Looking for a measure of variance of variance?

Question

I was hoping you would be able to help me identify the statistic I am looking for, or point me in the right direction.

Is there a statistical measure that will represent variance of variance (or s.d. of s.d.)? In the graphs attached, is there such a measure that will identify graphs 1 and 2 as similar because variance is pretty constant for all values of x, and show graphs 1 and 2 as different from graph 3 as the variance of graph 3 fluctuates greatly as we change x? I understand that I could split the x axis into bins, calculate the variances of each bin, then calculate the variance of the bin variances but is there a better way that wouldn't involve more decisions to be made (like bin size)?

Thank you for any help you can give me, Chrissy x

Answer 1

The term you are looking for is called heteroscedasticity.

As for testing if you have heteroscedasticity, the tests basically proceed as you guessed, except that often you use the fact that you know the underlying model (i.e. some type of regression model), and so you don't exactly "split" up the data. At the extreme, this is precisely what you do for grouped data. See the detection section of the wiki link for some common tests.

Looking at the test statistic of some of these tests should give you a sense of how to measure it. Again, given some model (such as a regression model), this is usually done by taking the residuals of your model and seeing if the variance of the residuals changes as a function of the regressors.

Hopefully this is enough to get you going!

Answer 2

In quant finance there are several models that incorporate such a concept. For instance, SABR model looks as follows in a simplest form: $$dF_t=\sigma_t \left(F_t\right)^\beta\, dW_t,$$ $$d\sigma_t=\alpha\sigma^{}_t\, dZ_t,$$ were, $\alpha$ is the variance of the variance of the changes in rate series $F_t$.

In order to estimate the variance of the variance it's best to have some kind of a model in mind. The variance is not directly observed, to start with. This makes measuring variance of variance even more difficult. If you have a model, then you can come up with a fitting approach to extract the parameter from the data.