In this article we provide a statistical procedure for the analysis of stock market prices that is robust toward departures from the normal distribution assumption and that can detect and evaluate a shift of parameters at an unknown time point. The method is an adaptation of a Bayesian inferential procedure developed by Box and Tiao that allows data to deviate moderately from the normal distribution model. It is applied to a set of U.S. stock market prices for 1971–1974. In addition to the detection of shift in distribution parameters, the procedure is also applied to the examination of shift of the “beta coefficients” that represent the degree of undiversifiable (systematic) risk of individual securities. Implications of the empirical findings for financial theories and their applications are sketched.