Prediction Intervals for ARIMA Models

As indicated by Chatfield (1993) in his comprehensive stateof-
the-art review, the construction of valid prediction intervals
(PI's) for time series continues to present considerable difficulties.
In particular, Chatfield noted a number of reasons why
PI's may be too narrow; these include the following:
I. Model parameters may have to be estimated.
2. Innovations may not be normally distributed.
3. There may be outliers in the data.
4. The wrong model may be identified.
5. The underlying model may change, either during the
period of fit or in the future.
In this article, we focus on the first of these issues. If the
uncertainty relating to parameter estimation is not allowed for
explicitly, the resulting PI's would be too narrow. Further,
the nonlinear nature of the parameter estimates in time series
makes the problem intractable as regards an exact analytic
solution, so we develop various approximate solutions, which
are then explored in a simulation study. Only when we are
confident of our ability to produce reliable PI's in the basic
case can we address the remaining issues. Thus, in this article,
we examine the construction of PI'S when the parameters
are unknown and the errors are assumed to be normal, leaving
the other issues to be addressed in further research.

We identify four approaches to the construction of PI's and
report on an extensive simulation study of these alternatives.
The particular model used in our simulations is the additive
Holt-Winters (HW) scheme; see Example 1.2. Yar and
Chatfield (1990) provided PI's for this scheme based on its
autoregressive integrated moving average (ARIMA) representation
and setting the parameter values equal to their estimates
(the "plug-in" approach). These authors found the method
to be superior to previous, albeit heuristic, approaches, and
the plug-in PI is one of the options considered in our study.