Food Prices and Overweight Patterns in Italy

^p

Xt = μ0 + μiT + ^φhDth + ^ AiXt-i + ^εt               (3)

i=1

where X_1t = (w_1t,, w_2t, ...w_n,_t) is the n * 1 vector of budget shares and X_2t =
(p_1t,p_2t, ...pn_t,y_t) is the n + 1 * 1 vector containing price indices and real expen-
diture. μ₀ is a n * 1 constant term, μ₁ is a n * 1 vector of coefficients related to
the deterministic trend T, Dth is a vector containing deterministic variables (in our
application centred seasonal dummies) and φ_h the corresponding n * h matrix of
parameters. A_i is a matrix of unknown parameters for the lags of X_t , ε_t is a Gaus-
sian white noise process with covariance matrix Ω and p the lag order of the VAR.
Equation (3) may be re-written in a VECM form as:

p-1

^δx∣. = μo ^{+ π}X_t*_-1 ⁺ ^ l'^X/-i ^{+ ε}t                  (4)

i=1

where Π = (∑P=₁ Ai — I_p), Γi = — ∑p=i₊₁ Aj,with j = 1, ....p — 1. The matrix of
parameters Π describes the long-run relationships of the VECM among the variables
in vector X_t*_-1 = [X_t-1; D_t; T]. Γi, with i = 1,...k — 1, is a vector of parameters which
refers to the short-run dynamics of the system ∆Xt-i . In known general conditions,
VECM equation (4) is formulated as:

_′           p-1

^δχ = μo + ^αβ * X_t*_-ι + ^ I'^Xt-i + ^εt                (5)

i=1

where α is a n * r matrix, β^* = (n + Υ) * r matrix and r(0 < r < Q = 2n) is the
cointegration rank of the demand system. Υ is the matrix containing deterministic
and seasonal components.

The hypothesis of stochastic trends in budget shares predicts a convergence (in
the long-run) towards steady state, and these values are proxied by their intercepts
(Pesaran and Shin, 2002).

Pesaran and Shin (2002) also show that, to recover exactly the long-run struc-
tural parameters of model (5), r restrictions on cointegrating relationships must be
imposed on each non-singular demand equation, expressed in budget shares. In this
context, the adding-up theoretical constraint executes a crucial role in identifying
the structural model, implying a further implicit restriction of the rank of cointegra-
tion of the VAR model, i.e., r = n — 1. Formally, disregarding deterministic terms,
the matrix of cointegration vectors for the demand system is specified as:

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