Testing for Cross-Sectional Dependence in a Random Effects Model

This paper extends and generalizes the works of [1,2] to allow for cross-sectional dependence in the context of a two-way error components model and consequently develops LM test. The cross-sectional dependence follows the first order spatial autoregressive error (SAE) process and is imposed on the remainder disturbances. It is important to note that this paper does not consider alternative forms of spatial lag dependence other than SAE. It also does not allow for endogeneity of the regressors and requires the normality assumption to derive the LM test.


Introduction
The standard error components model assumes, among others, spatial independence across cross-sectional units.However, this restrictive assumption may not hold for a lot of panel data applications.When one begins to look at a cross section of regions, states, countries, etc., these aggregate units may exhibit cross-sectional correlation that has to be dealt with (see [3]).Ignoring cross-sectional dependence when in fact it exists, results in biased, inconsistent and inefficient estimates of regression coefficients (see [1,4,5]).
In the literature, several test statistics have been developed for spatial econometrics however in the context of either cross sectional framework or one-way error components model. 1 The specification of cross-sectional dependence in linear regression models by most of these works follows either spatial autoregressive (SAR) process often defined as spatial lag dependence (see [6][7][8][9][10]); spatial moving average processes (SMA) often called spatial error dependence (see [11]); spatial autoregressive error process (SAE) (see [6,12,13]); SARMA (a combination of SAR and SMA) (see [2,4,14]); a combination of SAR and SARE (see [15]); direct representation form of cross-sectional dependence (see [16,17]) or spatial error component process (SEC) suggested by [18].Consequently, various tests as well as estimators were derived against these different specification forms using either the Maximum Likelihood (ML) approach (see [2,19]; for a survey of the literature) or Instrumental Variables (IV) and Generalized Method of Moments (GMM) (see [9,20,21]).
The present study develops LM test for cross-sectional dependence in the context of panel data framework.The latter is a two-way random effects model where the cross-sectional dependence follows the SAE and is imposed on the remainder disturbances.Prominent papers that have adopted the SAE include [2,4] in the context of cross-sectional framework, and [1,22,23] in the context of one-way error components framework.Thus, the main objective of this work is to extend and generalize the works of [1,2] to allow for cross-sectional dependence in the context of a two-way error components model.The panel data model considered here is the restricted twoway random effects model assuming no cross-sectional dependence in the remainder disturbances.Thus, the LM test will be similar to the one developed by [1] if we further modify the hypothesis to test for cross-sectional dependence assuming the presence of random individual effect only (while ignoring the presence of time effects).In the same vein, the LM test will be similar to [2,4] if the hypothesis is reconstructed to test for cross-sectional dependence ignoring the presence of both the random country and time effects.
In Section 2, the structure of the two-way error random effects model is described in the context of crosssectional dependence in the remainder disturbance term.Analyses of the LM test are provided in Section 3 and Section 4 concludes the paper.

The Model 1 k
We consider the following panel data regression model: where the index i denotes N regional units and the index t refers to the T observations of each region i.The i subscript, therefore, denotes the cross-sectional dimension whereas t denotes time-series dimension.The total number of observations is NT.ti is the observation on the region over the time period; ti x is the observation on k explanatory variables and ti is the regression disturbance term.The error term ti follows a two-way random effects with both regional specific and temporal effects; that is, where i  denotes regional specific effects, t denotes temporal effects and ti v  represents the remainder disturbance term.Stacking the N observations of each timeperiod t, Equation (2) may be written as: , , , where , is a vector of ones of N dimension, , , , , , Assumption 1: Both i  and t v are assumed independent and normally distributed according to, ; ~0, The remainder disturbance term t  is assumed to follow the first order spatial error correlation (see [2,3]), that is: where , , e e e    and . The term  is the scalar spatial autoregressive coefficient with 1   .The matrix W is an spatial weight matrix which represents the degree of potential interaction between neighboring locations whose diagonal elements are zero and off-diagonal elements are non-zero.Equation ( 5) can be further simplified as: Given Equation ( 6), the weight matrix W also satisfies the condition that is nonsingular for all 1   .t is also assumed to be independent and nor- mally distributed as: The ti process is also independent of the e i  and terms.

t v
The model ( 1) can be re-written in matrix notation as: where y is of dimension  vector, X is an NT k  matrix,  is  vector and u is vector.The matrix X is assumed to be of full column rank and its elements are assumed to be asymptotically bounded in absolute value.Given Equation (6), Equation (3) can be re-written as: We can write Equation (8) in vector from as: The variance-covariance (VCV) matrix of Equation (10) (that is, the unrestricted model) can be expressed as: where T T T J i i  and it is a matrix of ones of dimension T. To obtain the spectral decomposition of Equation ( 11), we use the [24]  and and consequently, we obtain 3 : Also, using the [25] method of inversion, Equation ( 12) can be expressed as:

Derivation of the LM Test
In this section, we derive the LM test for testing for no cross-sectional dependence in a two-way random effects model.We employ the Maximum Likelihood (ML) approach and consequently, the log-likelihood function.
The LM test derived is based on the idea that the score of the likelihood function evaluated under the null is equal to zero when the null hypothesis is true, so that a  test based on the square of the score divided by the appropriate element of the information matrix (since this is the variance of the score) can be constructed.The use of the normal likelihood function requires the assumption of normality of the error term.
Essentially, the derivation of the LM test involves the following steps: Step 1: Derive the VCV matrix for the unrestricted model; Step 2: Derive the VCV matrix for the restricted model; Step 3: Derive the spectral decomposition for the matrices obtained in steps 1 and 2; Step 4: Derive the inverse of the matrices obtained in steps 1 and 2 using the results from step 3; Step 5: Derive the general log-likelihood function; Step 6: Use the information in steps 1 -5 to derive the score functions of the likelihood evaluated from the restricted ML under 0 a H ; Step 7: Derive the information matrix and its inverse; Step 8: Use the results obtained in steps 6 and 7 to develop the LM test.
The log likelihood function, L under normality of disturbances is given as: where and the vector of parameters is denoted as , , where Since our test statistic requires information only on the vector of parameters  , consequently, information due to  is ignored.Following [26], the gradient of the log likelihood with respect to  can be expressed as: By further simplification, it is easy to show that: Equations ( 15) and ( 16) represent the score function and the information matrix respectively.The information matrix- is block diagonal.The LM statistic can, therefore, be written generally as: where   and I    are the score function and information matrix respectively evaluated at the null hypothesis.The LM test statistic expressed in ( 17) is distributed as 2 k   (i.e.chi-square distributed) with k  degrees of freedom, k  being the number of parameters in the vector  .Based on Equation (17), therefore, the following hypotheses can be tested in relation to cross-sectional dependence: This is a test of no cross-sectional dependence assumeing the presence of random individual and time effects.This is the null hypothesis this study sets out to test.
This hypothesis tests for cross-sectional dependence assuming the presence of random individual effect only (while ignoring the presence of time effects).This test is similar to [1] LM test for spatial error correlation as well as random country effects.
This hypothesis tests for cross-sectional dependence ignoring the presence of both the random country and time effects.This is similar to the LM test by [2,4].
We derive below the score function for the null hypothesis expressed in (18) above which is the focus of this paper; that is: Under the null hypothesis in (18), the VCV matrix reduces to:4 and, therefore, The Equation ( 21) is the VCV matrix for the restricted model.Using [25] Lemma 2.1, the inverse of Equation ( 21) can be expressed as: 22) is the reduced form of Equation (13) and is also the VCV matrix for the familiar two-way random effects error components model.In addition, it is a principal component required in the log-likelihood function to derive the LM test.In particular, both Equations ( 21) and ( 22) are required to derive the partial derivatives and information matrix for the LM test.
Using the general formulas on log likelihood differentiation, we derive its gradients evaluated at the restricted ML under 0 a H as follows: Recall Equation ( 15): T e e T T E BB Based on assumption 1, it is easy to establish from Equation (24) that: We also replace by T I where Copyright © 2012 SciRes.OJS   These derivatives are quite straightforward to show particularly using the information in proposition 1.
Proposition 3: Based on propositions 1 and 2 and assumptions 2 and 3, we can write the derivations of  , respectively, as:

Proof:
These derivatives are straightforward to show using the information in proposition 2.

Proof:
See the appendix for further simplifications and proofs of the partial derivatives.

Recall that we define
, , ,0 can be defined as the H , under solution obtained after maximization of the first order condition and MLE is the corresponding residual under . ., , ,0 were evaluated when This is because we are testing whether  is statistically different from zero.Thus, the partial derivatives under 0 a H are rewritten in vector form as: Also, using the method developed by [27], we obtain Copyright © 2012 SciRes.OJS 0 a the information matrix under H .The information matrix is given by: 2 1 2 Proposition 5: Using the formular expressed in Equation (29) and information in proposition 2, we can derive respective elements in I  a under 0 H for the vector of parameters 1 tr 2 1 tr 2 Given these information under H , the LM statistic is given by,6 Under 0

Decision Criteria:
The LM statistic is a scalar and the value obtained when the test is performed on the two-way error components model is compared with the critical value for the chi-squared distribution-1 2  .The intention is to ascertain whether to reject the null hypothesis, 0 a H , that there is no cross-sectional dependence problem in a two-way random effects model.Essentially, if LM  is less than the critical value for the chi-squared distribution, then, we do not reject the null hypothesis implying that there is no cross-sectional dependence; otherwise, we reject it.

Concluding Remarks
This paper provides a framework for testing for no crosssectional dependence assuming the presence of random individual and time effects.Thus, several important issues have not been incorporated.These include testing other hypotheses earlier specified, that is; which tests for cross-sectional dependence ignoring the presence of both the random country and time effects.Also, the empirical applications section involving Monte Carlo experiments is also not yet considered.These are some of the suggestions for future research.

Appendix (A) Derivation of the VCV Matrix for the Unrestricted Model
Here, 0 and the VCV matrix of u can be derived as follows.
Recall Equation (10), Using assumption (1), the VCV matrix can be expressed as: Let in this case be represented by , and by further simplification, (A.1) becomes: where T T T J i i  T and it is a matrix of ones of dimension T. To obtain the spectral decomposition of (A.3), we use the [24] method which involves replacing J by and 3).This is done as follows: 2 Using the [25] method of inversion, therefore, the inverse of Equation (A.5) can be expressed as: where V matrix can be ed as: The assumption (1), the VC express Thus, (A.4) under the unrestricted model reduces to: Just as before, we use the [24] method to spectral decomposition of (B.4) and following the same procedure as Appendix A, we have: Similarly, using the [25] method of inversion, 1 Ω (B.5)  can be expressed as: where By some algebraic simplifications, we can write that: Note further that:   ence, we can write (C.1) as: As a consequ

L tr u u tr E A A u E A J A u tr E A J A u E
Using the information leading to (C.2), we can prove that: And also with the representations that: in which case,   where

28) Proposition 2 :
Based on proposition 1 and assumptions 2 and 3, we can write the derivations of

Proposition 4 : 2 v
Following propositions 1 -3, we can easily calculate the partial derivates L , respectively, evaluated at the restricted MLE: for crosssectional dependence assuming the presence of random individual effect only (while ignoring the presence of time effects