Pivotal Variable Detection of the Covariance Matrix and Its Application to High-Dimensional Factor Models

Abstract

To estimate the high-dimensional covariance matrix, row sparsity is often assumed such that each row has a small number of nonzero elements. However, in some applications, such as factor modeling, there may be many non-zero loadings of the common factors. The corresponding variables are also correlated to one another and the rows are non-sparse or dense. This paper has three main aims. First,a detection method is proposed to identify the rows that may be non-sparse, or at least dense with many non-zero elements. These rows are called dense rows and the corresponding variables are called pivotal variables. Second, to determine the number of rows, a ridge ratio method is suggested, which can be regarded as a sure screening procedure. Third, to handle the estimation of high-dimensional factor models, a two-step procedure is suggested with the above screening as the first step. Simulations are conducted to examine the performance.