Pooling

Journal

of Econometrics

17 (1981) 21-49.

North-Holland

Publishing

Company

POOLING An Experimental

Study of Alternative Testing and Estimation Two-Way Error Component Model

Procedures in a

Badi H. BALTAGI* University

Received

of Houston, Houston, TX 77004, USA

March

1981, final version

received June 1981

This paper considers a two-way error component model with no lagged dependent variable and investigates the performance of various testing and estimation procedures applied to this model by means of Monte Carlo experiments. The following results were found: (1) The Chow-test performed poorly in testing the stability of cross-section regressions over time and in testing the stability of time-series regressions across regions. (2) The Roy-Zellner test performed well and is recommended for testing the poolability of the data. (3) The Hausman specification test, employed to test the orthogonality assumption, gave a low frequency of Type I errors. (4) The Lagrange multiplier test, employed to test for zero variance components, did well except in cases where it was badly needed. (5) The problem of negative estimates of the variance components was found to be more serious in the two-way model than in the one-way model. However, replacing the negative variance estimates by zero did not have a serious effect on the performance of the second-round GLS estimates of the regression coefficients. (6) As in the oneway model, all the two-stage estimation methods performed reasonably well. (7) Better estimates of the variance components did not necessarily lead to better second-round GLS estimates of the regression coefficients.

1. Introduction Economists have used an error component model to pool time-series of cross-section data; see Balestra and Nerlove (1966), Kuh (1959), Hoch (1962) and Mundlak (1963). The econometric theory was developed for a two-way layout with both time-period and cross-section effects; see Wallace and Hussain (1969), Amemiya (1971), Swamy (1971; 1974), Rao (1972), Maddala (1971), Nerlove (1971b), Swamy and Arora (1972), Swamy and Mehta (1973), *The author acknowledges with appreciation the financial support received from the Research Initiation Grant Program at the University of Houston. I would like to thank Jay Ferry and Angela Hsu for their able research assistance, also Omar Ashur for his helpful programming assistance. The author would also like to thank an anonymous referee for his valuable comments and suggestions. Any errors remaining are solely my own.

0165-7410/81/0000-0000/$02.75

0

1981 North-Holland

22

B.H. Baltagi, A two-way error component

model

and Fuller and Battese (1974), to mention only a few. Analytical studies were performed by Swamy (1971) and Swamy and Arora (1972). However, the Monte Carlo studies treated only the one-way layout and ignored the individual invariant time-period effects; see Nerlove (1967, 1971a), Arora (1973) and Maddala and Mount (1973). This paper is a Monte Carlo study aimed at remedying this situation. What distinguishes this study from previous ones is that (i) it studies a two-way model rather than a one-way model, (ii) it studies testing procedures as well as estimation procedures, (iii) it provides mean square error (MSE) tables for the variance components estimates,’ and (iv) it adds a new dimension to the problem of negative estimates of the variance components and this is contrasted with the results obtained for the one-way model. In the next section, we describe the model and the estimation methods to be compared. Section 3 gives the design of the experiment, and section 4 studies the different testing procedures. Section 5 considers the problem of negative estimates of the variance components. Section 6 considers the performance of the different estimation methods, and the last section provides a summary of the main results. 2. The model Consider

a balanced Yi, = @+

two-way

model with a single covariate

Pxi,+ uit>

(2.1) i=l,...,

with Ui*= pi + it + Vif,

N,

t=l,...,

T, (2.2)

where pi is an individual time-invariant cross-sectional effect, 1, is a period individual-invariant effect, and vit is a remainder peculiar to a particular cross-section time-period combination. ,LQ,R,, and vit are random variables having zero means, independent among themselves and with each other, with variances I$, a,2, and r~:, respectively. In vector form this can be written as y=ae,,+x/?+u=Zy+u,

(2.3)

where Z = [eNT x], y = (~1,b)‘, and U=Zpp+Z~~b+V,

(2.4)

‘These tables provide evidence on the relationship between better estimates of the variance components and better second round estimates of the regression coeffkients. This relationship is studied in section 6.

23

B.H. Baltagi, A two-way error component model

where Z,=I,@e,, Z,=e,@l,, I, and I, are identity matrices of order N and T, respectively. eN and er are vectors of ones of order N and T, respectively. $ = (pI, pLz,. . ., p,,), A’= (A,, A,, ., A,), and v’= (v, 1, v12,. . ., vErT). SZ=E(uu’)=a2[pA+oB+(1-p-o)I,,],

(2.5)

where A=I,Oe,e;. and B=e,eh@lT. w = of/C?. The Aitken estimator of p is

c2=$+oj+oz,

Km-‘(8,B,,+O,C,,+

&IS=WL+~,~,,+

p=o,2ja2,

Wxy),

and

(2.6)

where

B,,=Tx(xi.

-X..)‘,

C,,=N~(.f.,-x..)2,

I

w,,=c

c i

t

(Xi, -xi.-x.t

TX,= c 1 (xit -x i

+x..)2,

t

.)2,

f

are the familiar analysis of variance expressions for the between groups, between time-periods, within groups and total sums of squares. Similar expressions can be derived for y, and the cross-product terms between x and y. Also, Xi.=Cxir/T,

X.,=Cx,/N,

The estimation (1) Ordinary

methods

Least Squares

X..=CCX~~/NT.

L

t

considered

i

f

are:

(OLS) (2.7)

(2) Least Squares

with Dummy

Variables

(LSDk’)

(2.8) (3) GLS using the true variance

components

given in (2.6).

24

B.H. Baltagi, A two-way error component

model

(4) Six two-stage GLS methods where tJ1 and 8, are estimated in the first step and to,, is computed in the second step using the estimated 0’s. These methods are named WALHUS, AMEMIYA, SWAR, MINQUE, FUBA, and NERLOVE and were developed by Wallace and Hussain (1969), Amemiya (1971), Swamy and Arora (1972), Rao (1972), Fuller and Battese (1974), and Nerlove (1971a), respectively. WALHUS and AMEMIYA are analyses of variance type estimators of the variance components. The WALHUS method uses OLS residuals, while the AMEMIYA method uses LSDV residuals in place of the true disturbances. SWAR estimates the variance components using mean squares residuals of the between groups, between time periods and within groups regressions. MINQUE constructs minimum norm quadratic unbiased estimators of the variance components. FUBA is a Henderson Method III type estimator of the variance components. NERLOVE estimates C: as the sum of squares of the LSDV residuals divided by the number of observations, while 0,” and 0: are estimated as the sample variances of the estimated coefficients of the dummy variables with no correction for the degrees of freedom. To review some of the properties of these estimators, OLS is unbiased, but asymptotically inefficient, while LSDV is unbiased and asymptotically efficient. True GLS is BLUE, but the variance components are usually not known and have to be estimated. All of the feasible GLS estimators considered are asymptotically efficient except for WALHUS whose asymptotic properties depend on the relative speed of increase of N and T . ’ Moreover, Swamy and Arora (1972) proved the existence of a family of asymptotically efficient estimators of the regression coefficients. Therefore, based on asymptotics only, we cannot differentiate among these two-stage GLS estimators.3 This leaves undecided the question of which estimator is the best to use.

Three criteria are used by economists to determine which estimator to use; these are the following: (1) The simplicity of derivation and practicality of the estimator. According to this criterion, for example, LSDV would rank among the easiest to compute, while MINQUE would rank among the most difficult to compute. Early in the literature, Wallace and Hussain (1969) recommended the LSDV estimator for the practical researcher, based on theoretical considerations but more importantly for its ease of computation. The LSDV estimator is easily obtainable, and a two-stage estimation procedure is not required. The LSDV

*See Amemiya (1971) for a derivation of the asymptotic properties of the WALHUS estimator. 3Fuller and Battese (1974) derived sufficient conditions for a two-stage GLS estimator to have the same asymptotic distribution as that of true GLS.

B.H. Baltagi,

A two-way error component

model

25

estimator is unbiased. This is true whether or not prior information about the variance components is available. It is also asymptotically equivalent to the Aitken estimator in case of weakly non-stochastic exogenous variables. In Wallace and Hussain’s (1969, p. 66) words the ‘. . .covariance estimators come off with a surprisingly clear bill of health’. (2) Finite sample properties. Important analytical results were obtained by Swamy (1971) and Swamy and Arora (1972). These studies derived the relative efficiencies of (i) SWAR with respect to OLS, (ii) SWAR with respect to LSDV, and (iii) LSDV with respect to OLS. Then, for various values of N, T, the variance components and between groups, between time periods and within groups sums of squares of the independent variable, they, tabulated these relative efficiency values, see Swamy (1971, chs. II and III) and Swamy and Arora (1972, p. 272). Among their basic findings is the fact that, for small samples, SWAR is less efficient than OLS if ct and cz are small. Also, SWAR is less efficient than LSDV if 0,’ and 0: are large. The latter result was disconcerting, since LSDV which used only a part of the available data was more efficient than SWAR, a feasible Aitken estimator, which used all of the available data. Swamy and Mehta (1979) offered a modified estimator which would make the feasible Aitken estimator better than LSDV in cases that appear in practice. Given a priori information about the relative magnitudes of the variance components, Swamy and Mehta (1979, p. 10) offer a prescription for choosing an estimator of the regression coefficients from the a-class estimators of Swamy (1971, p. 361). Another finite sample study was done on the one-way model by Taylor (1980) showing that, for all but the fewest degrees of freedom, feasible GLS is always better than LSDV and its variance is never more than seventeen percent above the Cramer-Rao lower bound. Also, Taylor (1980, p. 203) showed that ‘. . . more efficient estimators for the variance-components do not necessarily yield more efficient feasible Gauss-Markov estimators’. This conclusion is in fact supported by the experiment results in section 6. (3) Monte Carlo studies. The first two Monte Carlo studies were performed by Nerlove. Nerlove (1967) considered a one-way model with a lagged dependent variable but no exogenous variables, while Nerlove (1971a) considered the one-way model with both exogenous and a lagged dependent variable. Among Nerlove’s basic findings was the following: Based on the minimum relative MSE criterion, the NERLOVE method compared favorably with OLS, LSDV and the maximum likelihood estimator (MLE). Another striking feature of this study was the poor performance of MLE. Utilizing Nerlove’s (1971a) model, Maddala (1971) confirmed some of Nerlove’s findings. In case a lagged dependent variable is present in the model, Maddala (1971) found that (i) MLE has its drawbacks, giving boundary solutions more often than in the case of purely exogenous variables, and (ii) the analysis of variance technique for estimating the

26

B.H. Baltagi, A two-way error component model

variance components cannot be relied upon since the between group and the within group mean squares are badly biased in this case. Maddala (1971) suggested NERLOVE’s estimator but warned that this method is biased towards LSDV. Arora (1973) considered a balanced one-way model with an intercept, a single covariate, and no lagged dependent variables. The estimation methods considered were OLS, LSDV, the between-groups estimator, SWAR, and true GLS. Based on both the minimum MSE and the unbiasedness criteria, the SWAR estimator, in the one-way context, compared favorably with all of the other estimators considered. Finally, Maddala and Mount (1973) considered the same model as Arora (1973) but without an intercept. The estimators compared were OLS, LSDV, the between-groups estimator, NERLOVE, AMEMIYA, WALHUS, MLE, SWAR, MINQUE, Henderson’s Method III, and true GLS. Among their findings are the following: (i) Besides being complicated to derive and difficult to compute, the MINQUE estimator of the slope coefficient, as far as the MSE criterion is concerned, performed no better than any of the twostage estimators described above. (ii) The problem of negative estimates of the variance components was found to occur in the same cases where OLS and the true GLS estimators of /I had very close MSE. (iii) The MLE as well as all the two-stage estimators of the slope coefficient performed equally well so that Maddala and Mount (1973, p. 324) concluded that ‘. . . in models with no lagged dependent variables there is nothing much to choose among these estimators’. Hence, Maddala and Mount (1973, p. 328) suggested ‘. . . that one should estimate the model by a couple of alternative methods say Maximum Likelihood and the Analysis of Variance procedures. If the estimates obtained by these different methods vary widely this should be taken as an indication that something is wrong with the specification of the model.’ A statistical test of significance based on this idea is given by Hausman (1978).4 Having summarized some of the main results in the literature, we now define the objectives of this study. This paper asks the following questions: (1) Do the previous results hold under the two-way model where now between time-periods, between cross-sections, and within group variations are present? (2) To what extent is the problem of negative variances important in a twoway setting? (3) How reliable are the tests performed on the error component model, and which of these tests are the proper ones to perform?

4The performance

of this test is studied

in section 4.

B.H. Baltagi, A two-way error component

model

27

(4) Is there evidence in the two-way model for or against the result that better estimates of the variance components give better second round estimates of the regression coefficients? In order to answer these questions, we embark on our Monte Carlo study. First, we describe how the data was generated, then we give the results in the remaining sections.

3. Design of the experiment The exogenous variable x was generated by a similar method to that of Nerlove (1971a) and Arora (1973). For this x series, B,,=832,196, C,, =494,883 and W,,= 100,198.5 Throughout the experiment a=& /?=O.S, N = 25, T = 10 and 0’ = 20. However, p and o varied over the set (0,0.01,0.2, 0.4, 0.6, 0.8) such that (1 -p-w) is always positive. In each experiment a hundred repetitions were performed. For every repetition (NT + N + T) IIN (0,l) random numbers were generated. This first N numbers were used to generate the ,u”;s as llN(O,o~). The second T numbers were used to generate the &‘s as IIN (O,o:), and the last NT numbers were used to generate the v,‘s as IIN (0,~:). Next, the +‘s were formed as in (2.2) and the Y’s were calculated from (2.1). Twenty-six experiments were performed. Each experiment contained a hundred replications. Each replication had a sample size of two hundred and fifty observations. In all experiments, the model is assumed to be properly specified. The only type of misspecification considered is where one of the variance components is actually equal to zero, i.e., the model is actually one-way but is estimated using two-way estimation procedures.

4. Alternative tests for pooling the data Is the data poolable? This question must be answered before the data is pooled and the model estimated. The most common test performed by economists in this situation is the Chow (1960) test.6 Since most economic situations are of the type where N, the number of cross-sections, is greater than T, the number of time-periods, we will formulate the null hypothesis as follows: Consider T behavioral economic relationships with N observations

‘Maddala and Mount (1973, p 326) gave an important warning which for the two-way model can be stated as follows: If WY, is large with respect to B,, and C,,, then GLS would be equivalent to LSDV and ‘. .errors in the estimation of the variance components would not be of much consequence for estimating the slope coefficient p’. 6For an excellent discussion on the Chow test, see Fisher (1970).

28

B.H. Baltagi, A two-way error component

model

on each variable, Yt =

z&J,+ 43

t = 1,2,. . ., T,

(4.1)

where yj = (yl,, yzt,. . .,yNt), Z, = CeN x,1, xl = h, The restricted model, implied by pooling relationships with one having the same regions, i.e.,

y=zy+u,

xZr,. . ., xd

and Y;= CT,P,).

the data, replaces these economic parameters over time and across

(4.2)

as described in (2.3). The null hypothesis in this case is H, : yt =y for all t.7 Not rejecting the null hypothesis is usually an indication to the researcher that the data is poolable. Using the sample design described above, data was generated from the latter model for different values of the variance components and the Chow test was subsequently applied for every data set generated. Table 1 gives the percentage of cases for which H, was rejected, at the five percent significance level. Since H, is true by construction, this table also gives the frequency of committing a Type I error by applying the Chow test as a criterion for pooling. 8 It is clear from this table that for w=O, 0.01 (i.e., a: = 0.0.2) the Chow test rejected H, in zero to nine percent of the cases, whereas for 020.2 (i.e., a,2 24), the Chow test rejected H, in at least ninety percent of the cases. The bad performance of the Chow test for G; 24 is justified once we realize that the Chow test assumes homoskedasticity of the error term when in fact the true covariance of the disturbance terms is given by (2.5). In fact, for non-zero variance-components, the Chow test is no longer a central F-statistic, and therefore, the researcher who applies the Chow test blindly to the data is using the wrong significance values.g Note that the performance of the Chow test is more dependent on the magnitude of 0: rather than 0:. This is because the null hypothesis was formulated in terms of the stability of cross-section regressions over time. If the null hypothesis was formulated in terms of the stability of time-series regressions ‘In testing H, :yl=y against H, :yI#y, a referee points out that the tests performed in this section may be inconsistent. The reason is that a, and j., may not be identifiable and the yt’s may not be consistently estimable if N is fixed and T+co; see the Uncertainty Principle of Swamy and Tinsley (1980, p. 117). Therefore, these tests may have poor power in small samples and may accept a false hypothesis even in large samples because the model is not identifiable when the alternative hypothesis, H, : yt# y, is true. ‘Recall that the model is properly specified by construction, hence the performance of this test is studied in terms of the frequency of committing a Type I error. ‘Toyoda (1974) showed that under heteroskedasticity the Chow statistic has an approximate F-distribution where the degree of freedom of the denominator depends upon the true variances. Hence, for specific values of these variances, Toyoda demonstrated how wrong it is to apply the Chow test in case we have heteroskedasticity. Two notes along the same lines as Toyoda (1974) are given by Jayatissa (1977) and Schmidt and Sickles (1977).

B.H. Baltagi,

A two-way error component

model

Table 1 Frequency of rejections of H, in 100 sample replications H,:y,=y

w

P

0

0

Chow central F

29

Chow’s test.’

for all t Strong MSE

First weak MSE

Better bounds on the first weak MSE

Second weak MSE -

0.2 0.4 0.6 0.8

9 9 3 0 0 0

8 7 1 0 0 0

0.01

0 0.01 0.2 0.4 0.6 0.8

6 6 2 1 0 0

6 5 2 1 0 0

0.2

0 0.01 0.2 0.4 0.6

100 100 96 92 90

98 97 95 91 89

86 86 83 81 75

86 86 83 80 75

72 71 66 59 52

0.4

0 0.01 0.2 0.4

100 100 100 100

100 100 100 100

100 100 100 99

100 100 100 99

100 100 100 95

0.6

0 0.01 0.2

100 100 100

100 100 100

100 100 100

100 100 100

100 100 100

0.8

0 0.01

100 100

100 100

100 100

100 100

100 100

0.01

“For all experiments in the table a = 5, B = 0.5, c? = 20, N = 25 and T = 10.

across regions, then as shown in table 5, the performance of the Chow test will be more dependent on the magnitude of 0: rather than cr:. Next, we consider the mean square error (MS) criteria developed by Toro-Vizcarrondo and Wallace (1968) and later by Wallace (1972). These MSE criteria test whether the pooled estimator is better than the unpooled estimator when both bias and variance are taken into account. The pooled estimator is unbiased only if H, is true whereas the unpooled estimator is unbiased whether Ho is true or false. Also, the pooled estimator has a smaller variance than the unpooled estimator whether H, is true or false. This motivated Toro-Vizcarrondo and Wallace (1968, p. 560) to write: ‘If one is willing to accept some bias in trade for a reduction in variance, then

30

B.H. Baltagi, A two-way error component

model

even if the restriction is not true one might still prefer the restricted estimator.’ Three MSE criteria were developed to take advantage of this tradeoff. For a summary table describing these MSE criteria, see Wallace (1972, p. 697). As shown in table 1, these MSE criteria will have a lower percentage of committing a Type I error than the Chow test, but this percentage is still high for all cases where cj 24. The fourth column of table 1 shows the results of using the lower bound developed by Yancey, Judge and Bock (1973) for the first weak MSE criteria. For these experiments, there was little gain from obtaining these better bounds, Table 5 gives similar results for the null hypothesis H, :yi=y for all i. Again, the MSE criteria performed poorly for of 2 4.” Given the poor performance of the Chow test under the error components model, we ask what is the right test to perform for pooling the data? In case 52 is known up to a scalar factor, the answer is simple. All we need to do is transform our model by Q2-* so that the transformed disturbances will have a homoskedastic variancecovariance matrix. Then apply the Chow test to the transformed model. The resulting Chow statistic is named the Roy statistic since it was derived by Roy (1957) using a slightly different approach. It turns out, that the Roy statistic is in fact the likelihood-ratio statistic.” Table 2 shows the frequency of committing a Type I error when using the Roy statistic to test Ho. As expected, this test performs well when compared with the Chow test. McElroy (1977) extended the strong and weak MSE criteria to the case of non-spherical disturbances. These MSE criteria performed well, with at most eight occurrences of a Type I error out of a 100 replications. In case Q is not known, Zellner (1962) suggested replacing Q in the Roy test statistic by a consistent estimator, say 8. For small samples, the distribution of the resulting statistic is approximated by the F-distribution of the Roy statistic knowing the true s2. Tables 3 and 4 correspond to the Roy test using the Wallace and Hussain (1969) and the Amemiya (1971) type estimates of the variance components, respectively. The frequency of committing a Type I error when using these two tests is at most four out of a hundred replications. In essence, these tables show that if we want to pool the data, knowing that the true underlying model is an error component model, then we should apply the Roy test using a consistent estimate of the variance components rather than the Chow test. Next, we applied Hausman’s (1978) specification test in order to test whether the unobservable individual and time-period effects are uncorrelated with the exogenous variable. This is referred to as the orthogonality condition. If it is violated, the GLS estimator is biased and inconsistent

“The only exception is the second weak MSE criterion “See Zellner (1962) for a proof of this result.

which performed

well for g: =4

B.H. Baltagi,

A two-way error component

model

31

Table 2 Frequency

of H, in 100 sample replications components.”

of rejections

~

Roy’s test using the true variance

HO:Y,=Y for all t

Roy’s test

Strong MSE

First weak MSE

Better bounds on the first weak MSE

9 9 6 6 5 6

8

0.01 0.2 0.4 0.6 0.8

5 5 5 5

0 0 0

0 0 0

2 0 0

2 0 0

0.01

0 0.01 0.2 0.4 0.6 0.8

5 5 4 3 3 3

5 5 2 3 3 2

1 0 0 1 1 1

1 0 0 1 1

0.2

0 0.01 0.2 0.4 0.6

5 5 4 2 2

4 4 1 2 2

0 0 1 2 1

0 0 0 0 0

0.4

0 0.01 0.2 0.4

5 4 6 2

3 3 3 1

0 0 2 1

0 0 0 0

0.6

0 0.01 0.2

3 3 3

3 3 2

0 0 2

0 0 0

0.8

0 0.01

3 3

3 3

0 0

0 0

w -

P

0

0

“For all experiments

Second weak MSE 0 0 0 0 0 0 0 0 0 0 0 0

in the table x = 5, fl= 0.5, uz = 20, N = 25 and T= 10.

whereas the LSDV estimator remains unbiased and consistent. The basic idea of the Hausman test is to form the difference between the GLS estimator and the LSDV estimator. Under the null hypothesis this difference has a zero asymptotic variancecovariance matrix with the GLS estimator. Also, if the null hypothesis is violated the probability limit of this difference is different from zero.12 The data was generated such that it satisfies this orthogonality assumption. At the five percent significance level, the Hausman “For (1978).

more details

on the construction

of the HausmanChi-squared

statistic,

see Hausman

32

B.H. Baltagi, A two-way error component

model

Table 3 Frequency of rejections of H, in 100 sample replications - Roy’s test using Wallace and Hussain type estimates of the variance components. H,,:y,=y

for all 1

Roy’s test

Strong MSE

First weak MSE

Second weak MSE

0.01 0.2 0.4 0.6 0.8

3 3 2 1 1 2

3 3 1 1 1 2

0 0 0 0 0 0

0 0 0 0 0 0

0.01

0 0.01 0.2 0.4 0.6 0.8

3 3 2 1 1 2

3 3 1 1 1 2

0 0 0 0 0 0

0 0 0 0 0 0

0.2

0 0.01 0.2 0.4 0.6

4 4 4 4 4

4 4 4 1 3

0 0 0 1 3

0 0 0 0 0

0.4

0 0.01 0.2 0.4

4 4 4 3

4 4 3 2

0 0 2 1

0 0 0 0

0.6

0 0.01 0.2

4 4 4

4 4 2

0 0 2

0 0 0

0.8

0 0.01

4 4

4 4

0 0

0 0

0

P

0

0

“For all experiments in the table x = 5, /3= 0.5, uz = 20, N = 25 and T= 10.

test gave a low frequency of committing a Type I error (at most two out of a 100 replications) whether we used the true variance components or their estimates according to Amemiya or Wallace and Hussain.13 Hence, the results of the experiments, given in table 6 are in favor of performing the Hausman test whenever the specification of the model is in question. Finally, we applied the Lagrange Multiplier (LM) test as described in Breusch and Pagan (1980) to test the following null hypotheses: (i) H,:ci=O

and a:=O,

(ii) HB:ci=O,

(iii) Hc:ai=O.

13The perf&ance of the Hausman test should also be studied for the case where the orthogonality assumption is violated.

B.H. Baltagi,

A two-way error component

33

model

Table 4 Frequency of rejections of H, in 100 sample replications - Roy’s test using Amemiya type estimates of the variance components.” HO:?,=7

for all t

Roy’s test

Strong MSE

First weak MSE

Second weak MSE

3 3 1 1 2 2

3 1 0 0 0 1

0 0

0

0.01 0.2 0.4 0.6 0.8 0 0.01 0.2 0.4 0.6 0.8

3 3 1 1 1 2

2 2 0 0 0 1

0

0 0.01 0.2 0.4 0.6

4 4 3 4 4

4 4 3 0 3

0 0

0

1 0 3

0 0 0

0 0.01 0.2 0.4

4 4 5 3

4 4 4 2

1 0 3 1

0

0.6

0 0.01 0.2

4 4 4

4 4 0

1 0 0

0 0 0

0.8

0 0.01

4 4

4 4

0 0

0 0

0

P

0

0

0.01

0.2

0.4

0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

0 0 0

“For all experiments in the table a = 5, B = 0.5, 6’ = 20, N = 25 and T = 10.

Table 7 gives the frequency of correct decisions in 100 sample replications. For HA, the LM test performs well except for the cases where both p and o are 0.01, or one of them is 0.01 while the other is 0. For H,, the LM test performs well except for cases where p=O.Ol (or equivalently ei =0.2), but its performance improves (i.e., the frequency of Type II error declines) as o increases. Similarly, for p=O (or equivalently c,” =0), the test performs poorly (i.e., the frequency of Type I error increases) as o increases. For Hc,the LM test performs well except for w =O.Ol (or equivalently 0: =0.2), no matter what 0; is. Although the overall performance of the LM test is good, those cases where the LM test performs poorly turn out to be important cases for the problem of negative variance components.

34

B.H. Baltagi,

A two-way error component

model

Table 5 Frequency H,:y,=Y

of rejections

of H, in 100 sample replications.”

for all i

Chow’s test

Roy’s test using

True variance

P

0

0

0

Chow central

F

Strong MSE

Roy’s central

F

Strong MSE

0 0 0 0 0

0.01 0.2 0.4 0.6 0.8 0.01

Second weak MSE

components Second weak MSE

1

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

0 0.01 0.2 0.4 0.6 0.8

2 3

0.2

0 0.01 0.2 0.4 0.6

92 92 86 73 64

92 91 82 69 63

14 12 10 9 12

0.4

0 0.01 0.2 0.4

100 100 100 98

100 100 100 98

90 90 83 71

0 0 0 0

0.6

0 0.01 0.2

100 100 100

100 100 100

100 100 99

0 0

0 0.01

100 100

100 100

100 100

0.8

“For all experiments

6 6 I

0 0 0 0 0

0

5 6

in the table a = 5, fl= 0.5, 6’ = 20, N = 25 and T= 10.

0 0

B.H. Bultagi, A two-way error component

model

35

Table 5 (continued) for all i

H,:y,=y

Roy’s test using Amemiya type estimates of the variance components Strong MSE

Second weak MSE

0

1 1 1

0 0 0 0 0 0

0 0.01 0.2 0.4 0.6 0.X

1 2 1 1 1 1

0 0 0 0 0 0

0 0.01 0.2 0.4 0.6

1 2 2 2 1

0 0 0 0 0

1

1 1

0 0 0 0 0

0.4

0 0.01 0.2 0.4

1 2 2 3

0 0 0 0

2 2 2 3

2 2 2 2

0 0 0 0

0.6

0 0.01 0.2

0 0 0

2 2 2

2 2 1

0 0 0

0.8

0 0.01

0 0

2 2

2 2

0 0

P

w

0

0

0.01 0.2 0.4 0.6 0.8 0.01

0.2

Roy’s central

Wallace and Hussain type estimates of the variance components

1 1

F

1

Roy’s central

2 1 1

F

Strong MSE

Second weak MSE

0

1 0 0 0 0

0 0 0 0 0 0

1 2 1 1 1 0

0 0 0 0 0 0

2

36

B.H. Baltagi, A two-way error component model

Table 6 Frequency of rejections of H, in 100 sample replications -

Hausman’s Test.’

H, : Effects are uncorrelated with the exogenous variable ___.___True variance components

Amemiya type estimates of the variance components

Wallace and Hussain type estimates of the variance components

2 2 2 2 2 2

1 1 1 1 0 0

2 2 2 2 2 2

0 0.01 0.2 0.4 0.6 0.8

2 2 2 2

1 1 1 0 0 0

2 2 2 2 2 1

0 0.01 0.2 0.4 0.6

2 2 1 1 1

2 2 0 0

2 2 1 1 1

0.4

0 0.01 0.2 0.4

2 2 0 1

2 2 0 0

2 2 1 1

0.6

0 0.01 0.2

2 2 0

2 1 0

2 2 1

0.8

0 0.01

2 2

2 1

2 2

W

P

0

0

0.01 0.2 0.4 0.6 0.8 0.01

0.2

L

1

0

--..

.-

‘For all experiments in the table c1= 5, /3= 0.5, cz = 20, N = 25 and T = 10.

To summarize the results of this section, the Chow test as well as the MSE criteria performed badly under the error component model simply because these tests assume that the underlying disturbances are homoskedastic when in fact they are not. Roy’s test, along with the MSE criteria that assume nonspherical disturbances performed well giving a low frequency of Type I error. Hence, if pooling is contemplated, and the model to be used is an error component model, then Roy’s test rather than Chow’s test should be performed. The Hausman specification test which tests the orthogonality between the exogenous variable and the unobservable time-period and crosssection effects performed well giving a low frequency of Type I error. Last, but not least, the Lagrange Multiplier test, which tests whether the variance

B.H. Baltagi,

A two-way error component

31

model

Table 7 Frequency

of correct

decisions

in 100 sample replications

-

Lagrange’s

w

P

H,:az=O

0

0

0.01 0.2 0.4 0.6 0.8

96 (6) 96 100 100 100

(6) 98 100 100 100

0.01

0 0.01 0.2 0.4 0.6 0.8

(4) (5) 9.5 100 100 100

94 (6) 98 100 100 100

0.2

0 0.01 0.2 0.4 0.6

100 100 100 100 100

94 (5) 95 100 100

100 100 99 98 95

0.4

0 0.01 0.2 0.4

100 100 100 100

(79) (12) 92 100

100 100 100 100

0.6

0 0.01 0.2

100 100 100

(53) (32) 87

100 100 100

0.8

0 0.01

100 100

(19) (65)

100 100

*For all experiments

and crf=d

H,:o;=O

multiplier

94

Hc:a:=O

test.” -

97 97 98 100 98 84 (5) (3) (2) (1) (1) (2)

in the table a = 5, /I =0.5, uz = 20, N = 25 and T = 10.

components are zero, performed well except in those cases where one of the variance components was equal to zero or relatively small but different from zero. 5. Negative estimates of the variance components Except for Nerlove’s (1971a) method which restricts the variance components to be positive, the two-stage GLS methods considered in this study may give negative estimates of the variance components. This is not a new problem for it has been plaguing the statistical literature for many years, see Searle (1971, p. 22) for some suggested remedies to this problem. Table 8 shows the results of our experiments. It is clear that for p or co equal to 0 or 0.01 we have the problem of negative estimates of the variance components regardless of the method of estimation used. However, if p and CIItake the

38

B.H. Baltagi, A two-way error component

model

Table 8 Percentage Method

of cases with negative variance components

estimates.’

of estimation

AMEMIYA

WALHUS

SWAR

MINQUE

FUBA

0

51

0.01 0.2 0.4 0.6 0.8

50 38 38 38 38

78 69 52 43 37 30

70 64 47 47 47 47

14 64 53 57 61 67

12 67 53 53 53 53

0.01

0 0.01 0.2 0.4 0.6 0.8

49 40 22 17 14 3

73 61 31 20 15 3

65 55 32 26 18 1

71 51 32 29 24 24

67 58 36 30 23 8

0.2

0 0.01 0.2 0.4 0.6

36 20 0 0 0

54 34 0 0 0

53 37 0 0 0

56 41 0 0 0

53 38 0 0 0

0.4

0 0.01 0.2 0.4

36 15 0 0

46 2s 0 0

53 30 0 0

60 40 0 0

53 31 0 0

0.6

0 0.01 0.2

36 14 0

38 17 0

53 23 0

64 46 0

53 25 0

0.8

0 0.01

36 8

29 6

53 13

71 54

53 14

P

w 0

“For all experiments

in the table cx= 5, b = 0.5, u2 = 20, N = 25 and T = 10.

values (0.2, 0.4, 0.6, 0.8) then we have no cases with negative variance components estimates. It is also true that for a fixed p =0 or 0.01, as w increases, the number of negative estimates is non-increasing. This is true for every method in the table except MINQUE. A similar result is obtained if we fixed w at 0 or 0.01 and allowed p to increase. The percentage of cases with negative variance components reaches an alarming rate, ranging from 57% for AMEMIYA to 78% for WALHUS, when both p and o are zero. But this is precisely the case where OLS is equivalent to GLS. In fact, table 9 gives the cases where OLS had a relatively close MSE to that of true GLS. For these cases, there is no need to perform the feasible two-stage GLS methods and hence no need to worry about the problem of negative variance components estimates. This is basically the result reported in the one-way

B.H. Baltagi, A two-way error component

model

39

Table 9 Relative

MSE of OLS and SWAR

in cases with negative variance components

w

a

P

y/, of cases with negative variances

SWAR

OLS

B

estimates

OL

P

Min-max

Cases where OLS had a close MSE to GLS 0

0.01

1.oo 1.009

1.000 1.015

1.040 1.053

1.050 1.062

57-78 5G69

0 0.01

0.997 1.011

0.993 1.009

1.053 1.056

1.061 1.063

40-61

0

0.01

Cases where there is gain in performing,feusible Misspecijied

49-13

GLS

model

0

0.2 0.4 0.6 0.8

1.233 1.365 1.441 1.492

1.721 3.151 6.043 14.736

1.018 1.002 0.997 0.997

1.056 1.043 1.034 1.027

38-53 38-57 37-61 3C-67

0.2 0.4 0.6 0.8

0 0 0 0

1.175 1.492 1.853 2.232

1.160 1.706 2.887 6.494

1.051 1.032 1.015 1.002

1.063 1.059 1.049 1.036

36-56 36-60 3664 29-7 1

Correctly

specified model

0.01

0.2 0.4 0.6 0.8

1.243 1.380 1.459 1.513

1.681 3.011 5.575 12.430

1.026 1.009 1.003 1.000

1.068 1.058 1.055 1.055

22-36 17-30 14-24 3-24

0.2 0.4 0.6 0.8

0.01 0.01 0.01 0.01

1.149 1.422 1.738 2.106

1.150 1.629 2.592 5.042

1.037 1.021 1.000 0.984

1.066 1.081 1.090 1.103

2&41 15-40 1446 6-54

model by Maddala and Mount (1973). But, also shown in table 9 is a list of cases where OLS does not necessarily have a small MSE relative to true GLS and yet there is a serious problem of negative variance components estimates.14 These cases are classified into two categories. The first category is that of a misspecified model, where the data is generated by a one-way model (i.e., one of the variance components is actually equal to zero) and the model is estimated using two-way GLS procedures. The second category is that of a correctly specified two-way model which has one of its variance components very small but not equal to zero. Note that the number of cases with negative variance estimates is smaller, on the average, for the correctly 14The gain from performing feasible GLS rather than relative MSE of SWAR as compared with that of OLS.

OLS

is given

here in terms

of the

40

B.H. Baltagi, A two-way error component model

specified model than for the misspecified model. This is due to the fact that p or w is equal to 0 for the latter case, whereas p or o is non-zero and equal to 0.01 for the former case. Hence, for all cases in table 9, the problem of negative variance components is serious, but it seems to indicate two possibilities, either that the model is misspecified or that one of the variance components is relatively small and close to zero. It is important for the researcher to be able to distinguish b.etween these two cases, i.e., perform a test to determine whether these variance components are actually zero or not. Unfortunately, the LM test results given in table 7 perform badly in exactly these cases, that is, cases where p or o equal 0 or 0.01. For example, when p = 0.6 and o = 0.01 we get a high frequency of correct decisions for H, and H,, but a low frequency of correct decisions for H,, i.e., the LM test does not reject that cr: =O, when in fact CT:=0.2, in 99 out of a 100 replications. Also, when p =0 and w = 0.8 the LM test gave a low frequency of correct decisions for H,, i.e., it rejected cr,‘=O, in 81 out of 100 replications, when in fact it was true. These examples show the inability of the LM test to distinguish between the two possibilities. Although negative variance estimates occur in precisely those cases where the LM test does not have a good performance we can still get more information on the problem by computing estimates of these variance components using other two-round GLS methods. After all, we suspect that the negative variance estimate occurred because the true variance component was actually zero or close to zero. If some of the other methods gave nonnegative estimates of this variance component, their average should give us a fair idea of the magnitude of this variance component. If, on the other hand, all the other methods gave negative estimates of this variance component, then the NERLOVE method will provide us with a positive estimate of this variance component. However, the Nerlove estimate of the corresponding variance component is, on the average, over-estimated (i.e., positively biased). For example, in case p =0.6 and CJJ=O.Ol, Nerlove’s estimate of cr: had a mean of 0.628 when in fact r~: = 0.2. Similarly, when p =0 and w =0.8, Nerlove’s estimate of 0; had a mean of 0.317 when in fact CJ~=O. So Nerlove’s method confirms that the variance components are relatively small, but does not distinguish between the above two cases either.15 The main point of this section is the following: For the two-way model, the researcher should not label the problem of negative variance estimates ‘not serious’. This is because we cannot distinguish between the case where the model is misspecified (i.e., with at least one of the variance components actually equal to zero) and the case where the model is properly specified

15Tables’for the mean and mean squared error of the variance components an appendix to this paper, and will be supplied upon request from the author.

are contained

in

B.H. Baltagi, A two-way errm component model

41

(i.e., with at least one of the variance components relatively small but different from zero). Another important reason is that we may not be able to distinguish between a case where OLS is equivalent to GLS according to the MSE criterion and a case where it is not. For these cases, the practical solution seems to be the replacement of a negative estimate by zero. Of course, this will affect the properties of the variance components estimates especially if the actual variances are different from zero. But as table 12 shows, the performance of the two-stage GLS methods will not be seriously affected by this substitution. Hence, the results of the experiments are in favor of performing the two-stage GLS procedures, even when these methods produce negative estimates of the variance components, and those negative estimates are in turn replaced by zero.

6. Alternative estimation

methods

Table 10 gives the means and relative MSE’s of 51 and p for one of the experiments performed. From this table it is clear that /I is unbiased Table 10 Means

and relative

MSE’s of z and b.” Relativeb

Method

Mean (cz)

Relativeb MSE (x)

Mean (B)

MSE (B)

True value 1. OLS 2. LSDV

5.000 4.961 5.136

0 1.707 1.068

0.500 0.501 0.499

0 2.893 1.735

Two-stage GLS 1. AMEMIYA 2. WALHUS 3. SWAR 4. MINQUE 5. NERLOVE 6. FUBA

5.082 5.073 5.075 5.074 5.086 5.076

1.030 1.037 1.037 1.037 1.024 1.035

0.500 0.500 0.500 0.500 0.500 0.500

1.131 1.133 1.159 1.138 1.145 1.150 _.

_

“For the experiment in the table N =25, T= 10, c2 =20, p =0.4, w=O.4. bRelative MSE with respect to true GLS.

regardless of the estimation method, however, more replications are needed to get rid of the bias in M which is at most 3 Y0 for LSDV. For j3, the mean square error of OLS is almost (2.89) three times as large as that of GLS, while that of LSDV is less than twice (1.74) as large. The mean square error of the various two-stage GLS methods are at most about 16% larger than that of exact GLS. For CI,the mean square of OLS is less than twice as large as that of GLS, while that of LSDV and the various two-stage GLS methods is very close to that of true GLS.

42

B.H. Baltagi,

A two-way error component

model

Table 11 gives the means and MSE’s of the variance components. Although we are not interested in the estimates of the variance components per se, we need to check whether they are non-negative and whether better estimates of these variance components imply better second round estimates of the regression coefficients. Negative estimates of the variance components were dealt with in section 3.16 In this section, we deal with the latter issue. From table 11, WALHUS has lower MSE for all variance components than AMEMIYA. However, as Table 10 shows, AMEMIYA has lower relative MSE for both regression coefficients than WALHUS. An opposite result is

Table

11

Means and MSE’s of the variance

components.”

Method

Mean (0:)

MSE (0:)

Mean (oj)

MSE (cs:)

Mean (0:)

MSE (CT:]

True value 1. AMEMIYA 2. WALHUS 3. SWAR 4. MINQUE 5. NERLOVE 6. FUBA

8.000 8.129 7.837 8.035 7.952 8.184 8.030

0 7.901 7.220 7.881 7.726 7.282 7.856

8.000 8.139 7.728 8.370 8.186 7.467 8.311

0 14.849 12.828 16.263 14.592 12.306 15.819

4.000 3.963 4.012 3.982 3.915 3.424 3.982

0 0.132 0.128 0.132 0.127 0.429 0.132

“For the experiment

in the table N = 25, T = 10, u2 = 20, p =0.4, w = 0.4.

obtained if we compare FUBA and SWAR in the same tables. For the variance components, FUBA has MSE’s no larger than those for SWAR, and for CI and p, FUBA has MSE’s lower than those for SWAR. This clearly shows that better estimates of the variance components do not necessarily imply better second round estimates of the regression coefficients. This extends Taylor’s (1980) results from the one-way to the two-way model. In fact, Taylor (1980, p. 126) showed analytically that ‘. . reduction in the mean square error (MSE) of the estimated variance components may produce either a larger or smaller MSE estimator for the slope coefficients.. .‘.l’ Table 12 gives the ratio of the MSE of two-stage GLS to that of true GLS. As is clear from this table, the NERLOVE method consistently gave one of the largest MSE’s for p. NERLOVE’s MSE of /I was 13 to 47 percent

‘% case of negative variance estimates, the corresponding variance components were replaced by zero, in order to obtain the second round GLS estimates. However, the MSE tables for the variance components were constructed using the actual variance components estimates whether negative or positive. “This fact is also implied by the way in which Swamy and Mehta (1979) used extra information to improve upon a feasible GLS estimator for the coefkients of a one-way model.

B.H. Baltagi, A two-way error component

model

43

larger than that of true GLS. Still, this method generates positive estimates of the variance components, and was shown by Nerlove (1971a) to perform well in dynamic models. In most cases there was no choice between the other two-stage GLS methods. Defining the ‘best’ two-stage GLS method as that which gives the lowest MSE for /?, not one method can claim to be best for all cases. The best method varies from case to case, with its MSE (fl) being only 3 to 14 percent larger than that of true GLS. In general, as long as one of the parameters p and o are not 0 or 0.01, tables 12 and 13 show that there is always gain according to the MSE criteria in performing two-stage GLS rather than OLS or LSDV. This is true for both regression coefficients a and p. Also, looking at table 13, we conclude that if either y or w is allowed to increase while the other is held constant, then the relative MSE of OLS will eventually increase while that of LSDV will eventually decrease. This shows that as the variance components increase, and we depart from homoskedasticity of the disturbances, the performance of OLS gets worse Again, these results are in complete and that of LSDV gets better.” agreement with the analytical results derived in Taylor (1980). In fact, Taylor (1980) found that for the one-way model, if N-k >9, where k is the number of independent variables, then this is a necessary and sufficient condition for SWAR to dominate LSDV.19 Also, that the variance of SWAR is never more than 17 percent above that of true GLS. Looking at table 12, SWAR, MINQUE, WALHUS, and FUBA satisfied this property. The weak performance of AMEMIYA relative to the other methods can be explained by the fact that it is dependent on LSDV residuals. In turn, LSDV performed poorly when compared to the other methods. The main reason is that W,, is small relative to C,, and B,,. If W,, had been large relative to would have performed better C,, and B,,, then LSDV and AMEMIYA relative to the other estimation methods.

7. Summary This paper considered a two-way error component model with no lagged dependent variables, and studied the performances of different estimation and testing procedures applied to this model by means of Monte Carlo experiments. We summarize the major results obtained in this study under three main headings: testing procedures, negative estimates of the variance components, and estimation methods. First, with regards to testing procedures: (1) Data was generated from a two-way error component model

“Note that LSDV is the correct within groups estimator if the true model is two-way, and the wrong one if the true model is one-way. “Swamy and Mehta (1979) found two feasible GLS estimators, one dominating LSDV if and only if N - k 2 5, and the other dominating LSDV if and only if N > max (2, k).

44

B.H. Baltagi, A two-way error component

model

Table 12 MSE using two-stage GLS procedures+ MSE using true GLS. AMEMIYA

WALHUS

SWAR

P

UJ

G(

B --

!Y.

P

c(

B

0

0 0.01 0.2 0.4 0.6 0.8

1.199 1.179 1.064 1.019 1.003 0.996

1.265 1.255 1.257 1.245 1.230 1.219

1.036 1.047 1.020 1.006 1.003 1.002

1.042 1.058 1.056 1.059 1.077 1.128

1.040 1.053 1.018 1.002 0.997 0.997

1.050 1.062 1.056 1.043 1.034 1.027

0.01

0 0.01 0.2 0.4 0.6 0.8

1.179 1.154 1.068 1.026 1.009 1.002

1.254 1.239 1.257 1.247 1.232 1.213

1.050 1.050 1.028 1.011 1.00s 1.003

1.053 1.060 1.067 1.068 1.078 1.107

1.053 1.056 1.026 1.009 1.003 1.000

1.061 1.063 1.068 1.058 1.055 1.055

0.2

0 0.01 0.2 0.4 0.6

1.073 1.059 1.065 1.038 1.020

1.185 1.188 1.178 1.169 1.146

1.047 1.039 1.055 1.036 1.022

1.048 1.065 1.116 1.129 1.137

1.051 1.037 1.060 1.036 1.019

1.063 1.066 1.135 1.148 1.155

0.4

0 0.01 0.2 0.4

1.050 1.034 1.055 1.030

1.213 1.223 1.155 1.131

1.041 1.026 1.065 1.037

1.041 1.069 1.142 1.133

1.032 1.021 1.071 1.037

1.059 1.081 1.170 1.159

0.6

0 0.01 0.2

1.025 1.006 1.037

1.24; 1.252 1.125

1.032 1.006 1.050

1.029 1.070 1.127

1.01s 1.000 1.054

1.049 1.090 1.158

0.8

0 0.01

0.997 0.976

1.281 1.242

1.020 0.987

1.036 1.086

1.002 0.984

1.036 1.103

’

and the Chow (1960) test was performed to test the null hypothesis that the data is poolable. This test gave a high frequency of rejecting the null hypothesis when it was true. The reason for the poor performance of the Chow-test is that it is applicable only under homoskedasticity of the disturbances. For example, in testing the stability of cross-section regressions over time, the high frequency of Type I error occurred whenever the variance component due to the time effects was not relatively small. Similarly, in testing the stability of time-series regressions across regions, the high frequency of Type I error occurred whenever the variance component due to the cross-section effects was not relatively small. Under this case of non-spherical disturbances, the proper test to perform is the Roy (1957) test. In fact, this test was shown to be the Likelihood Ratio test by Zellner (1962). Applying this test knowing the true variance

B.H. Baltagi,

A two-way error component

model

45

Table 12 (continued) -.----_-_._ NEKLOVE

MINQUE

._-

P

x

-

FUBA

B

0

0 0.01 0.2 0.4 0.6 0.8

1.040 1.050 1.018 1.004 1.000 0.999

1.048 1.062 1.053 1.047 1.051 1.068

1.325 1.296 1.126 1.055 1.020 1.002

1.432 1.430 1.456 1.469 1.459 1.384

1.053 1.059 1.022 1.004 0.998 0.997

I.070 1.078 1.069 1.053 1.044 1.035

0.01

0 0.01 0.2 0.4 0.6 0.8

1.053 1.050 1.027 1.010 1.005 1.003

1.060 1.064 1.064 1.056 1.056 1.067

1.266 1.246 1.114 1.050 1.018 1.002

1.390 1.388 1.419 1.427 1.405 1.356

1.063 1.059 I .028 1.009 1.003 1.000

1.077 1.076 1.078 1.066 1.062 1.064

0.2

0 0.01 0.2 0.4 0.6

1.049 1.039 1.055 1.036 1.024

1.050 1.067 1.119 1.129 1.139

1.098 1.079 1.064 1.036 1.017

1.267 1.268 1.224 1.210 1.173

1.053 ’ ,035 1.059 1.035 1.019

1.072 1.079 1.136 1.145 1.148

0.4

0 0.01 0.2 0.4

1.042 1.026 1.065 1.037

1.046 1.069 1.149 1.138

1.067 1.043 1.046 1.024

1.304 1.296 1.178 1.145

1.035 1.019 1.068 1.035

1.075 1.101 1.164 1.150

0.6

0 0.01 0.2

1.032 1.009 1.049

1.030 1.055 1.134

1.037 1.008 1.028

1.359 1.326 1.13s

1.018 1.000 1.049

1.072 1.115 1.148

0.8

0

1.021 0.997

1.027 1.027

0.995 0.969

1.410 1.308

1.002 0.982

1.065 1.124

components or using the Amemiya (1971) and the Wallace and Hussain (1969) type estimates of the variance components lead to low frequencies of committing a Type I error. Therefore, if pooling is contemplated using an error component model, then the Roy-Zellner test should be used rather than the Chow test.‘O (2) Alternative MSE criteria, developed by Toro-Vizcarrondo and Wallace (1968), and later by Wallace (1972), were applied to the error component model in order to choose between the pooled and the unpooled estimators. These weaker criteria gave a lower frequency of committing a Type I error ‘“One qualification to this result is that the Roy-Zellner test ignores the additional information contained in the difference between the within and between groups regression. A test statistic that incorporates this additional information is developed by Portnoy (1973). I would like to thank an anonymous referee for pointing out this fact. JOE-C

B.H. Baltagi, A two-way error component model

46

Table 13 MSE of OLS and LSDV sMSE OLS

of true GLS. LSDV

P

W

r

P

(x

B

0

0 0.01 0.2 0.4 0.6 0.8

1.000 1.009 1.233 1.365 1.441 1.492

1.000 1.015 1.721 3.151 6.043 14.736

8.894 7.836 3.323 2.141 1.560 1.211

10.970 10.562 8.498 8.183 8.069 8.019

0.01

0 0.01 0.2 0.4 0.6 0.8

0.997 1.011 1.243 1.380 1.459 1.513

0.993 1.009 1.681 3.011 5.575 12.430

8.191 7.251 3.168 2.058 1.505 1.175

10.487 10.078 7.997 7.538 7.140 6.335

0.2

0 0.01 0.2 0.4 0.6

1.175 1.149 1.263 1.435 1.591

1.160 1.150 1.380 2.140 3.829

4.208 3.751 1.920 1.374 1.095

6.750 6.361 3.865 2.991 2.154

0.4

0 0.01 0.2 0.4

1.492 1.422 1.450 1.707

1.706 1.629 1.622 2.893

2.848 2.516 1.364 1.068

5.609 5.100 2.480 1.735

0.6

0 0.01 0.2

1.853 1.738 1.843

2.887 2.592 2.516

2.008 1.753 1.059

5.080 4.327 1.674

0.8

0 0.01

2.232 2.106

6.494 5.042

1.368 1.191

4.780 3.443

than the Chow test, but their performance was still poor when compared to the Roy-Zellner test. McElroy (1977) extended these weaker MSE criteria to the case of non-spherical disturbances and these criteria performed well when compared with the Roy-Zellner test. (3) The Hausman (1978) specification test was applied to the error component model in order to test whether the unobservable individual and time-period effects are uncorrelated with the exogenous variable. This test gave a low frequency of committing a Type I error whether we used the true variance components or their estimates according to Amemiya or Wallace and Hussain.21 (4) The Lagrange Multiplier test, as described in Breusch and Pagan (1980), was applied to the error component model in order to test whether “Again, we note that the results obtained for the Chow test, Zellner-Roy test and the Hausman test all pertain to the probability of type I error and not the probability of Type II error, and any conclusion based on these results ought to be properly qualitied.

B.H. Baltagi, A two-way error component model

41

the variance components are zero. The test performed well except in cases where it was badly needed, i.e., in cases where one of the variance components was actually zero or relatively small but different from zero. A high frequency of correct decisions is needed in these cases because these are precisely the cases where negative estimates of the variance components occur. Second, with regards to negative estimates of the variance components: The problem of negative estimates of the variance components is more serious for the two-way model than for the one-way model. In the one-way model, Maddala and Mount (1973) found that negative variance estimates occurred in precisely those cases where OLS had a relatively close MSE to true GLS. However, in the two-way model, negative variance estimates also occurred in cases where OLS had a weak performance relative to true GLS, i.e., in cases where there was gain in performing feasible GLS rather than OLS. For all the experiments performed, negative variances occurred in two cases. The first is the case where the model is misspecified (i.e., with at least one of the variance components actually equal to zero), and the second is the case where the model is properly specified (i.e., with at least one of the variance components relatively small but different from zero). The Lagrange multiplier did not distinguish between these two cases, since the test performed poorly in precisely these cases. The practical solution in this case is to replace the negative estimate by zero. This is the proper solution in case the true variance component is zero. Otherwise, this may affect the properties of the two-stage GLS estimators. Fortunately, the results of the experiments show that the performance of the two-stage GLS methods are not seriously affected by this substitution. Finally,

with regards

to estimation

methods:

(1) As long as the variance components are not relatively small and close to zero, there is always gain according to the MSE criterion in performing feasible GLS rather than least squares or least squares with dummy variables. (2) All the two-stage GLS methods considered performed reasonably well according to the relative MSE criteria. However, none of these methods could claim to be the best for all the experiments performed. Most of these methods had relatively close MSE’s and therefore, made it difficult to choose among them. This same result was obtained in the one-way model by Maddala and Mount (1973). (3) Better estimates of the variance components do not necessarily give better second round estimates of the regression coefficients. This confirms the finite sample results obtained by Swamy and Mehta (1979) and Taylor (1980) and extends them from the one-way to the two-way model.

48

B.H. Baltagi, A two-way error component

model

Finally, the recommendation given in Maddala and Mount (1973) is still valid, i.e., always perform more than one of the two-stage GLS procedures and if the estimates obtained differ widely, then test the specification of the model using some of the tests described in section 4. A problem for further research is the derivation of the power of these specification tests, after all, if the power of these tests is poor, these tests would not be useful in testing the specification of the model.

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model

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Schmidt, P. and R. Sickles, 1977, Some further evidence on the use of the Chow test under heteroskedasticity, Econometrica 45, 1293-1298. Searle, S.R., 1971, Topics in variance components estimation, Biometrics 27, l-76, Swamy, P.A.V.B., 1971. Statistical inference in random coefficient regression models (SpringerVerlag, New York). Swamy, P.A.V.B., 1974, Linear models with random coeffients, m: P. Zarembka, ed., Frontiers in econometrics (Academic Press, New York). Swamy, P.A.V.B. and S.S. Arora, 1972, The exact finite sample properties of the esttmators of coefficients in the error components regression models, Econometrica 40, 261-275. Swamy, P.A.V.B. and J.S. Mehta, 1973, Bayesian analysis of error components regression models, Journal of the American Statistical Association 68, 648658. Swamy, P.A.V.B. and J.S. Mehta, lY79, Estimation of common coefficients in two regressions equations, Journal of Econometrrcs 10, 1-14. Swamy, P.A.V.B. and P.A. Tinsley, 1980, Linear prediction and estimation methods for regression models with stationary stochastic coefficients, Journal of Econometrics 12, 103142. Taylor, W.E., 1980, Small sample considerations in estimation from panel data, Journal of Econometrics 13, 203-223. Toro-Vizcarrondo, C. and T.D. Wallace, 1968, A test of the mean square error criterion for restrictions in linear regression, Journal of the American Statistical Association 63, 558-572. Toyoda, T., 1974, Use of the Chow test under heteroskedasticity, Econometrica 42, 601-608. Wallace, T.D., 1972, Weaker criteria and tests for linear restrictions in regression, Econometrica 40, 689-698. Wallace, T.D. and A. Hussain, 1969, The use of error components models in combining crosssection with time-series data, Econometrica 37, 55-72. Yancey, T.A., G.G. Judge and M.E. Bock, 1973, Wallace’s weak mean square error criterion for testing linear restrictions in regressions: A tighter bound, Econometrica 41, 1203-1206. Zellner, A., 1962, An efficient method of estimating seemingly’unrelated regression and tests for aggregation bias, Journal of the American Statistical Association 57, 348-368.