We examine the theory and behavior in practice of Bayesian and bootstrap methods for generating error bands on impulse responses in dynamic linear models. The Bayesian intervals have a firmer theoretical foundation in small samples, are easier to compute, and are about as good in small samples by classical criteria as are the best bootstrap intervals. Bootstrap intervals based directly on the simulated small-sample distribution of an estimator, without bias correction, perform very badly. We show that a method that has been used to extend to the overidentified case standard algorithms for Bayesian intervals in reduced form models is incorrect, and we show how to obtain correct Bayesian intervals for this case.
A model for U.S. macroeconomic time series that has been used for forecasting for several years is described in some detail. The model is a multivariate Bayesian autoregression, with allowance for conditional heteroskedasticity, stochastic time-variation in parameters, and non-normality of disturbances. It specifies the prior distribution in ways that improve on previous Bayesian vector autoregression specifications in realism and forecasting performance. The model’s record of forecasting in recent years is displayed and discussed.
Existing theory and evidence on the effects of monetary policy are reviewed. Substantial room for disagreement among economists remains. New evidence, based on multivariate time series studies of several countries, is presented. While certain patterns in the data consistent with effective monetary policy are strikingly similar across countries, others, particularly the tendency of interest rate increases to predict high inflation, are harder to reconcile with effective monetary policy.
In his paper “To Criticize the Critics” (1991), Peter Phillips discusses Bayesian methodology for time series models. The main point that Uhlig and I set out to make, however, was that careful consideration of the implications of the likelihood principle suggests that much of the recent work under the “unit root” label in the econometrics literature is being incorrectly interpreted in practice. We pointed out that time series models with possible unit roots are one of the few domains within which the implications of a likelihood principle approach to inference are difference, even in the large samples, from those of a classical hypothesis testing approach. Phillips addresses this part of our paper only indirectly.
Keywords: Bayesian analysis, time series, unit roots