The Annals of Statistics

Nonparametric methods for inference in the presence of instrumental variables

Peter Hall and Joel L. Horowitz

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We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the “difficulty” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter.

Article information

Ann. Statist., Volume 33, Number 6 (2005), 2904-2929.

First available in Project Euclid: 17 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Bandwidth convergence rate eigenvalue endogenous variable exogenous variable kernel method linear operator nonparametric regression smoothing optimality


Hall, Peter; Horowitz, Joel L. Nonparametric methods for inference in the presence of instrumental variables. Ann. Statist. 33 (2005), no. 6, 2904--2929. doi:10.1214/009053605000000714.

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  • Blundell, R. and Powell, J. L. (2003). Endogeneity in nonparametric and semiparametric regression models. In Advances in Economics and Econometrics: Theory and Applications (M. Dewatripont, L. P. Hansen and S. J. Turnovsky, eds.) 2 312–357. Cambridge Univ. Press.
  • Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843–874.
  • Darolles, S., Florens, J.-P. and Renault, E. (2002). Nonparametric instrumental regression. Working paper, GREMAQ, Univ. Social Science, Toulouse.
  • Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101–126.
  • Efromovich, S. and Koltchinskii, V. (2001). On inverse problems with unknown operators. IEEE Trans. Inform. Theory 47 2876–2894.
  • Florens, J.-P. (2003). Inverse problems and structural econometrics: The example of instrumental variables. In Advances in Economics and Econometrics: Theory and Applications (M. Dewatripont, L. P. Hansen and S. J. Turnovsky, eds.) 2 284–311. Cambridge Univ. Press.
  • Groetsch, C. (1984). The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, London.
  • Johnstone, I. M. (1999). Wavelet shrinkage for correlated data and inverse problems: Adaptivity results. Statist. Sinica 9 51–83.
  • Kress, R. (1999). Linear Integral Equations, 2nd ed. Springer, New York.
  • Mathé, P. and Pereverzev, S. V. (1999). Optimal discretization and degrees of ill-posedness for inverse estimation in Hilbert scales in the presence of random noise. Preprint No. 469, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin.
  • Nashed, M. Z. and Wahba, G. (1974). Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations. SIAM J. Math. Anal. 5 974–987.
  • Newey, W. K. and Powell, J. L. (2003). Instrumental variable estimation of nonparametric models. Econometrica 71 1565–1578.
  • Newey, W. K., Powell, J. L. and Vella, F. (1999). Nonparametric estimation of triangular simultaneous equations models. Econometrica 67 565–603.
  • O'Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems (with discussion). Statist. Sci. 1 502–527.
  • Tikhonov, A. and Arsenin, V. (1977). Solutions of Ill-Posed Problems. Winston, Washington.
  • Van Rooij, A. and Ruymgaart, F. H. (1999). On inverse estimation. In Asymptotics, Nonparametrics, and Time Series (S. Ghosh, ed.) 579–613. Dekker, New York.
  • Wahba, G. (1973). Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind. J. Approximation Theory 7 167–185.