Summary: 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 the Lagrange multiplier 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 is shown to perform well in finite sample situations both when the Kronecker product model is true and when it is not true.