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



As [1]Alley et al. (1992) show, the non-positivity of the diagonal elements
of the Slnstky matrix is only a necessary but not a sufficient condition for
the negative semidefiniteness of the Slustky matrix. A necessary and suf-
ficient condition for the negative semidefiniteness of the Slustky matrix is
the non-positiviy of all the eigenvalues of the matrix. An additional draw-
back of many demand studies is that the proof of negative semidefiniteness
of the Slustky matrix is only conducted at one data point, typically at the
mean of the price and expenditure observations. By contrast, we are calcu-
lating the eigenvalues of the Slutsky matrix at each data point of our sample.
Table 6 provides the percentage of observations which does not violate the
monotonicity and concavity conditions. The above described test of the con-
cavity condition is carried out both with the observed and the fitted values of
the budget shares
wi. On the first view, the results of the concavity test dis-
play a broad rejection of the concavity condition. A closer inspection of the
particular eigenvalues reveals that for the wide majority of the observations
only one of the
N-eigenvalues is not negative. So we can conclude similarly
to [1]Alley et al. (1992) that our cost functions have a “weakly” non-concave
shape. The rejection of the symmetry, homogeneity and concavity condi-
tions for most of the estimated models is a clear sign that the households
didn’t behave in the neoclassical way. This finding is in line with most of
the empirical demand analysis (as summarized for example by Cozzarin and
Gilmour (1998)) and was already expected by [11]Deaton and Muellbauer
(1980) when they formalized the AIDS. As already mentioned in the third
section, the AIDS is nested by the QUAIDS. So we can conduct a LR test
to see if the restricted model (AIDS) has the same goodness of fit as the
unrestricted model (QUAIDS). Table 7 shows that for all four data sets the
null hypotheses, that the AIDS has the same goodness of fit as the QUAIDS,
can be rejected on a level of significance smaller than 0.05.

With the estimated parameters of the AIDS and QUAIDS for 1988 we
calculate COLI time series from 1988 to 2009. The expenditure of the repre-
sentative household to attain the cost of living can be calculated by inserting
the price data for the nine commodities out of the official German consumer
price statistics from 1988 to 2009. For the same nine goods, a traditional
Laspeyres price index is calculated by using a weighting scheme obtained
from the 2581 household observations of 1988s EVS that we have used to
estimate the AIDS and QUAIDS. Table 8 presents the results of the AIDS-
and QUAIDS-COLI and the Laspeyres type CPI with base year 1988. To
compare our AIDS- and QUAIDS COLI with superlative index numbers, we

31



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