How to model a time series of volatility?

Question

I am working with data related to options and I have a series of implied volatility derived from it. I want to model this time series.

I know GARCH(1,1) and EWMA are used when we have volatility clustering, but that's when we are modeling volatility of assets and we have the return series corresponding to the changes in asset prices.

The key difference here is that the series itself is that of volatility. So I am thinking since there has to to be auto-correlation in the volatility, maybe an AR(1) model would be better than GARCH or EWMA? I am not sure if there is clustering in volatility of volatility.

Answer 1

An autoregressive model for volatility could make sense, but care should be taken to keep the fitted values in the positive territory; variance cannot be negative. For that matter you could consider applying an autoregressive (or more generally ARIMA) model on logarithms of volatility, which would be somewhat similar to an exponential GARCH model (with the exception that in a GARCH model the volatility equation is deterministic while in your model it will be stochastic; see my answers to this question). Alternatively, you could consider an ARIMA model with coefficient restrictions ensuring positivity of the fitted values; however, that may be tedious to implement and may yield poor fit.

A GARCH model applied on estimated volatility rather than on raw model residuals would clearly model something else than you are interested in, namely, the volatility of the volatility (fourth conditional central moment). So it should not be considered as a competitor but perhaps as a complement to the models mentioned above.