A size-corrected Wald test for Slutsky symmetry in systems of demand equations

327

5) 327-330

U in

eilfor v

ssion

ts on

t of t

t.

-1

s

it

is

of

T. G. Taylor, J.S. Shonkwrler Table 1 Rejections

of symmetry

n=5 PI=8 n =11 n=14 A Average

rejection

/ Sue-correcled

Wald test for Slutsky symmetry

329

at the 5% and 1% levels using W* and W’ Replications

Rejections W’

W*

W’

W*

400 200 100 100

10 23 55 93

4 3 2 0

3 12 39 84

2 2 1 0

a at 5%

Rejections

a at 1%

per 100 replications.

In order to maintain consistency with previous studies, we evaluate the performance of W* for testing symmetry using the Rotterdam model in conjunction with Dutch data. ’ The absolute price version of the Rotterdam model with homogeneity imposed is n-l

~;;Dq;,=8;DQ;+

c

71,i(DP;:-

DP,y:)+c;:.

!7!

;=1

where w,, denotes the arithmetic average of the i th budget share over periods t and t - 1, DP,, is the log change in the jth price and the r,, = T,~ are Slutsky coefficients for i = 1,. . . , n: t = 1,. . . , T. Following Laitinen (1978) and Meisner (1979) pseudo-normal variates with mean zero and covariance .E were generated and eq. (7) was constructed to be true by definition. * The system ’ was then estimated using four different values of n. Table 1 presents the number of rejections of symmetry per 100 replications using W* and IV’ = [(T - n)/T] W where W is defined as in (5). Generally, W* performs somewhat better than w’ for n less than 11. At the 5% level, however, for both n = 8 and 11, the number of rejections yielded by W* is slightly less than the 5% expected. At the 1% level, W* rejects the null hypothesis at a slightly higher rate than theoretically expected for n = 5 and 8 and exactly once for n = 11. For n = 14, an extreme situation develops. While w’ exhibits the expected bias towards rejection of symmetry, W* fails to reject the null hypothesis at the 5% and 1% levels. The results presented in table 1 seem to suggest that when n is moderate relative to the sample size, T, correcting for the bias in estimating X’ provides a size correction for the Wald test which performs reasonably well. The failure to reject the null hypothesis at the 5% and 1% levels for n = 14, is, however, disconcerting. Based on the results above, it appears that when the number of equations (n) relative to the sample size is such that T - 2n is of reasonable magnitude, W* appears to be a viable alternative to either W or w’ in tests of Slutsky symmetry. As T - 2n becomes small, however, W* appears biased towards non-rejection of the null hypothesis.

’ See Theil (1975, pp. 264-265) for the data. * See Laitinen (1978, p. 188) for the definition of X, 0, and QT,,, i. J =l..__. n, as well as the construction of the demand system for n = 5, 8 and 11. ’ Since the share components sum to one in eq. (7), the covariance matrix for the full system is singular. Thus, estimation is carried out by deleting the nth equation.

330

T.G. Ta.vlor, J.S. Shonkwiler

/ Sue-corrected

Waid test for S&sky

.syrmetry

References Bera, A.K.. R.P. Byron and C.M. Jarque. 1981, Further evidence on asymptotic tests for homogeneity and symmetry in large demand systems, Economics Letters 8. no. 2, 101-105. Bewley. R.A., 1983, Tests of restrictions in large demand systems, European Economic Review 20, 257-269. Evans. G.B.A. and N.E. Savin, 1982, Conflict among criteria revisited: The W, LR and LM tests. Econometrica 50, 737-748. Johnson. N.L. and S. Katz, 1972, Distributions in statistics: Continuous multivariate distributions. Vol. 4 (Wiley. New York). Laitinen, K., 1978, Why is demand homogeneity so often rejected?, Economics Letters 1. no. 3. 187-191. Meisfier . *_z I F _, 1970 , The .~rl .fate arwmntntir 91,,tcLv ..,L
and measurement

of consumer

demand,

Vol. 1 (North-Holland,

Amsterdam).