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Is the prediction of the variance the same as the variance of the prediction? I know the concept of prediction intervals, used to specificy the variance of the prediction. I also know (G)ARCH models, used to predict the variance.

How are the two concepts related?

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No, the two concepts are quite different.

If you have a process with a mean you are not so much interested in, you can predict the variance. An example is (G)ARCH modeling in financial applications, where we typically assume (by the efficient markets hypothesis) that the mean follows a random walk, and our interest shifts to predicting the volatility around this mean, i.e., the variance.

On the other hand, when you are predicting (and regardless of whether you are predicting the mean, or the variance, or something else!), if you re-run your prediction algorithm on a new sample of the underlying data generation process, you will get a different point prediction. That is, your prediction will vary. It will have a variance. And you can try to analyze the variance of your prediction. We typically want lower variance of predictions, because it is one component of the prediction error.

So, to tie the two concepts together, you might be interested in a prediction of the variance of a process... and then also in the variance of this prediction, i.e., in the variance of your variance prediction. (To take it one step further, this is again a prediction. A prediction of the variance of your (original) variance prediction.)

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  • $\begingroup$ But assuming I can forecast a point with full certainty based on an information set, i.e. I precisely know the data generating process. The variance of the prediction would then indicate how this prediction varies dependent on the different information sets, correct? If that is so, wouldn't this coincide with the future variance of the point itself? Because we know ex ante, that this variation is only due to different information sets (as we precisely know the data-generating process and we know that it does not change). This would then be similar to the variance of the prediction, right? $\endgroup$
    – shenflow
    Nov 23 '20 at 10:18
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    $\begingroup$ Yes, if you can forecast with certainty based on an information set (which in itself would be random), then the two notions coincide. $\endgroup$ Nov 23 '20 at 14:26

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