  This paper considers the maximum generalized empirical likelihood (GEL)
estimation and inference on parameters identified by high dimensional moment
restrictions with weakly dependent data when the dimensions of the moment
restrictions and the parameters diverge along with the sample size. The
consistency with rates and the asymptotic normality of the GEL estimator are
obtained by properly restricting the growth rates of the dimensions of the
parameters and the moment restrictions, as well as the degree of data
dependence. It is shown that even in the high dimensional time series setting,
the GEL ratio can still behave like a chi-square random variable
asymptotically. A consistent test for the over-identification is proposed. A
penalized GEL method is also provided for estimation under sparsity setting.
