Identification via completeness for discrete covariates and orthogonal polynomials

We solve a class of identification problems for nonparametric and semiparametric models when the endogenous covariate is discrete with unbounded support. Then we proceed with an approach that resolves a polynomial basis problem for the above class of discrete distributions, and for the distributions given in the sufficient condition for completeness in Newey and Powell (2003). Thus, in addition to extending the set of econometric models for which nonparametric or semiparametric identification of structural functions is guaranteed to hold, our approach provides a natural way of estimating these functions. Finally, we extend our polynomial basis approach to Pearson-like and Ord-like families of distributions.