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

TL;DR

This paper establishes the asymptotic normality and consistency of a two-stage generalized least squares estimator in a growth curve model, providing theoretical guarantees for large sample behavior.

Contribution

It introduces a two-stage GLS estimator for growth curve models and proves its asymptotic normality and consistency under general conditions.

Findings

Estimator converges in probability to true parameter as sample size increases.

Scaled estimator converges in distribution to a multivariate normal.

Variance estimator is consistent for the true covariance matrix.

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 $\bolds{\Sigma}$.