This paper proposes a new approach that allows to exploit the information emerging from the set of possible parameter histories in a time varying SVAR model of the economy. The analysis of the evolution of the parameters in a SVAR framework is likely to convey relevant information about the changes in the behavior of the variables of interest and in their relationships in the sample period under consideration. However, the techniques that are commonly adopted to estimate time varying SVARs do not allow for this type of analysis. Most studies make use of Bayesian estimation methods developed on the assumption of a random walk time variation of the covariances and log standard deviations of the innovations in a SVAR obtained imposing a specific ordering to the variables in the model. Orthogonal rotation matrices are subsequently used to generate a set of possible SVARs for the economy, which form the basis for the analysis to be performed. Because of the way in which the time variation is model, this procedure will deliver possible alternative values of the parameters of interest that are observationally equivalent in each period, but not across histories. More specifically, the different covariance matrices that define the set of possible SVAR models of the economy, will have the same conditional posterior distribution at each time t, but the posterior probabilities of their histories will typically not have the same functional form. This paper proposes an approach that avoids this issue by modeling the time variation directly in the covariance matrix of the reduced form vector of innovations. This method generates a set of possible parameter values for the SVAR model that are observationally equivalent in each period and in their time variation, thus allowing to use the additional information arising from the comparison of their histories. In addition to the description of the model and assumptions on the time variation of its parameters, this work also provides an illustration of the Bayesian techniques that can be employed for the empirical implementation of this method. Finally, the paper presents an application of this new approach to the study of the relationship between oil prices and US domestic variables with focus, in particular, on the interpretation of the changes in this relationship over time and on the role of monetary policy in these changes.