Comparing two samples of volatility data: Some complications

#1
Hi guys,

I'm having some problems establishing the best possible test for a comparison of two samples.

Background:

I'm investigating the volatility in returns of Exchange Traded Funds which trade non-synchronously with the market of the underlying basket. For example, an ETF for a basket of Japanese stocks may trade on the US market. However, because of time differences, the Japanese market may be closed while the US market is active. The Japanese ETF will, however, keep trading and seeing changes in price even with the Japanese market closed.

The Problem:

I have calculated the volatilities for a large range of ETFs for 15 minute time increments over a period of roughly 6 years ending 2014. The problem is I would like to compare the volatility series for the different pairs of ETFs, but:

1. The series are definitely not normal so I'll need to use a non-parametric test of some kind.

2. The series are not stationary (the data includes the 2008 crisis period).

3. The series are likely not independent of each other so I can't make that assumption.

I'm completely stumped as to what might be an appropriate test to use. Might anyone have any suggestions?