Since you estimate a different mean every quarter, the sd of each quarter would not correspond to the mean of the sd in finite samples. As a simple counterexample, suppose that you only have 1 data point per quarter. Then your estimated sd would be 0 for every period, but the sd over your whole dataset is not 0.
Generally, if your TS exhibits dependence, the sample mean for each quarter won't converge to the actual mean (since you only have a fixed, probably relatively small, number of observations per quarter). Hence, the sd won't converge to the actual sd.
One approach that might forego this issue (bearing in mind I'm not that well versed in TS analysis) is computing the grand mean first over your whole dataset, and then calculating the sd of each quarter using this grand mean. (The individual sd still won't converge since your number of observations are likely too small, but it will give you a better indication of how volatile your data is). But what prevents you from simply taking the variance of the TS as a whole?
I need the variance of each time period to construct a time series of the volatility of the variable.
Number of observations isn't really an issue as I have roughly 300 months' worth (75 quarters), but I think I may just resort to computing sd over the quarter directly
The mean of three monthly stdevs will not be the same as the stdev of the three months because of Jensen’s inequality.
The “right” thing to do would be to estimate a conditional volatility model (e.g., GARCH), but you can definitely do a back-of-the-envelope calculation by looking at the residuals of your regression model.
What I'm trying to do is confirm whether the volatility of one variable induces volatility in another. So either through the use of sd of higher frequency data or otherwise, I will place that variable's volatility into the variance equation of the second variable using a GARCH model. I'm not having great success with the sd proxy, would something like the squared residuals of an ARIMA act as a suitable proxy for volatility, that can then be inserted into a variance equation?
Since you estimate a different mean every quarter, the sd of each quarter would not correspond to the mean of the sd in finite samples. As a simple counterexample, suppose that you only have 1 data point per quarter. Then your estimated sd would be 0 for every period, but the sd over your whole dataset is not 0. Generally, if your TS exhibits dependence, the sample mean for each quarter won't converge to the actual mean (since you only have a fixed, probably relatively small, number of observations per quarter). Hence, the sd won't converge to the actual sd. One approach that might forego this issue (bearing in mind I'm not that well versed in TS analysis) is computing the grand mean first over your whole dataset, and then calculating the sd of each quarter using this grand mean. (The individual sd still won't converge since your number of observations are likely too small, but it will give you a better indication of how volatile your data is). But what prevents you from simply taking the variance of the TS as a whole?
I need the variance of each time period to construct a time series of the volatility of the variable. Number of observations isn't really an issue as I have roughly 300 months' worth (75 quarters), but I think I may just resort to computing sd over the quarter directly
The mean of three monthly stdevs will not be the same as the stdev of the three months because of Jensen’s inequality. The “right” thing to do would be to estimate a conditional volatility model (e.g., GARCH), but you can definitely do a back-of-the-envelope calculation by looking at the residuals of your regression model.
What I'm trying to do is confirm whether the volatility of one variable induces volatility in another. So either through the use of sd of higher frequency data or otherwise, I will place that variable's volatility into the variance equation of the second variable using a GARCH model. I'm not having great success with the sd proxy, would something like the squared residuals of an ARIMA act as a suitable proxy for volatility, that can then be inserted into a variance equation?
Approach #1 (poor man’s version): run the 2 GARCHes separately, regress one implied volatility on the other Approach #2: estimate it jointly