6.
Process or Product Monitoring and Control
6.4.
Introduction to Time Series Analysis
6.4.4.
Univariate Time Series Models
6.4.4.5.

BoxJenkins Models


BoxJenkins Approach

The BoxJenkins ARMA model is a combination of the
AR and MA
models (described on the previous page):
$$ \begin{eqnarray}
X_t & = & \delta + \phi_1 X_{t1} + \phi_2 X_{t2} + \cdots + \phi_p X_{tp} + \\
& & A_t  \theta_1 A_{t1}  \theta_2 A_{t2}  \cdots  \theta_q A_{tq} \, ,
\end{eqnarray} $$
where the terms in the equation have the same meaning as given for
the AR and MA model.

Comments on BoxJenkins Model

A couple of notes on this model.
 The BoxJenkins model assumes that the time series is
stationary. Box and Jenkins
recommend differencing nonstationary 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
the model.
 BoxJenkins 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 nonseasonal autoregressive
and moving average terms.
 The most general BoxJenkins 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
Reisel (1994),
Chatfield (1996),
or Brockwell and Davis
(2002).

Stages in BoxJenkins Modeling

There are three primary stages in building a BoxJenkins
time series model.
 Model Identification
 Model Estimation
 Model Validation

Remarks

The following remarks regarding BoxJenkins models should be noted.
 BoxJenkins 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)
recommends decomposition
methods for series in which the trend and seasonal components
are dominant.
 Building good ARIMA models generally requires more experience
than commonly used statistical methods such as regression.

Sufficiently Long Series Required

Typically, effective fitting of BoxJenkins models requires at
least a moderately long series.
Chatfield (1996)
recommends at least 50 observations. Many others would recommend
at least 100 observations.
