The realization of a random event (e.g. basin outflows, rainfall depths) may be represented by a deterministic link between forcings (input system series) and outputs. In those circumstances when such a non-linear link (e.g. rainfall-runoff models) is not known or when sufficient physical insight of the process is not available, non linear empirical models may be applied with fairly good chances of success. Moreover, even if knowledge of the process causality produces a deterministic relationship between input and output, randomness still remains in the realization of the event, e.g. streamflow events. NARMAX models, initially developed by Leontaritis and Billings (1985) are an example of non-linear models that may be used in this application. Herein the original and simplified version of NARMAX models (NARX) is introduced in a comparative study in the reproduction of daily streamflows data for two small catchments in the Ligurian region (Italy). NARMAX models, in general, yield an input-output representation of a non linear system where the current output a function of lagged inputs, outputs and noise. The influence of disturbances in rainfall-runoff modeling at daily scale using NARMAX models is evaluated in this paper. The corresponding NARX models are used when the focus is on the deterministic input- output relationship with a crude simplification of the disturbance model.