Joint LM Test for Homoskedasticity in a One-Way Error Component Model

This paper considers a general heteroskedastic error component model using panel data, and derives a joint LM test for homoskedasticity against the alternative of heteroskedasticity in both error components. It contrasts this joint LM test with marginal LM tests that ignore the heteroskedasticity in one of the error components. Monte Carlo results show that misleading inference can occur when using marginal rather than joint tests when heteroskedasticity is present in both components.

1 Introduction Mazodier and Trognon (1978) seem to be the …rst to deal with the problem of heteroskedasticity in panel data. Since then, several papers have followed, see Rao, Kaplan and Cochran (1981), Magnus (1982), Baltagi (1988), Baltagi and Gri¢ n (1988), Randolph (1988), Wansbeek (1989), Li and Stengos (1994), Lejeune (1996), Holly and Gardiol (2000), Roy (2002) and Phillips (2003). These papers usually consider a regression model with one-way error component disturbances: u it = i + v it , i = 1; :::; N , t = 1; :::; T , where the index i refers to the N individuals and the index t to the T time series observations. Both Mazodier and Trognon (1978) and Baltagi and Gri¢ n (1988) were concerned with the estimation of a model allowing for heteroskedasticity on the individual-speci…c error term, i.e., assuming that i 0; 2 i while v it IID (0; 2 v ) : In contrast, Rao, Kaplan and Cochran (1981), Magnus (1982), Baltagi (1988) and Wansbeek (1989) adopted a symmetrically opposite speci…cation allowing for heteroskedasticity on the remainder error term, i.e., assuming that i IID 0; 2 while v it 0; 2 v i : Randolph (1988) allowed for a more general heteroskedastic error component model assuming that both the individual and remainder error terms were heteroskedastic, i.e., i 0; 2 i and v it 0; 2 v it , with the latter varying with every observation over time and individuals. As Lejeune (1996) pointed out, if heteroskedasticity is say due to di¤erences in size across individuals, …rms or countries, then both error components are expected to be heteroskedastic and it may be di¢ cult to argue that only one component of the error term is heteroskedastic but not the other. Early on, Verbon (1980) derived a Lagrange multiplier test where the null hypothesis is that of a standard normally distributed homoskedastic model against the heteroskedastic alternative i 0; 2 i and v it 0; 2 v i : In Verbon's model, however, 2 i and 2 v i are, up to a multiplicative constant, identical parametric functions of a vector of time invariant exogenous variables z 0 i , i.e., 2 i = 2 f (z 0 i 2 ) and Lejeune (1996) on the other hand, dealt with maximum likelihood estimation and Lagrange multiplier testing of a general heteroskedastic one-way error components regression model assuming that i 0; 2 i and v it 0; 2 v it where 2 i and 2 v it are distinct parametric functions of exogenous variables z 0 it and f 0 i , i.e., 2 v it = 2 v h v (z 0 it 1 ) and 2 i = 2 h (f 0 i 2 ). In the context of incomplete panels, Lejeune (1996) derived two joint LM tests for no individual e¤ects and homoskedasticity in the remainder error term. The …rst LM test considers a random e¤ects one-way error component model with i IIN 0; 2 and a remainder error term that is het- The joint hypothesis 1 = 2 = 0 renders OLS the restricted MLE. Lejeune argued that there is no need to consider a variance function for i since one is testing 2 equal to zero. While the computation of the LM test statistic is simpli…ed under this assumption, i.e., i IIN 0; 2 ; this is not in the original spirit of Lejeune's ML estimation where both i and v it have general variance functions. Lejeune's second LM test considers a …xed e¤ects one-way error component model where i is a …xed parameter to be estimated and the remainder error The joint hypothesis is i = 1 = 0 for all i = 1; 2; ::; N: This renders OLS the restricted MLE.
With regards to estimation, Li and Stengos (1994) suggested an adaptive estimation procedure for an error component model allowing for heteroskedasticity of unknown form on the remainder error term, i.e., assuming Li and Stengos (1994) also suggested a robust version of the Breusch and Pagan (1980) LM test for no random individual e¤ects 2 = 0 by allowing for adaptive heteroskedasticity of unknown form on the remainder error term. Holly and Gardiol (2000) proposed a Rao score test for homoskedasticity assuming the existence of individual e¤ects. The unrestricted model assumes that i N 0; 2 i and v it IIN (0; 2 v ) where 2 i is a parametric function of exogenous variables f 0 i , i.e., 2 i = 2 h (f 0 i 2 ): Under the null hypothesis 2 = 0, with h (0) = 1 and the restricted model reverts to the homoskedastic one-way error component model. Roy (2002) dealt with adaptive estimation of an error component model assuming heteroskedasticity of unknown form for the individual-speci…c error term, i.e., assuming that i 0; 2 cently, Phillips (2003) followed Mazodier and Trognon (1978) in considering a one-way strati…ed error component model. As unobserved heterogeneity occurs through individual-speci…c variances changing across strata, Phillips provided an EM algorithm for estimating this model and suggested a bootstrap test for identifying the number of strata.
In the spirit of the general heteroskedastic model of Randolph (1988) and Lejeune (1996), this paper derives a joint Lagrange multiplier test for homoskedasticity, i.e., 1 = 2 = 0. Under the null hypothesis, the model is a homoskedastic one-way error component regression model and is estimated by restricted MLE. Note that this is di¤erent from Lejeune (1996), where under the null, 2 = 0, so that the restricted MLE is OLS and not MLE on a one-way homoskedastic error component model. Allowing for 2 > 0 is more likely to be the case in panel data where heterogeneity across the individuals is likely to be present even if heteroskedasticity is not. The model under the null is exactly that of Holly and Gardiol (2000) but it is more general under the alternative since it does not assume a homoskedastic remainder error term. We also derive an LM test for the null hypothesis of homoskedasticity of the individual random e¤ects assuming homoskedasticity of the remainder error term, i.e., 2 = 0 j 1 = 0. Not surprisingly, we get the Holly and Gardiol (2000) LM test. In addition, we derive an LM test for the null hypothesis of homoskedasticity of the remainder error term assuming homoskedasticity of the individual e¤ects, i.e., 1 = 0 j 2 = 0: The rest of the paper is organized as follows: Section 2 reviews the general heteroskedastic one-way error component model. Section 3 derives the marginal and joint Lagrange multiplier tests described above. Section 4 performs Monte Carlo simulations comparing the size and power of these LM tests. Section 5 concludes.
2 The general heteroskedastic one-way error component model We consider the following regression model: where y it , u it , i and v it are scalars, x 0 it is a (1 k ) vector of strictly exogenous regressors (the …rst element being a constant) and is a (k 1) vector of parameters. The index i refers to the N individuals and the index t to the T observations of each individual i. The total number of observations is N T . The error terms i and v it are assumed mutually independent and normally distributed according to: where h v (:) and h (:) are arbitrary non-indexed (strictly) positive twice continuously di¤erentiable functions 1 satisfying h v (:) > 0; h (::: denotes the …rst derivative of h (x) with respect to x: z 0 it and f 0 i are respectively (1 k 1 ) and (1 k 2 ) vectors of strictly exogenous regressors while 1 and 2 are respectively (k 1 1) and (k 2 1) vectors of parameters 2 . We will denote by ' = 2 v ; 2 ; 0 1 ; 0 2 0 ' p 0 ; p = 1; :::; 4 the vector of variance-speci…c parameters.
Staking the T observations of each individual i, (1) may be written as: where T is a (T 1) vector of ones, y i , u i and v i are (T 1) vectors and X i a (T k ) matrix of regressors. From (2), the (T T ) covariance matrix i of u i may be written as: where J T = T 0 T and Z i is a (T k 1 ) matrix of regressors with a typical row being z 0 it . Here, diag (h v (Z i 1 )) denotes a diagonal (T T ) matrix with its t-th diagonal element being the t-th element of the (T 1) vector h v (Z i 1 ) : Finally, stacking again the above vectors and matrices, we obtain the general matrix form of the model: y, u and v are (N T 1) vectors, is the (N 1) vector of individual e¤ects X; Z and F are respectively (N T k ), (N T k 1 ) and (N k 2 ) matrix of regressors and is the (N T N T ) block-diagonal covariance matrix of u. Here, diag (h v (Z 1 )) denotes a diagonal (N T N T ) matrix with its it-th diagonal element being the it-th element of the (N T 1) vector h v (Z 1 ) : Similarly, diag (h (F 2 )) denotes a diagonal (N N ) matrix with its i-th diagonal element being the i-th element of the (N 1) vector h (F 2 ) :

The marginal and joint Lagrange Multiplier tests
Based on equations (3)-(4), the log-likelihood function L may be written as: Following Magnus (1978) and Lejeune (1996), the information matrix is given by N T = E 0 [HjX; Z; F ] with E 0 [:] denoting expectation taken with respect to the true distribution. H is the hessian matrix with elements H = @ 2 L @ @ 0 ; The information matrix N T is block diagonal between the and ' parameters. Therefore, the Lagrange Multiplier (LM) statistic for testing, H 0 : ' = 0; may be written as: (see Breusch and Pagan (1980) p. 245) where g ' is the gradient of the loglikelihood with respect to ', N T '' is the ' block of the information matrix and e denotes quantities evaluated under the null. Under the null, this statistic is asymptotically distributed as a 2 with k ' degrees of freedom, k ' being the number of parameters in the vector ': 3.1 Marginal LM test for 2 = 0 assuming 1 = 0 For the Mazodier and Trognon (1978), Baltagi and Gri¢ n (1988) and Roy (2002) papers, the remainder error term is assumed to be homoskedastic, i.e., 1 is assumed to be zero and the covariance matrix i in (4) becomes: Testing for homoskedasticity in this model amounts to testing: the LM statistic for H a 0 is given by 3 : with e 2 1 = T e 2 + e 2 v where e 2 and e 2 v are the restricted ML estimates of 2 and 2 v , The e u i 's are vectors of restricted ML residuals obtained from a one-way error component model with no heteroskedasticity. These can be obtained from standard regression packages, for e.g., using xtreg in Stata with the mle option. F is the matrix of regressors (N k 2 ) with typical row f 0 i : Under the null H a 0 , this statistic is asymptotically distributed as 2 with (k 2 ) degrees of freedom. This statistic is exactly identical to the Rao score statistic proposed by Holly and Gardiol (2000). The LM statistic (9) is simply one half the explained sum of squares (ESS) from the arti…cial least squares regression of 1 e 2 1 S N on F : 3 Technical appendices giving the derivation of all LM tests considered in this paper are not reproduced here to save space and can be obtained upon request from the authors.

Marginal LM test for
For the Rao, Kaplan and Cochran (1981), Magnus (1982), Baltagi (1988), Wansbeek (1989) and Li and Stengos (1994) papers, the individual error component is assumed to be homoskedastic, i.e., 2 is assumed to be zero and the covariance matrix i in (4) becomes: Testing for homoskedasticity amounts to testing where stands for the Hadamard product, i.e., an element-by-element multiplication. Z is the matrix of (N T k 1 ) regressors. Under the null, the covariance matrix is: e = e 2 1 P + e 2 v Q where P = I N J T and Q = I N E T , see Baltagi (2005). P and Q are the Between and Within transformations and E T = I T J T while J T = J T =T . Under the null H b 0 , this statistic is asymptotically distributed as 2 with (k 1 ) degrees of freedom. The LM statistic (10)  As a special case of this heteroskedasticity speci…cation, one may only want the variance of v it to vary with i = 1; ::: vector of strictly exogenous regressors. So, the covariance matrix i becomes: Testing for homoskedasticity amounts to testing H b 0 ; but the alternative is now di¤erent. For this reason we call the null H b 0 0 to distinguish this special case from the more general alternative in H b 0 : The corresponding LM statistic is given by:

Joint LM test
For the general heteroskedastic one-way error component model described by equations (3)-(4), testing for homoskedasticity in this model amounts to jointly testing: Under the null H c 0 , the variance-covariance matrix of u i is given by: The corresponding LM statistic is given by: where G, Z, M; F and S were de…ned above. Also, Under the null H c 0 , this statistic is asymptotically distributed as 2 with (k 1 + k 2 ) degrees of freedom.
The expression (12) may be computationally cumbersome and the associated arti…cial regression is not easy to obtain 5 . But, if one ignores the last term in the expression for e , one gets and, as the two middle terms of (12) tend to zero, as T gets large, the joint LM test can be approximated by: i.e., the joint LM test can be approximated by the sum of the two marginal LM tests LM 1 =0 and LM 2 =0 . So, the joint LM test (12) is approximatively 6 the sum of one half the explained sum of squares from the arti…cial least squares regressions of 5 We cannot derive arti…cial regressions a la MacKinnon (1990, 2003) since we cannot derive the jacobian of the variance-covariance matrix i when 2 vit depends on t. In that case, and as already shown by Mazodier and Trognon (1978, p. 456), both characteristic roots and characteristic vectors of i do depend on the variances (the 2 's and the 2 vit 's) which are unknown. Moreover, this dependance is generally intricate, so that it would not help much in that respect if we knew 2 and 2 vit .

Monte Carlo results
The design of our Monte Carlo experiments follows closely that of Li and Stengos (1994) and Roy (2002) for panel data, which in turn adapted it from Rilstone (1991) and Delgado (1992) for cross-section data. Consider the following simple regression model: where where x i: is the individual mean of x it : Denoting the expected variance of i by 2 i and following Roy (2002), we …x the expected total variance 2 = 2 i + 2 v = 8 to make it comparable across the di¤erent data generating processes. We let 2 v take the values 2 and 6. For each …xed value of 2 v ; is assigned values 0, 1, 2 and 3 with = 0 denoting the homoskedastic individual speci…c error. For a …xed value of 2 v ; we obtain a value of 2 i = (8 2 v ) and using a speci…c value of ; we get the corresponding value for 2 from (16). Of course, we can choose a quadratic or an exponential heteroskedastic speci…cation for : For case 1, the appropriate hypothesis to test is H a 0 : 2 = 0 j 2 > 0; 2 v > 0; 1 = 0: Case 2: For the Li and Stengos (1994) set up, we generate i as i:i:d N 0; 2 and v it N 0; 2 v it where 8 < Denoting the expected variance of v it by 2 v it and following Li and Stengos (1994), we set the expected total variance 2 = 2 + 2 v it = 8 to make it comparable across the di¤erent data generating processes. We let 2 take the values 2 and 6. For each …xed value of 2 ; v is assigned values 0, 1, 2 and 3 with v = 0 denoting the homoskedastic remainder error term. For a …xed value of 2 ; we obtain a value of 2 v it = (8 2 ) and using a speci…c value of v ; we get the corresponding value for 2 v from (17). Again, we can choose a quadratic or an exponential heteroskedastic speci…cation for Case 3: For the special case where the variance of v it varies only with i = 1; 2; ::; N , we generate i as i:i: We let 2 take the values 2 and 6 and for each …xed value of 2 ; v is assigned values 0, 1, 2 and 3 with v = 0 denoting the homoskedastic remainder error term. For a …xed value of 2 ; we obtain a value of 2 v i = (8 2 ) and using a speci…c value of v ; we get the corresponding value for 2 v from (18). For case 3, the appropriate hypothesis to test is and we set the expected total variance 2 = 2 i + 2 v it = 8 to make it comparable across the di¤erent data generating processes. We let 2 i take the values 2 and 6. For each …xed value of 2 i ; is assigned values 0, 1, 2 and 3 with = 0 denoting the homoskedastic individual speci…c error term. For a …xed value of 2 i ; we obtain a value of 2 v it = (8 2 i ) and using a speci…c value of ; we get the corresponding value for 2 from (19). Second, for each …xed value of 2 i , we obtain the corresponding value for 2 v it . v is assigned values 0, 1, 2 and 3, with v = 0 denoting the homoskedastic remainder error term. For a speci…c value of v ; we get the corresponding value for 2 v from (19). For case 4, the appropriate hypothesis to test is H c 0 : 1 = 0 and 2 = 0 j 2 > 0; 2 v > 0.
For each replication, we compute the restricted one-way error component MLE. Using the e u's, i.e., the vectors of restricted ML residuals and e 2 and e 2 v which are the corresponding restricted ML estimates of 2 and 2 v , we compute the marginal and joint LM tests derived above for H a 0 , H b 0 , H b 0 0 and H c 0 . For each experiment, 5000 replications are performed and we obtain the empirical size for each test at the 5% level.
[Put Table 1] Table 1 reports the percentage of rejections of the null hypothesis based on nominal critical values in 5000 replications at the 5% signi…cance level. We …rst look at case 1, for (N; T ) = (50,5) and (E . This is the set up of Roy (2002) where homoskedasticity is assumed on the remainder error term, i.e., 1 = 0. The parameter ( ) determines the degree of heteroskedasticity on the random individual speci…c e¤ect i . When we test the null hypothesis H a 0 : 2 = 0 j 2 > 0; 2 v > 0; 1 = 0, we use the the marginal LM test de…ned in (9) and also derived by Holly and Gardiol (2000). When there is no heteroskedasticity, i.e., = 0, we get 2 = 0, and the null H a 0 is true. Table 1 shows that the empirical size of this test is not signi…cantly di¤erent from 5%. However, the power of this test is poor especially for small and the quadratic form of heteroskedasticity. For example, for a low degree of heteroskedasticity ( = 1), the power is 24:1% for the quadratic type of heteroskedasticity and 33:6% for the exponential type of heteroskedasticity. This power increases with the degree of heteroskedasticity and for = 3; it yields 41% rejection of the null of homoskedasticity for the quadratic form and 98:4% for the exponential form of heteroskedasticity. The power for the exponential form of heteroskedasticity is better than that of the quadratic case. Doubling the size of T from 5 to 10, does not help in improving the power of this test 7 . However, increasing N from 50 to 200 improves the power drastically, see Fig. 1. For case 1, H a 0 is the right test to perform, while testing for H b 0 or H b 0 0 is checking for the wrong form of heteroskedasticity on the wrong error component. As expected, the marginal tests derived in (10) and (11) yield a rejection rate for the null between 4:6% and 5:6% for H b 0 and for H b 0 0 : In other words, the power of these tests is close to the nominal size of 5%: If we interpret these results as not rejecting a homoskedastic error component model, we reach the wrong conclusion whenever is di¤erent from zero. If we perform the joint LM test de…ned in (12) for H c 0 : 1 = 0 and 2 = 0 j 2 > 0; 2 v > 0 under case 1, the size is not signi…cantly di¤erent from 5% and the power is a little lower than that of the right marginal test for H a 0 de…ned in (9). Note that, with the joint LM test, we are overtesting in case 1 and still performing a close second to the preferred marginal LM test. For example, when = 3; the joint LM test yield a rejection of the null in 31:8% for the quadratic form of heteroskedasticity and 96% for the exponential form of heteroskedasticity. These are below the 41% and 98:4% rejection rates reported above for the marginal test de…ned in (9).
For case 2, for (N; T ) = (50,5) and E ; this is the set up of Li and Stengos (1994) where homoskedasticity is assumed on the individual error term, i.e., 2 = 0. The parameter ( v ) determines the degree of heteroskedasticity on the remainder error term v it . When we test the null hypothesis H b 0 : 1 = 0 j 2 > 0; 2 v > 0; 2 = 0, we use the the marginal LM test de…ned in (10). When there is no heteroskedasticity, i.e., v = 0, we get 1 = 0, and the null H b 0 is true. Table 1 shows that the empirical size of this test is not signi…cantly di¤erent from 5%. Also, the power of this test is excellent, yielding 99:4 to 100% rejection rates for v 6 = 0 for the quadratic form as well as the exponential form of heteroskedasticity. For case 2, testing H b 0 is the right hypothesis test to perform, while testing for H a 0 is checking for the wrong form of heteroskedasticity on the wrong error component. As expected, the marginal test derived in (9) yields a rejection rate for the null between 4:1% and 4:7% for H a 0 under the quadratic form of heteroskedasticity and between 4:3% and 6:1% under the exponential form of heteroskedasticity. In other words, the power of this test is close to the nominal size of 5%, except when 2 = 2 and v is larger than 2 for the exponential form of heteroskedasticity. If we interpret these results as not rejecting a homoskedastic error component model, we reach the wrong conclusion whenever v is di¤erent from zero. If we perform the joint LM test for H c 0 : 1 = 0 and 2 = 0 j 2 > 0; 2 v > 0 under case 2, the size is not sig-ni…cantly di¤erent from 5% and the power is above 98:7%. Note that, with the joint LM test, we are overtesting in case 2 and still performing about the same as the preferred test (10). For case 2, the Monte Carlo design is more in favor of the test in (10) for H b 0 than the test in (11) for H b 0 0 . The latter test is still testing for heteroskedasticity in the remainder error form but misspeci…es that it does not vary over time but only over individuals. This test has empirical size that is not signi…cantly di¤erent from 5%. Its power is good exceeding 78:8% for the quadratic form of heteroskedasticity and 90:3% for the exponential form of heteroskedasticity.
For case 3, for (N; T ) = (50,5) and E . This is the set up in line with the heteroskedastic form of Baltagi (1988) and Wansbeek (1989) where homoskedasticity is assumed on the individual error term, i.e., 2 = 0 just like case 2. However, unlike case 2, the parameter ( v ) determines the degree of heteroskedasticity on the remainder error term v it which now varies only over i and not over t. In this case, v it N 0; 2 v i . The design is in ‡avor of testing the null hypothesis H b 0 0 with the LM test de…ned in (11). Table 1 shows that the empirical size of this test is 5:6% for T = 5 and 5:2% for T = 10, so it is close to its nominal size (5%). However, its power is good exceeding 83% for the quadratic form of heteroskedasticity and 94% for the exponential form of heteroskedasticity. For case 3, testing for H a 0 is checking for the wrong form of heteroskedasticity on the wrong error component. As expected, the marginal test derived in (9) yields a rejection rate for the null between 4:3% and 5:4% for H a 0 under the quadratic form of heteroskedasticity and between 4:3% and 7% under the exponential form of heteroskedasticity. In other words, the power of this test is close to the nominal size of 5%, except when 2 = 2 and v is larger than 1. If we interpret these results as not rejecting a homoskedastic error component model, we reach the wrong conclusion whenever v is di¤erent from zero. If we perform the joint LM test for H c 0 : 1 = 0 and 2 = 0 j 2 > 0; 2 v > 0 under case 3, the size is not signi…cantly di¤erent from 5%. However, the power is poor for the quadratic form of heteroskedasticity varying between 15% and 57%. The power is much higher for the exponential form of heteroskedasticity varying between 20% and 99:9%. Note that, with the joint LM test, we are overtesting in case 3 and not performing as well as the preferred marginal test (11). For case 3, the Monte Carlo design is more in favor of the test in (11) for H b 0 0 than the test in (10) for H b 0 . The latter test is still testing for heteroskedasticity in the remainder error form, but misspeci…es its form, assuming that it varies over time and individuals when it only varies over individuals. This test has empirical size not signi…cantly di¤erent from the 5% level. Its power is low for the quadratic form of heteroskedasticity varying between 19:8% and 68:4%, while it is better for the exponential form of heteroskedasticity varying between 26:4% and 99:9%. Table 1 also reports the percentage of rejections of the null hypothesis in 5000 replications at the 5% signi…cance level for case 4. This is the general heteroskedastic set up of Lejeune (1996). The design is in favor of testing the null hypothesis H c 0 : 1 = 0 and 2 = 0 j 2 > 0; 2 v > 0 with the joint LM test de…ned by (12). For (N; T ) = (50,5) and (E , the empirical size of this test is between 4:3% and 5:1%; so it is not signi…cantly di¤erent from the 5% level. Its power is excellent as long as v is di¤erent from zero. In fact, the power for v > 0 always exceeds 98:7%. Low power occurs for v = 0 and 6 = 0: This power can be as low as 18% for v = 0 and = 1 for the quadratic form of heteroskedasticity and 25:9% for the exponential form of heteroskedasticity. In general, the test has more power under the exponential form of heteroskedasticity rather than the quadratic form. The marginal LM tests are designed to test for heteroskedasticity in one error component assuming homoskedasticity on the other error component. Obviously, the design for case 4 is not in the marginal LM tests favor if the other error component is heteroskedastic and this shows in Table  1. When = 0, but v 6 = 0, the marginal test for H a 0 given by (9) yield power between 4:1% and 5:1% for the quadratic form of heteroskedasticity and between 4:3% and 6:1% for the exponential form of heteroskedasticity. When v = 0, but 6 = 0, the marginal test for H b 0 given by (10) and H b 0 0 given by (11) yield power between 4:5% and 5:1% for the quadratic form of heteroskedasticity and for the exponential form of heteroskedasticity.
[Put Fig. 1] Fig. 1 replicates Table 1 for T = 5 and N = 200. Larger N is more likely encountered in micro-panels. As clear from Fig. 1, the power of these tests improves drastically as we increase N . However, it is still the case that performing the wrong test for heteroskedasticity yields misleading results.

Conclusion
For the random error component model popular in panel data applications, the researcher does not know whether heteroskedasticity is absent from both error components or whether it is in one or both components. This paper derived a joint as well as marginal LM tests for homoskedasticity against heteroskedasticity in one or both error components. Monte Carlo experiments show that this joint LM test performs well when both error components are heteroskedastic, and it performs second best when one of the components is homoskedastic while the other is not. In contrast, the marginal LM tests perform best when heteroskedasticity is present in the right error component.