Inference on time series nonparametric conditional moment restrictions using nonlinear sieves

This paper studies estimation of and inference on dynamic nonparametric conditional moment restrictions of high dimensional variables for weakly dependent data, where the unknown functions of endogenous variables can be approximated via nonlinear sieves such as neural networks and Gaussian radial bases. The true unknown functions and their sieve approximations are allowed to be in general weighted function spaces with unbounded supports, which is important for time series data. Under some regularity conditions, the optimally weighted general nonlinear sieve quasi-likelihood ratio (GN-QLR) statistic for the expectation functional of unknown function is asymptotically Chi-square distributed regardless whether the functional could be estimated at a root-$n$ rate or not, and the estimated expectation functional is asymptotically efficient if it is root-$n$ estimable. Our general theories are applied to two important examples: (1) estimating the value function and the off-policy evaluation in reinforcement learning (RL); and (2) estimating the averaged partial mean and averaged partial derivative of dynamic nonparametric quantile instrumental variable (NPQIV) models. We demonstrate the finite sample performance of our optimal inference procedure on averaged partial derivative of a dynamic NPQIV model in simulation studies.