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

Abstract

Let $\mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}}$ be the 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 parametric transformation 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 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 $\lim_{n\to\infty}\mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator $\hat{\bolds{\Sigma}}(\mathbf{Y})$ defined in (6) is also proved to be consistent with the second-order parameter matrix $\bolds{\Sigma}$.