Pivotal variable detection of the covariance matrix and its application to high-dimensional factor models

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 nonzero 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 nonzero 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 of the new method and a real dataset is analyzed for illustration.