RegARIMA and ARMAX models both combine features of simple regression models with ARIMA modelling. BOXJENK supports both approaches. In RATS, both types of models are parameterized in exactly the same way—the differences lie in the estimation technique.

 

With ARMAX, the focus is on the ARMA process itself, with additional explanatory variables. For RegARIMA, the focus of the estimation is on the mean equation represented by the explanatory variables rather than the ARIMA model “noise” term. The output is switched around so the explanatory variables are listed first.

 

The RegARIMA approach is used in the X12-ARIMA methodology, supported in Professional versions of RATS through a combination of the BOXJENK and X11 instructions.

 

APPLYDIFFERENCES

The APPLYDIFFERENCES option is used to choose between the following two forms for a mixed model: the first for APPLYDIFFERENCES, the second for NOAPPLY:

\begin{equation} \left( {1 - L} \right)^d \left( {1 - L^s } \right)^e (y_t - X_t \beta ) = {\text{ARMA noise}} \label{eq:uvfore_applydifferences} \end{equation}

\begin{equation} \left( {1 - L} \right)^d \left( {1 - L^s } \right)^e y_t = X_t \beta + {\text{ARMA noise}} \label{eq:uvfore_noapplydiff} \end{equation}

In most applications, \eqref{eq:uvfore_applydifferences} is more natural, though with deterministic regressors such as trends and dummies, the model can generally be written either way. For instance, if we need to first difference, and the regression model in the first form has 1, t and the level shift dummy , then the corresponding regressors in the second form are (zero), 1 and the additive outlier dummy \(D_{t \ge T_0 } \), which you get by differencing the \(X_t\). The original intercept giving the overall level of the process is lost in moving between the two, and can be recovered only by referring back to the original data.

 

There’s a third possible way to add regressors to an ARMA model, which is

\begin{equation} \varphi (L)\left\{ {\left( {1 - L} \right)^d \left( {1 - L^s } \right)^e y_t } \right\} = X_t \beta + \theta (L)u_t \end{equation}

However, it’s harder to interpret the regression part in this form than in the similar form \eqref{eq:uvfore_noapplydiff}. The two are equivalent if there are no AR terms—if there are AR terms and you need to estimate a model like this, you can use ITERATE rather than BOXJENK.

 

REGRESSORS option

To estimate an ARMAX model in RATS, use the REGRESSORS option on BOXJENK, and follow the instruction with a supplementary card listing (in regression format) the variables in the regression model. The ARMA terms are handled using the standard AR and MA options. You can also use the INPUTS option, though that is designed primarily for implementing transfer function and intervention models.

 

The following is part of an example from Diebold (2004). The series of interest is liquor sales. It has both a strong trend and seasonal. The basic regression model is a full set of seasonals plus trend and trend squared:

 

seasonal seasons

linreg lsales 1968:1 1993:12 resids

# time time2 seasons{0 to -11}

 

From the residuals, an AR(3) model is identified for the noise term. The full model is estimated using the BOXJENK instruction shown.

 

boxjenk(regressors,ar=3) lsales 1968:1 1993:12 resids

# time time2 seasons{0 to -11}

 

RegARIMA Models

Use the GLS option, rather than REGRESSORS, to do RegARIMA estimation. As with REGRESSORS, you supply a list of explanatory variables on a supplementary card.  The output is switched around so the explanatory variables are listed first. GLS forces the use of maximum likelihood estimation and also includes the behavior of the APPLYDIFFERENCES option.

 

You can employ the automatic outlier detection as part of your estimation. This is done using the OUTLIER option which selects the types of outliers for which to scan. OUTLIER can take a while since it needs to estimate a model for each data point for each type of outlier.

 

The REGARIMA.RPF example does a preliminary analysis of calendar effects for a seasonally unadjusted series of appliance sales. (A series used as an example in Census X12-ARIMA).

Copyright © 2025 Thomas A. Doan