On Testing for Sphericity with Non-normality in a Fixed Effects Panel Data Model

Building upon the work of Chen et al. (2010), this paper proposes a test for sphericity of the variance-covariance matrix in a fixed effects panel data regression model without the normality assumption on the disturbances.


Introduction
This paper proposes testing the null of sphericity of the variance-covariance matrix in a …xed e¤ects panel data model which does not require the normality assumption on the disturbances.This builds on the paper by Chen et al. (2010) who use U -statistics to test for sphericity of the variance-covariance matrix in statistics.The null of sphericity means that the variance-covariance matrix is proportional to the identity matrix.Rejecting the null means having cross-sectional dependence among the individual units of observation or heteroskedasticity or both.In empirical economic studies, individuals are a¤ected by common shocks.For example, investors' decisions may be in ‡uenced by the way they interact with each other and also by common macro-economic shocks or public policies.These potentially cause cross-sectional dependence among the units.
In statistics, the n n sample covariance matrix S n is widely used for tests of sphericity since it is a consistent estimator for the variance-covariance matrix n .One could either use the likelihood ratio test, see Anderson (2003), or test the Frobenius norm of the di¤erence between S n and n , see John (1971John ( , 1972)).However, with panel data sets where n the number of individuals is larger than the time series dimension of the data T , the sample covariance matrix becomes singular.This causes problems for the likelihood ratio test which is based on the inverse of S n .Even when n is smaller than T , the sample covariance matrix S n is ill-conditioned as shown in the Random Matrix Theory (RMT) literature.In fact, the eigenvalues of the sample covariance matrix S n are no longer consistent for their population counterpart, see Johnstone (2001).Ledoit and Wolf (2004) show that the scaled Frobenius norm of S n does not converge to that of n with n=T !c 2 (0; 1).
As a result, John's test, see John (1971John ( , 1972)), is no longer applicable.Hence, Ledoit and Wolf (2002) propose a new test for the null of sphericity which could be applied even when n is relatively as large as T .However, these statistical tests for raw data are not directly applicable to testing sphericity in panel data regressions since the disturbances are unobservable.Baltagi et al. (2011) extend the Ledoit and Wolf (2002)'s John test to the …xed e¤ects panel data model and correct for the bias due to substituting within residuals for the actual disturbances.However, their test relies on the normality assumption and their simulation results show that the test has size distortion under non-normality of the disturbances.
To account for the possible "non-normality" of the disturbances as well as the "large n, small T " issues in testing the null of sphericity, Chen et al. (2010) propose a modi…ed John test by constructing U -statistics of observable samples for estimating tr 2 n and tr n .Based on their work, this paper proposes a new test for the null of sphericity of the disturbances in a …xed e¤ects regression panel data model.This test does not require the assumption of normality of the disturbances, and can be applied to the case where n is larger than T .The limiting distribution of this test statistic under the null is derived.Also, its …nite sample properties are studied using Monte Carlo simulations.
The paper is organized as follows.Section 2 speci…es the …xed e¤ects panel data regression model and the assumptions required.Section 3 introduces the test statistic.Section 4 derives the limiting distribution of this test statistic under the null and discusses its power properties.Section 5 reports the results of Monte Carlo simulations, while Section 6 concludes.All the proofs and technical details can be found in an Appendix available upon request from the authors.

The Model and Assumptions
Consider the following …xed e¤ects panel data regression model ; for i = 1; 2; : : : ; n; t = 1; 2; : : : ; T; (2:1) where i indexes the cross-sectional dimension and t indexes the time series dimension.y it is the dependent variable, x it denotes the k 1 vector of exogenous regressors, and is the corresponding k 1 vector of parameters.i denotes the time-invariant individual e¤ects which can be …xed or random and could be correlated with the regressors.De…ne the vector of disturbances v t = (v 1t ; : : : ; v nt ) 0 and its corresponding variance-covariance matrix n .The null hypothesis of interest is sphericity: (2:2) The alternative hypothesis allows cross-sectional dependence or heteroskedasticity or both.
For the panel data regression model, v it is unobserved, and the test statistic is based upon consistent estimates of variance-covariance matrix, denoted by S n or its correlation coe¢ cients matrix counterpart, see Breusch and Pagan (1980).Baltagi et al. (2011) extend the Ledoit and Wolf (2002) test to a …xed e¤ects panel data model with large n and large T .They show that the noise resulting from using within residuals rather than the actual disturbances accumulates and causes bias for the proposed test statistic.However, their simulations show that their test is oversized under non-normality of the disturbances.This paper extends Chen et al. (2010)  , and : The within estimator of given by
Assumption 2. The regressors x it ; i = 1; :::; n; t = 1; :::; T are independent of the idiosyncratic disturbances v it ; i = 1; : : : ; n; t = 1; : : : ; T .The regressors x it have …nite fourth mo- The asymptotics follow the framework employed by Chen et al. (2010).Assumption 3 requires tr( 4 n ) to grow at a slower rate than tr 2 ( 2 n ).This assumption is ‡exible.In fact, if all the eigenvalues of n are bounded away from zero and in…nity, tr( 4n )=tr 2 ( 2 n ) ! 0, is always true for any n as n ! 1.Moreover, this assumption allows n to be much larger than T , which is more suitable for micro-panel data.

J u Test
For testing the null hypothesis (2.2), the test statistic is based on the scaled distance measure between 2 v n and I n : where S n is the n n sample covariance matrix and I n is an n n identity matrix.With the normality assumption, John (1972) shows that for …xed n, and as T !1: But when n goes to in…nity, the test statistic diverges.Ledoit and Wolf (2002) propose a modi…ed test statistic under the null, as (n; T ) ! 1 and n=T !c 2 (0; 1): , then under the null J 0 !N (0; 1).However, this test cannot be used directly in a …xed e¤ects panel data regression model.The raw data sample covariance matrix S n is replaced by its counterpart Ŝn = 1 P T T t=1 v ^tv ^t 0 , where v ^t is the within residual given by (2.3).

2
The residual-based ^de… ^^ U 0 is ned as Baltagi et al. (2011) propose a bias correction: They show that in a …xed e¤ects panel data regression, as d (n; T ) ! 1 and n=T !c, J BF K !N (0; 1) under the null.However, their result relies on the normality assumption of v t .Without the normality assumption, the bias-corrected John test is not robust, see the simulations in Baltagi et al. (2011).Chen et al. (2010) propose a new test statistic for the sphericity of the variance-covariance matrix of the disturbances without the normality assumption and under much relaxed conditions where n could be much larger than T .They construct the U -statistics for estimating tr n and tr 2 n .Following their framework, we propose a residual-based test statistic for testing the null of sphericity described in (2.2) in a …xed e¤ects panel data model.De…ne observe the true v t , then R 1 , R 2 and M j;T ; for j = 1; 2; 3; 4; 5 are obtained similarly by replacing . Chen et al. (2010) show that as (n; T ) ! 1: Let J CZZ = T U T 2 , then under the null d J CZZ !N (0; 1).Following this framework, we propose the following test statistic: J u is the residual-based statistic corresponding to J CZZ .There are two important issues to be considered.First, whether the residual-based R1 and R2 are consistent estimates for tr n and tr 2 n under the null, respectively.Second, the asymptotics of the proposed test need to be derived.Both concerns are tackled in the next Section.

Asymptotics of the J u Test
In this Section, we prove that, under the null, 1 ^R v and n tr n v , respectively.Next, we show converges to N (0; 1) under the null and we discuss its power properties.To examine the asymptotics of J u , we rewrite it as The …rst term J CZZ is asymptotically standard normal under the null.The second term J u J CZZ is the scaled di¤erence between the residual-based ÛT and the true U T .From Section 3, this di¤erence can be rewritten as follows: From equation (4:2), it is clear that this term depends upon the two di¤erences: . Their asymptotic behavior is given in the following propositions: Proposition 1.Under Assumptions 1-2 and the null, (1) Proposition 2. Under Assumptions 1-2 and the null, (1) nT Propositions 1 and 2 show that the di¤erences Therefore, since estimates for 2 v and 4 v respectively.The following corollary gives these conclusions: Corollary 1.Under Assumptions 1-2 and the null, as (n; T ) ! 1, (1) Note that 1 R is a consistent estimator of 4 2 n v under the null with large n and large T .However, 1 tr Ŝ2 n n is not consistent, see Baltagi et al. (2011).
T ( ÛT U T ) Proposition 3.Under Assumptions 1-2 and the null, Propositions 1, 2 and 3 give the asymptotics of the bias term J u J CZZ .
depends upon how ~ : and x ~0 t ( ) accumulate in 1 v M 1 j;T M j;T ; n n for j = 1; 2; 3; 4; 5. Note that v : is an n dimensional vector, although each element of v : is O p ( p 1 ), v : may still accumulate in the above …ve terms as (n; T ) , depending upon the which is related to both n and T .We may nT expect its convergence speed p 1 to be fast enough so that x ~0 t( ) vanishes as (n; T ) nT ! 1.
More speci…cally, Proposition 2 shows the leading terms of 1 M 1 j;T M n j;T n , for j = 3; 4; 5 will not vanish if n T 2 does not converge to zero.These terms are caused by the accumulation of v : .However, due to the subtraction formulation of the test statistic, the leading terms cancel each other in both since their expressions are exactly the same.These cancellations lead us , and consequently ! 0 as (n; T ) ! 1 and we do not need to correct the bias in the …xed e¤ects panel data regression model.This result is based on our detailed calculation of how v : and 0 xt ( ) are accumulating in 1 M 1 j;T M n j;T n , for j = 1; 2; 3; 4; 5 and the special formulation of J CZZ .As discussed above, the convergence of J u is given by the following theorem:  2010), we consider an alternative: where 2 ( 1; 1) and = 0: 2 l = var(v lt ); which is uniformly bounded away from in…nity and zero, for l = 1; : : : ; n.Under this alternative, we can show that J u J CZZ = o p (1), which in turn implies that J u and t ) J CZZ have the same power properties.De…ne and 2;T = tr This satis…es the conditions of Theorem 4 in Chen et al. (2010).By using this Theorem, the corresponding power function P (J u z j where z is the upper quantile of N (0; 1).Let us consider a special case under this alternative.

Monte Carlo Simulations
We conduct Monte Carlo experiments to assess the empirical size and power of the J u test proposed in this paper.We follow the design of Baltagi et al. (2011) and assume homoskedasticity on the remainder error term.In this case, the J u test becomes a test for cross-sectional dependence.
We also report the performance of J BF K proposed by Baltagi et al. (2011) for comparison purposes.

Experiment Design
Consider the following data-generating process: 1; : : : ; n; t = 1; : : : ; T; (5.1) the results with v it coming from alternative non-normal distributions.The size of J u is close to 5% when n and T are large; for small n or small T , it is slightly oversized.However, J BF K is no longer robust to non-normality and su¤ers from size distortions.
Table 2 presents the size adjusted power of the tests under the alternative speci…cation of a factor model.Both tests have size adjusted power that is almost 1 when n and T are large with v it normally distributed.For small n and small T , the size adjusted power of J u works as well as J BF K .Note that the size adjusted power of J BF K is quite good even when n is a lot larger than T for the normal distribution scenario.However, for non-normal distributions, the size adjusted power of J u is 1 as n and T become large; and it is larger than the size adjusted power of J BF K for all (n; T ) combinations.
Table 3 reports the size adjusted power of both tests under the alternative speci…cation of SAR(1).The results are similar to the factor model.J u works as well as J BF K for the normal distribution scenario, but better for all combinations of n and T for non-normal distribution scenarios.

Conclusion
Though the John test proposed by Baltagi et al. (2011) has been shown to perform well for a large panel data regression model with …xed e¤ects, it relies heavily on the normality assumption.
This paper proposes a new test, J u ; for the null of sphericity of the disturbances which does not rely on the normality assumption.Instead of n=T !c, we allow n to be a larger order of T which is consistent with micro-panel data sets with "large n and small T " .Note: This table computes the size adjusted power for a factor structure model that allows for cross-sectional dependence in the error.We conduct the simulation with four distributions: normal, gamma, lognormal and chi-squared with mean 0, and variance 0.5.Note: This table computes the size adjusted power for a SAR(1) structure model that allows for cross-sectional dependence in the error.We conduct the simulation with four distributions: normal, gamma, lognormal and chi-squared with mean 0, and variance 0.5.
Notation: jjBjj = (tr(B 0 1=2 B)) is the Frobenius norm of a matrix B or the Euclidean norm of a vector B, and tr(B) is the trace of B. d !denotes convergence in distribution and p ! denotes convergence in probability.For two matrices B = (b ij ) and C = (c ij ), we de…ne B C = (b ij c ij ).
n. Similar cancellations occur for other terms which are O p T 2 ,

Table 1 :
Size of Tests Note: This table reports the size of Ju and J BF K with di¤erent error distribution speci…cation in a …xed e¤ects panel data model without cross-sectional dependence among the errors.The tests are one-sided and are conducted at the 5% nominal signi…cance level.We conduct the simulation with four distributions: normal, gamma, lognormal and chi-squared with mean 0, and variance 0.5.

Table 2 :
Size adjusted power of tests: factor model

Table 3 :
Size adjusted power of tests: SAR(1) model