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

Let $\mathbf{Y}=\mathbf{X}\text{bolds}\mathrm{\Theta }{\mathbf{Z}}^{\mathrm{\prime }}+\text{bolds}\mathcal{E}$ be the growth curve model with $\text{bolds}\mathcal{E}$ distributed with mean $\mathbf{0}$ and covariance ${\mathbf{I}}_n\otimes \text{bolds}\mathrm{\Sigma }$, where $\text{bolds}\mathrm{\Theta }$, $\text{bolds}\mathrm{\Sigma }$ are unknown matrices of parameters and $\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form $\bolds {\gamma}=$\mathbf{C}$\bolds{\Theta}$\mathbf{D}$‘$withgiven$$\mathbf{C}$$and$$\mathbf{D}$$,thetwo-stagegeneralizedleast-squaresestimator$\hat{\bolds \gamma}($\mathbf{Y}$)$definedin(7)convergesinprobabilityto$\bolds\gamma$asthesamplesize$n$tendstoinfinityand,further,$\sqrt{n}[\hat{\bolds{\gamma}}($\mathbf{Y}$)-\bolds {\gamma}]$ converges in distribution to the multivariate normal distribution $\ma thcal{N}($\mathbf{0}$,($\mathbf{C}$$\mathbf{R}$^{-1}$\mathbf{C}$’)\otimes(\mat hbf{D}($\mathbf{Z}$‘\bolds{\Sigma}^{-1}$\mathbf{Z}$)^{-1}$\mathbf{D}$’))$ under the condition that ${\lim }_{n\to \mathrm{\infty }}{\mathbf{X}}^{\mathrm{\prime }}\mathbf{X}\mathrm{/}n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator $\widehat{\text{bolds}\mathrm{\Sigma }}(\mathbf{Y})$ defined in (6) is also proved to be consistent with the second-order parameter matrix $\text{bolds}\mathrm{\Sigma }$.