A variance analysis of the parameters of a plant, belonging to a class of multivariable systems, estimated in closed-loop operation is performed. More specifically, having in mind the control applications where it is not desirable to excite all external reference inputs, the effect of the absence of one or more reference signals on the variance of the estimated parameters is investigated. The derived expressions are valid for a wide range of model structures including all conventional prediction error models. It is shown that, regardless of the parametrization, the absence of a reference signal never improves and, in most cases, impairs the accuracy of the parameter estimates. In other words, there is a price to pay when restrictions are imposed on the experimental conditions. The analytical results are illustrated by two simulation examples.