Database Reference
In-Depth Information
8.2 ARIMA Model
To fully explain an ARIMA (Autoregressive Integrated Moving Average) model, this
section describes the model's various parts and how they are combined. As stated in
the first step of the Box-Jenkins methodology, it is necessary to remove any trends
or seasonality in the time series. This step is necessary to achieve a time series
with certain properties to which autoregressive and moving average models can
be applied. Such a time series is known as a stationary time series. A time series,
,, is a
stationary time series
if the following three conditions
are met:
• (a)The expected value (mean) of
is a constant for all values of
t
.
• (b)The variance of
is finite.
• (c)The covariance of
depends only on the value of
,
…for all .
The covariance of
is a measure of how the two variables,
,
vary together. It is expressed in
Equation 8.1
.
If two variables are independent of each other, their covariance is zero. If the
variables change together in the same direction, the variables have a positive
covariance. Conversely, if the variables change together in the opposite direction,
the variables have a negative covariance.
For a stationary time series, by condition (a), the mean is a constant, say . So, for
a given stationary sequence,
, the covariance notation can be simplified to what's
shown in
Equation 8.2
.
By part (c), the covariance between two points in the time series can be nonzero, as
for
.
