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

Abstract

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