Identifying Heterogeneity in Economic Choice and Selection Models Using Mixtures

We show how to nonparametrically identify the distribution of heterogeneity using the linear independence of a class of structural economic choice models. We state an economic property known as reducibility and prove that reducibility ensures linear independence and hence identi cation. Reducibility makes verifying the identi cation of nonlinear models easy. We can allow for a nonparametric distribution over nonparametric functions of the data. We use our mixtures framework to prove identi cation in three classes of economic models: 1) continuous outcomes including triangular systems and simultaneous equations, 2) multinomial discrete choice, and 3) selection and mixed continuous-discrete choice. We rely on linear independence, not identi cation at in nity. For selection, we allow for essential heterogeneity in both the selection and outcome equations and fully identify the joint distribution of outcomes. 1 The Identi cation Problem Heterogeneity is important in many economic problems. Some consumers may value a product characteristic more than others, so that the consumers with higher values might have less elastic demands. Firms that bene t the most from adopting a new technology might adopt it sooner, so that the returns of early adopters exceed the returns of late adopters. Workers may pick jobs in the sector where they earn the most, so that the wages of a worker who loses his preferred position may go down substantially. Drivers that buy expensive cars may be likely to use them more, so a tax on gasoline will hurt those consumers the most. Firms that enter a market may ∗Thanks to Steven Durlauf, Azeem Shaikh, Morten Sørensen and Edward Vytlacil for helpful comments. Also thanks to seminar participants at Chicago, Cowles, Stanford and Wisconsin. Fox thanks the National Science Foundation, the Olin Foundation, and the Stigler Center for nancial support. Thanks to Chenchuan Li for research assistance. Our email addresses our [email protected] and [email protected].