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 $μ$. Testing linear restrictions of $μ$ 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.