  We propose a new estimator, the quadratic form estimator, of the Kronecker
product model for covariance matrices. We show that this estimator has good
properties in the large dimensional case (i.e., the cross-sectional dimension
$n$ is large relative to the sample size $T$). In particular, the quadratic
form estimator is consistent in a relative Frobenius norm sense provided
$\log^3n/T\to 0$. We obtain the limiting distributions of Lagrange multiplier
(LM) and Wald tests under both the null and local alternatives concerning the
mean vector $\mu$. Testing linear restrictions of $\mu$ is also investigated.
Finally, our methodology performs well in the finite-sample situations both
when the Kronecker product model is true, and when it is not true.
