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 non-seasonal ARIMA model can be written as \begin{equation}\label{eq:c} (1-\phi_1B - \cdots - \phi_p B^p)(1-B)^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)(1-B)^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 $(1-B)^d y_t$. R uses the parametrization of the second equation.

Thus, the inclusion of a constant in a non-stationary 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 $d-1$.) 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 $\hat{y}_{t+h|t}$ as a function of the forecast horizon $h$ as $h\rightarrow\infty$.

The constant $c$ has an important effect on the long-term 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 non-zero 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 non-seasonal differencing.