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Let $\mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}}$ bethe growth curve model with $\bolds{\mathcal{E}}$ distributed with mean$\mathbf{0}$ and covariance $\mathbf{I}_n\otimes\bolds{\Sigma}$, where$\bolds{\Theta}$, $\bolds{\Sigma}$ are unknown matrices of parameters and$\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametrictransformation of the form $\bolds{\gamma}=\mathbf{C}\bolds{\Theta}\mathbf{D}'$ with given $\mathbf{C}$ and$\mathbf{D}$, the two-stage generalized least-squares estimator $\hat{\bolds\gamma}(\mathbf{Y})$ defined in (7) converges in probability to $\bolds\gamma$as the sample size $n$ tends to infinity and, further,$\sqrt{n}[\hat{\bolds{\gamma}}(\mathbf{Y})-\bolds {\gamma}]$ converges indistribution 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 thecondition that $\lim_{n\to\infty}\mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for somepositive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariantquadratic estimator $\hat{\bolds{\Sigma}}(\mathbf{Y})$ defined in (6) is alsoproved to be consistent with the second-order parameter matrix$\bolds{\Sigma}$.

Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model