Abstract Model identification and parameter estimation in linear static systems, when more than one variable is subject to noise contamination, is the focus of this paper. When more than one variable is noisy, the appropriateness of a single-equation model versus a multi-equation model must be scrutinized. Linear regression models are the most common single-equation models; simultaneous-equations models typify the multiequation ones. When data covariance matrix ∑ is inverse positive, only a single linear relationship can be identified, and hence a single-equation model should be selected. Otherwise, a multi-equation model is most appropriate. In this paper, some new sufficient conditions for ∑ to be inverse positive are presented. Furthermore, for cases when ∑ is not inverse positive, techniques for identifying the subsets of variables within which linear relations can exist are presented. Finally, when some or all variables of a linear system are noisy, unique estimation of model parameters is difficult unless sufficient prior information about the noise variances is available. Here, a new weighted regression procedure enables the modeler to compute unique parameter estimates based only on computed upper bounds on the noise variances.