## The ARIMAX model muddle

There is often confusion about how to include covariates in ARIMA models, and the presentation of the subject in various textbooks and in R help files has not helped the confusion. So I thought I’d give my take on the issue. To keep it simple, I will only describe non-seasonal ARIMA models although the ideas are easily extended to include seasonal terms. I will include only one covariate in the models although it is easy to extend the results to multiple covariates. And, to start with, I will assume the data are stationary, so we only consider ARMA models.Let the time series be denoted by . First, we will define an ARMA model with no covariates:

### ARMAX models

An ARMAX model simply adds in the covariate on the right hand side:*not*the effect on when the is increased by one (as it is in regression). The presence of lagged values of the response variable on the right hand side of the equation mean that can only be interpreted conditional on the value of previous values of the response variable, which is hardly intuitive.

If we write the model using backshift operators, the ARMAX model is given by

### Regression with ARMA errors

For this reason, I prefer to use regression models with ARMA errors, defined as follows.Using backshift operators, this model can be written as

### Transfer function models

Both of these models can be considered as special cases of transfer function models, popularized by Box and Jenkins:Sometimes these are called “dynamic regression models”, although different books use that term for different models.

The method for selecting the orders of a transfer function model that is described in Box and Jenkins is cumbersome and difficult, but continues to be described in textbooks. A much better procedure is given in Pankratz (1991), and repeated in my 1998 forecasting textbook.

### Non-stationary data

For ARIMA errors, we simply replace with where denotes the differencing operator. Notice that this is equivalent to differencing both and before fitting the model with ARMA errors. In fact, it is necessary to difference all variables first as estimation of a model with non-stationary errors is not consistent and can lead to “spurious regression”.### R functions

The`arima()`

function in R (and `Arima()`

and `auto.arima()`

from the forecast package) fits a regression with ARIMA
errors. Note that R reverses the signs of the moving average
coefficients compared to the standard parameterization
given above.The

`arimax()`

function from the TSA package fits the transfer function model (but not the ARIMAX
model). This is a new package and I have not yet used it, but it is
nice to finally be able to fit transfer function models in R.
Sometime I plan to write a function to allow automated order
selection for transfer functions as I have done with `auto.arima()`

for regression with ARMA errors (part of the forecast package)
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