, and i ( ut ) =, for i = m + ; m + 2; :::;, we have m ( t ) m ( ut ) ; and m+ ( t ) m+ ( ut ) + ( et ) = ( et ) : Under the assumtion that m ( ut ) is bounded away from zero at rate, and noting that, under (3), ( et ) = O(), it follows that ( t ) ; :::; m ( t ) increase without bound as!, while m+ ( t ) ; :::; ( t ) satisfy the bounded eigenvalue condition. Most factor structures yield eigenvalues that increase at rate. But as shown by Kaetanios and Marcellino (28), it is ossible to devise factor models that generate eigenvalues that rise at rate d, for < d <. Remark 9 Our concets of weak and strong cross section deendence are related to the notion of diversi ability rovided by the asset ricing theory (Chamberlain (983)). In this context, t reresents the covariance matrix of a vector of random returns on di erent assets, and w it ; for i = ; 2; :::;, denotes the roortion of investor s wealth allocated to the i th asset. From De nition 2 it follows that the art of asset returns that is weakly (or semi-weakly) deendent will be fully diversi ed by ortfolios constructed using w t as the ortfolio weights, and as!. Suose that the asset returns fr it g have the factor structure r it = i;t + if t + e it ; i = ; 2; :::; ; where i;t is the conditional mean returns, f t is an m vector of unobserved factors, i is the associated m vector of factor loadings, and fe it g is a CWD rocess distributed indeendently of f t and i. It is assumed that for each i; e it is distributed indeendently of i, whilst f t follows a general time series rocess with the conditional m m covariance matrix, t, also distributed indeendently of e it. he return on a ortfolio constructed with the granular weights w it is given by t = P i= w itr it = w t t + w t f t + w te t ; where t = ( ;t ; 2;t ; :::; ;t ), e t = (e t ; e 2t ; :::; e t ), and = ( ; 2 ; :::; ). It is easily seen that V ar ( t ji t ) = w t t w t + V ar w te t ji t ; 22 Working Paer Series o

24 and since by assumtion fe it g is a CWD rocess, then lim V ar ( t ji t ) = lim!! w t t w t : First consider the case where the factors are weak or semi-weak, and note that w t t w t w tw t t w tw t k k k t k : Since m is nite then k t k k k K, and the ortfolio is fully diversi ed for all granular weights if his condition holds if k k = O( w tw t k k!. " ) for some ositive xed ", namely if the factors are weak or semi-weak. In general, however, the ortfolio is not fully diversi able if there is at least one strong factor (see heorem ). In the resence of strong factors full diversi cation is only ossible with ortfolio weights that are deendent on the factor loadings. One such ortfolio weights is given by w = h I M ( M ) i ; where M = I ( ), and = (; ; :::; ). It is easily seen that the weights w add u to unity and are granular in the sense that 7 w w = " + M #!, as! : It is also easily seen that w =. Hence, lim! V ar (w r t ji t ) =, as required. 5 CCE estimation of anel data models with in nite factors In this section we focus on consistent estimation of a regression model where the error term has a factor structure with in nite factors. Let y it be the observation on the ith cross section unit at time t, for i = ; 2; :::; ; and t = ; 2; :::;, and suose that it is generated as y it = id t + ix it + u it ; (37) where d t = (d t ; d 2t ; :::; d nt ) is a n vector of observed common e ects, and x it is a k vector of observed individual seci c regressors. he arameter of interest is the mean of individual sloe coe cients, = E( i ). 8 7 When the factors are strong and M are O(). If some of the factors are weak the columns of associated with the weak factors can be removed when constructing the weights, w. 8 We assume that individual sloe coe cients are drawn from common distribution with mean. In the case Working Paer Series o 23

25 he error term, u it, is given by the following general factor structure, m m 2 u it = i`f`t + i`g`t + e it, (38) `= `= where we distinguish between two tyes of unobserved common factors, f t = (f t ; :::; f m t) and g t = (g t ; :::; g m2 t). he former are factors that are ossibly correlated with regressors x it, while the latter are not correlated with the regressors. De ne for future reference the vectors of factor loadings i = i ; :::; im and i = ( i ; :::; im2 ). o model the correlation between the individual seci c regressors, x it, and the innovations u it, we suose that x it can be correlated with any of the factors in f t, x it = A id t + if t + v it ; (39) where A i and i are n k and m k factor loading matrices with xed comonents, and v it is the individual comonent of x it ; assumed to be distributed indeendently of the innovations u it, and of the common factors. Equations (37) and (39) can be written more comactly as z it = y it x it! = B id t + C if t + it ; (4) where B i = D i = i A i D i, C i = i i D i,!, it = ig t + e it + iv it v it i I k! : Similar anel data models have been analyzed by Pesaran (26), Kaetanios, Pesaran, and Yagamata (29), and Pesaran and osetti (29). Pesaran (26) introduced CCE estimators in a anel model where m is xed and m 2 =, and i f t reresents a strong factor structure. Contrary to what Bai (29) (see age 23) suggests, CCE estimators are valid even in the rank de cient case where m could be larger than k +. Kaetanios, Pesaran, and Yagamata (29) extended the results of Pesaran (26) by allowing unobserved common factors to follow unit root rocesses. In both aers, innovations fe it g are assumed to be cross sectionally indeendent although ossibly serially correlated. his assumtion is relaxed by Pesaran and osetti (29) who assume that fe it g is a weakly deendent rocess, which includes satial MA or AR rocesses considered in the literature as secial cases. In this aer, we focus exlicitly on cross-correlations modelled by general factor structures - weak, strong, or where is are assumed to be non-stochastic, the object of interest would be cross section mean of i, de ned by = lim! P i= i. 24 Working Paer Series o

26 somewhere in between. Our model is thus an extension of Pesaran (26) to in nite factor structures. he secial case where both m and m 2 are xed has already been analyzed in the above cited aers. he case where f t ; :::; f m t are strong factors and m = m ()! as!, is not that meaningful as the variances of u it rise with. However, it would be ossible to let m 2, the number of the weak factors, to rise with, whilst keeing m xed. We show below that the CCE estimators continue to be consistent and asymtotically normal under this tye of in nite-factor error structures. We make the following assumtions on the common factors and their loadings: Assumtion 8 (Common factors) he (n+m ) vector (d t; ft) is a covariance stationary rocesses, with absolute summable autocovariances, distributed indeendently of g it, e it and v it for all i; t and t. 9 For each i, common factor g it follows a linear stationary rocess with absolute summable autocovariances, zero mean, unit variance, and nite fourth moments. Individual factors collected in vector g t are distributed indeendently of each other and of e it and v it for all i; t and t. Assumtion 9 (Factor loadings) Factor loadings i, assume that the following conditions hold. i, and i are non-stochastic. In addition, we (a) he unobserved factor loadings, i and i. i are bounded, i.e. k i k 2 < K and k i k 2 < K, for all (b) he unobserved factor loadings i satisfy the following absolute summability condition for each individual unit, m 2 lim j i`j < K <, (4)! `= where m 2 = m 2 () is a nondecreasing function of and the constant K does not deend on i nor on. Remark Factor structure i f t could be strong, weak or neither strong nor weak. ote that the number of strong factors cannot increase with for variance of u it to exists as!. We do not imose that P `= i`g`t is a weak factor structure. Remark Condition (4) is required for V ar ig t to exist as!. ote that the matrix of factor loadings = ( ; 2 ; :::; ) is not required to have bounded column norm as!. Remark 2 It is straightforward to extend the analysis to stochastic factor loadings distributed indeendently of the errors e it, v it and the individual coe cients i. In case where factor loadings are non-stochastic, the following rank condition rank C = m for all, (42) 9 his assumtion can be relaxed to allow for unit roots in the common factors, along the lines shown in Kaetanios, Pesaran, and Yagamata (29). Working Paer Series o 25

27 where C = P i= C i, would have to hold for the consistent inference about. Regardless of whether the rank condition (42) holds or not, it is straightforward to show, along the same lines as in Pesaran (26), that the CCE estimators continues to be valid in the case when the factor loadings i, for i = ; ::;, are stochastic and distributed indeendently from the common factors with mean. Also see Kaetanios, Pesaran, and Yagamata (29). he remaining assumtions are similar to Pesaran (26): Assumtion (Errors) he individual-seci c errors e it and v jt are distributed indeendently for all i; j; t and t, and for each i; v it follows a linear stationary rocess with absolute summable autocovariances given by v it = i` i;t `; `= where for each i, it is a k vector of serially uncorrelated random variables with mean zero, the variance matrix I k ; and nite fourth-order cumulants. For each i, the coe cient matrices i` satisfy the condition where vi V ar(v it ) = i` i` = v i ; `= is a ositive de nite matrix, such that su i k vi k 2 < K: Errors e it ; for i = ; ::;, follow a linear stationary rocess with absolute summable autocovariances, " it = where is IID (; ) with nite fourth moments. a is i;t `, `= Assumtion (Random coe cients) he sloe coe cients follow the random coe cient model i = + i, i IID (; ), for i = ; ::;, where kk 2 < K, k k 2 < K, is symmetric non-negative de nite matrix, and the random deviations i are distributed indeendently of x jt, d t and u jt for all i; j and t. Assumtion 2 Consider the cross section averages of the individual seci c variables z it = (y it ; x it ), de ned by z t = P i= z it and let M = I H H H H, H = D; Z, where D and Z are, resectively, the matrices of observations on d t and z t. hen the following conditions hold: (a) he matrix lim! P i= v i is nite and nonsingular. (b) here exists and such that for all and, the k k matrices i M i and exist for all i, where Mg = I G (G G) G, with G = (D; F), F and i M g i 26 Working Paer Series o