Orthogonal polynomials for seminonparametric instrumental variables model

Kovchegov, Yevgeniy; Yıldız, Neşe

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 293-306

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, $Z_1$ and $Z_2$ are the instrumental variables (vectors), and $Z=\left(\genfrac{}{}{0pt}{}{Z_1}{Z_2}\right)$ . Now, assume that the conditional distributions of $X$ given $Z$ satisfy the conditions sufficient for solving the identification problem as in Newey and Powell [17] or as in Proposition 1.1 of the current paper. That is, for a function $\pi \left(z\right)$ in the image space there is a.s. a unique function $g(x,z_1)$ in the domain space such that

$$\begin{array}{c}\hfill E\left[g(X,Z_1)\right|Z]=\pi \left(Z\right)Z-\mathrm{a}.\mathrm{s}.\end{array}$$

In this paper, for a class of conditional distributions $X|Z$, we produce an orthogonal polynomial basis ${\left\{Q_j(x,z_1)\right\}}_{j=0,1,...}$ such that for a.e. $Z_1=z_1$, and for all $j\in {\mathbb{Z}}_{+}^d$, and a certain $\mu \left(Z\right)$,

$$\begin{array}{c}\hfill P_j\left(\mu \left(Z\right)\right)=E\left[Q_j(X,Z_1)\right|Z],\end{array}$$

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 $\pi \left(z\right)$, 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.

Classification : 33C45, 62, 62P20
Keywords: Orthogonal polynomials, Stein’s method, nonparametric identification, instrumental variables, semiparametric methods

A.D. Barbour, Stein’s method for diffusion approximations. Probab. Theory Relat. Fields 84 (1990) 297–322. | MR | Zbl

A.D. Barbour and L.H.Y. Chen, An Introduction to Stein’s Method. Singapore University Press (2005). | MR | Zbl

R. Blundell, X. Chen and D. Kristensen, Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75 (2007) 1613–1669. | MR | Zbl

X. Chen and D. Pouzo, Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80 (2012) 277–321. | MR | Zbl

X. Chen and M. Reiss, On rate optimality for Ill-posed inverse problems in econometrics. Econ. Theory 27 (2011) 497–521. | MR | Zbl

L.H.Y. Chen, L. Goldstein and Q.M. Shao, Normal Approximation by Stein’s Method. Springer (2011). | MR | Zbl

V. Chernozhukov, P. Gagliardini and O. Scaillet, Nonparametric Instrumental Variable Estimation of Quantile Structural Effects. Working Paper, HEC University of Geneva and Swiss Finance Institute (2008).

S. Darolles, J.P. Florens and E. Renault, Nonparametric instrumental regression. Econometrica 79 (2006) 1541–1565. | MR | Zbl

P. Hall and J.L. Horowitz, Nonparametric methods for inference in the presence of instrumental variables. Ann. Stat. 33 (2005) 2904–2929. | MR | Zbl

S. Hoderlein and H. Holzmann, Demand analysis as an ill-posed problem with semiparametric specification. Econ. Theory 27 (2011) 460–471. | MR | Zbl

L. Hörmander, An Introduction to Complex Analysis in Several Variables. In vol. 7, 2nd edition. North-Holland Mathematical Library (1973). | MR | Zbl

J.L. Horowitz and S. Lee, Uniform confidence bands for functions estimated nonparametrically with instrumental variables. J. Econometrics 168 (2012) 175–188. | MR | Zbl

N.L. Johnson, S. Kotz and A.W. Kemp, Univariate Discrete Distributions, 2nd edition. Wiley Series in Probability and Statistics (1992). | MR | Zbl

Y.V. Kovchegov and N. Yıldız, Identification via Completeness for Discrete Covariates and Orthogonal Polynomials. Oregon State University Technical Report (2011).

E.L. Lehmann, Testing Statistical Hypotheses. Wiley, New York (1959). | MR | Zbl

E.L. Lehmann, S. Fienberg (Contributor) and G. Casella, Theory of Point Estimation. Springer Texts Stat. Springer (1998). | MR | Zbl

W.K. Newey and J.L. Powell, Instrumental variable estimation of nonparametric models. Econometrica 71 (2003) 1565–1578. | MR | Zbl

W. Schoutens, Stochastic Processes and Orthogonal Polynomials. Vol. 146 of Lect. Notes Stat. Springer-Verlag (2000). | MR | Zbl

T.A. Severini and G. Tripathi, Some identification issues in nonparametric linear models with endogenous regressors. Econ. Theory 22 (2006) 258–278. | MR | Zbl

C. Stein, Approximate Computation of Expectations. Lect. Notes, Monogr. Series. Institute of Mathematical Statistics (1986) | MR | Zbl

G. Szegö, Orthogonal Polynomials, 4th edition. AMS Colloquium Publications (1975), Vol. 23. | MR

E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edition. Cambridge Mathematical Library (1975). | MR | Zbl