In this study, we compare the performance of various bivariate Markov switching~(MS) models, including models for the independence case, the perfect-synchronization case, and for the general case, in characterizing the international business cycles~(IBC) of the U.S. and Canada and that of the U.S. and Japan. This empirical study shows that, compared to the independence model and the perfect-synchronization model, the general model is capable of explaining the IBC in a better way. Meanwhile, this generalized model also permits us to define a time-varying synchronization index for the IBC by the smooth probability of the common recession or expansion state. Using this synchronization index, we find that the IBC of the U.S. and Canada has been further strengthened in the 1990's; in comparison, the IBC of the U.S. and Japan has been less synchronized in the same period.