Density Weighted Linear Least Squares

for an unknown vector of parameters β0 and an unknown univariate function τ(·). This model is implied by many important limited dependent variable and regression models, as discussed in Ruud (1986) and Stoker (1986). Consistent estimators for β0, up to an unknown scale factor, have been developed by Ruud (1986), Stoker (1986), Powell, Stock, and Stoker (1989), Ichimura (1993), and others. In this paper, we return to a type of estimator developed by Ruud (1986). He proposed an inverse-density-weighted quasi-maximum likelihood estimator. We consider least squares estimation that is weighted by the ratio of an elliptically symmetric density with compact support to a kernel estimator of the true density. We give conditions for √ n-consistency and asymptotic normality of the estimator, and derive a consistent estimator for the asymptotic variance. We also show that the first-order conditions for the scaled least squares coefficients has an analogous form to the efficient score for an index model. This form is used to suggest ways to choose weights that have high efficiency. Among the semi-parametric index estimators, the inverse-density-weighted least squares estimator is unique because it permits discontinuities in the transformation τ . Discontinuities in the conditional expectation of dependent variables arise in such economic problems as optimization over nonlinear budget sets and production frontiers. In labor supply for example, nonconvexities in