Homogeneity and symmetry conditions are frequently rejected in applied demand studies. Deaton(1972) argues that symmetry is "fundamentally a weak hypothesis" so that rejection of these restrictions is "intuitively implausible" (Bewley[1983]). It would appear that the reason for rejecting such restrictions would be
because of the inappropriate nature of the test statistics. Laitinen(1978) demonstrates that the standard test for homogeneity is seriously biased towards rejection. He argues that the large sample criteria is highly misleading and the test statistic is in fact distributed as Hotelling's T ^ , which is distributed as a multiple "[(n-1)(T-n-1)]/(T-2n+1)'' of F[(n-1 ),(T-2n+1)] degrees of freedom. Meisner(1979), using a simulation experiment, shows that the standard test statistic for Slutsky symmetry is also biased towards rejection of the null hypothesis, particularly for large demand systems. The bias as illustrated by Meisner(1979) is due to the use of the estimator Q. rather than the true yet unknown covariance matrix Q.
As asymptotic tests are biased toward over-rejection, a size correction has been suggested to adjust the critical values of asymptotic test. The idea is to center the distribution of the observed test statistic by multiplying a correction factor so that the test statistic coincides more closely with its hypothetical distribution. Arbitrarily chosen factors have been proposed by different people. For example, Bohm, Rieder and Tintner(1980) suggested the factor (T-k)/T where k is the number of parameters in each demand equation. The rationale is as follows :
Suppose that there is a linear system of n equations:
Y = X(3 + U ( 4 9 8 ) Y is ( r n ) , X is ( r k ) , (3 is (k*n) and U is ( r n ) where U ~ N(O,a0l^).
Imposing m linear restrictions R[3 = g
where R is (m*nk) and g is (m*1).
As pointed out by Bewley(1983), if the covariance matrix Q is known, the exact F test statistic is:
F = { tr[Q-''(U'U-U'U)]/m } / { tr[n-1 (0'U)]/(T-k)n } (4.99)
*
If a is unknown and replaced by the estimate n , the test statistic is:
W* = (T-k)
which is distributed as asymptotic x^ with m degrees of freedom. In *
fact, W can be defined with a small sample correction factor, (T-k)/T, times the Wald statistic W. Similarly, the correction factor can be applied to the LR and LM statistics and generate the
* *
corrected statistics LR and LM . It also follows that the inequality remain valid, ie,
w ' > LR' > LM*
Bera, Byron and Jarque(1981) argue that the size correction factor (T-k)/T is inadequate especially for large demand system.
They propose a factor which is equal to the ratio of the asymptotic 5% critical value to the empirical critical value. This method will be discussed in Chapter 6.
Later, Byron and Rosalsky(1985) proposed an Edgeworth correction method to approximate the true tests. The suggested method appears to be fairly useful, but the computational cost is quite heavy especially for large SUR systems with a large number of restrictions. In addition, the reliability of Edgeworth corrections has yet to be established.
Deaton(1972) also suggests another test statistic which approximate the F statistic in (4.99) with a known covariance matrix by
and formed an asymptotic statistic
D r = { (T-k) t r [ Q r - r ' ' ( S r - " ) ] } / n r [ Q r . r ^ S ] / n } (4.100)
It is based upon the constrained Iterative Zellner Efficient (IZEF) method, a terminology by Kmenta and Gilbert(1968). Say for the
model in (4.98), the constrained one-step GLS residuals U^ = Y - xp,
can be used to form a revised estimate of the covariance matrix Q.|
and subsequently a revised estimate of the constrained parameter
matrix p g . This r-step iterative p r o c e d u r e will generate a
constrained estimator which is identical to the ML estimator, and
the associated covariance matrix is (T-k)/T times the one from the ML estimator.
The limiting value of D,. is:
D* = LM* / { 1 - [ LM* / (T-k)n ] }
Bohm, Reider & Tintner(1980) noted an inequality of the form:
W * > D p > L M * (4.101)
but no general statement can be made about the alternative
magnitude of D^ and LR*.
The size corrected test statistics from Table 4.14 and Table 4.18 are presented in Table 4.22. All W*, LR*, LM* and D* statistics strongly rejected the demand theory except for the homogeneity c o n d i t i o n in the DLOG model. Homogeneity restrictions are
* •
marginally accepted using the LM and D statistics in the DLOG model. The relationship of the test statistics in (4.101) is also s a t i s f i e d .
4 . 1 0 S u m m a r y
Since the demand conditions are rejected by the maintained AIDS model, is it possible to conclude that consumers do suffer from money illusions and are irrational as they do not maximize their utility? Rejection of demand theory is not an uncommon finding in empirical research, but does it mean that the demand theory is in invalid and inappropriate in reality? Simmons and Weiserbs(1979) argue that given that the approximation of the true utility function is exact only at one point (the base point of approximation), it is only possible to test the true restrictions on the true utility at that point. But explicit restrictions on the approximated utility function can be tested over any subset of the data observed and will be considered at all points of the sample. Hence even though a restriction or a specification is rejected, this does not imply a conclusion can be reached concerning the validity of the theory. It does not mean that well behaved consumer preferences do not exist. Rather, it is probably the case that other specifications may perform better. This argument motivates the following analysis on diagnostic testing of the specification of the maintained model.