Working Papers 2000 – Abstracts
2/2000
An EM Algorithm for Modelling Variably-Aggregated Demand
Simone Grose and Keith McLaren
The response of consumer demand to prices, income, and other characteristics
is important for a range of policy issues. Naturally, the level of detail
for which consumer behaviour can be estimated depends on the level of disaggregation
of the available data. However, it is often the case that the available data
is differently aggregated in different time periods, with the information
available in later time periods usually being more detailed. The applied researcher
is thus faced with choosing between detail, in which case the more highly
aggregated data is ignored; or duration, in which case the data must be aggregated
up to the "lowest common denominator". Furthermore, since parametric demand
systems invariably involve a large number of parameters, with the number increasing
at least linearly with the number of expenditure categories, it may well be
that only the second option is feasible. That is, there is simply not enough
data available at the finer aggregation level for the chosen model to be estimated.
This paper develops an EM algorithm for the estimation of a consumer demand
system involving variably aggregated data. The methodology is based on the
observation that more highly aggregated data does in fact contain information
on the finer subcategories. It is therefore possible, under certain simplifying
assumptions, to derive the distribution of the unobserved fine-level expenditures
conditional on the observed but more highly aggregated data. The expectation
of the log-likelihood is then taken with respect to this conditional distribution.
Under the assumption of multivariate normality both these steps can be performed
analytically, resulting in an EM criterion that can be maximised iteratively
at comparatively little cost. The technique is applied to an ABS dataset containing
historical information relating to private final consumption expenditures
on up to 18 commodities.
Keywords: EM Algorithm, Singular demand systems, Linear expenditure
system, Missing data.
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