Estimation of Time-invariant Effects in Static Panel Data Models

Transcription

1 Estimation of Time-invariant Effects in Static Panel Data Models M. Hashem Pesaran University of Southern California, and Trinity College, Cambridge Qiankun Zhou University of Southern California September 5, 204 Abstract This paper proposes the Fixed Effects Filtered FEF and Fixed Effects Filtered instrumental variable FEF-IV estimators for estimation and inference in the case of time-invariant effects in static panel data models when is large and T is fixed. It is shown that the FEF and FEF-IV estimators are -consistent, and asymptotically normally distributed. The FEF estimator is compared with the Fixed Effects Vector Decomposition FEVD estimator proposed by Plumper and Troeger 2007 and conditions under which the two estimators are equivalent are established. It is also shown that the variance estimator proposed for FEVD estimator is inconsistent and its use could lead to misleading inference. Alternative variance estimators are proposed for both FEF and FEF-IV estimators which are shown to be consistent under fairly general conditions. The small sample properties of the FEF and FEF-IV estimators are investigated by Monte Carlo experiments, and it is shown that FEF has smaller bias and RMSE, unless an intercept is included in the second stage of the FEVD procedure which renders the FEF and FEVD estimators identical. The FEVD procedure, however, results in substantial size distortions since it uses incorrect standard errors. We also compare the FEF-IV estimator with the estimator proposed by Hausman and Taylor 98, when one of the time-invariant regressors is correlated with the fixed effects. Both FEF and FEF-IV estimators are shown to be robust to error variance heteroskedasticity and residual serial correlation. Keywords: static panel data models, time-invariant effects, Fixed Effects Filtered estimator, Fixed Effects Filtered instrumental variables estimator JEL classification: C0, C23, C33 We would like to thank Alex Chudik and Ron Smith for helpful suggestions. Corresponding Author: Department of Economics, University of Southern California, 3620 South Vermont Avenue, Kaprielian Hall 300, Los Angeles, CA 90089, USA, pesaran@usc.edu

2 Introduction Identification and estimation of the effects of time-invariant regressors, such as the effects of race or gender is often the focus of panel data analysis, yet estimation procedures such as fixed effects FE, that yield consistent estimates of the coeffi cients of time-varying regressors under fairly general conditions, cannot be used for estimation of the time-invariant effects, since the FE transformation eliminates all time-invariant regressors. As a result estimation of time-invariant effects has posed a challenge in panel data econometrics - namely how to carry out inference on time-invariant effects without making strong assumptions on the correlation between unobserved individual effects and the time-varying regressors. For the estimation of time-invariant effects, Plumper and Troeger 2007 PT propose the so called Fixed Effects Vector Decomposition FEVD through a three-step procedure. As we shall see, whilst the FEVD approach can be modified to yield consistent estimates of the time-invariant effects, the variance estimator proposed by PT for their estimator is not consistent. PT do not provide any formal statistical proofs to support their stated claims about the consistency of their estimator and its variance estimator. See Greene 20a. In the case where one or more of the time-invariant regressors are endogenous, an early pioneering contribution by Hausman and Taylor 98 HT propose using instrumental variables in the context of a pooled random coeffi cient panel data model. The instruments are obtained by assuming that known sub-sets of time-varying and time-invariant regressors are exogenous. HT also assumed that individual-specific effects as well as the idiosyncratic errors of the panel data model under consideration are serially uncorrelated and homoskedastic. Some of these assumptions are relaxed in the subsequent literature, but the main idea that sub-sets of time-varying and time invariant regressors are exogenous is typically maintained. See also Amemiya and MaCurdy 986, Breusch et al. 989, Im et al. 999 and Baltagi and Bresson 202. In this paper, we consider a general static panel data model, which allows for an arbitrary degree of correlation between the time-varying covariates and the individual effects, and propose the fixed-effects filtered FEF estimation for the coeffi cients of the time-invariant regressors when the cross-sectional observation,, is large and the time-series dimension, T, is small and fixed. Our proposed estimator has two simple steps. In the first step FE estimates are computed for the coeffi cients of the time-varying variables, and these estimates are used to filter out the time-varying effects. The residuals from the first stage panel regression are then averaged over time and used as a dependent variable in a cross-section OLS regression that includes an intercept and the vector of time-invariant regressors. Under the identifying assumption that the time-invariant regressors are uncorrelated with the individual effects and a number of other regularity conditions, it is shown that the FEF estimator is unbiased and consistent for a finite T and as. We derive the asymptotic distribution of the FEF estimator and propose a non-parametric estimator of its Plumper and Troeger 2007 s FEVD approach is very popular in political science, and there even is a STATA procedure for the implementation of the FEVD estimator. 2

3 covariance matrix, not known in the literature, which we show to be consistent in the presence of heteroskedasticity of the individual effects and performs well in the presence of residual serial correlation. Finally, we consider the case when one or more of the time-invariant variables are endogenous, and develop the FEF-IV estimator assuming there exist valid instruments. It is shown that the FEF- IV estimator is consistent and asymptotically normally distributed. A feasible variance estimator is also proposed for this FEF-IV estimator, which works well under heteroskedasticity and residual serial correlation. The main advantage of the proposed FEF-IV over the HT estimator lies in the fact that it does not require a sub-set of time-varying regressors to be exogenous, whilst at the same can use time averages of the time-varying regressors as instruments if it is known that such time averages are uncorrelated with the fixed effects, and at the same time correlated with the endogenous time-invariant regressors. 2 The second advantage of the FEF-IV estimator of timeinvariant effects is its robustness to residual serial correlation and error heteroskedasticity. We also contribute to the controversy over the FEVD estimator proposed by PT, discussed by Greene 20a and Breusch et al. 20b, and followed up with responses and rejoinders by Plumper and Troeger 20, Greene 20b, Breusch et al. 20a, and Beck 20. The FEVD estimator of PT is based on a three step procedure, we show that when an intercept is included in the second step of their procedure, then the FEVD estimator is identical to the FEF estimator. But if an intercept is not included in the second stage, the FEVD estimator is in general biased and inconsistent. The extent of the bias of the FEVD estimator depends on the magnitude of intercept and the mean of time-invariant variables. What is more important is that, even if an intercept is included in the second step of the FEVD procedure, inferences based on the FEVD estimators and their variances in the third step of PT s estimation procedure could be misleading since the variance of the FEVD estimator obtained in the third step is biased and most likely will result in over-rejection of the null. This is confirmed by the Monte Carlo simulations. The small sample properties of the FEF and FEF-IV estimators for static panel data model are investigated, using two sets of comprehensive Monte Carlo experiments including error variance heteroskedasticity and residual serial correlation. In one set we generate the time-invariant regressors as exogenous, whilst in the second set we allow one of the time-invariant regressors to be correlated with the fixed effects. In both sets of experiments we allow the time-varying regressors to be correlated with the fixed effects. We compare FEF and FEVD estimators using the first set of experiments only, since these procedures are not appropriate in the case of the second set of experiments where one of the time-invariant regressors is endogenous. We find that our proposed estimator has smaller bias and RMSE, unless an intercept is included in the second stage of the 2 It is important to note that the assumption of a zero correlation between time averages of time-varying regressors and the fixed effects is less restrictive than the assumption of zero correlation between time-varying regressors and the fixed effects. The latter implies the former but not the reverse. For example, suppose that the time-varying regressor, x it, is related to the fixed effects, α i, according to x it = α ig t + w it, for t =, 2,..., T, where w it is distributed independently of α i and T t= where x i = T T t= xit. gt = 0. In this setting Covxit, αi = gtv arαi 0 for each t, but Cov xi, αi = 0, 3

4 FEVD procedure which renders the FEF and FEVD estimators identical. However, as predicted by our theoretical derivations, the FEVD procedure results in substantial size distortions since it uses incorrect standard errors. In contrast, the use of the standard errors derived in this paper yields the correct size and satisfactory power in the case of all experiments, illustrating the robustness of our variance formula to heteroskedasticity and residual serial correlation. We also compare the FEF-IV estimator with the HT estimator using the second set of experiments where one of the time-invariant regressors is correlated with the fixed effects. The FEF-IV procedure performs well and has the correct size when an instrument is used for the endogenous time-invariant regressor. It is also robust to error variance heteroskedasticity and residual serial correlation. But a straightforward application of the HT procedure results in biased estimates and size distortions since it incorrectly assumes that one of the time-varying regressors is uncorrelated with the fixed effects. In such cases the HT procedure must be modified so that none of the time-varying regressors are used as instruments. The rest of the paper is organized as follows: Section 2 sets out the panel data model with time-invariant effects. Section 3 develops the FEF estimator, derives its asymptotic distribution, gives robust variance matrix estimator for the proposed FEF estimator, and provides a comparison of the FEF and FEVD estimators. Section 4 considers the FEF-IV estimator in the case where one or more of the time-invariant regressors are correlated with the errors. Section 5 discusses the HT estimator and derives its covariance matrix under a general specification of the error covariance matrix. The small sample properties of the FEF and FEF-IV estimators are then investigated in Section 6. The paper ends with some concluding remarks in Section 7. Some of the detailed mathematical proofs are provided in the Appendix. For any real-valued matrix A, we will use A to denote the Frobenius norm of matrix A defined as A = tr AA ] /2. Throughout, K denotes a generic non-zero positive constant that does not depend on. The symbols p and d are used to denote convergence in probability and in distribution, respectively. 2 Panel data models with time-invariant effects Consider the following panel data model that contains time-varying as well as time-invariant variables: y it = α i + z iγ + x itβ + ε it, i =, 2,..., ; t =, 2,..., T, where α i = α + η i, 2 x it is a k vector of time-varying variables, and z i is an m vector of observed individualspecific variables that only vary over the cross section units, i. In addition to z i, the outcomes, y it, are also governed by unobserved individual specific effects, α i. The focus of the analysis is on estimation and inference involving the elements of γ. It is clear that without further restrictions on 4

5 α i, γ cannot be identified even if β was known to the researcher. For example consider the simple case where β = 0, and assume that T is small. Then averaging across t we obtain ȳ i = α + z iγ + v i where ȳ i = T Σ T t= y it, v i = η i + ε i, and ε i = T Σ T t= ε it. It is clear that without specifying how v i and z i are related it will not be possible to identify the effects of z i. To deal with this problem, it is often assumed that there exists instruments that are uncorrelated with v i but at the same time are suffi ciently correlated with z i. Even if such instruments exist a number of further complications arises if β 0. In such a case the IV approach must be extended also to deal with the possible dependencies between η i and x it. In what follows we allow for η i and x it to have any degree of dependence, but initially assume that z i and v i are uncorrelated for identification of γ, and assume that x it and ε is are uncorrelated for all i,t and s, to identify β. This approach can be modified in cases where one or more instruments are available for z i and/or x it. 3 Fixed effects filtered FEF estimator of time-invariant effects 3. FEF estimator Under the assumption that x it and ε is are uncorrelated for all i, t and s, as it is well known β can be estimated consistently under fairly general assumptions on temporal dependence and crosssectional heteroskedasticity of ε it, and the distribution of the fixed effects, α i. Denoting the FE estimator of β by ˆβ, γ can then be estimated by the regression of ȳ i ˆβ x i on an intercept and z i. We denote this estimator by ˆγ F EF and refer to it as the fixed effects filtered FEF estimator of γ. Formally, the FEF estimator can be computed using the following two-step procedure: Step : Using model, compute the fixed-effects estimator of β, denoted by ˆβ, and the associated residuals û it defined by û it = y it ˆβ x it. 3 Step 2: Compute the time averages of these residuals, û i = T T t= ûit, and regress û i on z i with an intercept to obtain ˆγ F EF, namely ] ˆγ F EF = z i z z i z z i z û i û, 4 and ˆα F EF = û ˆγ F EF z, 5 where û = ûi. The use of the FE residuals, û it, for consistent estimation of γ is not new and has been used in the literature extensively starting with the pioneering contribution of Hausman and Taylor 98. The 5

6 FEVD procedure proposed by Plumper and Troeger 2007 also makes use of the FE residuals. see Section 3.4. The main contribution of this paper lies in development of the asymptotic distribution of ˆγ F EF and its IV version, ˆγ F EF IV introduced in Section 4 under fairly general conditions on the error processes ε it, and η i, and alternative assumptions concerning the correlation of z i and η i + ε i. We also derive conditions under which the covariance matrix of ˆγ F EF and ˆγ F EF IV can be consistently estimated. 3.2 Asymptotic Properties of the FEF Estimator of γ We examine the asymptotic properties of the FEF estimator of γ, ˆγ F EF, defined by 4, under the following assumptions: Assumption P: Eε it x is = 0, for all i, t and s, and E ε 4 it < K <, for all i and t. Assumption P2: E ε it ε js X = 0, for all i j, and all t and s, where X = x it ; i =, 2,..., ; t =, 2,..., T. Assumption P3: The errors, ε it, are heteroskedastic and temporally dependent, namely E ε it ε is X = γ i t, s, for all t and s, where 0 < γ i t, t = σ 2 i, and γ it, s < K, for all i, t and s. Assumption P4 : The regressors, x it, have either bounded supports, namely x it < K <, or satisfy the moment conditions E x it x 4 < K <, and E x i x 4 < K <, for all i and t, where x i = T T t= x it, and x = x i. Assumption P5: The k k matrices Q p,t and Q F E,T defined by Q p,t = T Q F E,T = T t= t= x it x x it x, 6 x it x i x it x i, 7 converge in probability to Q p,t and Q F E,T for a fixed T and as tends to infinity, λ min Q F E,T > /K and λ min Q p,t > /K, for all and T where K is a finite, non-zero constant. Assumption P6: The m m matrix, Q zz,, and the m k matrix Q z x, defined by Q zz, = z i z z i z, 8 Q z x, = z i z x i x, 9 converge in probability to the non-stochastic limits Q zz and Q z x, and λ min Q zz, > /K, for all > m. 6

7 Assumption P7: The time-invariant regressors, z i, are independently distributed of v j = η j + ε j, for all i and j, and η i and ε i are independently distributed. Also, z i either have bounded support, namely z i < K, or satisfy the moment conditions E z i z 4 < K, for all i. Remark ote that since x it x i = x it x x i x x it x + x i x, then any order moment conditions on x i x and x i x imply the same order moment conditions on x it x i. The boundedness of x it x and x i x are also suffi cient for the boundedness of x it x i. Remark 2 Assumptions P5 and P6 ensure that there exists a finite 0 such that for all > 0, Q zz, and Q F E,T are positive definite and converge in probability to the fixed matrices Q zz and Q F E, respectively. But using the results in lemma A. in the Appendix, one can then relax the conditions λ min Q zz, > /K and λ min Q F E,T > /K by requiring λ min Q zz > 2/K and λ min Q F E > 2/K. Under our assumptions the latter conditions ensure that the former conditions hold with probability approaching one. Remark 3 Although, our focus is on fixed T and large panels, we shall also discuss conditions under which our analysis will be valid when both T and are large. To derive the asymptotic distribution of ˆγ F EF, we first note that the FE estimator of β is given by T ] T ˆβ = x it x i x it x i x it x i y it ȳ i. 0 t= t= Under the above assumptions, ˆβ is unbiased and consistent for any fixed T and as, and where V F E,T = T t= V ar ˆβ X = T Q F E,T V F E,T Q F E,T, σ 2 i x it x i x it x i + T t s γ i t, s x it x i x is x i. 2 In the standard case where ε it IID0, σ 2, we obtain the more familiar expression V ar ˆβ X = T σ 2 Q F E,T. Also, for a fixed T and as, we have the following limiting distribution ˆβ β d 0,T Ωˆβ, 3 where Ωˆβ = Q F E,T V F E,T Q F E,T, 4 7

8 and Q F E,T is defined in Assumption P5, and V F E,T = p lim V F E,T. Consider now the FEF estimator of γ defined by 4 and note that ˆβ û i û = η i η + ε i ε + z i z γ x i x β. Using this result in 4 we now have noting that z i z ε + η = 0 where Q zz, is defined by 8 and ˆγ F EF γ = Q zz, z i z ζ i ], 5 ζ i = η i + ε i x i x ˆβ β. 6 Let Z = z,..., z, X = x it ; i =, 2,..., ; t =, 2,..., T, and η = η, η 2,..., η, and note that ] E z i z ζ i Z, X, η ] = z i z η i z i z x i x E ˆβ X β = z i z η i. Also under Assumption P7 we have E z i z η i ] = 0 for all i, and using 5 it follows that E ˆγ F EF = γ, which establishes that ˆγ F EF is an unbiased estimator of γ. Consider now the consistency and the asymptotic distribution of ˆγ F EF. To this end we first note that z i z ζ i = z i z η i + ε i ] z i z x i x ˆβ β. Also under Assumptions P6 and P7, z i z x i x p Q z x, /2 z i z η i + ε i d 0, ω 2 it Q zz, 7 where ω 2 it = σ 2 η + σ2 i T + T 2 γ i s, t 8 s t 8

9 with γ i s, t = E ε is ε it, and z i z ζ i = O p ˆβ β + O p /2. ow using 5, and since Q zz, p Q zz, which is a non-singular matrix, then we also have Therefore, in view of 9 we obtain ˆγ F EF γ = O p /2 + O p ˆβ β. 9 ˆγ F EF γ = O p /2, which establishes that ˆγ F EF, is a consistent estimator of γ. To derive the asymptotic distribution of ˆγ F EF, we first note that z i z ζ i = z i z η i + ˆβ ] z i z ε i x i x β, 20 and consider the limiting distribution of the two terms of 20 and their covariance. We first note that the second term of the above can be written as ˆβ ε i x i x β = T = T ε it T t= ε it t= x i x Q F E,T T w ij,t ε jt, t= j= t= j= x jt x j ε jt where 3 w ij,t = x i x Q F E,T x jt x j. 2 Hence, = = ˆβ ] z i z ε i x i x β z i z T ξ i,, ε it t= w ij,t ε jt j= 3 ote that w ij,t w ji,t. 9

10 where and Using these results in 20 now yield ξ i, = T d z,it ε it, 22 t= d z,it = z i z z i z ζ i = z j z w ji,t. 23 j= z i z η i + ξ i,. 24 However, = Cov i,j z i z η i, E z i z η i ξ j, ] = i,j ξ i, { ]} E E η i z i z ξ j, Z, X, η = 0, and it is suffi cient to derive the asymptotic distributions of the two terms in 24, separately. this end we note that under Assumptions P6 and P7, and using standard central limit theorems it follows that z i z η i d 0, σ 2 ηq zz. 25 Consider now the second term in 24 and note that under Assumption P-P3 and P7, w ji,t and z i are distributed independently of ε is, for all i, j, t, and s, and hence conditional on Z and X, ξ i, have zero means, and are cross sectionally independently distributed noting that by Assumption P2, ε it are assumed to be cross-sectionally independent. But since the terms, ξ i,, in 24 vary with it suffi ces to show that the following Liapunov condition, see Davidson 994, p. 373 is satisfied. lim E /2 ξ 2+δ i, = 0 for some δ > The validity of this condition is established under Assumptions P-P7 in Section A. of the Appendix. Hence ξ i, = T T t= d z,it ε it d 0, Ω ξ where Ω ξ = lim Ω ξ,, and since ε it are assumed to be cross-sectionally independent Ω ξ, = T 2 T t,s= 0 To 27 d z,it d z,ise ε it ε is. 28

11 The above results are summarized in the following theorem: Theorem Consider the FEF estimator ˆγ F EF of γ in the panel data model defined by 4, and suppose that Assumptions P-P7 hold. Then ˆγ F EF is an unbiased and a consistent estimator of γ, and ˆγF EF γ d 0, ΩˆγF EF, 29 where ΩˆγF EF = Q zz σ 2 η Q zz + Ω ξ Q zz. 30 Q zz is defined in Assumption P6, Ω ξ = lim Ω ξ,, with Ω ξ, is defined by 28, and σ 2 η is the variance of the fixed effects defined by Consistent estimation of V ar ˆγ F EF In order to estimate ΩˆγF EF, it is helpful to begin with the following proposition regarding Ω ξ, defined by 28, which enters the expression for ΩˆγF EF. Proposition Let V zz, = ω 2 it z i z z i z, 3 where ω 2 it = σ2 η + κ 2 it, κ 2 it = σ2 i T + T 2 γ i s, t, 32 γ i t, s = E ε it ε is, and σ 2 i = γ i t, t. Then σ 2 ηq zz + Ω ξ,, with Ω ξ, defined by 28, can be written as s t σ 2 ηq zz + Ω ξ, = V zz, + Q z x, V ar ˆβ Q z x, ξ, + ξ,, 33 where Q z x, is defined in 9, and ξ, = Q z x, Q F E,T T 2 T t,s= γ i t, s x it x i z i z. 34 Proof. A proof is provided in Section A.2 in the Appendix. Proposition 2 Under Assumptions P-P7, and if T 2 t,s= γ i t, s x it x i z i z = o p, 35

12 then the variance of the FEF estimator 4, can be consistently estimated for a fixed T and as, by ˆΩˆγF EF = V ar ˆγ F EF = Q zz, ˆV zz, + Q z x, V ] arˆβ Q z x, Q zz,, 36 where Q zz, and Q z x, are defined by 8 and 9, respectively, V arˆβ = x i x i x i e i e ix i x i x i, 37 where x i = x i x i, x i2 x i,..., x it x i denotes the demeaned vector of x it, and the t-th element of e i is given by e it = y it ȳ i x it x i ˆβ, and ˆV zz, = ˆςi ˆς 2 zi zz i z, 38 where ˆς i ˆς = ȳ i ȳ x i x ˆβ zi z ˆγ F EF. 39 Proof. A proof is provided in Section A.3 of the Appendix. Condition 35 is not as restrictive as it may appear at first, and holds under a number of still fairly general assumptions regarding the error processes, ε it. To see this, first not that T 2 t,s= γ i t, s x it x i z i z = T X i M T Γ i τ T z i z ], T where X i = x i,..., x it, Γ i = γ i t, s, and τ T is a T vector of ones. Also, T Γ i τ T = γ i, γ i2,..., γ it, where γ it = T T s= γ it, s. Then condition 35 is met exactly if γ it = c i for all t. Since in such a case T Γ i τ T = c i τ T, and T X i M T Γ i τ T z i z = c i X i M T τ T z i z = 0. Condition γ it = c i is clearly met if γ i t, s = 0 for all t s, and γ i t, t = Eε 2 it = σ2 i. After extensive simulations including cases where there are significant variations over time in γ it, we find that the effect of ξ, is negligible and the use of 36 for inference seems to be justified more generally. 4 Furthermore, the quality of approximating the variance of ˆγ F EF by 36 tends to improve with T so long as T T t,s= γ it, s < K. 3.4 Comparison of FEF and FEVD estimators In this section, we will compare the FEF estimator with the FEVD proposed by Plumper and Troeger The FEVD procedure is based on the following three steps: by 3. Step : The fixed effects approach is applied to, to compute the FE residuals, û it, defined 4 The results of these simulations are available upon request. 2

13 Step 2: In the second step, PT regress û i on z i where û i = T T t= ûit = ȳ i x iˆβ, ȳ i = T T t= y it and x i = T T t= x it. To obtain equivalence between the FEVD and FEF estimators of γ, we modify this regression by also including an intercept in the regression and hence define the residuals from the second stage by where â = û z ˆγ, û = ûi, and ĥ i = û i â z iˆγ, 40 ] ˆγ = z i z z i z z i z û i û. 4 which is exactly the same as our FEF estimator. Using the above results we now have ĥ i ĥ = û i û z i z ˆγ = ȳ i ȳ x i x ˆβ z i z ˆγ, 42 where ĥ = ĥi. Also, from the normal equations of this step, note that ĥ = 0, ĥi ĥ z i z = Step 3: The third step uses ĥi computed in the earlier stage, as defined by 40, and estimates the following panel regression by pooled OLS y it = a + x itβ + z iγ + δĥi + ε it. 44 These estimators are the modified FEVD estimators which we shall denote by γ, δ and β, and as before we denote the FE estimator of β by ˆβ, and the estimator of γ obtained in the second step of FEVD approach by ˆγ which is identical to the FEF estimator if an intercept is included in the second step. The original FEVD estimators proposed by PT are based on the same pooled OLS regression, but do not include an intercept in the second stage regression that computes the ĥi. 5 As we shall see this makes a great deal of difference to the resultant estimators. To investigate the relationship between FEF and FEVD estimators we first introduce the fol- 5 For example, equations 4 and 5 on p28 of Plumper and Troeger

14 lowing notations q p,t = T Q h x, = Q hȳ, = t= x it x y it ȳ, Q zȳ, = ĥi ĥ x i x, Q hz, = ĥi ĥ ȳ i ȳ, Q hh, = z i z ȳ i ȳ, 45 ĥi ĥ z i z, 46 2 ĥi ĥ, 47 where ĥ, x, x i, z, ȳ, and ȳ i are defined as before. Using these additional notations, the normal equations of the pooled OLS regressions for the panel data model defined by 44 are given by q p,t = Q p,t β + Q z x, γ + Q h x, δ, Q zȳ, = Q z x, β + Q zz, γ + Q hz, δ, Q hȳ, = Q h x, β + Q hz, γ + Q hh, δ. Also, when an intercept is included in the second step of FEVD we have Q hz, = 0 = Q hz,, see 43 and 46, and the normal equations reduce to q p,t = Q p,t β + Q z x, γ + Q h x, δ, 48 Q zȳ, = Q z x, β + Q zz, γ, 49 Q hȳ, = Q h x, β + Q hh, δ. 50 The FEVD estimator of γ, namely γ, can now be obtained using the above system of the equations. The results are summarized in the following proposition. Proposition 3 Consider the panel data model 44, and suppose that Q p,t and Q zz, are nonsingular, and Q hh, > 0. Let Q T = Q p,t Q z x,q zz, Q z x, Q h x, Q hh, Q h x,, 5 and suppose also that Q T is non-singular. Then the FEVD estimators proposed by PT and FEF estimators proposed in this paper are identical if an intercept is included in the second step regression of the FEVD procedure, namely γ = ˆγ, and β = ˆβ. Furthermore δ, the FEVD estimator of δ in the third step of the FEVD procedure, is identically equal to unity. Proof. See Section A.4 in the Appendix for a proof. Proposition 4 Suppose that the three-step FEVD estimators denoted as before by β, γ and δ are computed without including an intercept in the regression in the second step. In this case we 4

15 continue to have β = ˆβ and δ =, but for γ we obtain γ = γ, where γ is the OLS estimator of the coeffi cient of z i in the OLS regression of û i on z i, without an intercept, and γ is biased and inconsistent unless αe z iz i ] = 0. Proof. See Section A.5 in the Appendix for a proof. z i It is also interesting to compare the covariance of the FEF given by 30, with the one that is obtained when the standard formula for the variance of the pooled OLS estimators is applied to the third step of the FEVD procedure as proposed by PT. Recall that the FEVD estimator of γ coincides with ˆγ if an intercept is included in the second step of the procedure, and pooled OLS applied to the third step will result in a valid inference only if the variance obtained using the FEVD procedure also coincides with ΩˆγF EF. To simplify the comparisons suppose that ε it IID0, σ 2 for all i and t, and note that in this simple case V zz and Ωˆβ given by 3 and 4 reduce to V zz = Ωˆβ = σ2 T σ 2η + σ2 T lim Q zz, T t= ] x it x i x it x i = σ2 T Q p Q x x, and we have 6 ΩˆγF EF = Q zz = σ 2η + σ2 T σ 2η + σ2 Q zz + Q z x ΩˆβQ xz ] Q zz T Q zz + σ2 T Q zz Q z x Q p Q x x Q xz Q zz. Under the same model specifications the covariance of the FEVD estimator also scaled by is given by ΩˆγF EV D = σ 2 ε Q zz + Q zz Q z x Q Q z xq ] zz, where as before Q = Q p Q z xq zz Q z x Q h x Q hh Q h x, σ 2 ε = lim T = lim T T t= T ε 2 it t= ] 2 y it α x it β z i γ δĥi ], and ε it = y it ȳ x it x ˆβ z i z ˆγ ĥi. 6 ote that since in the present case ε its are serially uncorrelated then ξ = 0. 5

16 To derive σ 2 ε note that ĥi = ȳ i ȳ x i x ˆβ zi z ˆγ, and hence ε it = y it ȳ x it x ˆβ z i z ˆγ ĥi = y it ȳ i x it x i ˆβ ˆβ = ε it ε i x it x i β. ˆβ ] Also, in the case where ε it IID0, σ 2, we have Cov ε it ε i, x it x i β = 0, and hence E ε 2 it = σ 2 T + k. A comparison of the expressions derived above for ΩˆγF EF and ΩˆγF EV D clearly shows that they differ irrespective of whether T is fixed or T. Remark 4 Plumper and Troeger 2007 argue that the necessity of third step is to correct standard errors of γ Plumper and Troeger 2007, p29, however, as shown above and in the simulations below, the variance term calculated in the third step of FEVD does not fully correct the bias of the variance estimator in the second step. 7 4 FEF-IV estimation of time-invariant effects The FEF procedure assumes that the time-invariant regressors, z i, are distributed independently of the individual-specific effects η i + ε i. However, it is relatively straight forward to modify the FEF estimator to allow for possibly endogeneity of the time-invariant regressors, if there exists a suffi cient number of valid instruments. In particular, it is possible to derive an IV version of FEF, which we denote by FEF-IV, under the following assumptions: Assumption P8: There exists the s vector of instruments r i for z i, i =, 2,...,, where r i is distributed independently of η j and ε j for all i and j, and s m, and satisfy the moment condition E r i r 4 < K <, if it has unbounded support. Assumption P9: Let Z = z, z 2,..., z, R = r, r 2,..., r. Consider the s m matrix Q rz,, the s k matrix Q rx, = r i r x i x, and the s s matrix Q rr, defined by Q rz, = r i rz i z, Q r x, = r i r x i x, Q rr, = r i rr i r, 52 where r = r i. Q rz, and Q rr, are full rank matrices for all > r, and have finite probability limits as given by Q rz and Q rr, respectively. Matrices Q r x, and Q zz, have finite probability limits given by Q r x and Q zz, respectively, and in cases where x it and z i are 7 PT state "...only the third stage allows obtaining the correct SE s.", P.29. 6

17 stochastic with unbounded supports, then λ min Q rr, > /K, for all, and as, with probability approaching one. Under the above assumptions and maintaining Assumptions P-P6 as before, a consistent twostage estimator of γ can be obtained as follows 8 ˆγ F EF IV = where Q zr, and Q rr, are defined by 52, Q zr, Q rr, Q zr, Q zr, Q rr, Q rû,, 53 Q rû, = r i r û i û, û = ûi, û i = ȳ i x i x ˆβ, and ˆβ is the FE estimator of β from the first stage. It then follows that ˆγF EF IV γ = Q zr, Q rr, Q zr, Q zr, Q rr, r i r ζ i with ζ i = η i + ε i x i ˆβ β, as before see 6. Following a similar line of proof as in the case with exogenous z i, under Assumptions P-P6 and P8-P9, it can be shown that r i r ζ i ζ d 0, σ 2 ηq rr + Ω ψ,, where Ω ψ = lim T 2 T t,s= d r,it d r,ise ε it ε is, 54 where d r,it = r i r j= r j rw ji,t, and w ij,t = x i x Q F E,T x jt x j, as before. Moreover, we note that under Assumption P9 where Q zr, Q rr, Q zr, p Q zr Q rr Q zr, and Q zr, Q rr, p Q zr Q rr. Using the above results and Slutsky s theorem now yields ˆγF EF IV γ d 0, ΩˆγF EF IV, ΩˆγF EF IV = Q zr Q rr Q zr Qzr Q rr σ 2 η Q rr + Ω ψ Q rr Q zr Qzr Q rr Q zr A derivation is available upon request. 7

18 For convenience, the above results are summarized in the following theorem. Theorem 2 Suppose that Assumptions P-P6, P8 and P9 hold, and let the FEF-IV estimator be defined as in 53. Then we have where ΩˆγF EF IV is given by 55. ˆγF EF IV γ d 0, ΩˆγF EF IV, The variance of ˆγ F EF IV can now be estimated along similar lines as in Section 3.3. We have V ar ˆγ F EF IV = H zr, ˆV rr, + Q r x, V ] arˆβ Q r x, H zr,, 56 where H zr, = Q zr, Q rr, Q zr, Qzr, Q rr,, Q r x, = r i r x i x, ˆV rr, = r i rr i r ˆυ i ˆυ 2, ˆυ i ˆυ = ȳ i ȳ x i x ˆβ zi z ˆγ F EF IV, and as before V arˆβ is given by 37. It can be shown that V ar ˆγF EF IV is a consistent estimator of ΩˆγF EF IV defined by 55 if condition 35 is met. Our simulation results suggest that the above variance estimator performs well even if condition 35 is not satisfied. 5 Hausman and Taylor 98 estimation procedure Hausman and Taylor 98 approach the problem of estimation of the time-invariant effects in the panel data model,, by assuming that x it and z i can be partitioned into two parts as x,it, x 2,it and z,i, z 2,i, respectively, such that E x,itη i = 0, E z,i η i = 0, E x 2,itη i 0, E z 2,i η i 0. To compute the HT estimator the panel data model is first written as y i = X i β + z iγ + α + η i τ T + ε i, for i =, 2,...,, 57 8

19 where X i = x i,..., x it, y i = y i, y i2,..., y it, and ε i = ε i, ε i2,..., ε it. Then the following two-step procedure is used 9 : Step of HT: As in our approach β is estimated by ˆβ, the FE estimator, the deviations ˆd i = ȳ i x iˆβ, i =, 2,...,, are used to compute the 2SLS or IV estimator ˆγ IV = Z P A Z Z P Aˆd, 58 where ˆd = ˆd, ˆd 2,..., ˆd, Z = z, z 2,..., z = Z, Z 2, and P A = A A A A is the orthogonal projection matrix of A = τ, X, Z, where X = X, X 2, and X = x, x 2,..., x, x i = x i,, x i,2. Using these initial estimators of β and γ, the error variances σ 2 ε and σ 2 η are estimated as ˆσ 2 η = s 2 ˆσ 2 ε, ˆσ 2 ε = s 2 = T T t= y i X iˆβ F E MT y i X iˆβ F E, y it ˆµ x itˆβ F E z iˆγ IV 2, Step 2 of HT : In the second step the equations in 57 are stacked to obtain y = Wθ + η τ T + ε, where W = τ τ T, X, Z τ T ], θ = α, β, γ, y = y, y 2,..., y, η = η, η 2,..., η, and ε = ε, ε 2,..., ε. Under the assumptions that the errors are cross-sectionally independent, serially uncorrelated and homoskedastic we have Ω = V ar η τ T + ε] = σ 2 η I τ T τ T + σ 2 ε I I T, which can be written as Ω = σ 2 ε + T σ 2 η PV + σ 2 εq V, where P V = I I T M T and Q V = I M T. It is now easily verified that Ω /2 = σ ε ϕp V + Q V, where ϕ = σ ε / σ 2 ε + T σ 2 η. Then the transformed model can be written as Ω /2 y = Ω /2 Wθ + Ω /2 η τ T + ε]. 59 To simplify the notations we assume that the first column of Z is τ, and then write the infeasible HT estimator as, ˆθ HT = W Ω /2 P A Ω /2 W W Ω /2 P A Ω /2 y, 60 9 As noted in the Introduction, HT procedure is further developed and extended in the papers by Amemiya and MaCurdy 986, Breusch et al. 989, Im et al. 999 and Baltagi and Bresson

20 where P A = A A A A, A = τ τ T, Q V X, X, Z τ T, X = X,, X,2,..., X,, with X,i = x,i,..., x,it, and x,it contains the regressors that are uncorrelated with η i. 0 The variance covariance matrix of ˆθ HT in the general case where the fixed effects, η i, are heteroskedastic and possibly cross-sectionally correlated is given by ] V ar ˆθ HT = Q T Vη + σ 2 ε + T σ 2 Q W Ω /2 P A σ 2 ηi η T τ T τ T P A Ω /2 W Q, 6 where Q = W Ω /2 P A Ω /2 W, and V η represents the covariance matrix of η. V ar ˆθ HT reduces to Q in the standard case where η i s are assumed to be homoskedastic and crosssectionally independent, namely when V η = σ 2 ηi. To our knowledge the above general expression for V ar ˆθ HT is new. Remark 5 The HT approach assumes that the errors η i and ε it are both homoskedastic, serially uncorrelated and cross-sectionally independent. However, simulations to be reported below suggest that the HT estimator works well even in cases of heteroskedasticity and serially correlated errors, if the orthogonality conditions of HT estimator are satisfied. But it is important to bear in mind that if the orthogonality conditions are not met the HT approach breaks down and must be modified. Remark 6 In the case where the eff ects of the time-invariant regressors are exactly identified, then the HT estimator of γ, ˆγ HT, is identical to the first stage estimator of γ, given by 58. See Baltagi and Bresson 202. It is also easily seen that in such a case, ˆγ HT is also identical to the FEF-IV estimator. 6 Monte Carlo Simulation In order to evaluate the performance of the FEF and the FEF-IV estimators proposed in this paper, we conducted two sets of simulations. One set with exogenous time-invariant regressors and a second set where one of the time-invariant regressors is correlated with the fixed effects. In both sets of experiments the data generating process DGP include two time-varying and two time-invariant regressors, and allow for error heteroskedasticity and residual serial correlation. DGP: Initially, we consider the following data generating process y it = + α i + x,it β + x 2,it β 2 + z i γ + z 2i γ 2 + ε it, i =, 2,..., ; t =, 2,..., T, 0 See Amemiya and MaCurdy 986 and Breusch et al. 989 for discussion on the choice of instruments for HT estimation. See Section A.6 of the Appendix for a derivation. 20

21 with β = β 2 = and γ = γ 2 =. We generate the fixed effects as α i 0.5 χ 2 2 2, for i =, 2,...,. For the time varying regressors we consider the following relatively general specifications x,it = + α i g t + ω it,, x 2,it = + α i g 2t + ω it,2, where the time effects g t and g 2t for t =, 2,..., T, are generated as U 0, 2 and are then kept fixed across the replications. The stochastic components of the time varying regressors ω it, and ω it,2 are generated as heterogenous AR processes ω it,j = µ ij ρ ω,ij + ρ ω,ij ω it,j + ρ 2 ω,ij ɛ ωit,j for j =, 2, where ɛ ωit,j IID0, σ 2 ɛi, for all i, j and t, σ 2 ɛi IIDχ 2 2, ω i0,j IIDµ i, σ 2 ɛi, for all i, j, ρ ω,ij IIDU0, 0.98], µ ij IID0, σ 2 µ, σ 2 µ = 2, for all i, j. The time-invariant regressors are generated as z i + 0,, for i =, 2,...,, z 2i IU 7, 2], for i =, 2,...,, where IU 7, 2 denotes integers uniformly drawn within the range 7, 2]. DGP2: In this case the DGP is the same as DGP, except that the second time-invariant regressor, z 2i, is generated to depend on the fixed effects, α i, namely z 2i = r i + α i, for i =, 2,...,, r i IU 7, 2] for i =, 2,...,. 62 We use r i as the instrument for z 2i in the FEF-IV estimation procedure. For each of the above two baseline DGPs, we generate ε it according to Case : Homoskedastic errors: ε it IID0,, for i =, 2,..., ; t =, 2,..., T. Case 2: Heteroscedastic errors: ε it IID0, σ 2 i, i =, 2,..., ; t =, 2,..., T, 2

22 where σ 2 i IIDχ2 2 for all i. Case 3: Serially correlated and heteroscedastic errors: ε it = ρ εi ε i,t + ρ 2 εi v it, where ε i0 = 0 for all i, v it IID0, σ 2 vi, for all i and t, σ 2 vi IIDχ 2 2, ρ εi IIDU0, 0.98], for all i, for t = 49, 48,..., 0,, 2,..., T, with u i, 49 = 0, for all i. The first 50 observations are discarded, and the remaining T observations are used in the experiments. Remark 7 The two DGPs are intended to capture the different features of the estimators proposed in this paper and in the literature. DGP allows for an arbitrary degree of dependence between the time-varying regressors and the fixed eff ects, but assumes that the time-invariant regressors are exogenous. This DGP is designed to be applicable to the FEVD and FEF estimators. DGP2 allows one of the time-invariant regressors to be correlated with the fixed effects, and is used to evaluate the small sample properties of FEF-IV in the presence of heteroscedastic and serially correlated errors. We use, 000 replications for each experiment, and report bias, root mean squared error RMSE, size and power for different estimators of γ, namely FEVD with and without intercepts in the second step, and the FEF estimator proposed in this paper for DGP. We also consider HT and FEF-IV estimators in the case of DGP2 where all the time varying regressors are correlated with the errors. The results of FEF for DGP are summarized in Tables -6, and clearly show that the FEF estimator performs well in all experiments, even when the errors are serially correlated and/or heteroscedastic. It has much lower bias and RMSE as compared to the FEVD estimator proposed by PT. However, in accordance with our theoretical findings, the FEF and FEVD estimators become identical when an intercept is included in the second stage of the PT estimation procedure. However, even after this correction the FEVD approach continues to exhibit substantial size distortions due to the use of incorrect standard errors in the third stage of the procedure see Section 3.4. The results for DGP2 are summarized in Tables 7-2. The FEF-IV estimator is computed using r i defined by 62 as an instrument for the endogenous time-invariant regressor, z 2i. The HT estimator uses time averages of the time-varying regressors, x i and x 2i, as well as z i, as instruments. The FEF-IV procedure performs well in all cases, irrespective of whether the errors are heteroskedastic and/or serially correlated. In particular the size of the FEF-IV estimator is very 22

23 close to the 5% nominal value, with the power rising steadily in. This suggests that the variance estimator for the FEF-IV, 56, is valid in the case of error heteroskedasticity and performs well even the errors are serially correlated for example, see Tables -2. In contrast, the application of the standard HT procedure yields biased estimates and significant size distortions, particularly in the case of γ 2, the coeffi cient of the endogenous time-invariant regressor, z 2i. Perhaps this is not surprising, considering that in these experiments both of the included time-varying regressors are correlated with the fixed effects, and neither cannot be used as valid instruments. 2 It is possible to modify the HT procedure by including r i as an additional instrument, in which case the HT estimator will become identical to the FEF-IV estimator considering that under DGP2 the parameters of the panel data model are exactly identified. 7 Conclusion In this paper, we propose the FEF and FEF-IV estimators for panel models with time-invariant regressors. The FEF estimator is computed using a two-step procedure, where in the first step the fixed effects estimators are used to filter the effects of time-varying regressors. In the second step, time averages of the residuals are used in cross-section regressions to estimate the coeffi cients of time-invariant regressors. We also develop the asymptotic distribution for the FEF, and show that it s unbiased, consistent and asymptotically normally distributed. The FEF estimator is suffi - ciently robust and allows for cross-sectional heteroskedasticity and serial correlation. An alternative variance estimators of the FEF estimator is also proposed in this paper. Moreover, when there is correlation between the time-invariant variables and individual effects, we propose the FEF-IV estimator, which can also be calculated by a two step procedure. The first step of FEF-IV is similar to FEF, but in the second step, we use the instrument variable estimation for the time-invariant regressors. We also show that this FEF-IV estimator is consistent and asymptotically normally distributed. Similar to the FEF estimator, the FEF-IV estimator is suffi ciently robust and allows for cross-sectional heteroskedasticity and serial correlation. An alternative variance estimator of the FEF-IV estimator is also proposed in this paper. By simulations, we find both the FEF and FEF-IV have better small sample performance in terms of bias and RMSE, and most importantly has the correct size in the presence of correlation of arbitrary degree between the time-varying regressors and the individual effects. Furthermore, we also contribute to the debate on the FEVD estimator proposed by Plumper and Troeger We show that the FEVD estimator is exactly the same as our FEF estimator if an intercept is included in the second step of PT s procedure, but the FEVD estimator is inconsistent in general if no intercept is included in the second stage see equation 5 in PT. Furthermore, even if the FEVD estimator is computed using an intercept in the second stage, it will still lead to misleading inference since contrary to what is claimed by PT, the standard errors computed in the 2 In implemetation of the HT procedure we use Q V X the demeaned time varying regressors, see 60 as instruments. 23

24 third stage of the PT procedure are not valid. Overall, our Monte Carlo simulations suggest that FEF and FEF-IV estimators proposed in this paper perform well in terms of bias, RMSE, size and power. The simulation results also confirm our theoretical derivations showing that in general the FEVD estimator suffers from size distortions. Finally, in cases where none of the time-varying regressors is uncorrelated with the fixed effects, the use of standard HT procedure can lead to bias and significant size distortions. In such cases a modified version of the HT procedure can be considered if there exists a suffi cient number of instruments for the endogenous time-invariant regressors. But it is not clear what advantages there might be in following such a modified HT procedure as compared to the FEF-IV estimator proposed in this paper. Appendix: Mathematical Derivations Lemma A. Suppose that A is a p p symmetric where p is fixed, λ min A 2/K, and λ max A K/2, with K being a fixed, non-zero positive constant. Consider now the stochastic matrix Â, viewed as an estimator of A, such that  A p 0. Then with probability approaching one λ min  /K and λ max  K. Source: Lemma A0 in the mathematical supplement to ewey and Windmeijer Lemma A.2 Given the cross-product sample moments defined by 45, 46 and 47, and in view of 42 and 43, we have Q hȳ, = Q h x, ˆβ + Qhh,, A. q p,t = Q p,t ˆβ + Q z x, ˆγ + Q h x,, A.2 Q zȳ, = Q z x, ˆβ + Qzz, ˆγ, A.3 Q zz, ˆγ γ + Q z x, ˆβ β = 0, A.4 δ Q hh, = Q h x, ˆβ β. A.5 Proof. Using 42 and 43 we first note that Q hȳ, = = ĥi ĥ ȳ i ȳ ĥi ĥ x i x ˆβ + = Q h x, ˆβ + Q hh,. ĥi ĥ z i z ˆγ + ĥi ĥ 2 Similarly q p,t = = T T t= t= x it x y it ȳ x it x x it x ˆβ + T = Q p,t ˆβ + Q z x, ˆγ + Q h x,, t= x it x û it û 24

25 where T t= x it x û it û = T x it x i û it û i + t= = x i x û i û = x i x z i z ˆγ + = Q z x, ˆγ + Q h x,, x i x û i û x i x ĥi ĥ and Q zȳ, = z i z ȳ i ȳ = z i z x i x ˆβ + z i z û i û = Q z x, ˆβ + Q zz, ˆγ. Using this result together with 49 now yields Q zz, ˆγ γ + Q z x, ˆβ β = 0. Finally, A.5 follows immediately using A. and 50. A. Proof of Liapunov condition 26 The Liapunov condition 26, for δ = 2 can be written as lim 2 E ξi, 4 = 0. A.6 From 22 we first note that and by Holder inequality ξi, T d z,it ε it, t= T /4 T 3/4 d z,it ε it d z,it 4 ε it 4/3, t= t= t= and hence T ξi, 4 T 3 T 4 d z,it 4 ε it 4/3. t= t= But under Assumptions P3 and P7, x it and z i are distributed independently of ε it, and it follows that E ξi, 4 T T 3 T E d z,it 4 E T ε it 4/3. t= t= 25

26 But under Assumption P, E ε it 4 < K, and since T is finite and for each i, ε it are serially independent, then for some positive finite constant K we have E T T t= 3 ε it 4/3 K T T E ε it 4 < K K. t= Hence, ow recall that 2 ξi, 4 T ] < K KT 2 E d z,it 4 t= d z,it = z i z z j z w ji,t, j= A.7 where w ji,t = x j x Q F E,T x it x i. Then d z,it = z i z z j z x j x Q F E,T x it x i j= = z i z A z x, x it x i, A.8 where A z x, = Q z x, Q F E,T. But3 A z x, 2 = tr Q F E,T Q z x, Q z x, Q F E,T λ max Q 2 F E,T tr Q z x, Q z x, = λ 2 max Q F E,T Q z x, 2 = λ 2 min QF E,T Qz x, 2 Q z x, 2 = λ 2, min QF E,T = tr Q 2 F E,T Q z x, Q z x, and noting that under Assumption P5, λ min QF E,T > /K, then Az x, K Qz x,. A.9 Also, it is easily seen that d z,it 2 = d z,it d z,it = z i z 2 2 z i z A z x, x it x i + x it x i A z x, A z x, x it x i, and d z,it 4 = z i z z i z A z x, x it x i x it x i A z x, z i z + x it x i A z x, A z x, x it x i x it x i A z x, A z x, x it x i 4 z i z 2 z i z A z x, x it x i + 2 z i z 2 x it x i A z x, A z x, x it x i 4 z i z A z x, x it x i x it x i A z x, A z x, x it x i. 3 ote that for any p p matrices A and B such that A is symmetric and B positive semi-definite, then tr AB λ max A tr B. 26

27 Hence d z,it 4 z i z z i z 2 x it x i 2 A z x, 2 + x it x i 4 A z x, zi z 3 x it x i A z x, +2 z i z 2 x it x i 2 Az x, zi z x it x i 3 Az x, 3, and using A.9 we have d z,it 4 z i z 4 + 4K 2 Q z x, 2 ] z i z 2 x it x i 2 +K 4 Q z x, 4 ] x it x i 4 ] + 4K Qz x, z i z 3 x it x i +4K 3 Qz x, 3 ] z i z x it x i 3. Using this result in A.7 we now obtain ξi, 4 K K z i z 4 + W + W 2 + W 3 + W 4, where W = 4KK 3 Q z x, 2 T W 2 = KK 5 Qz x, 4 T ] z i z 2 x it x i 2, t= ] x it x i 4, t= W 3 = 4KK 2 ] Qz x, T z i z 3 x it x i, t= W 4 = 4KK 4 Q z x, 3 ] z i z x it x i 3. To investigate the limiting property of ξi, 4, we first note that by Assumption P6 Qz x, p c, as, where c is a finite constant, and by Slutsky s theorem as, for a fixed T we have W p4kk 3 c2 W 2 pkk 5 c4 W 3 p4kk 2 c W 4 p4kk 4 c 3 lim T lim T lim T lim E z i z 2 x it x i 2]], t= ] E x it x i 4, t= E z i z 3 x it x i ] ], t= ] z i z x it x i 3. 27

28 Hence lim E ξi, 4 K lim +4KK 3 c2 +KK 5 c4 +4KK 2 c +4KK 4 c 3 E z i z 4 lim T lim T lim T lim E z i z 2 x it x i 2]] t= ] E x it x i 4 t= ] ] E z i z 3 x it x i t= ] z i z x it x i 3. Therefore, E ξi, 4 is bounded and converges to a finite limit as irrespective of whether T is fixed or tends to infinity if the following conditions hold lim E z i z 4 < K, lim lim T and T E x it x i 4 < K, when T is fixed E x it x i 4 < K t= lim lim T and T t= E z i z 2 x it x i 2] < K, when T is fixed, A.0 E z i z 2 x it x i 2] < K, lim lim T and T t= ] E z i z 3 x it x i ] E z i z 3 x it x i < K, when T is fixed, A. < K, lim lim T and T t= E z i z x it x i 3] < K, when T is fixed, A.2 E z i z x it x i 3] < K, The above conditions are clearly satisfied if x it and z i have bounded supports. In the case where x it and z i do not have bounded supports the following moment conditions are suffi cient to ensure that lim 2 E ξi, 4 = 0, as required 28