I have a conceptual question that I haven't managed to grasp yet and is most likely a econometrics 101 question by here it goes:
If we estimate a GARCH model for a time series, how do we then use this in my model for the returns? For example; I have the return data of an index. I know that I have volatility clustering in this data. I find a suitable GARCH model for the volatility (variance). Now, if I model the returns an a suitable model, i.e. a regression model, and look at the coefficients and the p-values that it spits out, these values are still based on the regular OLS assumptions right? How do I make use of the GARCH in this model so that I can get coefficients and p-values that have accounted for the conditional heteroscedastic variances in the time series?
Best Answer
Suppose the conditional mean of returns is constant. A GARCH model gives you a fitted value of the conditional variance for each data point. These fitted values can be used to weight the data points to construct an efficient estimate of the mean (e.g. using weighted least squares); data points with high fitted conditional variance would be down-weighted relative to data points with low fitted conditional variance.
Now suppose the conditional mean of returns is not constant. Then you would build a model for the conditional mean simultaneously with a GARCH model for the conditional variance. The effect of the GARCH model would again be similar to the case discussed above. The data points with high fitted conditional variance would be down-weighted relative to the points that have low fitted conditional variance when estimating the model for the conditional mean.
One example given by @CadgasOzgenc is an ARMA-GARCH model. A rich choice of specifications of ARIMAX models and different versions of (G)ARCH models can be implemented using "rugarch" package in
R(functionsugarchspec,ugarchfit).Simultaneous estimation is efficient, but two-stage estimation could be done, too, if you can consistently estimate the conditional mean model in presence of conditionally heteroskedastic errors. First you would estimate the conditional mean model ignoring that the errors have a GARCH structure. Second, you would estimate a GARCH model on the residuals from the conditional mean model. Then you would reestimate the conditional mean model using the fitted conditional variances to weight the data points as discussed above. That could be done iteratively until convergence. For example, an AR-GARCH model could be estimated that way as an AR(p) model can be estimated consistently even in presence of GARCH errors. However, estimating an AR-GARCH model in one stage (simultaneously) would be more efficient.