# Defining SARIMA Equation in a "simpler" way.

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.

• 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.
– vvg
Jan 4, 2023 at 2:26
• Explain "who is" SARIMA or at least give a reference. Jan 4, 2023 at 9:26
• No answer to my comment. Do you really believe that this acronym is well known ? Jan 4, 2023 at 22:02
• 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. Jan 6, 2023 at 6:40
• @JeanMarie: ARIMA is Autoregressive Integrated Moving Average model used in timeseries forecasting. SARIMA is a variant that includes seasonal parametrization.
– vvg
Jan 6, 2023 at 11:36