Orthogonal polynomials for seminonparametric instrumental variables model^∗

¹ Department of Mathematics, Oregon State University, Kidder Hall, Corvallis, OR 97331, USA
kovchegy@math.oregonstate.edu
² Department of Economics, University of Rochester, 231 Harkness Hall, Rochester, NY 14627, USA
nese.yildiz@rochester.edu

Received: 12 February 2013
Revised: 4 September 2014

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 [17], where the endogenous covariate is continuous. Suppose X is a d-dimensional endogenous random variable, Z₁ and Z₂ are the instrumental variables (vectors), and $\mathit{Z}\mathrm{=}{{{}^{\mathrm{(}}\mathit{Z}_{\mathrm{2}}}_{{\mathit{Z}}_{\mathrm{1}}}^{}}^{\mathrm{)}}$ . Now, assume that the conditional distributions of X given Z satisfy the conditions sufficient for solving the identification problem as in Newey and Powell [17] 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 $\begin{array}{ccc}\mathit{E}\mathrm{\left[}\mathit{g}\mathrm{\right(}\mathit{X,}{\mathit{Z}}_{\mathrm{1}}\mathrm{)}\mathrm{|}\mathit{Z}\mathrm{]}\mathrm{=}\mathit{\pi }\mathrm{(}\mathit{Z}\mathrm{)} \mathit{Z}\mathrm{-}\mathrm{a}\mathit{.}\mathrm{s}\mathit{.}& & \end{array}$ In this paper, for a class of conditional distributions X | Z, we produce an orthogonal polynomial basis { Q_j(x,z₁) } _{j = 0,1,...} such that for a.e. Z₁ = z₁, and for all $\mathit{j}\mathrm{\in }{Z}_{\mathrm{+}}^{\mathit{d}}$, and a certain μ(Z), $\begin{array}{ccc}{\mathit{P}}_{\mathit{j}}\mathrm{\left(}\mathit{\mu }\mathrm{\right(}\mathit{Z}\mathrm{\left)}\mathrm{\right)}\mathrm{=}\mathit{E}\mathrm{\left[}{\mathit{Q}}_{\mathit{j}}\mathrm{\right(}\mathit{X,}{\mathit{Z}}_{\mathrm{1}}\mathrm{)}\mathrm{|}\mathit{Z}\mathrm{]}\mathit{,}& & \end{array}$ where P_j 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.