Constants and ARIMA models in R

This post is from my new book Forecasting: principles and practice, available freely online at

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


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.


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.


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.

    • 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?




    • 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?

    • 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?


  • 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..

  • 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

    • 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.