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

Let Y = X Theta Z' + epsilon be the growth curve model with epsilon distributed with mean 0 and covariance I-n circle times Sigma, where Theta, Sigma are unknown matrices of parameters and X, Z are known matrices. For the estimable parametric transformation of the form gamma = C Theta D' with given C and D, the two-stage generalized least-square estimator (gamma) over cap (Y) defined in (7) converges in probability to gamma as the sample size n tends to infinity and, further, root n|(gamma) over cap (Y) - gamma| converges in distribution to the multivariate normal distribution N(0, (CR-1C') circle times (D(Z'Sigma(-1)Z) 1D')) under the condition that lim(n ->infinity) X'X/n = R for some positive definite matrix R. Moreover, the unbiased and invariant quadratic estimator (Sigma) over cap (Y) defined in (6) is also proved to be consistent with the second-order parameter matrix Sigma.