Statistical Inferences Using Large Estimated Covariances for Panel Data and Factor Models

Abstract

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.

Econometric Analysis of Large Factor Models

“…They assume Σ e is sparse and use the regularization method to jointly estimate Λ and Σ e . Consistency is established for the estimated Λ and Σ e , but the limiting distributions remain unsolved, though the limiting distributions are conjectured to be the same as the two-step feasible GLS estimator in Bai & Liao (2013) under large N and T .…”

“…This is a feasible GLS estimator of f t in the model X t = µ+Λf t +e t . The estimated factor loadings have the same asymptotic distributions for the three different estimation methods (PC, GPCE, MLE) under large N and large T , But the estimated factors are more efficient under GPCE and MLE than standard PC (see Choi 2010;Bai & Li 2012;Bai & Liao 2013). If time series heteroskedasticity is of more concern, and especially when T is relatively small, then the role of F and Λ (also T and N ) should be switched.…”

“…So the high dimensionality of Σ e makes a fundamental difference in terms of inference. Bai & Liao (2013) show that the true matrix Σ e has to be sparse and its estimates should take into account the sparsity assumption. Sparsity does not require many zero elements, but many elements must be sufficiently small.…”

“…Under the sparsity assumption, a shrinkage estimator of Σ e (for example, a hard thresholding method based on the residuals from the standard PC method) will give a consistent estimation of Σ e . Bai & Liao (2013) derive the conditions under which the estimated Σ e can be treated as known.…”

“…The estimatedΣ e can be used to obtain more efficient estimation of Λ and F based on the generalized principal components method (Bai & Liao, 2013). Alternatively, the unknown parameters of Λ and Σ e can be jointly estimated by the maximum likelihood method (Bai & Liao, 2016).…”