The stochastic specification of demand share equations: restricting predicted budget shares to the unit simplex

In the context of estimating systems of demand share equations, the argument that "the component (shares) predicted or implied by the model should be non-negative and sum to (unity)" (Bewley (1980), p.2) is not new. As a result, the aggregation restrictions should play a central role in the analysis of demand· share equations.The traditional approach to estimating such systems of demand share equations is to append a multivariate normal stochastic component and carry out the estimation. However, such a specification does not exploit the aggregation restriction to its full extent. That is, although we may specify a deterministic component that is restricted to the (0,1) interval, the use of an additive multivariate normal error term means that there is, at least in theory, a non-zero probability that the shares implied or predicted by the model will be outside the(0,1) interval. [...]

Author requested conversion to open access 04 Apr 2022