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Would it make sense to run an AR(1) regression to estimate a beta and then estimate the half life as -ln(2)/beta?

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First, we need to start with how to properly interpret the VIX itself and the correct answer might shock many people here. No, it's not an indicator of anticipated realized volatility. It is slightly different.

The most up-to-date commentary on the VIX can be found in a paper by Ian Martin published in the QJE in 2017 here. As a small recap for interested readers, the VIX is computed for the S&P500 index and its formula is a discretization of the theoretical fair value of a variance swap. The idea is that you can get a sense of Q-anticipated realized volatility over the next month by forming a suitably weighted portfolio of options maturing in a month, but over a whole continuum of strikes if the underlying price process is a diffusion. If you allow for jumps, the replication argument fails. In the paper above, Martin shows that the VIX formula actually capture the conditional entropy under Q when you allow for jump. You can express this as an expansion in the conditional cumulants under Q meaning jumps "pollutes" the estimate of future realized Q.

Now that this part is clear, we can talk about the "half-life" of the VIX. I am not sure exactly what you seek to capture here, but I can try to provide some guidance. Setting aside the interpretation for now, it is true that if you model the dynamics of the VIX as an AR(1) process, this is indeed the half-life of the process. Two issues then arise:

  1. Why do you limit yourself to an AR(1)? You can solve for $h$ in $E_t(y_{t+h})=y_t/2$ for other models too. If you want to stick to simple stuff, stay with the class of ARMA models, but check if you can have a better fit with an ARMA(1,1) or using more lags in your autoregression. To be entirely fair, it's possible nonlinear models would fit better, but then you'd have to solve for $h$ numerically in many cases.

  2. Then comes the issue of why you want to do this? The half-life is the point on the impulse response function for which the response has been halved, so you're capturing a sort of belief persistance with regards to the anticipations of investors, under Q, of the conditional entropy of returns. How is that going to be useful exactly? It might be useful, but you probably see how much of a mouthful it is.

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