  While most of the convergence results in the literature on high dimensional
covariance matrix are concerned about the accuracy of estimating the covariance
matrix (and precision matrix), relatively less is known about the effect of
estimating large covariances on statistical inferences. We study two important
models: factor analysis and panel data model with interactive effects, and
focus on the statistical inference and estimation efficiency of structural
parameters based on large covariance estimators. For efficient estimation, both
models call for a weighted principle components (WPC), which relies on a high
dimensional weight matrix. This paper derives an efficient and feasible WPC
using the covariance matrix estimator of Fan et al. (2013). However, we
demonstrate that existing results on large covariance estimation based on
absolute convergence are not suitable for statistical inferences of the
structural parameters. What is needed is some weighted consistency and the
associated rate of convergence, which are obtained in this paper. Finally, the
proposed method is applied to the US divorce rate data. We find that the
efficient WPC identifies the significant effects of divorce-law reforms on the
divorce rate, and it provides more accurate estimation and tighter confidence
intervals than existing methods.
