Estimation for Ultra-high Dimensional Factor Model: A Pivotal Variable Detection-based Approach

Abstract

To estimate the high dimensional covariance matrix, row sparsity is often assumed such that each variable is mainly related with a small number of other variables. However, in some applications such as factor modelling, the common factors may lead some variables to strongly associate with many others and thus row sparsity does not hold. These variables are called the pivotal variables. In this paper, a novel pivotal variable detection approach is developed. As an application, estimating the high dimensional covariance matrix in factor model is investigated, and two-stage estimation procedures are proposed to handle ultra-high dimensionality. In these procedures, pivotal variable detection is performed as a screening step and then existing approaches are applied to refine the working model. The estimation efficiency can be promoted under weaker assumptions on the model structure. Simulations are conducted to examine the performance of the new method and a real dataset is analysed for illustration.