Constants and ARIMA models in R

Date

6 June 2012

Topics
forecasting 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 (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 or equivalently as (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, 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 \ne 0 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 \ne 0 and d = 0, the EFF will go to the mean of the data.

  • If c \ne 0 and d = 1, the EFF will follow a straight line with slope equal to the mean of the differenced data.

  • If c \ne 0 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.