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.

  • $\begingroup$ You say (1,0,2)(1,1,1) BUT you specify (0,1,2)(1,1,1) in your command line (last line ).. Which is it ? $\endgroup$
    – IrishStat
    Commented Nov 29, 2018 at 18:04
  • $\begingroup$ Sorry for the confusion ,I had mistakenly typed (0,1,2) in the command line .It was meant to be (1,0,2) .@IrishStat $\endgroup$
    – Noob101
    Commented Nov 29, 2018 at 18:07
  • 2
    $\begingroup$ There have been tons of questions like yours before. Have you checked any of them? $\endgroup$ Commented Nov 29, 2018 at 20:19
  • $\begingroup$ @RichardHardy Yes I did check many of them , however I could not find where any model had explained the use of multiple regressors with seasonal ARIMA. It will be great if you can link me to a question that discusses this . $\endgroup$
    – Noob101
    Commented Nov 29, 2018 at 21:04

1 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.

enter image description here 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

enter image description here

  • $\begingroup$ Thank you for the answer , I am a little confused in how have you used the seasonal difference in order to account for the seasonal order where (P,D,Q) = (1,1,1). It will be great you can just type the equation in latex maybe $\endgroup$
    – Noob101
    Commented Nov 29, 2018 at 18:59
  • $\begingroup$ I don't use Latex. Why don't you contact me offline and I will try and help some more. $\endgroup$
    – IrishStat
    Commented Nov 29, 2018 at 19:48

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