I was delighted by yesterday’s announcement that Angus Deaton has been awarded the Nobel Prize in Economic Science this year. His contributions have have been many, fundamental, and varied, and I certainly won’t attempt to summarize them here. Suffice to say that the official citation says that the award is “for his contributions to consumption, poverty, and welfare”.
In this earlier post I made brief mention of Deaton’s path-breaking work, with John Muellbauer, that gave us the so-called “Almost Ideal Demand System”. The AIDS model took empirical consumer demand analysis to a new level. It facilitated more sophisticated, and less restrictive, econometric analysis of consumer demand behaviour than had been possible with earlier models. The latter included the fundamentally important Linear Expenditure System (Stone, 1954), and the Rotterdam Model (Barten, 1964; Theil, 1965).
I thought that readers may be interested in an empirical exercise with the AIDS model. Let’s take a look at it.
First of all,the theoretical model needs to be explained.Let total expenditure on the n goods in the system be
M = Σi (pi qi) , (1)
wi = (pi qi) / M ; i = 1, …, n (2)
denote the “budget share” for the ith good.The system of demand equations itself is:
wi = αi + βi [log(M) – log(P)] + Σj γij log(pj) + εi ; i = 1, …., n (3)
The overall price index, P, is defined by the following translog specification:
log(P) = α0 + Σj αj log(pj) + 0.5 Σi Σj γij log(pi) log(pj) . (4)
(All of the summations in equations (1) to (4) run from i = 1 to n; or from j = 1 to n.)Notice that once (4) is substituted into (3), each of the n equations in the latter system is highly non-linear in the parameters of the model.
In practice, a value for α0 is usually pre-assigned, and there are various ways of choosing an “optimal” value (see Michalek and Keyzer, 1992).
One of the things that I like about empirical exercises such as the one that follows is that they illustrate how the underlying microeconomic theory can be incorporated explicitly into the formulation of the econometric model, and the subsequent estimation and testing.
(This stands in contrast with a lot of other empirical work that we encounter – see this post.)
Specifically, for the AIDS model there are various restrictions on the parameters that we have to consider:
Engel aggregation requires that
Σk αk = 1 ; Σk βk = 0 ; Σk γkj = 0 ; for all j = 1, …., n (5)
These restrictions will be satisfied automatically, a long as the individual expenditures add up to total expenditure in the sample.Homogeneity requires that
Σkγik = 0 ; for all i = 1, …., n ; (6)
and Slutsky symmetry requires that
γij = γji ; for all i, j = 1, …., n . (7)
(In equations (5) and (6), the summations run from k = 1 to n.)
The homogeneity and symmetry restrictions are testable, and can be imposed, as appropriate.
We’re going to estimate an AIDS model for beer, wine, and spirits (numbered in that order), using annual time-series data for the U.K. over the period 1955 to 1985 inclusive. The data are available on the data page for this blog, and they come from Selvanathan, 1995, p.124). I’m going to use the ‘micEconAids’ package for R (Henningsen, 2015) for estimation and hypothesis testing, and my R code is on the code page for this blog.
In the following application, we’ll assume weak separability of the underlying utility function, so “Total Expenditure” (M) will be the total expenditure on the three types of alcoholic beverages. ….