Food Prices and Overweight Patterns in Italy

APPENDIX B. Identification of long-run AI

The cointegration relationships in the VECM equation (6), subject to reduced rank
restrictions on the Π = αβ* matrix, are not identified. Following Pesaran and Shin
(2002), the identification of the long-run parameters in β* requires the imposition
of r restrictions on each cointegrating vector, although a necessary and sufficient
condition (order condition) for the identification is that the number of the identifying
restrictions, k, should be at least equal to r² .

In order to explain these fundamental identifying conditions in our demand sys-
tem with three categories of goods, we note first that adding up reduces the rank to
two, i.e. r = (n - 1). As a formal extension of the ECM vectors in equation (6), let
us consider two non-identified cointegrating vectors made up of the variables w1t ,
w2t , lnP1t, lnP2t, lnP3t, ln(yt/pt) and the intercept. The associated parameters are:

{ⁱ'l I   ^β21  ^β31  ^β41  ^β51  ^β61  ^β71  I                    ,    ,

(B.1)

β12  β22  β32  β42  β52  β62  β72 ^I

The exact identifying restrictions r² = (n - 1)² = 4 assume a diagonal structure
because theory suggests that budget shares responds mainly respond to own and
cross-price changes and income impulses, but not to (endogenous) changes in other
budget shares. Formally,

^β11 = ^{- 1, β}12 = ⁰ I

β21 = 0,   β22 = -1 ^I

so that the cointegrating vectors may be written as

^-1   0   β31  β41  β51  β61  β71

0   ^-1  β32  β42  β52  β62  β72

In order to test theoretical restrictions, long-run parameter restrictions should be
included. As discussed in the text, the property of symmetry may be imposed as
a cross-equation restriction, β32 = β41. The cointegrating vectors thus assume the
following structure:

^{-1   0   β}31   *   ^β51  ^β61  ^β71

0   ^-1  β32  β42  β52  β62  β72

Estimations of cointegrating vectors subject to symmetry is tested by the LR

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