Macroeconomic Forecasting and Variable Ordering in Multivariate Stochastic Volatility Models

Abstract

We document five novel empirical findings on the well-known potential ordering drawback associated with the time-varying parameter vector autoregression with stochastic volatility developed by Cogley and Sargent (2005) and Primiceri (2005), CSP-SV. First, the ordering does not affect point prediction. Second, the standard deviation of the predictive densities implied by different orderings can differ substantially. Third, the average length of the prediction intervals is also sensitive to the ordering. Fourth, the best ordering for one variable in terms of log-predictive scores does not necessarily imply the best ordering for another variable under the same metric. Fifth, the best ordering for variable x in terms of log-predictive scores tends to put the variable x first while the worst ordering for variable x tends to put the variable x last. Then, we consider two alternative ordering invariant time-varying parameter VAR-SV models: the discounted Wishart SV model (DW-SV) and the dynamic stochastic correlation SV model (DSC-SV). The DW-SV underperforms relative to each ordering of the CSP-SV. The DSC-SV has an out-of-sample forecasting performance comparable to the median outcomes across orderings of the CSP-SV.

Keywords: Vector Autoregressions, Time-Varying Parameters, Stochastic Volatility, Variable Ordering, Cholesky Decomposition, Wishart Process, Dynamic Conditional Correlation, Out-of-sample Forecasting Evaluation

JEL Classification: C8, C11, C32, C53