  General nonlinear sieve learnings are classes of nonlinear sieves that can
approximate nonlinear functions of high dimensional variables much more
flexibly than various linear sieves (or series). This paper considers general
nonlinear sieve quasi-likelihood ratio (GN-QLR) based inference on expectation
functionals of time series data, where the functionals of interest are based on
some nonparametric function that satisfy conditional moment restrictions and
are learned using multilayer neural networks. While the asymptotic normality of
the estimated functionals depends on some unknown Riesz representer of the
functional space, we show that the optimally weighted GN-QLR statistic is
asymptotically Chi-square distributed, regardless whether the expectation
functional is regular (root-$n$ estimable) or not. This holds when the data are
weakly dependent beta-mixing condition. We apply our method to the off-policy
evaluation in reinforcement learning, by formulating the Bellman equation into
the conditional moment restriction framework, so that we can make inference
about the state-specific value functional using the proposed GN-QLR method with
time series data. In addition, estimating the averaged partial means and
averaged partial derivatives of nonparametric instrumental variables and
quantile IV models are also presented as leading examples. Finally, a Monte
Carlo study shows the finite sample performance of the procedure
