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?
Best Answer
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.