Constants and ARIMA Models in R
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Join For FreeThis post is from my new book Forecasting: principles and practice, available freely online at OTexts.com/fpp/.
A nonseasonal ARIMA model can be written as
(1)
(2)
where is the backshift operator, and is the mean of . R uses the parametrization of equation (2).
Thus, the inclusion of a constant in a nonstationary ARIMA model is equivalent to inducing a polynomial trend of order in the forecast function. (If the constant is omitted, the forecast function includes a polynomial trend of order .) When , we have the special case that is the mean of .
Including constants in ARIMA models using R
arima()
By default, the arima()
command in R sets when and provides an estimate of when . The parameter
is called the “intercept” in the R output. It will be close to the
sample mean of the time series, but usually not identical to it as
the sample mean is not the maximum likelihood estimate when .
The arima()
command has an argument include.mean
which only has an effect when and is TRUE
by default. Setting include.mean=FALSE
will force .
Arima()
The Arima()
command from the forecast package provides more flexibility on the inclusion of a constant. It has an argument include.mean
which has identical functionality to the corresponding argument for arima()
. It also has an argument include.drift
which allows when . For ,
no constant is allowed as a quadratic or higher order trend is
particularly dangerous when forecasting. The parameter is called the “drift” in the R output when .
There is also an argument include.constant
which, if TRUE
, will set include.mean=TRUE
if and include.drift=TRUE
when . If include.constant=FALSE
, both include.mean
and include.drift
will be set to FALSE
. If include.constant
is used, the values of include.mean=TRUE
and include.drift=TRUE
are ignored.
When and include.drift=TRUE
, the fitted model from Arima()
is
In this case, the R output will label as the “intercept” and as the “drift” coefficient.
auto.arima()
The auto.arima()
function automates the inclusion of a constant. By default, for or , a constant will be included if it improves the AIC value; for the constant is always omitted. If allowdrift=FALSE
is specified, then the constant is only allowed when .
Eventual forecast functions
The eventual forecast function (EFF) is the limit of as a function of the forecast horizon as .
The constant has an important effect on the longterm forecasts obtained from these models.
 If and , the EFF will go to zero.
 If and , the EFF will go to a nonzero constant determined by the last few observations.
 If and , the EFF will follow a straight line with intercept and slope determined by the last few observations.
 If and , the EFF will go to the mean of the data.
 If and , the EFF will follow a straight line with slope equal to the mean of the differenced data.
 If and , the EFF will follow a quadratic trend.
Seasonal ARIMA models
If a seasonal model is used, all of the above will hold with replaced by where is the order of seasonal differencing and is the order of nonseasonal differencing.Published at DZone with permission of Rob J Hyndman, DZone MVB. See the original article here.
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