# How would you convert an $ARIMA(0,1,1)(0,1,1)_{12}$ model to equation form?

I'm still having some difficulty understanding how an $ARIMA(p,d,q)(P,D,Q)_m$ model is translated into an equation. Also, given some output (for example in R), how do the coefficients relate to the equation?

In ordinarily least squared regression, it's pretty straight forward where B1, B2, etc... are the coefficients to their respective regressors. With time series it is a bit harder.

Any resources for this?

• otexts.org/fpp2/arima.html Commented Apr 24, 2018 at 0:04
• This is pretty good too (same guy): robjhyndman.com/talks/RevolutionR/10-Seasonal-ARIMA.pdf Commented Apr 24, 2018 at 1:22
• All you ever have to do is write down the standard multiplicative form with your p's and d's and q's plugged in, multiply out all the terms, and then rearrange as you like. Commented Apr 24, 2018 at 1:25

Here's the example you ask for in your title question. I'm doing this purely from memory, which will either prove that this is actually easy, or that my memory is lousy:

$ARIMA(0, 1, 1)(0, 1, 1)_{12}$ has the form

$(1 - L)(1 - L^{12}) y_t = c + (1 + \theta L)(1 + \Theta L^{12}) \epsilon_t$

where $L$ is the lag operator. Multiply the terms out to get

$(1 - L - L^{12} + L^{13}) y_t = c + (1 + \theta L + \Theta L^{12} + \theta \Theta L^{13}) \epsilon_t$

$y_t - y_{t-1} - y_{t-12} + y_{t-13} = c + \epsilon + \theta \epsilon_{t-1} + \Theta \epsilon_{t-12} + \theta \Theta \epsilon_{t-13}$

$y_t = c + y_{t-1} + y_{t-12} - y_{t-13} + \epsilon + \theta \epsilon_{t-1} + \Theta \epsilon_{t-12} + \theta \Theta \epsilon_{t-13}$

It's the same principle to get any multiplicative SARIMA model. From what I remember, a good reference on this is Box, Jenkins et al. Time Series Analysis: Forecasting and Control.

• Might you be missing an L towards the end of your 2nd equation? Commented Apr 24, 2018 at 12:19