After an SARIMA transformation, is $\epsilon_t$ equal to the difference in observed original $y_t$ from its estimate or the equivalent quantity for transformed $y_t$, $y'_t$?
The motivation: I am trying to build a STAN model (called from R) and want to move from existing frequentist SARIMAX models to a broader state-space perspective. First, though, I want to reproduce the model and I want to code it manually so that I understand the guts. This leads me to my question:
Say that I have an $ARIMA(0, 1, 1)(0, 1, 1)_{52}$ model with regression coefficients (holidays, traffic on a given day, etc...). I know that I need to take seasonal and first differences for all covariates, leading to
$y'_t = \beta X'_t + \eta_t$, where
$(1-B)(1-B^{52})\eta_t = (1-\phi B)(1-\phi_{52} B_{52})\epsilon_t$
Perhaps naively, I plan to forecast using $\epsilon_t = y_{actual,t} - \hat{y_t}$. Should I use my original $y_t$ or my transformed $y'_t$? Does $\epsilon_t = y_{actual,t} - \hat{y_t}$ or $y'_{actual,t} - \hat{y'_t}$?
This question may just be clarification, but I've consulted Hyndman's ARIMAX blog post (here), his textbook chapter on regression with ARIMA errors, his slides, the STAN manual ARMA section, and these helpful [questions][7].
