Testing the distribution of error components in panel data models

Economics Letters 77 (2002) 47–53 www.elsevier.com / locate / econbase

Testing the distribution of error components in panel data models Scott Gilbert* Department of Economics, Southern Illinois University at Carbondale, Mailcode 4515, Carbondale, IL 62901 -4515, USA Received 14 August 2001; received in revised form 19 February 2002; accepted 12 March 2002

Abstract We propose tests for normality of the error components in panel data models, using estimates of skewness and kurtosis in the components.  2002 Published by Elsevier Science B.V. Keywords: Error component; Normality; Skewness; Kurtosis JEL classification: C33

1. Introduction The one-way error components model with individual effects is: y it 5 x 9it b 1 u it , i 5 1, . . . ,n, t 5 1, . . . ,T

(1)

where y it is the dependent variable for individual i at time t, and x it is a vector of observables for which we assume that x i , i 5 1, . . . ,n, are independent and identically distributed (IID) with finite eighth moments and non-singular variance–covariance matrix. The variable u it 5 mi 1 nit is the model’s error, composed of two components. The unobserved individual effects m1 , . . . , mn are the permanent component, random and IID with zero population mean and finite eighth moments. The unobserved random variables nit are the transitory component, zero-mean and IID across i and t, with finite eighth moments. Also, mi is independent of ni for all i, and both mi and nit are independent of x it for all i and t. We are interested in the distribution of the error components. The normal distribution is usually * Tel.: 11-618-453-5065; fax: 11-618-453-2717. E-mail address: [email protected] (S. Gilbert). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00111-8

 2002 Published by Elsevier Science B.V.

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invoked to obtain the exact null distribution of hypothesis tests for the parameters b, hence departures from normality are clearly an important issue for economic inference when samples are sufficiently small. Normality is also invoked to obtain the exact distribution of tests for individual effects, and ´ ´ (1996) examine the consequences of non-normal error components for the Blanchard and Matyas performance of such tests. As noted by Baltagi (1998), in economic panel data modelling it can be important to adjust for departures of error components from normality. Inference methods for b, including (quasi) maximum likelihood, can be usefully adjusted for error component non-normality. Moreover, for constructing prediction intervals for y it , conditional on specified regressor values, one clearly needs information about the error component distributions. An interesting example is Horowitz and Markatou (1996), who study the dynamics of worker earnings. For this purpose they need to estimate the joint distribution of the T-vector y i , conditional on the T-vector x i , and for this they need estimates of the error component distributions. Their methods simplify considerably when the transitory component is prescribed as normal, and testing for component normality is then a useful supplementary procedure. The present work proposes tests for normality of the error component distributions. We obtain formulas for the skewness and kurtosis of the error components, as functions of the moments and cross-moments of the errors. We use these formulas to estimate skewness and kurtosis in the error components, and to test for non-normality. The tests are straightforward to implement and have convenient chi-square asymptotic distributions.

2. Moments of error components Denoting by Fm and Fn the (cumulative) distribution functions for the two unobserved error components, we compute higher order moments of these distributions, up to fourth order. Since, by assumption, the first moments of m and n are both zero, e.g. e z dFm (z) 5 e z dFn (z) 5 0, it follows that, for each such distribution F, the quantity fk ; e z k dF(z) is, for k $ 2, the centered k-th moment. For positive integers j and k we let:

cj ; Eu jit , cj,k ; Eu isj u itk

(2)

for any given i and distinct s and t. Due to the assumed structure of the error components, cj and cj,k are invariant to the choice of i, s and t. The second moments of the error components, which are well known functions of the error distribution (see Hsiao, 1986; Baltagi, 1995), can be expressed as: fm ,2 5 c1,1 and fn,2 5 c2 2 c1,1 . For the third moments of the error components, we have first:

fm ,3 5 c1,2

(3) 2

which is true since c1,2 5 Eu is u it 5 Es mi 1 nisds mi 1 nitd 2 , and applying the zero-mean and independence properties of the error components, c1,2 5 Em 3i 5 fm ,3 . For the remaining error component, we have:

fn,3 5 c3 2 c1,2

(4) 3

To verify this, we have c3 ; Eu it 5 Es mi 1 nitd 3 , and applying the zero-mean and independence

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properties of error components, we deduce that Eu 3it 5 fm ,3 1 fn,3 , and combining this fact with (3) yields (4). To further interpret skewness in the error components, we note first from (3) that for each individual i, non-zero skewness fm ,3 in the permanent component induces correlation between the error u is at time s and the squared errors u 2it at remaining times. A measure of this nonlinear ]]]] dependence is the correlation c1,2 /œc2sc4 2 c 22d between u is and u 2it . This correlation is obviously determined by c1,2 , c2 and c4 , and we have c1,2 5 fm ,3 , c2 5 fm ,2 1 fn,2 , and c4 5 fm ,4 1 fn,4 1 6fn,2 fm ,2 . Hence, while the skewness fm ,3 of the permanent component contributes to nonlinear dependence via c1,2 , the skewness fn,3 of the transitory component has no effect on this dependence. We also note that, since third moments can be negative, and since the sum of error component third moments equals the error third moment, it is possible to have skewness in the error components but no skewness in the error itself. For the fourth moments, we have first:

fm ,4 5 c1,3 2 3c1,1sc2 2 c1,1d

(5)

This holds since c1,3 5 Esu is u it3 d, and by expanding terms and applying zero-mean and independence assumptions, we obtain c1,3 5 Es m 4i d 1 3Es m 2i d Esn 2itd, and hence conclude (5). For the second error component:

fn,4 5 c4 2 c1,3 2 3c1,1sc2 2 c1,1d 4

(6) 4

2

2

which is true since, first, c4 5 Em i 1 En it 1 6Em i En it , and combining this with (5) yields (6). To further interpret the fourth moments and kurtosis in the error components, we recall that, for normally distributed variables z, fz,4 5 3f 2z,2 and so the kurtosis fz,4 /sfz,2d 2 equals 3. Non-normal levels of kurtosis are those less than three (platykurtotic) and greater than three (leptokurtotic), and we deduce from (5) that the permanent component m has normal kurtosis if and only if:

c1,3 2 3c2 c1,1 5 0

(7)

while from (6) we deduce that the transitory component n has normal kurtosis if and only if:

c4 2 c1,3 2 3c2sc2 2 c1,1d 5 0

(8) 3

According to (7), if m has normal kurtosis then the covariance c1,3 between u is and u it is positive and given by the product of variances for the error and its permanent component. According to (8), with normal kurtosis in n there may or may not be positive covariance between u is and u 3it , and the sign of the covariance is determined by the relative magnitudes of the quantities c2 , c4 and c1,1 .

3. Normality tests To test for the normality of error components, we first obtain estimates of the moments cj and cross-moments cjk for the relevant j and k. With bˆ any weakly consistent estimator of b (such as those described in Hsiao, 1986 and Baltagi, 1995), the residuals are uˆ it 5 y it 2 x it9 bˆ , and we obtain:

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OO

1 n T j ˆ cj 5 ] uˆ nT i 51 t51 it

(9)

and

OO

n 1 cˆ j,k 5 ]]] uˆ j uˆ k nT(T 2 1) i51 s±t is it

(10)

for each j and k. We require estimates of the variances and covariances for the statistics cˆ j and cˆ ij , and in the interest of simplicity we propose the following form:

Osa 2 a¯ dsb 2 b¯d n

Ca;b 5 n

21

i

i

(11)

i51

21 T j 21 j k where a i and b i are quantities of type T o t51 uˆ it or (T(T 2 1)) o s±t uˆ is uˆ it , and a¯ and b¯ are the sample averages of a i and b i . As a and b are each identified by a specific j or ( j, k), we use the notation Cj a ; j b , Cj a ; j a , j b , etc., for Ca ;b . To test for skewness in the error components we introduce the ratios:

cˆ 1,2 cˆ 3 2 cˆ 1,2 ]]] ]]]]]]]] zm ,1 5 ]] , zn,1 5 ]]]]]]] œC1,2;1,2 œC3;3 1 C1,2;1,2 2 2C3;1,2

(12)

where the numerator of the ratio zm ,1 estimates the skewness (3) of the error’s permanent component, and the denominator estimates (consistently) the standard deviation of the statistic defined by the numerator. The numerator of zn,1 estimates the skewness (4) of the transitory component, and the denominator, once again, estimates (consistently) the standard deviation of the statistic defined by the numerator. To test for non-normal kurtosis, we introduce the ratios:

cˆ 1,3 2 3 cˆ 2 cˆ 1,1 z 3 5 ]]]]]]]]]]]]]]]]]]]]]]]] ]]]]]]]]]]]]]]]]]]]]]]] 2 2 œC1,3;1,3 1 9scˆ 2 C1,1;1,1 1 cˆ 1,1 C2;2 1 2cˆ 2 cˆ 1,1 C2;1,1d 2 6scˆ 2 C1,3;1,1 1 cˆ 1,1 C1,3;2d

(13)

cˆ 4 2 cˆ 1,3 2 3 cˆ 2scˆ 2 2 cˆ 1,1d z 4 5 ]]]]]]]] Œ]h

(14)

where h 5 C4;4 1 C1,3;1,3 2 2C4;1,3 2 2 1 9sscˆ 2 2 cˆ 1,1d C2;2 1 cˆ 2sC2;2 1 C1,1;1,1 2 2C2;1,1d 1 2 cˆ 2scˆ 2 2 cˆ 1,1dsC2;2 2 C2;11dd 2 6scˆ 2s2sC2;4 2 C2;1,3d 1 C1,3;1,1 2 C4;1,1d 1 cˆ 1,1sC2;1,3 2 C2;4dd In (5) and (6), the numerator of the ratio zm ,2 estimates a quantity which equals 0 if and only if the error’s permanent component has normal kurtosis, while the numerator of zn,2 estimates a quantity which equals 0 if and only if the transitory component has normal kurtosis. The denominators of these ratios estimate (consistently) the standard deviation of the statistics defined by the numerators.

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We have four null hypotheses of interest, denoted Hm ,1 , Hm ,2 , Hn,1 , Hn,2 , the first two signifying zero skewness and normal kurtosis, respectively, in m, and the second two signifying zero skewness and normal kurtosis in n. Under Hm ,1 , the statistic zm ,1 is unit normal asymptotically, and under Hm ,2 the statistic zm ,2 is asymptotically unit normal, with analogous results for zn,1 and zn,2 . Facilitating these facts are the consistency of the estimator bˆ and the consistency of the variance estimators appearing in the denominators of ratios zm ,1 ,zm ,2 ,zn,1 ,zn,2 . We obtain the consistent variance estimators using standard asymptotic approximations for smooth functions of statistics converging to constants (as for example in van der Vaart, 1998, p. 33). To test each of the four null hypotheses, we propose to use the relevant z 2 statistic, rejecting if the statistic exceeds the chi square critical value. Asymptotically, the statistics zm ,1 and zm ,2 are independent under joint hypothesis Hm ,1 > Hm ,2 , while zn,1 and zn,2 are independent under the joint hypothesis Hn,1 > Hn,2 (verified using standard asymptotic approximations to smooth functions of normal statistics). To test the first joint hypothesis we use z m2 ,1 1 z 2m2 , asymptotically chi square (with 2 degrees of freedom), and we proceed analogously with the second joint null hypothesis. Interestingly, the statistics zm ,1 and zn,1 are not asymptotically independent under Hm ,1 > Hn,1 . We have performed simulations to examine test performance. We let n 5 2, T [ h25, 50, 100, 250, 500, 1000j, K 5 2, x ij 5 (1,hij ) with hij IID standard normal. For the null (normality) hypothesis we let m,n1 ,n2 be IID standard normal. For the alternative, we let m,n1 ,n2 be IID, each having the distribution of j 5 (1 1 2z )(v 1 v 2 2 1), where z and v are independent standard normal variables. Using the first eight moments of the standard normal distribution (0, 1, 0, 3, 0, 15, 0, 105) we obtain Ew 5 0, Ew 2 5 15, Ew 3 5 182, Ew 4 5 8979, in which case the skewness of w is 182 /(15)3 / 2 5 3.133, and the kurtosis is 8979 / 15 2 5 39.91. Table 1 shows rejection rates, at the 5 and 1% nominal significance levels, for 10 000 simulations (carried out in Eviews 4.0) of the null and alternative hypotheses. As indicated, the rejection frequency under the null can differ from nominal size, but the discrepancy is mostly modest. The Table 1 Rejection rates, error component normality tests Sample size

Skewness Perm.

Kurtosis Trans.

Perm.

Skewness

Kurtosis

Trans.

Perm.

Trans.

Perm.

Trans.

0.05 0.06 0.07 0.07 0.07 0.06

0.00 0.01 0.01 0.01 0.01 0.01

0.01 0.01 0.01 0.01 0.01 0.01

0.00 0.01 0.01 0.02 0.02 0.02

0.01 0.01 0.01 0.02 0.02 0.02

Panel B: Rejection rates under alternative hypothesis 25 0.04 0.02 0.07 0.04 50 0.04 0.03 0.05 0.06 100 0.03 0.08 0.05 0.12 250 0.07 0.31 0.12 0.34 500 0.22 0.51 0.27 0.53 1000 0.47 0.76 0.50 0.72

0.01 0.01 0.00 0.01 0.03 0.15

0.00 0.00 0.01 0.06 0.20 0.48

0.01 0.01 0.00 0.01 0.04 0.17

0.01 0.01 0.01 0.06 0.20 0.40

Panel A: Rejection rates under null hypothesis 25 0.04 0.05 0.04 50 0.04 0.05 0.05 100 0.04 0.05 0.07 250 0.05 0.05 0.07 500 0.04 0.05 0.08 1000 0.05 0.05 0.07

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largest discrepancy (3%) is for the kurtosis tests, and this is diminished in the largest sample. The normality tests for the transitory component correctly reject more often than do tests of the permanent component. In all cases, much higher rejection rates could be achieved (as we have confirmed, via simulation) if one could apply standard normality tests (e.g. the Jarque–Bera test) directly to the component series, but this is not possible since the components are unobserved.

4. Example To illustrate the proposed methods we give a numerical example, taken from Biddle and Hamermesh (1990) (see also Wooldridge, 2000, p. 424). A panel of 239 people is surveyed in the years 1975 and 1981. The dependent variable y it is the average number of hours that person i slept per day (‘slpnap’) in year t. We let x it consist of six variables, the first is a dummy (‘d81’) indicating the year 1981, the second is the number of hours worked per day (‘totwrk’), the third measures education (‘educ’), the fourth is a dummy (‘marr’) for married persons, the fifth (‘yngkid’) is a dummy for having a young child in the household, and the sixth (‘gdhlth’) is a dummy variable indicating good health. To examine error component distributions, we first estimate the model (1) by (pooled) ordinary least squares, and obtain residuals e it , i 5 1, . . . 239, t 5 1,2. Denoting by e75 and e81 the residual series for the two years, panel A of Table 2 reports the sample skewness and kurtosis for these series, as well as the Jarque–Berra normality test and P value. As indicated, both error series are consistent with non-normality, and the kurtosis values in particular appear substantially higher from the theoretical value (53) prescribed by normality. Panel A likewise reports on the time-differenced errors, which in the model (1) exclude the permanent component; the statistics indicate non-normality and excess kurtosis. Panel B of Table 1 reports estimates and tests of skewness and kurtosis of error components. Neither error component has statistically significant skewness, and only the transitory component shows significant kurtosis. Adding together the transitory component’s skewness and kurtosis test statistics we get 7.775, and a P value50.02 from the chi-square (2 df) distribution. Hence, for the Table 2 Normality tests, errors and their components Panel A: errors Residual series

Skewness

e75 e81 e81–e75

0.872 20.342 0.705

Kurtosis 6.51 5.343 4.739

Panel B: error components Skewness Permanent Estimate Test statistic P value

1.695 1.561 0.212

Jarque–Bera test

P value

152.958 59.342 49.919

0.000 0.000 0.000

Kurtosis Transitory 20.042 0.007 0.933

Permanent 8.954 1.145 0.285

Transitory 7.429 6.630 0.010

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permanent component the normal distribution appears to be a reasonable assumption, but not for the transitory component. This information could not be obtained from the report in panel A, but instead required a detailed analysis of the error components.

5. Conclusion The present work provides estimators and tests of skewness and kurtosis of the error components in panel data models. Because the error components are unobserved, direct application of popular normality tests is not possible, and we instead exploit various sample moments of the error components. Our proposed tests are convenient, with chi-square asymptotic distributions, and in a simulation we find generally good agreement between nominal and exact test size. Simulation also suggests that normality tests of the transitory error component are likely more powerful than those of the permanent component, consistent with the greater number of independent realizations comprising the transitory component.

Acknowledgements This research was supported by Office of Naval Research Grant N000-14-99-1-0722.

References Baltagi, B.H., 1995. Econometric Analysis of Panel Data. Wiley, New York. Baltagi, B.H., 1998. Panel data methods. In: Ullah, P., Giles, F. (Eds.), Handbook of Applied Economic Statistics, pp. 291–323, Chapter 8. Biddle, J.E., Hamermesh, D.S., 1990. Sleep and the allocation of time. Journal of Political Economy 98, 922–943. ´ ´ L., 1996. Robustness of tests for error component models to non-normality. Economics Letters 51, Blanchard, P., Matyas, 161–167. Horowitz, J.L., Markatou, M., 1996. Semiparametric estimation of regression models for panel data. Review of Economic Studies 63, 145–168. Hsiao, C., 1986. Analysis of Panel Data. Cambridge University Press, Cambridge. van der Vaart, A.W., 1998. Asymptotic Statistics. Cambridge University Press, Cambridge. Wooldridge, J.M., 2000. Introductory Econometrics: A Modern Approach. South-Western College Publishing, Cincinnati, OH.