variables of the demand system, irrespective of the price index used. Thus, by
assuming that r=2, the long-run model (5) provides a consistent representation for
assessing the significance of the effects of substitution between healthy and unhealthy
food expenditure categories.
The parameter estimates of the demand model, which aggregates non-durable
goods according to the regional price index (RPI), are listed in Table 2a; those
obtained from the data aggregated by the national price index (NPI) are shown in
Table 2b10 .
Although almost all the parameters of the two models estimated with imposed
homogeneity and symmetry restrictions are significant with similar size after their
imposition, we find that the theoretical restrictions are not jointly rejected at 1 per-
cent only for the data obtained by RPI11 . Thus we proceed below to the estimation
of elasticities of the long-run demand system by using variables obtained from the
regional price index.
From the stationary and error serial correlation criticisms of Lewbel and Ng
(2005), generally found in static demand models estimated with aggregate data,
show the patterns of estimated error vectors and related residual serial correlation
tests of VECM. Figure 5 plots the resulting estimated disequilibrium errors (w∙-
wi), with i = 1, 2. Consistent with the assumptions of stationarity of errors, a stable
dynamic is found. Furthermore, both Q-statistics, adjusted Q-statistics and multi-
variate LM statistics, Table 3, indicate the absence of any significant autocorrelation
in the vector of the errors.
Table 4 lists the estimated compensated price and expenditure elasticities com-
puted at the sample means12 . A few aspects of these estimations should be noted.
We estimate negative and large own price-elasticities. These results show that there
are no violations of concavity and that consumers’ demand for food responds to
price changes. It is worth noting that the smaller compensated price elasticity in
the residual component is strongly biased downwards by the inclusion of the expen-
diture categories for bread, pasta and olive oil (although the size is reduced when
we compare it with the uncompensated price elasticities). We will return to the
10 The third cointegrating vector for other foods and non-durables is then recovered by the adding-up constraint.
11The differences in the results of the theoretical restriction tests are emphasized when small sample statistics are
performed. In this case, although the model estimated with a national price index is still rejected at one percent,
data which use a regional price index do not reject homogeneity and symmetry at five percent. These results are
available upon request.
12 Typically, one chooses this point to hold concavity because it is the point with the highest sample ‘’information”
and the data are scaled consequently. Asymptotic standard errors of elasticities and confidence intervals are derived
from bootstrap replications of the estimated parameters and their standard errors.
18