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Vector autoregressions (VAR) combined with Minnesota-type priors are widely used for macroeconomic forecasting. The fact that strong but sensible priors can substantially improve forecast performance implies VAR forecasts are sensitive to prior hyperparameters. But the nature of this sensitivity is seldom investigated. We develop a general method based on Automatic Differentiation to systematically compute the sensitivities of forecasts – both points and intervals – with respect to any prior hyperparameters. In a forecasting exercise using US data, we find that forecasts are relatively sensitive to the strength of shrinkage for the VAR coefficients, but they are not much affected by the prior mean of the error covariance matrix or the strength of shrinkage for the intercepts.

We present Bayesian models for finding the longitudinal causal effects of a randomized two-arm training program when compliance with the randomized assignment is less than perfect in the training arm (but perfect in the non-training arm) for reasons that are potentially correlated with the outcomes. We deal with the latter confounding problem under the principal stratification framework of Sommer and Zeger (1991) and Frangakis and Rubin (1999), and others. Building on the Bayesian contributions of Imbens and Rubin (1997), Hirano et al. (2000), Yau and Little (2001) and in particular Chib (2007) and Chib and Jacobi (2007, 2008), we construct rich models of the potential outcome sequences (with and without random effects), show how informative priors can be reasonably formulated, and present tuned computational approaches for summarizing the posterior distribution. We also discuss the computation of the marginal likelihood for comparing various versions of our models. We find the causal effects of the observed intake from the predictive distribution of each potential outcome for compliers. These are calculated from the output of our estimation procedures. We illustrate the techniques and ideas with data from the 1994 JOBS II trial that was set up to test the efficacy of a job training program on subsequent mental health outcomes.

Bayesian Econometrics is a volume in the series Advances in Econometrics that illustrates the scope and diversity of modern Bayesian econometric applications, reviews some recent advances in Bayesian econometrics, and highlights many of the characteristics of Bayesian inference and computations. This first paper in the volume is the Editors’ introduction in which we summarize the contributions of each of the papers.