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 is a -dimensional endogenous random variable, and are the instrumental variables (vectors), and . Now, assume that the conditional distributions of given 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 in the image space there is a.s. a unique function in the domain space such that
Assuming the knowledge of and an inference of , our approach provides a natural way of estimating the structural function of interest . Our polynomial basis approach is naturally extended to Pearson-like and Ord-like families of distributions.
DOI : 10.1051/ps/2014025
Mots-clés : Orthogonal polynomials, Stein’s method, nonparametric identification, instrumental variables, semiparametric methods
@article{PS_2015__19__293_0, author = {Kovchegov, Yevgeniy and Y{\i}ld{\i}z, Ne\c{s}e}, title = {Orthogonal polynomials for seminonparametric instrumental variables model}, journal = {ESAIM: Probability and Statistics}, pages = {293--306}, publisher = {EDP-Sciences}, volume = {19}, year = {2015}, doi = {10.1051/ps/2014025}, mrnumber = {3412647}, zbl = {1332.33016}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2014025/} }
TY - JOUR AU - Kovchegov, Yevgeniy AU - Yıldız, Neşe TI - Orthogonal polynomials for seminonparametric instrumental variables model JO - ESAIM: Probability and Statistics PY - 2015 SP - 293 EP - 306 VL - 19 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2014025/ DO - 10.1051/ps/2014025 LA - en ID - PS_2015__19__293_0 ER -
%0 Journal Article %A Kovchegov, Yevgeniy %A Yıldız, Neşe %T Orthogonal polynomials for seminonparametric instrumental variables model %J ESAIM: Probability and Statistics %D 2015 %P 293-306 %V 19 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2014025/ %R 10.1051/ps/2014025 %G en %F PS_2015__19__293_0
Kovchegov, Yevgeniy; Yıldız, Neşe. Orthogonal polynomials for seminonparametric instrumental variables model. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 293-306. doi : 10.1051/ps/2014025. http://archive.numdam.org/articles/10.1051/ps/2014025/
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
Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75 (2007) 1613–1669. | MR | Zbl
, and ,Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80 (2012) 277–321. | MR | Zbl
and ,On rate optimality for Ill-posed inverse problems in econometrics. Econ. Theory 27 (2011) 497–521. | MR | Zbl
and ,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).
Nonparametric instrumental regression. Econometrica 79 (2006) 1541–1565. | MR | Zbl
, and ,Nonparametric methods for inference in the presence of instrumental variables. Ann. Stat. 33 (2005) 2904–2929. | MR | Zbl
and ,Demand analysis as an ill-posed problem with semiparametric specification. Econ. Theory 27 (2011) 460–471. | MR | Zbl
and ,L. Hörmander, An Introduction to Complex Analysis in Several Variables. In vol. 7, 2nd edition. North-Holland Mathematical Library (1973). | MR | Zbl
Uniform confidence bands for functions estimated nonparametrically with instrumental variables. J. Econometrics 168 (2012) 175–188. | MR | Zbl
and ,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
Instrumental variable estimation of nonparametric models. Econometrica 71 (2003) 1565–1578. | MR | Zbl
and ,W. Schoutens, Stochastic Processes and Orthogonal Polynomials. Vol. 146 of Lect. Notes Stat. Springer-Verlag (2000). | MR | Zbl
Some identification issues in nonparametric linear models with endogenous regressors. Econ. Theory 22 (2006) 258–278. | MR | Zbl
and ,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
Cité par Sources :