Possibly Ill-behaved Posteriors in Econometric Models

Highly non-elliptical posterior distributions may occur in several econometric models, in particular, when the likelihood is allowed to dominate and information in the data is weak. This latter feature occurs frequently in empirical econometric analysis. Well-known cases are: instrumental variable models with weak instruments like the income-education models; vector autoregressive models with co-integration restrictions, widely used for the analysis of macroeconomic and financial time series; and mixture processes where one component is nearly non-identified like business cycle models with recessions and expansions as components of the mixture. We explain the issue of highly non-elliptical posteriors in the context of a simple model for the effect of education on income using data from the wellknown Angrist and Krueger (1991) study and discuss how a so-called Information Matrix or Jeffreys’ prior may be used as a ‘regularization prior’ that in combination with the likelihood function yields posteriors with desirable properties. We also illustrate that the IV model and the vector autoregressive model with co-integration restrictions have a similar mathematical structure and thus this leads to similar posterior shapes. In order to perform a Bayesian posterior analysis using simulation techniques in these models, one has to face the issue of finding a good candidate density ∗Preliminary versions of this paper were presented at the 2007 ISI Conference in Lisbon, the 2008 MCMSki meeting in Bormio, and at the University of Montreal, Harvard University and Louisiana State. Helpful comments of several participants led to substantial improvements. The authors further thank David Ardia for useful suggestions. The second author gratefully acknowledges the hospitality of Harvard’s Economics department where part of this paper was written and financial assistance from the Netherlands Organization of Research (grant 400-07-703). †Econometric and Tinbergen Institutes, Erasmus University Rotterdam, The Netherlands.