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The class of semi-nonparametric conditional moment restrictions with possibly different information sets contains many important economics and structural econometrics models as special cases. In this talk, I shall present some existing and new results about optimal estimation and inference on functionals of semi-nonparametric conditional moment restrictions containing unknown functions of endogenous variables. I shall present two examples in details: (1) rate-adaptive minimax estimation of a quadratic functional of a nonparametric instrumental variables (NPIV); and rate-adaptive minimax optimal testing of NPIV; (2) optimal estimation and inference on weighted average derivatives of nonparametric quantile instrumental variables. The first example highlights the difficulty of inference on nonlinear functionals of linear ill-posed inverse problems with unknown operator. The second example highlights the difficulty of inference on linear functionals of nonlinear ill-posed inverse problems with unknown operator.
The talk is based on joint research with Tim Christensen, Christoph Breunig, Demian Pouzo and others.