A DIRECT TEST FOR CROSS-SECTIONAL CORRELATION IN PANEL DATA MODELS

Transcription

1 A DIREC ES FOR CROSS-SECIOAL CORRELAIO I PAEL DAA MODELS ED JUHL Abstract. his paper proposes a test for cross-sectional correlation for use in panel data models. I propose a test that uses squares of the estimates of covariances between cross-sectional units. he test requires both the cross section dimension and but the asymptotic theory is not sequential and there are no restrictions on the diagonal asymptotic path. Moreover, the test does not require the assumption of normally distributed error terms. I show that the test is valid for dynamic panel data models and many models that can be estimated via instrumental variables methods. I provide two additional versions of the test. One version can be used to test the null hypothesis of no cross-sectional correlation in a two-way model employing both individual and time effects. Finally, a version of the test can be used in a two-way fixed effects spatial model to test for the null of no remaining cross-sectional correlation after accounting for potential spatial matrices. I provide a Monte Carlo experiment that explores the size and power of the new tests in several cases for a variety of sample sizes. 1. Introduction Panel data models are often employed as a device to combine data from many cross sectional units in order to improve the efficiency of estimates of slope parameters in regression models. One of the assumptions typically used when conducting inference is the absence of correlation in the cross-sectional error terms. Early exceptions are models of spatial correlation that make use of some known structure of the possible cross sectional correlation. Recently, Baltagi, Song, and Koh 2003) developed a test for spatial correlation in a random components model. heir test is based on comparison of a model with no cross-sectional correlation versus a model which employs a known spatial matrix. As an alternative to assuming cross-sectional independence, factor models have gained recent attention as a way to model cross sectional correlation in panel data models. Procedures for inference about the regression parameters have been developed in Pesaran 2006) for panel models with heterogeneous slopes, and Bai 2009) for fixed effects type panel data Version: ovember 11, I thank seminar participants at Indiana University, participants at ew York Camp Econometrics V, Badi Baltagi, and Zhijie Xiao for comments on an earlier version of this paper. ed Juhl, Department of Economics, University of Kansas, 415 Snow, Lawrence, KS 66045; el 785) ; juhl@ku.edu). 1

2 2 models. For example, Bai 2009) lists several possible covariance matrix estimators for potential cross-sectional correlation, heteroskedasticity, and serial correlation combinations for models estimated with interactive fixed effects. A popular technique in time series models is to estimate static models and allow for unknown forms of serial correlation and heteroskedasticity in the error terms. he covariance matrix is estimated using a heteroskedasticity and autocorrelation consistent HAC) estimator such as Andrews 1991). Kiefer, Vogelsang, and Bunzel 2000) KVB) explore an alternative to the standard t-statistic that is pivotal. For panel data models that have an unknown form of cross-sectional dependence, Vogelsang 2008) developed an analogue of the KVB type statistic that is valid if one assumes exogenous regressors. Using Vogelsang s 2008) statistic does not require researchers to take a stand on the form of cross-sectional dependence, yet inference on regression parameters remains valid. Given the possibility of modeling cross-sectional correlation directly or using inference procedures that allow for its existence, it is natural to use some diagnostic to test for crosssectional correlation in panel data models. In particular, it is useful to have tests that make no assumptions about the potential form of cross sectional dependence. Pesaran, Ullah, and Yamgagata 2008) develop such a test that is valid for heterogeneous static panels with normally distributed errors. heir approach uses results concerning moments of functions of normally distributed variables recently developed by Ullah 2004) to adjust for biases in estimated correlations. Hsiao, Pesaran, and Pick 2009) applied the CD test of Pesaran 2004) to nonlinear panel data models with a specific example using panel probit models. Baltagi, Feng, and Kao 2010b) modify the John 1972) test to make it applicable to fixed effects panel data models. heir test is derived under the assumption of normally distributed errors with c, where c is some constant. In addition, Baltagi, Feng, and Kao 2010a) develop a variant of the LM test that is modified to subtract the bias when estimating a fixed effects model. In this paper, I propose a test for cross-sectional correlation in panel data models. he test is applicable for panel data models when the regression parameters are estimated at rate. Construction of the test is based on estimating squares of covariances between each cross-sectional pair of residuals. Each estimated squared covariance is adjusted for its bias

3 and the entire sum is standardized. he resulting statistic has a normal distribution under the null hypothesis of no cross-sectional correlation. he limiting distribution is developed using U-statistics, and I derive a new central limit theorem for degenerate U-statistics with heterogeneous data. he limiting theory does not require sequential asymptotics, where is required to go to infinity first and then. Instead, the only requirement is that as. Moreover, there is no link between the rates that and approach infinity. Variants of this test can allow for two-way models and a model of spatial correlation with known spatial matrices. he structure for the remainder of the paper is as follows. he one-way panel model with fixed effects is discussed in Section 2, along with the development of the test and its limiting distribution. Section 3 extends the model to account for two-way models where a time effect is added. he model with spatial dependence is discussed in Section 4. I conduct a Monte Carlo experiment in Section 5 to examine size and power of the test relative to some existing tests. Section 6 concludes. 3 he basic model is given as 2. Model 2.1) y it = µ i + x itθ + v it, for i = 1,..., ; t = 1,..., where i is an index of the cross sectional observations and t represents the time series observation. he hypothesis that we wish to test is H 0 : Ev it v jt ) = γ ij = 0 for all i j. he null hypothesis is true if and only if j 1 γ2 ij = 0. his quantity will be the basis for my test statistic. he Breusch and Pagan 1980) Lagrange multiplier LM) statistic is based on a similar quantity given by j 1 ρ2 ij where ρ ij is the pairwise correlation rather than the covariance that I use. If one allows for group-wise heteroskedasticity, where σ 2 i vary with different cross sectional units, this distinction becomes very important. For example, Pesaran, Ullah, and Yamagata 2008) develop a test based on j=2 j 1 i=1 ρ2 ij by modifying the LM statistic. By assuming normality of the error terms, they are able to calculate the moments of ˆρ 2 ij and subtract the bias. he asymptotic theory needed to justify a normal limiting distribution involves sequential asymptotic theory where first, and then

4 4. he normality is required to be able to find the bias of the ˆρ 2 ij of Ullah 2004). using the results o motivate the proposed statistic, I consider the infeasible estimator for γ ij assuming we can view the 1 vectors v i and v j ) given by γ 2 ij = 1 ) 2 v it v jt t=1 = 1 2 v i v j v i v j Since I do not divide by σ i σ j, the bias term is easily calculated regardless of the distribution of v it. he expectation is given by E γ ij) 2 = 1 [ ] 2 E tracev i vi v j vj ) = 1 2 σ2 i σj 2 tracei ) = 1 σ2 i σ 2 j In light of this bias term, I propose a statistic based on all pair-wise covariances squared which is adjusted for bias: 1 v i v jv i v j v i v i vj v ) j As it stands, the proposed statistic is not operational since we do not observe v it. Let ˆθ be a consistent estimator of θ and define the group demeaned residuals as where ṽ it = y it ȳ i ) x it x i ) ˆθ ȳ i = 1 = v it v i ) + x it x i ) θ ˆθ) y it and x i = 1 t=1 he direct statistic based on model 2.1) is given by D = 1 σ D 1 x it. t=1 [ ṽi ṽjṽ i ṽj 1) ṽ i ṽi 1) ṽ j ṽj 1) ]

5 5 where σ 2 D = 1 2 [ ṽi ṽjṽ i ṽj 1) ṽ i ṽi 1) ṽ j ṽj 1) ] 2 he assumptions needed for the main theorem are discussed below. Assumption 1. v i are independent across i and there is no serial correlation across t. Assumption 2. θ is consistently estimated so that θ ˆθ) = O p 1/2 1/2 ) Assumption 3. he elements of the matrix ) E v i vi v j vj X i, X j are bounded by a function hx i, X j ) such that E hx i, X j ) <. Assumption 1 is no cross-sectional correlation or serial correlation. Assumption 2 is used to put an explicit rate of consistency of ˆθ. Such a rate is possible in dynamic panel models as well. For example, Alvarez and Arellano 2003) illustrate several cases of dynamic models and estimators where such a rate of consistency is obtained. he conditions are purposely left at a high level as there are many possible models, including those estimated via instrumental variables, that are covered by this rate so we avoid listing all possible requirements on estimation methods and assumptions about the regressors. Assumption 3 allows for heteroskedasticity conditional on X i but ensures that there are bounds on conditional second moments of v i. We require an assumption to limit the amount of time series dependence in x it as well as the amount of feedback from v it to future values of x it. o accomplish this, we will use the concept of β mixing with an adjustment that allows the time series dependence to vary over cross-sectional units. o this end, we define Z ijt = x it v it x jt v jt

6 6 Let M t ij,s be the sigma algebra generated by Z ijs,..., Z ijt ) for s t. hen we define the β τ coefficient as β τ = sup i,j [ sup E s sup A M ij,s+τ [ P A M s ij, ) P A) ])] Assumption 4. Suppose that for each i and j that Z ijt is strictly stationary with for δ > 0 and In addition suppose that max We state the main theorem below. [ ] Evit), 8 sup E x is x js 41+δ) < P <. i,j Ev it Z ijs ) = 0 for s t. δ τ=1 β 1+δ τ < heorem 2.1. Suppose that Assumptions 1-4 hold, and that as. hen under the null hypothesis of γ ij = 0 for all i j, D d 0, 1). he theorem applies to fixed effects models, dynamic panel data models, and some models estimated with instrumental variables, so long as θ is estimated at rate. he result does not require normality of v it nor sequential asymptotics. For example, using the LM test of Bruesch and Pagan 1980), one can first let to obtain the intermediate result that each estimated correlation is asymptotically normal and uncorrelated with each other. hen, the statistic is analyzed by letting to obtain normality. he above theorem does not require such sequential asymptotics. Instead, it contains a condition that goes to infinity as does. his condition specifies that we must take a diagonal path to infinity. However, this condition is very general since we do not specify any restriction on the path. his is analagous to requiring = g) for some function g. he condition is analogous to the types of conditions on a bandwidth in nonparametric estimation. hat is, one typically requires that the in the neighborhood of a specified point, the number of observations becomes infinite, and, at the same time, we use a smaller window to calculate the nonparametric estimate of a regression curve or density estimate). he link of to is needed to apply the version of the martingale central limit theorem applied to U-statistics

7 derived in the appendix, which relies on. he central limit theorem for U-statistics is derived in the appendix and follows the conditional arguments in Hall 1984). his variant is well suited for application to the D statistic wo Way Model A two-way model is popular alternative to standard fixed effects model. his model includes a term to represent a common shock that affects all cross-sectional units in the same way. he model is given as 3.1) y it = µ i + x itθ + λ t + v it, for i = 1,..., ; t = 1,.... We can modify the statistic to allow for this alternative model in the following way. In order to remove both the individual effect, µ i as well as the time effect, λ t, the typical transformation of the data employed is given by where y it ȳ i ȳ t + ȳ, ȳ t = 1 ȳ = 1 i=1 y it i=1 t=1 Using this idea, we define a new set of residuals as y it. v it = y it ȳ i ȳ t + ȳ ) x it x i x t + x ) ˆθ = v it v i v t + v ) + x it x i x t + x ) θ ˆθ) he direct statistic based on model 2.1) is given by [ D2 = 1 1 v i v j v i v j σ D2 1) v i v i 1) v j v ] j 1) 1 An alternative central limit theorem for U-statistics with heterogeneous data is given in De Jong 1987). However, the version in the appendix gives conditions on conditional moments of the terms in the sum as opposed to the unconditional moments in De Jong 1987).

8 8 where σ 2 D2 = 1 2 [ v i v j v i v j 1) v i v i 1) v j v j 1) ] 2 heorem 3.1. Suppose that Assumptions 1-4 hold, model 3.1) holds, and that as, and / 0. hen under the null hypothesis of γ ij = 0 for all i j, D2 d 0, 1). From the statement of the theorem, there is an additional requirement that / 0. his condition removes a bias term that results from the estimation of λ t for each t. We explore this condition in the Monte Carlo experiment in Section 5. From a practical standpoint, the D2 statistic is particular useful since if one rejects the null of no cross-sectional correlation with the D statistic, the two-way model may be the next model that a researcher would consider. If the D2 statistic fails to reject, the two-way model would effectively account for cross-sectional correlation. If D2 rejects, one may consider alternative models allowing for cross-sectional correlation. One class of models is the spatial model considered in the next section. 4. Spatial Model Estimation of a spatial model is one method for dealing with cross-sectional dependence. In particular, spatial models treat cross-sectional dependence as a known with a spatial matrix used to represent the dependence. he spatial autoregressive SAR) model of Cliff and Ord 1973) treated by Anselin and Bera 1998) is one such model. Consider the SAR model with SAR error structure with two-way effects written in vector form. 4.1) 4.2) y t = µ + λw y t + X t θ + λ t ι + u t, for i = 1,..., u t = ρm u t + v t where the matrices W and M are known and potentially the same matrix, y t = y 1t, y 2t,..., y t ), µ = µ 1, µ 2,..., µ ), X t is an k matrix with rows given by x it, and ι is a 1 vector of ones. he W and M matrices represent the spatial dependence between elements within y t and elements within u t respectively. Recently, Lee and Yu 2010) provide an estimation strategy where, given W and M, one can estimate the parameters θ, λ,

9 and ρ at rate. We want to be able to test whether error terms v it have cross-sectional correlation. Absence of cross-sectional correlation in v it could be used to justify the use of spatial matrices W and M as opposed to some other arbitrary choice. o this end, we can use the parameter estimates ˆθ, ˆλ, ˆρ) to find new set of residuals. Let the 1 vector y 1t ȳ 1 ȳ t + ȳ y 2t ȳ 2 ȳ t + ȳ y t =. y t ȳ ȳ t + ȳ, with X t an k matrix defined similarly. hen the 1 vectors ü t = y t ˆλW y t X t ˆθ v t = ü t ˆρM ü t so that v t has typical element v it. ow, we construct the 1 vector v i and the direct statistic based on model 4.1) is given by [ DS = 1 1 v i v j v i v j σ DS 1) v i v i 1) where σ 2 DS = 1 2 [ v i v j v i v j 1) v i v i 1) v j v ] j 1) v j v j 1) his statistic has an asymptotically normal distribution as in the last theorem, so long as ˆθ, ˆλ, ˆρ) is consistent at rate as in Lee and Yu 2010). he proof is nearly identical to heorem Monte Carlo We consider several specifications for our Monte Carlo experiments. he static model is given by y it = µ i + x it β + v it where µ i are drawn from a 0, 1) distribution. he independent variables were generated by x it = µ i + ɛ it where ɛ it 0, 1). here are three separate specifications for v it. First, I use v it 0, 1) i.i.d.. errors. Finally, I generate v it = 0, σ 2 i ) with σ2 i ext, v it 3/5t 5 so that the variance is still one for the ] U0, 1) so that there is group-wise heteroskedasticity. I generate samples for various sizes of and using 2000 replications 9

10 10 for each experiment. he tests that are compared are the test by Pesaran, Ullah, and Yamagata 2008) which is the bias adjusted version of the LM test. his test is denoted AdjLM in the tables. In addition, I include the adjusted LM test that uses panel estimation of θ, denoted AdjLM P, recently explored by Baltagi, Feng, and Kao 2010a). he direct test is denoted as D in the tables and graphs. he size results appear in ables 1-3 for a nominal size of 5 %. In able 1, we see that for = 5, the statistics AdjLM and D perform similarly. Each test is close to the nominal size of 5% but both are slightly oversized when = 200. he AdjLMP test of Baltagi, Feng, and Kao 2010a) has very good size properties for all values of. In able 2, we allow for group-wise heteroskedasticity σi U0, 1)). Again, the statistics AdjLM and D perform similarly. he AdjLMP test performs very well, as in able 1. In able 3, we use t 5 errors. Both the AdjLM and AdjLMP tests are derived assuming normality. However, neither test seems to be affected by non-normality in this case whent errors are distributed as t 5. he D test is slightly undersized for larger values of. For the two-way model, the data is generated using normal errors. However, I include λ t 0, 1) in the data generating process. he only statistic that is designed to work in this model is the D2 but I include all the other statistics for comparison. he results appear in able 4. he D2 statistic has very good size properties for smaller values of. However, as increases, the size is poor for small values of. his feature is expected since heorem 3.1 requires / 0 as. he performance of D is reasonable for all values of when = 100. he other tests, including D are oversized, as they are interpreting the λ t as cross-sectional dependence. In practice, if D fails to reject and the other tests reject, we should employ a two-way model as a simple way to account for cross-sectional dependence. For the final size experiment, I consider a dynamic model given by y it = µ i + φy i,t 1 + v it. he error observations are distributed as v it 0, 1), phi = 0.5, and I use 50 startup observations for the autoregressive process. Both the AdjLM and AdjLM P statistics are

11 not designed for dynamic models. However, Pesaran, Ullah, and Yamagata 2008) include dynamic models in their simulations and find that the AdjLM test works well in some cases. he D statistics is calculated using residuals where the parameters are estimated using Arellano and Bond 1991). he results appear in able 5. When = 5, the D test is severely undersized. he AdjLM test can t be calculated since the bias calculation is not possible for = 5 in a dynamic model. For = 10 both tests can be calculated. As increases, size is better for both tests. Power against factor model y it = α i + x it β + v it v it = c 1 γ i f t + c 2 ɛ it such that c 2 1 1/12) + c2 2 = 1. he loading γ i U 0.5, 0.5) and the factor f t 0, 1). he values of c 1 and c 2 are chosen such that v it will have different variances for each i but the average variance is one. As c 1 gets larger, correlation between units is larger since each error v it will have a larger fraction of its variation coming from the term f t which is common to each i. I use 10,000 replications using = 10 and = 100, and also = 5 and = 100. he power graphs are shown in Figures 1 and 2. We see that the both the AdjLMP test and D test have more power than the AdjLM since both tests are based on the common slope coefficients, whereas the the AdjLM test allows for different slopes for each cross-sectional unit. 6. Conclusion I propose a test for cross-sectional dependence in this paper that is based on the squared covariances of all possible pairs in the cross-sectional dimension. he advantage of the formulation of the statistic using covariances is that the bias terms take a very simple form that can be easily estimated, regardless of the distribution of the error terms. Moreover, the test applies to dynamic models, two-way models, and spatial two-way models. From a technical standpoint, the asymptotics are of the diagonal path variety, where both and go to infinity simultaneously. In particular, the test does not rely on sequential asymptotic theory, where goes to infinity first, followed by. In addition, the new test puts limited restrictions on the distribution of the errors rather than requiring normality. 11

12 12 As it stands, the test is useful in that it will reject for general alternatives with crosssectional dependence. From a practical point of view, if one rejects using the basic version of the test, D, yet fails to reject using the version of the test for the two-way model, D2, it is reasonable to conclude that the cross-sectional dependence can be dealt with simply by specifying a two-way model. Similarly, one may use the DS version of the test as a diagnostic to check the adequacy of some spatial matrices. However, if we reject using all of the tests, a natural model to consider is the interactive fixed effects model of Bai 2009). A useful avenue for future research would be to design a statistic to check if such a strategy effectively models the cross-sectional dependency. References Andrews, D. W. K. 1991): Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica, 59, Anselin, L., and A. K. Bera 1998): Spatial Dependence in Linear Regression Models with an Introduction to Spatial Econometrics, in Handbook of Applied Economic Statistics, ed. by A. Ullah, and D. Giles. Marcel Dekker, ew York. Arellano, M., and J. Alvarez 2003): he ime Series and Cross-Section Asymptotics of Dynamic Panel Data Estimators, Econometrica, 71, Arellano, M., and S. Bond 1991): Some ests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations, Review of Economic Studies, 58, Bai, J. 2009): Panel Data Models with Interactive Fixed Effects, Econometrica, 77, Baltagi, B. H., Q. Feng, and C. Kao 2010a): A Lagrange Multiplier est for Cross-Sectional Dependence in a Fixed Effects Panel Data Model, Discussion paper, Syracuse University. 2010b): esting for Sphericity in a Fixed Effects Panel Data Model, he Econometrics Journal, forthcoming. Baltagi, B. H., S. Song, and W. Koh 2003): esting Panel Data Regression Models with Spatial Error Correlation, Journal of Econometrics, 117, Breusch,. S., and A. R. Pagan 1980): he Lagrange Multiplier est and Its Application to Model Specication in Econometrics, Review of Economic Studies, 47, Cliff, A., and J. Ord 1973): Spatial Autocorrelation. London: Pion. DeJong, P. 1987): A Central Limit heorem for Generalized Quadratic Forms, Probability heory and Related Fields, 75, Hall, P. 1984): A Central Limit heorem for Integrated Square Error of Multivariate onparametric Density Estimators, Journal of Multivariate Analysis, 14, 1 16.

13 13 Hsiao, C., M. H. Pesaran, and A. Pick 2008): Diagnostic ests of Cross Section Independence for onlinear Panel Data Models, Discussion paper, University of Southern California. John, S. 1972): he Distribution of a Statistic Used for esting Sphericity of ormal Distribution, Biometrika, 59, Kiefer,.,. J. Vogelsang, and H. Bunzel 2000): Simple Robust esting of Regression Hypotheses, Econometrica, 68, Lee, L. F., and J. Yu 2010): Estimation of Spatial Autoregressive Panel Data Models with Fixed Effects, Journal of Econometrics, 154, Pesaran, M. H. 2004): General Diagnostic ests for Cross Section Dependence in Panels, Discussion paper, Cambridge University. 2006): Estimation And Inference In Large Heterogeneous Panels With A Multifactor Error Structure, Econometrica, 74, Pesaran, M. H., A. Ullah, and. Yamagata 2008): A Bias-Adjusted Lm est Of Error Cross Section Independence, he Econometrics Journal, 11, Ullah, A. 2004): Finite Sample Econometrics. ew York: Oxford University Press. Vogelsang,. J. 2008): Heteroskedasticity, Autocorrelation, and Spatial Correlation Robust Inference in Linear Panel Models with Fixed-Effects, Discussion paper, Michigan State University.

14 14 Appendix A. Proofs Lemma A.1. Lemma 1, Yoshihara 1976)) Let x t1, x t2,..., x tk with t 1 < t 2 < < t k ) be absolutely regular random vectors with mixing coefficients β. Let hx t1, x t2,..., x tk ) be a Borel measurable function and let there be a δ > 0 such that P = max{p 1, P 2 } < where P 1 = P 2 = hx t1, x t2,..., x tk ) 1+δ df x t1, x t2,..., x tk ) hx t1, x t2,..., x tk ) 1+δ df x t1,..., x tj )df x tj+1,..., x tk ). hen hx t1, x t2,..., x tk )df x t1, x t2,..., x tk ) for all τ = t j+1 t j. hx t1, x t2,..., x tk )df x t1,..., x tj )df x tj+1,..., x tk ) 4P 1 δ 1+δ β 1+δ τ Lemma A.2. Suppose that U = j 1 i=1 j=2 H Z i, Z j ). Assume H is symmetric with E [H Z i, Z j ) Z i ] = 0 almost surely and E [ H Z i, Z j ) 2] <. Suppose that Z i are independent. Define G i Z j, Z k ) = E [H Z i, Z j )H Z i, Z k ) Z j, Z k ]. If 4 sup i,j,k E [ G i Z j, Z k ) 2] + 3 sup i,j E [ H Z i, Z j ) 4] s 2 ) 2 0 as, then s 1 U d 0, 1) where s 2 = j 1 i=1 j=2 E [ H Z i, Z j ) 2]. Proof: Following Hall 1984), we verify the two conditions required for a Martingale central limit theorem but now the Z i are heterogeneous and independent. he first is a Lindeberg condition A.1) s 2 E [ Wi1 W 2 i > ηs ) ] 0 i=2 as 0 for all η > 0, where W i = i 1 j=1 H Z i, Z j ). he second condition is A.2) s 2 V 2 p 1

15 15 where V 2 = i=2 EW i 2 Z 1, Z 2,..., Z i 1 ). For the first condition, EWi) 4 c sup E [ H Z i, Z j ) 4] + 2 sup E [ H Z i, Z j ) 2 H Z i, Z k ) 2]) i,j i,j,k c 2 sup E [ H Z i, Z j ) 4]) i,j for some generic constant c. hen and A.3) so that A.1) holds. s 4 i=2 i=2 EWi) 4 c 3 sup E [ H Z i, Z j ) 4] i,j EW 4 i) c 3 sup i,j E [ H Z i, Z j ) 4] s 2 ) 2 ext, let e i = EWi 2 Z 1, Z 2,..., Z i 1 ). hen V 2 = i=2 e i and [ ] EV) 4 = E e k i=2 e i k=2 i 1 i 1 = E G i Z j, Z l ) i=2 j=1 l=1 k 1 k=2 j =1 l =1 k 1 G k Z j, Z l ) he term E [ G i Z j, Z l )G k Z j, Z l ) ] will be non-zero whenever at least two pairs of the indices in j, l, j, l match. If j = l = j = l, If j = l j = l, If j = j l = l E [ G i Z j, Z l )G k Z j, Z l ) ] E [G i Z j, Z j ) 2 ] E [G k Z j, Z j ) 2 ] sup E [ G i Z j, Z j ) 2] i,j E [ G i Z j, Z j )G k Z j, Z j ) ] = E [G i Z j, Z j )] E [ G k Z j, Z j ) ] E [G i Z j, Z l )G k Z j, Z l )] sup E [G i Z j, Z j )]) 2 i,j E [G i Z j, Z l ) 2 ] E [G k Z j, Z l ) 2 ] sup E [ G i Z j, Z l ) 2] i,j,l

16 16 Hence, we have and A.4) so that A.2) holds. E V 2 s 2 s 2 EV) 4 4 sup E [ G i Z j, Z l ) 2] i,j,l ) 2 c 4 sup i,j,k E [ G i Z j, Z k ) 2] s 2 ) 2. Lemma A.3. Suppose that Assumption 1 holds and Evit 8 ) is finite. hen j 1 1 vi v jvi v j v i v i vj v ) j d 0, σ 2 u) i=1 j=2 as and, where σu 2 = 1 j 1 2 E i=1 j=2 v i v jv i v j v i v i vj v ) 2 j Proof: In what follows, we let = to simplify notation. Let H Z i, Z j ) = 1 vi v jvi v j v i v i vj v ) j which we will denote as H ij. We must show that the conditions of Lemma A.2) hold. By Loève s c r inequality, we have EHij) 4 = E 1 vi v jvi v j 4 [ E v i v i [ ) ) 4 )] 4 E v i v j vi v j = E v it v jt v iu v ju = E t=1 vj v ) 4 j ) 4 vi v j vi 1 ) ] 4 v j E vi v i vj v j u=1 t=1 u=1 r=1 s=1 p=1 q=1 m=1 n=1 v it v jt v iu v ju v ir v jr v is v js v ip v jp v iq v jq v im v jm v in v jn here are four classes of index cases where Ev it v jt v iu v ju v ir v jr v is v js v ip v jp v iq v jq v im v jm v in v jn ) is nonzero. In case 1, we have t = u = r = s = p = q = m = n so that the number of terms will be of order. In case 2, we have t = u = r = s p = q = m = n so that the number of terms in this case will be of order 2. In case 3, we have t = u = r = s p = q m = n

17 with the number of these terms being of order 3. Finally, for case 4, we have t = u r = s p = q m = n. he number of terms in case 4 is of order 4. Hence, for the order, we have E v i v j v i v j ) 4 max [ sup i,j Similarly, we have Ev 8 it)ev 8 jt), 2 sup i,j ) 4 E v i v i vj v j = E aking the first term separately gives ) 4 E v it = E t=1 max r=1 s=1 t=1 u=1 sup i Ev 4 it)ev 4 jt), 3 sup i,j vit 2 vju 2 t=1 u=1 ) 4 4 = E vit) 2 E v 2 irv 2 isv 2 itv 2 iu t=1 ) u=1 v 2 ju ) 4 17 Evit)σ 4 i 2 σj 2, 4 sup σi 2 σj 2 i,j [ Ev 8 it ), 2 Ev 4 it)ev 4 it), 2 Ev 6 it)σ 2 i, 3 Ev 4 it)σ 2 i ) 2, 4 σ 2 i ) 4]) 4 he above result squared implies that E vi v ivj j) v = O 4 ) so that EH ij ) 4 = O 4 ) Define ] G i = E H ij H ik v j, v k ) = 1 v 2 E i v j vi v j 1 v 2 E i v j vi v j 1 2 E vi v kvi v k E vi v kvi v ) k v j, v k v i v i v i v i vi v i vj v j vi v i Consider the first term of G i which can be written as v k v k ) v j, v k ) vj v j v j, v k vk v k v j, v k E tr v i v j v j v i v i v k v k v i v j, v k ) = E [ vecv k v k ) v i v i v i v i )vecv j v j ) v i, v j ] )

18 18 he other three terms have a similar representation. Let D i = Ev i v i have G i = vecv kv k ) D i vecv j v j ) vecv k v ki ) D i vecv j v j ) vecv kv k ) D i vecv j v j I ) vecv k v ki ) D i vecv j v j I ) = 1 [ 2 2 vecv k vk ) ˆσ2 k veci )] Di vecv j vj ) 1 [ 2 2 vecv k vk k] ) ˆσ2 Di veci )ˆσ j 2 = G i1 + G i2 v i v i ). hen we where ˆ sigma 2 k = 1 Let e p be a 1 vector with a one in the pth entry and zeros elsewhere. t=1 v 2 kt hen, for p q, the p, q) block of D i is e p e q + e q e p )σ 4 i. For p = q, the p, p) block of D i is I e p e p )σi 4 + e pe p Evit 4 ). We have p=1 Ev2 ip v2 i1 )v2 jp 2σi 4 v j1v j2. 2σi 4v j1v j 2σi 2v j2v j1 p=1 Ev2 ip v2 i2 )v2 jp 2σi 4 D i vecv j vj v j2v j3 ) =. 2σi 4 v j2v j. 2σi 4 v j v j1. 2σi 4v j v j, 1 p=1 Ev2 ip v2 i )v2 jp

19 19 so that G i1 = m=1 p=1 = G i11 + G i12 Evipv 2 im)v 2 jpv 2 km 2 ˆσ2 k ) + 2σ4 i 2 2 m=1 p m v jm v jp v km v kp Similarly, we have By Loève s inequality, G i2 = ˆσ2 j 2 2 m=1 p=1 EG 2 i ) 2 [ EG 2 i1) + EG 2 i2) ] Evipv 2 im)ˆσ 2 k 2 v2 km ) 4 [ EG 2 i11) + EG 2 i12) ] + 2EG 2 i2) ow EG 2 i2) = Eˆσ4 j ) ote that for m n, m=1 p=1 n=1 q=1 E [ ˆσ 2 k v2 km )ˆσ2 k v2 kn )] = Ev4 kt ) = Ev4 kt ) Evipv 2 im)ev 2 iqv 2 in)e 2 [ ˆσ k 2 v2 km )ˆσ2 k v2 kn )] + 1)σ4 k 2 + σ4 k 2 1) σk 4 2 Ev4 kt ) + σ4 k Hence, EG 2 i2 ) = O 4 1 ). he result for EG 2 i11 ) is similar. ow for EG2 i12 ) we have EG 2 i12) = 4σ8 i 4 4 E v jm v jp v km v kp v jn v jq v kn v kq = 8σ8 i 4 4 E = O 4 2 ) m=1 p m n=1 q n m=1 p m v 2 jmv 2 jpv 2 km v2 kp so that EG 2 i ) = O 4 1 ). σ 2 u = O1) so that the conditions required in Lemma A.2 hold.

20 20 Proof of heorem 2.1: I omit the M 0 term as it does not change the order of the statistics. Again, denote =. hen A.5) 1 A.6) A.7) A.8) A.9) A.10) A.11) A.12) A.13) A.14) A.15) A.16) A.17) A.18) A.19) ṽ i ṽjṽ i ṽj + 2 ṽ i ṽi ) ṽj ṽj = 1 vi v jvi X j + v i v jv j X i + 2 θ ˆθ) j 1 v i v j Xi X j v i v jv i v j ) θ ˆθ) + 2 θ ˆθ) j 1 X i v j vi X ) j θ ˆθ) + 1 θ ˆθ) j 1 X j v i vi X ) j θ ˆθ) + 1 θ ˆθ) j 1 X i v j vj X ) i θ ˆθ) 4 θ ˆθ) j 1 X j v j vi X ) i 2 θ ˆθ) 1 θ ˆθ) 1 θ ˆθ) j 1 v i v i Xj X ) j θ ˆθ) 2 j 1 v j v j Xi X ) i θ ˆθ) 2 v i v iv j X j 2 vi X j θ ˆθ)θ ˆθ) Xi X j θ ˆθ) vj X i θ ˆθ)θ ˆθ) Xi X j θ ˆθ) v i v i θ ˆθ) Xi X j θ ˆθ)θ ˆθ) Xi X j θ ˆθ) vi X i θ ˆθ)θ ˆθ) Xi X j θ ˆθ) vj X j θ ˆθ)θ ˆθ) Xi X j θ ˆθ) θ ˆθ) Xi X i θ ˆθ)θ ˆθ) Xj X j θ ˆθ) vj v ) j v j v jv i X i 2 ) θ ˆθ)

21 For the right hand size of the first term, A.5), we apply Lemma A.2 to obtain normality. ow consider A.6). Without loss of generality, assume that k = 1 so that X i is 1 with entries x it. aking the expectation of the square of the double sum, we have sixteen terms. he first term is of the form k 1 k=2 l=1 ) E vi v j vi X j vk v lvk X l he expectation is nonzero if i = k and j = l, and then the expectation is given by E v is v js v it x jt v ir v jr v iq x jq = E i=1 j=1 s=1 t=1 r=1 q=1 i=1 j=1 s=1 t=1 v 2 isv 2 jsv 2 itx 2 jt which is O 2 2 ) so that dividing by 2 2 gives an O1) term. All of the remaining fifteen terms are similarly O1) so that A.6) is O p 1/2 1/2 ). For A.7), we have E v i v j Xi X ) j 2 2 E vi v jxi X j i=1 ) 2 E v it v jt x is x js j=2 t=1 s=1 21 Consider the highest order cases in E t=1 s=1 If t = t s s, we have ) 2 v it v jt x is x js t=1 s=1 t =1 s =1 E v it v jt x is x js v it v jt x is x js ) t=1 s s E v 2 itv 2 jtx is x js x is x js ) = O 3 ) If t < t < s < s, we have t<t <s<s E vit v jt x is x js v it v jt x is x js ) t<t <s<s = O 3 ) 4P δ 1+δ βs t ) δ 1+δ

22 22 where the first inequality comes from an application of Lemma A.1 and the last line is from the summability of the β coefficients. Combining the above results, we have E v i v j Xi X ) j = O 2 1/2 ) Since θ ˆθ) = O p 1/2 1/2 ), we have A.7) = O p 1/2 ). Similarly, the terms A.8) through A.10) are also of the same order. A.8), hence lower order. For A.12), note that E v i v ix j X j 2 A.11) has an extra 1 term compared to t=1 s=1 = E v2 it x2 js 2 = O1) so that A.12) is O p 1 ). he same argument holds for A.13). It is easy to show that the remaining terms are at most O p 1/2 1/2 ). he proof that σd 2 is consistent is similar. Proof of heorem 3.1: Again, denote =. We have [ D2 = 1 1 v i v j v i v j σ D2 1) v i v i 1) where v j v ] j 1) v it = y it ȳ i ȳ t + ȳ ) x it x i x t + x ) ˆθ = v it v i v t + v ) + x it x i x t + x ) θ ˆθ) Similar to the proof of heorem 2.1, the terms involving ˆθ θ are o p 1) under the conditions of the theorem so that the dominant term is 1 where [ v i ˇv) M 0 v j ˇv) v i ˇv) M 0 v j ˇv) 1) ˇv = 1 p=1 v p v i ˇv) M 0 v i ˇv) 1) ] v j ˇv) M 0 v j ˇv) 1) he new terms in this version of the theorem involve ˇv, and all other terms remain the same as in heorem 2.1. Consider the term 1 ˇv M 0ˇvˇv M 0ˇv 1)

23 ote that ˇv M 0ˇvˇv ) M 0ˇv P 1) η E ˇv M 0ˇvˇv M 0ˇv ) η 1) = E 1 4 p=1 v p M 0 q=1 v q r=1 v r M 0 s=1 v s η 1) here are three non-zero cases in the numerator of the right hand side. First if p = q = r = s, we have If p = q r = s, we have 1 4 p r 1 4 Evp M 0 v p vp M 0 v p ) p=1 Ev p M 0 v p vr 1)2 M 0 v r ) = 4 If p = r q = s or p = s q = r), we have 1 Ev 4 p M 0 v q vp M 0 v q ) = 1 4 p q herefore, the order is so that 1 = p q 1) 4 E ˇv M 0ˇvˇv M 0ˇv ) 1) ˇv M 0ˇvˇv M 0ˇv 1) he other terms involving ˇv are also the same order. p r σ 2 pσ 2 r. traceevp v p M 0 v q v q M 0 ) p q σ 2 pσ 2 q ) = O 2 = O p ) ) 23

24 24 able 1. Size AdjLM AdjLMP D = = 5 = = = = = 10 = = = = = 20 = = = = = 30 = = = = = 50 = = =

25 25 able 2. Size with Group-wise Het σ 2 i = 1 + U0, 1) AdjLM AdjLMP D = = 5 = = = = = 10 = = = = = 20 = = = = = 30 = = = = = 50 = = =

26 26 able 3. Size with t5 Errors AdjLM AdjLMP D = = 5 = = = = = 10 = = = = = 20 = = = = = 30 = = = = = 50 = = =

27 27 able 4. Size with wo-way AdjLM AdjLMP D D2 = = 5 = = = = = 10 = = = = = 20 = = = = = 30 = = = = = 50 = = =

28 28 able 5. Size Dynamic Models D AdjLM = 5 = A = 5 = A = 10 = = 10 = = 20 = = 20 = = 30 = = 30 = = 50 = = 50 =

29 29 Figure 1 Size Adjusted Power Against Factor Model: =10, = AdjLM AdjLMP Dn 60 Power c 1 2

30 30 Figure 2 Size Adjusted Power Against Factor Model: =5, = AdjLM AdjLMP Dn Power c 1 2