T ime-varying parameter vector autoregressions (TVP-VARs) have become an increasingly popular tool for analyzing the behavior of macroeconomic time series. TVP-VARs di¤er from more standard …xed-coe¢ cient VARs in that they allow for coe¢ cients in an otherwise linear VAR model to vary over time following a speci…ed law of motion. In addition, TVP-VARs often include stochastic volatility (SV), which allows for time variation in the variances of the error processes that a¤ect the VAR.The attractiveness of TVP-VARs is based on the recognition that many, if not most, macroeconomic time series exhibit some form of nonlinearity. For instance, the unemployment rate tends to rise much faster at the start of a recession than it declines at the onset of a recovery. Stock market indices exhibit occasional episodes where volatility, as measured by the variance of stock price movements, rises considerably. As a third example, many aggregate series show a distinct change in behavior in terms of their persistence and their volatility around the early 1980s when the Great In ‡ation of the 1970s turned into the Great Moderation, behavior that is akin to a structural shift in certainWe are grateful to Pierre-Daniel Sarte, Daniel Tracht, John Weinberg, and Alex Wolman, whose comments greatly improved the exposition of this paper. The views expressed in this paper are those of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve System.