How to compare different volatility measures?

Question

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.

Answer 1

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)

Answer 2

The term RV is the “answer” to the multiple regression. It provides an estimate of the explanatory power of the variables. Your interlocutor assumed you knew the mechanics of regression.
He or she is correct that the central task facing you is to define a set of known outcomes (which can be pricing changes or other metrics) and to then test each of your candidate models to see which best fits the data. This testing is easier if you have a definition (OLS or more sophisticated) of what constitutes fit.
I personally think throwing all the models into one MR will be less useful and less discriminatory (discrimination is a good thing) because the MR will systematically test each X term while holding all other X terms at their mean. This is to reduce autocorrelation between variables, and it helps but the various R^2 values are not as trustworthy as a clean and sequential testing of each explanatory model in linear regressions on the same data set. Make sure you print the plots to get an intuition of what is happening.