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.

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