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Suppose I have a time-series prediction problem, where the loss between the model's prediction and the true outcome is some custom loss function $\ell(\hat{y}, y)$

Is there some theory of how the standard ARMA / ARIMA models should be modified? For example, if $\ell$ is not measuring the additive deviation, the "right" error term in the MA part of ARMA may not be additive, but something else. Is it also not obvious what would be the generalized counterparts of the standard conditions, like stationarity, in this setting.

I was looking for literature, but the only thing I found was a theory specially tailored towards Poisson time series. But nothing for more general cost functions.

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  • $\begingroup$ You may want to look at this paper on Loss Functions in Time Series Forecasting. $\endgroup$ Commented Apr 3 at 10:46
  • $\begingroup$ Unfortunately, there is no discussion on the meaning of "error" and its effect on the model. $\endgroup$ Commented Apr 10 at 6:18
  • $\begingroup$ Right in the second paragraph, the paper says "The loss function of the forecast error $e_{t+h} =Y_{t+h} − f_{t,h}$, is denoted as $c(Y_{t+h}, f_{t,h})$". $\endgroup$ Commented Apr 10 at 7:37
  • $\begingroup$ Yes. And it's unreasonable. If my model is learning in log space, such as Poisson maximum likelihood loss, it's obviously unreasonable di define a difference as the error. $\endgroup$ Commented Apr 10 at 19:18

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