Learning about the across-regime correlation in switching regression models

Vijverberg (1993) considers the extended Roy/switching regression model of selectivity, focusing attention on the nonidentified correlation between the regime disturbances and describing how the positive definiteness of the covariance matrix implies that it is possible to learn about this covariance. In this paper, we show that this learning derives from prior dependence between identified and nonidentified parameters. Even though beliefs about the nonidentified covariance are updated, we show under reasonable a priori independence assumptions that beliefs about the partial correlation between disturbances, control of the switching index, are not updated. Empirical illustrations show how an exact Bayesian analysis can be carried out using Gibbs sampling and related techniques.