Mathematical Equation for Seasonal Arima Model with external Regressors

Question

I am trying to write the mathematical ARIMA equation for the following - A seasonal ARIMA(1,0,2)(1,1,1) with quarterly data using two external explanatory variables aside from the explained variable(also of the same frequency) . I have an idea on how to write the ARIMA equation for a standard ARIMA model which does not involve seasonal differences and with just one explanatory variable.

This question stems from the fact that I was trying to understand how the R function arima from the stats package would work under the following command

arima(y, xreg = cbind[x1,x2],order = c(1,0,2),seasonal = c(1,1,1)), where x1 and x2 are the external regressors.

Answer 1

Caveat: I do not use the function and can only surmise what it should do.

The only possible alternative is that the seasonal differencing factor is ALSO applied to the two input series.

You might reach out to the author and ask which of these two interpretations is correct i.e. stationary X1 and X2 or quarterly differences of X1 and X2. I would guess stationary ( no differences of the X's ).

Also note I elected to include a constant which would/should be an option.

I took your specification (1,0,2)(1,1,1)4

and created some dummy data and estimated . I selected to ASSUME that X1 AND X2 entered without any differencing with no lags . I estimated without testing significance and without examining the residuals for pulses,seasonal pulses, step/level shifts or local trends and without examining the residuals for either error autocorrelation or cross-correlation with the 2 X's ..

A second alternative which upon reflection is probably the one that is used by your software. Note that if you divide by [1-B**4] this effectively cancels the seasonal differencing in the X's BUT does give you the seasonal differencing for the ARIMA structure