I'm writing my master's thesis and looking to see if there exists fractional co-integration between the volatility of some large stock-indices.
My estimates of realized volatility are based on the sum of the log intra-day returns. However, as you might have guessed, the distribution of the intra-day volatilities (variance) is highly right-skewed with some major outliers corresponding to events during 2007-09.
As a way of making the data easier to work with when testing for the parameter $d$ of integration, would it make sense to log-transform the realized volatility estimates? It kind of goes against what I've learned regarding co-integration since we are interested in the levels of the variables that we are testing. But I'm a bit afraid that I might estimate my $d$ parameter wrong.