Ensuring positive shares in homothetic demand system estimation

Question

I am performing a demand system estimation, and a counter-factual exercise which involves estimating demands at prices quite far from the observed ones. I know a priori that preferences are homothetic, but I want full flexibility given this restriction.

(See Barnett and Serletis (2008; JoE) for a review of the demand system estimation literature.)

Given these constraints, one possibility would be to use the homothetic translog specification in which the shares are given by:

$$s_i=\alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j},$$

where $\sum_{i=1}^n {\alpha_i}=1$ and $\sum_{i=1}^n {\beta_{ij}}=0$ for all $j$, and where $\beta_{ij}=\beta_{ji}$ for all $i,j$.

However, this can easily produce negative shares in counter-factual exercises, when demand is evaluated far from the observed prices.

Is there an approach for homothetic demand system estimation that ensures positivity of the shares?

The logit approach seems to have issues with symmetry (see e.g. Dumagan and Mound (1996; EM)). The homothetic normalized quadratic flexible functional forms from Diewert and Wales (1988; JBES) do not seem to guarantee positive shares. The miniflex Laurent (Barnett and Lee 1985; ECMA) does not seem to have a natural homothetic special case. (I imagine it is not valid just to set income to one in the share equations, replacing $v_i$ by $p_i$?)

Any suggestions would be appreciated!

Answer 1

If your objective is to enforce that shares are comprised between zero and one, there are several nonlinear transformations of your linear model available, as for instance $$s_i= \min\{ \max\{ \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j}, 0 \}, 1\} ,$$ $$s_i= \big( \sin( \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j} ) + 1 \big) / 2,$$ $$s_i= F( \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j} ) ,$$ where $F$ represents any cdf of your choice. However, usually violations of the theoretical properties indicate that your model is misspecified, and you should consider either to remove your homotheticity assumption, or include fixed (individual and time) effects or generalize the specification of your error term.

Answer 2

Usually these systems approximate the "true" demand model at a given point (obs. Shares and prices), and the approximation becomes worse the further you are away from this point. I don't think there's a demand system which provides a global approximation for the "true" demand system.