Estimation of covariance matrix with ARMA structure through quadratic loss function

In this paper we propose a novel method to estimate the high-dimensional covariance matrix with an order-1 autoregressive moving average process, i.e. ARMA(1,1), through quadratic loss function. The ARMA(1,1) structure is a commonly used covariance structures in time series and multivariate analysis but involves unknown parameters including the variance and two correlation coefficients. We propose to use the quadratic loss function to measure the discrepancy between a given covariance matrix, such as the sample covariance matrix, and the underlying covariance matrix with ARMA(1,1) structure, so that the parameter estimates can be obtained by minimizing the discrepancy.
Simulation studies and real data analysis show that the proposed method works well in estimating the covariance matrix with ARMA(1,1) structure even if the dimension is very high.