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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.

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  • $\begingroup$ How would you interpret the transformed "volatility" when it is negative?? Could you clarify what you mean by "log-transform" them? $\endgroup$
    – whuber ♦
    Mar 3, 2019 at 21:59
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    $\begingroup$ The intra-day "volatility" (i.e. sum of squared intra-day returns) cannot be negative. So log-transforming then would just be log(volatility estimate) $\endgroup$
    – Quantdaddy
    Mar 3, 2019 at 22:20
  • $\begingroup$ That's exactly my point: how much sense would that make when the volatility estimate is less than 1, making its logarithm negative? $\endgroup$
    – whuber ♦
    Mar 3, 2019 at 22:21
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    $\begingroup$ @whuber, perhaps the interpretation of log(volatility) is not all that important if this is just an intermediate step in the analysis and is used only for finding $d$. The more relevant question is, does it make sense to look for cointegration in logs rather than levels, since the relationship implied by cointegration has a different interpretation in each case. $\endgroup$
    – Richard Hardy
    Mar 4, 2019 at 7:54

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