Unpredicted and unmeasured disturbances impose practical difficulties on the identification of discrete multiple-input/multiple-output parametric models from plant tests. A proper solution to such difficulties is not yet available. This article establishes an identification method based on Laguerre models with adjustable time-scaling factors. A linear regression equation is derived to incorporate the terms concerning initial states as well as disturbances occurring before and after the start of the identification test. The resulting least-squares estimator of the Laguerre coefficients is thus employed to develop process and disturbance models efficiently and accurately. The idea of augmented order is introduced to account for distinct load dynamics. To improve identification under deterministic disturbances, two error criteria are developed to infer the proper values of the time-scaling factor, the load entering time, and process time delays. In the presence of a stochastic disturbance, another error criterion is presented to determine the time-scaling factor that rejects the most deteriorating effect of the stochastic disturbance on parameter estimation. It is demonstrated that the proposed method is reliable against multifarious characteristics of deterministic and stochastic disturbances.