# Orthogonal Polynomials for Seminonparametric Instrumental Variables Model

## Author Info

Listed author(s):
• Yevgeniy Kovchegov
• Nese Yildiz
Registered author(s):

## 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.

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File URL: http://arxiv.org/pdf/1409.1620

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1409.1620.

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## References

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1. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(03), pages 497-521, June.
2. Hoderlein, Stefan & Holzmann, Hajo, 2011. "Demand Analysis As An Ill-Posed Inverse Problem With Semiparametric Specification," Econometric Theory, Cambridge University Press, vol. 27(03), pages 609-638, June.
3. Horowitz, Joel L. & Lee, Sokbae, 2012. "Uniform confidence bands for functions estimated nonparametrically with instrumental variables," Journal of Econometrics, Elsevier, vol. 168(2), pages 175-188.
4. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
5. Severini, Thomas A. & Tripathi, Gautam, 2006. "Some Identification Issues In Nonparametric Linear Models With Endogenous Regressors," Econometric Theory, Cambridge University Press, vol. 22(02), pages 258-278, April.
6. Xiaohong Chen & Demian Pouzo, 2012. "Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals," Econometrica, Econometric Society, vol. 80(1), pages 277-321, 01.
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