- Invited talk (.pdf):
- “Statistik Seminar”, Departement of Economics, Rheinische Friedrich-Wilhelms-Universität in Bonn, Germany
- Presented by:
- Jan JOHANNES
- Title:
- Adaptive nonparametric instrumental regression
- Abstract:
- We consider the nonparametric estimation of the structural function in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The theory in this presentation covers both the estimation of the structural function or its derivatives (global case) as well as the estimation of a linear functional of the structural function (local case). We propose an estimator of the structural function which is based on a dimension reduction and additional thresholding. Moreover, in the local case replacing the unknown structural function by this estimator we obtain a plug-in estimator of the value of a linear functional evaluated at the structural function. Assuming an independent and identically distributed (iid.) sample of (Y,Z,W) it is shown that in both the global and the local case the estimator can attain minimax optimal rates of convergence. The estimator of the structural function, however, requires an optimal choice of a tuning parameter with regard to certain characteristics of the structural function and the conditional expectation operator of Z given W. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski’s method and is inspired by the recent work of Goldenshluger and Lepski [2011]. It is shown that given an iid. sample the data-driven estimator can attain the lower minimax risk bound in the global case up to a constant and in the local case up to a logarithmic factor. Finally, dismissing the independence and assuming sufficiently weak dependent observations characterized by fast decreasing mixing coefficients the data-driven estimator still can attain the iid. global lower risk bound up to a constant.
- References:
- N. Asin and J. Johannes. Adaptive non-parametric instrumental regression in the presence of dependence. Technical report, Université catholique de Louvain, 2016. arXiv:1604.01992.
- C. Breunig and J. Johannes. Adaptive estimation of functionals in nonparametric instrumental regression. Econometric Theory, pages 1–43, 2015.
- A. Goldenshluger and O. Lepski. Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. The Annals of Statistics, 39:1608–1632, 2011.
- J. Johannes and M. Schwarz. Partially adaptive nonparametric instrumental regression. Journal of the Indian Statistical Association, 49(2):149–175, 2011.
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