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At the below content, I learned that; "an unbiased variance estimator's square root doesn't imply being an unbiased estimator of the standard deviation".

Comparison of daily fitted volatility and observed absolute daily return

So, by the light of that information, I have some doubts about model comparison criteria for volatility models.

Some softwares of packages uses "mean square deviations between estimated conditional standart deviations and absolute returns(or square root of return squares)." for model comparison:

$\sum_{i=1}^n\frac{(\sigma_i-|r_i|)^2}{n}$

Some softwares of packages uses- which may be better to use- "mean square deviations between estimated conditional variance and squared returns." for model comparison:

$\sum_{i=1}^n\frac{(\sigma_i^2-r_i^2)^2}{n}$

So, Is the second criteria may be a better choice for model comparison, since being the squared return is an unbiased estimator of the estimated conditional variance?

I will be very glad for any help. Thanks a lot.

Richard Hardy
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oercim
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  • Hi: maybe someone can say something more useful but, as long as you consistently use one or the other, it's probably not going to matter. You're most likely dealing with a sample size that is going to make the bias of the standard deviation estimate tiny anyway. – mlofton May 26 '20 at 22:24
  • @mlofton, the problem is not the consistency of the estimators. Namely, sample size is not the main problem. The main problem is that, under the assumption of used the volatility process(ex:GARCH) consistently estimates the variance; while the two elements of the second criteria have the same expected value; the two elements of the first criteria have different expected values. Namely, absolute value of the return's expected vlaue is not equal to the process's conditional standard deviation. – oercim May 27 '20 at 20:33
  • Right. I understand what you're saying. I guess you can compute both and see what happens. But, my question is the following: you're computing some performance criterion right. The first computes how well the standard deviation estimate does and the second one computes how well the variance estimate does. But don't you need to be comparing what you get to something else ? Otherwise, what is the criterion being used for ? – mlofton May 28 '20 at 22:45
  • @mlofton, you right. Both criteria are comparing different things. But, most volatility models like GARCH estimates conditional variance since conditional variance is linear to dependent variables which makes things easier. So, conditional variance estimates are mostly unbiased and consistent ıf an appropriate is used. However, volatility iş mostly interpreter as conditional standard deviation(not conditional variance). Because of that after estimating the conditional variance is estimated, they are converted into standard deviations. These converted standard deviations are biased. – oercim May 28 '20 at 23:04
  • So by definition of volatility, sometimes conditional standard deviations are compared to select the proper model where the interested estimates are biased. Sometimes conditional variances are used where they are consistent and unbiased, but which is not directly defined as volatility. At practice most people will need standard deviations rather than variances for making analysis like confidence intetval estimations. – oercim May 28 '20 at 23:10
  • gotcha and thanks for explanation. all I'm saying is the following. Suppose you have model1 and model2 and pc_std and pc_var. ( performance criterion using the std and var respectively ). So, just calculate pc_std for model1 and model2 and pc_var for model1 and model2 and see if you get any conflicting results. If not, then you're probably fine to use either one. What matters is the COMPARISON rather than the absolute value of the performance criterion. – mlofton May 30 '20 at 00:54

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