Unconstrained models for the covariance structure of multivariate longitudinal data

The constraint that a covariance matrix must be positive definite presents difficulties for modeling its structure. Pourahmadi (1999, 2000) [18,19] proposed a parameterization of the covariance matrix for univariate longitudinal data in which the parameters are unconstrained, which is based on the modified Cholesky decomposition of the covariance matrix. We extend this approach to multivariate longitudinal data by developing a modified Cholesky block decomposition that provides an alternative unconstrained parameterization for the covariance matrix, and we propose parsimonious models within this parameterization. A Fisher scoring algorithm is developed for obtaining maximum likelihood estimates of parameters, assuming that the observations are normally distributed. The asymptotic distribution of the maximum likelihood estimators is derived. The performance of the estimators for finite samples is investigated by simulation and compared with that of estimators obtained under a separable (Kronecker product) covariance model. Estimation and model selection are illustrated using bivariate longitudinal data from a study of poplar growth.