Estimation of a multiplicative correlation structure in the large dimensional case

We propose a Kronecker product model for correlation or covariance matrices in the large dimensional case. The number of parameters of the model increases logarithmically with the dimension of the matrix. We propose a minimum distance (MD) estimator based on a log-linear property of the model, as well as a one-step estimator, which is a one-step approximation to the quasi-maximum likelihood estimator (QMLE). We establish rates of convergence and central limit theorems (CLT) for our estimators in the large dimensional case. A specification test and tools for Kronecker product model selection and inference are provided. In a Monte Carlo study where a Kronecker product model is correctly specified, our estimators exhibit superior performance. In an empirical application to portfolio choice for S&P500 daily returns, we demonstrate that certain Kronecker product models are good approximations to the general covariance matrix.