Let $\mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}}$ be the growth curve model with $\bolds{\mathcal{E}}$ distributed with mean 0 and covariance In⊗Σ, where Θ, Σ are unknown matrices of parameters and X, Z are known matrices. For the estimable parametric transformation of the form γ=CΘD' with given C and D, the two-stage generalized least-squares estimator γ̂(Y) defined in (7) converges in probability to γ as the sample size n tends to infinity and, further, $\sqrt{n}[\hat{\bolds{\gamma}}(\mathbf{Y})-{\bolds{\gamma}}]$ converges in distribution to the multivariate normal distribution $\mathcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mathbf{D}(\mathbf{Z}'\bolds{\Sigma }^{-1}\mathbf{Z})^{-1}\mathbf{D}'))$ under the condition that limn→∞ X'X/n=R for some positive definite matrix R. Moreover, the unbiased and invariant quadratic estimator Σ̂(Y) defined in (6) is also proved to be consistent with the second-order parameter matrix Σ.