Bayesian testing and testing Bayesians

This paper surveys Bayesian procedures of testing along with different attitudes of econometricians facing testing problems with some sympathy for Bayesian ideas. In the first part, the general Bayesian testing procedures are structured along two main axes. The first one presents the usual procedures based on the posterior probabilities of models, or of hypothesis, and relying mainly on a comparison of predicti ve distributions relative to each model, or hypothesis. The second one has been developped more recently and is based on a Bayesian extension of the encompassing principle. This approach basically relies on a comparison of posterior distributions on parameters of interest conditional on each hypothesis. In the second part, different attitudes of testing Bayesians is considered for the testing of two important problems in econometrics. The first one, testing the significance of a regression coefficient and chosing the regressors, is used to exemplify the flexibility of the Bayesian approach whereas the second one, testing for unit root, is used to exemplify how Bayesian thinking may be useful to shed new insights on controversial issues. A final section provides offers a short survey of Bayesian contributions on other testing problems.