In this paper, we study the finite sample properties of maximum likelihood estimator (MLE) of generalized multivariate analysis of variance-multivariate analysis of variance (GMANOVA-MANOVA) model with the first order-autoregressive ($AR(1)$) type covariance structure. We provide the necessary condition for the existence of maximum likelihood estimator in GMANOVA-MANOVA models and a sufficient condition for the uniqueness of the maximum likelihood estimator is also studied. Under the provided sufficient condition, we show that the exact distribution of the maximum likelihood estimator of the correlation coefficient only depends on the true value of $\rho$. In addition, we propose a simple hypothesis test for testing $H_0: \rho =0$ v.s. $H_1: \rho \neq 0$, which does not require any iteration procedures. Simulation shows that the proposed hypothesis test is unbiased and has very comparable power to that of the likelihood ratio test.