  We develop an approach that resolves a {\it 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 (2003), where the
endogenous covariate is continuous. Suppose $X$ is a $d$-dimensional endogenous
random variable, $Z_1$ and $Z_2$ are the instrumental variables (vectors), and
$Z=\left(\begin{array}{c}Z_1 \\Z_2\end{array}\right)$. Now, assume that the
conditional distributions of $X$ given $Z$ satisfy the conditions sufficient
for solving the identification problem as in Newey and Powell (2003) or as in
Proposition 1.1 of the current paper. That is, for a function $\pi(z)$ in the
image space there is a.s. a unique function $g(x,z_1)$ in the domain space such
that $$E[g(X,Z_1)~|~Z]=\pi(Z) \qquad Z-a.s.$$ In this paper, for a class of
conditional distributions $X|Z$, we produce an orthogonal polynomial basis
$Q_j(x,z_1)$ such that for a.e. $Z_1=z_1$, and for all $j \in \mathbb{Z}_+^d$,
and a certain $\mu(Z)$, $$P_j(\mu(Z))=E[Q_j(X, Z_1)~|~Z ],$$ where $P_j$ is a
polynomial of degree $j$. This is what we call solving the {\it polynomial
basis problem}.
  Assuming the knowledge of $X|Z$ and an inference of $\pi(z)$, our approach
provides a natural way of estimating the structural function of interest
$g(x,z_1)$. Our polynomial basis approach is naturally extended to Pearson-like
and Ord-like families of distributions.
