Explaining volatility with GARCH: one-step or two-step approach? I am interested in seeing what could explain the volatility of certain financial series. I know that one can get estimates of conditional variance by running a GARCH model. My question is whether the one-step estimate of conditional variance could be used in a second regression where the square root of variance would be the dependent variable which I would regress on some regressors I am interested in?
Or would it be better just to compare the result of univariate GARCH to multivariate GARCH which includes some other variables that I think can explain the variance? 
 A: Both approaches have been used before, so it is a good question which one should be preferred. 
In the two-step approach, the conditional variance is first defined (and estimated) as being conditional on the information in the variable of interest itself, particularly the past shocks and the past values of conditional variance. In the second step, additional variables are brought in to explain the variation in the conditional variance.
In the one-step approach, the conditional variance is defined and estimated as depending on the exogenous regressors in addition to the past shocks and past values of the conditional variance. There, the regressors both explain and actually help define the conditional variance (affect the values of the estimated conditional variance). 
These two approaches are fundamentally different in how the conditional variance is defined. It is up to you to decide whether you are interested in defining the conditional variance with conditioning only on past shocks and past own values or also on the exogenous variables. The difference between the two should be justified by subject-matter arguments rather than statistical ones. 
Note that the two-step approach seems to have the following advantage over the one-step approach: you can assess the goodness of fit of the equation where the fitted conditional variance is being explained by the exogenous regressors. There, you can see how much of the variation these variables explain. But actually you can also run a similar regression in the case of the one-step approach, too. Hence, the advantage is only apparent. 
