In nite-dimensional VARs and Factor Models

Transcription

1 In nite-dimensional VARs and Factor Models Alexander Chudik y Euroean Central Bank and CIMF M. Hashem Pesaran z Cambridge University, CIMF and USC January 22, 2 Abstract his aer rooses a novel aroach for dealing with the curse of dimensionality in the case of vector autoregressive (VAR) models with a large number of variables or units (). It is assumed that each unit is related to a small number of neighbors and a large number of non-neighbors. he neighbors could be individual units or, more generally, linear combinations of units. he neighborhood e ects are xed and do not change with, but the coe cients corresonding to the non-neighboring units are restricted to vanish in the limit as tends to in nity. he conditions under which such an in nite-dimensional VAR (or IVAR) can be arbitrarily well characterized by a large number of nite-dimensional models are derived. Problems of estimation and inference in a stationary IVAR model with an unknown number of unobserved common factors are also investigated. A cross section augmented least squares (CALS) estimator is roosed and its asymtotic distribution is derived. Satisfactory small samle roerties for the CALS estimator are documented by Monte Carlo exeriments. An emirical illustration shows the statistical signi cance of dynamic sill-over e ects in modelling of U.S. real house rices across the neighboring States. Keywords: Large and Panels, Weak and Strong Cross Section Deendence, VARs, Satial Models, Factor Models. JEL Classi cation: C, C33, C5 We are grateful to Elisa osetti for helful comments and suggestions. his version has also bene ted from comments by the seminar articiants at the University of California, San Diego, University of Southern California, Columbia, Leicester, Euroean University Institute, McGill, Princeton, Pennsylvania, Stanford; as well as by the conference articiants in the ordic Econometric Meeting held at Lund University, and in the conference on the Factor Structures for Panels and Multivariate ime Series Data held at Maastricht University. We would also like to acknowledge akashi Yamagata for carrying out the comutation of the results reorted in Section 6. y Euroean Central Bank, Kaiserstrasse 29, 63 Frankfurt am Main, Germany. alexander.chudik@ecb.int. z Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, UK. mh@econ.cam.ac.uk; htt://

2 Introduction Vector autoregressive models (VARs) rovide a exible framework for the analysis of comlex dynamics and interactions that exist between economic variables. he traditional VAR modelling strategy ostulates that the number of variables,, is xed and the time dimension,, tends to in nity. But since the number of arameters to be estimated grows at a quadratic rate with, the alication of the aroach in ractice is often limited to a handful of variables. he objective of this aer is to consider VAR models where both and are large. In this case, arameters of the VAR model can no longer be consistently estimated unless suitable restrictions are imosed to overcome the dimensionality roblem. wo di erent aroaches have been suggested in the literature to deal with this curse of dimensionality : (i) shrinkage of the arameter sace, and (ii) shrinkage of the data. Satial and/or satiotemoral literature shrinks the arameter sace by using a riori given satial weights matrices that restricts the nature of the links across the units. Alternatively, rior robability distributions are imosed on the arameters of the VAR such as the Minnesota riors roosed by Doan, Litterman, and Sims (984). his class of models is known as Bayesian VARs (BVAR). he second aroach is to shrink the data, along the lines of index models. Geweke (977) and Sargent and Sims (977) introduced dynamic factor models, which have more recently been generalized to allow for weak cross sectional deendence by Forni and Lii (2), Forni et al. (2) and Forni et al. (24). Emirical evidence suggests that few dynamic factors are needed to exlain the co-movements of macroeconomic variables. 2 his has led to the develoment of factor-augmented VAR (FAVAR) models by Bernanke, Bovian, and Eliasz (25) and Stock and Watson (25), among others. Alied researchers are often forced to imose arbitrary restrictions on the coe cients that link the variables of a given cross section unit to the current and lagged values of the remaining units, mostly because they realize that without such restrictions the model is not estimable. his aer Other tyes of riors have also been considered in the literature. See, for examle, Del egro and Schorfheide (24) for a recent reference. In most alications, BVARs have been alied to relatively small systems (e.g. Leeer, Sims, and Zha (996) considered 3- and 8-variable BVAR; a few excetions include Giacomini and White (26) and De Mol, Giannone, and Reichlin (28)), with the focus being mainly on forecasting. Bayesian VARs are known to roduce better forecasts than unrestricted VARs or structural models. See Litterman (986) and Canova (995) for further references. 2 Stock and Watson (999), Stock and Watson (22), Giannone, Reichlin, and Sala (25) conclude that only few, erhas two, factors exlain much of the redictable variations, while Bai and g (27) estimate four factors and Stock and Watson (25) estimate as much as seven factors.

3 rooses a novel way to deal with the curse of dimensionality by shrinking art of the arameter sace in the limit as the number of variables () tends to in nity. An imortant examle would be a VAR model where each unit is related to a small number of neighbors and a large number of nonneighbors. he neighbors could be individual units or, more generally, linear combinations of units (satial averages). he neighborhood e ects are xed and do not change with, but the coe cients corresonding to the remaining non-neighbor units are small, of order O. Such neighborhood and non-neighborhood e ects could be motivated by theoretical economic considerations, or could arise due to the mis-seci cation of satial weights. Although under this set-u each of the non-neighboring coe cients is small, sum of their absolute values in general does not tend to zero and the aggregate satiotemoral non-neighborhood e ects could be large. his aer shows that under weak cross section deendence, the sillover e ects from non-neighboring units are neither articularly imortant, nor estimable. 3 But the coe cients associated with the neighboring units can be consistently estimated by simly ignoring the non-neighborhood e ects that are of second order imortance in. On the other hand, if the units are cross sectionally strongly deendent, then the sillover e ects from non-neighbors are in general imortant, and ignoring such e ects can lead to inconsistent estimates. Another model of interest arises when in addition to the neighborhood e ects, there is also a xed number of dominant units that have non-negligible e ects on all other units. In this case the limiting outcome is shown to be a dynamic factor model. 4 Accordingly, the aer rovides a link between data and arameter shrinkage aroaches to mitigating the curse of dimensionality. By imosing limiting restrictions on some of the arameters of the VAR we e ectively end u with a data shrinkage. o distinguish high dimensional VAR models from the standard seci cations we refer to the former as the in nite dimensional VARs or IVARs for short. he aer also establishes the conditions under which the Global VAR (GVAR) aroach roosed by Pesaran, Schuermann, and Weiner (24) is alicable. 5 In articular, the IVAR featuring all macroeconomic variables could be arbitrarily well aroximated by a set of nite- 3 Concets of strong and weak cross section deendence, introduced in Chudik, Pesaran, and osetti (29), will be alied to VAR models. 4 he case of IVAR models with a dominant unit is studied in Pesaran and Chudik (2). 5 GVAR model has been used to analyse credit risk in Pesaran, Schuermann, reutler, and Weiner (26) and Pesaran, Schuermann, and reutler (27). Extended and udated version of the GVAR by Dées, di Mauro, Pesaran, and Smith (27), which treats Euro area as a single economic area, was used by Pesaran, Smith, and Smith (27) to evaluate UK entry into the Euro. Global dominance of the US economy in a GVAR model is considered in Chudik (28). Further develoments of a global modelling aroach are rovided in Pesaran and Smith (26). Garratt, Lee, Pesaran, and Shin (26) rovide a textbook treatment of GVAR. 2

4 dimensional small-scale models that can be consistently estimated searately in the sirit of the GVAR. A second contribution of the aer is the develoment of aroriate econometric techniques for estimation and inference in stationary IVAR models with an unknown number of unobserved common factors. his extends the analysis of Pesaran (26) to dynamic models where all variables are determined endogenously. A simle cross sectional augmented least-squares estimator (or CALS for short) is roosed and its asymtotic distribution derived. Small samle roerties of the roosed estimator are investigated through Monte Carlo exeriments. As an illustration of the roosed aroach we consider an extension of the emirical analysis of real house rices across the 49 U.S. States conducted recently by Holly, Pesaran, and Yamagata (29), and show statistically signi cant dynamic sillover e ects of real house rices across the neighboring States. he remainder of the aer is organized as follows. Section 2 introduces the IVAR model. Section 3 investigates cross section deendence in IVAR models. Section 4 focusses on estimation of a stationary IVAR model. Section 5 discusses the results of the Monte Carlo exeriments, and Section 6 resents the emirical results. he nal section o ers some concluding remarks. Proofs are rovided in the Aendix. A brief word on notations: j (A)j j 2 (A)j ::: j n (A)j are the eigenvalues of A 2 M nn, where M nn P is the sace of real-valued n n matrices. kak max n jn i= ja ijj denotes the P maximum absolute column sum matrix norm of A, kak max n in j= ja ijj is the absolute row sum matrix norm of A. kak = % (A A) is the sectral norm of A; % (A) max fj i (A)jg is in the sectral radius of A. 6 All vectors are column vectors and the i th row of A is denoted by a i. a n = O(b n ) denotes the deterministic sequence fa n g is at most of order b n. x n = O (y n ) states that the random variable x n is at most of order y n in robability. is the set of natural numbers, and Z is the set of integers. We use K and to denote ositive xed constants that do not vary with or. Convergence in distribution and convergence in robability is denoted by d! and!, resectively. Symbol q:m:! reresents convergence in quadratic mean. (; )! j denotes joint asymtotic in and ; with and!, in no articular order. 6 ote that if x is a vector, then kxk = % (x x) = x x corresonds to the Euclidean length of vector x. 3

5 2 In nite-dimensional Vector Autoregressive Models Suose we have time series observations on cross section units indexed by i 2 S () f; ::; g. Individual units could be households, rms, regions, or countries. Both dimensions, and, are assumed to be large. For each oint in time, t, and for each 2, the cross section, observations are collected in the vector x ();t = x ();t ; :::; x ();t and it is assumed that x ();t follows the VAR() model x ();t = () x ();t + u ();t, () u ();t = R () " ();t. (2) () and R () are coe cient matrices that cature the dynamic and contemoraneous deendencies across the units, and " ();t = (" t ; " 2t ; :::; " t ) is an vector of white noise errors with mean and the covariance matrix, I. VAR models have been extensively studied when is small and xed, and is large and unbounded. his framework, however, is not aroriate for many emirical alications of interest. his aer aims to ll this ga by analyzing VAR models where both and are large. he sequence of models () and (2) with dim(x ();t ) =! will be referred to as the in nitedimensional VAR model, or IVAR for short. he extension of the IVAR() to IVAR() where is xed, is relatively straightforward and will not be attemted in this aer. he analysis of deendence over time is simli ed by the fact that ordering of observations along the time dimension (t = ; 2; :::; ) is immutable and the arrival of new observations cannot change ast realizations, namely bygones are bygones. As a consequence for any given, i, and j, the cross time covariance function, cov x ();it ; x ();j;t ` ; does not change with and will deend only on ` if the time series rocesses are covariance stationary. However, since it can not be assumed that an immutable ordering necessarily exists with resect to the cross section dimension, addition of new cross section units to an existing set can otentially alter the air-wise cross section covariances of all the units. For instance in models of oligooly, where rms strategically interact with each other, new entries can change the relationshi between the existing rms. Similarly, introduction of a new asset in the market can change the correlation of returns on the existing assets. In what follows, to simlify the notations, the exlicit deendence of x t and u t and the related 4

6 arameter matrices on will be suressed with ()-(2) written as x t = x t + u t ; (3) and u t = R" t : (4) Clearly, it is not ossible to estimate all the 2 elements of the matrix when both and are large. Only a small ( xed) number of unknown coe cients can be estimated er equation and some restrictions on must be imosed. In order to deal with the dimensionality roblem, we assume that for a given i 2, it is ossible to classify cross section units a riori into neighbors and non-neighbors. o restrictions are imosed on neighbors, but the non-neighbors are assumed to have only negligible e ects on x it that vanish at a suitable rate with. he number of neighbors of unit i, collected in the index set i, is assumed to be small ( xed). eighbors of unit i can have non-negligible e ects that do not vanish even if!. A similar classi cation is followed in the satial econometrics literature, where the non-neighborhood e ects are set to zero for all and the non-zero neighborhood e ects are often assumed to be homogenous across i. In this sense our analysis can also be seen as an extension of satial econometric models. Subject to the above classi cation, equation for the unit i can be written as x it = X j2 i ij x j;t {z } eighbors he coe cients of the neighboring units, ij vary with. he remaining coe cients, ij + X j2 c i ij x j;t + u it. (5) {z } on neighbors j2 i, are the arameters of interest and do not j2 c i, tend to zero for each i as!, where c i f; ::; g n i is the index set of non-neighbors. ote that the non-neighbors are unordered. More seci cally, ij K for any 2 and any j 2 c i. (6) Individually the coe cients of non-neighbors are asymtotically negligible, but as we argue below it is not clear if the same alies to their aggregate e ects on the i th unit, namely P j2 c i ijx j;t. 5

7 P he bounds in (6) ensures that lim! j= ij < K. We refer to this as the cross section absolute summability condition, which is distinct from the absolute summability condition used in the time series literature where the same idea is alied to the coe cients of current and ast innovations. A similar constraint is used in Lasso and Ridge regression shrinkage methods. he Lasso estimation rocedure alied to (3) involves minimizing P t= u2 it for each i subject to ij K. Under the Ridge regression the minimization is carried out subject to the weaker P j= constraint, P j= 2 ij K. 7 In alication of shrinkage methods it is necessary that K is seci ed a riori, but no knowledge of the ordering of the units along the cross section dimension is needed. In our aroach we do not need to secify the value of K. Sum of the coe cients of the non-neighboring units, P j2 c i ij, does not necessarily tends to zero as!, which imlies that the non-neighbors can have a large aggregate satiotemoral imact on the unit i, as!. he question that we address is whether it is ossible to estimate neighborhood coe cients ij j2 i without imosing further restrictions. As it turns out, the answer deends on the stochastic behavior of P j2 c i ijx j;t, which in turn deends on the strength of cross section deendence in fx it g. If fx it g is weakly cross sectionally deendent then P j2 c i ijx j;t q:m:!, and the sillover e ects from non-neighboring units are neither articularly imortant nor estimable. But the coe cients associated with the neighboring units can be consistently estimated by simly ignoring the non-neighborhood e ects that are of second order imortance in. If on the other hand fx it g is strongly cross sectionally deendent, then P lim! V ar j2i c ijx j;t is not necessarily zero, and the sillover e ects from non-neighbors are in general O () and imortant. herefore, ignoring the non-neighborhood e ects can lead to inconsistent estimates. he concets of weak and strong cross section deendence have been introduced in Chudik, Pesaran, and osetti (29) and these concets are alied to the IVAR model in the next section. Our aroach to dealing with the curse of dimensionality can be motivated with several examles. In economic alications interactions across agents often deends on the number of agents, with the degree of air-wise interactions tyically declining in the number of units. Consider, for examle, the outut and ricing decisions of rms in an industry with rms. When is small (cases of duooly or oligooly) ricing and outut decisions are inter-related through the way rms 7 See Section of Hastie, ibshirani, and Friedman (2) for detailed descrition of the Lasso and Ridge regression shrinkage methods. 6

8 form exectations about the reactions of other rms, known as conjectural variations. But as becomes large such conjectural variations become relatively unimortant and in the cometitive case where is su ciently large conjectural variations are tyically set to zero. Another imortant examle is rovided by the Arbitrage Pricing heory (AP) originally develoed by Ross (976). Under aroximate ricing the conditional mean returns of risky assets, t, is modelled in terms of a xed number (k) of factor risk remia, t, and an vector of ricing errors, v t, namely t = B t + v t ; where B is an k matrix of factor loadings. In the absence of arbitrage oortunities we must have v t = when is xed, or vtv t = O () as!. (see Hubermann (982) and Ingersoll (984)). It is clear that any air-wise deendence of ricing errors must vanish as!, otherwise there will be unbounded ro table oortunities. he third examle relates to a multi-country DSGE model discussed in Chudik (28). he country interactions need not be symmetric. evertheless, as long as foreign trade weights are granular, the equilibrium solution of such a multi-country DSGE model has a similar structure to the basic IVAR model set out in the aer. eighbors in this setu could be identi ed in terms of the trade shares, for examle. For instance, US would be Canada s neighbor considering that 8% of Canada s trade is with the US, although using the same metric Canada might not qualify as a neighbor of the US. In some cases the strict division of individual units into neighbors and non-neighbors might be considered as too restrictive. In the assumtion below we consider a slightly more general set u where the neighborhood e ects are charachterized in terms of local averages de ned by S i x t, where S i is a known satial or neighborhood weight matrix. ASSUMPIO Let K be a non-emty index set. For any i 2 K, the row i of coe cient matrix, denoted by i, can be divided as i = ai + bi ; (7) where k bi k = max j2f;::;g bij K <, (8) 7

9 ai = S i i, (9) k i k < K, i is an h i dimensional vector containing the unknown coe cients to be estimated for unit i, which do not change with, h i < K, h i is xed and generally small, and S i is a known h i satial weight matrix such that ks i k < K. Assuming K and stacking (7)-(9) for i = ; 2; :::;, we have = a + b, = D S + b, () where a = ( a ; a2 ; :::; a ), b = ( b ; b2 ; :::; b ), D = h 2...., () C A h = P i= h i, and S is a known h matrix de ned by S = (S ; S 2 ; :::; S ). ote also that by assumtion the individual elements of b are uniformly O. Examle An examle of a is given by a = 2 : : : : : : : : : : : : ; 2 ; ; : : : ;, (2) C A where the nonzero elements are xed coe cients that do not change with. his reresents a bilateral satial reresentation where each unit, excet for the rst and the last units, has one left and one right neighbor. In contrast the individual elements of b are of order O( ), in articular bij < K for any 2 and any i; j 2 f; ::; g. he equation for unit i 2 f2; ::; g can be 8

10 written as x it = i;i x i ;t + ii x i;t + i;i+ x i+;t + bi x t + u it. (3) Section 3 shows that under weak cross section deendence of errors fu it g, bi x t q:m:!, while Section 4 considers roblem of estimation of the individual-seci c arameters i;i ; ii ; i;i+. We refer to this model as a two-neighbor IVAR model which we use later for illustrative uroses as well as in the Monte Carlo exeriments. Examle 2 As a simle examle consider the model x t = x S x x t + u t, (4) u t = u S u u t + " t, (5) where x and u are scalar unknown coe cients, and S x and S u are known satial weights matrices. his model can be obtained from ()-(2) by setting R = (I u S u ), i = x for i 2 f; ::; g, S = S x, and b =. 3 Cross Sectional Deendence in Stationary IVAR Models his section investigates the correlation attern of fx it g, over time, t, and along the cross section units, i. We follow Chudik, Pesaran, and osetti (29) and de ne covariance stationary rocess fx it g to be cross sectionally weakly deendent (CWD), if for all weight vectors, w = (w ; :::; w ), satisfying the granularity conditions kwk = O w j kwk = O 2 2, (6) for any j, (7) we have lim V ar! w x t =, for all t: 9

11 fx it g is said to be cross sectionally strongly deendent (CSD) if there exists a sequence of weight vectors, w; satisfying (6)-(7) and a constant K such that lim V ar! w x t K >. (8) ecessary condition for covariance stationarity for xed is that all eigenvalues of lie inside of the unit circle. For a xed, and assuming that max i j i ()j <, the Euclidean norm of ` de ned by r `` =2! exonentially in `; and the rocess xt = P `= `u t ` will be absolute summable, in the sense that the sum of absolute values of the elements of `, for ` = ; ; :::, converges. Observe that as!, V ar (x it ) need not necessarily be bounded in even if max i j i ()j < (and bounded away from ). For examle, consider the IVAR() model with = ' ' ' ', C A and assume that var (u it ) is uniformly bounded away from zero as!. It is clear that all the eigenvalues of are inside the unit circle if and only if j'j <, regardless of the value of the neighborhood coe cient,. Yet the variance of x t increases in without bounds at an exonential rate for j j > j'j. 8 herefore, a stronger condition than stationarity for each is required to revent the variance of x it from exloding as!. his is set out in the following assumtions. ASSUMPIO 2 he elements of the double index rocess f" it ; i 2 ; t 2 Zg are indeendently distributed random variables with zero means and unit variances on the robability sace (; F; P ). ASSUMPIO 3 (CWD Errors) Matrix R has bounded row and column matrix norms. 8 It can be shown that V ar fx tg = X j= 2( j) where k` = kq 2 (` + k j) for k > and (k )! ` =. j= X `= 2 j+;`' 2`,

12 ASSUMPIO 4 (Stationarity and bounded variances) here exists a real, in the range < <, such that 9 kk. (9) Remark Assumtions 2 and 3 imly fu it g is CWD, since for any weight vector, w, satisfying (6) we have V ar (w u t ) kwk 2 krk krk! as!. For future reference de ne covariance matrix = V ar (u t ) = RR and denote the i th diagonal element of by 2 ii = V ar (u it). ote also that kk krk krk < K, which as shown in Pesaran and osetti (29) includes all commonly used rocesses in the satial literature, such as satial autoregressive and satial error comonent models ioneered by Whittle (954), and further develoed by Cli and Ord (973), Anselin (988), and Kelejian and Robinson (995). Remark 2 It is not necessary that roximity is measured in terms of hysical sace. Other measures such as economic (Conley (999), Pesaran, Schuermann, and Weiner (24)), or social distance (Conley and oa (22)) could also be emloyed. All these are examles of deendence across nodes in a hysical (real) or logical (virtual) networks. In the case of the IVAR model, de- ned by (3) and (4), such contemoraneous deendence can be modelled through the network toology matrix R., Remark 3 he IVAR model when combined with u t = R" t yields an in nite-dimensional satiotemoral model. he model can also be viewed more generally as a dynamic network, with R and caturing the static and dynamic forms of inter-connections that might exist in the network. Remark 4 (Eigenvalues of ) Assumtion 4 imlies olynomial (L) is invertible (for any 2 ) and % (), (2) which is a su cient condition for covariance stationarity. Assumtion 4 also delivers a bounded variance for x it, as!. 9 Our assumtions concerning coe cient matrix can be relaxed so long as they hold for all (where is a xed constant that does not deend on ). But in order to kee notations and exosition simle, we simly state that Assumtions and 4 hold for any value of. A network toograhy is usually reresented by grahs whose nodes are identi ed with the cross section units, with the airwise relations catured by the arcs in the grah. It is also ossible to allow for time variations in the network matrix, R, to cature changes in the network structure over time. However, this will not be ursued here.

13 Proosition Consider model () and suose that Assumtions 2-4 hold. hen for any arbitrary sequence of xed weights w satisfying condition (6), and for any t 2 Z, lim V ar (x wt) =. (2)! Assumtions 2-4 are thus su cient conditions for weak deendence. Proosition has several interesting imlications. Suose that we can imose limiting restrictions given by Assumtion. Corollary Consider model () and suose Assumtions -4 hold. hen, for any i 2 K, lim V ar x it aix t u it =. (22)! Remark 5 It is also ossible to establish (22) under the following conditions: k bi k = O 2 ; (23) kk = O ; (24) which are less restrictive than condition (8) and the Assumtion 3 on the boundedness of the column and row norms of matrix R. hese stronger conditions are needed for establishing the asymtotic roerties of the CALS estimator to be roosed below in Section IVAR Models with Strong Cross Sectional Deendence he IVAR model can generate observations with strong cross section deendence if the boundedness assumtion on the column and row norms of R and are relaxed. he analysis of this case is beyond the scoe of the resent aer and is considered in Pesaran and Chudik (2). But even if the boundedness assumtions on R and are maintained, it is still ossible for x it to show strong cross section deendence if the IVAR model is augmented with common factors. he basic IVAR model, (3), can be augmented with exogenously seci ed common factors in a number of di erent ways. Here we consider two imortant ossibilities. First, a nite number of common factors can be added to the vector of the error terms, de ned by (4). his is equivalent to assuming that a nite number of the columns (or linear combinations of the columns) of R have unbounded norms. his comounding of the satial (weak) cross section deendence with the strong factor deendence 2

14 comlicates the analysis unduly and will not be ursued here. A more attractive alternative would be to assume that (L) (x t f t ) = u t, for t = ; 2; :::; ; (25) where (L) = I L, = ( ; :::; ) is an vector of xed e ects, f t is an m vector of unobserved common factors (m is xed but otherwise unknown), = ( ; 2 ; :::; ) is the m matrix of factor loadings, and as before u t = R" t. Under this seci cation the strong cross section deendence of x it due to the factors is exlicitly searated from other sources of cross deendence as embodied in and R. 4 Estimation of a Factor Augmented Stationary IVAR Model We now consider the roblem of estimation and inference in the case of the factor augmented IVAR model as set out in (25), as both and tend to in nity. We focus on arameters of the i th equation and assume that i (the i th row of matrix ) can be decomosed as in Assumtion. See (7)-(9). As an imortant examle we consider the two-neighbor IVAR model de ned in Examle, where the arameters of interest is given by the elements of the i th row of matrix a given by (2). In what follows we set it = S i x t; where S i is de ned by (9), and note that it reduces to (x i ;t ; x it ; x i+;t ) in the case of the two-neighbor IVAR model. We suose that the following assumtions hold. ASSUMPIO 5 (Available observations) Available observations are x ; x ; :::; x with the starting values x = P `= `R" ` + + f. ASSUMPIO 6 (Common factors) he unobserved common factors, f t ; f 2t ; :::; f mt ; are covariance stationary and follow the general linear rocesses: f st = s (L) " fst ; for s = ; 2; :::; m, (26) where s (L) = P `= s`l` with absolute summable coe cients that do not vary with, the factor innovations, " fst, are indeendently distributed over time with zero means and a constant variance, 2 " fs ; that do not vary with. " fst s are also distributed indeendently of the idiosyncratic errors, " it, for any i 2, any t; t 2, and any s 2 f; ::; mg. E (f t f t) exists and is a ositive de nite 3

15 matrix. ASSUMPIO 7 (Existence of fourth order moments) here exists a ositive real constant K such that E " 4 fst < K and E " 4 it < K for any s 2 f; ::; mg, any t 2 and any i 2. ASSUMPIO 8 (Bounded factor loadings and xed e ects) For any i 2, i and i do not change with, k i k < K; and j i j < K. We follow Pesaran (26) and introduce the following vector of cross section averages x W t = W x t, where W = (w ; w 2 ; :::; w ) and fw j g j= are m w dimensional vectors. Subscrits denoting the number of grous are again omitted where not necessary, in order to kee the notations simle. Matrix W does not corresond to any satial weights matrix. It is any arbitrary matrix of re-determined weights satisfying the following granularity conditions kwk = O kw j k kwk = O 2 2, (27) for any j. (28) Multilying (25) by the inverse of olynomial (L) and then by W yields x W t = W + W f t + W t, (29) where W = W, W = W, W t = W t, and X t = `u t `. (3) Under Assumtions 2-3, fu t g is weakly cross sectionally deendent and kv ar ( W t )k = `= X W ` `W, `= kwk 2 kk X ` 2, `= = O, (3) where kwk 2 = O by condition (27), kk = O () by Assumtion 3 (see Remark ) and P ` `= P kk` `= = O () under Assumtion 4. his imlies W t = O 2 and the 4

16 unobserved common factors can be aroximated as W W W (x W t W ) = f t + O 2, (32) rovided that the matrix W W is non-singular. It can be inferred that the full column rank of W is imortant for the estimation of unit-seci c coe cients. Pesaran (26) shows that the full column rank condition is not, however, necessary if the object of the interest is the cross section mean of the arameters, E( i ), as oosed to the unit-seci c arameters, i ; which are the focus of the current aer. Using (25), the equation for unit i 2 K can be written as: x it i if t = is i (x t f t ) + i;t + u it, (33) where it = ib t = O 2, (34) since by Assumtion ib satis es condition (27). It follows from (29) that if t ia f t = b ix W t + b i2x W;t (b i + b i2 ) W b i W t b i2 W;t, (35) where b i = i W W yields W and b i2 = is i W W W. Substituting (35) into (33) x it = is ix t + b ix W t + b i2x W;t + c i + u it + q it, (36) where c i = i ia (b i + b i2 ) W, and q it = i;t b i W t b i2 W;t = O 2. (37) Consider now the following auxiliary regression based on (36): x it = g it i + it, (38) where it = u it + q it, i = i; b i ; b i2 ; c i is the k i vector of coe cients associated with the 5

17 , regressors g it = i;t ; x W t ; x W;t ; and ki = h i + 2m w +. he arameters of interest, i, can now be estimated using the cross section augmented regression de ned by (38). We refer to such an estimator of i as the cross section augmented least squares estimator (or CALS for short), and denote it by b i;cals. We have b i = b i;cals ^b i ^b i2 = C A! X X g it git g it x it. (39) t= t= ^c i Also using artitioned regression formula, b i;cals = Z im H Z i Z i M H x i, (4) where M H = I H H H + H ; H = X W ; X W ( ) ; ; (4) Z i = i ( ) ; i2 ( ) ; :::; ihi ( ) ; (42) ir ( ) = ir ; :::; i;r;, for r 2 f; ::; hi g ; is a vector of ones, X W = (x W ; :::; x W mw), X W ( ) = [x W ( ) ; :::; x W mw ( )], x W s = (x W s ; :::; x W s ), x W s ( ) = (x W s ; :::; x W s; ) ; for s 2 f; ::; m w g, and x i = (x i ; :::; x i ), For future reference also let v it = S i t = it S i f t S i ; Q = [F; F ( ) ; ] ; (43) and A = (2m+)(2m w+) W W W mmw C A, (44) mmw W where F = (f ; :::; f m ), F ( ) = [f ( ) ; :::; f m ( )], f r = (f r ; :::; f r ) and f r ( ) = (f r ; :::; f r; ) for r 2 f; ::; mg. 6

18 First we consider the asymtotic roerties of b i (and b i;cals ) as (; ) j! ; in the case where the number of unobserved common factors equals to the dimension of x W t (m = m w ), and make the following additional assumtion. ASSUMPIO 9 (Identi cation of i ) here exists and such that for all ; and for any i 2 K, P t= g itgit exists, C();i = E (g it git ) is ositive de nite, and C < K. ();i Remark 6 Assumtion 9 imlies W is a square, full rank matrix and, therefore, the number of unobserved common factors is equal the number of columns of the weight matrix, W (m = m w ). In cases where m < m w, full augmentation of individual models by (cross sectional) averages is not necessary. heorem Let x t be generated by model (25), Assumtions -9 hold, and W is any arbitrary (re-determined) matrix of weights satisfying conditions (27)-(28), and Assumtion 9. hen for any i 2 K and as (; ) j!, b i de ned in equation (39) has the following roerties. a) b i i! : b) If in addition =! {, with { <, ();ii C 2 ();i (b i i ) D! (; I ki ), (45) where 2 ();ii = V ar (u it) = E (e i RR e i ), and C 2 ();i is the square root of the ositive de nite matrix C ();i = E (g it git ). Also c) C ();i b C();i!, and ();ii b ();ii! ; where bc ();i = X g it git; b 2 ();ii = t= X bu 2 it, (46) t= and bu it = x it g it b i. 7

19 Remark 7 Suose that in addition to the assumtions of heorem, the limits of C ();i and 2 ();ii, as!, exist and are given by C();i, and 2 ();ii, resectively.2 hen (45) yields (bi i )! D ; 2 ();ii C ();i. (47) Consider now the case where the number of unobserved common factors is unknown, but it is known that m w m. Since the auxiliary regression (38) is augmented ossibly by a larger number of cross section averages than the number of unobserved common factors, we have otential roblem of multicollinearity (as! ). But this does not a ect the estimation of i so long as the sace sanned by the unobserved common factors including a constant and the sace sanned by the vector (; x W t ) are the same as!. his is the case when W has full column rank. For this more general case we relace Assumtion 9 with the following, and suress the subscrit to simlify the notations. ASSUMPIO (Identi cation of i ) here exists and such that for all ; and for any i 2 K, Z i M HZ i exists, W is a full column rank matrix, vi = E (v it vit ) = P `= S i `RR `S i is ositive de nite, and = O (). vi heorem 2 Let x t be generated by model (25), Assumtions -8, and hold, and W is any arbitrary (re-determined) matrix of weights satisfying conditions (27)-(28) and Assumtion. hen for any i 2 K, and if in addition (; )! j such that =! {, with { <, the asymtotic distribution of b i;cals de ned by (4) is given by. ii 2 vi bi;cals i D! (; Ihi ), (48) where 2 ii = V ar (u it), vi = E (v it v it ) and v it = S i t = P `= S i `u t `. Remark 8 As before, we also have bi;cals D! i ; 2 ();ii v();i ; where v();i = lim! vi, and 2 ();ii = lim! 2 ii, assuming limits exist. 2 Su cient condition for lim! C ();i to exist is the existence of the following limits (together with Assumtions -8): lim! S i, lim! S i, lim! W, lim! W, and lim! P `= S i`rr `S i. 8

20 5 Monte Carlo Exeriments: Small Samle Proerties of CALS Estimator 5. Monte Carlo Design In this section we reort some evidence on the small samle roerties of the CALS estimator in the resence of unobserved common factors and weak error cross section deendence and comare the results with standard least squares estimators. Objectives of the exeriments are twofold. First, we would like to investigate how well the CALS estimator erforms in the resence of unobserved common factors. Second, we would like to nd out the extent to which cross section augmentation a ects the small samle roerties of the estimator when the cross section deendence is weak, and therefore cross section augmentation is asymtotically unnecessary. he focus of our analysis will be on the estimation of the individual-seci c arameters in an IVAR model that also allows for other inter-deendencies that are of order O( ). he data generating rocess (DGP) used is given by x t f t = (x t f t ) + u t, (49) where f t is the only unobserved common factor considered (m = ), and = ( ; :::; ) is the vector of factor loadings. We consider two sets of factor loadings to distinguish the case of weak and strong cross section deendence. Under the former we set = ; and under the latter we generate the factor loadings i ; for i = ; 2; :::;, from a stationary satial rocess in order to show that our estimators are invariant to ossible cross section deendence in the factor loadings. Accordingly, the factor loadings are generated by the following bilateral Satial Autoregressive Model (SAR) rocess i = a 2 i + i+ a + i, < a < ; (5) where i IID ; 2. As established by Whittle (954), the unilateral SAR(2) scheme i = i + 2 i 2 + i, (5) 9

21 with = 2b ; 2 = b 2 and b = q a 2 =a, generates the same autocorrelations as the bilateral SAR() scheme (5). he factor loadings are generated using the unilateral scheme (5) with 5 burn-in data oints (i = 49; :::; ) and the initialization, 5 = 5 =. We set a = :4, =, and choose 2 such that V ar ( i ) =. 3 he common factors are generated according to the AR() rocess f t = f f t + ft, ft IID ; 2 f, with f = :9. In line with the theoretical analysis, the autoregressive arameters are decomosed as = a + b, where a catures own and neighborhood e ects as in a = ' 2 ' ' ' ' ; C A and the remaining elements of, de ned by b, are generated as 8 >< i! ij for j =2 fi ; i; i + g bij = >: for j 2 fi ; i; i + g, where i IIDU ( :; :2) and! ij = & ij P j= &, (52) ij with & ij IIDU (; ). his ensures that bij = O ( ), and lim! E bij =, for all i and j. With a as seci ed above, each unit i, excet the rst and the last, has two neighbors: the 3 he variance of factor loadings is given by 2 =

22 left neighbor i and the right neighbor i +. he DGP for the i th unit can now be written as x t = ' x ;t + x 2;t + b x t + f t f t + u t, x it = ' i x i;t + i (x i ;t + x i+;t ) + bi x t + i f t i f t + u it ; i 2 f2; ::; g, x t = ' x ;t + x ;t + b; x t + f t f t + u t. o ensure that the DGP is stationary we generate ' i IIDU (:4; :6), and i IIDU ( :; :) for i 6= 2. We choose to focus on the equation for unit i = 2 in all exeriments and we set ' 2 = :5 and 2 = :. his yields kk :9, and together with f < it is ensured that the DGP is stationary and the variance of x it is bounded in. he cross section averages, x wt ; are constructed as simle averages, x t = P j= x it. he -dimensional vector of error terms, u t ; is generated using the following SAR model: u t = a u u 2t + " t, u it = a u 2 (u i ;t + u i+;t ) + " it, i 2 f2; ::; g u t = a u u ;t + " t, for t = ; 2; ::;. We set a u = :4 which ensures that the errors are cross sectionally weakly deendent, and draw " it, the i th element of " t, as IID ; 2 ". We set 2 " = =tr (R u R u) so that on average V ar(u it ) =, where R u = (I a u S) ; and the satial weights matrix S is S = (53) C A In order to minimize the e ects of the initial values, the rst 5 observations are droed. 2 f25; 5; 75; ; 2g and 2 f25; 5; 75; ; 2g. For each, all arameters were set at the beginning of the exeriments and 2 relications were carried out by generating new innovations 2

23 " it, ft and i. he focus of the exeriments is to evaluate the small samle roerties of the CALS estimator of the own coe cient ' 2 = :5 and the neighboring coe cient 2 = :; in the case of the second cross section unit. 4 he cross-section augmented regression for estimating ( 2 ; 2 ) is given by x 2t = c (x ;t + x 3;t ) + ' 2 x 2;t + 2; x t + 2; x t + 2t. (54) We also reort results of the Least Squares (LS) estimator comuted using the above regression but without augmentation with cross-section averages. he corresonding CALS estimator and non-augmented LS estimator are denoted by b' 2;CALS and b' 2;LS (own coe cient), or b 2;CALS and b 2;LS (neighboring coe cient), resectively. o summarize, we carry out two di erent sets of exeriments, one set without the unobserved common factor ( = ), and the other with the unobserved common factor ( 6= ). here are many sources of interdeendence between individual units: satial deendence of innovations fu it g, satiotemoral interactions due to coe cient matrices a and b, and nally in the case where 6= the cross section deendence also arises via the unobserved common factor, f t ; and the cross-sectionally deendent factor loadings, i. Additional intermediate cases are also considered, the results of which are available in a Sulement from the authors, uon request Monte Carlo Results ables and 2 give the bias () and RMSE () of CALS and LS estimators as well as size and ower of tests based on them at the 5% nominal level. Results for the estimated own coe cient, b' 2;CALS and b' 2;LS, are reorted in able. he to anel of this table resents the results for the exeriments with an unobserved common factor ( 6= ). In this case, fx it g is CSD and the standard LS estimator without augmentation by cross section averages is not consistent. he bias of b' 2;LS is indeed quite substantial for all values of and and the tests based on b' 2;LS are grossly oversized. CALS, on the other hand, erforms well for and all values of. For smaller values of, there is a negative bias, and the test based on b' 2;CALS is slightly oversized. 4 Similar results are also obtained for other cross section units. 5 he sulement resents the results for the exeriments with all combination of zero and/or non-zero coe cient matrix b, zero or non-zero factor loadings, and low or high cross section deendence of errors (a u = :4 or a u = :8). 22

24 his is the familiar time series bias where even in the absence of any cross section deendence the LS estimator of the autoregressive coe cient is biased downward (when ' 2 > ) in small samles. Moving on to the exeriments without a common factor (given at the bottom half of the table), we observe that the LS estimator only slightly outerforms the CALS estimator. In the absence of common factors, fx it g is weakly cross sectionally deendent and therefore the augmentation with cross section averages is (asymtotically) innocuous. Distortions coming from cross section augmentation are in this case very small. ote that the LS estimator is not e cient because the residuals are cross sectionally deendent. Augmentation by cross-section averages hels to reduce art of this deendence. evertheless, the reorted RMSE of b' 2;CALS does not outerform the RMSE of b' 2;LS. he estimation results for the neighboring coe cient, 2 ; are resented in able 2. hese are qualitatively similar to the ones reorted in able. Cross section augmentation is clearly needed and very helful when common factors are resent. But in the absence of such common e ects, the resence of weak cross section deendence, whether through the dynamics or error rocesses, does not ose any di culty for the least squares and the CALS estimators so long as is su ciently large. Finally, not surrisingly, the estimates are subject to the small bias irresective of the size of or the degree of cross section deendence. Figure lots the ower of the CALS estimator of the own coe cient, b' 2;CALS, (to chart) and the neighboring coe cient, b 2;CALS, (bottom chart) for = 2 and two di erent values of 2 f; 2g. hese charts rovide a grahical reresentation of the results reorted in ables -2, and also suggest signi cant imrovement in ower as increases for a number of di erent alternatives. 6 An Emirical Illustration: a satiotemoral model of house rices in the U.S. In a recent study Holly, Pesaran, and Yamagata (29), hereafter HPY, consider the relation between real house rices, it ; and real er caita ersonal disosable income y it (both in logs) in a anel of 49 US States over 29 years (975-23), where i = ; 2; :::; 49 and t = ; 2; :::;. Controlling for heterogeneity and cross section deendence, they show that it and y it are cointegrated with 23

25 coe cients (; ), and rovide estimates of the following anel error correction model: it = c i +! i ( i;t y i;t ) + i i;t + 2i y it + it. (55) o take account of unobserved common factors, HPY augmented (55) with simle cross section averages, t = 49 i= it=49, y t = 49 i= y it=49, and t y t = 49 i= ( i;t y i;t )=49, and obtained common correlated e ects mean grou and ooled estimates (denoted as CCEMG and CCEP) of f! i ; i ; 2i g which we reroduce in the left anel of able 3. HPY then showed that the residuals from these regressions, ^ it, dislay a signi cant degree of satial deendence. Here we exloit the theoretical results of the resent aer and consider the ossibility that dynamic neighborhood e ects are artly resonsible for the residual satial deendence reorted in HPY. o this end we considered an extended version of (55) where the lagged satial variable s i;t = P j= s ij j;t is also included amongst the regressors, with s ij being the (i; j) element of a satial weight matrix, S, namely it = c i +! i ( i;t y i;t ) + i i;t + i s i;t + 2i y it + it. (56) Here we consider a simle contiguity matrix s ij = when the States i and j share a border and zero otherwise, with s ii =. Possible strong cross section deendence is again controlled for by augmentation of the extended regression equation with t, y t, and t y t. Estimation results are reorted in the right anel of able 3. he dynamic satial e ects are found to be highly signi cant, irresective of the estimation method, increasing R 2 of the rice equation by 6-9%. he dynamics of ast rice changes are now distributed between own and neighborhood e ects giving rise to much richer dynamics and sill over e ects. It is also interesting that the inclusion of the satiotemoral variable s i;t of the coe cient of the real income variable, 2i. in the model has had little imact on the estimates 7 Concluding Remarks his aer has roosed restrictions on the coe cients of in nite-dimensional VAR (IVAR) that are binding only in the limit as the number of cross section units (or variables in the VAR) tends to in nity to circumvent the curse of dimensionality. he roosed framework relates to the various 24

26 aroaches considered in the literature. For examle when modelling individual households or rms, aggregate variables, such as market returns or regional/national income, are treated as exogenous. his is intuitive as the imact of a rm or household on the aggregate economy is small, of the order O. his aer formalizes this idea in a satiotemoral context. he aer establishes that in the absence of common factors and when the degree of cross section deendence is weak, then equations for individual units decoule as! ; and can be consistently estimated by running searate regressions. In the resence of observed and/or unobserved common factors, individual-seci c VAR models can still be estimated searately if they are conditioned on the common factors. Unobserved common factors can be aroximated by cross sectional averages, following the idea originally introduced by Pesaran (26). he aer shows that the global VAR aroach of Pesaran, Schuermann, and Weiner (24) can be motivated as an aroximation to an IVAR model featuring all the macroeconomic variables. Asymtotic distribution of the cross sectionally augmented least-squares (CALS) estimator of the arameters of the unit-seci c equations in the IVAR model is established both in the case when the number of unobserved common factors is known, and when it is unknown but xed. Small samle roerties of the roosed CALS estimator were investigated through Monte Carlo simulations, and an emirical illustration shows the statistical signi cance of dynamic sill-over e ects in modelling of U.S. real house rices across the neighboring States. oics for future research include estimation and inference in the case of IVAR models with dominant individual units, analysis of large dynamic networks with and without dominant nodes, and an examination of the relationshis between IVAR and dynamic factor models. 25

27 Own Coefficient, =2 % 9% 8% 7% 6% 5% 4% 3% 2% % % = = eighboring Coefficient, =2 % 9% 8% 7% 6% 5% 4% 3% 2% % % = = Figure : Power Curves for the CALS t-tests of Own Coe cient, ' 2 (the uer chart) and the eighboring Coe cient, 2 (the lower chart), in the Case of Exeriments with 6=. 26

28 able : MC results for the own coe cient ' 2. Bias () Root Mean Square Errors () Size (5% level, H : ' 2 = :5) Power (5% level, H : ' 2 = :6) (,) Exeriments with nonzero factor loadings LS estimator not augmented with cross section averages, b' 2;LS CALS estimator b' 2;CALS Exeriments with zero factor loadings LS estimator not augmented with cross section averages,b' 2;LS CALS estimator b' 2;CALS otes: ' 2 = :5, 2 = :, a = au = :4, and V ar ( i ) =. he DGP is given by 2-neighbor IVAR model (49) where the equation for unit i 2 f2; ::; g is xit = ' i xi;t + i (xi ;t + xi+;t ) + bi xt + i ft i ft + uit. he CALS estimator of the own coe cient ' 2 and the neighboring coe cient 2 is comuted using the following auxiliary regression, x2t = c2 + 2 (x;t + x3;t ) + ' 2 x2;t + 2;xt + 2;xt + 2t. Estimators b' 2;LS and b 2;LS are comuted from the auxiliary regressions not augmented with cross section averages. Unobserved common factor ft is generated as stationary AR() rocess, and factor loadings and innovations fuitg are generated according to stationary satial autoregressive rocesses. Please refer to Section 5 for detailed descrition of Monte Carlo design. 27