Optimal design for multivariate observations in seemingly unrelated linear models

The concept of seemingly unrelated models is used for multivariate observations when the components of the multivariate dependent variable are governed by mutually different sets of explanatory variables and the only relation between the components is given by a fixed covariance within the observational units. A multivariate weighted least squares estimator is employed which takes the within units covariance matrix into account. In an experimental setup, where the settings of the explanatory variables may be chosen freely by an experimenter, it might be thus tempting to choose the same settings for all components to end up with a multivariate regression model, in which the ordinary and the least squares estimators coincide. However, we will show that under quite natural conditions the optimal choice of the settings will be a product type design which is generated from the optimal counterparts in the univariate models of the single components. This result holds even when the univariate models may change from component to component. For practical applications the full factorial product type designs may be replaced by fractional factorials or orthogonal arrays without loss of efficiency.