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Minimum Integrated Distance Estimation in Simultaneous Equation Models

This paper considers estimation and inference in semiparametric econometric models. Standard procedures estimate the model based on an independence restriction that induces a minimum distance between a joint cumulative distribution function and the product of the marginal cumulative distribution functions. This paper develops a new estimator which generalizes estimation by allowing endogeneity of the weighting measure and estimating the optimal measure nonparametrically. The optimality corresponds to the minimum of the integrated distance. To accomplish this aim we use Kantorovich's formulation of the optimal transportation problem. The minimizing distance is equivalent to the total variation distance and thus characterizes finer topological structures of the distributions. The estimation also provides greater generality by dealing with probability measures on compact metric spaces without assuming existence of densities. Asymptotic statistics of the empirical estimates have standard convergent results and are available for different statistical analyses. In addition, we provide a tractable implementation for computing the estimator in practice.