The problem of endogeneity in statistics and econometrics is often handled by introducing instrumental variables (IV) which fulfill the mean independence assumption, i.e. the unobservable is mean independent of the instruments. When full independence of IV’s and the unobservable is assumed, nonparametric IV regression models and nonparametric demand models lead to nonlinear integral equations with unknown integral kernels. We prove convergence rates for the mean integrated square error of the iteratively regularized Newton method applied to these problems. Compared to related results we derive stronger convergence results that rely on weaker nonlinearity restrictions. We demonstrate in numerical simulations for a nonparametric IV regression that the method produces better results than the standard model.
The author would like to thank Thorsten Hohage and Johannes Schmidt-Hieber for interesting and fruitful discussions on this topic. He also would like to thank two anonymous referees for valuable comments that improved the paper.
"Adaptive estimation for some nonparametric instrumental variable models with full independence." Electron. J. Statist. 15 (2) 6151 - 6190, 2021. https://doi.org/10.1214/21-EJS1938