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I am very confused about why different volatility estimators (RV, RAV, BPV, etc) exist. If the goal is to find the best estimator for stock volatility, and volatility is latent, how do I know which estimator performs the best when the mathematical definition for these estimators are all different?

Realized variance = sum[(return at t)^2] in a day
Realized absolute value = sum|return at t| in a day
Bipower variation = sum |return at t-1|*|return at t| in a day

If I use a linear model and substitute each of these estimators as the regressor, doesn't this mean I can't compare the results directly, but I can only compare how well the linear model is able to predict itself based on MSE, R^2, and so on?

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"Performs the best" in what sense? –  whuber Jan 29 '13 at 23:08
    
Performs the best as in being able to predict future volatility values. I've been reading a paper and I'm not sure how that applies, as the regressors (as defined above) all have different mathematical formulae, yet they use MSE and R^2 to compare. –  Louise Jan 30 '13 at 4:52
    
OK, so you use historical RV to predict future RV, historical RAV to predict future RAV, etc. There doesn't seem to be any problem with that. You therefore appear to be asking something nebulous about what "volatility" is. This question appears to be analogous to a forester asking why different measures of harvestable timber exist: board-feet, number of trees, total mass of trees, acreage, etc. That suggests you are likely to find a more interested audience for your question on the finance site. Would you like to migrate the question? –  whuber Jan 30 '13 at 14:36
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