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

Abstract

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

Published in at http://dx.doi.org/10.3150/08-BEJ128 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)