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

The traditional approach to estimating systems of share equations is to append a multivariate normal stochastic component to the deterministic component. The adding up restrictions are then exploited to delete one equation and estimation is carried out by maximum likelihood. Such an approach leads to an implied model which violates economic theory, as there remains a nonzero probability of shares being negative or greater than unity. In this paper we propose the use of a method from the statistics literature, compositional data analysis, and a distribution familiar to statisticians, the additive logistic normal, for the estimation of a system of demand share equations. Key words: Stochastic; Demand share; Simplex; Additive logistic normal; Dirichlet distribution; Multivariate normality