  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 SP500 daily returns, we
demonstrate that certain Kronecker product models are good approximations to
the general covariance matrix.
