This study considers third-order linear systems and a two mode linear system as candidates for equivalence to the bilinear hysteretic system subjected to a Gaussian white noise excitation. In each linear system some parameters are chosen to match stationary mean squared levels of both displacement and velocity response to those of the bilinear hysteretic system (using empirical values for the nonlinear system). The particular types of equivalence being sought here involve the power spectral density (psd) of stationary response and transient buildup of mean squared response. The values predicted by the linear models are compared with empirical data for the nonlinear system. The bilinear hysteretic systems considered have the slope ratio between post-yielding and pre-yielding stiffnesses of a - 1/2 (moderately nonlinear system) and a = 1/21 (nearly elastoplastic system). Two ways of choosing the parameters of the third-order linear system are considered. One method involves choosing two damping factors for equivalence and the other involves choosing one damping factor and one spring stiffness for equivalence. The two mode linear system is a model with two uncorrelated modes whose undamped natural frequencies correspond to the initial stiffness and the reduced stiffness of the bilinear hysteretic system, and with two damping factors chosen for equivalence. Based on the comparisons of both psd and transient response, it is concluded that the third-order system is the better equivalent linearization overall (arbitary a), although the two mode linear system is somewhat better in the particular case of a nearly elasto-plastic system.