Iterative regularisation in nonparametric instrumental regression

• Published in: Journal of statistical planning and inference Vol. 143; no. 1; pp. 24 - 39
• Main Authors: Johannes, Jan, Van Bellegem, Sébastien, Vanhems, Anne
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-01-2013
• Subjects: Estimation by projection; Ill-posed inverse problem; Instrumental variable; Iterative method; Nonparametric estimation; Instrumental variable; Nonparametric estimation; Iterative method; Estimation by projection; Ill-posed inverse problem
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2012.07.010

This paper considers the nonparametric regression model with an additive error that is correlated with the explanatory variables. Motivated by empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. However, the estimation of a nonparametric regression function by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function that is based on projection onto finite dimensional spaces and that includes an iterative regularisation method (the Landweber–Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both strong and weak source conditions. A Monte Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.