Long Memory / Scaling Laws in Return Volatility

The 25-year accumulation of evidence for long memory / fractional integration / self-similarity / scaling laws in financial asset return volatility continues unabated.  For the latest see this nice new paper from Bank of Portugal, in particular its key Table 6. Of course the interval estimates of the fractional integration parameter "d" are massively far from both 0 and 1 -- that's the well-known long memory. But what's new and interesting is the systematic difference in the intervals depending on whether one uses absolute or range-based volatility. The absolute d intervals tend to be completely below 1/2 (0<d<1/2 corresponds to covariance-stationary dynamics), whereas the range-based d intervals tend to include 1/2 (1/2<d<1 corresponds to mean-reverting but not covariance- stationary dynamics, due to infinite unconditional variance). 

Realized vol based on the range is less noisy than realized vol based on absolute returns. But least noisy of all, and not considered in the paper above, is realized vol calculated directly from high-frequency return data (HFD-vol), as done by numerous authors in recent decades. Interestingly, recent work for HFD-vol also reports d intervals that tend to poke above 1/2. See this earlier post.