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## LONG MEMORY IS A RED HERRING IT IS OBVIOUS IN FINANCE

The Efficient Market Hypothsis is a non sequitor.  It’s proponents have attacked people for whom long memory in financial time series is not even a question.  Take the autocorrelations of either $r_t^2$ or $|r_t|$ and
>>> from meixnermle import *
>>> r=get_returns(‘DPZ’)
>>> import numpy as np
>>> c=np.correlate(r,r,mode=”full”)
>>> c
array([ 0.00010266, -0.00070106, 0.00011973, …, 0.00011973,
-0.00070106, 0.00010266])
>>> import matplotlib.pyplot as plt
>>> plt.plot(c)
[<matplotlib.lines.Line2D object at 0x107d02cd0>]
>>> plt.show()

will show you that the autocorrelations of stock returns exist.  The real issue is not whether long memory exists in financial series; that is obvious, and the discussion of efficient market and random walks is stupid.  Mandelbrot was right in 1968 about this.  The real issue is WHAT IS THE FUNDAMENTAL SCIENTIFIC MODEL for firnancial data that actually has some merit?  So Hurst exponents $H$ that are different from 1/2 was dealt by Mandelbrot by the fractional Brownian motion, and estimated Hurst exponents are almost never quite 1/2.  This does not mean that there is a long memory by itself, as hurst-exponents-for-markov-processes constructs diffusion processes without long memory with Hurst different from 1/2.  But the right perspective is not to worry at all about long memory and consider the issues through the possible space-time equations: consider the deviation from a standard diffusion equation by fractional powers in both space and time:

$( (d/dt)^\gamma + (d/dx)^\delta ) u(t,x) = F(u(t,x))$

When $\gamma =1$ the time component is local and when it is fractional, it is nonlocal (long memory).  When $F=0$ there is no turbulence possible, and there is when $F \ne 0$.  Note thar Mandelbrot’s concrete implementation of the fractional Brownian motion is the Riemann-Liouville fractional integration modified to avoid problems at $t=0$.  The issue of whether or not long memory exists is stupid.  The real task in my view is to find out something about the ‘Fokker-Planck equation’ that can describe the underlying continous time return ‘evolution’ as you won’t get a Markov process if $\gamma$ is nonintegral.  Intuitively, when there is a vast variance of estimates of the Hurst exponent of actual series, we are basically looking at variation in $\gamma$ in these fractional equations.