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In this article, we consider a class of discrete choice models in which consumers care about a finite set of product characteristics. These models have been used extensively in the theoretical literature on product differentiation and the goal of this article is to translate them into a form that is useful for empirical work. Most recent econometric applications of discrete choice models implicitly let the dimension of the characteristic space increase with the number of products (they have “tastes for products”). The two models have different theoretical properties, and these, in turn, can have quite pronounced implications for both substitution patterns and for the welfare impacts of changes in the number and characteristics of the goods marketed. After developing those properties, we provide alternative algorithms for estimating the parameters of the pure characteristic model and compare their properties to those of the algorithm for estimating the model with tastes for products. We conclude with a series of Monte Carlo results. These are designed to illustrate: (i) the computational properties of the alternative algorithms for computing the pure characteristic model, and (ii) the differences in the implications of the pure characteristic model from the models with tastes for products.