Multivariate data collected over time on the same experimental unit, referred to as multivariate longitudinal data, are typical of many agricultural, biological, clinical and medical studies. One way to account for the correlations that exist both within and across time is to express the variance-covariance matrix as the Kroneeker product of two matrices. These matrices, denoted by [Delta] and [Omega], reflect the characteristic and time dimensions underlying multivariate longitudinal data. The purpose of this thesis is to investigate the asymptotic relative efficiency (ARE) of hypothesis tests in the linear model for multivariate longitudinal data, evaluated through the trace asymptotic relative efficiency (TARE) and curvature asymptotic relative efficiency (CARE). The gain in efficiency from exploiting a Kronecker product covariance structure when it is appropriate is investigated. To estimate the TARE and CARE, a Monte-carlo simulation study is conducted. The loss of efficiency from imposing a Kronecker product model when it is not appropriate is also considered. Using a class of non-Kroneeker product covariance matrices and an index, which quantifies how far a given matrix departs from Kronecker product structure, a Monte-carlo simulation study is conducted. Ordinary least squares and generalised least squares procedures were also compared under a Kronecker product model. For the designs and covariance matrices considered, the gain in efficiency from exploiting the Kronecker product covariance structure is most pronounced when there is high correlation across time. For the class of non-Kronecker product covariance matrices defined, a noticeable loss of efficiency occurs when the covariance matrix is far from Kronecker product structure, in particular when there is a moderate departure from the null hypothesis under consideration. The use of ordinary least squares, which ignores cross-sectional and longitudinal correlations, is shown to be inefficient, especially when these correlations are high in absolute value