Estimation of Coherent Demand Systems with Many Binding Non-Negativity Constraints

Two econometric issues arise in the estimation of complete systems of producer or consumer demands when many non-negativity constraints are binding for a large share of observations, as frequently occurs with micro-level data. The first is computational. The econometric model is essentially an endogenous switching regimes model which requires the evaluation of multivariate probability integrals. The second is the relationship between demand theory and statistical coherency. If the indirect utility or cost function underlying the demand system does not satisfy the regularity conditions at each observation, the likelihood is incoherent in that the sum of the probabilities for all demand regimes is not unity and maximum likelihood estimates are inconsistent. The solution presented is to use the Gibbs Sampling technique and data augmentation algorithm and rejection sampling, to solve both the dimensionality and coherency problem. With rejection sampling one can straightforwardly impose only the necessary conditions for coherency, coherency at each data point rather than global coherency. The method is illustrated with a series of simulated demand systems derived from the translog indirect random utility function. The results highlight the importance of imposing regularity when there are many non-consumed goods and the gains from imposing such conditions locally rather than globally.