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Wo r k i n g Pa p e r S e ri e s n o 9 9 8 / J a n ua Ry 2 0 0 9 InfInIte-dImensIonal VARS AND FACTOR MODELS by Alexander Chudik and M. Hashem Pesaran WO R K I N G PA P E R S E R I E S N O 9 9 8 / J A N U A RY 20 0 9 INFINITE-DIMENSIONAL VARS AND FACTOR MODELS 1 by Alexander Chudik 2 and M. Hashem Pesaran 3 In 2009 all ECB publications feature a motif taken from the €200 banknote. This paper can be downloaded without charge from http://www.ecb.europa.eu or from the Social Science Research Network electronic library at http://ssrn.com/abstract_id=1316219. 1 We are grateful to Elisa Tosetti for helpful comments and suggestions. This version has also benefited from comments by the seminar participants at University of California, San Diego, University of Southern California, Columbia, Leicester, McGill, Princeton, Pennsylvania, and Stanford. We would also like to acknowledge Takashi Yamagata for carrying out the estimation of the results reported in Section 6. The views expressed in this paper do not necessarily reflect the views of the European Central Bank or the Eurosystem. 2 University of Cambridge, Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom and European Central Bank, Kaiserstrasse 29, 60311 Frankfurt am Main, Germany; e-mail: [email protected]; http://www.econ.cam.ac.uk/phd/ac474/ 3 University of Cambridge, CIMF and USC; Faculty of Economics, Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom; e-mail: [email protected]; http://www.econ.cam.ac.uk/faculty/pesaran/ © European Central Bank, 2009 Address Kaiserstrasse 29 60311 Frankfurt am Main, Germany Postal address Postfach 16 03 19 60066 Frankfurt am Main, Germany Telephone +49 69 1344 0 Website http://www.ecb.europa.eu Fax +49 69 1344 6000 All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the author(s). The views expressed in this paper do not necessarily reflect those of the European Central Bank. The statement of purpose for the ECB Working Paper Series is available from the ECB website, http://www.ecb.europa. eu/pub/scientific/wps/date/html/index. en.html ISSN 1725-2806 (online) CONTENTS Abstract 4 Non-technical summary 5 1 Introduction 7 2 Infinite-dimensional vector autoregressive models 10 3 Cross sectional dependence in stationary IVAR models 3.1 Contemporaneous dependence: spatial or network dependence 3.2 IVAR models with strong cross sectional dependence 12 16 17 4 Estimation of a stationary IVAR 21 5 Monte Carlo experiments: small sample properties of CALS estimator 5.1 Monte Carlo design 5.2 Monte Carlo results 28 28 32 6 An empirical application: a spatiotemporal model of house prices in the U.S. 33 7 Concluding remarks 34 Figures and tables 35 References 39 Appendix 42 European Central Bank Working Paper Series 61 ECB Working Paper Series No 998 January 2009 3 Abstract This paper introduces a novel approach for dealing with the ‘curse of dimensionality’in the case of large linear dynamic systems. Restrictions on the coe¢ cients of an unrestricted VAR are proposed that are binding only in a limit as the number of endogenous variables tends to in…nity. It is shown that under such restrictions, an in…nite-dimensional VAR (or IVAR) can be arbitrarily well characterized by a large number of …nite-dimensional models in the spirit of the global VAR model proposed in Pesaran et al. (JBES, 2004). The paper also considers IVAR models with dominant individual units and shows that this will lead to a dynamic factor model with the dominant unit acting as the factor. The problems of estimation and inference in a stationary IVAR with unknown number of unobserved common factors are also investigated. A cross section augmented least squares estimator is proposed and its asymptotic distribution is derived. Satisfactory small sample properties are documented by Monte Carlo experiments. Keywords: Large N and T Panels, Weak and Strong Cross Section Dependence, VAR, Global VAR, Factor Models JEL Classi…cation: C10, C33, C51 4 ECB Working Paper Series No 998 January 2009 Non-technical summary Vector autoregressive models (VARs) provide a ‡exible framework for the analysis of complex dynamics and interactions that exist between variables in the national and global economy. However, the application of the approach in practice is often limited to a handful of variables which could lead to misleading inference if important variables are omitted merely to accommodate the VAR modelling strategy. Number of parameters to be estimated grows at the quadratic rate with the number of variables, which is limited by the size of typical data sets to no more than 5 to 7. In many empirical applications, this is not satisfactory. The objective of this paper is to analyze large linear dynamic systems of endogenously determined variables. In particular, we study VAR models where both the number of variables (N ) and the number of time periods (T ) tend to in…nity. In this case, parameters of the VAR model can no longer be consistently estimated unless suitable restrictions are imposed to overcome the dimensionality problem. Two di¤erent approaches have been suggested in the literature to deal with this ‘curse of dimensionality’: (i) shrinkage of the parameter space and (ii) shrinkage of the data. This paper proposes a novel way to deal with the curse of dimensionality by shrinking part of the parameter space in the limit as the number of endogenous variables (N ) tends to in…nity. An important example would be a VAR model where each unit is related to a small number of neighbors and a large number of non-neighbors. The neighborhood e¤ects are …xed and do not change with N , but the coe¢ cients corresponding to the remaining units are small, of order O N 1 . 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 the case where the VAR contains neighborhood e¤ects our speci…cation would converge to a spatiotemporal as N ! 1. Finally, when the VAR includes dominant units the limiting outcome will be a dynamic factor models. Such VAR models of growing dimension (N ! 1) are referred in the paper as the in…nite-dimensional VARs, or IVARs. The analysis of the paper also formally establishes the conditions under which the Global VAR (GVAR) approach proposed by Pesaran, Schuermann and Weiner (JBES, 2004) is applicable. In particular, the IVAR featuring all macroeconomic variables could be arbitrarily well approximated by a set of …nite-dimensional small-scale models that can be consistently estimated separately in ECB Working Paper Series No 998 January 2009 5 the spirit of the GVAR. Besides the development of an econometric approach for the analysis of groups that belong to a large interrelated system, the second main contribution of the paper is in considering the problems of the estimation and inference in stationary IVAR models with known as well as an unknown number of unobserved common factors. A simple cross sectional augmented least-squares estimator is proposed and its asymptotic distribution derived. Satisfactory small sample properties are documented by Monte Carlo experiments. As an illustration of the proposed approach we follow the recent empirical analysis of real house prices across the 49 U.S. States by Holly, Pesaran and Yamagata (2008) and show statistically signi…cant dynamic spill over e¤ects of real house prices across the neighboring states. 6 ECB Working Paper Series No 998 January 2009 1 Introduction Vector autoregressive models (VARs) provide a ‡exible framework for the analysis of complex dynamics and interactions that exist between economic variables. The traditional VAR modelling strategy postulates that the number of variables, denoted as N , is …xed and the time dimension, denoted as T , tends to in…nity. The number of parameters to be estimated grows at the quadratic rate with N and consequently the application of the approach in practice is often limited (by the size of typical datasets) to a handful of variables. The objective of this paper is to analyze VAR models where both N and T tend to in…nity. In this case, parameters of the VAR model can no longer be consistently estimated unless suitable restrictions are imposed to overcome the dimensionality problem. Two di¤erent approaches have been suggested in the literature to deal with this ‘curse of dimensionality’: (i) shrinkage of the parameter space and (ii) shrinkage of the data. Spatial and/or spatiotemporal literature shrinks the parameter space by using the concept of spatial weights matrix, which links individual units with the rest of the system. Alternatively, one could use techniques whereby prior distributions are imposed on the parameters to be estimated. Bayesian VAR (BVAR) proposed by Doan, Litterman and Sims (1984), for example, use what has become known as ‘Minnesota’ priors to shrink the parameters space.1 In most applications, BVARs have been applied to relatively small systems2 (e.g. Leeper, Sims, and Zha, 1996, considered 13- and 18-variable BVAR), with the focus mainly on forecasting.3 The second approach to mitigating the curse of dimensionality is to shrink the data, along the lines of index models. Geweke (1977) and Sargent and Sims (1977) introduced dynamic factor models, which have more recently been generalized to allow for weak cross sectional dependence by Forni and Lippi (2001) and Forni et al. (2000, 2004). Empirical evidence suggests that few dynamic factors are needed to explain the co-movement of macroeconomic variables: Stock and Watson (1999, 2002), Giannoni, Reichlin and Sala (2005) conclude that only few, perhaps two, factors explain much of the predictable variations, while Bai and Ng (2007) estimate four factors and Stock and Watson (2005) estimate as much as seven factors. This has led to the development 1 Other types of priors have also been considered in the literature. See, for example, Del Negro and Schorfheide (2004) for a recent reference. 2 A few exceptions include Giacomini and White (2006) and De Mol, Giannone and Reichlin (2006). 3 Bayesian VARs are known to produce better forecasts than unrestricted VARs and, in many situations, ARIMA or structural models. See Litterman (1986) and Canova (1995) for further references. ECB Working Paper Series No 998 January 2009 7 of factor-augmented VAR (FAVAR) models by Bernanke, Boivin, and Eliasz (2005) and Stock and Watson (2005), among others. This paper proposes a novel way to deal with the curse of dimensionality by shrinking part of the parameter space in the limit as the number of endogenous variables (N ) tends to in…nity. An important example would be a VAR model where each unit is related to a small number of neighbors and a large number of non-neighbors. The neighborhood e¤ects are …xed and do not change with N , but the coe¢ cients corresponding to the remaining units are small, of order O N 1 . 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. This set-up naturally arises in the context of global macroeconomic modelling. When all economies are small and open, using a multicountry DSGE model Chudik (2008) shows that the coe¢ cients of the foreign variables in the rational expectations solution are all of order O N 1 . In the case where the VAR contains neighborhood e¤ects our speci…cation would converge to a spatiotemporal as N ! 1. Finally, when the VAR includes dominant units the limiting outcome will be a dynamic factor models. Such VAR models will be referred as the in…nite-dimensional VARs, or IVARs. The analysis of the paper also provides a link between data and parameter shrinkage approaches to mitigating the curse of dimensionality. By imposing limiting restrictions on some of the parameters of the VAR we e¤ectively end up with a data shrinkage. We apply the concept of strong and weak Cross Section (CS) dependence (introduced by Pesaran and Tosetti, 2007) in the context of IVARs and show that only strong CS dependence can be ‘transmitted’through O N 1 coe¢ - cients. This …nding links our analysis to the factor models by showing that dominant unit becomes (in the limit) a dynamic common factor for the remaining units in a large system of endogenously determined variables. Static factor models are also obtained as a special case of IVAR. Last but not least, this paper formally establishes the conditions under which the Global VAR (GVAR) approach proposed by Pesaran, Schuermann and Weiner (2004) is applicable.4 In particular, the IVAR featuring all macroeconomic variables could be arbitrarily well approximated by a set of …nite-dimensional small-scale models that can be consistently estimated separately in the spirit of 4 GVAR model has been used to analyse credit risk in Pesaran, Schuermann, Treutler and Weiner (2006) and Pesaran, Schuerman and Treutler (2007). Extended and updated version of the GVAR by Dees, di Mauro, Pesaran and Smith (2007), which treats Euro area as a single economic area, was used by Pesaran, Smith and Smith (2007) to evaluate UK entry into the Euro. Global dominance of the US economy in a GVAR model is explicitly considered in Chudik (2007). Further developments of a global modelling approach are provided in Pesaran and Smith (2006). Garratt, Lee, Pesaran and Shin (2006) provide a textbook treatment of GVAR. 8 ECB Working Paper Series No 998 January 2009 the GVAR. Besides the development of an econometric approach for the analysis of groups that belong to a large interrelated system, the second main contribution of the paper is in considering the problems of the estimation and inference in stationary IVAR models with known as well as an unknown number of unobserved common factors. Our set-up extends the analysis of Pesaran (2006) to dynamic models where all variables are determined endogenously. A simple cross sectional augmented leastsquares estimator (or CALS for short) is proposed and its asymptotic distribution derived. Small sample properties of the proposed estimator are investigated through Monte Carlo experiments. As an illustration of the proposed approach we follow the recent empirical analysis of real house prices across the 49 U.S. States by Holly, Pesaran and Yamagata (2008) and show statistically signi…cant dynamic spillover e¤ects of real house prices across the neighboring states. The remainder of this paper is organized as follows. Section 2 outlines IVAR model, introduces limiting restrictions, and provides few examples, which link IVAR with the literature. Section 3 investigates cross section dependence in IVAR models where key asymptotic results are provided. Section 4 focusses on estimation of a stationary IVAR. Section 5 presents Monte Carlo evidence and a spatiotemporal model of the US house prices is presented in Section 6. The …nal section o¤ers some concluding remarks. Proofs are provided in the Appendix. are the eigenvalues of A 2 Mn n , where Pn max Mn n is the space of real-valued n n matrices. kAkc i=1 jaij j denotes the maximum 1 j n Pn max j=1 jaij j is the absolute row-sum matrix absolute column sum matrix norm of A, kAkr 1 i n p max fj i (A)jg is the spectral norm of A.5 kAk = % (A0 A) is the spectral norm of A; % (A) A brief word on notation: j 1 (A)j ::: j n (A)j 1 i n radius of A.6 Row i of A is denoted by a0i and the column i is denoted as ai . All vectors are column vectors. Row i of A with the ith element replaced by 0 is denoted as a0 i . Row i of A 2 Mn the element i and the element 1 replaced by 0 is a0 1; i = (0; ai2 ; :::; ai;i n with 1 ; 0; ai;i+1 ; :::; ai;iN ). The matrix constructed from A by replacing its …rst column with a column vector of zeros is denoted _ by A 1. j Joint asymptotics in N; T ! 1 are represented by N; T ! 1. an = O(bn ) states the deterministic sequence an is at most of order bn . xn = Op (yn ) states random variable xn is at most of order yn in probability. N is the set of natural numbers, and Z is the set of integers. We use K 5 The maximum absolute column sum matrix norm and the maximum absolute row sum matrix norm are sometimes denoted in the literature as k k1 and k k1 ,p respectively.p 6 Note that if x is a vector, then kxk = % (x0 x) = x0 x corresponds to the Euclidean length of vector x. ECB Working Paper Series No 998 January 2009 9 and to denote large and small positive constants that do not vary with i; t; N or T . Convergence d p q:m: in distribution and convergence in probability is denoted by ! and !, respectively. Symbol ! represents convergence in quadratic mean. 2 In…nite-Dimensional Vector Autoregressive Models Suppose there are N cross section units indexed by i 2 S f1; ::; N g N. Depending on empirical application, units could be households, …rms, regions, countries, or macroeconomic indicators in a given economy. Let xit denote the realization of a random variable belonging to the cross section unit i in period t, and assume that xt = (x1t ; :::; xN t )0 is generated according to the following stationary structural VAR model A0 xt = A1 xt 1 + A2 "t , (1) where one lag is assumed for the simplicity of exposition, A0 , A1 and A2 are N unknown coe¢ cients, and innovations collected into N N matrices of 1 vector "t = ("1t ; :::; "N t )0 are IID (0; IN ). The model (1), for example, arises as the rational expectations solution of a multi-country DSGE model. (See, for example, Pesaran and Smith, 2006). Assuming matrix A0 is invertible, the reduced form of structural model (1) is: xt = xt 1 + ut , (2) where the vector reduced-form errors ut is given by the following ‘spatial’model ut = R"t , (3) = A0 1 A1 , and R = A0 1 A2 . Focus of this paper is on a sequence of reduced-form models (2) of growing dimension (N ! 1), where the elements of and R (and hence the variance matrix of ut ) depend on N . But to simplify the notations we do not show this dependence explicitly, although it will be understood throughout that all the parameters and the dimension of the random variables xt and ut vary with N , unless otherwise stated. The sequence of models (2) and (3) with dim(xt ) = N growing will be referred to as the in…nite-dimensional VAR(1) model. To allow for neighborhood e¤ects it is convenient to decompose 10 ECB Working Paper Series No 998 January 2009 into two components: a sparse N N matrix matrix, a, with …xed elements (that do not vary with N ) which captures b, the neighborhood e¤ects, and a complement matrix, so that = a + b. An example of a B 11 B B B 12 B B B 0 =B B B 0 B B B B @ 0 is given by a 0 characterizing the remaining interactions, 12 0 0 0 22 23 0 0 32 33 34 0 43 0 0 44 .. 1 . 0 .. . .. . N 1;N N 1;N NN C C C C C C C C, C C C C C C A (4) where the nonzero elements are …xed coe¢ cients that do not change with N . This represents an ‘approximate line’ model where each unit, except the …rst and the last unit has one left and 1 ), in particular for any i; j 2 f1; ::; N g and any N 2 N. Equation for unit i 2 f2; ::; N 1g can be one right neighbor. In contrast the individual elements of bij < K N b are of order O(N written as xit = i;i 1 xi 1;t 1 + ii xi;t 1 + i;i+1 xi+1;t 1 + 0 bi xt 1 Next section shows that under weak CS dependence of errors fuit g, considers problem of estimation of the individual-speci…c parameters + uit . q:m: 0 bi xt 1 ! i;i 1 ; ii ; (5) 0, and Section 4 i;i+1 . We refer to this model as a two-neighbor IVAR model which we use later for illustrative purposes as well as in the Monte Carlo experiments. The above decomposition of matrix is a pivotal example of limiting restrictions developed in this paper. More generally, we have b; =D S+ where as before, the individual elements of matrix 0 B B B B D=B B B @ 0 1 0 0 .. . 0 2 0 0 b (6) are (uniformly) of order O N 0 .. . 1 , 1 C C C C C, C 0 C A (7) 0 N ECB Working Paper Series No 998 January 2009 11 1 dimensional vector containing the unknown coe¢ cients to be estimated for unit P N matrix partitioned as i 2 f1; ::; N g, hi is bounded in N , h = N i=1 hi , and S is a known h i is an hi S = (S1 ; S2 ; :::; SN )0 , with Si being the N hi selection matrix, which de…nes the neighbors for unit i as in the example above. S could also be related to a spatial weights matrix as in the following example. Example 1 Consider the following spatiotemporal model where Sx and Su are N xt = x Sx xt 1 ut = u Su ut + ut , (8) + "t , (9) N spatial weights matrices. Spatiotemporal model (8)-(9) is a special case of the model (2)-(3) by setting u Su ) R = (I 1 , i = x for i 2 f1; ::; N g , S = Sx , and b = 0. Remark 1 Note, however, that not all types of structural models (1) have reduced forms that satisfy the restrictions given by (6). For example, consider the spatiotemporal model: xt = Sx xt + xt Assuming matrix (I (2) with = (I x Sx ) x Sx ) and unknown parameters x 1 + "t . is invertible, the reduced form of spatiotemporal model (10) is model 1 and R = (I x Sx ) 1 . For the known spatial weights matrix Sx and , the reduced form coe¢ cient matrix in equation (6), where the matrix S is assumed to be known, f i g and elements of 3 b (10) are uniformly O N 1 cannot be decomposed as b are unknown, and the . Cross Sectional Dependence in Stationary IVAR Models Here we investigate the correlation pattern of fxit g, over time, t, and across the cross section units, i. Unlike the time index t which is de…ned over an ordered integer set, the cross section index, i, refers to an individual unit of an unordered population distributed over space or more generally over networks. To avoid having to order the cross section units we make use of the concepts of weak 12 ECB Working Paper Series No 998 January 2009 and strong cross section dependence recently developed in Pesaran and Tosetti (2007, hereafter PT). A process fxit g is said to be cross sectionally weakly dependent (CWD) with respect to a pre-determined information set, It 1, if for all weight vectors, wt = (w1t ; :::; wN t )0 , satisfying the ‘granularity’conditions7 kwt k = O N wjt kwt k = O N 1 2 , 1 2 (11) for any j N, (12) we have lim V ar wt0 N !1 1 xt j It 1 = 0, for all t 2 T . Since we will be dealing with stationary processes in what follows we con…ne our analysis to time invariant weight vectors, w, and information sets, I = ;. Accordingly, we adopt the following concept. De…nition 1 Stationary process fxit ; i 2 S; t 2 T ; N 2 Ng, generated by the IVAR model (2), is said to be cross sectionally weakly dependent (CWD), if for any sequence of non-random vectors of weights w satisfying the granularity conditions (11)-(12), lim V ar (xwt j I) = lim V ar (xwt ) = 0, N !1 (13) N !1 where xwt = w0 xt . fxit g is said to be cross sectionally strongly dependent (CSD) if there exists a sequence of weights vectors w satisfying (11)-(12) and a constant K such that lim V ar (xwt ) N !1 K > 0. (14) Necessary condition for covariance stationarity for …xed N is that all eigenvalues of of the unit circle. For a …xed N , and assuming that maxi j ` de…ned by T r ` `0 1=2 lie inside i( )j < 1, the Euclidean norm of P `u ! 0 exponentially in `; and the process xt = 1 t ` will `=0 be absolute summable, in the sense that the sum of absolute values of the elements of 7 `, for Condition (12) is understood as wjt kwt k K p , for any j 2 f1; ::; N g and any N 2 N, N where constant K < 1 does not depend on N nor on j. ECB Working Paper Series No 998 January 2009 13 ` = 0; 1; ::: converge. Observe that as N ! 1, V ar (xit ) need not necessarily be bounded in N if maxi j i( )j < 1 . For example, consider the IVAR(1) model with 0 B B B B B =B B B B B @ ' 0 0 ' 0 0 .. . 0 ' .. . 0 .. 1 C C 0 C C C 0 C C, C C 0 C A ' . 0 and assume that var (uit ) is uniformly bounded away from zero as N ! 1. It is clear all eigenvalues of are inside the unit circle if and only if j'j < 1, regardless the value of the neighboring coe¢ cient . Yet the variance of xN t increases in N without bounds at an exponential rate for j j > 1 j'j.8 Therefore, a stronger condition than stationarity is required to rule out variances of xit exploding as N ! 1. This is set out in the following assumptions. ASSUMPTION 1 Individual elements of double index process of errors fuit ; i 2 S; t 2 T g are random variables de…ned on the probability space ( ; F; P ). ut is independently distributed of ut0 , for any t 6= t0 2 T . For each t 2 T , ut has mean and variance, E (ut ) = 0, E ut u0t = where is an N 2 ii i 2 S and = V ar (uit ) is the i-th diagonal element of covariance matrix , k k = O N1 V ar fxN t g = where 14 k` = 1 (k 1)! ECB Working Paper Series No 998 January 2009 kQ2 j=0 (` + k 1 . (15) , (16) > 0 is an arbitrarily small positive constant. It can be shown that < K < 1 for any and CWD ut ) k k<1 8 2 ii N symmetric, nonnegative de…nite matrix, such that 0 < ASSUMPTION 2 (Coe¢ cients matrix where , N X 2(N j) j=1 j) for k > 1 and 1 X `=0 1` = 1. 2 N j+1;` ' 2` , Remark 2 Assumption 1 and equation (16) of Assumption 2 imply fuit g is CWD. Remark 3 Condition (15) of Assumption 2 is a su¢ cient condition for covariance stationarity and also delivers bounded variance of xit , as N ! 1. Note that Assumption 2 also rules out cases where strong cross sectional dependence arises due to a particular unit (or units) since both p p k kc < N k k = O N and k kc cannot diverge to in…nity at the rate N . Proposition 1 Consider model (2) and suppose that Assumptions 1 and 2 hold. Then for any arbitrary sequence of …xed weights w satisfying condition (11), and for any t 2 T , lim V ar (xwt ) = 0. (17) N !1 Proposition 1 has several interesting implications. Suppose that unit i has a …xed number of neighbors, j = 1; 2; ::; p, for which coe¢ cients ij = O (1), while the in‡uence of each of the remaining units on the unit i through coe¢ cient matrix following decomposition of the ith row of matrix 0 ai j=1 0 bi : 11=2 2 A bij =O N Remark 4 Obvious examples of the decomposition of =( i1 ; :::; i;i 1 ; 0; hood model where with i;i 1 ; ii and 0 ai i;i+1 ; :::; iN ), = (0; :::; 0; i;i+1 into a possibly sparse vector S be a non-empty index set. For any i 2 K, 0 N X k bi k = @ 0 bi 0 i, , denoted as and the remaining coe¢ cients collected into vector ASSUMPTION 3 Let K is small. In particular, consider the where i;i 1 ; ii ; ii i 1 2 i = ai + bi , where . is when (18) 0 ai = (0; :::; 0; ii ; 0; :::; 0), and does not depend on N , and the left-right neigbour- i;i+1 ; 0:::; 0) and 0 bi =( i1 ; ::: i;i 2 ; 0; 0; 0; i;i+2 :::; , iN ) being …xed parameters that do not vary with N . Remark 5 As we shall see in Section 4, for estimation and inference the following slightly stronger condition on the row norm of bi will be needed. k bi kr =O N Remark 6 Assumption 3 implies that for i 2 K, 1 PN j=1 . bij k bi kc = O (1). Therefore, it is possible for the dependence of each individual unit on the rest of the units in the system to be large ECB Working Paper Series No 998 January 2009 15 even if ai = 0. However, as we shall see below, in the case where fxit g is a CWD process, vector does not play a role in the model for the ith cross section unit as N ! 1. bi Corollary 1 Consider model (2) and suppose Assumptions 1-3 hold. Then, lim V ar xit N !1 Observe that if ii 0 ai xt 1 uit = 0, for i 2 K. is the only nonzero element of ai , (19) then the regression model for unit i completely de-couples from the rest of the system as N ! 1, in the sense that lim V ar (xit N !1 ii xi;t 1 uit ) = 0. The above corollary in e¤ect states that in econometric modelling of xit one can ignore the e¤ects of those cross section units that have zero entries in ai as N becomes large, so long as xt is a CWD process.9 3.1 Contemporaneous Dependence: Spatial or Network Dependence An important form of cross section dependence is contemporaneous dependence across space. The spatial dependence, pioneered by Whittle (1954), models cross section correlations by means of spatial processes that relate each cross section unit to its neighbor(s). Spatial autoregressive and spatial error component models are examples of such processes. (Cli¤ and Ord, 1973, Anselin, 1988, and Kelejian and Robinson, 1995). However, it is not necessary that proximity is measured in terms of physical space. Other measures such as economic (Conley, 1999, Pesaran, Schuermann and Weiner, 2004), or social distance (Conley and Topa, 2002) could also be employed. All these are examples of dependence across nodes in a physical (real) or logical (virtual) networks. In the case of the IVAR model, de…ned by (2) and (3), such contemporaneous dependence can be modelled through an N 9 4. N network topology matrix R.10,11 For example, in the case of a …rst order spatial j Appropriate rates for N ,T ! 1 needed for inference about the nonzero parameters in 10 ai are established Section A network topography is usually represented by graphs whose nodes are identi…ed with the cross section units, with the pairwise relations captured by the arcs in the graph. 11 It is also possible to allow for time variations in the network matrix to capture changes in the network structure over time. However, this will not be pursued here. 16 ECB Working Paper Series No 998 January 2009 moving average model, R would take the form 0 RSM A where s B B B B B B B = IN + s B B B B B B B @ 0 1 0 0 ::: 0 0 1=2 0 1=2 0 ::: 0 0 0 .. . 1=2 .. . 0 .. . 1=2 : : : .. .. . . 0 .. . 0 .. . 0 0 0 0 : : : 1=2 0 0 0 0 0 ::: 0 1 1 0 C C 0 C C C C 0 C C .. C ; . C C C 1=2 C C A 0 is the spatial moving average coe¢ cient. The contemporaneous nature of dependence across i 2 S is fully captured by R. As shown in PT the contemporaneous dependence across i 2 S will be weak if the maximum absolute column and row sum matrix norm of R are bounded, namely if kRkc kRkr < K < 1. It turns out that all spatial models proposed in the literature are in fact examples of weak cross section dependence. More general network dependence such as the ‘star’ network provides an example of strong contemporaneous dependence that we shall consider below. The form of R for a typical star network is given by 0 RStar where N X 1 0 B B B r21 1 B B =B B r31 0 B . .. B . B . . @ rN 1 0 0 0 .. . 1 C C 0 0 C C C 0 0 C C; C C 1 0 C A 0 1 rj1 = O(N ). j=2 The IVAR model when combined with ut = R"t yields an in…nite-dimensional spatiotemporal model. The model can also be viewed more generally as a ‘dynamic network’, with R and capturing the static and dynamic forms of inter-connections that might exist in the network. 3.2 IVAR Models with Strong Cross Sectional Dependence Strong dependence in IVAR model could arise as a result of CSD errors fuit g, or could be due to dominant patterns in the coe¢ cients of , or both. Strong cross section dependence could also arise in the case of residual common factor models where the weighted averages of factor loadings ECB Working Paper Series No 998 January 2009 17 do not converge to zero.12 Section 4 considers estimation and inference in the case of stationary CSD IVAR models with unobserved common factors and/or deterministic trends. An example of a stationary IVAR model where the column corresponding to unit i = 1 in matrices , and R is dominant is provided below. The following assumption postulates that for any i, coe¢ cient vector into a sparse vector 0; i2 ; :::; i;i 1 ; 0; ai = ( i;i+1 ; :::; ASSUMPTION 4 Let of 0 ii ; 0; :::; 0) i1 ; 0; :::; 0; iN = 0 and a vector i can be decomposed = 1; i 1i ; :::; 0 N i) bi where 1; i = . PN i=1 0 i ei 0 1 e1 = + _ 1 where i =( is the ith column ; ei is an N 1 selection vector for unit i, with the ith element of ei being one and the remaining elements zero. Denote by _ the matrix constructed from by replacing its …rst column with a PN vector of zeros, and note that _ 1 = i=2 i e0i . Suppose as N ! 1 1 1 r = O (1) . (20) Further, suppose that 1; i r =O N 1 uniformly for all i 2 N, (21) namely there exists a constant K such that 1; i r < K for any i 2 S and any N 2 N. N ASSUMPTION 5 (Stationarity) k kr < < 1 for any N 2 N. ASSUMPTION 6 The N 1 vector of errors ut is generated by the ‘spatial’model (3). E (ut u0t ) = P 0 th 0 0 _ = RR0 is time invariant, where R = N i=1 ri ei = r1 e1 + R 1 , and ri = (r1i ; :::; rN i ) is the i column of matrix R. Suppose as N ! 1 2 _ R 1 = O N1 kr1 kr = O (1) , 12 18 See Pesaran and Toesetti (2007, Theorem 16). ECB Working Paper Series No 998 January 2009 , (22) (23) and lim kr N !1 where is an arbitrarily small constant, r 1; i k 1; i = 0 for any i 2 N, = (0; ri1 ; :::; ri;i (24) 0 1 ; 0; ri;i+1 ; :::; riN ) , and rij denotes the (i; j) element of matrix R. Remark 7 Assumptions 4 and 6 imply matrix has one dominant column and matrix R has at least one dominant column, but the absolute column sum for only one column could rise with N at the rate N . Part (21) of Assumption 4 allows the equation for unit i 6= 1 to de-couple from the equations for units j 6= 1, for any j 6= i, as N ! 1. Remark 8 Using the maximum absolute column/row sum matrix norms rather than eigenvalues in principle allows us to make a distinction between cases where dominant e¤ ects are due to a particular unit (or a few units), and when there is a pervasive unobserved factor that makes all column/row sums unbounded. Eigenvalues of the covariance matrix will be unbounded in both cases and it will not be possible from the knowledge of the rate of the expansion of the eigenvalues of , and/or R to known which one of the two cases are in fact applicable. Remark 9 As it will become clear momentarily, conditional on x1t and its lagged values, process fxit g become cross sectionally weakly dependent. We shall therefore refer to unit i = 1 as the dominant unit. Remark 10 It follows that under Assumptions 5 and 6 the IVAR model speci…ed by (2) and (3) is stationary for any N , and the variance of xit will be uniformly bounded. Proposition 2 Under Assumptions 4-6 and as N ! 1, equation for the dominant unit i = 1 in the IVAR model de…ned by (2) and (3) reduces to x1t where # (L; e1 ) = condition (11), P1 `=0 e01 `r 1 q:m: # (L; e1 ) "1t ! 0, (25) L` . Furthermore, for any …xed sequence of weights w satisfying xwt q:m: # (L; w) "1t ! 0. (26) ECB Working Paper Series No 998 January 2009 19 The model for unit i = 1 can be approximated by an AR(p1 ) process, which does not depend on the realizations from the remaining units as N ! 1. Let the lag polynomial a (L; p1 ) be an approximation of # 1 # 1 (L; e1 ) (27) (L; e1 ). Then equation for unit i = 1 can be written as a (L; p1 ) x1t "1t . (28) The following proposition presents mean square error convergence results for the remaining cross section units. Proposition 3 Consider system (2), let Assumptions 4-6 hold and suppose that the lag polynomial # (L; e1 ) de…ned in Proposition 2 is invertible. Then as N ! 1, equations for cross section unit i 6= 1 in the IVAR model de…ned by (2) and (3) reduce to (1 ii L) xit q:m: i (L) x1t rii "it ! 0; for i = 2; 3; ::: (29) where i (L) = i1 L + ri1 + # L; 1; i L # 1 (L; e1 ) ; and # L; 1; i = 1 X 0 1; i ` r1 L` , for i > 1. `=0 Remark 11 Exclusion of the current value of x1t from (29) is justi…ed only if ri1 = 0. But even in this case xit will depend on lagged values of x1t . Remark 12 Cross section unit 1 becomes (in the limit) a dynamic common factor for the remaining units in the IVAR model. Note that setting x1t = ft , (29) can be written as13 (1 ii L) xit Remark 13 Conditional on fx1t ; x1;t 13 rii "it + 1 ; x1;t 2 ; ::::g, i (L) ft , for i > 1. (30) the process fxit g for i > 1 is CWD. x1t could be equivalently approximated by cross sectional weighted averages of xt and its lags, namely xwt ; xw;t 1 ; :::. 20 ECB Working Paper Series No 998 January 2009 Remark 14 For 1 = 0 and i = 0, we obtain from (29) the following static factor model as a special case (1 ii L) xit rii "it + ri1 r11 ft , for i > 1, (31) where ft = x1t . We now turn our attention to the problems of estimation and inference in IVAR models. In what follows we consider the relatively simple case where there are no dominant units, but allow for the possibility of unobserved common factors. The analysis of IVAR models featuring both unobserved common factors, ft , and matrices with unbounded maximum absolute column sum matrix norms is provided in a supplement, which is available from the authors on request. 4 Estimation of a Stationary IVAR Assume xt = (x1t ; :::; xN t )0 is generated according to the following factor-augmented IVAR(1): (L) (xt f t ) = ut , (32) for t = 1; 2; :::; T , where the vector of errors ut is generated by spatial model (3), namely ut = R"t , (L) = IN L, is N N dimensional matrix of unknown coe¢ cients, 1 dimensional vector of …xed e¤ects, ft is m N factors (m is …xed but otherwise unknown), factor loadings with its i-th row denoted as =( 0, i =( 1 ; :::; 0 N) is 1 dimensional vector of unobserved common 1; 2 ; :::; 0 N) is N m dimensional matrix of and "t = ("1t ; "2t :::; "N t )0 is the vector of error terms assumed to be independently distributed of ft0 8t; t0 2 f1; ::; T g. Without major di¢ culties, one could also add observed common factors and/or additional deterministic terms to the equations in (32), but in what follows we abstract from these for expositional simplicity. System (32) models deviations of endogenous variables from common factors in a VAR. Alternatively, one could introduce common factors directly in the residuals. This extension is pursued in Pesaran and Chudik (2008), who focus on estimation of IVARs with dominant units.14 De…ne the following vector of weighted averages xW t = W0 xt , where W = (w1 ; w2 ; :::; wN )0 and fwj gN j=1 are mw 1 dimensional vectors. Subscripts denoting the number of groups are again 14 This extension is not straightforward as it introduces in…nite-lag polynomials in the corresponding auxiliary cross-section augmented regressions for the individual units. ECB Working Paper Series No 998 January 2009 21 omitted where not necessary, in order to keep the notations simple. Matrix W does not correspond to any spatial weights matrix. It is any arbitrary matrix of pre-determined weights satisfying the following granularity conditions15 kWk = O N kwj k kWk = O N 1 2 , 1 2 (33) for any j N. (34) We consider the problem of estimating the parameters of equation i 2 N in a non-nested sequence of models (32) as both N and T tend to in…nity, where =D S+ b. can be decomposed as See (6). As an important example we consider the two-neighbor IVAR model de…ned by (5). In the case of this model the vector of unknown coe¢ cients of interest for the ith equation is on the ith row of D, de…ned by (7) namely i hi = 3, and the corresponding N 1 ; ei ; ei+1 ) 3 matrix Si = (ei = i;i 1 ; ii ; i;i+1 0 for i 2 = f1; N g, with in S = (S1 ; S2 ; :::; SN )0 ; which selects the unit i and the left and the right neighbors of unit i.16 In what follows we set and note that it reduces to (xi 0 1;t ; xit ; xi+1;t ) it = S0i xt ; in the case of the two-neighbor IVAR model. We suppose that the following assumptions hold for any N 2 N and i 2 f1; ::; N g, unless otherwise stated. ASSUMPTION 7 (General limiting restrictions) The ith row of matrix i = Si i + bi , can be decomposed as, (35) where k Si is predetermined and known N bi kr = max j2f1;::;N g bij < K , N (36) hi dimensional matrix, kSi kc < K, and hi < K. The unknown coe¢ cients and the …xed e¤ ects are bounded, namely k i k < K and j i j < K. For any i 2 N, there exists constant N0 2 N such that the vector of unknown coe¢ cients 15 do not change with N > N0 . Condition (34) is understood as kwj k kWk K p , for any j 2 f1; ::; N g and any N 2 N, N where constant K < 1 does not depend on N nor on j. 16 The …rst and the last unit has only one neighbor. 22 i ECB Working Paper Series No 998 January 2009 ASSUMPTION 8 (Stationarity) k k < < 1. ASSUMPTION 9 (Weakly dependent errors with …nite fourth moments) Innovations f"jt gN j=1 are identically and independently distributed with mean 0, unit variances and …nite fourth moments. Furthermore, matrix R has bounded row and column matrix norms. ASSUMPTION 10 (Available observations) Available observations are x0 ; x1 ; :::; xT with the P ` R" ( `) + + f 0 .17 starting values x0 = 1 `=0 ASSUMPTION 11 (Common factors) Unobserved common factors f1t ; :::; fmt follow stationary MA(1) processes: fst = where polynomials s (L) = P1 `=0 s (L) "f st ; ` s` L for s = 1; ::; m, (37) are absolute summable, "f st IID 0; 2 "f s , and the fourth moments of "f st are bounded, E "4f st < 1. "f st is independently distributed of "t0 for any t; t0 2 T , and any s 2 f1; ::; mg. Polynomials s (L) and variances 2 "f s , for s 2 f1; ::; mg, do not change with N and the covariance matrix E (ft ft0 ) is positive de…nite. ASSUMPTION 12 (Bounded factor loadings) k i k < K. Remark 15 (Eigenvalues of ) Assumption 8 implies polynomial (L) is invertible (for any N 2 N) and %( ) < < 1. (38) This is in line with the …rst part of Assumption 2 and is therefore su¢ cient for stationarity of xt for any N 2 N. Also, as noted in Section 3, this assumption rules out explosive variance of individual p N k k, Assumption 8 rules elements of the vector xt as N ! 1. Furthermore, since k kc out cases where k kc diverges to in…nity at the rate N . Hence the dominance of a particular unit or units due to the coe¢ cient matrix is also ruled out by this assumption. Remark 16 The spectral norm of covariance matrix E (ut u0t ) = sumption 9, since kRR0 k kRR0 kr is bounded in N under As- kRkr kRkc .18 ut is therefore a cross sectionally weakly dependent process, which, as shown in Pesaran and Tosetti (2007), includes all commonly used spatial processes in the literature. 17 We use notation " ( `) instead of " ` in order to avoid possible confusion with the notation used in previous sections. p 18 kAkc kAkr for any matrix A. kAk ECB Working Paper Series No 998 January 2009 23 (L) and then by W0 yields Multiplying system (32) by the inverse of polynomial xW t = where xW t = W0 xt , W = W0 , W = W0 , W t = + Wt 1 X + W ft W t, = W0 ` t, (39) and ut ` . (40) `=0 Under Assumption 9, fut g is weakly cross sectionally dependent and therefore kV ar ( 1 X W t )k = ` W0 0` W , `=0 kWk2 k k = O N 1 , 1 X ` 2 , `=0 (41) where kWk2 = O N 1 by condition (33), k k = O (1) by Assumption 9 (see Remark 16) and P1 P1 1 ` ` 2 and the `=0 `=0 k k = O (1) under Assumption 8. This implies W t = Op N unobserved common factors can be approximated as provided that the matrix W 1 0 W W 0 W W 0 W (xW t W) = ft + Op N 1 2 , (42) is nonsingular. It can be inferred that the full column rank of is important for the estimation of unit-speci…c coe¢ cients. Pesaran (2006) shows that the full column rank is, however, not necessary if the object of interest is a panel estimation of the common mean of the individual coe¢ cients as opposed to the consistency of individual-speci…c estimates. Using system (32), equation for unit i can be written as: xit i 0 i ft = 0 0 i Si (xt 1 ft 1) + i;t 1 + uit , (43) where it 24 ECB Working Paper Series No 998 January 2009 = 0 ib t = Op N 1 2 , (44) since the vector ib 0 i ft satis…es condition (33) under Assumption 7. It follows from equation (39) that 0 ia 0 W 0 i where bi1 = ft 1 = b0i1 xW t + b0i2 xW;t 1 W 0 W (bi1 + bi2 )0 1 0 W 0 0 i Si and bi2 = b0i1 W 1 W 0 W. Wt b0i2 W;t 1 , (45) Substituting equation (45) into equation (43) yields xit = ci + where ci = (bi1 + bi2 )0 0 ia i 0 0 i Si xt 1 qit = i;t 1 W, + b0i1 xW t + b0i2 xW;t 1 + uit + qit , (46) 1 2 (47) and b0i1 Wt b0i2 W;t 1 = Op N . Consider the following auxiliary regression based on the equation (46): 0 xit = git where it = uit + qit , regressors git = 1; estimator of i 0 0 0 0 i ; bi1 ; bi2 is ki 0 0 0 0 i;t 1 ; xW t ; xW;t 1 , and ki i = ci ; : T X bi = t=1 We denote the estimator of coe¢ cients i i + it , (48) 1 vector of coe¢ cients associated to the vector of = 1 + hi + 2mw . Let b i be the least squares (LS) 0 git git ! 1 T X git xit . (49) t=1 given by the corresponding elements of the vector b i as the cross section augmented least squares estimator (or CALS for short), denoted as bi;CALS . Asymptotic properties of b i (and bi;CALS in the case where the number of unobserved common factors is unknown) are the objective of this analysis as N and T tend to in…nity. First we consider the case where the number of unobserved common factors equals to the dimension of xW t (m = mw ), and make the following additional assumption. ASSUMPTION 13 (Identi…cation of N N0 and for any i 2 f1; ::; N g, T and CiN1 = O (1). 1 i) PT There exists T0 and N0 such that for all T 0 t=1 git git 1 T0 ; 0 ) is positive de…nite, exists, CiN = E (git git ECB Working Paper Series No 998 January 2009 25 Remark 17 Assumption 13 implies 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 = mw ). In cases where m < mw , full augmentation of individual models by (cross sectional) averages is not necessary. Theorem 1 Let xt be generated by model (32), Assumptions 7-13 hold, and W is any arbitrary (pre-determined) matrix of weights satisfying conditions (33)-(34) and Assumption 13. Then as j N; T ! 1 (in no particular order), the estimator b i de…ned in equation (49) has the following properties. a) bi b) If in addition T =N ! {, with 0 p 2 ii;N ! 0: { < 1, T ii;N where p i 1 2 CiN (b i i) D ! N (0; Ik1 ) , (50) 1 2 = V ar (uit ) = E (e0i RR0 ei ), and CiN is square root of positive de…nite matrix 0 ). Also CiN = E (git git c) CiN where p b iN ! C 0, and p ii;N bii;N ! 0; T T X 1X 2 0 b iN = 1 git git ; b2ii;N = u bit , C T T t=1 and u bit = xit 0 b . git i (51) t=1 Remark 18 Suppose that in addition to the assumptions of Theorem 1, the limits of CiN1 and 2 ii;N , as N ! 1; exist and are given by Ci11 , and p 19 T (b i i) D 2 ii;1 ; ! N 0; respectively.19 Then (50) yields 1 2 ii;1 Ci;1 . (52) Su¢ cient condition for limN !1 CiN to exist is the existence of the following with Assumptions P limits0 (together ` 7-12): limN !1 S0i , limN !1 S0i , limN !1 W0 , limN !1 W0 , and limN !1 1 RR0 0` Si . `=0 Si 26 ECB Working Paper Series No 998 January 2009 Consider now the case where the number of unobserved common factors is unknown, but it is known that mw m. Since the auxiliary regression (48) is augmented possibly by a larger number of cross section averages than the number of unobserved common factors, we have potential problem of multicollinearity (as N ! 1). But this observation has no bearings on estimates of i so long as the space spanned by the unobserved common factors including a constant and the space spanned by the vector (1; x0W t )0 are the same as N ! 1. This is the case when W has full column rank. Using partition regression formula, the cross sectionally augmented least squares (CALS) estimator 1 dimensional vector of hi i in the auxiliary regression (48) is where xi = (xi1 ; :::; xiT )0 , Zi = r 2 f1; ::; hi g ; MH = IT 1 bi;CALS = Z0 MH Zi i i1 ( 1) ; i2 ( H (H0 H)+ H0 , H = 1) ; :::; Z0i MH xi , ihi ( 1) , (53) ir ; XW ; XW ( 1) , ( 1) = is T ir0 ; :::; i;r;T 1 0 for 1 dimensional vector of ones, XW = (xW 1 ; :::; xW mw ), XW ( 1) = [xW 1 ( 1) ; :::; xW mw ( 1)], xW s = (xW s1 ; :::; xW sT )0 and xW s ( 1) = (xW s0 ; :::; xW s;T S0i t = it S0i f t 0 1) for s 2 f1; ::; mw g. De…ne for future reference vector vit = S0i , and the following matrices. Q = [ ; F; F ( 1)] , and 0 (54) 0 W B 1 B 0 =B A 0 W (2m+1) (2mw +1) B @ 0 0m mw 0 W 0m mw 0 W 1 C C C, C A (55) where F = (f1 ; :::; fm ), F ( 1) = [f1 ( 1) ; :::; fm ( 1)], fr = (fr1 ; :::; frT )0 and fr ( 1) = (fr0 ; :::; fr;T 0 1) for r 2 f1; ::; mg. For this more general case we replace Assumption 13 with the following (and suppress the subscript N to simplify the notations) ASSUMPTION 14 (Identi…cation of i) There exists T0 and N0 such that for all T 1 N vi exists, N0 and for any i 2 f1; ::; N g, T 1 Z0i MH Zi P1 0 ` 0 )= = E (vit vit RR0 0` Si is positive de…nite and `=0 Si W 1 vi T0 ; is full column rank matrix, = O (1). ECB Working Paper Series No 998 January 2009 27 Theorem 2 Let xt be generated by model (32), Assumptions 7-12, and 14 hold, and W is any arbitrary (pre-determined) matrix of weights satisfying conditions (33)-(34). Then if in addition j { < 1, the asymptotic distribution of bi;CALS de…ned by N; T ! 1 such that T =N ! {, with 0 (53) is given by. p T ii where 2 ii = V ar (uit ), vi 1 2 vi D bi;CALS ! N (0; Ihi ) , i 0 ) and v = S0 = E (vit vit it i t = P1 0 `=0 Si (56) `u t `. Remark 19 As before, we also have p where vi;1 = limN !1 vi , T bi;CALS and 2 ii;1 D i ! N 0; 2 ii;1 1 vi;1 ; 2. ii = limN !1 Extension of the analysis to a IVAR(p) model is straightforward and it is relegated to a Supplement available from the authors upon request. 5 Monte Carlo Experiments: Small Sample Properties of CALS Estimator 5.1 Monte Carlo Design In this section we report some evidence on the small sample properties of the CALS estimator in the presence of unobserved common factors and weak error cross section dependence and compare the results with standard least squares estimators. The focus of our analysis will be on the estimation of the individual-speci…c parameters in an IVAR model that also allows for other interactions that are of order O(N 1 ). The data generating process (DGP) is given by xt ft = (xt 1 ft 1) + ut , where ft is the only unobserved common factor considered (m = 1), and N (57) =( 1 ; :::; 0 N) is the 1 vector of factor loadings. We consider two sets of factor loadings to distinguish the case of weak and strong cross section dependence. Under the former we set 28 ECB Working Paper Series No 998 January 2009 = 0; and under the latter we generate the factor loadings i; for i = 1; 2; :::; N , from a stationary spatial process in order to show that our estimators are invariant to the cross section dependence of the factor loadings. The following bilateral Spatial Autoregressive Model (SAR) is considered. = i where IIDN 0; i 2 1 = 2b ; 2 i 1 + a i+1 + i, (58) . As established by Whittle (1954), the unilateral SAR(2) scheme i with a 2 = 1 i 1 q = b2 and b = 1 + + 2 i 2 i, (59) a2 =a , generates the same autocorrelations as 1 the bilateral SAR(1) scheme (58). The factor loadings are generated using the unilateral scheme (59) with 50 burn-in data points (i = a = 0:4, 2 = 1, and choose 49; :::; 0) and the initializations 51 = 50 = 0. We set such that V ar ( i ) = 1.20 The common factors are generated according to the AR(1) process ft = with f f ft 1 + f t, 2 f IIDN 0; 1 ft , = 0:9. In line with the theoretical analysis the autoregressive parameters are decomposed as b, where a a+ capture own and neighborhood e¤ects as in 0 a 20 = 0 0 1 B '1 B B 0 B 2 '2 2 B B B 0 3 '3 3 =B B B 0 0 4 '4 B B .. B . B @ 0 0 0 0 0 0 .. . .. . N 1 'N N The variance of factor loadings is given by 2 = 1+ 1 2 1 2 1 C C C C C C C C; C C C C C C A 2 2 1 . 2 ECB Working Paper Series No 998 January 2009 29 and the remaining elements of , de…ned by = bij 8 > < i ! ij > : 0 are generated as for j 2 = fi 1; i; i + 1g for j 2 fi 1; i; i + 1g , where & ij IIDU ( 0:1; 0:2) and ! ij = PN i j=1 & ij IIDU (0; 1). This ensures that with & ij b, bij = Op (N 1 ), , (60) and limN !1 E bij = 0, for all i and j. With a as speci…ed above, each unit i, except the …rst and the last, has two neighbors: the ‘left’neighbor i 1 and the ‘right’neighbor i + 1. The DGP for the ith unit can now be written as x1t = '1 x1;t xit = 'i xi;t 1 1 xN t = 'N xN;t + + 1 + + 0 b1 xt 1 i (xi 1;t 1 + xi+1;t 1 x2;t 1 N xN 1;t 1 + + 1) 0 1 1 ft 0 bi xt 1 + 0 b;N xt 1 + ft 1 + 0 i i ft 0 N N ft To ensure the DGP is stationary we generate 'i + u1t , ft 1 ft 1 + uit ; i 2 f2; ::; N 1g , + uN t . IIDU (0:4; 0:6), and i IIDU ( 0:1; 0:1) for i 6= 2. We choose to focus on the equation for unit i = 2 in all experiments and we set '2 = 0:5 and 2 = 0:1. This yields k kr 0:9, and together with f < 1 it is ensured that the DGP is stationary and the variance of xit is bounded in N . The cross section averages, xwt ; are constructed P as simple averages, xt = N 1 N j=1 xit . The N -dimensional vector of error terms, ut ; is generated using the following SAR model: u1t = au u2t + "1t , uit = au (ui 2 uN t = au uN 1;t 1;t + ui+1;t ) + "it , i 2 f2; ::; N 1g + "N t , for t = 1; 2; ::; T . We set au = 0:4 which ensures that the errors are cross sectionally weakly dependent, and draw "it , the ith element of "t , as IIDN 0; 30 ECB Working Paper Series No 998 January 2009 2 " . We set 2 " = N=tr (Ru R0u ) so that on average V ar(uit ) is 1, where Ru = (I 0 B 0 B B 1 B 2 B B B 0 S=B B B B B B B @ 0 au S) 1 ; and the spatial weights matrix S is 1 0 0 0 1 2 0 1 2 0 .. . 1 2 0 .. . .. . 1 2 0 0 1 1 0 C C 0 C C C C 0 C C. C C C C 1 C 2 C A 0 (61) In order to minimize the e¤ects of the initial values, the …rst 50 observations are dropped. N 2 f25; 50; 75; 100; 200g and T 2 f25; 50; 75; 100; 200g. For each N , all parameters were set at the beginning of the experiments and 2000 replications were carried out by generating new innovations "it , ft and i. The focus of the experiments is to evaluate the small sample properties of the CALS estimator of the own coe¢ cient '2 = 0:5 and the neighboring coe¢ cient regression for estimating coe¢ cients f 2; 2g 2 = 0:1. The cross-section augmented in the case of the second cross section unit is given by (similar results are also obtained for other cross section units) x2t = c2 + 2 (x1;t 1 + x3;t 1) + '2 x2;t 1 + 2;0 xt + 2;1 xt 1 + 2t . (62) We also report results of the Least Squares (LS) estimator computed using the above regression but without augmentation with cross-section averages. The corresponding CALS estimator and b 2;LS (own coe¢ cient), or b 2;CALS and non-augmented LS estimator are denoted by ' b 2;CALS and ' b 2;LS (neighboring coe¢ cient), respectively. To summarize, we carry out two di¤erent sets of experiments, one set without the unobserved common factor ( = 0), and the other with unobserved common factor ( 6= 0). There are many sources of interdependence between individual units: spatial dependence of innovations fuit g, spatiotemporal interactions due to coe¢ cient matrices with a and b, and …nally in the latter case 6= 0 the cross section dependence also arises via the unobserved common factor ft and cross-sectionally dependent factor loadings. Additional intermediate cases are also considered, the results of which are available in a Supplement from the authors on request.21 21 Supplement presents experiments with all combination of zero or nonzero coe¢ cient matrix b, zero or nonzero ECB Working Paper Series No 998 January 2009 31 5.2 Monte Carlo Results Tables 1-2 give the bias ( 100) and RMSE ( 100) of CALS and LS estimators as well as size and power of tests at the 5% nominal level. Results for the estimated own coe¢ cient, ' b 2;CALS and ' b 2;LS , are reported in Table 1. The top panel of this table presents the results for the experiments with an unobserved common factor ( 6= 0). In this case, fxit g is CSD and the standard LS estimator without augmentation with cross section averages is not consistent. The bias of ' b 2;LS is indeed quite substantial for all values of N and T and the tests based on ' b 2;LS are grossly oversized. CALS, on the other hand, performs well for T 100 and all values of N . For smaller values of T , there is a negative bias, and the test based on ' b 2;CALS is slightly oversized. This is the familiar time series bias where even in the absence of cross section dependence the LS estimator of autoregressive coe¢ cients will be biased in small T samples. Moving on to the experiments without a common factor (given at the bottom half of the table), we observe that the LS estimator slightly outperforms the CALS estimator. In the absence of common factors, fxit g is weakly cross sectionally dependent and therefore the augmentation with cross section averages is (asymptotically) redundant. Note that the LS estimator is not e¢ cient because the residuals are cross sectionally dependent. Augmentation by cross-section averages helps to reduce part of this dependence. Nevertheless, the reported RMSE of ' b 2;CALS does not outperform the RMSE of ' b 2;LS . The estimation results for the neighboring coe¢ cient, 2; are presented in Table 2. These are qualitatively similar to the ones reported in Table 1. Cross section augmentation is clearly needed when common factors are present. But in the absence of such common e¤ects, the presence of weak cross section dependence, whether through the dynamics or error processes, does not pose any di¢ culty for the least squares estimates so long as N is su¢ ciently large. Finally, not surprisingly, the estimates are subject to the small T bias irrespective of the size of N or the degree of cross section dependence. Figure 1 plots the power of the CALS estimator of the own coe¢ cient, ' b 2;CALS , (top chart) and the neighboring coe¢ cient, b 2;CALS , (bottom chart) for N = 200 and two di¤erent values of T 2 f100; 200g. These charts provide a graphical representation of the results reported in Tables 1-2, and suggest signi…cant improvement in power as T increases for a number of di¤erent factor loadings 32 ECB Working Paper Series No 998 January 2009 , and low or high cross section dependence of errors (au = 0:4 or au = 0:8). alternatives. 6 An Empirical Application: a spatiotemporal model of house prices in the U.S. In a recent study Holly, Pesaran and Yamagata (2008, HPY) consider the relation between real house prices, pit ; and real per capita personal disposable income yit (both in logs) in a panel of 49 US States over 29 years (1975-2003), where i = 1; 2; :::; 49 and t = 1; 2; :::; T . Controlling for heterogeneity and cross section dependence, they show that pit and yit are cointegrated with coe¢ cients (1; 1), and provide estimates of the following panel error correction model: pit = ci + ! i (pi;t 1 yi;t 1) + pi;t 1i 1 + 2i yit + it . (63) To take account of unobserved common factors HPY augmented (63) with cross section averages and obtained common correlated e¤ects mean group and pooled estimates (denoted as CCEMG and CCEP) of f! i ; 1i ; 2i g which we reproduce in the left panel of Table 3. HPY then showed that the residuals from these regressions, ^ it , display a signi…cant degree of spatial dependence. Here we exploit the theoretical results of the present paper and consider the possibility that dynamic neighborhood e¤ects are partly responsible for the residual spatial dependence reported in HPY. To this end we considered an extended version of (63) where the lagged spatial variable psi;t 1 = PN j=1 sij pj;t 1 is also included amongst the regressors, with sij being the (i; j) element of a spatial weight matrix, S, namely pit = ci + ! i (pi;t 1 yi;t 1) + 1i pi;t 1 + i psi;t 1 + 2i yit + it . (64) Here we consider a simple contiguity matrix sij = 1 when the states i and j share a border and zero otherwise, with sii = 0. Possible strong cross section dependence is again controlled for by augmentation of the extended regression equation with cross section averages. Estimation results are reported in the right panel of Table 3. The dynamic spatial e¤ects are found to be highly signi…cant, irrespective of the estimation method, increasing R2 of the price equation by 6-9%. The dynamics of past price changes are now distributed between own and neighborhood e¤ects ECB Working Paper Series No 998 January 2009 33 giving rise to much richer dynamics and spill over e¤ects. It is also interesting that the inclusion of the spatiotemporal variable psi;t 1 coe¢ cient of the real income variable, 7 in the model has had little impact on the estimates of the 2i . Concluding Remarks This paper has proposed restrictions on the coe¢ cients of in…nite-dimensional VAR (IVAR) that bind 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. The proposed framework relates to the various approaches considered in the literature. For example when modelling individual households or …rms, aggregate variables, such as market returns or regional/national income, are treated as exogenous. This is intuitive as the impact of a …rm or household on the aggregate economy is small, of the order O N 1 . This paper formalizes this idea in a spatio-dynamic context. It was established that, under certain conditions on the order of magnitudes of the coe¢ cients in a large dynamic system, and in the absence of common factors, equations for individual units decouple as N ! 1 and can be estimated separately. In the presence of a dominant economic agent or unobserved common factors, individual-speci…c VAR models can still be estimated separately if conditioned upon observed and unobserved common factors. Unobserved common factors can be approximated by cross sectional averages, following the idea originally introduced by Pesaran (2006). The paper shows that the GVAR approach can be motivated as an approximation to an IVAR featuring all macroeconomic variables. This is true for stationary models as well as for systems with integrated variables of order one. Asymptotic distribution of the cross sectionally augmented least-squares (CALS) estimator of the parameters of the unit-speci…c models was established both in the case where the number of unobserved common factors is known and in the case where it is unknown but …xed. Small sample properties of the proposed CALS estimator were investigated through Monte Carlo simulations, and an empirical application was provided as an illustration of the proposed approach. Topics for future research could 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 a closer examination of the relationships between IVAR and dynamic factor models. 34 ECB Working Paper Series No 998 January 2009 Figure 1: Power Curves for the CALS t-tests of Own Coe¢ cient, '2 (the upper chart) and the Neighboring Coe¢ cient, 2 (the lower chart), in the Case of Experiments with 6= 0. ECB Working Paper Series No 998 January 2009 35 36 ECB Working Paper Series No 998 January 2009 21.15 21.70 20.40 20.80 21.30 7.20 9.45 8.80 9.35 10.10 6.95 6.85 7.15 6.40 7.35 7.05 7.65 7.95 7.10 8.35 13.20 13.80 13.85 12.65 13.50 13.30 14.35 14.00 13.20 15.45 8.40 9.10 8.35 8.75 8.95 9.35 10.90 10.20 10.55 10.55 6.47 6.56 6.80 6.83 6.80 6.67 6.74 6.56 6.61 6.62 6.43 6.69 6.57 6.63 6.67 6.85 6.35 7.00 7.15 6.10 6.65 6.00 6.80 5.95 5.20 7.60 7.95 8.10 8.70 7.55 27.70 30.30 27.50 28.40 28.75 4.80 5.80 6.90 5.65 5.65 5.35 5.75 6.25 5.25 5.00 7.00 6.15 6.80 6.50 6.70 33.25 32.35 32.05 32.80 32.70 4.85 6.40 5.25 6.00 6.30 5.45 6.85 5.20 6.30 6.30 5.80 5.30 6.00 6.55 6.25 45.55 42.65 43.60 44.25 42.85 Size (5% level, H0 : '2 = 0:50) 25 50 75 100 200 17.63 17.24 17.55 17.63 17.18 100) 200 18.20 19.45 19.75 18.90 19.30 15.80 17.25 17.35 16.70 17.30 23.80 24.85 26.05 25.85 27.00 16.85 16.00 16.45 14.90 15.30 19.50 20.20 21.05 18.35 21.80 19.25 18.85 19.70 17.75 20.80 22.90 26.90 26.35 26.80 26.40 19.80 20.85 18.45 18.05 20.95 21.00 23.85 22.95 24.65 24.90 21.70 22.60 22.90 24.25 24.15 27.10 28.10 28.00 28.40 30.00 23.55 26.80 24.60 23.55 26.35 23.85 25.95 28.65 27.85 25.65 25.00 25.45 27.65 26.05 25.10 31.20 33.50 33.25 31.50 33.85 29.20 29.50 31.10 30.35 29.00 39.50 39.90 42.50 42.80 41.90 42.60 41.20 42.25 43.40 41.00 41.00 45.15 46.55 47.80 47.80 42.00 39.20 41.85 43.65 41.45 Power (5% level, H1 : '2 = 0:60) 25 50 75 100 200 2 i according to stationary spatial autoregressive processes. Please refer to Section 5 for detailed description of Monte Carlo design. + xit = 'i xi;t 1 Notes: '2 = 0:5, = 0:1, a = au = 0:4, and V ar ( i ) = 1. The DGP is given by 2-neighbor IVAR model (57) where the equation for unit i 2 f2; ::; N 1g is 0 (xi 1;t 1 + xi+1;t 1 ) + 0bi xt 1 + i ft 2 is computed using i ft 1 + uit . The CALS estimator of the own coe¢ cient '2 and the neighboring coe¢ cient the following auxiliary regression, x2t = c2 + 2 (x1;t 1 + x3;t 1 ) + '2 x2;t 1 + 2;0 xt + 2;1 xt 1 + 2t . Estimators ' b 2;LS and b 2 are computed from the auxiliary regressions not augmented with cross section averages. Unobserved common factor ft is generated as stationary AR(1) process, and factor loadings and innovations fuit g are generated Bias ( 100) Root Mean Square Errors ( (N,T) 25 25 50 50 75 75 100 100 200 Experiments with nonzero factor loadings LS estimator not augmented with cross section averages, ' b 2;LS 25 -4.34 25.78 18.82 5.46 17.48 7.34 17.49 8.76 11.02 50 -3.04 25.14 18.83 4.39 18.37 8.21 17.45 8.61 10.65 75 -2.94 25.20 18.50 4.58 17.55 6.99 17.47 8.36 10.59 100 -2.67 24.19 18.27 4.89 17.65 7.60 17.74 8.94 10.74 200 -3.71 24.59 18.90 4.11 17.98 7.79 17.62 8.84 10.26 b 2;CALS CALS estimator ' 25 -18.43 29.04 15.52 -7.63 12.14 -5.22 -3.38 9.86 -0.72 50 -19.65 29.40 16.73 -9.58 12.44 -5.69 10.13 -4.77 -1.76 75 -19.76 29.83 16.56 -9.36 12.41 -5.82 10.35 -4.41 -2.30 100 -20.19 29.94 16.82 -9.31 12.60 -6.00 10.18 -4.29 -2.30 200 -20.96 -10.17 -6.27 -4.85 30.33 17.17 12.27 10.41 -2.37 Experiments with zero factor loadings LS estimator not augmented with cross section averages,b '2;LS 25 -13.10 24.20 14.80 -6.17 11.45 -3.89 9.35 -2.60 -1.59 50 -13.32 24.79 14.98 -6.16 11.51 -4.22 9.45 -2.76 -1.35 75 -12.74 24.15 14.97 -6.19 11.60 -4.14 9.97 -3.50 -1.37 100 -12.36 23.69 14.63 -5.65 11.87 -4.41 9.46 -3.15 -1.72 200 -13.30 24.49 15.03 -6.42 11.43 -4.54 9.47 -2.79 -1.46 b 2;CALS CALS estimator ' 25 -14.43 25.84 15.23 -6.22 11.36 -3.58 9.22 -2.21 -0.86 50 -15.38 27.13 15.61 -6.87 11.77 -4.54 9.48 -2.81 -1.10 75 -15.03 26.69 15.72 -6.93 11.95 -4.48 10.14 -3.64 -1.34 100 -14.90 26.49 15.40 -6.51 12.15 -4.78 9.70 -3.48 -1.75 200 -15.67 26.94 15.92 -7.51 11.91 -5.14 9.75 -3.23 -1.61 Table 1: MC results for the own coe¢ cient '2 . ECB Working Paper Series No 998 January 2009 37 7.67 7.14 7.03 6.99 7.21 7.47 7.22 6.31 6.69 6.45 7.02 6.33 6.55 6.67 6.76 1.18 0.88 0.87 0.71 1.14 1.17 0.89 0.52 0.55 0.72 0.78 0.43 0.64 0.58 0.46 18.01 16.96 17.27 17.00 18.06 22.64 21.44 21.46 21.54 21.37 10.04 10.12 10.16 10.14 10.56 10.69 10.56 10.37 10.74 10.55 16.86 16.83 16.11 16.33 17.61 7.98 7.89 8.01 7.94 7.99 8.20 8.39 8.25 8.23 8.30 15.10 15.24 14.87 15.07 15.20 6.58 6.81 6.82 6.77 6.75 6.99 6.79 6.82 6.72 6.88 14.62 14.83 13.71 14.15 14.18 2;LS 4.66 4.80 4.67 4.72 4.65 4.93 4.73 4.79 4.60 4.57 13.27 13.19 13.48 13.20 13.26 0.77 0.67 0.98 0.52 1.11 0.51 0.41 0.60 0.53 0.48 0.39 0.33 0.54 0.68 0.40 0.09 0.36 0.09 0.30 0.14 2.15 1.92 1.15 2.07 1.31 1.02 0.91 1.05 0.58 1.17 0.66 0.47 0.64 0.58 0.54 0.58 0.43 0.61 0.82 0.43 0.23 0.46 0.13 0.38 0.14 CALS estimator b 2;CALS 2.04 1.74 1.22 1.94 1.38 17.64 16.88 16.48 17.40 16.80 16.10 15.76 15.49 16.19 15.63 10.47 10.48 10.43 10.42 10.90 8.29 8.08 8.12 8.18 8.13 6.70 6.92 6.98 6.84 6.87 4.63 4.84 4.69 4.77 4.68 LS estimator not augmented with cross section averages, b 2;LS with zero factor loadings 1.70 1.52 1.35 1.26 1.22 See the notes to Table 1. 25 50 75 100 200 25 50 75 100 200 7.88 8.06 7.62 7.35 8.28 CALS estimator b 2;CALS 9.51 8.04 8.49 7.75 8.52 25 2.31 50 2.20 75 2.71 100 1.82 200 2.55 Experiments 25 50 75 100 200 Bias ( 100) Root Mean Square Errors ( 100) (N,T) 25 25 50 50 75 100 200 75 100 200 Experiments with nonzero factor loadings LS estimator not augmented with cross section averages, b 2. 8.25 7.50 7.00 8.00 7.10 8.20 7.05 6.80 7.25 7.15 9.10 8.15 8.05 8.10 9.00 19.70 18.60 17.40 18.25 18.85 6.35 6.70 6.60 6.15 7.40 5.55 6.00 6.10 6.00 6.45 6.75 6.85 6.75 6.90 6.45 27.30 27.45 26.60 27.05 29.50 5.85 5.90 5.40 5.80 5.40 6.15 5.65 5.30 5.00 5.10 5.75 7.40 6.05 6.75 6.50 36.35 35.60 36.00 35.85 36.75 5.00 5.90 5.60 5.60 5.95 5.60 5.75 5.55 5.60 6.05 6.70 5.95 5.70 5.45 6.00 43.50 44.10 41.85 42.30 41.00 4.70 5.75 5.30 5.50 4.85 5.20 5.90 5.00 5.25 5.05 7.30 6.05 6.25 5.25 5.55 60.30 59.25 58.25 59.15 60.40 Size (5% level, H0 : 2 = 0:10) 25 50 75 100 200 Table 2: MC results for the neighboring coe¢ cient 11.80 10.00 11.45 12.20 11.70 11.55 10.75 11.10 12.25 11.30 12.65 11.60 11.45 11.00 11.95 16.70 15.85 17.30 17.50 16.50 17.45 16.30 16.20 17.65 16.85 17.40 17.25 17.45 18.45 18.45 15.75 16.00 15.60 17.00 17.80 23.15 24.80 22.20 24.45 25.30 23.55 23.30 22.25 23.70 23.05 23.10 23.90 22.85 24.05 24.05 21.85 24.60 23.35 24.20 21.95 28.65 29.95 30.25 30.20 29.65 28.20 29.95 30.20 27.50 30.05 30.05 31.80 30.70 29.20 30.65 28.10 29.55 30.70 30.50 29.90 33.05 34.45 34.40 34.00 34.90 56.35 54.30 56.20 53.50 57.35 58.05 55.95 57.15 54.70 56.65 52.80 57.35 53.85 53.90 55.70 43.10 44.80 47.00 45.35 44.10 Power (5% level, H1 : 2 = 0:20) 25 50 75 100 200 Table 3: Alternative Average Estimates of the Error Correction Models for House Prices Across 49 U.S. States over the Period 1975-2003 yi;t 1 Regressions augmented with without dynamic spatial e¤ects dynamic spatial e¤ects MG pit pi;t Holly et al. (2008) regressions 1 0:105 (0:008) pi;t 1 yit psi;t 0:524 (0:030) 0:500 1 R2 Average Cross Correlation CCEMG 0:183 (0:016) 0:449 (0:038) 0:277 CCEP 0:171 (0:015) 0:518 (0:065) 0:227 (0:040) (0:059) (0:063) - - - 0:54 0:70 0:284 0:005 MG 0:095 (0:009) 0:296 (0:060) 0:497 (0:040) 0:331 CCEMG 0:154 (0:018) 0:188 (0:049) 0:284 (0:059) 0:350 CCEP 0:152 (0:018) 0:272 (0:082) 0:201 (0:088) 0:431 (0:066) (0:085) (0:105) 0:66 0:60 0:79 0:72 0:016 0:267 0:012 0:016 Coe¢ cients ( ^ ) Notes: MG stands for Mean Group estimates. CCEMG and CCEP signify the Common Correlated E¤ects Mean Group and Pooled estimates de…ned in Pesaran (2006). Standard errors are in parentheses. ^ denotes the average pair-wise correlation of the residuals from the cross-section augmented regressions across the 49 U.S. States. 38 ECB Working Paper Series No 998 January 2009 References [1] Anselin, L., 1988, Spatial econometrics: Methods and models. Kluwer Academic Publishers, Dordrecht, the Netherlands. [2] Bai, J. and S. 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[39] Whittle, P., 1954, On stationary processes on the plane. Biometrika, 41, 434-449. ECB Working Paper Series No 998 January 2009 41 Appendix A Lemmas and Proofs Proof of Proposition 1. For any N 2 N, the variance of xt is =V ar (xt ) = 1 X 0` ` , (65) `=0 and, k k is under Assumptions 1-2 bounded by k k k k 1 X `=0 k k2` = O N 1 . (66) It follows that for any arbitrary nonrandom vector of weights satisfying granularity condition (11), V ar w0 xt where % ( ) = k k = O N 1 O N = w0 w % ( ) w0 w , and w0 w = kwk2 = O N , (67) by condition (11). This implies kV ar (w0 xt )k = 1 and limN !1 kV ar (w0 xt )k = 0. Proof of Corollary 1. Assumption 3 implies that for i 2 K, vector bi satis…es condition (11). It follows from Proposition 1 that 0 bi xt lim V ar N !1 = 0 for i 2 K. (68) System (2) implies 0 a xt 1 xit uit = 0 b xt 1 ; for any i 2 S and any N 2 N: (69) Taking variance of (69) and using (68) now yields (19). Proof of Proposition 2. Solving (2) backwards yields xt = 1 X ` R "t `, (70) `=0 _ where R = r1 e01 + R 1. Hence 1 X x1t e01 ` r1 "1;t ` 1 X = `=0 e01 ` _ R "t 1 `. (71) `=0 Under Assumptions 4-6, V ar 1 X `=0 e01 ` _ R 1 "t ` ! = 1 X e01 ` _ R _0 1R 1 _ e01 R _0 1 R 1 e1 + 1 X `=1 42 ECB Working Paper Series No 998 January 2009 0` e1 , `=0 e01 ` _ R _0 1R 1 0` e1 . (72) But _ e01 R lim N !1 under Assumption 6. Set a0` e01 ` and let a`; _0 1 R 1 e1 1 2 1k = lim kr N !1 (73) =0 = (0; a`2 ; :::; a`N )0 . Note that under Assumptions 4-5: ` ka` kc ka`; , (74) a`1 = O (1) , 1 kr = O N (75) 1 , (76) for ` = 0; 1; 2; :::. Result (74) follows from Assumption 5 by taking the maximum absolute row-sum matrix norm of a0` = e01 ` 22 . Results (75)-(76) follow by induction directly from Assumptions 4-5. Using (74)-(76), we have 1 X e01 ` _ R _0 1R 1 0` 1 X = e1 `=1 _ a0` R _0 1 R 1 a` , `=1 1 X a2`1 r0 1 r 1 + `=1 kr 1 X a0`; _ _0 1 R 1 R 1 a`; 1 , `=1 2 1k 1 X 2` 2 _ + R 1 `=1 1 X `=1 ka`; 1 kr ka`; (77) 1 kc 2 P 2` _ 1 = O N1 where as before limN !1 kr 1 k2 = 0 under Assumption 6, 1 = O (1) by Assumption 5, R `=1 P1 P 2 1 by properties (74)-(76). It follows that by Assumption 6, and 1 `=1 ka`; 1 k `=1 ka`; 1 kr ka`; 1 kc = O N hP i P1 0 ` 1 0 0` 0 ` _ 1R _ 1 _ 1 "t ` = 0, we have R R limN !1 e1 = 0. Noting that E `=1 e1 `=0 e1 1 X e01 ` _ R 1 q:m: "t ` `=0 ! 0, as N ! 1. (78) This completes the proof of equation (25). To prove (26), we write 1 X xwt w0 ` r1 "1;t ` `=0 Since the vectors w0 ` = 1 X w0 ` _ R 1 "t `. (79) `=0 have the same properties as vectors fa` g in equations (74)-(76), it follows that (using the same arguments as above), 1 X w0 ` _ R 1 "t q:m: ` `=0 ! 0, as N ! 1. (80) This completes the proof of equation (26). Proof of Proposition 3. xit The vector r 1; i 0 ii xi;t 1 1; i xt 1 ka` kc e01 ri1 "1t rii "it = r0 1; i "t (81) satis…es equation (24) of Assumption 6, V ar ("t ) = IN , and E ("t ) = 0, which implies r0 22 i1 x1;t 1 ` r ke01 kr k k`r ` 1; i "t q:m: ! 0: (82) . ECB Working Paper Series No 998 January 2009 43 (6DF=E 2?5 :?G6CE:3:=:EJ @7 [email protected][email protected]>:2= :>A=:6D $! 5' ' :?2==J D:?46 D2E:DL6D [email protected]?5:E:@? F?56C DDF>AE:@? 6BF2E:@? :>A=:6D ' $! ' )F3DE:EFE:?8 C6DF=ED :[email protected] 6DE23=:D96D 6BF2E:@? *11& +: 99;36:/549 '4* .52* '4* 9;6659+ ': '4? 8':+ #.+4 ,58 '4? - . '4* ,58 '4? 9+7;+4)+9 5, 4548'4*53 <+):589 '4* 9;). :.': '4* =+ .'<+ ( # '# ' '4* # ( '# '$ '# '$ ' =.+8+ :.+ 685)+99 ' /9 *+A4+* /4 +7;':/54 ;8:.+8358+ /, :.+4 '4* ( # ' ' # ( '# '$ '# '$ ' 633+ "6E 36 2?J [email protected]?564C62D:?8 :?E686CG2=F65 7F?4E:@? @7 DF49 E92E @?D:56C E96 [email protected][email protected]:?8 [email protected]:>6?D:@?2= 2CC2J ' ' ' 56L?65 3J ' '# H96C6 E96 DF3D4C:AE :D FD65 [email protected] 6>A92D:K6 E96 ?F>36C @7 [email protected] D64E:@? F?:ED 2?5 ' [email protected] 2? :?4C62D:?8 D6BF6?46 @7 L6=5D ' ' DF49 E92E ' :?4=F56D 2== :[email protected]>2E:@? 2G2:=23=6 2E E:>6 1 2?5 ' :D >62DFC23=6 H:E9 C6DA64E [email protected] ' [email protected] 2?J "6E " ' ' 36 [email protected]:>6?D:@?2= 2CC2J @7 [email protected]?DE2?ED 2?5 D6E " ' [email protected] 2== 1 2?5 [email protected] E92E ' '" " ' # # m ( " [email protected] E92E [email protected] ' 2?5 492?86 H:E9 2D H6== 3FE E96 DF3D4C:AE :D @>:EE65 96C6 [email protected] <66A E96 [email protected]:@? D:>A=6 44 ECB Working Paper Series No 998 January 2009 H96C6 m "# + - 2?5 " m # ( +D:?8 DDF>AE:@?D 2?5 " 92D E96 [email protected][email protected]:?8 [email protected]:6D 2?5 " 2D + 1 J ":[email protected] :?6BF2=:EJ ' '" ' '" 2G:[email protected]? *[email protected]> E [email protected][email protected] E92E E96 [email protected]:>6?D:@?2= 2CC2J ' ' ' :D >:I:?82=6 H:E9 C6DA64E [email protected] E96 [email protected]?DE2?E 2CC2J " ' BF2 E:@?D 2?5 6DE23=:D9 2CC2J ' " ' :D F?:[email protected]>=J [email protected]?565 :? [email protected]> *9:D :>A=:6D F?:[email protected]> :?E68C23:=:EJ [email protected] E92E ( ' ( ( ' ( ' " ' " ' ' & *[email protected] 2CC2J ' ' H9:49 :>A=:6D ' ' ' D2E:DL6D [email protected]?5:E:@?D @7 2 >:I:?82=6 H62< =2H :6 ( '# ' 2D 2E 2?J C2E6 @?G6C86?46 :? [email protected]> :>A=:6D [email protected]?G6C86?46 :? [email protected]:=:EJ *9:D [email protected]>A=6E6D E96 [email protected]@7 @7 E96 C6DF=E +?56C E96 [email protected]?5:E:@? C6DF=E [email protected][email protected] [email protected]> C6DF=E 3J [email protected]:?8 E92E (6DF=E :D 6DE23=:D965 :? 2 D:>:=2C 72D9:@? @?D:56C E96 [email protected][email protected]:?8 [email protected]:>6?D:@?2= 2CC2J ' ' ' 56L?65 3J ' '# '$ '# '$ H96C6 2D [email protected] :D 2?J [email protected]?564C62D:?8 :?E686CG2=F65 7F?4E:@? @7 DF49 E92E )6E " ' [email protected] 2== 1 2?5 [email protected] E92E ' '" " ' " ( &# &# ( ( '& ( $ ' '" # '# '$ $ , &# '& $ ' &# '& $ ' &m m -6 FD6 DF3>F=E:A=:42E:G6 [email protected] @7 >2EC:I [email protected]>D [email protected] 2?J >2EC:46D DF49 E92E :D H6== 56L?65 2?5 E96 724E E92E E96 DA64EC2= >2EC:I [email protected]> :D D6=725;@:?E :6 [email protected] [email protected] E92E DDF>AE:@? & :>A=:6D )FQ4:6?E [email protected]?5:E:@? [email protected] F?:[email protected]> :?E68C23:=:EJ :D F?:[email protected]> [email protected]?565?6DD [email protected] 2?J 2G:[email protected]? *[email protected]> D [email protected] ' :D 2? :?4C62D:?8 D6BF6?46 @7 L6=5D ' ' DF49 E92E ' :?4=F56D 2== :[email protected]>2E:@? 2G2:=23=6 2E E:>6 1 2?5 ' :D >62DFC23=6 H:E9 C6DA64E @7 ' [email protected] 2?J ECB Working Paper Series No 998 January 2009 45 "6E & & 2?5 ' '" " ' ( ( ( ( &# '& $ ' # ' $ ' &m m m m " ( ( &# '& $ ' # &m m +D:?8 E96 :?56A6?56?46 @7 ' 2?5 ' 7[email protected] 2?J 1 1 DDF>AE:@? H6 92G6 ( ( &# '& $ ' ( &m m # $ # $ " H96C6 " # " $ " ( 2?5 - + . . - + - - . + . 3J DDF>AE:@? & 3J DDF>AE:@? 2?5 " 92D E96 [email protected][email protected]:?8 [email protected]:6D 2?5 " 2D + ):>:=2C=J FD:?8 E96 :?56A6?56?46 @7 ' 2?5 ' [email protected] 2?J 1 1 [email protected]=2E65 :? DDF>AE:@? E96 LCDE E6C> @? E96 C:89E D:56 @7 6BF2E:@? :D [email protected]?565 3J E96 [email protected][email protected]:?8 FAA6C [email protected]?5 " " ! ( #$ " # $ " # " $ ( H96C6 - + . + - + . :D >2EC:I H:E9 E96 6=6>6?E & ' 6BF2= [email protected] 2?5 :D >2EC:I @7 [email protected] >@>6?ED H:E9 E96 6=6>6?E + 0 6BF2= [email protected] 2' 2' 2"' 2&' E [email protected][email protected] [email protected]> DDF>AE:@?D &# '& $ ' # ' $ ' :D [email protected][email protected] @?=J :7 @?6 @7 E96 [email protected][email protected]:?8 [email protected] 42D6D & 0 ' # && 0 ' 2?5 ' # &&& 0 ' ' 2?5 # @C &3 0 # # 2?5 ' 46 ECB Working Paper Series No 998 January 2009 8 and 9 that & b;n has following properties2 9 h E E Nt cN t j Ft n i2 & b;0 = O (1) , and & b;n ! 0 as n ! 1. (95) is therefore bounded by & n = & a;n + & b;n . Equations (94) and (95) establish & 0 = O (1) , & n ! 0 as n ! 1. By Liapunov’s inequality, E jE ( Nt j Ft q E [E ( n )j that the two-dimensional array f 1 1 , N t ; Ft gt= 1 N =1 thermore, (96) establishes array f N t =cN t g 0 j Ft 2 n )] (Davidson, 1994, Theorem 9.23). It follows is L1 -mixingale with respect to a constant array fcN t g. Fur- is uniformly bounded in L2 norm. This implies uniform integrability.3 0 Since also equations (91) and (92) hold, array f P N L1 which implies Tt=1 N t ! 0, i.e.: T 1 X T t=1 Nt (96) 0 t p' 1 1 N t ; Ft gt= 1 N =1 t q E 0 satis…es conditions of a mixingale weak law,3 1 0 t p' L !1 0, t q j as N; T ! 1 at any rate. Convergence in L1 norm implies convergence in probability. This completes the proof of p 1 result (86). Under the condition k k = O N 2 , result (88) follows from result (86) by noting that N = O (1). j Lemma 2 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for and ' with growing dimension N any p; q 2 f0; 1g, and for any sequences of non-random vectors 1 such that k kc = O (1) and k'kc = O (1), we have T 1 X T t=1 29 = = max n2f1;::;N g max n2f1;::;N g max n2f1;::;N g max N X s=1 N X s=1 N X s=1 n2f1;::;N g RR0 31 xt p E Matrix B is symmetric by construction. Therefore kBk kBkr 30 0 2 r k 0 xt p p (97) ! 0, p kBkr kBkc = kBkr , where ns k max i2f1;::;N g max i2f1;::;N g N X N X s=1 `0 =0 N X N X j=1 `=0 jri` rj` rs` rn` j N N X X j=1 `=0 ! jrs`0 rn`0 j kRk2r kRk2c = O (1) jri` rj` j max N X `0 =0 i2f1;::;N g ! jrs`0 rn`0 j N X N X j=1 `=0 ! jri` rj` j Su¢ cient condition for uniform integrability is L1+" uniform boundedness for any " > 0. Davidson (1994, Theorem 19.11). ECB Working Paper Series No 998 January 2009 47 and T 1 X T t=1 0 xt 0 p ' xt q 0 E p 0 p ' xt q xt (98) ! 0. Furthermore, for k k = O (1) and k'kc = O (1) we have T 1 X T t=1 where t 0 0 t p' p ft (99) ! 0, q is de…ned in equation (40). Proof. Let TN = T (N ) be any non-decreasing integer-valued function of N such that limN !1 TN = 1. Consider the following two-dimensional array f 1 1 , N t ; Ft gt= 1 N =1 Nt 1 TN = 0 where fFt g denotes an increasing sequence of -…elds (Ft time t and Nt independence of ft and t0 1 TN ft q, Ft ) such that Ft includes all information available at 1 for any t; t0 2 Z, we have where & n = sup N 2N k k2 = O (1), k k < ( 1 1 N =1 fcN t g1 t= be two-dimensional array for all t 2 Z and N 2 N. Using submultiplicative property of matrix norm, and E q 0 t p' is measurable with respect to Ft for any N 2 N. Let of constants and set cN t = ft de…ned by 2 ( 2 E Nt j Ft cN t 2 maxf0;n pg k k k kk k `=0 by Assumption 8, and k k is covariance stationary and k'0 0 1 X n ) 2` k k E & n, n 0 E ' ft q j Ft 2 n o ) . p k kc k kr = O (1) by Assumption 9. Furthermore, since 'k = O (1) (by condition k'kc = O (1) and Assumption 12), we have E n E '0 f t q j Ft 2 n o = O (1) . It follows that & n has following properties & 0 = O (1) and & n ! 0 as n ! 1. Array f N t =cN t g is thus uniformly bounded in L2 norm. This proves uniform integrability of array f Furthermore, using Liapunov’s inequality, two-dimensional array f 1 1 N t ; FN t gt= 1 N =1 N t =cN t g. is L1 -mixingale with respect to constant array fcN t g. Noting that equations (91) and (92) hold, it follows that the array f N t ; Ft g satis…es P N L1 conditions of a mixingale weak law,3 2 which implies Tt=1 N t ! 0. Convergence in L1 norm implies convergence in probability. This completes the proof of result (99). Assumption 11 implies that sequence 0 (as well as '0 ) is deterministic and bounded. Vector of endogenous variables xt can be written as xt = 32 48 Davidson (1994, Theorem 19.11) ECB Working Paper Series No 998 January 2009 + ft + t. Process ft is independent of t. j Suppose N; T ! 1 at any rate. Processes f in mean by Lemma 1 since k k 0 t pg and f 0 0 t p' t qg are ergodic k kc = O (1). Furthermore, T 1 X T t=1 0 p ft 0 E (ft ) ! 0, q 0 E ft ft0 and T 1 X T t=1 since ft is covariance stationary m 0 f t '0 f t 0 q p ' ! 0, 1 dimensional process with absolute summable autocovariances (ft is ergodic in mean as well as in variance), and 0 0 0 0 ' ' 2 = O (1) , = O (1) , by Assumption 12, conditionk kc = O (1) and condition k'kc = O (1). Sum of bounded deterministic process and independent processes ergodic in mean is a process that is ergodic in mean as well. This completes the proof. j Lemma 3 Let xt be generated by model (32), Assumptions 7-12 hold and N; T ! 1 at any rate. Then for any p; q 2 f0; 1g, for any sequence of non-random matrices of weights W of growing dimension N mw satisfying conditions (33)-(34), and for any r 2 f1; ::; mw g, p N T p N T p N T p N T where the process t T X p W0 t p W0 t p xW;t q W0 t p xi;t q t=1 T X (100) ! 0, p t=1 T X t=1 T X t=1 (101) ! 0, p (102) ! 0, p (103) git qit ! 0, is de…ned in equation (40), vector git = 1; 0 0 0 0 i;t 1 ; xW t ; xW;t 1 and qit is de…ned equation (47). Proof. Let wr for r 2 f1; ::; mw g denote the rth column vector of matrix W. Noting that granularity condition (33), result p T N X 0 wr T t=1 p t p p N wr = O (1) by (104) !0 follows directly from Lemma 1, equation (87). This completes the proof of result (100). Let ' be any sequence of non-random N We have p T N X 0 wr T t=1 1 dimensional vectors of growing dimension such that k'kc = O (1). 0 t p ' xt q = p T N X 0 wr T t=1 0 t p' ( + ft q + t q) . (105) ECB Working Paper Series No 998 January 2009 49 Since p N wr = O (1) for any r 2 f1; ::; mw g by condition (33), we can use Lemma 1, result (88), which implies p T N X 0 wr T t=1 0 t p' E wr0 t q 0 t p' p t q (106) ! 0. Sequence f'0 g is deterministic and bounded in N , and therefore it follows from Lemma 1, result (87), that p T N X 0 wr T t=1 p 0 t p' (107) ! 0. Similarly, Lemma 2 equation (99) implies p T N X 0 wr T t=1 0 t p' p ft q (108) ! 0. Results (106), (107) and (108) establish p T N X 0 wr T t=1 p 0 t p ' xt q ! (109) 0. Result (101) follows from equation (109) by setting '= wl for any l 2 f1; ::; mw g. Result (102) follows from equation 1 dimensional selection vector for the ith element. (109) by setting ' = ei where ei is N Finally, the result (103) directly follows from results (100)-(102). This completes the proof. j Lemma 4 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for any sequence of non-random matrices of weights W of growing dimension N T 1 X 0 git git T t=1 0 ) and vector git = 1; where matrix Ci = E (git git mw satisfying conditions (33)-(34), p (110) Ci ! 0, 0 0 0 0 i;t 1 ; xW t ; xW;t 1 . Proof. Result (110) directly follows from Lemmas 1, 2 and 3. j Lemma 5 Let xt be generated by model (32), Assumptions 7-12 hold, and N; T ! 1 at any rate. Then for any sequence of non-random matrices of weights W of growing dimension N for any …xed p 0, T 1 X 0 W T t=1 where the process mw satisfying conditions (33)-(34), and t t p uit p (111) ! 0, is de…ned in equation (40). If in addition T =N ! {, with 0 T 1 X 0 p W T t=1 t p uit p { < 1, (112) ! 0. Proof. Let TN = T (N ) be any non-decreasing integer-valued function of N such that limN !1 TN = 1 and limN !1 TN =N = { < 1, where { 0 is not necessarily nonzero. De…ne N it 50 ECB Working Paper Series No 998 January 2009 1 = p TN W0 t p uit E W0 t p uit , (113) where the subscript N is used to emphasize the number of cross section units.3 3 Let fFt g denote an increasing sequence of -…elds (Ft Ft ) with 1 i 2 N, the vector array f N it measurable with respect of Ft . First it is established that for any …xed 1 1 N it =cN t ; Ft gt= 1 N =i 1 N TN is uniformly integrable, where cN t = p . For p > 0, we can write 0 N it N it 2 cN t E = N = E 2 ii N " 1 X W 0 ` ` p uit ut `=0 1 X W0 0` ` ! 1 X W 0 ` ut ` p uit `=0 !0 # , W , `=0 N 1 kWk2 k k O (1) , = where kWk2 = O N 2 ii 1 X ` 2 , `=0 by condition (33), k k = O (1) by Assumption 9, and 8. For p = 0, we have E 0 N it N it 2 cN t = is N and f 2 kWk k ii k + 2 2 N ii ` ut kWk k k O (1) , N it =cN t g N it =cN t g. E jE ( W 0 ` uit ` 2 `=0 ! 1 X = O (1) by Assumption , ` 2 +O N 1 `=1 ! , N symmetric matrix with the element (n; s) equal to E (uit uit unt ust ). Therefore for p the two-dimensional vector array f f 1 X `=1 = ii W ut uit + N V ar N where as before 0 P1 N it 1 1 N it ; FN t gt= 1 N =i j Ft n )j = 8 < : 0, is uniformly bounded in L2 norm. This proves uniform integrability of for any n > 0 and any …xed p 0 mw cN t O (1) for n = 0 and any …xed p 0 , (114) 0 is L1 -mixingale with respect to constant array fcN t g.3 4 Note that lim N !1 TN X cN t = lim N !1 t=1 TN X t=1 1 p = lim N !1 N TN r p TN = { < 1, N and lim N !1 TN X t=1 c2N t = lim N !1 TN X t=1 1 1 = lim = 0. N !1 N TN N Therefore for each …xed i 2 N, each of the mw two-dimensional arrays given by the elements of vector array 33 Note thatW and t p change with N , but as before we ommit subscript N here to keep the notation simple. The last equality in equation (114) takes advatage of Liapunov’s inequality. mw is mw 1 dimensional vector of ones. 34 ECB Working Paper Series No 998 January 2009 51 f 1 1 N it ; Ft gt= 1 N =i satis…es conditions of a mixingale weak law3 5 , which implies TN 1 X 0 p W TN t=1 p TN E W0 t p uit t p uit L !1 0. But p TN E W0 t p uit = c p TN E W0 ut uit = c p TN O 1 N ! 0, since limN !1 TN =N = { < 1. Convergence in L1 norm implies convergence in probability. This completes the proof of result (112). Result (111) is established in a very similar fashion. De…ne new vector array qN it = p1 TN N it where N it is array de…ned in (113) and i 2 N is …xed. Let TN = T (N ) be any non-decreasing integer-valued function of N such that such n p o1 1 TN qN it =cN t ; Ft t= 1 is uniformly that limN !1 TN = 1. Notice that for any …xed i 2 N, vector array N =i integrable because f 1 N it =cN t ; Ft gt= 1 1 N =i is uniformly integrable. Furthermore, p1 cN t mixingale with respect to the constant array since f TN 1 1 N it ; Ft gt= 1 N =i fqN it ; Ft g1 t= 1 1 N =i is L1 - is L1 mixingale with respect to the constant array fcN t g. Note that lim N !1 TN X t=1 TN X 1 1 1 p p = lim p = 0, cN t = lim N !1 N !1 TN T N N t=1 N and lim N !1 TN X t=1 1 p cN t TN 2 = lim N !1 Therefore for any …xed i 2 N, a mixingale weak law TN X t=1 36 TN X 2 1 p TN N t=1 = lim N !1 1 = 0. TN N implies L qN it !1 0 as N ! 1. (115) Since also E W0 t p uit =O N 1 , it follows T 1 X 0 W T t=1 t p uit L !1 0, j as N; T ! 1 at any rate. Convergence in L1 norm implies convergence in probability. This completes the proof of result (111). j Lemma 6 Let xt be generated by model (32), Assumptions 7-12 hold and N; T ! 1 such that T =N ! {, with { < 1. Then for any sequence of non-random matrices of weights W of growing dimension N 0 conditions (33)-(34), we have, 35 36 52 See Davidson (1994, Theorem 19.11). See Davidson (1994, Theorem 19.11). ECB Working Paper Series No 998 January 2009 mw satisfying a) under Assumption 13, 1 Ci T 1 X D p eit uit ! N (0; Iki ) , g T t=1 1 2 ii 0 eit = 1; eit where Ci = E (e git g ) and g 0 0 W ; ft 1 0 0 i;t 1 ; ft 0 0 W (116) . b) under Assumption 14, 1 p ii T where matrix vi 1 2 vi T X 1 uit t=1 P1 0 = E (vit vit ) and vector vit = S0i Proof. Let a be any ki vi;t ` `=0 D ! N (0; Ihi ) , ut (117) `. 1 dimensional vector such that kak = 1 and de…ne Nt 1 = p TN 1 2 a0 Ci ii eit uit , g where TN = T (N ) is any non-decreasing integer-valued function of N such that limN !1 TN = 1 and limN !1 TN =N = { < 1, where 0 a0 Ci 1 2 { < 1. Array f N t ; Ft g is a stationary martingale di¤erence array.3 7 Lemmas 1 and 2 imply eit is ergodic in variance, in particular g TN 1 X 0 a Ci TN t=1 1 2 0 eit g eit g Ci 1 2 p a ! 1. eit and uit are independent and the fourth moments of uit are …nite. Therefore a0 Ci g and TN X t=1 Furthermore, E ii 1 0 a Ci 1=2 eit uit g p 2 Nt ! 1 2 eit uit is ergodic in variance g (118) 1. 4 = O (1) and therefore lim N !1 TN X 4 Nt E = 0. t=1 Using Liapunov’s theorem (Davidson, 1994, Theorem 23.11), Lindeberg condition3 8 holds, which in turn implies max j 1 t TN Ntj p ! 0 as N ! 1. (119) Results (118), (119) and the martingale di¤erence array central limit theorem (Davidson, 1994, Theorem 24.3) establish TN X t=1 Since equation (120) holds for any ki Nt 1 = p TN a0 Ci ii 1 2 TN X t=1 D eit uit ! N (0; Iki ) g (120) 1 dimensional vector a such that kak = 1, result (116) directly follows from equation (120) and Davidson (1994, Theorem 25.6). Result (117) can be established in the same way as the result (116), but this time we set 37 Ft . 38 As before, let fFt g denote an increasing sequence of -…elds (Ft 1 Ft ) with Nt Nt 1 a0 TN ii = p 1 2 vi vi;t 1 uit , measurable with respect of See Davidson (1994, Condition 23.17). ECB Working Paper Series No 998 January 2009 53 where a is any hi 1 dimensional vector such that kak = 1. j Lemma 7 Let xt be generated by model (32), and suppose Assumptions 7-12 hold and N; T ! 1 at any rate. Then for any arbitrary matrix of weights W satisfying conditions (33)-(34), and for any p; q 2 f0; 1g : T 1 X T t=1 T 1 X T t=1 T 1 X T t=1 T 1 X T t=1 W;t p 0 W;t p ft q W;t p , , (122) 1 p N = op 0 W;t q (121) 1 p N = op 0 W;t q i;t p 1 p N = op , 1 p N = op (123) , (124) 0 iQ = op (1) . T (125) Furthermore, H0 Q Q0 Q = A0 + op T T 1 p N , (126) Z0i H Z0 Q = i A + op T T 1 p N , (127) H0 H Q0 Q = A0 A + op T T 1 p N , (128) H0 ui Q0 ui = A0 + op T T 1 p N , (129) where T vit = S0i P1 `=0 ` ut `, i hi = (vi0 ; vi1 ; :::; vi;T 0 1) , (130) matrices H and Zi are de…ned below equation (53), and matrices Q, F and A are de…ned in equations (54)-(55). Proof. Result (121) follows directly from equation (87) of Lemma 1 since the spectral norm of any column vector of the matrix W is O N of W;t 1 2 . Result (122) follows from result (121) by noting that ft is independently distributed P p and all elements of the variance matrix of ft are …nite. Furthermore, since (by Lemma 1) T1 Tt=1 vit ! 0, equation (125) follows. Results (123) and (124) follows directly from equation (88) of Lemma 1 by noting that p NE i;t p 0 W;t q =O 1 p N W;t p 0 W;t q =O 1 p N (131) as well as3 9 p NE . In order to prove equations (126)-(129), …rst note that the row t of the matrix H 39 (132) QA is 0; 0 W t; 0 W;t 1 . Results (131) and (132) are straightforward to establish by taking the row norm and by noting that the granularity conditions (33)-(34) imply kWkr = O N 1 . 54 ECB Working Paper Series No 998 January 2009 Using results (121)-(124), we have (H QA)0 Q T Z0i (H QA) T H0 (H QA) T (H QA)0 (H T QA) = = = = 1 T 20 T 6B X 6B 6B [email protected] t=1 2 C C C A Wt W;t 1 2 T 6 1 X6 6 T t=1 6 4 1 0 i;t 1 0 0 B B B @ Wt W;t 1 0 1 0 T 6 B xW t 1 X6 [email protected] AB B 6 T t=1 4 x @ W;t 1 20 T 6B 1 X6 6B B T t=1 6 [email protected] ft0 , 1, 0 Wt W;t 1 CB CB CB [email protected] Wt W;t 1 1 10 3 C7 C7 = op C7 A7 5 10 0 ft0 0 Wt W;t 1 3 7 7 7 = op 5 1 p N 10 3 C7 C7 = op C7 A7 5 10 3 C7 C7 = op C7 A7 5 1 p N , , (133) (134) 1 p N , (135) 1 p N , (136) Equations (133)-(134) establish results (126) and (127). Note that H0 H T = = = H0 (H QA) H0 (QA) + , T T 0 (H QA)0 Q H (H QA) Q0 Q + A + A0 A, T T T 0 1 QQ , A+op p A0 T N where the last equality uses equations (133) and (135). This completes the proof of result (128). Equation (115) (see proof or Lemma 5) implies T 1 X T t=1 W;t p uit E( W;t p uit ) p NE ( j as N; T ! 1 at any rate. Result (129) follows by noting that p ! 0, W;t p uit ) =O N 1 2 . This completes the proof. j Lemma 8 Let xt be generated by model (32), suppose Assumptions 7-12, 14 hold, and N; T ! 1 at any rate. Then for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have Q0 Q p ! T QQ Q , (137) is nonsingular, and 0 i T p i vi ! 0, (138) ECB Working Paper Series No 998 January 2009 55 where Q f (`) = E (ft ft0 ` ), vi 0 1 0 B B =B 0 @ 0 1 0 f (0) f f (1) f C C (1) C , A (0) = E (vi vi0 ), matrix Q is de…ned in equation (54), and matrix Proof. Assumption 11 implies matrix Q i = (vi0 ; vi1 ; :::; vi;T 0 1) . is nonsingular. Result (137) directly follows from the ergodicity properties of the covariance stationary time-series process ft . j Consider now asymptotics N; T ! 1 at any rate. Lemma 1 implies that hi 1 dimensional vector vit = S0i P p is ergodic in variance, in particular T1 Tt=1 S0i t 0t Si E (S0i t 0t Si ) ! 0.4 0 This completes the proof. t j Lemma 9 Let xt be generated by model (32), suppose Assumptions 7-12 and 14 hold, and N; T ! 1 at any rate. Then for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have Z0i MH Zi T Z0i MQ Zi T Z0i MH ui p T vi 1 p N , (139) p = ! T N op 0 i MQ ui p T = is de…ned in Assumption 14, MH = IT (140) ! 0, r vi Z0i MH Q p T where Z0i MQ Zi + op T = , (141) r + op T N ! , (142) + H (H0 H) H0 , matrices H and Zi are de…ned below equation (53), matrices Q and F are de…ned in equation (54), and matrix i = (vi0 ; vi1 ; :::; vi;T 0 1) . Proof. Z0i MH Zi Z0 Zi = i T T Z0i H T H0 H T + H 0 Zi . T (143) Results (127)-(128) of Lemma 7 imply Z0i H T H0 H T + H 0 Zi Q0 Q Z0 Q = i A A0 A T T T + A0 Q0 Zi + op T p1 N . (144) Using de…nition of the Moore-Penrose inverse, it follows A0 Multiply equation (145) by Q0 Q T Q0 Q A T 1 A0 (AA0 ) 1 Q0 Q A T A0 Q0 Q A T A0 = Q0 Q A . T A from the left and by A0 (AA0 ) A A0 40 + Q0 Q A T + A0 = Q0 Q T 1 Q0 Q T (145) 1 from the right to obtain4 1 1 . (146) kSi kc = O (1) by Assumption 7. Note that plimT !1 T1 Q0 Q is nonsingular by Lemma 8, equation (137). AA0 is nonsingular, since matrix A has full row-rank by Assumption 14. 41 56 ECB Working Paper Series No 998 January 2009 Equations (146) and (144) imply Z0i H T H0 H T + H 0 Zi Z0 Q = i T T 1 Q0 Q T Q0 Zi + op T p1 N . (147) Result (139) follows from equations (147) and (143). System (32) implies Zi = 0 i Si 0 i Si + F ( 1) + i. (148) Since Q = [ ; F; F ( 1)], it follows Z0i MQ Zi = T 0 i MQ i T 0 i = i T 0 iQ + Q0 Q T T 1 Q0 T i . (149) Using equations (125), (137) and (138), result (140) follows directly from (149). Results (126)-(128) of Lemma 7 imply Z0i H T + H0 H T + H0 Q Q0 Q Z0 Q = i A A0 A T T T A0 Q0 Q + op T p1 N . (150) Substituting equation (146), it follows Z0i H T + H0 H T H0 Q Z0 Q = i T T 1 Q0 Q T Equation (151) implies r Z0i MH Q Z0 MQ Q p = ip + op T T T N ! Q0 Q + op T = op r T N p1 N ! . (151) . This completes the proof of result (141). Results (127)-(129) of Lemma 7 imply Z0i H T H0 H T + + Q0 Q H0 ui Z0 Q = i A A0 A T T T A0 Q0 ui + op T p1 N . Substituting equation (146), it follows Z0i H T Noting that MQ ( 0 i Si +F 0 i Si ) H0 H T + H0 Q Z0 Q = i T T Q0 Q T 1 Q0 ui + op T p1 N . (152) = 0 since Q = [ ; F; F ( 1)], equations (152) and (148) imply Z0i MH ui p T 0 = Zi MQ ui p + op T 0 = i MQ ui p T + op ! T , N r ! T . N r This completes the proof. j Lemma 10 Let xt be generated by model (32), and suppose Assumptions 7-12 and 14 hold, and N; T ! 1 at any ECB Working Paper Series No 998 January 2009 57 rate. Then for any arbitrary matrix of weights W satisfying conditions (33)-(34) and Assumption 14, we have Z0i MH i ( 1) T Z0i MH ui p T = 1 p N op , 0 ui pi + op T = where matrices MH ,and Zi are de…ned below equation (53), i;0 ; :::; i;T 1 0 (153) r T N ! + op (1) , (154) = (vi0 ; vi1 ; :::; vi;T i 1) and vector i ( 1) = . Proof. Z0i H0 k ib kr =O N 1 ( 1) T i = i ( 1) T = " T 1 X xi;t 1 T t=1 20 T 1 X [email protected] xW t T t=1 x W;t 1 p T = = Q0 u i p T 1 X ` ut ` 1 `=0 1 A 0 ib 1 X !0 # ` ut , ` 1 `=0 !0 3 5. by Assumption 7, therefore result (153) directly follows from equations (134) and (135). 0 i MQ ui where 0 ib 0 0 ui Q Q0 Q pi + i T T T 0 u i pi + op (1) , T = Op (1), plimT !1 T1 Q0 Q is nonsingular by Lemma 8, and 1 Q0 ui p , T (155) 0 iQ T = op (1) by Lemma 7, equation (125). Substituting (155) into equation (142) implies result (154). This completes the proof. Proof of Theorem 1. a) Substituting for xit in equation (49) yields bi j i = T 1 X 0 git git T t=1 ! 1 T T 1 X 1 X git qit + git uit T t=1 T t=1 ! . (156) With N; T ! 1 in any order, Lemma 5 yields4 2 T 1 X p git uit ! 0. T t=1 (157) Also using Lemmas 3 and 4 we have T 1 X p git qit ! 0, T t=1 and 42 1 T PT T 1 X 0 git git T t=1 p 58 ECB Working Paper Series No 998 January 2009 (159) CiN ! 0, p xj;t 1 uit ! 0 since xjt is ergodic in mean by Lemma 2 and uit is independent of xj;t P p f1; ::; N g and any N 2 N. Furthermore, using similar arguments, T1 Tt=1 ft uit ! 0. t=1 (158) 1 for any j 2 respectively. Assumption 13 postulates that the matrix CiN is invertible for any N N0 and CiN1 is bounded in N . It follows from equation (159) that T 1 X 0 git git T t=1 ! 1 p CiN1 ! 0. (160) p Result b i i ! 0 directly follows from equations (157), (158) and (160). b) Multiplying equation (156) by p T (b i p T yields i) T 1 X 0 git git T t=1 = ! 1 T T 1 X 1 X p git qit + p git uit T t=1 T t=1 ! . (161) j With N; T ! 1 such that T =N ! { < 1, Lemma 3 can be used to show that T 1 X p p git qit ! 0. T t=1 (162) Since CiN1 = O (1), equations (160) and (162) now yield T 1 X 0 git git T t=1 ! 1 T 1 X p p git qit ! 0. T t=1 (163) Lemma 5 establishes T 1 X p T t=1 W;t p uit p ! 0 for p 2 f0; 1g . (164) It follows from equation (164) that eit = 1; where g 0 0 i;t 1 ; ft 0 0 W ; ft 1 0 0 W T 1 X p (git T t=1 p eit ) uit ! 0, g (165) . Lemma 6 establishes that T 1 1 X D eit uit ! N (0; Iki ) , g CiN2 p ii;N T t=1 1 (166) Equations (160), (163), (165) and (166) imply result (50). c) Lemma 4 establishes equal to u bit = uit where 1 T 1 T PT t=1 0 git (b i i ), T T 1 X 2 1 X 2 u bit = uit T t=1 T t=1 PT t=1 u2it 2 ii;N p 0 git git CiN ! 0. The estimated residuals from auxiliary regression (48) are which implies is established in Lemma 4, and 1 T T 1 X git uit + (b i T t=1 p p ! 0, b i 0 i) 2 (b i i PT t=1 0 i) T 1 X 0 git git T t=1 ! 0 is established in part (a) of this proof, p 1 T ! (b i PT t=1 i) , 0 git git (167) p CiN ! 0 git uit ! 0 is established in equation (157). This completes the proof. ECB Working Paper Series No 998 January 2009 59 Proof of Theorem 2. Vector xi can be written, using system (32), as xi = where i ( 1) = i0 ; :::; i;T 0 0 i Si i 1 0 + Zi i +F Si i + i ( 1) + ui , (168) . Substituting equation (168) into the partition least squares formula (53) and noting that by Lemma 9, Z0i MH Q p = op T it follows p T bi 0 F ( 1) i i = 1 Z0i MH Zi T " r T N ! , Z0i MH (ui + p T (169) i ( 1)) + op r T N !# . (170) Lemma 9 also establishes that Z0i MH Zi T where vi p vi j ! 0, as N; T ! 1 at any rate, (171) 0 = E (vit vit ) is nonsingular by Assumption 14. j Consider now asymptotics N; T ! 1 such that T =N ! { < 1. Lemma 10 establishes Z0i MH i ( 1) p p ! 0, T and 0 Z0i MH ui ui p = pi + op T T where i = (vi0 ; :::; vi;T 0 1) . 1 2 vi T X t=1 The desired result (56) now follows from (170)-(174). 60 T N ! + op (1) , (173) Also from Lemma 6 1 p ii T ECB Working Paper Series No 998 January 2009 r (172) vi;t 1 uit D ! N (0; Ihi ) . (174) European Central Bank Working Paper Series For a complete list of Working Papers published by the ECB, please visit the ECB’s website (http://www.ecb.europa.eu). 944 The New Area-Wide Model of the euro area: a micro-founded open-economy model for forecasting and policy analysis” by K. Christoffel, G. Coenen and A. Warne, October 2008. 945 “Wage and price dynamics in Portugal” by C. Robalo Marques, October 2008. 946 “Macroeconomic adjustment to monetary union” by G. Fagan and V. Gaspar, October 2008. 947 “Foreign-currency bonds: currency choice and the role of uncovered and covered interest parity” by M. M. Habib and M. 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