β ' =[-Wn-ι,B] (6)
where W is the n — 1 * n — 1 budget share matrix, and B the matrix of the
parameters for log prices and real income. The long-run identifying restrictions can
be imposed in the W matrix. A necessary and sufficient condition indicates that
the number of identifying conditions k, is at least equal to r2 , and that the exact
identification for a long-run AI demand system requires r2 = (n — 1)2 restrictions.
This implies that, for a demand system with three good categories, a diagonal
framework for identification may be imposed as:
β' =[—In-1, B] (7)
where In-1 is the unit matrix, and k — r2 over-identifying restrictions can be
imposed and tested directly on cointegrating vectors. In the demand system con-
text, the restrictions are derived from the theory and concern the hypotheses of
homogeneity and symmetry (Deaton and Muellbauer, 1980). Appendix B lists the
matrices of the dynamic demand system described in (6) and (7) and those with
imposed the theoretical over-identifying restrictions.
The maximum likelihood estimations of cointegration matrix β' are carried out
by the ML estimator, which is super-consistent and mixed normal. This allows
us to test over-identifying restrictions by a log-likelihood ratio statistic which is
asymptotically distributed as an χ2 , with degrees of freedom equal to the number
of over-identifying restrictions imposed.
As already observed, the cointegration rank for identifying the patterns of ad-
justment in each budget share implies r = n — 1. Thus, the rank condition excludes
all the cases in which r < n — 1. Although in the empirical analysis of demand
systems it may happen that the rank of cointegration is higher than the number
of cointegration relations, i.e., r > n — 1, in our study this hypothesis is econom-
ically inadmissible. This should imply additional cointegration relations involving
price variables in the cointegration space. However, the consequence of this over-
dimensionality in equilibrium relationships is in contrast with the causal impact of
relative prices on the evolution of budget share consumption and in the measure of
effects of substitution.
13
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