The state space representation and estimation of a time-varying parameter VAR with stochastic volatility

To capture the evolving relationship between multiple economic variables, time variation in either coefficients or volatility is often incorporated into vector autoregressions (VARs). However, allowing time variation in coefficients or volatility without restrictions on their dynamic behavior can increase the number of parameters too much, making the estimation of such a model practically infeasible. For this reason, researchers typically assume that time-varying coefficients or volatility are not directly observed but follow random processes which can be characterized by a few parameters. The state space representation that links the transition of possibly unobserved state variables with observed variables is a useful tool to estimate VARs with time-varying coefficients or stochastic volatility. ; In this paper, we discuss how to estimate VARs with time-varying coefficients or stochastic volatility using the state space representation. We focus on Bayesian estimation methods which have become popular in the literature. As an illustration of the estimation methodology, we estimate a time-varying parameter VAR with stochastic volatility with the three U.S. macroeconomic variables including inflation, unemployment, and the long-term interest rate. Our empirical analysis suggests that the recession of 2007-2009 was driven by a particularly bad shock to the unemployment rate which increased its trend and volatility substantially. In contrast, the impacts of the recession on the trend and volatility of nominal variables such as the core PCE inflation rate and the ten-year Treasury bond yield are less noticeable.