I am trying to understand how to write a SARIMA equation in a way that is more easily understood, for me. The only specification I can find for the model looks like:

$(1-\phi_1)(1-\Phi_1^4)(1-B)(1-B^4)y_t = (1+\theta_1B)(1+\Theta_1B^4)\epsilon_t$,

for a SARIMA(1,1,1)(1,1,1)4 model. My preferred way of writing these equations are like:

$y_t = \beta_0 + \beta_1 y_{t-1} + \beta_2 y_{t-2} + \beta_3 y_{t-3} + \varepsilon_t$

Here for an AR(3) model. My question is, how would this first equation look if it was written in the same way as the second equation?

Thank you.

  • $\begingroup$ See online.stat.psu.edu/stat510/lesson/4/4.1/#paragraph--292 for an example of how to convert the model parameters into an equation of the form you require. Hint: Simple polynomial multiplication. $\endgroup$
    – vvg
    Jan 4, 2023 at 2:26
  • $\begingroup$ Explain "who is" SARIMA or at least give a reference. $\endgroup$
    – Jean Marie
    Jan 4, 2023 at 9:26
  • $\begingroup$ No answer to my comment. Do you really believe that this acronym is well known ? $\endgroup$
    – Jean Marie
    Jan 4, 2023 at 22:02
  • $\begingroup$ Hi, didn't have time to check my notifications. In my world I thought this was well not among mathematicians, but I guess I was wrong. @vvg solved my problem though, wish it would have been an answer so I could accept it as the solution. $\endgroup$ Jan 6, 2023 at 6:40
  • 1
    $\begingroup$ @JeanMarie: ARIMA is Autoregressive Integrated Moving Average model used in timeseries forecasting. SARIMA is a variant that includes seasonal parametrization. $\endgroup$
    – vvg
    Jan 6, 2023 at 11:36


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