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
\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.


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  • syazreen

    Hi Prof,
    auto.arima function is great, the best model can automatically be fitted .by having smallest error.
    Hpwever my model required a random walk with drift model.
    How to use arima() to include drift term.
    arima(x,c(0,1,0)) will not give the same model as I want it to be; that is random walk with drift term is the mean.

    • http://robjhyndman.com/ Rob J Hyndman

      auto.arima() returns the fitted model.

      If you need to fit it again, use
      Arima(x, order=c(0,1,0), include.drift=TRUE)

      • syazreen

        Thanks so much!
        It works exactly as what I want.

  • Brian

    Is it possible the behavior of Arima has changed?

    install.packages(“fma”)
    library(fma)

    mod2.

    Thanks!

    • http://robjhyndman.com/ Rob J Hyndman

      Well spotted. The drift term is actually redundant and non-identifiable — note the size of the standard error on the drift coefficient. So it is not actually fitting a cubic drift. I’ll fix the function so it doesn’t return the parameter.

  • sunsetter

    Can auto.arima() be set to never allow non-zero mean, even when d=0?

    • http://robjhyndman.com/ Rob J Hyndman

      No. You would have to do your own modification of it if you wanted to do that.

  • Débora Spenassato

    Hi prof. Hyndman,
    my model using the function auto. arima() is ARIMA(1,1,0) with drift (AR = 0.208 and drift =2.531). I’m having difficulty to form the equation. would be deltaYt=2.531+0.208 deltaYt-1 + et?

    Thanks,

  • Shraddha Panda

    Hello Professor, Could you please have a look at this question here and share your inputs? I am trying to fit auto.arima for longitudnal data by grouping different regions.. http://stackoverflow.com/questions/25036986/auto-arima-using-xreg-and-forecasting-several-ts-together

  • Joshua Makubu

    Hi Prof
    Arima() is not a function in R is the feedback i get when i try to model with drift. Please educate me

    • http://robjhyndman.com/ Rob J Hyndman

      load the forecast package first.

      • Joshua Makubu

        Prof .Good evening, Am using r version 3.1.2 but still cant get the function auto.arima or the function arima after installing the forecast package.
        is there anything am not doing right?

        • Mathijs

          You’ve likely not loaded the package, load the package by entering library(forecast). After that, loading the help files by entering ?auto.arima (use R Studio!) and the examples in Prof. Hyndman’s book will help you get there.

  • JJJJ

    Does the same apply when i include dummies in my sarima model? My model seems to get a trend when i include dummies (and differencing).

  • JJJJJ

    Does the same apply when i include dummy variables? My model seems to get a trend when i include dummy variables (together with my differencing).

    • http://robjhyndman.com/ Rob J Hyndman

      The dummy variables should be differenced before use. Otherwise they will induce a trend.

      • Alex Mustermann

        thanks!

  • Alex Mustermann

    Does the same apply when i include dummy variables? I seem to get a trend when i include dummies together with my differencing.