Orthogonal Polynomials for Seminonparametric Instrumental Variables Model

TL;DR

This paper introduces a method using orthogonal polynomials to solve the polynomial basis problem in econometric models with endogenous variables, enabling better estimation of structural functions.

Contribution

It develops an orthogonal polynomial basis approach for models with discrete and continuous endogenous covariates, extending to Pearson-like and Ord-like distribution families.

Findings

Constructed orthogonal polynomial basis for specific conditional distributions

Provides a natural estimation method for structural functions

Extends approach to Pearson-like and Ord-like distribution families

Abstract

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.