The State Space Representation and Estimation of a Time-Varying Parameter VAR with Stochastic Volatility

Abstract

To capture the evolving relationship between multiple economic variables, time variation in either coefficients or volatility is often incorporated into vector autoregressions (VARs). 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 US 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 10-year Treasury bond yield are less noticeable.