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:
The overall price index, P, is defined by the following translog specification:
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.
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
(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.
Here is my R code to obtain a value for α0, and to estimate the 3-equation system with both the homogeneity and symmetry restrictions imposed on the parameters.
Here are the basic results, which are self-explanatory:
We see that the total expenditure (“income”) elasticities suggest that beer is a necessity, while spirits and (especially) wine are both luxury goods. It would be a good idea to recall that the data are for U.K. for the period 1955 to 1985! Each good has own-price elasticities of demand that are negative, as expected. Beer is own-price inelastic, while wine and spirits are own-price elastic. The compensated price elasticities suggest, among other things, that each of the three beverages are substitutes.
The “summary” command in my code yields some additional results:
So, these test results suggest that:
- We should reject the symmetry restrictions when they are added to the homogeneity restrictions. ( p = 0.0003).
- We should reject the homogeneity restrictions against the alternative of no restrictions, other than Engel aggregation. (p = 4*10-6).
- We should reject the (joint) symmetry and homogeneity restrictions in favour of no restrictions. other than Engel aggregation. (p = 3*10-8).