Process or Product Monitoring and Control
Introduction to Time Series Analysis
Univariate Time Series Models
The Box-Jenkins ARMA model is a combination of the
AR and MA
models (described on the previous
where the terms in the equation have the same meaning as given for
the AR and MA model.
Comments on Box-Jenkins Model
A couple of notes on this model.
- The Box-Jenkins model assumes that the time series is
stationary. Box and Jenkins
recommend differencing non-stationary series one or more times
to achieve stationarity. Doing so produces an ARIMA model,
with the "I" standing for "Integrated".
- Some formulations transform the series by subtracting the
mean of the series from each data point. This yields a
series with a mean of zero. Whether you need to do this
or not is dependent on the software you use to estimate
- Box-Jenkins models can be extended to include
seasonal autoregressive and seasonal
moving average terms. Although this complicates the
notation and mathematics of the model, the underlying
concepts for seasonal autoregressive and seasonal moving
average terms are similar to the non-seasonal autoregressive
and moving average terms.
- The most general Box-Jenkins model includes difference
operators, autoregressive terms, moving average terms,
seasonal difference operators, seasonal autoregressive
terms, and seasonal moving average terms. As with
modeling in general, however, only necessary terms should
be included in the model. Those interested in the
mathematical details can consult
Box, Jenkins and
or Brockwell and Davis
Stages in Box-Jenkins Modeling
There are three primary stages in building a Box-Jenkins
time series model.
- Model Identification
- Model Estimation
- Model Validation
The following remarks regarding Box-Jenkins models should be noted.
- Box-Jenkins models are quite flexible due to the inclusion of both
autoregressive and moving average terms.
- Based on the Wold decomposition thereom (not discussed in
the Handbook), a stationary process can be approximated by
an ARMA model. In practice, finding that approximation may
not be easy.
- Chatfield (1996)
methods for series in which the trend and seasonal components
- Building good ARIMA models generally requires more experience
than commonly used statistical methods such as regression.
Sufficiently Long Series Required
Typically, effective fitting of Box-Jenkins models requires at
least a moderately long series.
recommends at least 50 observations. Many others would recommend
at least 100 observations.