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There are lot of articles out there that talks about why extrapolation is a bad thing to do.

My question is if the above is true , how are forecasting methods like forecasting the trend based on some time series statistically significant at all ?

Could someone explain why we still use forecasting techniques like regression , ARIMA knowing that we shouldn't extrapolate?

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    $\begingroup$ I'm reminded of a quotation of Michel Foucault where he remarked, in a somewhat different context, "My point is not that everything is bad, but that everything is dangerous, which is not exactly the same as bad." $\endgroup$
    – Sycorax
    Commented Apr 8, 2021 at 18:05
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    $\begingroup$ The better articles show quantatively how prediction error blows up quickly as a function of how much you are extrapolating. Thus, they point out the potential problems but do not categorically state it's a "bad thing to do." $\endgroup$
    – whuber
    Commented Aug 19, 2023 at 19:54
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    $\begingroup$ extrapolation is not a bad thing to do. it's just more difficult than interpolation. forecasting is done out of necessity not for its virtues. whether we're good or not we need to forecast, so we do. $\endgroup$
    – Aksakal
    Commented Aug 19, 2023 at 20:17

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or "all models are wrong, some models are useful." When they say extrapolation is dangerous they are talking about cross sectional regression not time series. There are dangers when you estimate outside the range of your data. But if you don't want to extrapolate you can't do time series at all so its a meaningless point (well for those who do times series to predict the future, others use it for existing data notably in economics). One should not confuse the assumptions and purpose of cross sectional analysis with time series. When you say say something is "statistically significant" you are really saying would what I find in this sample apply outside the sample to the population. That is not what you are doing in time series where there is no sample or population (you are predicting something that does not yet exist). I don't think most who do time series to predict worry if its statistically meaningful or not and few pay attention to regression assumptions generally. They worry if the prediction will be correct.

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I believe that there is a misunderstanding in this question.

When people mean extrapolation is bad, what they are talking about is using a model outside the range of the input (training) data.

So yes, extrapolating a trend is so 'bad' that its almost a meme of financial journalism. But note that extrapolating a trend is modelling a function of time, and no one really believes time causes these effects.

arima and other time series models do not explicitly model time, and as such they are not extrapolating outside the range of the input data (as their inputs are eg industrial production values etc, rather than time)

There is a different issue, though, that to forecast twenty steps into the future, we have to use predictions (including errors) of the previous timestep as inputs to making predictions of the next, so the errors accumulate. Nevertheless, these models allow us to 'extrapolate' a few steps in time.

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