Common Correlated Effects Estimation of Heterogeneous Dynamic Panel Data Models with Weakly Exogenous Regressors *

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1 Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paer o. 46 htt:// Common Correlated Effects Estimation of Heterogeneous Dynamic Panel Data Models with Weakly Exogenous Regressors * Alexander Chudik Federal Reserve Bank of Dallas, CAFE and CIMF M. Hashem Pesaran University of Southern California, CAFE, USA, and rinity College, Cambridge, UK Aril 203 Abstract his aer extends the Common Correlated Effects CCE aroach develoed by Pesaran 2006 to heterogeneous anel data models with lagged deendent variable and/or weakly exogenous regressors. We show that the CCE mean grou estimator continues to be valid but the following two conditions must be satisfied to deal with the dynamics: a sufficient number of lags of cross section averages must be included in individual equations of the anel, and the number of cross section averages must be at least as large as the number of unobserved common factors. We establish consistency rates, derive the asymtotic distribution, suggest using co-variates to deal with the effects of multile unobserved common factors, and consider jackknife and recursive de-meaning bias correction rocedures to mitigate the small samle time series bias. heoretical findings are accomanied by extensive Monte Carlo exeriments, which show that the roosed estimators erform well so long as the time series dimension of the anel is sufficiently large. JEL codes: C3, C33 * Alexander Chudik, Research Deartment, Federal Reserve Bank of Dallas, Pearl Street, Dallas, X alexander.chudik@dal.frb.org. M. Hashem Pesaran, Deartment of Economics, University of Southern California, 3620 South Vermont Avenue, Los Angeles, CA esaran@usc.edu. We are grateful to Ron Smith, Vanessa Smith, akashi Yamagata and Qiankun Zhou for helful comments. In writing of this aer, Chudik benefited from the visit to the Center for Alied Financial Economics CAFE. Pesaran acknowledges financial suort from ESRC grant no. ES/I03626/. he views in this aer are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.

2 Introduction In a recent aer, Pesaran 2006 roosed the Common Correlated Effects CCE aroach to estimation of anel data models with multi-factor error structure, which has been further develoed by Kaetanios, Pesaran, and Yagamata 20, Pesaran and osetti 20, and Chudik, Pesaran, and osetti 20. he CCE method is shown to be robust to different tyes of cross section deendence of errors, ossible unit roots in factors, and sloe heterogeneity. However, the CCE aroach as it was originally roosed does not cover the case where the anel includes a lagged deendent variable and/or weakly exogenous variables as regressors. his aer extends the CCE aroach to allow for such regressors. his extension is not straightforward because coeffi cient heterogeneity in the lags of the deendent variable introduces infinite order lag olynomials in the large relationshis between cross-sectional averages and the unobserved factors Chudik and Pesaran, 203a. Our focus is on stationary heterogenous anels with weakly exogenous regressors where the cross-sectional dimension and the time series dimension are suffi ciently large. We focus on estimation and inference of the mean coeffi cients, and consider the alication of bias correction techniques to deal with the small bias of the estimators. Recent literature on large dynamic anels focuses mostly on how to deal with cross-sectional CS deendence assuming sloe homogeneity. Estimation of anel data models with lagged deendent variables and cross-sectionally deendent errors has been considered in Moon and Weidner 200a and 200b, who roose a Gaussian quasi maximum likelihood estimator QMLE. 2 Moon and Weidner s analysis assumes homogeneous coeffi cients, and therefore is not alicable to dynamic anels with heterogenous coeffi cients. 3 Similarly, the interactive-effects estimator IFE develoed by Bai 2009 also allows for cross-sectionally deendent errors, but assumes homogeneous sloes. 4 Song 203 extends the analysis of Bai 2009 by allowing for a lagged deendent variable as well as coeffi cient heterogeneity, but rovides results on the estimation of cross-section secific coeffi cients only. his aer rovides an alternative CCE tye estimation aroach to Song s extension of the See Everaert and Groote 202 who derive asymtotic bias of CCE ooled estimators in the case of dynamic homogeneous anels. 2 See also Lee, Moon, and Weidner 20 for an extension of this framework to anels with measurement errors. 3 Pesaran and Smith 995 show that in the resence of coeffi cient heterogeneity ooled estimators are inconsistent in the case of anel data models with lagged deendent variables. 4 Earlier literature on large anels tyically ignores cross section deendence of errors, including ooled mean grou estimation roosed by Pesaran, Shin, and Smith 999, fully modified OLS estimation by Pedroni 2000 or the anel dynamic OLS estimation by Mark and Sul hese aers can also handle anels with nonstationary data. here is also a large literature on dynamic anels with large but finite, which assumes homogeneous sloes.

3 IFE estimator. In addition, we roose a mean grou estimator of the mean coeffi cients, and show that CCE tyes estimators once augmented with a suffi cient number of lags and cross-sectional averages erform well even in the case of dynamic models with weakly exogenous regressors. We also show that the asymtotic distribution of the CCE estimators develoed in the literature continue to be alicable to the more general setting considered in this aer. Our method could extend to Song s IFE and we also investigate the erformance of the mean grou estimator based on Song s unit-secific coeffi cient estimates. More secifically, in this aer we considered estimation of autoregressive distributed lagged ARDL anel data models where the deendent variable of the i th cross section unit at time t, y it, is exlained by its lagged values, current and lagged values of k weakly exogenous regressors, x it, m unobserved ossibly serially correlated common factors, f t, and a serially uncorrelated idiosyncratic error. In addition to the regressors included in the anel ARDL model, following Pesaran, Smith, and Yamagata 203 we also assume that there exists a set of additional covariates, g it, that are affected by the same set of unobserved common factors, f t. his seems reasonable considering that agents in making their decisions face a common set of factors such as technology, institutional set us and general economic conditions, which then get manifested in many variables, whether included in the anel data model under consideration or not. Similar arguments also underlie forecasting using a large number of regressors oularized recently in econometrics by Stock and Watson 2002 and Forni et al A necessary condition for the CCE mean grou CCEMG estimator to be valid in the case of ARDL anel data models is that the number of cross-sectional averages based on x it and g it must be at least as large as the number of unobserved common factors minus one m. In ractice, where the number of unobserved factors is unknown, it is suffi cient to assume that the number of available cross-sectional averages is at least m max, where m max denotes the assumed maximum number of unobserved factors. In most economic alications m max is likely to be relatively small. 5 We also reort on the small samle roerties of CCEMG estimators for anel ARDL models, using a comrehensive set of Monte Carlo exeriments. In articular, we investigate two bias correction methods, namely the half-anel jackknife due to Dhaene and Jochmans, 202, and the recursive mean adjustment due to So and Shin, 999. We find that the roosed estimators have 5 Stock and Watson 2002, Giannone, Reichlin, and Sala 2005 conclude that only few, erhas two, factors exlain much of the redictable variations, while Bai and g 2007 estimate four factors and Stock and Watson 2005 estimate as many as seven factors. 2

4 satisfactory erformance under different dynamic arameter configurations, and regardless of the number of unobserved factors, so long as they do not exceed the number of cross-sectional averages, and the time dimension is suffi ciently large. We comare the erformance of CCEMG with the mean grou estimator based on Song s IFE, and also with Moon and Weidner s QMLE, Bai s IFE estimators develoed for sloe homogeneous ARDL anels. We find that jackknife bias correction is more effective in dealing with the small samle bias than the recursive mean adjustment rocedure. Also, the bias correction seems to be helful only for the coeffi cients of the lagged deendent variable. he uncorrected CCEMG estimators of the coeffi cients of the regressors, x it, seem to work fine even in the case of anels with a relatively small time dimension. he remainder of the aer is organized as follows. Section 2 extends the multifactor residual anel data model considered in Pesaran 2006 by introducing lagged deendent variables and allowing the regressors to be weakly exogenous. Section 3 develos a dynamic version of the CCEMG estimator for anel ARDL models. Section 4 discusses the jackknife and recursive demeaning bias correction rocedures. Section 5 introduces the mean grou estimator based on Song s individual estimates, describes the Monte Carlo exeriments, and reorts the small samle results. Mathematical roofs are rovided in the Aendix and additional Monte Carlo findings are rovided in a Sulement. 2 Panel ARDL Model with a Multifactor Error Structure Suose that the deendent variable, y it, the regressors, x it, and the covariates, g it, are generated according to the following linear covariance stationary dynamic heterogenous anel data model, y it = c yi + φ i y i,t + β 0ix it + β ix i,t + u it, u it = γ if t + ε it, 2 and ω it = x it g it = c ωi + α i y i,t + Γ if t + v it, 3 for i =, 2,..., and t =, 2,...,, where c yi and c ωi are individual fixed effects for unit i, x it is k x vector of regressors secific to cross-section unit i at time t, g it is k g vector of covariates 3

5 secific to unit i, k x + k g = k, f t is an m vector of unobserved common factors, ε it are the idiosyncratic errors, Γ i is an m k matrix of factor loadings, α i is a k vector of unknown coeffi cients, and v it is assumed to follow a general linear covariance stationary rocess distributed indeendently of the idiosyncratic errors, ε it. he rocess for the exogenous variables, 3, can also be written equivalently as a anel ARDL model in ω it. But we have chosen to work with this articular secification as it allows us to distinguish between cases of strict and weak exogeneous regressors in terms of the feed-back coeffi cients, α i. he case of strictly exogenous regressors, covered in Pesaran 2006, refers to the secial case when α i = 0 k. As in the earlier literature, the above secification also allows the regressors to be correlated with the unobserved common factors. Lags of x it and g it are not included in 3, but they could be readily included. In order to kee the notations and exosition simle we also abstract from observed common effects, additional lags of the deendent variable, and other deterministic terms in and 3. Such additional regressors can be readily accommodated at the cost of further notational comlexity. In the above ARDL formulation, we secify the same lag orders for y it and x it because it is desirable in emirical alications to start with a balanced lag order to avoid otential roblems connected with ersistent regressors. It is also worth noting that a number of anel data models investigated in the literature can be derived as secial cases of -3. he analysis of Moon and Weidner 200a and 200b assumes that β i0 = β 0, β i = β and φ i = φ. Bai 2009 assumes β i0 = β 0, β i = β and φ i = 0. Under the restriction β i = φ i β 0i, 4 we have y it θ ix it = c yi + φ i yi,t θ ix it + uit, where θ i = β i /φ i, which in turn can be written as assuming that φ i < y it = c yi + θ ix it + γ i f t + ε it, 5 where c yi = c yi / φ i, ε it = φ i L ε it is a serially correlated error term, and f t is a new set of unobserved common factors. Estimation and inference in anel model 5 have been 4

6 studied by Pesaran 2006 who introduced the CCE aroach. his aroach has been shown to be robust to an unknown number of unobserved common factors Pesaran, 2006, and Chudik, Pesaran, and osetti, 20, ossible unit roots in factors Kaetanios, Pesaran, and Yagamata, 20, serial correlation of unknown form in ε it Pesaran, 2006, satial or other forms of weak cross-sectional deendence in ε it Pesaran and osetti, 20, and Chudik, Pesaran, and osetti, 20. However, if the restrictions set out in 4 on β 0i and β i do not hold then the CCE aroach is no longer alicable and the standard CCE estimators could be seriously biased, even asymtotically. 6 Our objective in this aer is to consider estimation and inference in the anel ARDL model -3, where the arameter restrictions 4 do not necessarily hold, and the sloe coeffi cients π i = φ i, β i0, β i are allowed to vary across units. For future reference, artition matrix Γ i = Γ xi, Γ gi into m k x and m k g matrices Γ xi and Γ gi, vector α i = α xi gi, α into kx and k g vectors α xi and α gi, and similarly v it = v xit, v git into k x and k g vectors v xit and v git. 3 Estimation Let z it = y it, x it, g it and write -3 comactly as A 0i z it = c i + A i z i,t + C i f t + e it, 6 where c i = c yi, c ωi, C i = γ i, Γ i, A 0i = β 0i 0 k g 0 k x 0 k g I kx 0 k x k g 0 k g k x I kg, A i = φ i β i 0 k g α xi 0 k x k x 0 k x k g α gi 0 k g k x 0 k g k g, and e it = ε it, v it is a serially correlated error rocess. A 0i is invertible for any i and multilying 6 by A 0i, we obtain the following reduced form VAR reresentation of z it with serially correlated errors, z it = c zi + A i z i,t + A 0i C if t + e zit, 6 See Everaert and Groote 202 for derivation of asymtotic bias of CCE ooled estimators in the case of dynamic homogeneous anels. 5

7 where c zi = A 0i c i, e zit = A 0i e it, and A i = A 0i A i. We ostulate the following assumtions for the estimation of the short-run coeffi cients. ASSUMPIO Individual Secific Errors he individual secific errors ε it and v jt are indeendently distributed for all i, j, t and t. he vector of errors ε t = ε t, ε 2t,..., ε t is satially correlated according to ε t = Rς εt, 7 where the matrix R has bounded row and column matrix norms, namely R < K and R < K, resectively, for some constant K <, which does not deend on, diagonal elements of RR are bounded away from zero, ς εt = ς εt, ς ε2t,..., ς εt, and ς εit, for i =, 2,..., and t =, 2,..,, are indeendently and identically distributed IID with mean 0, unit variances, and finite fourth-order moments. For each i =, 2,...,, v it follows a linear stationary rocess with absolute summable autocovariances uniformly in i, v it = S il ς v,i,t l, 8 l=0 where ς vit is a k vector of IID random variables, with mean zero, variance matrix I k and finite fourth-order moments. In articular, V ar v it = S il S il K <, 9 for i =, 2,...,, where A is the sectral norm of the matrix A. l=0 ASSUMPIO 2 Common Effects he m vector of unobserved common factors, f t = f t, f 2t,..., f mt, is covariance stationary with absolute summable autocovariances, distributed indeendently of the individual secific errors ε it and v it for all i, t and t. Fourth moments of f lt, for l =, 2,..., m, are bounded. ASSUMPIO 3 Factor Loadings he factor loadings γ i, and Γ i are indeendently and identically distributed across i, and of the common factors f t, for all i and t, with mean γ and Γ, resectively, and bounded second moments. In articular, γ i = γ + η γi, η γi IID 0 m, Ω γ 6, for i =, 2,...,,

8 and vec Γ i = vec Γ + η Γi, η Γi IID 0, Ω Γ, for i =, 2,...,, km where Ω γ and Ω Γ are m m and k m k m symmetric nonnegative definite matrices, γ < K, Ω γ < K, Γ < K, and Ω Γ < K. ASSUMPIO 4 Heterogenous Coeffi cients 2k x + dimensional vector of coeffi cients π i = φ i, β 0i, β i follows the random coeffi cient model π i = π + υ πi, υ πi IID 0 2k x+, Ω π, for i =, 2,...,, 0 where π = φ, β 0, β, π < K, Ωπ < K, Ω π is 2k x + 2k x + symmetric nonnegative definite matrix, and the random deviations υ πi are indeendently distributed of γ j, Γ j, ε jt, v jt, and f t for all i,j, and t. Furthermore, the suort of φ i lies strictly inside the unit circle, and E c i < K for all i. ASSUMPIO 5 Regressors and Covariates Regressors and covariates in ω it = x it, g it are either strictly exogenous and generated according to the canonical factor model 3 with α i = 0 k, or weakly exogenous and generated according to 3 with α i, for i =, 2,...,, IID across i and indeendently distributed of υ πj, γ j, Γ j, ε jt, v jt, and f t for all i, j and t. In the case where the regressors are weakly exogenous we also assume: i the suort of λ A i lies strictly inside the unit circle, for i =, 2,...,, where A i = A 0i A i, and λ A i denotes the largest eigenvalue in absolute value of A i ; and ii the inverse of olynomial Λ L = l=0 Λ ll l, where Λ l = E A l i A 0i, exists and has exonentially decaying coeffi cients. Let w = w, w 2,..., w be an vector of non-stochastic or re-determined weights that satisfies the following granularity conditions w = O w i w = O 2 2, uniformly in i, 2 7

9 and the normalization condition w i =. 3 he weights vector w deends on, but we suress the subscrit to simlify notations. ext, we derive a large reresentation for cross-sectional averages of z it following Chudik and Pesaran 203a. Since the suort of the eigenvalues of A i is assumed to lie strictly inside the unit circle, z it is an invertible covariance stationary rocess and can be written as z it = l=0 A l i czi + A 0i C if t l + e z,i,t l, for i =, 2,...,. aking weighted cross-sectional averages of the above and making use of the fact that under our assumtions the elements of e zit are weakly cross-sectionally deendent, together with the random coeffi cients Assumtions 3-5, we have l=0 w i A l ie z,i,t l = O /2. Since under Assumtions 3-5 A i and A 0,i are indeendently distributed of C i, and A i, A 0,i and C i are indeendently distributed across i, we have l=0 w i A l ia 0,i C if t l = l=0 E A l ia 0,i C i f t l + O /2, = Λ L Cf t + O /2, where C = E C i = γ, Γ. hus, yielding the following large reresentation z wt = Λ L Cf t + O /2, 4 where z wt = z wt c zw is k + dimensional vector of de-trended cross section averages, z wt = ȳ wt, x wt, ḡ wt = w iz it is k + dimensional vector of cross section averages, and c zw = w i I k+ A i c zi. Multilying 4 by the inverse of Λ L now yields the following large exression for a linear 8

10 combination of the unobserved common factors, Cf t = Λ L z wt + O /2. 5 Consider now the secial case where α i = 0 k, and the regressors are strictly exogenous. In this case the regressors are indeendently distributed of the coeffi cients in π i = φ i, β 0,i, β,i, which simlifies the derivation of the large reresentation for z wt. In articular, φ i L is invertible for any i =, 2,..., under Assumtion 4, and multilying by φ i L we have y it = φ l ic yi + φ l iβ 0ix i,t l + φ l iβ ix i,t l + φ l iγ if t l + φ l iε i,t l. 6 l=0 l=0 l=0 aking weighted cross-sectional averages, under Assumtions -5, and assuming α i = 0, we k obtain y wt = c yw + a L γ f t + a L β 0 + β L x wt + O /2, 7 l=0 l=0 and ω wt = c ωw + Γ f t + O /2, 8 where c yw = w ic y,i φ i, c ωw = w ic ωi, and a L = l=0 a ll l with its elements given by the moments of φ i, namely a l = E φ l i, for l = 0,, 2,... ote that under Assumtion 4, which constraints the suort of φ i to lie strictly inside the unit circle, the rate of decay of the coeffi cients in a L is exonential. his restriction on the suort of φ i also ensures the existence of all moments of φ i. he rate of decay of the coeffi cients of a L will not necessarily be exonential if the suort of φ i covered, and deending on the roerties of the distribution of φ i in the neighborhood of, a L need not be absolute summable, in which case y wt could converge in a quadratic mean to a long memory rocess as. Such ossibilities are ruled out by Assumtion 4. However, under Assumtion 4 and By Lemma A. of Chudik and Pesaran 203b, the inverse of a L exists and has exonentially decaying coeffi cients. Pre-multilying both sides of 7 by b L = a L, we obtain γ f t = b L y wt b c yw β 0x wt β x w,t + O /2. 9 9

11 Stacking equations 8 and 9, we obtain 5 with Λ L reduced in the strictly exogenous case to Λ L = b L β 0 β L 0 k g 0 k x 0 k g I kx 0 k x k g 0 k g k x I kg. 20 It follows from 5 that when rank C = m and regardless of whether the regressors are weakly or strictly exogenous, de-trended cross section averages z wt and their lags can be used as roxies for the unobserved common factors, assuming that is suffi ciently large, namely we have f t = G L z wt + O /2, 2 where G L = C C C Λ L. ote that the coeffi cients of the distributed lag function, G L, decay at an exonential rate. In articular, in the case of strictly exogenous regressors where α i = 0 k, the decay rate of the coeffi cients in G L is given by the decay rate of the coeffi cients in b L, see 20 and 23. As established by Lemma A. of Chudik and Pesaran 203b, the decay rate of the coeffi cients in b L is exonential under Assumtion 4, which confines the suort of φ i to lie strictly within the unit circle. In the case of weakly exogenous regressors, an exonential rate of decay of the coeffi cients in Λ L is ensured by Assumtion 5-ii. he full column rank of C ensures that C C is invertible and this rank condition is required for the estimation of unit-secific coeffi cients. In contrast, the rank condition is not always necessary for estimation of the cross-sectional mean of the coeffi cients, as we shall see below. ASSUMPIO 6 k + m dimensional matrix C = γ, Γ has full column rank. obtain Substituting the large reresentation for the unobserved common factors 2 into, we y it = c yi + φ i y i,t + β 0ix it + β ix i,t + δ i L z wt + ε it + O /2, 22 0

12 where δ i L = δ il L l = G L γ i, 23 l=0 and c yi = c yi δ i c zw. Consider now the following cross-sectionally augmented regressions, based on 22, y it = c yi + φ i y i,t + β 0ix it + β ix i,t + l=0 δ il z w,t l + e yit, 24 where is the number of lags assumed to be the same across units, for the simlicity of exosition. he error term, e yit, can be decomosed into three arts: an idiosyncratic term, ε it, an error comonent due to the truncation of ossibly infinite olynomial distributed lag function, δ i L, and an error comonent due to the aroximation of unobserved common factors, namely e yit = ε it + δ il z w,t l + O /2. l= + ote that the coeffi cients of the distributed lag function, δ i L = γ ig L, decay at an exonential rate. Let ˆπ i = ˆφi, ˆβ 0i, ˆβ i be the least squares estimates of π i based on the cross-sectionally augmented regression 24. Also consider the following data matrices Ξ i = y i x i, + x i y i, + x i, +2 x i, +..., Q w = z w, + z w, z w, z w, +2 z w, + z w,2...., 25 y i, x i x i, z w, z w, z w, and the rojection matrix M q = I Q w Q w Q w + Q w, where I is a dimensional identity matrix, and A + denotes the Moore- Penrose generalized inverse of A. Matrices Ξ i, Q w, and M q deend also on, and, but we omit these subscrits to simlify notations. We summarize and introduce additional notations that will be useful for roofs in Aendix A..

13 ˆπ i can now be written as π i = Ξ i M q Ξ i Ξ i M q y i, 26 where y i = y i, +, y i, +2,..., y i,. he mean grou estimator of π = E π i = φ, β 0, β is given by π MG = ˆπ i. 27 In addition to Assumtions -6 above, we shall also require the following further assumtion. ASSUMPIO 7 a Denote the t -th row of matrix Ξ i = M h Ξ i by ξ it = ξit, ξ i2t,..., ξ i,2kx+,t, where M h is defined in the Aendix by A.4. Individual elements of ξ it have uniformly bounded fourth moments, namely there exists a ositive constant K such that E ξ4 ist < K for any t = +, + 2,...,, i =, 2,..., and s =, 2,..., 2k x +. b here exists 0 and 0 such that for all 0, 0, 2k x + 2k x + matrices Ψ Ξ,i = Ξ i M q Ξ i / exist for all i. c 2k x + 2k x + dimensional matrix Σ iξ defined in A.4 in the Aendix is invertible for all i and Σ < K < for all i. iξ his assumtion lays a similar role as Assumtion 4.6 in Chudik, Pesaran, and osetti 20 and ensures that π i, π MG and their asymtotic distributions are well defined. First, we establish suffi cient conditions for the consistency of unit-secific estimates. heorem Consistency of π i Suose y it, for i =, 2,..., and t =, 2,..., is given by the anel ARDL model -3, and Assumtions -7 hold. hen, as,, j, such that 3 / κ, 0 < κ <, we have πi πi 0 2k x+, 28 where π i = φi, β 0i, β i is given by 26. o restrictions on the relative exansion rates of and to infinity are required for the consistency of π i in the theorem above, but the number of lags needs to be restricted so that there are suffi cient degrees of freedom for consistent estimation i.e. the number of lags is not too large, in articular it is required that 2 / 0 and the bias due to the truncation of ossibly infinite 2

14 lag olynomials is suffi ciently small i.e. the number of lags is not too small, in our case ρ 0 for some ositive constant ρ <. Letting 3 / κ, 0 < κ <, as, ensures that these conditions are met. 7 he rank condition in Assumtion 6 is also necessary for the consistency of π i. his is because the unobserved factors are allowed to be serially correlated as well as being correlated with the regressors. 3. Consistency and asymtotic distribution of π MG Consistency of the unit-secific estimates π i is not always necessary for the consistency of the mean grou estimator of π = Eπ i, which is established next. heorem 2 Consistency of π MG Suose y it, for i =, 2,..., and t =, 2,..., is given by the anel data model -3, and Assumtions -5 and 7 hold, and,, j, such that 3 / κ, 0 < κ <. hen, i if Assumtion 6 also holds, where π MG = φmg, β 0MG, β MG is given by 27; π MG π 0 2k x+, 29 ii if Assumtion 6 does not hold but f t is serially uncorrelated, π MG π 0 2k x+. heorem 2 establishes that π MG is consistent as and tend jointly to infinity at any rate, regardless of the rank condition when factors are serially uncorrelated, although they can still be correlated with the regressors. When the factors are serially correlated, then the rank condition is required for the consistency of π MG. As we have seen, full column rank of C is suffi cient for aroximating the unobserved common factors arbitrarily well by cross section averages and their lags. In this case, the serial correlation of factors and correlation of factors and regressors do not ose any roblems. When the rank condition does not hold, but factors are serially uncorrelated, then π i could be inconsistent due to the correlation of x it and f t, but the asymtotic bias of π i π i is cross-sectionally weakly deendent with zero mean and consequently the mean grou estimator is consistent. he following theorem establishes the asymtotic distribution of π MG. 7 See also a related discussion in Berk 974, Chudik and Pesaran 203b and Said and Dickey 984 on the truncation of infinite olynomials in least squares regressions. 3

15 heorem 3 Asymtotic distribution of π MG Suose y it, for i =, 2,..., and t =, 2,..., are generated by the anel ARDL model -3, Assumtions -5 and 7 hold, and,, j such that / κ and 3 / κ 2, 0 < κ, κ 2 <. hen, i if Assumtion 6 also holds, we have πmg π d 0 2k x+, Ω π, 30 ii if Assumtion 6 does not hold, but f t is serially uncorrelated, we have πmg π d 0, Σ MG, 3 2k x+ where π MG = in the Aendix. φ MG, β 0MG, β MG is given by 27 and Σ MG is given by equation A.84 In both cases, the asymtotic variance of π MG can be consistently estimated nonarametrically by Σ MG = π i π MG π i π MG. 32 he convergence rate of π MG is due to the heterogeneity of the coeffi cients. heorem 3 shows that the asymtotic distribution of π MG differs deending on the rank of the matrix C in Assumtion 6. If C has full column rank, then the unit secific estimates π i are consistent, Σ MG reduces to Ω π, and the asymtotic variance of the mean grou estimator is given by the variance of π i alone. If, on the other hand, C does not have the full column rank and factors are serially uncorrelated then the unit-secific estimates are inconsistent since f t is correlated with x it, but π MG is consistent and asymtotically normal with variance that deends not only on Ω π but also on other arameters including the variance of factor loadings. Pesaran 2006 did not require any restrictions on the relative rate of convergence of and for the asymtotic distribution of the common correlated mean grou estimator. his is no longer the case in our model due to O time series bias of π i and π MG that arises from the resence of lagged values of the deendent variable. his bias dates back to at least to Hurwicz 950 and it has been well documented in the literature. heorem 3 requires / κ for the derivation of the asymtotic distribution of 4

16 π MG due to the time series bias, and it is therefore unsuitable for anels with small relative to. 4 Bias-corrected CCEMG estimators In this section we review the different rocedures roosed in the literature for correcting the small samle time series bias of estimators in dynamic anels and consider the ossibility of develoing bias-corrected versions of CCEMG estimators for dynamic anels. Existing literature focuses redominantly on homogeneous anels, where several different ways to correct for O time series bias have been roosed. his literature can be divided into the following broad categories: i analytical corrections based on an asymtotic bias formula Bruno, 2005, Bun, 2003, Bun and Carree, 2005 and 2006, Bun and Kiviet, 2003, Hahn and Kuersteiner, 2002 and 20, Hahn and Moon, 2006, Hahn and ewey, 2004, Kiviet, 995 and 999, and ewey and Smith, 2004; ii bootstra and simulation based bias corrections Everaert and Ponzi, 2007, Phillis and Sul, 2003 and 2007, and iii other methods, including jackknife bias corrections Hahn and ewey, 2004, and Dhaene and Jochmans, 202 and the recursive mean adjustment correction rocedures So and Shin, 999. In contrast, bias correction for dynamic anels with heterogenous coeffi cients have been considered only in few studies. Hsiao, Pesaran, and ahmiscioglu 999 investigate bias-corrected mean grou estimation, where Kiviet and Phillis 993 bias correction is alied to the individual estimates of short-run coeffi cients. Hsiao, Pesaran, and ahmiscioglu 999 roose also a Hierarchical Bayesian estimation of short-run coeffi cients, which they find to have good small samle roerties in their Monte Carlo study. 8 Pesaran and Zhao 999 investigate bias correction methods in estimating long-run coeffi cients and consider, in articular, two analytical corrections based on an aroximation of the asymtotic bias of long-run coeffi cients, a bootstra bias-corrected estimator, and a "naive" bias-corrected anel estimator comuted from bias-corrected short-run coeffi cients using a result derived by Kiviet and Phillis, Zhang and Small 2006 further develos the hierarchical Bayesian aroach of Hsiao, Pesaran, and ahmiscioglu 999 by imosing a stationarity constraint on each of the cross section units and by considering different ossibilities for starting values. Bayesian aroach has also been develoed by Canova and Marcet 999 to study income convergence in a dynamic heterogenous anel of countries, and by Canova and Ciccarelli 2004 and 2009 to forecast variables and turning oints in a anel VAR. Forecasting with Bayesian shrinkage estimators have also been considered by Garcia-Ferrer, Highfield, Palm, and Zellner 987, Zellner and Hong 989 and Zellner, Hong, and ki Min 99. 5

17 4. Bias corrected versions of π MG All the bias correction rocedures reviewed above are develoed for anel data models without unobserved common factors, and are not directly alicable to π MG. his alies to bootstraed based corrections, as well as the analytical corrections based on asymtotic bias formulae such as the one derived by Kiviet and Phillis 993. he develoment of analytical or bootstraed bias correction rocedures for dynamic anel data models with a multifactor error structure is beyond the scoe of the resent aer and deserve searate investigations of their own. Instead here we consider the alication of jackknife and recursive mean adjustment bias correction rocedures to π MG that do not require any knowledge of the error factor structure and are articularly simle to imlement. 4.. Jackknife bias correction Jackknife bias correction is oular due to its simlicity and wide alicability. Jackknife bias correction can be alied to the anel mean grou estimator, or at the level of unit-secific estimates. Since the mean grou estimator is a linear function of the unit-secific estimators, alying the correction to π MG or to the unit-secific estimates, π i, yields numerically identical results. We consider the "half-anel jackknife" method discussed by Dhaene and Jochmans 202, which corrects for O bias. Jackknife bias-corrected CCEMG estimators are constructed as: π MG = 2 π MG 2 π a MG + π b MG, where π a MG denotes the CCEMG estimator comuted from the first half of the available time eriod, namely over the eriod t =, 2,..., [/2], where [/2] denotes the integer art of /2, and π b MG is the CCEMG estimators comuted using the observations over the eriod t = [/2] +, [/2] + 2,..., Recursive mean adjustment he second bias-correction is based on the recursive mean adjustment method roosed by So and Shin 999, who advocate demeaning variables using the artial mean based on observations u 6

18 to the time eriod t. In articular, we let ỹ it = y it t y is, t s= and ω it = ω it t ω is, t s= for i =, 2,..., and t = 2, 3,...,, where ω it = x it, g it. We then comute bias-adjusted CCE mean grou estimator based on the recursive demeaned variables ỹ it and ω it with available time eriods, t = 2, 3,...,. 5 Monte Carlo Exeriments Our main objective is to investigate the small samle roerties of the CCEMG estimator and its bias corrected versions in anel ARDL models under different assumtions concerning the arameter values and the degree of cross-sectional deendence. We also examine the robustness of the quasi maximum likelihood estimator QMLE develoed by Moon and Weidner 200a and 200b and the interactive-effects estimator IFE roosed by Bai 2009 to coeffi cients heterogeneity, and include an alternative MG estimator based on Song s extension of Bai s IFE aroach denoted as ˆπ s MG and investigate its erformance as well. We start with the descrition of the data generating rocess in subsection 5., followed by a summary account of the different estimators being considered in subsection 5.2, before roviding a summary of our main findings in the final subsection. 5. Data Generating Process We set k x = k g = and write -3 as y it = c yi + φ i y i,t + β 0i x it + β i x i,t + u it, u it = γ if t + ε it, 33 and x it g it = c xi c gi + α xi y i,t + γ xi f t + α gi γ gi v xit v git. 34 7

19 he unobserved common factors in f t and the unit-secific comonents v it = v xit, v git are generated as indeendent stationary AR rocesses: f tl = ρ fl f t,l + ς ftl, ς ftl IID 0, ρ 2 fl, 35 v xit = ρ xi v xi,t + ς xit, ς xit IID 0, σ 2 vxi, 36 v git = ρ gi v gi,t + ς git, ς git IID 0, σ 2 vgi 37 for i =, 2,...,, l =, 2,.., m, and for t = 99,..., 0,, 2,..., with the starting values f l, 00 = 0, and v xi, 00 = v gi, 00 = 0. he first 00 time observations t = 99, 48,..., 0 are discarded. We generate ρ xi and ρ gi, for i =, 2,... as IIDU [0.0.95], and consider two values for ρ fl, reresenting the case of serially uncorrelated factors, ρ fl = 0, for l =, 2,..., m, and the case of the serially correlated factors ρ fl = 0.6, for l =, 2,..., m. We set σ 2 vxi = σ2 vgi = σ2 vi and allow σ vi to be correlated with β 0i and set σ vi = β i0 [E ρ xi ] 2. As before, we let z it = y it, x it, g it, and write the data generating rocess for z it more comactly as see 6, where c zi = c yi + β 0i c xi, c xi, c gi, z it = c zi + A i z i,t + A 0i C if t + A 0i e it, 38 A i = φ i + β 0i α xi β i 0 α xi 0 0 α gi 0 0, A 0i = β 0i , C i = γ i, γ xi, γ gi, and e it = ε it + β 0i v xit, v xit, v git is a serially correlated error vector. We generate z it for i =, 2,...,, and t = 99,..., 0,, 2,..., based on 38 with the starting values z i, 00 = 0, and the first 00 time observations t = 99, 48,..., 0 are discarded as burn-in relications. he fixed effects are generated as c iy IID,, c xi = c yi + ς cxi, and c gi = c yi + ς cgi, where ς cxi, ς cgi IID 0,, thus allowing for deendence between x it, g it and c yi. For each i the rocess {z it } is stationary if f t and e it are stationary and the eigenvalues of A i lie inside the unit circle. More secifically the arameter choices for λ A i < have to be such that 2 φ i + α xi β 0i ± φ i + α xi β 0i 2 + 4β i α xi <. 8

20 Suose now that we only consider ositive values of φ i, α xi and β 0i, such that φ i + α xi β 0i < 2. hen it is easily seen that suffi cient stationary conditions are β 0i + β i α xi < φ i, β i β 0i α xi < + φ i. Accordingly, we set β i = 0.5 for all i, and generate β 0i as IIDU0.5,. When α xi > 0, we need to generate α xi such that 0.5α xi < φ i. We consider two ossibilities for φ i : Low values where φ i are generated as IIDU0, 0.8 and α xi as IIDU0, High values where use the draws, φ i IIDU0.5, 0.9 and α xi IIDU0, 0.5. hese choices ensure that the suort of λ A i lies strictly inside the unit circle, as required by Assumtion 5. Values of α gi do not affect the eigenvalues of A i and are generated as α gi IIDU0,. he above DGP is more general than the other DGPs used in other MC exeriments in the literature and allows for weakly exogenous regressors. he factors and regressors are allowed to be correlated and ersistent, and correlated fixed effects are included. All factor loadings are generated indeendently as γ il = γ l + η i,γl, η i,γl IID 0, σ 2 γl, γ xil = γ xl + η i,γxl, η i,γxl IID 0, σ 2 γxl, γ gil = γ gl + η i,γgl, η i,γgl IID 0, σ 2 γgl for l =, 2,.., m, and i =, 2,...,. Also, without loss of generality, the factor loadings are calibrated so that V arγ i f t = V ar γ xi f t = V ar γ gi f t =. We also set σ 2 γl = σ2 γxl = σ2 γgl = 0.2 2, γ l = b γl, γ xl = lb xl and γ gl = 2l b gl, for l =, 2,..., m, where b γ = /m σ 2 γl, b x = 2/ [m m + ] 2/ m + σ 2 xl and b g = /m 2 σ 2 gl /m, for l =, 2,..., m. his ensures that the contribution of the unobserved factors to the variance of y it does not rise with m. We consider m =, 2 or 3 unobserved common factors. Finally, the idiosyncratic errors, ε it, are generated to be heteroskedastic and weakly crosssectionally deendent. Secifically, we adot the following satial autoregressive model SAR to generate ε t = ε t, ε 2t,..., ε t : ε t = a ε S ε ε t + e εt, 39 9

21 where the elements of e εt are drawn as IID 0, 2 σ2 i, with σ 2 i obtained as indeendent draws from χ 2 2 distribution, S ε = , and the satial autoregressive arameter is set to a ε = 0.4. ote that {ε it } is cross-sectionally weakly deendent for a ε < 0.5. In addition to these exeriments, we also consider ure anel autoregressive exeriments where we set β 0i = β i = 0, for all i. able summarizes the various arameter configurations of all the different exeriments. In total, we conducted 24 exeriments covering the various cases: with or without regressors in the equation for the deendent variable, low or high values of φ = E φ i, m =, 2, or 3 common factors, and ersistent or serially uncorrelated common factors. We consider the following combinations of samle sizes:, {40, 50, 00, 50, 200}, and set the number of relications to R = 2000, in the case of all exeriments. 5.2 Estimation techniques he focus of the MC results will be on the estimates of the average arameter values φ = E φ i and β 0 = E β 0i, in the case of exeriments with regressors, x it. But before resenting the outcomes we briefly describe the comutation of the alternative estimators being considered Dynamic CCE mean grou estimator We base the CCE mean grou estimator on the following cross-sectionally augmented unit-secific regressions, y it = c iy + φ i y i,t + β 0i x it + β i x i,t + l=0 δ il z t l + e yit, 40 codes. 9 We are grateful to Jushan Bai, Hyungsik Roger Moon, and Martin Weidner for roviding us with their Matlab 20

22 for i =, 2,...,, where z t = z it = ȳ t, x t, ḡ t. We set equal to the integer art of /3, denoted as = [ /3]. his gives the values of = 3, 3, 4, 5, 5 for = 40, 50, 00, 50, 200, resectively. he CCE mean grou estimator of φ and β 0 is then obtained by arithmetic averages of the least squares estimates of φ i and β 0i based on 40. We also comuted bias-corrected versions of the CCEMG estimator using the half-anel jackknife and the recursive mean adjusted estimators as described in Section QMLE estimator by Moon and Weidner We deal with fixed effects by de-meaning the variables before imlementing the QMLE estimation rocedure. Denote the demeaned variables as ẏ it = y it y it, and ẋ it = x it t= t= x it, 4 for s =, 2 and i =, 2,...,. We comute the bias-corrected QMLE estimator defined in Corollary 3.7 in Moon and Weidner 200a using ẏ it as the deendent variable and the vector ż it = ẏ i,t, ẋ it, ẋ i,t as the vector of exlanatory variables. wo otions for the number of unobserved factors are considered: the true number of factors and the maximum number, 3, of unobserved factors Interactive-effects estimator by Bai We deal with the fixed effects in the same way as before. In articular, we use the demeaned variables ẏ it, and ẋ it,s for s =, 2, to comute the interactive-effects estimator as the solution to the following set of non-linear equations: ˆπ b = Ξ im ˆF Ξ i Ξ im ˆF ẏ i, 42 ẏ i Ξ iˆπ b ẏ i Ξ iˆπ b ˆF = ˆF ˆV, 43 where ˆπ b = ˆφb, ˆβ 0b, ˆβ b is the interactive-effects estimator, M ˆF = I ˆF ˆFˆF ˆF, ˆV is a diagonal matrix with the m largest eigenvalues of the matrix ẏ i Ξ iˆπ b ẏ i Ξ iˆπ b 2

23 arranged in decreasing order, ẏ i = ẏ i2, ẏ i3,..., ẏ i and Ξ i = ẏ i ẋ i2 ẋ i ẏ i,2 ẋ i3 ẋ i2... ẏ i, ẋ i ẋ i,. he system of equations is solved by an iterative method. Bai 2009 does not allow for a lagged deendent variable in the derivation of the asymtotic results for the interactive-effects estimator, but considers this ossibility in Monte Carlo exeriments and concludes that arameters are well estimated also for the DGP with a lagged deendent variable. As in the case of the QMLE estimator, we consider Bai s estimates based on the true number of factors, and on the maximum number of factors, namely Mean Grou estimator based on Song s extension of Bai s IFE aroach Song 203 extends Bai s IFE aroach by allowing for coeffi cient heterogeneity and lags of the deendent variable. Song focuses on the estimates of individual coeffi cients obtained from the solution to the following system of nonlinear equations, which as he shows minimizes the sum of squared errors, ˆπ s i = Ξ im ˆF Ξ i Ξ im ˆF ẏ i, for i =, 2,...,, 44 ẏ i Ξ iˆπ i ẏ i Ξ iˆπ i ˆF = ˆF ˆV. 45 Similarly to Bai s IFE rocedure, we use demeaned observations to deal with the resence of fixed effects and the system of equations is solved numerically by an iterative method. Song 203 establishes consistency rates of individual estimates ˆπ s i under asymtotics, j such that / 2 0. Given our random coeffi cient assumtion on π i, we adot the following mean grou estimator based on Song s individual estimates, ˆπ s MG = ˆπ s i, 22

24 and investigate the erformance of ˆπ s MG with its variance estimated nonaremetrically by Σ s MG = π s i π s MG π s i π s MG. ote that since ˆπ s i π i = O uniformly in i as, j such that / 2 0 see Song, 203, heorem 2, it readily follows that also see Assumtion 4 ˆπ s MG π = υ πi + O. However, suffi cient conditions for ˆπ s MG π d 0, Ω π as, j remains to be investigated and this is outside the scoe of the resent aer. 6 Monte Carlo findings In this section we reort some of the main findings, and direct the reader to an online Sulement where the full set of results can be accessed. able 2 summarizes the results for the bias 00 and root mean square error RMSE, 00 in the case of the exeriment with regressors, φ = E φ i = 0.4, and one serially correlated unobserved common factor Exeriment 4 in able. he first anel of this table gives the results for the fixed effects estimator FE which rovides a benchmark against three sources of estimation bias: the time series bias of order, the bias from ignoring a serially correlated factor, and the bias due to coeffi cient sloe heterogeneity. he latter two biases are not diminishing in and we see that their combined effect remains substantial even for = 200. ext consider the QMLE estimator due to Moon and Weidner, which allows for unobserved factors, but fails to account for coeffi cient heterogeneity. As can be seen, this estimator still suffers from a substantial degree of heterogeneity bias which does not diminish in. his is in line with the theoretical results derived in Pesaran and Smith 995, where it is shown that in the resence of sloe heterogeneity ooled least squares estimators are inconsistent in the case of anel data models with lagged deendent variables. his would have been the case even if the unobserved factors could have been estimated without any samling errors. Initially, for = 40, negative time series bias hels the erformance of QMLE in our design, but as increases, the time series bias 23

25 diminishes and the ositive coeffi cient heterogeneity bias dominates the outcomes. he bias for = 200 ranges between 0.07 to 0.0 which amounts to 20 25% of the true value. Inclusion of 3 as oosed to unobserved common factor imroves the erformance but does not mitigated fully the consequences of coeffi cient heterogeneity. Results for Bai s IFE aroach are similar to those of QMLE and are therefore reorted only in the online Sulement to save sace. In contrast the CCEMG estimator deals with the resence of ersistent factors and coeffi cient heterogeneity, but fails to adequately take account of the time series bias. As can be seen from the results, the uncorrected CCEMG estimator suffers from the time series bias when is small, with the bias diminishing as in increased. he sign of the bias is negative, which is in line with the existing literature. hee bias of the CCEMG estimator is around 0.2 for = 40, and declines to around 0.02 when = 200. Both bias correction methods considered are effective in reducing the time series bias of the CCEMG estimator, but the jackknife bias correction method turns out to be more successful overall. It is also interesting that the jackknife correction tends to slightly over-correct whereas the RMA rocedure tends to under-correct. Both bias-correction methods also reduced the overall RMSE for all values of and considered. he mean grou estimator based on Song s individual estimates erforms slightly worse than the jackknife bias-corrected CCEMG, but overall its erformance in terms of bias and RMSE seems to be satisfactory. he knowledge of the true number of factors, however, lays a very imortant role in imroving the erformance of this estimator. able 3 reorts findings for estimation of β 0 in the same exeriment. As before, the FE and QMLE estimators continue to be biased even when is large. he selection of the number factors seems to be quite imortant for the bias of QMLE estimator and also Bai s IFE estimator reorted in the Sulement. he bias of CCEMG estimators is, in contrast, very small, between 0.0 to 0.02 for all values of and. Bias correction does not seem to matter for the CCEMG estimation of β 0. he small samle time series O bias for the estimation of β 0 is much smaller as comared to the bias of the autoregressive coeffi cient. Bias correction seems therefore not so imortant for the estimation of β 0, and the uncorrected version of CCEMG estimator erforms better in terms of RMSE comared to its bias corrected versions. ˆπ s MG also erforms well although its RMSE is, in the majority of cases, slightly worse than RMSE of the uncorrected CCEMG estimator. 24

26 An imortant question is how robust are the various estimators to the number of unobserved factors. he MC results with more than one factor are summarized in ables 4-7, and show that the CCEMG estimator continues to work well regardless of the number of factors and whether the factors are serially correlated. For m = 2 or 3, the erformance of the CCEMG estimator and its bias-corrected versions is qualitatively similar to the case of m = discussed above. Only a slight deterioration in bias and RMSE is observed when m is increased to 3, most likely due to the increased comlexity encountered in aroximating the sace sanned by the unobserved common factors. o check the validity of the asymtotic distribution of the CCEMG and other estimators, we now consider the size and ower erformance of the different estimators under consideration. We comute the size 00 at 5% nominal level and the ower 00 for the estimation of φ and β 0 with the alternatives H : φ = 0.5 and H : φ = 0.8, associated with the null values of φ = 0.4 and 0.7, resectively, and the alternative of H : β 0 = 0.85, associated with the null value of β 0 = he results for size and ower in the case of the Exeriments 4, 6 and 8 are summarized in ables 8-3. As can be seen the tests based on FE and QMLE estimators and Bai s IFE reorted in the Sulement are grossly oversized irresective of whether the arameter of interest is φ or β 0. In contrast the CCEMG estimator and the MG estimator based on Song s individual estimates have the correct size if one is interested in making inference about β 0, but both estimators tend to be over-sized if the aim is to make inference about φ. hese results are in line with our theoretical findings and largely reflect the time series bias of order O which is resent in the MG tye estimators of φ. he bias-corrected versions of the CCEMG estimator erform much better, with the jackknife bias-correction method generally outerforming the RMA rocedure. he condition / κ, 0 < κ <, in heorem 3 lays an imortant role in ensuring that the tests based on the CCEMG estimator of φ have the correct size. In articular, the size worsens with an increase in the ratio /, esecially when = 40. Relatively good size 7%-9% is achieved only when > 00. As already noted, the size of the tests based on the CCEMG estimator of β 0, ables 9, and 2 is strikingly well behaved in all exeriments and is very close to 5 ercent for all values of and, which is in line with low biases reorted for this estimator. Similar results also hold 25