Constants and ARIMA Models in R

DZone's Guide to

Constants and ARIMA Models in R

Rob J Hyndman

Oct. 10, 12 · Big Data Zone ·

Free Resource

Like (0)

Comment (0)

Save

{{ articles[0].views | formatCount}} Views

Hortonworks Sandbox for HDP and HDF is your chance to get started on learning, developing, testing and trying out new features. Each download comes preconfigured with interactive tutorials, sample data and developments from the Apache community.

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) $\begin{equation*} (1-\phi_1B - \cdots - \phi_p B^p)(1-B)^d y_t = c + (1 + \theta_1 B + \cdots + \theta_q B^q)e_t \end{equation*}$

or equivalently as

(2) $\begin{equation*} (1-\phi_1B - \cdots - \phi_p B^p)(1-B)^d (y_t - \mu t^d/d!) = (1 + \theta_1 B + \cdots + \theta_q B^q)e_t, \end{equation*}$

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 equation (2).

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.

Hortonworks Community Connection (HCC) is an online collaboration destination for developers, DevOps, customers and partners to get answers to questions, collaborate on technical articles and share code examples from GitHub. Join the discussion.

DOWNLOAD

Topics:

Like (0)

Comment (0)

Save

{{ articles[0].views | formatCount}} Views

Published at DZone with permission of Rob J Hyndman , DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

Constants and ARIMA Models in R

Constants and ARIMA Models in R