Not quite. Volatility, implied or actual, is not independent even measured over non-overlapping intervals. There are clearly periods of higher and lower volatility.
If you divide return by trailing volatility, as nodoodahs describes, you eliminate much of the non-Normality in unconditional returns. Thus you can separate non-Normality from heteroskedasticity.
My experience matches nodoodahs in that dividing by a short-term trailing volatility eliminates much of the non-Normality, but not all. Also, the standard deviation of your adjusted series will be greater than one.
More complicated volatility prediction models can get rid of all the non-Normality, but have too many parameters to be stable and practical.
Normality test
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nodoodahs
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Normality test
and those periods of higher/lower are (for stocks and for stock indices) trend-dependent.
Did a Chi^2 on the dailies for the SPY based on a 2-moving-average definition of 'bull' or 'bear' market (shorter MA over longer = bull). Checked to see if the distributions were the same - for SPY, got 97% confidence they are NOT. Even the scaled returns have different distributions in 'bull' and 'bear' markets.
Had done that years ago with raw returns and got a much stronger significant result, did it today with returns scaled by trailing vol and got the result above.
Did a Chi^2 on the dailies for the SPY based on a 2-moving-average definition of 'bull' or 'bear' market (shorter MA over longer = bull). Checked to see if the distributions were the same - for SPY, got 97% confidence they are NOT. Even the scaled returns have different distributions in 'bull' and 'bear' markets.
Had done that years ago with raw returns and got a much stronger significant result, did it today with returns scaled by trailing vol and got the result above.
I haven’t seen a beatin’ like that since somebody stuck a banana in my pants and turned a monkey loose.