Estimating Discrete Joint Probability Distributions for Demographic Characteristics at the Store Level Given Store Level Marginal Distributions and a City-Wide Joint Distribution

This paper provides a solution to the problem of estimating a joint distribution using the associated marginal distributions and a related joint distribution. The particular application we have in mind is for estimating joint distributions of demographic characteristics corresponding to market areas for individual retail stores. Marginal distributions are generally available at the census tract level, but joint distributions are only available for Metropolitan Statistical Areas which are generally much larger than the market for a single retail store. Joint distributions over demographics are an important input into mixed logit demand models for aggregate data. Market shares that vary systematically with demographics are essential for relieving the restrictions imposed by the Independence of Irrelevant Alternative property of the logit model. We approach this problem by formulating a parametric function that incorporates both the city-wide joint distributional information and marginal information specific to the retail store's market area. To estimate the function, we form moment conditions equating the moments of the parametric function to observed data, and we input these into a GMM objective. In one of our illustrations we use four marginal demographic distributions from each of eight stores in Dominick's Finer Foods data archive to estimate a four dimensional joint distribution for each store. Our results show that our GMM approach produces estimated joint distributions which emit marginals that closely match the observed marginal distributions. Mixed logit demand estimates are also presented with show the estimates to be sensitive to the formulation of the demographics distribution