Sign Restrictions, Structural Vector Autoregressions, and Useful Prior Information

It has become common in empirical macroeconomics to use numerical Bayesian methods to form structural inference in vector autoregressions that are identified solely on the basis of sign restrictions. Because sign restrictions only provide set-identification of structural parameters, over certain regions of the parameter space all that these procedures can do is reproduce the researcher's implicit prior distribution. In this paper we characterize these regions, explicate the prior that is implicit in popular methods, provide an analytical characterization of the full posterior distribution for arbitrary priors, and analyze the asymptotic properties of this posterior distribution. We show that in a simple bivariate supply and demand example, if the population correlation between reduced-form residuals is negative, then even if one has available an infinite sample of data, any inference about the supply elasticity is simply a restatement of the implicit prior distributions. More generally, the asymptotic posterior distribution of contemporaneous coefficients in an n-variable VAR is confined to the set of values that orthogonalize the population variance-covariance matrix of OLS residuals, with the height of the posterior proportional to the height of the prior at any point within that set. We suggest that researchers should use explicit rather than implicit prior distributions and should routinely report the difference between prior and posterior distributions for key magnitudes of interest. We illustrate these methods with a simple macroeconomic model.

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