Food Prices and Overweight Patterns in Italy

4 Econometric framework, data issues and elasticities

This subsection provides econometric support for modelling a long-run demand sys-
tem which includes a gradual adjustment over time of consumption in response to
shifts in relative prices. However, when time-series have a significant dimension,
empirical demand system studies suffer from severe econometric flaws, because the
time-series of budget shares, prices and real income are non-stationary.

One way of solving these issues is to use linear model cointegration methods
(Attfield, 1997, 2004) although they may not be completely consistent, since errors
in demand systems tend to be autocorrelated (Lewbel and Ng, 2005). Standard
asymptotic theory may provide a poor guide to finite-sample inference when the
errors are persistent in a cointegrated demand system. As one aim of the empirical
strategy, we complement statistical analysis by investigating and testing the non-
stationarity behaviours of the time-series residuals.

There is also a profound policy interest in obtaining parameter estimations from
a cointegration framework. They are inextricably linked with the notion of long-run
estimation (Pesaran, 1997). Because we are specifically interested in analysing sub-
stitutability effects of healthy foods with respect to unhealthy ones, an important
question for policy-makers is whether these trends will continue in the future. In-
deed, a measure of the long-run elasticity of substitution is in being a powerful tool in
assessing potential government intervention in preventing obesity by taxes imposed
on unhealthy foods or subsidies applied to healthy ones (Powell and Chaloupka,
2009). In subsection 4.1, we report the conditions for the identification of a long-run
demand system based on the cointegration rank of a vector autoregressive (VAR)
model under the theoretical constraint of adding-up. We then show that, in this
framework, the other theoretical restrictions of homogeneity and symmetry can be
imposed and tested, and the estimated parameters recovered to calculate the price
elasticities.

4.1 Methods

We formalise the specification of the equations of the demand system in (1) as a coin-
tegrated demand system. Firstly, we consider the vector autoregressive (VAR) for-
mulation of a demand system and describe the corresponding vector error correction
(VECM) representation, following Johansen (1995). Formally, the data-generating
process for X_t = (X_1t, X_2t) is assumed to belong to the class of VAR models:

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