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I read the Euan Sinclair's book (Volatility trading) in which he suggests different volatility estimators (Close-to-close, Parkinson, Garman-Klass, ...).

I am inquiring about what is the best stock volatility measure to explain, let's say, the stock price returns.

Let's suppose that I want to test the effect of stock market volatility changes on stock market returns using the following model:

R(t) = α + β*VOL(t) + ε(t)

where R(t) is the stock returns in time t and VOL(t) is the volatility in time t. As suggested by Sinclair, there're different volatility measures. My question is what proxy measure I choose among all to describe in the best way this relationship among volatility and stock returns.

Does it makes sense to regress the stock returns on the single volatility measures and so to compare the related t-statistics (or p-values) and the R^2 of each model?

Or does there exist a statistical technique to do this?

Thanks for all the help you'll give me.

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You need to define what you mean by best and explain more precisely to get an germane answer. – Brian B Apr 4 '14 at 12:53
I'm sorry @Brian for the bad way in which I asked the question for. I modified it and I hope it can result clearer. – Quantopic Apr 4 '14 at 16:24
up vote 1 down vote accepted

In most of the literature on the information content of various volatility estimator the relevant question is whether a particular estimator can predict (is correlated) with future realized volatility. Hence, the testing regression would be $$ RV(t,T) = \alpha + \beta VOL(t) + \epsilon(t) $$ where RV(t,T) is an estimate of the realized volatility from t to T, usually either from 5 minute tick data or daily closing prices. This regression is a backtest whether your VOL estimator at time t is actually predictive of subsequent volatility.

A different test would be to run a multiple regression $$ RV(t,T) = \alpha + \beta_1 VOL_1(t) + \dots + \beta_K VOL_K(t) + \epsilon(t) $$ with K different volatility estimators to test if one (or some) VOL estimators subsume the information of others.

Testing whether volatility is able to predict returns is a different question, either related to the leverage effect of volatility feedbacks. A nice paper in this regard is Bekaert and Wu (2000)

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Thanks a lot @pbr142! your have really been a help to me. – Quantopic Apr 5 '14 at 14:24
But what is your RV estimate? The RV function that is closest to some $VOL_i$ will make the slope coefficient larger for that particular $i$... – Jase Apr 6 '14 at 6:05
What do you suggest about RV(t) estimate Jase? How may I define RV? – Quantopic Apr 8 '14 at 17:18

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