I have a time series of daily returns and and I computed the conditional variances by means of a Garch model. Now I would like to built a regression with some other monthly data and the previously computed daily variances. What is the best way to deal with it? How would I convert the daily into monthly variances?
One of the approaches in Literature to aggregate volatility estimates from "weak" GARCH processes is the Drost Nijman scaling as presented in http://finance.martinsewell.com/stylized-facts/dependence/DrostNijman1993.pdf.
Though the square root scaling approach (based on an i.i.d. assumption for returns) is probably often used, refer the paper by Diebold for some weaknesses. http://www.ssc.upenn.edu/~fdiebold/papers/paper18/dsi.pdf. This paper also contains a discussion on the Drost Nijman volatility scaling.