| Abstract
| - We develop an approach that resolves a polynomial basis problem for a class of models with discrete endogenous covariate, and for a class of econometric models considered in the work of Newey and Powell [ ], where the endogenous covariate is continuous. Suppose X is a d-dimensional endogenous random variable, Z1 and Z2 are the instrumental variables (vectors), and \hbox{$Z=\binom{Z_1}{Z_2}$} . Now, assume that the conditional distributions of X given Z satisfy the conditions sufficient for solving the identification problem as in Newey and Powell [ ] or as in Proposition 1.1 of the current paper. That is, for a function π( z) in the image space there is a.s. a unique function g( x,z1) in the domain space such that \begin{eqnarray*} E[g(X,Z_1)~|~Z]=\pi(Z) \qquad Z-{\rm a.s.} end{eqnarray*} In this paper, for a class of conditional distributions X | Z, we produce an orthogonal polynomial basis { Qj( x,z1) } j = 0 ,1 ,... such that for a.e. Z1 = z1, and for all \hbox{$j \in \mathbb{Z}_+^d$}, and a certain μ( Z) , \begin{eqnarray*} P_j(\mu(Z))=E[Q_j(X, Z_1)~|~Z ], end{eqnarray*} where Pj is a polynomial of degree j. This is what we call solving the polynomial basis problem. Assuming the knowledge of X | Z and an inference of π( z) , our approach provides a natural way of estimating the structural function of interest g( x,z1) . Our polynomial basis approach is naturally extended to Pearson-like and Ord-like families of distributions.
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