Singular Vector Autoregressions

This paper develops methods for the empirical analysis of singular processes, particularly in macroeconomics. A strong rationale, a well-developed theoretical framework and, as we show, empirical support exist for multivariate time series with a singular spectral density. A singular spectral density is consistent with economic theory underlying, for example, DSGE models in which the number of variables is greater than the number of structural shocks. This assumption guarantees the existence of a finite order VAR representation but there does not exist a unique probability density function with respect to the Lebesgue measure. We therefore define a density on a compact linear submanifold with respect to the Hausdorff measure and, in a Bayesian framework, develop an HMC algorithm that jointly samples coefficients, lag length, and the number of shocks. We use the proposed framework to carry out structural analysis on the US macroeconomy with fewer shocks than variables.