Constants and ARIMA models in R

This post is from my new book Forecasting: principles and practice, available freely online at OTexts.com/fpp/.

A non-seasonal ARIMA model can be written as

(1)  

or equivalently as

(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 non-stationary 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 long-term forecasts obtained from these models.

• If and , the EFF will go to zero.
• If and , the EFF will go to a non-zero 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 non-seasonal differencing.