Problems of operationalizing the concept of a cost-of-living index



As described in the previous section, we have four cross-section data sets
(EVS data from 1988, 1993, 1998 and 2003) available for our AIDS and
QUAIDS estimation. For lack of space we can’t display all estimation and test
results of the four data sets. Our focus will lay on the 1988 results, because
they enable us to calculate long COLI- time-series. A selection of output
tables can be found in the appendix of this paper, further results are available
upon request. After having estimated the AIDS and QUAIDS budget share
equations (34) and (42) using the EVS data set from 1988, 1993, 1998 and
2003, we first want to test the symmetry and homogeneity restrictions. As
the fulfillment of the symmetry restriction ensures also the fulfillment of the
homogeneity restriction because otherwise the adding-up condition would be
violated, we have to estimate the AIDS and the QUAIDS once with only the
homogeneity restrictions imposed (model 2) and once completely without
any restrictions (model 3) imposed. Now we can calculate the Likelihood-
Ratio (LR) test statistic to compare the fit of the unrestricted model (model
2 or model 3) and the restricted model (the primary model specification with
symmetry and homogeneity restrictions imposed, called model 1) which is a
special case of the other. As table 4 and table 5 show, the null hypotheses (the
restricted model has the same goodness of fit as the unrestricted model) can
be rejected for the AIDS and the QUAIDS for all years (except for the 1993
QUAIDS comparison of model 1 and model 2) with a level of significance
smaller than 0.05. This means that the observed household data sets are
not allowing the conclusion that the neoclassical assumptions of demand
functions that are homogeneous of degree 0 in prices and expenditure and
symmetric Slutsky matrices were fulfilled. So the behaviour of the households
is not free from money illusion and for the Hicks cross-price elasticities the
relation
∂qf1 ∕∂pj = ∂qjl∕∂pi is not valid.

To complete the test of the theoretical restrictions of the AIDS and
QUAIDS, we have to test the monotonicity condition, which requires a cost
function that is monotonically increasing in prices and the concavity condi-
tion that the cost function is concave. [10]Chalfant et al. (1991) show that
the monotonicity condition is fulfilled if the fitted budget shares of all
N -
goods of the model are laying in the interval between 0 and 1. Testing the
concavity condition is by far more complex. A cost function is concave at a
certain data point, if the matrix of the second-order derivatives of the cost
function - the so called Slutsky matrix - is negative semi-definite. Most of
the demand studies proof the negative semidefiniteness of the Slustky matrix
by only checking the sign of the diagonal elements of the Slustky matrix.

30



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