Common Correlated Effects Estimation of Dynamic Panels with Cross-Sectional Dependence

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1 Common Correlated Effects Estimation of Dynamic Panels with Cross-Sectional Dependence om De Groote and Gerdie Everaert SHERPPA, Ghent University Preliminary, February 00 Abstract his paper studies estimation of dynamic panels with error cross-sectional dependence generated by unobserved common stochastic components using the common correlated effects pooled CCEP estimator suggested by Pesaran [Econometrica, 006]. Pesaran shows that for a static model, the CCEP estimator is consistent for fixed and N. We now analyse the properties of this estimator in a dynamic setting, allowing for dynamics in both the model and in the common components. We first show that consistency of the CCEP estimator requires that next to N also should tend to infinity in this setting. Second, the asymptotic bias, for fixed, of the CCEP estimator is larger than that of the infeasible pooled OLS estimator, which assumes that the common components are known. hird, we propose a restricted CCEP estimator which is also consistent for both and N tending to infinity but has an asymptotic bias, for fixed, that is smaller than that of the unrestricted CCEP estimator and similar to that of the infeasible pooled OLS estimator. Monte Carlo experiments demonstrate that also the small sample properties of the restricted CCEP estimator are superior to that of the unrestricted CCEP estimator and not much worse than those of the infeasible pooled OLS estimator for not too small. JEL Classification: C3, C5, C3 Keywords: Cross-Sectional Dependence; Dynamic Panel; Common Correlated Effects Introduction In its influential paper, Nickell 98 demonstrated that, when applied to a dynamic panel data model with individual effects, the within groups WG estimator is inconsistent for fixed and N. Nowadays various alternative estimators are available ranging from, among others, general method of moments GMM estimators Arellano and Bond, 99; Blundell and Bond, Corresponding author: weekerkenstraat, B-9000 Gent, Belgium. tom.degroote@ugent.be. Website:

2 998, over analytical bias corrected within estimators Kiviet, 995; Bun and Carree, 005 to a bootstrap based bias corrected within estimator Everaert and Pozzi, 007. However, new challenges arise when it comes to the estimation of dynamic panel data models. he recent panel data literature shifted its attention to the estimation of models with error cross-sectional dependence. A particular form that has become popular is a common factor error structure with a fixed number of unobserved common factors with individual-specific factor loadings see e.g. Coakley et al., 00; Phillips and Sul, 003. For example, Serlenga and Shin 007 estimate a gravity equation of bilateral trade among 5 European countries. heir model introduces common time-specific effects that capture determinants such as the business cycle or deal with globalization issues. his introduces error cross-sectional dependence of the factor-type. heir empirical results show that this approach provides more sensible results than the traditional approach using fixed time dummies. he most obvious implication of error cross-sectional dependence is that standard panel data estimators such as WG are inefficient and estimated standard errors are biased and inconsistent. Phillips and Sul 003 for instance show that if there is high cross-sectional correlation there may not be much to gain from pooling the data. However, cross-sectional dependence can also induce a bias and even result in inconsistent estimates. In general, inconsistency arises when the observed explanatory variables are correlated with the unobserved common factors see e.g. Pesaran, 006, i.e. this is an omitted variables bias which does not disappear as, N or both. In dynamic panel data models more specifically, Phillips and Sul 007 show that the unobserved common factors introduce additional small sample bias and variability in the inconsistency of the WG estimator as N for fixed even under the assumption of temporarily independent factors such that these are not correlated with the lagged dependent variable. his bias disappears as. Sarafidis and Robertson 009 show that also standard dynamic panel data IV and GMM estimators either in levels or first-differences are inconsistent as N for fixed as the moment conditions used by these estimators are invalid under error cross-sectional dependence. In this paper we further analyse the impact of error cross-sectional dependence in the context of a linear dynamic panel data model. In order to keep the bias formulae as simple as possible, we do not include individual effects and consider a single unobserved factor. We first extend the work of Phillips and Sul 007 by deriving explicit bias formulae for the pooled OLS POLS estimator in the case of a temporally dependent factor. In line with Phillips and Sul, we find that under such circumstances the probability limit of the POLS estimator is a random variable rather than a constant for fixed and N. However, the POLS estimator is now also found to be inconsistent for as the temporal dependence in the unobserved factor implies that it is correlated with the lagged dependent variable. As a benchmark, we also present asymptotic bias formulae for an infeasible POLS estimator, which includes the unobserved factor as an explanatory variable. his infeasible POLS estimator is inconsistent for fixed and N but consistent Both restrictions will be relaxed in future versions of the paper.

3 for. Second, we extend the work of Pesaran 006 by analysing the asymptotic behaviour of the common correlated effects pooled CCEP estimator in a dynamic panel data setting. he basic idea of CCEP estimation is to filter out the common components by orthogonalizing on the cross-section averages of both regressand and regressors such that as N the differential effects of the common components are eliminated. Contrary to the static model, the CCEP estimator is no longer consistent for N and fixed in a dynamic panel data model. In fact, the inconsistency of the CCEP estimator is random for N and fixed but disappears as N, jointly. Furthermore, the asymptotic bias term for fixed is bigger than that of the infeasible POLS estimator. Note that the standard CCEP estimator ignores the restrictions on the individual-specific factor loadings as implied by the derivation of the specification of the model augmented with cross-sectional averages. Imposing these restrictions, the restricted CCEP estimator is inconsistent for fixed and N, but its asymptotic bias is smaller than that of the unrestricted CCEP estimator and similar to that of the infeasible POLS estimator. hus, the restricted CCEP is to be preferred over its unrestricted version. We demonstrate the small sample properties of the POLS and CCEP estimators using Monte Carlo simulations. First, the naive POLS estimator is biased for temporally dependent factors. Second, the infeasible POLS is biased for small, with the bias increasing in the degree of temporal dependence in the common factor. hird, both the unrestricted and the restricted CCEP have a higher bias than the infeasible POLS estimator for small values of but the restricted CCEP clearly outperform the unrestricted CCEP and is not much worse than the infeasible POLS estimator for moderate values of. Finally, the Monte Carlo simulation also illustrates the inefficiency of the naive POLS estimator and the substantial bias and inconsistency of its estimated standard errors. he remainder of this paper is organized as follows: Section sets out the basic model and its assumptions. Section 3 explores the asymptotic bias of, respectively a naive POLS, an infeasible POLS, the unrestricted and restricted CCEP in a dynamic context with error cross-sectional dependence. Section 4 contains the results of the Monte Carlo experiments. Section 5 concludes and outlines some future challenges. Model and assumptions We consider the following first-order autoregressive panel data model y it = ρy i,t ν it, ρ, i =,..., N; t =,...,, with y it being the observation of the dependent variable of the ith cross-sectional unit at time t. For notational convenience we assume y i0 is observed. We further assume: 3

4 Assumption A. Cross-section dependence he error term ν it has a single-factor structure ν it = γ i F t ε it, F t = θf t µ t, θ, 3 where F t is an individual-invariant time-specific unobserved effect with µ t i.i.d. 0, σµ. he individual-specific factor loadings γ i are nonrandom parameters satisfying γi = m N γ. ε it satisfies A. he restriction of a single-factor structure is for expositional purposes only. Assumption A. Error condition ε it i.i.d. 0, σ ε across i and t and is independent of Fs for all i, t, s. he model in equations -3 can be written in a convenient component form as y it = y it γ if t, y it = ρy i,t ε it, F t = ρl F t = ρ θ F t ρθf t µ t. 4 For further discussion, stacking - for each i yields y i = ρy i, γ i F ε i, 5 where y i = y i,..., y i, y i, = y i0,..., y i,, F = F,..., F and ε i = ε i,..., ε i. 3 Estimators In this section we analyse the properties of various estimators for the model in equations -3. We start with two extreme approaches, i.e. the naive pooled OLS POLSn estimator which ignores cross-sectional dependence and the infeasible pooled OLS POLSi estimator which adds the unobserved common factor F t as an explanatory variable to the model. Next, we analyse an unrestricted and a restricted version of the CCEP estimator suggested by Pesaran 006 and compare their properties to those of the POLSn and POLSi estimators. 3. Naive POLS We first analyse the asymptotic properties of the POLSn estimator, which is given by ρ POLSn = /N N y i, y i /N N y i, y i, = ρ /N N y i, γ if ε i /N N y i, y. 6 i, It is convenient to use sequential asymptotics with N followed by. All proofs are in the appendix. 4

5 Proposition. In model with errors having the factor structure -3 and satisfying A-A, the POLSn estimator is inconsistent as N and ρ POLSn ρ = m γ t= F t F t σε ρ m γ t= F. 7 t he inconsistency in 7 has the following asymptotic representation as ρ POLSn ρ = θ ρ θ θρ σε θρ θρ m γσµ o p. 8 Equation 7 shows that ρ POLSn is inconsistent as N with the inconsistency depending on the persistence θ of the common factor and on the variance ratio σ ε / m γ σ µ. For a temporally independent factor, the asymptotic bias will be small. Importantly, the asymptotic bias is random for fixed and depends on the particular values of F t. Equation 8 shows that the POLSn estimator is inconsistent even for both N and when θ 0. Essentially, this is an omitted variables bias as θ 0 implies E y i,t F t 0 such that omitting F t from the regression results in an inconsistent estimator for ρ. 3. Infeasible POLS he POLSi estimator, including F t as a regressor, is given by ρ POLSi = /N N y i, M F y i /N N y i, M F y i, = ρ where M F = I F F F F. N /N y i, M F ε i /N N y i, M F y i,, 9 Proposition. In model with errors having the factor structure -3 and satisfying A-A, the POLSi estimator is inconsistent as N and t= ρ POLSi ρ = ρ t= ρt g F,t ρ t= ρt g F,t, 0 m γ σ h ε F where g F,t = s=t τ s,s t with τ s,s t being the s, s t th element in F F F F and h F = F t t= FtF t. t= F t t= F t he inconsistency in 0 has the following asymptotic representation as ρ POLSi ρ = θ ρ θρ m γ θρ σ µ σ ε o p. hus, even with no fixed effects and observing the factor F t, ρ POLSi is inconsistent for N and fixed, with the asymptotic bias depending on the degree of inertia in F t, being on average 5

6 negative positive for positive negative values of θ. Importantly, the asymptotic bias is random as it depends on the particular realization of the process F t. For large values of or a temporally independent factor θ = 0, the bias will be small. 3.3 CCEP he CCEP estimator suggested by Pesaran 006 eliminates the unobserved common factors by including cross-section averages of the dependent and the explanatory variables. aking crosssectional averages of equation gives N y it = ρ N y i,t F t N γ i N ε it, y t = ρy t γf t ε t, which can be solved for F t as F t = γ y t ρy t ε t. 3 Inserting this expression in equation yields the following augmented form y it = ρy i,t γ i γ y t ρy t ε t ε it, = ρy i,t γ i y t γ i y t, ε it, 4 with γ i = γ i /γ, γ i = ργ i and ε it = ε it γ i ε t Unrestricted CCEP he unrestricted CCEP estimator, i.e. ignoring the restriction on γ i, for ρ in equation 4 is given by ρ CCEPu = /N N y i, M Gy i /N N y i, M Gy i, = ρ where M G = I GG G G and G = y, y. N /N y i, M G ε i /N N y i, M Gy i,, 5 heorem. In model with errors having the factor structure -3 and satisfying A-A, the CCEPu estimator is inconsistent as N and ρ CCEPu ρ = ρ t= ρt g F,t ρ t= ρt g F,t, 6 See Phillips and Sul 007 or an expression with multiple factors. 6

7 where g F,t = s=t τ s,s t with τ s,s t being the s, s t th element in H H H H and H = F, F. he inconsistency 6 has the following asymptotic representation as ρ CCEPu ρ = θ ρ θρ ρ o p. 7 he implication of heorem is that the CCEPu estimator is consistent for N, seq but has a different asymptotic bias term compared to the POLSi estimator. More specifically ρ CCEPu ρ POLSi = ρ θρ σ ε m γσ µ θ ρ θρ o p, 8 which implies that for the relevant case of both ρ > 0 and θ > 0, the CCEPu estimator has a relatively bigger downward asymptotic bias compared to the POLSi estimator Restricted CCEP he restricted CCEP estimator, i.e. taking into account the restriction on γ i, for ρ in equation 4 can be obtained by minimizing the objective function S N ρ, F = N y i ρy i, M F y i ρy i,. 9 Although F is not observed when estimating ρ and similarly, ρ is not observed when estimating F, we can replace the unobserved quantities by initial estimates and iterate until convergence. he continuously-updated estimator for ρ, F is defined as More specifically, ρ CCEPr, F = argmin S N ρ, F 0 ρ,f ρ CCEPr, F is the solution to the following two equations ρ CCEPr = /N N y i, M F y i /N N y i, M F y i, = ρ N /N y i, M F ε i /N N y i, M F y i,, F = y ρ CCEPr y where M F = I F F F F. heorem. In model with errors having the factor structure -3 and satisfying A-A, the CCEPr estimator is consistent as N, seq, ρ CCEPr p ρ, 3 7

8 with large expansion given by ρ CCEPr ρ POLSi = o p. 4 he theorem shows that like POLSi and CCEPu, the CCEPr estimator is consistent as N, seq but unlike the CCEPu estimator, its asymptotic bias as N is similar to that of the POLSi estimator. 4 Monte Carlo simulation In this section we investigate the small sample properties of the POLSn, POLSi, CCEPu and CCEPr estimators under cross-sectional dependence. We are interested in the effects of i the extent of cross-sectional dependence, ii the degree of inertia in both the factor and dependend variable and iii the relative importance of the variance of the factor loadings and the idiosyncratic errors. In order to correctly interpret the results, comparability between the different experiments is necessary. o this extent, we take into consideration the remarks made by Kiviet 995 on i the relative impact of the various error terms and ii the importance of the signal-to-noise ratio. Sarafidis et al. 009 extend the Monte Carlo design of Kiviet 995 towards an error structure containing common components. Our design slightly deviates from theirs since our model does not include individual effects, nor a vector of regressors x it. 4. Experimental design Data are generated according to -3 where ε it, µ t and γ i are randomly drawn in each replication from i.i.d.n 0, σ ε, i.i.d.n 0, σ µ and i.i.d.u [γ L, γ U ], respectively. y i, 49 is set to zero and the first 50 observations are discarded. We control for the relative impact of the common correlated effects γ i F t and the idiosyncratic error terms ε it on y it. Using A- and noting that V arγ i F t = m γσf, the variance of y it rewrites to V ary it = V ary it V arγ if t, = σ ε ρ θρm γσ µ ρ θ θρ, σ ε = ω ρ, 5 where ω = m γ σ µ / θ θρ σε /θρ. We therefore choose σµ by setting σ µ = θ θρ θρ ω σε m, 6 γ 8

9 where ω controls for the degree of cross-sectional dependence. In this setting, changing the specifications of the distribution of γ i only affects the amount of heterogeneity since an increase in m γ is offset by a decrease of σµ. Higher values of ω correspond to a higher amount of error cross-sectional dependence. Furthermore, we want to keep the signal to noise, σs, fixed over the various experiments. Using 5 and 6, the signal-to-noise ratio is given by σs = V ary it ν it, = ρ ω ρ θρ σ ε θρ ρ. 7 For a given ω, σ s and m γ, the values for σ ε and σ µ follow from 7 and 6. We conduct experiments for combinations of the following parameter values: ρ {0.4; 0.6; 0.8}, θ {0; 0.4; 0.8} and ω {0.5; ; }. Experiment serves as a point of reference and has the following settings: ρ = 0.6, θ = 0.4, ω =, σ s = 3 and γ i.i.d.u0.5,.. he other six simulations verify the impact on the small sample properties of the estimators for different degrees of error cross section dependence and a higher persistence in both y it and the common factors. he signal to noise ratio is set to 3. γ i is drawn from i.i.d.u [0.5,.]. and N respectively take the following values:{5;0;0;30;40;50} and {0;50}. All experiments are based on 5000 iterations. 4. Simulation Results First we examine the performances of the two extreme estimators, namely the naive and infeasable POLS. Experiment indicates that the naive POLS is biased in a small sample. Furthermore, this bias persists as or N increases. Proposition reveals that the bias emerges from the persistence in the common factors. Simulation results for experiment 5, which generates the common factors by drawing them randomly from a normal distribution, establish that the bias of the naive POLS stems from the inertia residing within the factors. he naive POLS exhibits a small bias for experiment 5, whereas, for the other experiments, it is severly biased. Experiment 4, 6 and 7 illustrate that increasing the level of persistency in the factors or the degree of error cross-sectional dependence, aggravates the bias. Increasing the dynamics of y it on the other hand, has the opposite effect see experiment and 3. In addition, doubling ω reduces the efficiency of the naive POLS. he standard errors substantially underestimate the standard deviations in all experiments. 9

10 able : Small sample properties of POLSn, POLSi, CCEPu and CCEPr POLSn POLSi CCEPu CCEPr, N bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse Experiment : ρ = 0.6, θ = 0.4, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Experiment : ρ = 0.8, θ = 0.4, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Continued on next page 0

11 Continued from previous page POLSn POLSi CCEPu CCEPr, N bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse Experiment 3: ρ = 0.4, θ = 0.4, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Experiment 4: ρ = 0.6, θ = 0.8, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Continued on next page

12 Continued from previous page POLSn POLSi CCEPu CCEPr, N bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse Experiment 5: ρ = 0.6, θ = 0, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Experiment 6: ρ = 0.6, θ = 4, ω =, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , , Continued on next page

13 Continued from previous page POLSn POLSi CCEPu CCEPr, N bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse bias stde stdv rmse Experiment 7: ρ = 0.6, θ = 0, ω = 0.5, σ s = 3 and γ i.i.d.u0.5,. 5, , , , , , , , , , , ,

14 Moving on to the infeasable POLS. For small, the experiments report a significant bias. Experiment 5 forms the exception. Introducing inertia into the data generating process of F t, thus causes the POLSi estimator to become biased for fixed. However, contrary to the naive POLS, the bias does disappear as increases. his is in line with our findings in proposition. he size of N is irrelevant for the bias. Higher degrees of inertia in F t, such as in experiment 4, render a higher bias. Experiments and 6 demonstrate that increasing the persistency of y it or the degree of error cross-sectional dependence, lowers the bias. Comparing the two POLS estimators, the infeasable POLS outperforms the naive POLS in terms of both bias and efficiency. For each experiment, the Root Mean Squared Error RMSE of the infeasable POLS is smaller than the one of the naive POLS. Next, turning to the unrestricted CCEP. We notice a significant small sample bias throughout all experiments. For relatively small, the unrestricted CCEP performs even worse than the naive POLS in terms of RMSE. Contrasted to our benchmark, the infeasable POLS, the small sample bias is larger. his fits with our previous findings expressed in 8, which implies that for relevant range of parameter values θ and ρ, the unrestricted CCEP has a larger asymptotic bias than the infeasable POLS. he small sample bias reduces as becomes larger, whereas the size of N only has a marginal impact. Experiment 5 reveals that, in contrast to the POLS estimators, the small sample bias of CCEPu for fixed, persists even when the common factors are randomly drawn from a normal distribution. herefore, the bias also directly depends on ρ. Finally, the simulations indicate that the restricted CCEP estimator exhibits a bias in a small sample, which diminishes for greater. Again, the size of N does not seem to matter. Setting both CCEP estimators side by side, we notice that the restricted CCEP outperforms its unrestricted counterpart in terms of bias for small and N. his is in accordance with our earlier statement, that the restricted CCEP is preferable over the unrestricted version. Higher or lower degrees of error cross section dependence, as in experiment 6 and 7, have only a minor effect on the performances of both CCEP estimators. Increasing the persistence of the common factors or y it, on the other hand, worsens the asymptotic bias. Experiments -5 demonstrate this. 4

15 5 Conclusion his papers examines the effects of error cross-sectional dependence on POLS and CCEP estimators in a linear dynamic panel data model. he naive POLS turns out to be inconsistent for N. In accordance with Phillips and Sul, we find that, for fixed, the asymptotic bias depends on the particular realisation of F t. he inconsistency remains, even for both and N, due to persistence in the common factors. he infeasable POLS estimator, which we use as a point of reference, is inconsistent for fixed and N. Just as for the naive POLS estimator, the inertia in F t plays an important role in the inconsistency. As, the asymptotic bias disappears. he unrestricted CCEP estimator, as proposed by Pesaran 006 is no longer consistent for fixed and N. In addition, the size of the inconsistency for fixed is bigger than the one of the infeasable POLS estimator. For both and N, this estimator becomes consistent. Contrary to the standard, unrestricted CCEP estimator, the restricted CCEP takes into account the restrictions on the individual-specific factor loadings as implied by the derivation of the specification of the model augmented with cross-sectional averages. his restricted version is, like the unrestricted CCEP, consistent for both and N. However, its asymptotic bias is similar to that of the infeasible POLS. herefore, when faced with error cross-sectional dependence, the restricted CCEP is the more appropriate estimator. Future work will extend the current framework by introducing individual effects, an additional variable x it and allowing for multiple factors. It is then necessary to apply an estimation procedure which can deal with both error cross-sectional dependence due to common factors and individual effects in a linear dynamic panel data model. 5

16 References Arellano, M., Bond, S., 99. Some tests of specification for panel data: Monte carlo evidence and an application to employment equations. Review of Economic Studies 58, Blundell, R., Bond, S., 998. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87, Bun, M., Carree, M., 005. Bias-corrected estimation in dynamic panel data models. Journal of Business & Economic Statistics 3, Coakley, J., Fuertes, A.-M., Smith, R., 00. A principal components approach to cross-section dependence in panels. Unpublished Manuscript, Birkbeck College, University of London. Everaert, G., Pozzi, L., 007. Bootstrap-based bias correction for dynamic panels. Journal of Economic Dynamics & Control 3 4, Kiviet, J., 995. On bias, inconsistency, and efficiency of various estimators in dynamic panel data models. Journal of Econometrics 68, Nickell, S., 98. Biases in dynamic models with fixed effects. Econometrica 49 6, Pesaran, M., 006. Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74 4, Phillips, P. C. B., Sul, D., 003. Dynamic panel estimation and homogeneity testing under cross section dependence. Econometrics Journal 6, Phillips, P. C. B., Sul, D., 007. Bias in dynamic panel estimation with fixed effects, incidental trends and cross section dependence. Journal Of Econometrics 37, Sarafidis, V., Robertson, D., 009. On the impact of error cross-sectional dependence in short dynamic panel estimation. Econometric Journal, 6 8. Sarafidis, V., Yamagata,., Robertson, D., 009. A test of cross section dependence for a linear dynamic panel model with regressors. Journal of Econometrics 48, Serlenga, L., Shin, Y., 007. Gravity models of intra-eu trade: Application of the ccep-ht estimation in heterogeneous panels with unobserved common time-specific factors. Journal of Applied Econometrics,

17 Appendices Appendix A Proofs Lemma A-. Under Assumption A we have from 4 λ 0 = E F t = ρ θ λ ρθλ σ µ, λ = E F t F t = ρ θ λ0 ρθλ, λ s = E F t F t s = ρ θ λs ρθλ s, s A- which can be solved to obtain θρ λ 0 = θρ θ ρ θρ θ ρ θρ σ µ, θρ = θρ ρ θ σ µ, A- θ ρ λ = θρ ρ θ σ µ. A-3 Next, using that F t = F t ρf t we have E F t F t = E F t F t ρe F t = λ ρλ 0 = Proof of Proposition. Suppose Assumptions A-A hold, then: /N N ρ POLSn ρ = y i, γ if ε i /N N y i, y, i, /N = = = t= /N t= N θ θ θρ σ µ. A-4 y i,t γ if t γi F t ε it t= N t= t=, y i,t γ if t y i,t γ if t γi F t ε it t= N N, y i,t γ if t γi F t F t y i,t γ i F, t 7

18 Letting and using Lemma A-, we have t= t= m γ t= = F tf t σε ρ m γ t= F. A-5 t F t = E F θρ t Op = θρ ρ θ σ µ O p, F t F t = E F t F t θ Op = θ θρ σ µ O p, hus, taking limits as N followed by an expansion as, 7 is given by ρ POLSn ρ = m θ γ θ θρ σ µ o p as, σε ρ m θρ γ θρ θ ρ σ µ o p ρ θ = o p as. θρ θ θρ σ ε m γ σ µ Proof of Proposition. First, the probability limit as N of the numerator of equation 9 is given by N = y i, M F ε i = N t= N t= y i,t ε it y i,t ε it s= τ st N τ st ε is, s= y i,t ε is, A-6 where τ st = F s F t / t= F t. Using that N y i,t ε i,t s = σερ s s, A-7 equation A-6 can be written as = 0 s <, A-8 N y i, M F ε i = σ t ε τ st ρ st = σ t ε τ t,t s ρ s, t= s= t= s= = σε τ t,t ρ τ t,t ρ τ t,t 3... ρ τ,, t= = σε ρ t t= s=t t=3 t=4 τ s,s t = σε ρ t g F,t, t= A-9 where g F,t = s=t τ s,s t = s=t F sf s t / t= F t. 8

19 Second, the probability limit as N of the denominator of equation 9 is given by N = N = y i, M F y i, = N t= t= N y i,t yi,t t= N t= s= y i,t y i,t t= s= τ st τ st y i,s, s= τ st y i,t y i,s, N y i,t y i,s. A-0 Since N where m γ = N = = σ ε ρ = σ ε ρ = σ ε ρ y i,t y i,s = N = N y i,t γ if t y i,s γ if s, y i,t y i,s N γi F t F s, = ρ t s ρ σ ε m γf t F s, A- γi, equation A-0 can be written as σ ε ρ m γ F t ρ t s τ ts ρ σ ε m γf t F s, t= t= s= τ ts ρ t s m γ F t τ st F t F s, t= s= t= t= s= τ tt ρ t τ t,t s ρ s t= t= s= m γ F t= s= t F tf s F t F s t= t= F t t= F, t ρ ρ t g F,t m γh F, A- t= where h F = t= F t t= F tf t F. t t= F t t= Deviding A-9 by A- yields the result in equation 0. 9

20 σ ε and Next, letting and using Lemma A- we have g F,t = t E F s F s t E Ft = t θt O p O p,, F E F t F h F = E t t F E Ft E t = Hence, as we have θρ ρ σ µ O p, O p, ρ t g F,t = σ t ε ρ t θt o p, t= t= = σε θ ρθ t ρ t ρθ t o p, t= t= = σε θ θρ θρ θρ ρ θρ θρ θρ o p θρ [ = σε θ θ ρ ] o p, θρ θρ = σε θ θρ o p, A-3 σ ε ρ ρ ρ t g F,t m γh F, t= = σ ε ρ θ ρ θρ = σ ε σ µ [ θ ρ ] θρ σ µ m γ θρ ρ o p, ρ m γ θρ ρ o p. A-4 hus, taking limits as N followed by an expansion as, equation 0 is given by ρ POLSi ρ = σε θ θρ o p = θ ρ θρ σε m γ θρ σ µ σ ε ρ m γσµ θρ ρ o p, o p. A-5, 0

21 Proof of heorem. First note that ε t = 0, y t = y t γf t A-6 = γf t, A-7 such that the probability limit as N of the numerator of equation 5 is given by N y i, M G ε i = N y i, M H ε i, A-8 with M H = I H H H H and H = F, F. Defining τ s,t to be the s, t th element in H H H H and using A-7-A-8, A-8 can be written as N = y i, M G ε i = N t= = σε N t= y i,t ε it y i,t ε it s= t τ stρ st = σε t= s= = σε ρ t t= s=t τ stε is, s= τ st N y i,t ε is, t τ t,t sρ s, t= s= τ s,s t = σε ρ t g F,t, t= A-9 where g F,t = s=t τ s,s t. Second, as A-7 implies that for N M G F = 0 such that M G y i, = M G y i,, the probability limit as N of the denominator of equation 5 is given by N y i, M G y i, = N = N = N = = = σ ɛ ρ = σ ε ρ t= t= y i, M Gy i,, t= N y i,t y i,t y i,t y i,t s= N t= s= τ sty i,s, t= s= τ st τ sty i,t y i,s, N σ ɛ ρ ρ τ t s ts ρ σ ε, t= t= s= τ tsρ t s, t= s= τ tt ρ t τ t,t sρ s, t= t= s= y i,t y i,s,

22 = σ ε ρ ρ ρ t g F,t. A-0 t= Deviding A-9 by A-0 yields the result in equation 6. In order to derive the the expression for, first note that τ st = α 0 α F t F s α F t F s α 3F t F s α F t F s, A- with α 0 = α α 3 α, α = t= F, t α = t= F t F t and α 3 = t= F. t Using Lemma A- and letting we therefore have g F,t = t ω t ω ω t ω ω t ω O p, A- with ω t = λ t /λ 0. Hence as, the numerator of 6 is given by ρ t g F,t = t= t= θρ = θ ρ = θρ = t= θ θρ ρ ρ t= t ρ t ω t ω ω t ω ω t ω o p, ρ t t ρ t t θ ρ ω ω t θρ ω ω t o p, ωt ρω t ρ ω t θω t θ o p, o p, A-3 where use is made of ω t = θ ρ ω t θρω t t and t ρ t ω t = θ ρ t= t= = θ ρ ρω ρ θρ = θ ρ ρω ρ ρ 3 t= θρ ρ ρ ω ρ t t= ρ t ω t θρ t= t ρ t ω t ρ ω ρ t ρ t ω t, t= = θ ρ ρω θρ ρ ρ ω θ ρ ρ θρ 3 O = ρ θ ρ θ ρ ρ θ ρ θ ρ t= t t ρ t ω t O, O, t ρ t ω t, ρ t ω t, t= ρ t ω t, t= ρ t ω t,

23 such that and t ρ t ω t = ω t= t= t= t ρ t ω t = ρ t ρ t ω t = θ ρ ρ θ ρ ρ θ ρ O t= ω θρ = ρ ρ θρ O t ρ t ω t = Similarly, the denominator of 6 as is given by ρ ρ ρ t g F,t t= ρ t ω t, t= θ ρ θρ ρ ρ θ ρ O = ρ O,.. A-4 hus, taking limits as N followed by an expansion as, equation 7 is given by ρ CCEPu ρ = [ θ θρ ρ ρ = θ ρ θρ ρ o p ] ρ o p,. A-5 Proof of heorem. Letting ρ 0, F 0 denote the true parameter ρ and the true factor F respectively such that, after centering, the objective function in 9 is given by S N ρ, F = N y i ρy i, M F y i ρy i, N ε im F 0ε i, = ρ 0 ρ y i, λ i F 0 ε i MF ρ 0 ρ y i, λ i F 0 ε i ε N N im F 0ε i, = S ρ 0 ρ N ρ, F y N i, M F ε i λ i F 0 M F ε i ε i M F M F 0 ε i, N N where ρ S 0 ρ N ρ, F = N y i, M F y i, N ρ λ i F 0 M F F 0 0 ρ N y i, M F F 0 λ i. Using that for N, N N y i, M F ε i = N λ i F 0 M F ε i = N y i, ε i y i, F F F N λ i F 0 ε i λ i F 0 F F F N F ε i = o p, F ε i = o p, 3

24 N ε i M F M F 0 ε i = N ε i F 0 F 0 F 0 F 0 ε i N ε i F F F F ε i = o p. we have S N ρ, F = S N ρ, F o p, A-6 uniformly over ρ and F. First note that as M F 0F 0 = 0, SN ρ 0, F 0 = 0. Second, we show that for any ρ, F ρ 0, F 0, S N ρ, F > 0; thus S N ρ 0, F 0 attains its unique minimum value at ρ, F = ρ 0, F 0. Define hen A = N y i, M F y i, ; B = N λ i ; C = N λ i M F y i,. S N ρ, F = ρ 0 ρ A F 0 M F BM F F 0 ρ 0 ρ C M F F 0, = ρ 0 ρ A C B C F 0 M F ρ 0 ρ C B B M F F 0 B C ρ 0 ρ, = ρ 0 ρ D F θ Bθ, 0, since D F = A C B C and B are both positive definite, where θ = M F F 0 B C ρ 0 ρ. Note that S N ρ, F > 0 if either ρ ρ 0 or F F 0. his implies that ρ CCEPr is consistent for ρ. Next, letting N and using the same derivations as to obtain 0 together with A-6 and A-7, we have from where ρ CCEPr ρ = g F,t = h F = s=t t= ρ F s Fs t / F t, t= F t t= ρt g F,t ρ t= ρt g F,t F t= t F t F, t t= F t t=, A-7 m γ σ h ε F with F t = y t ρ CCEPr y t = ρ 0 ρ CCEPr y t γf t o p. Letting also, we have s=t g F,t = F sf s t t= F o p = g F,t o p, A-8 t 4

25 h F = t= F t t= F tf t F o p = h F o p t t= F t t=, A-9 where use is made of the consistency of ρ CCEPr which implies that ρ 0 ρ CCEPr = o p, such that F = γf t o p. Substituting A-8-A-9 in A-7 implies that for N followed by an expansion as ρ CCEPr ρ = ρ which from using Proposition implies t= ρt g F,t ρ t= ρt g F,t o p m γ σ h ε F, A-30 ρ CCEPr ρ POLSi = o p. A-3 5