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It seems like almost every other day now, we get a new post with somebody asking the question: "My forecasts look very good, they almost follow the actual to the letter, except that they are shifted by one period. How can I fix that?". They don't realize that their $SARIMA(5,3,4)(1,1,0)_{12}$ or 35 layered LSTM is really just copying the value from one week to the next, what I think of as the "Random Walk Trap".

Should we have (is there already?) a community wiki that explains this problem, complete with a couple of illustrations and some pointers on how to avoid it)

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    $\begingroup$ You don't necessarily need a community wiki unless you expect others to edit it; just a single canonical version of the question with a good answer would suffice. Surely there are some good candidates already. If none are sufficiently generic, one could be chosen and edited to be made more generic -- or if substantial changes would be needed, a new one written and posted. [Whoever does the work to make it can get points for it as long as they properly credit anything they use. This is the model the network operates on. If you want to make it CW, you can, but I don't see it as required.] $\endgroup$ – Glen_b Jun 18 at 7:10
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    $\begingroup$ Go ahead! Please ... $\endgroup$ – kjetil b halvorsen Jun 18 at 19:39
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    $\begingroup$ @Glen_b: do you want to post your comment(s) as an answer? Better to have a short answer than no answer at all. Anyone who has a better answer can post it. $\endgroup$ – Stephan Kolassa Jun 20 at 10:36
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You don't necessarily need a community wiki unless you expect others to edit it; just a single canonical version of the question with a good answer would suffice.

Surely there are some good candidates already present as answers.

If none are sufficiently generic, one could be chosen and edited to be made more generic -- or if substantial changes would be needed, a new one written and posted.

In that instance, whoever does the work to make a new one can get points for it as long as they properly credit anything they use. This is the model the network operates on.

[If you want to make it CW, you can, but I don't see it as required.]

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I say go ahead, but would be cautious about what you put about this. Because this is not true for all advanced LSTM time series models. And whatever answer you put is probably not going to be true in the next 2-3 years.

Unlike SARIMA, SARCH and GARCH assume a 1 step difference, they don't require it. Please see this quote below from chapter 6 of this book the very last paragraph that talks about how this generalized assumption about step intervals might not always be valid: https://christophm.github.io/interpretable-ml-book/influential.html#disadvantages-of-identifying-influential-instances

The influence measures only take into account the deletion of individual instances and not the deletion of several instances at once. Larger groups of data instances may have some interactions that strongly influence model training and prediction. But the problem lies in combinatorics: There are n possibilities to delete an individual instance from the data. There are n times (n-1) possibilities to delete two instances from the training data. There are n times (n-1) times (n-2) possibilities to delete three … I guess you can see where this is going, there are just too many combinations.

Recall that H in ARCH and SARCH stands for Heteroscedasticity. Heteroscedasticity is a type of influence measure as we can see from the author's example of Cook's Distance. As that chapter mentions, Heteroscedasticity is part of a broader concept called Influence (or influence measures) which is in turn part of a broader concept called Anomalies (or anomaly detection). Now, look at the quote. The reason we make the assumption of minus one actually has to do more with limitations of computation power than the limitations of statistics that you learned in college. In fact, the computation power of a SARCH with a 2 step difference is O(n*(n-1))~O(n^2) of the regular SARIMA is O(n). For a 3 step difference, it is O(n^3). For a week difference, it would be O(n^7).

In relativity, we used to have the same issue with the 2-body black hole problem. However, about 5 years ago, we overcame that issue and it leads to the discovery of the first black hole. Someday, I believe people who do forecasting will overcome the limitations of basic forecasting.

In fact, considering the issues with predicting a model like bitcoin which has 100 times the volatility of the stock market, there could be applications of an L-step SARCH (where L is the period).

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