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

Let Y =XΘZ′ + E be the growth curve model with E distributed with mean 0 and covariance In ⊗Σ, where Θ, Σ are unknown matrices of parameters and X, Z are known matrices. For the estimable parametric transformation of the form γ =CΘD′ with given C and D, the twostage generalized least-squares estimator γ̂(Y) defined in (7) converges in probability to γ as the sample size n tends to infinity and, further, √ n[γ̂(Y)− γ] converges in distribution to the multivariate normal distributionN (0, (CRC)⊗(D(ZΣZ)D)) under the condition that limn→∞X ′ X/n =R for some positive definite matrix R. Moreover, the unbiased and invariant quadratic estimator Σ̂(Y) defined in (6) is also proved to be consistent with the second-order parameter matrix Σ.