# Constants and ARIMA Models in R

# Constants and ARIMA Models in R

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**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)

(2)

where is the backshift operator, and is the mean of . 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 in the forecast function. (If the constant is omitted, the forecast function includes a polynomial trend of order .) When , we have the special case that is the mean of .

### Including constants in ARIMA models using R

#### arima()

By default, the `arima()`

command in R sets when and provides an estimate of when . The parameter 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 .

The `arima()`

command has an argument `include.mean`

which only has an effect when and is `TRUE`

by default. Setting `include.mean=FALSE`

will force .

#### 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 when . For , no constant is allowed as a quadratic or higher order trend is particularly dangerous when forecasting. The parameter is called the “drift” in the R output when .

There is also an argument `include.constant`

which, if `TRUE`

, will set `include.mean=TRUE`

if and `include.drift=TRUE`

when . 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 and `include.drift=TRUE`

, the fitted model from `Arima()`

is

In this case, the R output will label as the “intercept” and as the “drift” coefficient.

#### auto.arima()

The `auto.arima()`

function automates the inclusion of a constant. By default, for or , a constant will be included if it improves the AIC value; for the constant is always omitted. If `allowdrift=FALSE`

is specified, then the constant is only allowed when .

### Eventual forecast functions

The eventual forecast function (EFF) is the limit of as a function of the forecast horizon as .

The constant has an important effect on the long-term forecasts obtained from these models.

- If and , the EFF will go to zero.
- If and , the EFF will go to a non-zero constant determined by the last few observations.
- If and , the EFF will follow a straight line with intercept and slope determined by the last few observations.
- If and , the EFF will go to the mean of the data.
- If and , the EFF will follow a straight line with slope equal to the mean of the differenced data.
- If and , the EFF will follow a quadratic trend.

### Seasonal ARIMA models

If a seasonal model is used, all of the above will hold with replaced by where is the order of seasonal differencing and 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.

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

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