Bayesian Comparison of ARIMA and Stationary ARMA Models

John Marriottl and Paul Newbold2
' ~ e ~ a r t m eonf tMathematics, Statistics and Operational Research, Nottingham Trent University,
Nottingham NG1 4BU, UK. 2~epartmenotf Economics, University of Nottingharn, Nottingharn
NG7 2RD. UK
Summary
Time series analysts have long been concerned with distinguishing stationary "generating processes"
from processes for which differencing is required to induce stationarity. In practical applications, this
issue is addressed almost invariably through formal hypothesis testing. In this paper, we explore some
aspects of the Bayesian approach to the problem, leading to the calculation of posterior odds ratios.
Interesting features arise in the simplest possible variant of the problem, where a choice has to be made
between a random walk and a stationary first order autoregressive model. We discuss in detail the analysis
of this case, and also indicate how our approach extends to the more general comparison of an ARIMA
model with a stationary competitor.
Key words: Model comparison; Posterior odds; Random walk; Time series models; Unit autoregressive roots

In modern time series analysis, considerable attention is paid to the question of the number of
times an individual series must be differenced to achieve stationarity. For the ARIMA ( p , d , q)
class of models, introduced by Box & Jenkins (1970), the question then is of the number d of unit
autoregressive roots in the generating process. Most often in practical applications the choice is
between d = 0 and d = 1, though of course choice between d = 1 and d = 2 can be made along
identical lines by working with the first differences of the original time series.