Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T17:11:02.388Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC IDENTIFICATION USING INSTRUMENTAL VARIABLES: SUFFICIENT CONDITIONS FOR COMPLETENESS

Published online by Cambridge University Press:  19 June 2017

Yingyao Hu  and
Ji-Liang Shiu
Yingyao Hu*
Affiliation:
Johns Hopkins University
Ji-Liang Shiu
Affiliation:
Jinan University
*
*Address correspondence to Yingyao Hu, Department of Economics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides sufficient conditions for the nonparametric identification of the regression function $m\left( \cdot \right)$ in a regression model with an endogenous regressor x and an instrumental variable z. It has been shown that the identification of the regression function from the conditional expectation of the dependent variable on the instrument relies on the completeness of the distribution of the endogenous regressor conditional on the instrument, i.e., $f\left( {x|z} \right)$. We show that (1) if the deviation of the conditional density $f\left( {x|{z_k}} \right)$ from a known complete sequence of functions is less than a sequence of values determined by the complete sequence in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$, then $f\left( {x|z} \right)$ itself is complete, and (2) if the conditional density $f\left( {x|z} \right)$ can form a linearly independent sequence $\{ f( \cdot |{z_k}):k = 1,2, \ldots \}$ in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$ and its relative deviation from a known complete sequence of functions under some norm is finite then $f\left( {x|z} \right)$ itself is complete. We use these general results to provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable $z.$

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 3 , June 2018 , pp. 659 - 693
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank Peter C.B. Phillips, two associate editors, and two anonymous referees for their time and helpful comments. We also thank the participants at the International Economic Association 16th World Congress, Beijing, China, and the 2013 Annual Meeting of American Economic Association, San Diego, USA, for various suggestions regarding this paper. The usual disclaimer applies.

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71(6), 1795–1843.Google Scholar
An, Y. & Hu, Y. (2012) Well-posedness of measurement error models for self-reported data. Journal of Econometrics 168(2), 259–269.CrossRefGoogle Scholar
Andrews, D. (2012) Examples of ${L^2}$-complete and boundedly-complete distributions. Cowles Foundation for Research in Economics.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75(6), 1613.CrossRefGoogle Scholar
Canay, I., Santos, A., & Shaikh, A. (2013) On the testability of identification in some nonparametric models with endogeneity. Econometrica 81(6), 2535–2559.Google Scholar
Carroll, R., Chen, X., & Hu, Y. (2010) Identification and estimation of nonlinear models using two samples with nonclassical measurement errors. Journal of Nonparametric Satistics 22(4), 379–399.CrossRefGoogle ScholarPubMed
Chen, X. & Hu, Y. (2006) Identification and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors. Cowles Foundation Discussion paper no. 1590.Google Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73(1), 245–261.CrossRefGoogle Scholar
Chernozhukov, V., Imbens, G., & Newey, W. (2007) Instrumental variable estimation of nonseparable models. Journal of Econometrics 139(1), 4–14.CrossRefGoogle Scholar
Darolles, S., Fan, Y., Florens, J., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79(5), 1541–1565.Google Scholar
D’Haultfoeuille, X. (2011) On the completeness condition in nonparametric instrumental problems. Econometric Theory 27(3), 1–12.CrossRefGoogle Scholar
Dostanić, M. (1990) On the completeness of the system of functions $\left\{ {{e^{ - \alpha {\lambda _n}x}}{\rm{sin}}{\lambda _n}x} \right\}_{n = 1}^\infty$. Journal of Mathematical Analysis and Applications 150(2), 519–527.CrossRefGoogle Scholar
Erdös, P. & Straus, E. (1953) On linear independence of sequences in a banach space. Pacific Journal of Mathematics 3(4), 689–694.Google Scholar
Florens, J., Mouchart, M., & Rolin, J. (1990) Elements of Bayesian Statistics. Marcel Dekker.Google Scholar
Gurarij, V. & Meletidi, M. (1970) The stability of the completeness of sequences in banach spaces. Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences, Mathematiques, Astronomiques et Physiques 18(9), 533–536.Google Scholar
Hall, P. & Horowitz, J. (2005) Nonparametric methods for inference in the presence of instrumental variables. The Annals of Statistics 33(6), 2904–2929.CrossRefGoogle Scholar
Hoderlein, S., Nesheim, L., & Simoni, A. (2012) Semiparametric Estimation of Random Coefficients in Structural Economic Models. Cemmap Working papers.Google Scholar
Horowitz, J. (2011) Applied nonparametric instrumental variables estimation. Econometrica 79(2), 347–394.Google Scholar
Horowitz, J. & Lee, S. (2007) Nonparametric instrumental variables estimation of a quantile regression model. Econometrica 75(4), 1191–1208.CrossRefGoogle Scholar
Hu, Y. & Schennach, S. (2008) Instrumental variable treatment of nonclassical measurement error models. Econometrica 76(1), 195–216.CrossRefGoogle Scholar
Hu, Y. & Shum, M. (2012) Nonparametric identification of dynamic models with unobserved state variables. Journal of Econometrics 171(1), 32–44.CrossRefGoogle Scholar
Lehmann, E. (1986) Testing Statistical Hypotheses, 2nd ed. Wiley.CrossRefGoogle Scholar
Mattner, L. (1993) Some incomplete but boundedly complete location families. The Annals of Statistics 21(4), 2158–2162.CrossRefGoogle Scholar
Mattner, L. (1996) Complete incomplete but boundedly complete location families. The Annals of Statistics 24(3), 1265–1282.Google Scholar
Newey, W. & Powell, J. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71(5), 1565–1578.CrossRefGoogle Scholar
Phillips, P.C.B. & Su, L. (2011) Nonparametric regression under location shifts. Econometrics Journal 14(3), 457–486.CrossRefGoogle Scholar
Rudin, W. (1987) Real and Complex Analysis. McGraw-Hill.Google Scholar
San Martin, E. & Mouchart, M. (2007) On joint completeness: Sampling and bayesian versions, and their connections. The Indian Journal of Statistics 69(4), 780–807.Google Scholar
Shiu, J. & Hu, Y. (2013) Identification and estimation of nonlinear dynamic panel data models with unobserved covariates. Journal of Econometrics 175(2), 116–131.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2007) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25(3), 710–738.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009) Structural nonparametric cointegrating regression. Econometrica 77(6), 1901–1948.Google Scholar
Wang, Q. & Phillips, P.C.B. (2016) Nonparametric cointegrating regression with endogeneity and long memory. Econometric Theory 32(2), 359–401.CrossRefGoogle Scholar
Young, R. (2001) An Introduction to Nonharmonic Fourier Series. Academic Press.Google Scholar
Yu, P. & Phillips, P.C.B. (2017) Threshold regression with endogeneity. Journal of Econometrics, forthcoming.Google Scholar
Zimmer, R. (1990) Essential Results of Functional Analysis. University of Chicago Press.Google Scholar