IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1409.1620.html
   My bibliography  Save this paper

Orthogonal Polynomials for Seminonparametric Instrumental Variables Model

Author

Listed:
  • Yevgeniy Kovchegov
  • Nese Yildiz

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.

Suggested Citation

  • Yevgeniy Kovchegov & Nese Yildiz, 2014. "Orthogonal Polynomials for Seminonparametric Instrumental Variables Model," Papers 1409.1620, arXiv.org.
  • Handle: RePEc:arx:papers:1409.1620
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1409.1620
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    as
    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, September.
    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, January.
    Full references (including those not matched with items on IDEAS)

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1409.1620. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators). General contact details of provider: http://arxiv.org/ .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.