Fast, Order-Invariant Bayesian Inference in VARs using the Eigendecomposition of the Error Covariance Matrix

Bayesian inference in Vector Autoregressions (VARs) involves manipulating large matrices which appear in the posterior (or conditional posterior) of the VAR coefficients. For large VARs, the computational time involved with these manipulations becomes so large as to make empirical work impractical. In response to this, many researchers transform their VARs so as to allow for Bayesian estimation to proceed one equation at a time. This leads to a massive reduction in the computational burden. This transformation involves taking the Cholesky decomposition for the error covariance matrix. However, this strategy implies that posterior inference depends on the order the variables enter the VAR. In this paper we develop an alternative transformation, based on the eigendecomposition, which does not lead to order dependence. Beginning with an inverse-Wishart prior on the error covariance matrix, we derive and discuss the properties of the prior it implies on the eigenmatrix and eigenvalues. We then show how an extension of the prior on the eigenmatrix can allow for greater exibility while maintaining many of the benefits of conjugacy. We exploit this exibility in order to extend the prior on the eigenvalues to allow for stochastic volatility. The properties of the eigendecomposition approach are investigated in a macroeconomic forecasting exercise involving VARs with 20 variables.