# Market data structure functions

I recently got some minute level stock market data from The Bonnot Gang for some data analytic (and stat arb design) purposes, when I noticed some funny behavior in the data structure function. Now the concept of a structure function may not be very widely known with quants/ data analysts/ economists, so here's a definition:

Suppose there's a time series

where for a given sample of data you just replace the ensemble expectation

These types of structure functions have been studied for some time now in finance in the context of similarities between financial markets and **hydrodynamic turbulence**. I think it all started in 1996 with the paper Turbulent cascades in foreign exchange markets by Ghashghaie et al. They computed the structure functions for some FX market data, and found a scaling relation

So I did some of my own data analysis with the Bonnot Gang data (I hope it's not bad data!). Here's a few plots of the structure functions, first for

Then

This is close to linear, i.e.

Clearly you can't fit a power law in all of this, but there seems to be clear power law regimes divided by about 6, 18, 60 and 180 minutes! I don't know the reason for this, but if I had to guess, I'd say it's because of traders/ algorithms operating w.r.t different data timeframes... or maybe it's because of the finite tick size...

Anyway, I don't have time to get to the bottom of this, but maybe someone else will... so if you see this stuff on a paper someday, **you saw it first here!!** ;)

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