Asymptotics for panel models with common shocks

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(1)UNIVERSITA’ DEGLI STUDI DI BERGAMO DIPARTIMENTO DI INGEGNERIA GESTIONALE E DELL’INFORMAZIONE° † QUADERNI DEL DIPARTIMENTO. Department of Management and Information Technology Working Paper Series “Economics and Management”. n. 15/EM – 2006. Asymptotics for panel models with common shocks by. Chihwa Kao, Lorenzo Trapani and Giovanni Urga. ° †. Viale Marconi. 5, I – 24044 Dalmine (BG), ITALY, Tel. +39-035-2052339; Fax. +39-035-562779 Il Dipartimento ottempera agli obblighi previsti dall’art. 1 del D.L.L. 31.8.1945, n. 660 e successive modificazioni..

(2) COMITATO DI REDAZIONE§ Series Economics and Management (EM): Stefano Paleari, Andrea Salanti Series Information Technology (IT): Stefano Paraboschi Series Mathematics and Statistics (MS): Luca Brandolini, Alessandro Fassò §. L’accesso alle Series è approvato dal Comitato di Redazione. I Working Papers ed i Technical Reports della Collana dei Quaderni del Dipartimento di Ingegneria Gestionale e dell’Informazione costituiscono un servizio atto a fornire la tempestiva divulgazione dei risultati dell’attività di ricerca, siano essi in forma provvisoria o definitiva..

(3) ASYMPTOTICS FOR PANEL MODELS WITH COMMON SHOCKS∗ Chihwa Kao Syracuse University cdkao@maxwell.syr.edu Lorenzo Trapani Cass Business School and Universita’ di Bergamo L.Trapani@city.ac.uk Giovanni Urga Cass Business School G.Urga@city.ac.uk November 3, 2006. Abstract This paper develops a novel asymptotic theory for panel models with common shocks. We assume that contemporaneous correlation can be generated by both the presence of common regressors among units and weak spatial dependence among the error terms. Several characteristics of the panel are considered: cross sectional and time series dimensions can either be fixed or large; factors can either be observable or unobservable; the factor model can describe either cointegration relationship or a spurious regression, and we also consider the stationary case. We derive the rate of convergence and the distribution limits for the ordinary least squares (OLS) estimates of the model parameters under all the aforementioned cases. ∗ We wish to thank participants in the ”1st Italian Congress of Econometrics and Empirical Economics” (Universita’ Ca’ Foscari, Venice, 24-25 January 2005), the 9th World Congress of the Econometric Society (London, 19-24 August 2005), in particular Peter Pedroni and Yixiao Sun, the North American Summer Meeting of the Econometric Society (Minneapolis, MN, 22-25 June 2006) and the 13th Conference on Panel Data (Cambridge, 7-9 July, 2006), the Monday Workshop of the Faculty of Finance at Cass Business School and the Lunch Seminars of the Department of Management and Information Technology at Bergamo University for useful comments. The usual disclaimer applies. This paper originated while Chihwa Kao was visiting the Centre for Econometric Analysis at Cass (CEA@Cass). Financial support from City University 2005 Pump Priming Fund and CEA@Cass is gratefully acknowledged. Lorenzo Trapani ackowledges financial support from Cass Business School (RAE Development Funds Scheme) and ESRC Postdoctoral Fellowship Scheme (PTA-026-27-1107).. 1.

(4) JEL Classification: C13, C23. Keywords: Panel data, common shocks, cross-sectional dependence, asymptotics.. 2.

(5) 1. INTRODUCTION. There is a growing body of literature dealing with limit theory for nonstationary panels. While the first generation of these contributions assumed independence across units (see for instance Phillips and Moon (1999), Kao (1999)), in the second generation this assumption is relaxed, and hypothesis testing and estimation methods are evaluated assuming alternative degrees of cross dependence (see Bai (2003, 2004), Bai and Ng (2002, 2004), Stock and Watson (2002)). We can distinguish the case where regressors are cross-sectionally dependent (see Donald and Lang (2004), Moulton (1990)) from the case where it is the error terms across unit to be dependent (see for instance Bai and Kao, 2005; Moon and Perron, 2004) or both (see for instance Ahn, Lee and Schmidt (2001), Pesaran (2006)). The main aim of this paper is to propose a novel asymptotic theory for panels with common shocks. We generalize the limit theory developed by Phillips and Moon (1999) and Andrews (2005) by employing and extending the theory for factor models in Bai (2003, 2004) and Bai and Ng (2004). Phillips and Moon (1999) analyze nonstationary panels when both n and T are large. They derive the seminal result that as n → ∞ a long-run average relationship between two nonstationary panel vectors exists even when the single units do not cointegrate. A similar result is also reported in Kao (1999). However, the asymptotics derived in Phillips and Moon (1999) is based on the assumption of cross section independence though the authors point out that their results still hold when certain degree of weak dependence among panel units is allowed. Thus, the case of Phillips and Moon (1999) with arbitrary dependence amongst units remains largely unexplored, and it is likely to lead to different asymptotics. Asymptotic normality may not hold, for example, when all or part of the regressors are aggregates, and may result in mixed asymptotic normality, as Andrews (2005) has demonstrated in a cross-sectional context.. 1. Andrews (2005, see Theorem 4, p. 1567) proves that the presence of common 1 See. also the discussion in Moon and Perron (2004).. 3.

(6) factors among the cross-sectional units makes the limiting distribution of the least squares estimator of β mixed normal and not normal as in the classical regression analysis. Note that in this case mixed normality of the least squares estimator of β holds even if regressors are I (0) and independent of errors. This finding is also obtained in our paper when studying the distribution limit for the least square estimator of β for the T fixed case (see equation 24 in Theorem 2 below), while when we consider the T → ∞ case, not explored by Andrews (2005), we show that in the stationary case as T → ∞ the least squares estimator of β is normally distributed. In this paper we consider the following panel regression model with common shocks yit = αi + β 0 Ft + uit ,. (1). i = 1, ..., n, t = 1, ..., T , where β is a k × 1 vector of slope parameters and Ft = (F1t , ..., Fkt )0 is a k × 1 vector of common shocks, Ft = Ft−1 + εt . Equation (1) could be either a spurious regression or a cointegration relationship depending on whether uit is I(1) or I (0), respectively. When common shocks are not observable, we assume that a set of exogenous variables, zit , is observable such that zit = λ0i Ft + eit. (2). where λi is a vector of factor loadings and eit is an idiosyncratic component. For the sake of the simplicity of the notation, we assume throughout the paper that the number of the zit s is the same as that of the yit s. However, the panel dimensions of yit s and the zit s may be different, for example yit s may refer to individuals while zit s may index several macro variables. To extend our results to the stationary panel model case, we also consider the first-differenced form of model (1), ∆yit = β 0 ∆Ft + ∆uit .. 4. (3).

(7) Our asymptotic theory considers several features of the underlying model. First, we assume that contemporaneous correlation can be generated by both the presence of common regressors (e.g. macro shocks, aggregate fiscal and monetary policies) among units and weak spatial dependence among the error terms. Second, the common shocks can either be known or unobservable. Classical examples of observed common shocks are index models such as those used in national trade, labor economics, urban regional, public economics and finance literature. Most often, shocks are unknown, as in the cases of index extraction and indicators aggregation in economics (Quah and Sargent (1993), Forni and Reichlin (1998), Bernanke and Boivin (2000)), while in finance the seminal multifactor framework of the arbitrage pricing theory has generated huge number of contributions in the attempt to identify the unobserved factors underlying the behavior of asset returns. Factor models are useful for forecasting purposes, as is well documented by Stock and Watson (1999, 2005). Bai (2003, 2004), Bai and Ng (2002, 2005) and Boivin and Ng (2005) discuss numerous areas of research where factor models could be employed and some applications in macro and finance. Third, regression model (1) may describe either a cointegration relationship or a spurious regression. Fourth, the time series dimension T and the cross-sectional dimension n can be either fixed or large. We develop our limit theory by considering cases where the time series dimension T and the number of units n are large and we also include the case of when either n or T is fixed2 . A short overview of the results we find under the conditions mentioned above is reported in Table 1.. The remainder of the paper is organized as follows. Section 2 introduces and comments on the main assumptions. In Section 3, we report the asymptotic 2 It is important to notice that the notion of fixed or ”small” n or T is not well specified. Pesaran (2005) cites n < 10 as the case when the number of cross sectional units is small. More generally, one could think as fixed n or T a number of cross sectional units or time series observations such that the cross-sectional or the time series average is still far away from the asymptotic limit, but such definition depends on the degree of cross sectional dependence or serial correlation in the panel and is therefore of scarce operational use.. 5.

(8) ˆ Table 1: Consistency (C) and Limiting Distribution (LD) of β OLS : Ft known (n, T ) F ixed n T →∞ F ixed T n→∞ (n, T ) → ∞. F ixed n T →∞ F ixed T n→∞ (n, T ) → ∞. Ft unknown (n, T ) Cointegration: uit ∼ I(0). C. LD. C. LD. Yes. Mixed Normal (Eq.10). Yes. Non Standard (Eq. 47). Yes. Mixed Normal (Eq.14). Yes. Mixed Normal (Eq.14). Yes. Mixed Normal (Eq.18). √ Yes √n/T → 0 T /n → 0 Yes Spurious Regression: uit ∼ I(1). Mixed Normal (Eq. 32) Non Standard (Eq. 34). No. Non Standard (Eq. 12). No. Non Standard (Eq. 49). Yes. Non Standard (Eq. 16). Yes. Non Standard (Eq. 16). Yes. Non Standard (Eq. 20). Yes. Non Standard (Eq. 36). Yes. Non Standard (Eq. 34). First Differences:. F ixed n T →∞ F ixed T n→∞ (n, T ) → ∞. yit = αi + β 0 Ft + uit ,. ˆF D β OLS. √ n/T √ →0 T / n→ 0 √ 2 n/T →0 √ T 2 / n→ 0 0. : ∆yit = β ∆Ft + ∆uit .. Yes. Normal (Eq. 22). No. Degenerate (Eq. 51). Yes. Mixed Normal (Eq. 24). Yes. Mixed Normal (Eq. 24). Yes. Normal (Eq. 26) Yes Yes. Degenerate (Eq. 38) Degenerate (Eq. 40). n/T → 0 T /n → 0. 6.

(9) theory of the least square estimators of β in models (1) and (3). We analyze both the cases of known factors (Section 3.1) and unknown factors (Section 3.2), and we distinguish the cases of large n and T , finite T and large n and finite n and large T . Section 4 concludes. Proofs are reported in the Appendix. Notation is fairly standard. Throughout the paper, kAk denotes the Euclidp ean norm of matrix A, tr (A0 A), ”→” the ordinary limit, ”⇒” weak converp. gence, ”→” convergence in probability. Stochastic processes such as B (r) on R1 R [0, 1] are usually written as B, integrals such as 0 B (r) dr as B and stochastic R1 R integrals such as 0 B (r) dB (r) as BdB.. 2. MODEL AND ASSUMPTIONS. We assume that yit is generated as follows yit. = αi + β 0 Ft + uit ,. Ft. = Ft−1 + εt .. zit. = λ0i Ft + eit. (4). i = 1, ..., n; t = 1, ..., T ; β is a (k × 1) vector of slope parameters; Ft = 0. (F1t , ..., Fkt ) is a k ×1 vector of common shocks; uit may be I(1) or I (0) (spurious regression or cointegration relationship); zit , is a set of observed exogenous variables. The following set of assumptions are used throughout the paper: Assumption 1: (a) either (i ) (cointegration case) uit = Di (L) η it , or (ii) (spurious regression case) ∆uit = Fi (L) η it with Fi (1) 6= 0 and such that ¡ ¢ P 8 2 i uit ∼ I (1); for both cases, η it ∼ iid 0, σ η over t and i, with E |η it | < M , P∞ P∞ 2 2 2 2 j=0 j |Dij | < M , j=0 j |Fij | < M and Di (1) σ η > 0, Fi (1) σ η > 0; (b). (time series and cross sectional correlation) letting E (uit ujs ) = τ ij,ts = τ ij,|t−s|. and E (∆uit ∆ujs ) = γ ij,ts = γ ij,|t−s| , as n → ∞ a Law of Large Numbers and a P P Central Limit Theorem hold for the quantities n−1/2 i uit and n−1/2 i ∆uit . Assumption 2: εt = C (L) wt where C (L) =. 7. P∞. j=0. Cj Lj ; (a) wt ∼ iid (0, Σu ).

(10) P 0 with E kwt k4+δ ≤ M for some δ > 0; (b) V ar (∆Ft ) = Σ∆F = ∞ j=0 Cj Σu Cj is P∞ a positive definite matrix; (c) j=0 j kCj k < M and (d) C (1) has full rank. Assumption 3: E kF0 k4 ≤ M and E |ui0 |4 ≤ M .. Assumption 4: The loadings λi are non random quantities such that (a) P P kλi k ≤ M ; (b) either n−1 ni=1 λi λ0i = ΣΛ if n is finite, or limn→∞ n−1 ni=1 λi λ0i =. ΣΛ , if n → ∞; in both cases, the matrix ΣΛ is positive definite and such 1/2. 1/2. that the eigenvalues of the matrix ΣΛ Σ∆F ΣΛ 1/2 R 1/2 ΣΛ Bε Bε0 ΣΛ are distinct with probability 1.. and of the stochastic matrix. ¡ ¢ 8 Assumption 5: eit = Gi (L) ν it where (a) ν it ∼ iid 0, σ 2vi , E |vit | < M , P∞ Pn 2 2 j=0 j |Gij | < M and Gi (1) σ vi > 0; (b) E (ν it ν jt ) = τ ij with i=1 |τ ij | ≤ ¯ −1/2 Pn ¯4 ¯ ¯ M for all j; (c) E n ≤ M for every (t, s); (d) i=1 [eis eit − E (eis eit )] ¯ ¯ ¯ £ −1 Pn ¤ PT PT ¯¯ −1 ¯ ¯ ¯ E n i=1 eit eis = γ s−t , γ s−t ≤ M for all s and T s=1 t=1 γ s−t ≤ M ; (e) E |ei0 |4 ≤ M .. Assumption 6: {εt }, {uit } and {eit } are three independent groups. Assumption 1(a) considers the possibility that equation (1) is either a cointegration or a spurious regression. Processes uit and ∆uit are assumed to be invertible MA processes as in Bai (2004) and Bai and Ng (2004), in a similar fashion to processes εt and eit . Assumption 1(b) also considers the presence of some, limited, cross sectional dependence among the uit s or the ∆uit s and therefore it rules out the possibility that all the cross sectional dependence is taken into account by the common factors Ft - see the related work by Conley (1999). Even if it refers to a different framework (panel data with common shocks as opposed to factor models), we take a position similar to that in Bai (2003, 2004) and Bai and Ng (2002, 2004). Using the factor models terminology, this means having a model with an ”approximate factor structure”, e.g., see the discussion in Chamberlain and Rotschild (1983) and Onatski (2005) - which differs from. 8.

(11) a strict common factor model where the uit s are assumed to be independent across i. The amount of cross sectional dependence we allow for in Assumption 1(b) is anyway limited, since we require that it allows the Law of Large Numbers Pn and a Central Limit Theorem to hold for the (rescaled) sequences i=1 uit and Pn i=1 ∆uit . Assumption 2 allows for some weak serial correlation in the dynamics of. εt . This process can be described as invertible MA process, implied by the absolute summability conditions. Both the short run and the long run variance of ∆Ft are positive definite (Assumptions 2(b) and 2(d), respectively). Note that Assumption 2(d) rules out the possibility that the (common) regressors Ft in model (1) are cointegrated. This requirement is standard in cointegration analysis to have non-degenerate limiting distributions 3 , Assumption 3 is a standard initial condition requirement. Assumption 4 serves to identify the factors, which, merely for the purpose of a concise discussion, are assumed to be non random. This requirement could be relaxed, as in Bai (2003, 2004) and Bai and Ng (2004), assuming that the λi s are randomly generated and independent of εt and eit , and our results would keep holding. Assumption 4(b) ensures that the factor structure is identifiable. Note that it would be possible to relax this assumption by constraining the minimum eigenPn value of i=1 λi λ0i to tend to infinity as n → ∞, as pointed out by Onatski (2005). This structure would allow factors to be less pervasive than in our. framework, thereby allowing the idiosyncratic component eit in equation (2) to have a greater impact in explaining the contemporaneous correlation among the zit s. Nonetheless, this would be made at the price of losing the possibility to model the zit as a serially correlated process, whilst in our framework some limited time series and cross sectional dependence in model (2) is allowed for - as one could realize from Assumption 5. As pointed out in Bai (2003), 3 Note that Bai and Ng (2004) allow for factors to be cointegrated given that they consider an approximate factor model with Ft s as common factors. In our paper Ft s denote instead a set of regressors. Therefore, we need to rule out cointegration among regressors to have non-degenerate limiting distributions.. 9.

(12) the conditions in Assumption 5 are fairly general and allow for consistency and distribution results to hold for the principal component estimator. Assumption 6 also rules out the existence of any form of dependence between factors Ft and uit . Therefore, it is a stronger requirement than the simple lack of correlation, and we need it in order to prove the main results in our paper. The following definitions are employed throughout the paper.Bε is the Brownian motion associated with the partial sums of εt with covariance matrix Ωεε , ¯ε (r) is the demeaned Brownian motion associated to the partial sums of Ft , B R ¯ε (r) = Bε (r) − 1 Bε (r) dr. Let hi (h∆ ) and hij (h∆ ) be the long run i.e., B i ij 0. variance for uit (∆uit ) and the long run covariance between processes uit and ujt PT PT PT PT (∆uit and ∆ujt ) - we have hij = t=1 s=1 τ ij,ts and h∆ ij = t=1 s=1 γ ij,ts . P P P Pn n n n −1 ∆ ¯ = limn→∞ n−1 ¯∆ Also, let h i=1 j=1 hij and h = limn→∞ n i=1 j=1 hij . Last, the following variances arising from cross sectional aggregation of the uit s Pn Pn and the ∆uit s are often used in our results: τ¯ts = limn→∞ n−1 i=1 j=1 τ ij,ts , P P and γ¯ ts = limn→∞ n−1 ni=1 nj=1 γ ij,ts .. 3. ASYMPTOTICS. The main objective of this paper is to derive the rate of convergence and limiting ˆ and β ˆ F D defined where the OLS estimator for β in equation distribution of β (1) is given by: " n T #−1 n T XX¡ XX¡ ¢¡ ¢0 ¢ ˆ= β Ft − F¯ Ft − F¯ Ft − F¯ yit i=1 t=1. P where F¯ = T −1 Tt=1 Ft , or, when using equation (3), by: ˆ β. FD. =. " n T XX. (5). i=1 t=1. ∆Ft ∆Ft0. i=1 t=1. #−1 " n T XX i=1 t=1. #. ∆Ft ∆yit .. (6). We consider several features of (1) and (3): 1. the shocks Ft can either be known or (more likely) unobservable. The ˆ and β ˆ F D are affected by the estimation errors if we asymptotics of β replace Ft by its estimate Fbt ;. 10.

(13) 2. the relationship described by equation (1) can be either a cointegration relationship or a spurious regression. As pointed out by Kao (1999) and √ Phillips and Moon (1999), convergence is obtained at rate n in panel √ spurious regression models and nT for panel cointegrated models. In this paper, we are going to face a similar issue, which is compounded by the presence of common shocks in the panel regression (1); 3. the time series dimension T and the cross-sectional dimension n can be either fixed or large. Asymptotics are likely to change depending on whether one considers either dimension T or n large, keeping the other one fixed, or whether both n and T are allowed to tend to infinity. We first start with the exploration of the case of known common shocks (Section 3.1) and then move to the case of unknown common shocks (Section 3.2).. 3.1. Observable Ft. In the case when Ft is known we have: ˆ−β = β where Wt = Ft − F¯ , and ˆF D − β = β. " n T XX. Wt Wt0. i=1 t=1. " n T XX. #−1 " n T XX. Wt uit ,. i=1 t=1. ∆Ft ∆Ft0. i=1 t=1. #. #−1 " n T XX i=1 t=1. (7). #. ∆Ft ∆uit .. (8). ˆ are now stated in The convergence rate and the limiting distribution for β the following theorem. Theorem 1 Suppose Assumptions 1-6 hold, and let Z ∼ N (0, Ik ) be independent of the σ-field generated by the common shocks Ft . For fixed n and T → ∞. ¡ ¢ ˆ − β = Op T −1 β 11. (9).

(14) ³. ´. ˆ−β ⇒ 1 T β n. µZ. ¯ε0 ¯ε B B. ¶−1/2. ⎛ ⎞1/2 n n X X ⎝ hij ⎠ Z. (10). i=1 j=1. if equation (1) is a cointegration relationship, and ˆ − β = Op (1) , β. (11). ⎞1/2 ⎛ µZ ¶−1 µZ ¶ X n X n ³ ´ ˆ−β ⇒ ¯ε B ¯ε Bu ⎝ ¯ε0 ⎠ β B h∆ B ij. (12). i=1 j=1. if (1) is a spurious regression.. For fixed T and n → ∞, we have. ³ ´ ˆ − β = Op n−1/2 , β. ´ √ ³ ˆ−β ⇒ n β. Ã T X. Wt Wt0. t=1. (13). !−1 Ã T T XX. Wt Ws0 τ¯ts. t=1 s=1. !1/2. Z,. (14). if (1) is a cointegration regression, whilst if it is a spurious relationship we have ³ ´ ˆ − β = Op n−1/2 , β ´ √ ³ ˆ−β ⇒ n β where u ¯t = limn→∞ n−1/2. Pn. i=1. Ã T X. Wt Wt0. t=1. (15). !−1 Ã T X. !. Wt u ¯t ,. t=1. (16). uit .. When (n, T ) → ∞, one has. ³ ´ ˆ − β = Op n−1/2 T −1 , β. (17). ¶−1/2 p µZ ´ √ ³ ˆ−β ⇒ ¯ ¯ε B ¯ε0 nT β B hZ,. (18). ³ ´ ˆ − β = Op n−1/2 , β. (19). if equation (1) is a cointegration relationship and. ´ √ ³ ˆ−β ⇒ n β. µZ. ¯ε B ¯ε0 B. if it is a spurious regression.. 12. ¶−1 µZ. ¯ε Bu B. ¶p ¯ ∆, h. (20).

(15) Proof. See Appendix. Equations (9)-(12) are the standard superconsistency and inconsistency results in the literature. With respect to the speed of convergence, when (n, T ) → ∞ our results in equations (17) and (19) lead to the same orders as in Phillips and Moon (1999) and Kao (1999) for both the cointegration and the spurious regression case. Consistency is achieved under the spurious regression case as √ well, where the rate of convergence is n. This result, which follows the seminal contributions of Kao (1999) and Phillips and Moon (1999), is reinforced for the case when T is fixed and n → ∞. Equations (13) and (15) prove that irrespective of model (1) to be a cointegration regression or a spurious regression, large n allows for consistency to hold. It is worth observing the complicated distribution that arises when T is finite; this is essentially due, as outlined in the proof, to the presence of serial correlation in the uit s. ˆ is the same For the case of n and T large, the rate of convergence for β as in Phillips and Moon (1999) under the case of contemporaneous independence across units, but the limiting distributions in equations (18) and (20) differ and are mixed normal rather than normal as in the Phillips and Moon (1999) case. The mixed normality is due to both Ft being nonstationary and common across units, as can be seen by considering equation (14) for T → ∞. ¡ ¢−1 Pn PT PT 0 −2 0 The design matrix nT 2 i=1 t=1 Ft Ft = T t=1 Ft Ft converge in disR ¯ε B ¯ε0 , rather than to a constant. Of tribution to a random matrix, namely B ¡ 2 ¢−1 Pn PT 0 course, nT i=1 t=1 Ft Ft would converge to a constant (in probability). if Ft were not common shocks, i.e., if Ft were replaced by, say, Fit .. ˆ F D are reported The convergence rates and the limiting distributions for β in the following theorem. Theorem 2 Suppose Assumptions 1-6 hold and let Z ∼ N (0, Ik ) be independent of the σ-field generated by ∆Ft . For fixed n and T → ∞. ³ ´ ˆ F D − β = Op T −1/2 , β 13. (21).

(16) ⎛ ⎞1/2 n n X ´ X √ ³ FD −1/2 ˆ ⎠ Z. T β − β ⇒ n−1 Σ∆F ⎝ h∆ ij. (22). ³ ´ ˆ F D − β = Op n−1/2 , β. (23). i=1 j=1. For T fixed and n → ∞, we have. ´ √ ³ FD ˆ n β −β ⇒. Ã T X t=1. When (n, T ) → ∞, one has. ∆Ft ∆Ft0. !−1 Ã T T XX. ∆Ft ∆Fs0 γ¯ ts. t=1 s=1. ³ ´ ˆ F D − β = Op n−1/2 T −1/2 , β. ´ p √ ³ FD −1/2 ˆ ¯ ∆ Z. nT β − β ⇒ Σ∆F h. !1/2. Z.. (24). (25) (26). Proof. See Appendix. The results were derived for the case of no serial correlation. The presence of time dependence in general involves a more complicated expression of the limiting distributions, but rates of convergence would not be affected. Note also that since the first differenced model is always stationary, irrespective of whether equation (1) is a cointegration equation or a spurious regression, one ˆ F D − β; this can always apply the CLT to obtain the limiting distribution of β is indirectly shown by the rate of convergence for the case when (n, T ) → ∞, √ equal to nT . It is worth noticing the remarkable result in equation (26): one would expect ˆ F D − β to be mixed normal given the strong depenthe limiting distribution of β dence across units due to the terms ∆Ft ∆uit sharing the common element ∆Ft across i, as we showed in (24) with large n and fixed T . However, the common shocks are found not to play any role in the case of large n and large T . This result is discussed thoroughly in the proofs of Theorems 1 and 2. and can also be seen in equation (24) which gives the limiting distribution for T fixed and PT n → ∞. The design matrix T −1 t=1 ∆Ft ∆Ft0 is a random matrix for all finite values of T . However, standard application of the LLN (its validity is ensured by Assumption 2) shows that the design matrix converges to a constant matrix 14.

(17) as T → ∞. Therefore, the mixed normality arising for finite T is wiped away by the smoothing over time as well. Asymptotic normality is therefore determined PT merely by design matrix T −1 t=1 ∆Ft ∆Ft0 being constant asymptotically.. 3.2. Unobservable Ft. We turn now to the case when common shocks are unknown and thus they need ˆ and β ˆ F D are affected by the errors in to be estimated. The asymptotics of β estimating shocks Ft .. ˆ t = Fˆt − T −1 PT Fˆt . EstiLet Fˆt be an estimate of the shock. Denote W t=1. ˆ or first differences (β ˆ F D ) respectively mations of β using the model in levels (β) are now given by: ˆ= β. " n T XX i=1 t=1. and ˆF D = β. " n T XX i=1 t=1. ˆ tW ˆ t0 W. #−1 " n T XX i=1 t=1. ∆Fˆt ∆Fˆt0. # ˆ Wt yit ,. #−1 " n T XX i=1 t=1. # ˆ ∆Ft ∆yit ,. with estimation errors: #−1 ( n T " n T ∙³ ¸) ´0 XX XX 0 ˆ−β = ˆ t β + uit ˆ tW ˆ ˆ t Wt − W , W W β t. i=1 t=1. ˆ β. FD. −β =. " n T XX i=1 t=1. (27). (28). (29). i=1 t=1. ∆Fˆt ∆Fˆt0. #−1 ( n T XX i=1 t=1. ∙³ ¸) ´0 ∆Fˆt ∆Ft − ∆Fˆt β + ∆uit .. (30). In what follows, for the purpose of a concise discussion, we assume the number of shocks k to be known4 . We like to emphasize that this is does not 4 An issue of importance that arises within this framework and that needs tackling prior to estimating the common components Ft is to determine their number, k. In light of some recent contributions, e.g., see Bai and Ng (2002) and Onatski (2005), it is natural to refer to model (2) in order to extract both the common factors Ft and their number k. It is worth pointing out though that determining k crucially depends on whether both n and T are large or if either dimension is fixed. Under all cases, the literature provides methodologies to estimate ˆ such that, as either (n, T ) → ∞ or, alternatively, k consistently, i.e. to obtain an estimate k k l k l ˆ = k = 1 and P k ˆ 6= k = op (1). max {n, T } → ∞ and min {n, T } is fixed, it holds that P k. Most often these methods treat estimation of k as a either model selection or a rank estimation. 15.

(18) lead to any loss of generality since the distribution of the estimated shocks does not depend on whether k is known or estimated, and therefore the estimation error that arises from using kˆ instead of k does not play any role as long as kˆ is consistent, e.g., see Bai (2003, p. 143, note 5) for an elegant proof of this statement. 3.2.1. The case of n and T large. In this section, we estimate the common shocks Ft using the principal component estimator. This means minimizing either Vb (k) =. T n ¢2 1 XX¡ zit − λ0i Ft , nT i=1 t=1. when considering Ft in levels, or Va (k) =. T n ¢2 1 XX¡ ∆zit − λ0i ∆Ft nT i=1 t=1. when estimating shocks ∆Ft from the first differenced version of model (2). Consider the T × n matrix Z = (z1 , ..., zT )0 , and the T × k matrix of shocks. F = (F1 , F2 , ..., FT )0 . Then each objective function Va (k) or Vb (k) can be. minimized by concentrating out λ and obtaining estimates ∆Fˆ and Fˆ using the normalizations ∆Fˆ 0 ∆F ˆ/T = Ik or Fˆ 0 Fˆ /T 2 = Ik . The estimated shock matrices √ ∆Fˆ and Fˆ are T times eigenvectors corresponding to the k largest eigenvalues of the T × T matrices ∆Z∆Z 0 or ZZ 0 . It is well known that the solutions to the above minimization problems are not unique, e.g., when estimating shocks ∆Ft and Ft , these are not directly identifiable even though they are up to a transformation. In our setup, the knowledge of H1 ∆Ft , H1 Ft and H2 λi is as good as knowing ∆Ft , Ft , and λi . For sake of notational simplicity, in what follows we shall assume that H1 (k × k) and H2 (n × n) are identity matrices. problem, thereby employing some information criteria. For the case of either n or T fixed, the contributions by Lewbel (1991), Donald (1997) and Cragg S and Donald (1997) ensure consistent estimation of the the rank of the n × n matrix t zt zt0 with zt ≡ [z1t , ..., znt ]0 or of the S either 0 0 T × T matrix i zi zi with zi ≡ [zi1 , ..., ziT ] , depending on whether n or T is fixed. When ˆ and (n, T ) → ∞, the aforementioned procedures are no longer usable to obtain a consistent k Bai and Ng (2002) propose a consistent estimator for k - see also Onatski (2005). Note that assumptions 2-6 in our settings ensure the applicability of these methods to equation (2), as it can be immediately verified.. 16.

(19) ˆ are in the following The convergence rate and the limiting distribution for β theorem. Theorem 3 Suppose Assumptions 1-6 hold. Let equation (1) be a cointegration relationship: √ if n/T → 0 ³ ´ ˆ − β = Op n−1/2 T −1 , β. (31). µZ ¶−1/2 ∙p ¸ q ´ √ ³ 0˜ 0 0 ˆ ¯ ¯ ˜ ¯ nT β − β ⇒ Bε Bε hZ1 + β QB ΓQB βZ2 ;. √ if T / n → 0. ¡ ¢ ˆ − β = Op T −2 , β ∙Z ¸−1 ³ ´ 1 2 2 ˆ 0 ¯ ¯ T β − β ⇒ σe Ωεε ; Bε Bε 2. (32). (33) (34). where Z1 ∼ N1 (0, Ik ) and Z2 ∼ N2 (0, Ik ) are independent, the random matrix ˜ B is defined as Q. T −2. T X t=1. and. Γ = lim n−1 n→∞. ˜B , ˆ t Wt0 ⇒ Q W. n n X X. λi λ0j E (eit ejt ) ,. i=1 j=1. n. 1X 2 σ ei , n→∞ n i=1. σ 2e = lim. where σ 2ei is the long run variance of process {eit }. Let equation (1) be a spurious regression: √ √ √ if n/T → 0, or T / n → 0 and n/T 2 → 0. ³ ´ ˆ − β = Op n−1/2 , β. ´ √ ³ ˆ−β ⇒ n β. µZ. ¯ε B ¯ε0 B. √ if T 2 / n → 0, then (33) and (34) hold.. 17. ¶−1 µZ. ¯ε Bu B. (35) ¶p ¯ ∆; h. (36).

(20) Proof. See Appendix. √ Consistency is ensured in both cases, even though T / n → 0 results in a √ slower (than in the case of n/T → 0) rate of convergence and in a degenerate ˆ − β. This is anyway not surprising given that behavior of the numerator of β. the shocks estimation errors (see Bai and Ng (2002), and Bai (2004) can be decomposed in several terms of different asymptotic stochastic magnitude, which have an impact only on the numerator. Notice the consequence of equation (1) being a spurious regression: as long as the number of cross sectional units is √ not exceedingly large, the classical n consistency holds, and we have the same limiting distribution as in equation (20). When n is far larger than T , we have the same result as if relationship (1) were a cointegration relationship. √ See below for the case when n/T tends to a constant. ˆ F D are in the folThe convergence rate and the limiting distribution for β lowing theorem. Theorem 4 Suppose Assumptions 1-2 and 4-6 hold. If. n T. →0. ³ ´ ˆ F D − β = Op n−1/2 T −1/2 , β ´ p √ ³ FD −1 ˆ nT β − β → Σ−1 β ∆F QV. (37) (38). where V is the probability limit of the diagonal matrix consisting of the first k eigenvalues of (nT )−1 ∆Z∆Z 0 in decreasing order, and " n # T T X X X¡ ¢ −3/2 0 −1 ˆ ˜ eit eis − γ s−t . ∆Ft ∆Fs n Q = p lim T s=1 t=1. If. T n. →0. i=1. ¡ ¢ ˆ F D − β = Op T −1 , β ³ FD ´ p ˆ ¯ e V −1 β, T β −β →h. ¯ e is the long-run variance of the process limn→∞ n−1/2 Pn eit . where h i=1 Proof. See Appendix.. 18. (39) (40).

(21) Notice that in this case we have degenerate limiting distributions, despite having consistent estimates. The condition n/T → 0 means that T is much larger than n, which in turn implies a panel where time series observations outnumber the cross sectional units. In such a case, we still have consistency. The condition T /n → 0 implies that the number of units n is far larger than T . In such a case, consistency is ensured, even though at a ”slow” rate, given by T . In this case the impact of n becomes ineffective, just as in Bai (2003, 2004) and Bai and Ng (2002, 2004), where consistency depends on the minimum between T and n or some functions of them. Further, the distribution limit is degenerate, and therefore convergence ¡ ¢ in distribution can be achieved at a slower speed than Op T −1 . Finally, it can be observed that in both Theorems 3 and 4, the boundary √ cases n/T → τ or n/T → τ 0 (for τ and τ 0 are constants), are implicitly analyzed. In these cases, the limit distributions are given by the sum of the limit distributions in equations (32)-(34) and (38)-(40), respectively.. 3.2.2. The case of T fixed and n large. When T is fixed and n is large, consistent estimation of shocks is still possible, see e.g. Connor and Korajzcyk (1986) and Bai (2003). However, the following restriction is necessary: Assumption 7: E (eit eis ) = 0 for all t 6= s. Assumption 7 rules out the possibility of serial correlation in the data generating process of the eit , and therefore this is a constraint on Assumption 5(d). However, contemporaneous correlation and cross sectional heteroscedasticity are preserved. Under Assumptions 4-7, we know that shocks estimation is i.e. we have both. ³ ´ Fˆt − Ft = Op n−1/2. 19. √ n consistent,.

(22) and. ³ ´ ∆Fˆt − ∆Ft = Op n−1/2. for all t.. Theorems (5) and (6) do not anyway require. √ n consistency, since they. ˆ and β ˆ F D for any consistent ensure the consistency of the OLS estimates β estimate of the shocks, irrespective of the rate of convergence. Theorem 5 Suppose Assumptions 1-7 hold; then for every consistent estimator Fˆt of Ft and for fixed T and n → ∞ we have the same results as in equations (13)-(16). Proof. See Appendix. Theorem 6 Suppose Assumptions 1-7 hold; then for every consistent estimator ∆Fˆt of ∆Ft and for fixed T and n → ∞ we have the same results as in equations (23) and (24). Proof. See Appendix. In both cases we have the same results as we would have if the Ft s were observable. Therefore, when T is fixed, having large n makes it indifferent to use observed or estimated shocks as long as shocks are estimated consistently. 3.2.3. The case of n fixed and T large. In what follows, we provide a new inferential theory for the case when shocks are unknown and the cross-sectional dimension n is finite. This case has not been explored in the literature, the only exception being Gonzalo and Granger (1995). Our contribution is aimed at making the estimated shocks usable in a regression framework. Rewriting model (2) in the vector form, one gets: zt = ΛFt + et , 0. 0. (41) 0. where zt = (z1t , ..., znt ) , et = [e1t , ..., ent ] , and Λ = (λ1 , λ2 , ..., λn ) . Here too one can estimate Λ using the principal components estimator. A feasible 20.

(23) ˆ is given by the √n times the eigenvectors corresponding estimator of Λ, Λ, to the k largest eigenvalues of Z 0 Z. Notice that this estimator exploits the ˆ 0 Λ/n ˆ normalization Λ = Ik , and it turns out to be computationally convenient for the case of n < T . For sake of the notation, and without loss of generality, Pn from Assumption 4 we assume henceforth that n−1 i=1 λi λ0i = Ik . The following theorem characterizes consistency and limiting distribution of. ˆ Λ. Proposition 1 Under Assumptions 3-6 we have ¡ ¢ ˆ − Λ = Op T −1 , Λ. (42). ∙ ¸ µZ ¶ µZ ¶−1 Z ³ ´ ˆ −Λ T Λ ⇒ In − n−1 Λ Bε Bε0 Λ0 dWe Bε0 Bε Bε0 µZ ¶ −n−1 Λ dWe Bε0 Λ ∙ ¸ Z +n−1 In − 2n−1 Λ Bε Bε0 Λ0 Ωe Λ, (43) where We is the Wiener process associated to the partial sums of et and Ωe = E (et e0t ). Proof. See Appendix. Note that in this case we have a T -consistent estimate of the shock loadings, even though the principal component estimator of Ft is not consistent (see Bai (2004) and Proposition 2 below) when n is finite. Henceforth, for sake of notation, we refer to the limiting distribution of ³ ´ ³ ´ ˆ − Λ as D1 , i.e. T Λ ˆ − Λ ⇒ D1 . Given the restriction Λ ˆ 0 Λ/n ˆ T Λ = Ik , Λ Λ the OLS estimator of Ft , obtained regressing the zt s on the estimated loadings. ˆ is Λ, ˆ 0 zt . Fˆt = n−1 Λ The following Proposition characterizes (the inconsistency of) this estimator:. 21.

(24) ˆ 0 zt , and also the first difference estimator, Proposition 2 Consider Fˆt = n−1 Λ ˆ 0 ∆zt . Then ∆Fˆt = n−1 Λ ° ° ° ° max °Fˆt − Ft ° = Op (1) ,. (44). 1≤t≤T. and. ° ° ° ° max °∆Fˆt − ∆Ft ° = Op (1). (45). 1≤t≤T. uniformly in t.. Proof. See Appendix. From Proposition 2 we note that the estimates of the shocks and of their first difference are inconsistent. However this inconsistency has no impact on ˆ and β ˆ F D , though it affects their asymptotic law. See the the consistency of β proofs of Theorems 7 and 8. ˆ are in the following The convergence rate and the limiting distribution for β theorem. ˆ we have: Theorem 7 For the estimator β, ¡ ¢ ˆ − β = Op T −1 , β. (46). ⎧ R ³ ´1/2 ¸−1 ⎪ ∙Z ¯ε dBu Pn Pn hij ⎨ ³ ´ B i=1 j=1 ¤ £R ˆ−β ⇒ ¯ε0 ¯ε B T β B ¯e0 Λβ + n−1 Λ0 Σe Λβ ¯ ε dB −n−1 R B ⎪ ¡ ¢ ⎩ ¯ε B ¯ε0 D10 Λ − Λ0 D1 β −n−1 B Λ Λ. ⎫ ⎪ ⎬ ⎪ ⎭. ,. (47). ¯e is the demeaned Brownian motion associated to the partial sums of et where B and Σe = V ar (et ). When this is a spurious relationship, one gets ˆ − β = Op (1) , β ˆ−β ⇒ β. µZ. ¯ε0 ¯ε B B. ¶−1 µZ. ¯ε Bu B. Proof. See Appendix.. ¶. (48) ⎞1/2 ⎛ n X n X ⎠ . ⎝ h∆ ij. (49). i=1 j=1. ˆ Note that even though common shocks cannot be estimated consistently, β is consistent when (1) represents a cointegration relationship but inconsistent 22.

(25) when (1) represents a spurious regression. shock estimation has an impact on ˆ − β when equation (1) is a cointegration regression the limit distribution of β see equation (47) above. On the other hand, it does not affect the asymptotic distribution when equation (1) is a spurious regression - see equation (49). This can be seen comparing the two distribution limits with equations (10) and (12) respectively, where shocks are assumed to be known. Equations (47) and (49) show an important common feature of this theoretical framework. Only the numerators of equation (47) and (49) depend on whether equation (1) is a cointegrating or spurious regression, whilst the denominators are not affected. This is due to the fact (detailed in the proof) P ˆ ˆ0 that though Fˆt is not a consistent estimator for Ft , the quantity Ft Ft is a P ˆ Ft Ft0 for any consistent estimator of the loadings Λ. consistent estimator for. ˆ F D are in the folThe convergence rate and the limiting distribution for β. lowing theorem. ˆ F D , we have: Theorem 8 For the first difference estimator β ˆ F D − β = Op (1) , β. (50). p −1 ˆF D − β → β −β + n [Λ0 Σ∆z Λ] Σ∆F β,. (51). and. where Σ∆e = V ar (∆et ) and Σ∆z = ΛΣ∆F Λ0 + Σ∆e . Proof. See Appendix. ˆ F D is inconsistent. As detailed in the proof, this is due to The estimator β P P the two terms ∆Ft ∆Ft0 and ∆et ∆e0t having the same asymptotic order, rather than to the shock estimates being inconsistent. Also, this hold for any ˆ (see discussion in the proof). consistent estimator Λ Theorems 7 and 8 hold if equation (2) represents a cointegration relationship. We now turn to evaluate the case of eit ∼ I (1).. 23.

(26) 4. CONCLUSION. This paper developed limiting theory for the OLS estimator for panel models with common shocks, where contemporaneous correlation is generated by both the presence of common regressors (e.g. macro shocks, aggregate fiscal and monetary policies) among units and weak spatial dependence among the error terms. We derived rates of convergence and limiting distributions under a comprehensive set of alternative characteristics of panels: several combinations of the cross-sectional dimension n and the time series dimension T ; shocks being either observable or unobservable; and stationary and nonstationary panel models, the latter representing either a cointegrating equation or a spurious regression. When the common shocks are observable, the OLS estimator always provides consistent estimates of the β, the case of spurious regression with fixed n being the only exception. Consistency holds for all possible combinations of the dimensions of n and T , including the case of n fixed, which so far has not been addressed in the literature on nonstationary panel factor models. We extend the study of consistency of OLS estimators to the case when the shocks are unobservable and we prove that consistency always holds, the cases of spurious regression and stationary regression when n is fixed being the only exceptions. A central result is represented by the limiting distributions derived under the strong cross-sectional dependence induced by the presence of common shocks. In this case, we obtained a mixed normality as consequence of the common shocks being nonstationary, while when shocks are stationary, normal distributions are obtained. In this paper, we consider a panel regression model with only latent shocks Ft as regressors. This formulation can be extended to a more general framework containing also idiosyncratic regressors, i.e. yit = αi + β 0 Ft + γ 0 xit + uit . Another important extension is to relax the exogeneity hypothesis in Assumption 6(a). In this case, fully modified OLS (Phillips and Hansen, 1990) and/or instrumental variable estimators may be employed.. 24.

(27) These interesting issues are beyond the scope of the present paper, and we leave them for future studies.. 25.

(28) Appendix To prove the theorem, we refer to equation (7) ˆ −β = [P P Wt W 0 ]−1 [P P Wt uit ]. The that contains the estimation error β t i t i t P P proof be derived splitting this quantity into the denominator i t Wt Wt0 and P P the numerator i t Wt uit , and analyzing the asymptotic behavior of both Proof of Theorem 1.. quantities separately.. Let us start considering the denominator. P P i. t. Wt Wt0 . When T → ∞. and n is fixed, we have from Assumptions 2 and 3 that under both the cases that equation (1) is a spurious regression or a cointegrating one it holds that ¡ 2¢ P P 0 and i t Wt Wt = Op T Z n T 1 XX 0 ¯ε B ¯ε0 . Wt Wt ⇒ B nT 2 i=1 t=1. As n → ∞, and for fixed T , we have. P P i. t. Wt Wt0 = Op (n). n T T 1 X 1 XX 0 Wt Wt = 2 Wt Wt0 , nT 2 i=1 t=1 T t=1. whilst as both n and T are large we have. (52). P P i. t. (53). ¡ ¢ Wt Wt0 = Op nT 2. Z n T 1 XX 0 ¯ε B ¯ε0 . Wt Wt ⇒ B nT 2 i=1 t=1. (54). As far as the numerator is concerned, the proof be derived with respect to three separate cases, following the same structure as in the theorem. We firstly P P derive the rate of convergence and the limiting distribution of i t Wt uit for. the case when T is large and n is fixed; we then study the opposite case, when. T is fixed and n is large; last, we analyze the case when both T and n are large. Case 1: large T and fixed n We firstly focus our attention to the case where equation (1) is a cointegration relationship. Denote ξ nt = T −1 Wt. Ã n X i=1. 26. uit. !.

(29) and ξ nT =. T X. ξ nt .. t=1. Assumption 6 ensures that Ft and the uit s are independent. Also, according to P Assumption 1(a), the process i uit has covariance structure given by "Ã n !Ã n !# n X n X X X E uit uis τ ij,ts . = i=1. i=1. i=1 j=1. Then the absolutely summability condition on τ ij,ts over time implied in Assumption 1(b), and Assumptions 2 and 3 ensure that a FCLT holds such that Z ¯ε dW, ξ nT ⇒ B where W is a Brownian motion with variance n n T n T n X X 1 XXXX τ ij,ts = hij . T →∞ T t=1 s=1 i=1 j=1 i=1 j=1. lim. An alternative way to write the limiting distribution of ξ nT is ξ nT. ⎛ ⎞1/2 µZ ¶1/2 n n X X ¯ε B ¯ε0 ⇒⎝ hij ⎠ Z, B i=1 j=1. where Z ∼ N (0, Ik ).. Hence we have a twofold result. First, the rate of convergence of the nuˆ − β is Op (T ); therefore, given equation (52) that ensures that merator of β ¡ ¢ ¡ ¢ ˆ − β is Op T 2 , we have that β ˆ − β = Op T −1 , proving the denominator of β. equation (9). As far as the distribution limit is concerned, we know, combining the asymptotic law of ξ nT with equation (52), we have that. ". T n 1 XX Wt Wt0 T 2 i=1 t=1. #−1 ". ⎛ ⎞1/2 # ¶−1/2 X µZ T n n n X 1 XX 1 ¯ε0 ¯ε B ⎝ Wt uit ⇒ hij ⎠ Z, B T i=1 t=1 n i=1 j=1. ¯ε is which proves equation (10). Note that independence between Z and B ensured by Assumption 6.. We can now consider the case when equation (1) is a spurious regression and therefore uit ∼ I (1). 27.

(30) P PT S Define first ξ Snt = T −2 Wt ( ni=1 uit ) and ξ SnT = t=1 ξ nt . The process Pn Pn Pn ∆ i=1 uit is still a unit root process with long run variance given by i=1 j=1 hij .. Therefore, a FCLT, which follows from Assumptions 1(a), 2 and 3, ensures that. ˆ − β = Op (1), as ξ SnT = Op (1). Together with equation (52), this proves that β reported in equation (11). As far as the limiting distribution is concerned, here ˆ − β is given by the asymptotic law of the numerator of β ξ SnT. T 1 X = 2 Wt T t=1. Ã n X. uit. i=1. !. ⇒. µZ. ¯ε Bu B. ¶. ⎛ ⎞1/2 n n X X ⎝ ⎠ . h∆ ij i=1 j=1. Combining this with the asymptotic law of the denominator given in equation (52), we get equation (12). Case 2: large n and fixed T . In this case the same approach as in the previous case be followed to prove the main results in the theorem.. ¡ ¢ P Consider first the cointegration case. Define ˜ξ nt = Wt n−1/2 ni=1 uit and ˜ξ = nT. T X t=1. Ã. −1/2. Wt n. n X i=1. !. uit .. P Assumption 1(a) ensures that a CLT holds for n−1/2 ni=1 uit , so that as n → P ∞ we have that, for every t, n−1/2 ni=1 uit ⇒ u ¯t , where u ¯t is a normally distributed, zero mean random variable with, after Assumption 1(b) E [¯ ut u ¯s ] = τ¯ts . ¯t are mixed normals random variables (due to the Therefore, the quantities Wt u randomness of Wt ) and it ultimately holds that ". ˜ξ ∼ N 0, nT. T T X X t=1 s=1. #. Wt Ws0 τ¯ts =. Ã T T XX t=1 s=1. Wt Ws0 τ¯ts. !1/2. Z,. where Z ∼ N (0, Ik ); Assumption 6 ensures independence between Z and the PT PT random variable t=1 s=1 Wt Ws0 τ¯ts .. ˆ − β is Therefore, in this case the rate of convergence of the numerator of β √ Op ( n). Combining this with the rate of convergence of the denominator, given 28.

(31) ¡ ¢ ˆ − β = Op n−1/2 , thereby proving equation by equation (53), we have that β. (13). As far as the distribution limit is concerned, combining the asymptotic law of ˜ξ nT with equation (53), we ultimately obtain (14).. ¡ ¢ Pn S Under the spurious regression case, define ˜ξ nt = Wt n−1/2 i=1 uit and ˜ξ S = PT ˜ξ S . Assumption 1(a) ensures the validity of the CLT for n−1/2 Pn uit , nT t=1 nt i=1 Pn −1/2 so that uniformly in t we have, as n → ∞, n ¯t . The process i=1 uit ⇒ u u ¯t is an aggregation of unit root processes, and in light of Assumption 1(a) it is. ¯∆. a unit root process with long run variance which by definition is equal to h S From this we have that ˜ξ nT = Op (1), and combining this with equation (53),. ˆ − β = Op (1) as reported in equation (15). As far as the limiting we obtain β PT S S distribution is concerned, since ˜ξ nT is a finite sum, we have ˜ξ nT ⇒ t=1 Wt u ¯t. as n → ∞. Combining this with equation (53), we prove the validity of equation (16). Case 3: large n and large T .. Let us start with the case where equation (1) is a cointegration relationship. ¡ ¢ P P Define ˇξ nt = T −1 Wt n−1/2 ni=1 uit , and let ˇξ nT = Tt=1 ˇξ nt . After AsP sumption 1(b), we know that the process n−1/2 ni=1 uit has zero mean and. covariance structure given by "Ã !Ã !# n n n n X X 1 XX −1/2 −1/2 E n uit uis τ ij,ts . n = n i=1 j=1 i=1 i=1 Pn Moreover, Assumption 6 ensures that n−1/2 i=1 uit and Wt are independent.. Hence, in light of Assumptions 1(a), 2 and 3, the FCLT ensures that Z ˇξ ⇒ B ¯ε dW, nT where the Brownian motion W has variance equal to n−1. Pn. i=1. Pn. j=1. hij . An al-. ternative formulation for the asymptotic distribution of ˇξ nT , as it is well known, ⎛ ⎞1/2 µZ ¶1/2 n n X X 1 0 ˇξ ⇒ ⎝ ¯ ¯ ⎠ hij Z, Bε Bε nT n i=1 j=1 ¯ε and Z are independent. As n → ∞ we have where Z ∼ N (0, Ik ) and B n. n. 1 XX ¯ hij = h, n→∞ n i=1 j=1 lim. 29.

(32) and therefore, as n → ∞ ˇξ ⇒ nT. ¶1/2 p µZ 0 ¯ ¯ ¯ Z. h Bε Bε. Hence, as far as the rate of convergence of ˇξ nT is concerned, we have ˇξ nT = ¡ ¢ ˆ − β = Op n−1/2 T −1 , Op (1). Combining this with equation (54), we get that β proving equation.(17). As far as the distribution limit is concerned, combining the asymptotic law of ˇξ nT with equation (54), we have: #" # " ¶−1/2 n T T n p µZ 1 XX 1 XX 0 0 ¯ ¯ ¯ √ B W W W u Z, ⇒ h B t t t it ε ε nT 2 i=1 t=1 nT i=1 t=1 which corresponds to equation (18). We now turn to the case when equation (1) is a spurious regression. Let ¡ ¢ P P S S = T −2 Wt n−1/2 ni=1 uit and ˇξ nT = Tt=1 ˇξ nt . For fixed n the process P n−1/2 ni=1 uit is still a unit root process with long run variance given by P P n−1 ni=1 nj=1 h∆ ij . Therefore, for fixed n and as T → ∞, a FCLT, which. ˇξ S nt. follows from Assumptions 1(a), 2 and 3, ensures that ξ SnT = Op (1). This result, ¡ ¢ ˆ − β = Op n−1/2 , as reported in together with equation (54), proves that β equation (19). As far as the limiting distribution is concerned as T → ∞ we. have ˇξ S nT. T 1 X = 2 Wt T t=1. Ã. n. 1 X √ uit n i=1. !. ⇒. taking the limit for n → ∞ leads to µZ. ¯ε Bu B. ¶. µZ. ¯ε Bu B. ¶. ⎞1/2 n n X X 1 ⎝ h∆ ⎠ ; n i=1 j=1 ij ⎛. ⎞1/2 ¶ n X n p µZ X 1 ∆⎠ ∆ ¯ ¯ ⎝ h → h Bε Bu . n i=1 j=1 ij ⎛. (55). Combining this result with the one reported in equation (54), we ultimately get equation (20). Proof of Theorem 2. To prove the theorem, we refer to equation (8) that ˆ F D − β = [P P ∆Ft ∆F 0 ]−1 [P P ∆Ft ∆uit ]. contains the estimation error β t i t i t P P The proof be derived splitting this quantity into the denominator i t ∆Ft ∆Ft0 30.

(33) and the numerator. P P i. t. ∆Ft ∆uit , and analyzing the asymptotic behavior of. both quantities separately. Let us start considering the denominator. P P i. t. ∆Ft ∆Ft0 . When T → ∞. and n is fixed, we have from Assumption 2 and the Law of Large Numbers that under both the cases that equation (1) is a spurious regression or a cointegrating P P one it holds that i t ∆Ft ∆Ft0 = Op (T ) and n T 1 XX p ∆Ft ∆Ft0 → Σ∆F . nT i=1 t=1. As n → ∞, and for fixed T , we have n. T. (56). T. X 1 XX ∆Ft ∆Ft0 = ∆Ft ∆Ft0 , n i=1 t=1 t=1. (57). whilst as both n and T are large we have. with. Pn. i=1. PT. t=1. n T 1 XX p ∆Ft ∆Ft0 → Σ∆F , nT i=1 t=1. (58). ∆Ft ∆Ft0 = Op (nT ).. As far as the numerator is concerned, as in the case of Theorem 1 the proof be derived with respect to three separate cases, following the same structure as in the theorem. We firstly derive the rate of convergence and the limiting P P distribution of i t ∆Ft ∆uit for the case when T is large and n is fixed; we. then study the opposite case, when T is fixed and n is large; last, we analyze the case when both T and n are large. The proofs for each of the three cases are along the same lines as in Theorem 1. It is worth noticing though that both under the case when equation (1) is a cointegration relationship and when it is a spurious regression, ∆uit is a stationary process. Therefore, there is no need to distinguish between the two cases unlike in Theorem 1. Case 1: large T and fixed n Denote ξ∆ nt. =T. −1/2. ∆Ft. Ã n X i=1. and ξ∆ nT =. T X t=1. 31. ξ∆ nt .. ∆uit. !.

(34) Assumption 6 ensures that ∆Ft and the ∆uit s are independent. Also, according P to Assumption 1(b), the process i ∆uit has zero mean and covariance structure !Ã n !# "Ã n n X n X X X ∆uit ∆uis γ ij,ts . = E i=1. i=1. i=1 j=1. Therefore the process ξ ∆ nt has zero mean and covariance structure given by ⎞ ⎛ n X n i h X 1⎝ ∆0 ⎠ E (∆Ft ∆Fs0 ) . γ E ξ∆ nt ξ nt = T i=1 j=1 ij,ts. After Assumption 1(b) and 2, that ensure weak dependence over time a CLT holds. Therefore, as T → ∞, we have ⎞ ⎡ ⎤1/2 ⎛ n n X X 1⎝ ξ∆ ⇒ ⎣ lim γ ij,ts ⎠ E (∆Ft ∆Fs0 )⎦ Z nT T →∞ T i=1 j=1 =. (59). ⎞1/2 ⎛ n X n X ⎠ Σ1/2 Z, ⎝ h∆ ij ∆F. i=1 j=1. where Z ∼ N (0, Ik ). Hence we have a twofold result. First, the rate of con³√ ´ ˆ F D − β is Op vergence of the numerator of β T ; therefore, given equation ˆ F D − β is Op (T ), we have that (56) that ensures that the denominator of β ¡ ¢ ˆ F D − β = Op T −1/2 , proving equation (21). As far as the distribution limit β. is concerned, combining the asymptotic law of ξ ∆ nT with equation (56), we have. that ". T n 1 XX ∆Ft ∆Ft0 T i=1 t=1. #−1 ". ⎞1/2 ⎛ # T n n n 1 XX 1 ⎝X X ∆ ⎠ −1/2 √ ∆Ft ∆uit ⇒ h Σ∆F Z, n i=1 j=1 ij T i=1 t=1. which proves equation (22).. Case 2: large n and fixed T . In this case the same approach as in the previous case be followed to prove the main results in the theorem. ¡ ¢ P ∆ Define ˜ξ nt = ∆Ft n−1/2 ni=1 ∆uit and ˜ξ ∆ = nT. T X t=1. 32. ˜ξ . nt.

(35) P Assumption 1(a) ensures that a CLT holds for n−1/2 ni=1 ∆uit , so that as Pn n → ∞ we have that, for every t, n−1/2 i=1 ∆uit ⇒ u ¯t , where ∆¯ ut is a. normally distributed, zero mean random variable with covariance structure E [∆¯ ut ∆¯ us ] =. T X T X. γ¯ ts .. t=1 s=1. ∆ Hence, in light of Assumption 6, ˜ξ nt is a zero mean random variable whose. covariance structure is given by (after Assumption 1(a)) T X T h ∆ ∆0 i X E ˜ξ nt ˜ξ ns = γ¯ ts E (∆Ft ∆Fs0 ) . t=1 s=1. ∆. Since ˜ξ nT is a finite sum of normally distributed random variables, we have that ˜ξ ∆ nT. ∼. Ã T T XX. ∆Ft ∆Fs0 γ¯ ts. t=1 s=1. !1/2. Z,. where Z ∼ N (0, Ik ); Assumption 6 ensures independence between Z and the P P random variable t s ∆Ft ∆Fs0 γ¯ ts . Therefore, in this case the rate of conˆ F D − β is Op (√n). Combining this with the vergence of the numerator of β rate of convergence of the denominator, given by equation (57), we have that ¡ ¢ ˆ F D − β = Op n−1/2 , thereby proving equation (23). Also, combining this β with equation (57), we ultimately obtain (24).. Case 3: large n and large T . ¡ ¢ Pn PT ∆ ∆ ∆ Define ˇξ nt = T −1/2 ∆Ft n−1/2 i=1 ∆uit , and let ˇξ nT = t=1 ˇξ nt . In light. ∆ of the passages derived above, the ˇξ nt s are random variables with zero mean and. covariance structure given by ⎞ ⎛ n n X h ∆ ∆0 i X 1 1 ⎠ E (∆Ft ∆Fs0 ) . E ˇξ nt ˇξ ns = ⎝ γ T n i=1 j=1 ij,ts. From equation (59) we know that, for fixed n and as T → ∞ ˇξ ∆ nT. ⎛. ⎞1/2 n n X X 1 1/2 ⇒⎝ h∆ ⎠ Σ∆F Z, n i=1 j=1 ij 33.

(36) with Z ∼ N (0, Ik ). As n → ∞ we have ⎛. ⎞1/2 n n X p X 1 1/2 ¯ ∆ Σ1/2 . hij ⎠ Σ∆F → h lim ⎝ ∆F n→∞ n i=1 j=1. ∆ ∆ Hence, as far as the rate of convergence of ˇξ nT is concerned, we have ˇξ nT = ¡ ¢ ˆ F D −β = Op n−1/2 T −1/2 , Op (1). Combining this with equation (58), we get that β. proving equation.(23). As far as the distribution limit is concerned, we know that ˇξ ∆ ⇒ nT. p ¯ ∆ Σ1/2 Z, h ∆F. as (n, T ) → ∞ with Z ∼ N (0, Ik ). Combining this result with equation (58), we have: #−1 " # " T T n n p 1 XX 1 XX 0 ¯ ∆ Σ−1/2 Z, √ ∆Ft ∆Ft ∆Ft ∆uit ⇒ h ∆F nT i=1 t=1 nT i=1 t=1 which corresponds to equation (24). Lemma 1 Let Assumptions 1-6 hold. Then the following results hold for the estimated shocks Fˆt when (n, T ) → ∞: 1. ³ ´ VnT Fˆt − Ft = T −1. T X. Fˆs γ s−t + T −1. s=1. ¤ £ where γ s−t = E n−1 e0t es ,. T X. Fˆs ζ st + T −1. s=1. ζ st =. T X s=1. Fˆs η st + T −1. T X. Fˆs ξ st ,. s=1. e0t es − γ s−t , n. 1 ∆Fs0 Λ0 et , n 1 = ∆Ft0 Λ0 es , n. η st = ξ st. and VnT is a diagonal matrix containing the largest k eigenvalues of (nT )−1 ZZ 0 in decreasing order; 34.

(37) √ 2. Denote CnT = min { n, T }. Consistency of Fˆt is expressed as ° ° ¡ ¢ ¡ ¢ ° ° (a) max °Fˆt − Ft ° = Op T −1 + Op n−1/2 T 1/2 and 1≤t≤T. (b). ° °2 ¡ ¢ °ˆ ° −2 F − F ; ° ° = Op T CnT t t t=1. PT. 3. It holds that: (a) (b) (c). PT. t=1. PT. t=1. PT. t=1. 4. When. √ n T. ³. Fˆt − Ft. ³. Fˆt − Ft. ³. Fˆt − Ft. ´0. ¡ −2 ¢ et = Op T CnT ;. ´0. ¡ ¢ −2 . Fˆt = Op T CnT. ´0. ¢ ¡ ¡ ¢ −2 Ft = Op (1) + Op n−1/2 T = Op T CnT ;. → 0 as (n, T ) → ∞, we have. n T ´ X X √ ³ ˆt = 1 ˆ s W 0 √1 n Wt − W λi eit , W s T 2 s=1 n i=1. with n−1/2. n X i=1. where Zt v N (0, Γ) and T −2 pendent - see Bai (2004).. PT. λi eit ⇒ Zt ,. s=1. ˜B ; Q ˜ B and Z are indeˆ s Ws ⇒ Q W. Proof. See Bai (2004). Lemma 2 Lemma 1 ensures that 1. T −2. PT. t=1. ¡ ¢ ˆ tW ˆ t0 = T −2 PT Wt Wt0 + op T −1/2 C −1 ; W nT t=1. ¡ ¢ ˆ t uit = n−1/2 T −1 Pn PT Wt uit + Op C −1 ; W nT i=1 t=1 ´ ³ ´ ¡ −2 ¢ PT ˆ ³ PT −1 0 ˆ ˆ 3. T −1 t=1 W W − F F + Op CnT ; t Ft − Ft = T t t t t=1. 2. n−1/2 T −1. Pn. i=1. PT. t=1. 35.

(38) Proof. Proof is as follows: T 1 X ˆ ˆ0 Wt Wt T 2 t=1. =. =. T ´³ ´0 1 X³ ˆ ˆ + W − W + W − W W W t t t t t t T 2 t=1 T T ³ ´0 1 X 1 X 0 ˆ W W + W − W W t t t t t T 2 t=1 T 2 t=1. T T ´ ´³ ´0 1 X³ ˆ 1 X³ ˆ 0 ˆ t − Wt + 2 Wt − Wt Wt + 2 Wt − Wt W T t=1 T t=1. = I + II + III + IV.. Consider II and III. Using the Cauchy-Schwartz inequality we get straightforwardly ¶ µ µ ¶ µ ¶ T ³ ´0 1 1 1 X ˆ t − Wt = Op √1 √ O W (1) = o o . W t p p p T 2 t=1 CnT T T CnT Consider now IV. In this case, Lemma 1.2.(b) states that T −2. T ³ ´³ ´0 X ¡ ¢ ˆ t − Wt W ˆ t − Wt = Op T −1 C −2 . W nT t=1. Then µ ¶ µ ¶ T T 1 X 1 1 1 X ˆ ˆ0 0 √ Wt Wt + op + Op , Wt Wt = 2 2 T 2 t=1 T t=1 T CnT T CnT which proves part 1 of the Lemma. Consider now part 2: n n n T T T ´ 1 XX 1 XX³ ˆ 1 XX ˆ √ Wt − Wt uit = I+II. Wt uit + √ Wt uit = √ nT i=1 t=1 nT i=1 t=1 nT i=1 t=1. For term I, Theorem 1 ensures that n−1/2 T −1. Pn. i=1. PT. t=1. Wt uit = Op (1). As. far as II is concerned, using the Cauchy-Schwartz inequality we get ° ° n X T ³ ° 1 X ´ ° ° ˆ t − Wt uit ° W °√ ° ° nT ° i=1 t=1 ° ° T ³ n °1 X ° ´ 1 X ° ° ˆ = ° uit ° Wt − Wt √ °T ° n t=1 i=1 ⎛ ° °2 ⎞1/2 Ã ! µ ¶ T n ° n ° °2 1/2 1 X X 1 X° 1 1 ° ° °ˆ ° ⎝ ≤ uit ° ⎠ = Op °√ °Wt − Wt ° ° n ° T T CnT t=1. i=1. 36. i=1.

(39) given that T −1 Hence,. ° °2 ¡ −2 ¢ Pn °ˆ ° W − W and n−1/2 i=1 uit = Op (1). ° ° = Op CnT t t t=1. PT. µ ¶ n n T T 1 XX ˆ 1 XX 1 √ Wt uit + Op , Wt uit = √ CnT nT i=1 t=1 nT i=1 t=1. proving part 2 of the Lemma. To prove part 3, we note that. T T T ³ ´ ´ 1X ´0 ³ ´ 1 X 0³ 1 X ˆ 0³ ˆ t − Wt Wt Ft − Fˆt + Wt Ft − Fˆt = W Ft − Fˆt = I+II. T t=1 T t=1 T t=1. ¡ −1 ¢ Term I is bounded by Op CnT - see Lemma 1.3.(c) - whilst II is bounded by Ã. ! Ã ! T T ° °2 1/2 1 X °2 1/2 1 X° °ˆ ° ° ° ˆ °Wt − Wt ° °Ft − Ft ° T t=1 T t=1 µ ¶ µ ¶ µ ¶ 1 1 1 = Op Op = Op . 2 CnT CnT CnT. Hence,. µ ¶ µ ¶ T ´ 1X ˆ ³ 1 1 + Op . Wt Ft − Fˆt = Op 2 T t=1 CnT CnT. Proof of Theorem 3. According to equation (29) ˆ−β = β. " n T XX i=1 t=1. ˆ tW ˆ t0 W. #−1 ( n T XX i=1 t=1. ∙³ ¸) ´0 ˆ t β + uit ˆ t Wt − W . W. Let us first consider the denominator of this expression. Assumption 3 and Lemma 2.1 imply that T n X X i=1 t=1. and. ¡ ¢ ˆ tW ˆ t0 = Op nT 2 , W. T n X ¡ 2 ¢−1 X ˆ tW ˆ t0 ⇒ W nT i=1 t=1. Z. ¯ε0 ; ¯ε B B. (60). (61). this holds under both the cases of cointegration and spurious regression. We now prove Theorem 3 for the case when equation (1) is a cointegration ˆ − β is given by relationship. The numerator of β n. T X t=1. T n X ³ ´0 X ˆt β + ˆ t Wt − W ˆ t uit = I + II. W W i=1 t=1. 37.

(40) Let us consider II. We know from Theorem 1 and Lemma 2.2 that, as far as II is concerned T T n n 1 XX ˆ 1 XX √ Wt uit + op (1) = Op (1) , Wt uit = √ nT i=1 t=1 nT i=1 t=1. and therefore. µZ ¶1/2 p T n 1 XX ˆ ¯ ¯ε B ¯ε0 √ B Wt uit ⇒ hZ, nT i=1 t=1. (62). where Z ∼ N (0, Ik ).As far as term I is concerned, two cases are possible: 1. if. √ n/T → 0, we know from Lemma 1.4 that T n ´ X X √ ³ ˆt = 1 ˆ s Ws0 √1 n Wt − W λi eit , W T 2 s=1 n i=1. and. n. 1 X √ λi eit ⇒ Zt , n i=1. with Zt v N (0, Γ) for every t. Therefore the asymptotic magnitude of √ term I is the same as that of term II and equal to Op ( nT ). This proves equation (31). As far as the distribution limit is concerned, we can write. =. T ´0 1X √ ³ ˆt β Wt n Wt − W T t=1 " Ã !#0 T n T 1 X ˆ 1 X 1X 0 Wt λi eit β + op (1) , Ws Ws √ T t=1 T 2 s=1 n i=1. and since by definition T −2. PT. s=1. ˜ B , we have ˆ s Ws0 ⇒ Q W. " Ã !#0 ¶1/2 ³ µZ T T n ´1/2 1X 1 X ˆ 1 X 0 0 ¯ ˜ 0B β ¯ ˜ B ΓQ √ β0Q Wt W λ e β ⇒ B Z, W B s s i it ε ε 2 T t=1 T s=1 n i=1 with Z ∼ N (0, Ik ). Combining this with the asymptotic law of II and with equation (61), we obtain equation (32); √ 2. if T / n → 0, after Lemma 1.3.(c), we have T X t=1. ´ ³ ´0 ³ ˆ t = Op n−1/2 T + Op (1) , ˆ t Wt − W W 38.

(41) and the term that dominates is the one with asymptotic magnitude Op (1). ´0 Pn PT ˆ ³ ˆ Therefore, I = = Op (n), and term II = i=1 t=1 Wt Wt − Wt Pn PT ˆ √ i=1 t=1 Wt uit = Op ( nT ) is dominated. The order of magnitude of the numerator is now Op (n), and combining this with equation (60) we. have. ¡ ¢ ˆ − β = Op T −2 , β. which proves equation (33). As far as the limiting distribution is concerned, following Bai (2004), we can write T T X ³ ´0 X ˆt = 1 ˆ t Wt − W Wt Ws0 W 2 T t=1 s=1 t=1. T X. Ã. n. 1X eit eis n i=1. !. + op (1) ,. and asymptotically we have: Ã n ! Ã !Ã ! n T T T T 1X 1X 1 XX 1X 1X 0 0 Wt Ws eit eis = Wt eit W eis . T 2 s=1 t=1 n i=1 n i=1 T t=1 T s=1 s We know that T −1. PT. t=1. Wt eit ⇒. R. ¯ε dBei , where Bei (r) is the Brownian B. motion associated to the partial sums of eit with long run variance σ 2ei ; therefore, applying a LLN, the limit for n → ∞ is given by ∙µZ µZ ¶ µZ ¶¸ ¶ n n 1X 1X 0 ¯ ¯ ¯ lim E Bε dBei V ar Bε dBei . Bε dBei = lim n→∞ n n→∞ n i=1 i=1 Since we have that V ar. ¡R. ¡R ¢ ¢ ¯ε dBei = σ 2 E ¯ε B ¯ε0 , it holds that B B ei. ¶ µZ n 1X ¯ V ar Bε dBei lim n→∞ n i=1 ! µZ Ã ¶ n 1X 2 ¯ε0 = 1 σ 2e Ωεε , ¯ε B σei E B = lim n→∞ n 2 i=1 given the definition of σ 2e and that E. ¡R. equation (61), equation (34) is proved.. ¢ ¯ε B ¯ε0 = 1/2Ωεε . Combining this B. We now prove results when we equation (1) is a spurious regression. Here, P P ˆ t uit in equation (29) is concerned, we have as far as term ni=1 Tt=1 W T n X X i=1 t=1. ˆ t uit = W. T n X X. Wt uit +. i=1 t=1. T ³ n X ´ X ˆ t − Wt uit . W i=1 t=1. 39.

(42) After equation (19) we know T n X X. Wt uit = Op. i=1 t=1. As per. Pn. i=1. ity leads to. PT. t=1. ³. ¡√ 2 ¢ nT .. ´ ˆ t − Wt uit , application of the Cauchy-Schwartz inequalW n X T ³ ´ X ˆ t − Wt uit W i=1 t=1. ! Ã n T ³ ´ X X ˆ = uit Wt − Wt t=1. and therefore. Pn. i=1. Hence, we have. i=1. ° ⎤1/2 " T # ⎡ T ° n °X °2 °2 1/2 X X° ° ° °ˆ ° ⎣ ≤ uit ° ⎦ ° °Wt − Wt ° ° ° t=1 t=1 i=1 ³√ ´ ¡√ ¢ −1 = Op T CnT Op nT , PT. t=1. ³. ´ ˆ t uit . ˆ t − Wt uit is always dominated by Pn PT W W i=1 t=1. T T n n 1 XX ˆ 1 XX √ 2 Wt uit + op (1) , Wt uit = √ 2 nT i=1 t=1 nT i=1 t=1. and, after equation (55), we have µZ ¶p n T 1 XX ˆ ¯∆. ¯ √ 2 Wt uit ⇒ Bε Bu h nT i=1 t=1 Consequently, there are two possibilities for the rate of convergence and the limiting distribution of the numerator: • when —. √ n/T 2 → 0, two subcases are possible:. ´0 Pn PT ˆ ³ √ ˆ n/T → 0, and in such case we have i=1 t=1 W β= t Wt − Wt Pn PT ˆ √ Op ( nT ); therefore the term that dominates is i=1 t=1 Wt uit ; combining this with equations (60) and (61), equations (35) and (36) can be obtained;. 40.

(43) ´0 Pn PT ˆ ³ √ √ ˆ — T / n → 0 and n/T 2 → 0, and here i=1 t=1 W β= t Wt − Wt Pn PT ˆ Op (n); in this case, again the term that dominates is i=1 t=1 Wt uit combining this with equations (60) and (61), equations (35) and (36) hold; ³ ´0 P P √ ˆ t β = Op (n), ˆ t Wt − W • when T 2 / n → 0, we have that ni=1 Tt=1 W. and this is the dominating term. This leads to the same results as in equations (33) and (34).. Lemma 3 Let Assumptions 1-2 and 4-6 hold. Then, for the estimated shocks ∆Fˆt , it holds that 1. ³ ´ VnT ∆Fˆt − ∆Ft = T −1. T X s=1. where γ s−t. ∆Fˆs γ s−t + T −1. T X. ∆Fˆs ζ st + T −1. s=1. n X i=1. ∆Fˆs ηst + T −1. s=1. £ ¤ P = E n−1 ni=1 eit eis , ζ st = n−1. T X. T X. ∆Fˆs ξ st ,. s=1. eit eis − γ s−t ,. ηst = n−1 ∆Fs0 Λ0 et , ξ st = n−1 ∆Ft0 Λ0 es , −1. and VnT is a diagonal matrix containing the largest k eigenvalues of (nT ) in decreasing order; n√ √ o 2. Denote δ nT = min n, T . Consistency of ∆Fˆt is ensured by ° ° ¡ ¢ ¡ ¢ ° ° (a) max °∆Fˆt − ∆Ft ° = Op T −1/2 + Op n−1/2 T 1/2 ; 1≤t≤T °2 ¡ ¢ PT ° ° ˆ ° ∆ F − ∆F (b) ° ° = Op T δ −2 t t nT ; t=1. 3. It holds that:. 41. ∆Z∆Z 0.