### Improving Bayesian VAR density forecasts through autoregressive Wishart Stochastic Volatility

*Paul Karapanagiotidis*

Last modified: 2012-07-12

#### Abstract

Dramatic changes in macroeconomic time series volatility pose a challenge to contemporary vector autoregressive (VAR) forecasting models. Traditionally, the conditional volatility of such models had been assumed constant over time or allowed for breaks across long time periods. More recent work, however, has improved forecasts by allowing the conditional volatility to be completely time variant by specifying the VAR innovation variance as a distinct discrete time process. For example, Clark (2011) specifies the volatility process as an independent log random walk for each time series in the VAR. Unfortunately, there is no empirical reason to believe that the VAR innovation volatility process of macroeconomic growth series follow log random walks, nor that the volatility of each series is independent of the others. This suggests that a more robust specification on the volatility process---one that both accounts for co-persistence in conditional volatility across time series and exhibits mean reverting behaviour---should improve density forecasts, especially over the long run forecasting horizon. In this respect, I employ a latent Inverse-Wishart autoregressive stochastic volatility specification on the conditional variance equation of a Bayesian VAR, given U.S. macroeconomic time series data, in evaluating Bayesian forecast efficiency against a competing log random walk specification by Clark (2011). In other words, this paper tackles the problem of macroeconomic VAR innovation heteroskedasticity by proposing a key change in the volatility process of VARs. Instead of assuming cross-series independent log random walk processes on the volatilities of the VAR shocks, as is popular amongst empirical forecasters, I employ an Inverse-Wishart process where the scale matrix parameter is an autoregressive function of past VAR innovation covariance matrices. Pros and cons to this modification are discussed and estimation steps are provided which employ MCMC methods and the Gibbs sampler, where conditional posterior distributions are available for most parameters. A Bayesian approach, incorporating log-predictive likelihoods, is also taken in comparing density forecasts between the benchmark model, Clark (2011), and my competing Inverse-Wishart autoregressive volatility modification. Results suggest that incorporating the more sophisticated Inverse-Wishart autoregressive volatility process improves density forecasts in both the short and long run forecasting horizons, with larger improvements seen as the horizon increases--despite a small sample size and increased parameterization of the model. Finally, as data I employ four major U.S. macroeconomic time-series: GDP growth, the inflation rate, the interest rate, and unemployment rate. Data employed is the same as that in Clark (2011) to enable direct comparison of results.