Constants and ARIMA models in R
This post is from my new book Forecasting: principles and practice, available freely online at OTexts.org/fpp/.
A nonseasonal ARIMA model can be written as \begin{equation}\label{eq:c} (1\phi_1B  \cdots  \phi_p B^p)(1B)^d y_t = c + (1 + \theta_1 B + \cdots + \theta_q B^q)e_t \end{equation} or equivalently as \begin{equation}\label{eq:mu} (1\phi_1B  \cdots  \phi_p B^p)(1B)^d (y_t  \mu t^d/d!) = (1 + \theta_1 B + \cdots + \theta_q B^q)e_t, \end{equation} where $B$ is the backshift operator, $c = \mu(1\phi_1  \cdots  \phi_p )$ and $\mu$ is the mean of $(1B)^d y_t$. R uses the parametrization of the second equation.
Thus, the inclusion of a constant in a nonstationary ARIMA model is equivalent to inducing a polynomial trend of order $d$ in the forecast function. (If the constant is omitted, the forecast function includes a polynomial trend of order $d1$.) When $d=0$, we have the special case that $\mu$ is the mean of $y_t$.
Including constants in ARIMA models using R
arima()
By default, the arima()
command in R sets $c=\mu=0$ when $d>0$ and provides an estimate of $\mu$ when $d=0$. The parameter $\mu$ 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 $p+q>0$.
The arima()
command has an argument include.mean
which only has an effect when $d=0$ and is TRUE
by default. Setting include.mean=FALSE
will force $\mu=0$.
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 $\mu\ne0$ when $d=1$. For $d>1$, no constant is allowed as a quadratic or higher order trend is particularly dangerous when forecasting. The parameter $\mu$ is called the “drift” in the R output when $d=1$.
There is also an argument include.constant
which, if TRUE
, will set include.mean=TRUE
if $d=0$ and include.drift=TRUE
when $d=1$. 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 $d=0$ and include.drift=TRUE
, the fitted model from Arima()
is $$(1\phi_1B  \cdots  \phi_p B^p) (y_t  a  bt) = (1 + \theta_1 B + \cdots + \theta_q B^q)e_t.
$$
In this case, the R output will label $a$ as the “intercept” and $b$ as the “drift” coefficient.
auto.arima()
The auto.arima()
function automates the inclusion of a constant. By default, for $d=0$ or $d=1$, a constant will be included if it improves the AIC value; for $d>1$ the constant is always omitted. If allowdrift=FALSE
is specified, then the constant is only allowed when $d=0$.
Eventual forecast functions
The eventual forecast function (EFF) is the limit of $\hat{y}_{t+ht}$ as a function of the forecast horizon $h$ as $h\rightarrow\infty$.
The constant $c$ has an important effect on the longterm forecasts obtained from these models.

If $c=0$ and $d=0$, the EFF will go to zero.

If $c=0$ and $d=1$, the EFF will go to a nonzero constant determined by the last few observations.

If $c=0$ and $d=2$, the EFF will follow a straight line with intercept and slope determined by the last few observations.

If $c\ne0$ and $d=0$, the EFF will go to the mean of the data.

If $c\ne0$ and $d=1$, the EFF will follow a straight line with slope equal to the mean of the differenced data.

If $c\ne0$ and $d=2$, the EFF will follow a quadratic trend.
Seasonal ARIMA models
If a seasonal model is used, all of the above will hold with $d$ replaced by $d+D$ where $D$ is the order of seasonal differencing and $d$ is the order of nonseasonal differencing.
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