This paper provides a Bayesian analysis of Autoregressive Fractionally
Integrated Moving Average (ARFIMA) models. We discuss in detail infer
ence on impulse responses, and show how Bayesian methods can be used t
o (i) test ARFIMA models against ARIMA alternatives and (ii) take mode
l uncertainty into account when making inferences on quantities of int
erest. Our methods are then used to investigate the persistence proper
ties of real U.S. GNP.