Hurst Exponent
by theorangedog on Dec.19, 2007, under Skills
Thanks to aiQUANT for his post on the Hurst Exponent. A copy of the paper he references is available here.
I tried to reproduce the results in excel but ran into a roadblock, mostly in understanding the log transformation. I wasn’t able to access the link referenced in the paper, but aiQUANT mentioned that there is a file on the Mathworks website, so I will check that out.
[Update1]
Bear Cave, I think a site[article] maintained[penned] by the author of the referenced Hurst paper, has a page about the Hurst exponent and its calculation. That information is available here.
Qian’s site also has the MATLAB code for the calculation. You can access the MATLAB file here.
I redid some of the calculations in the excel file, and now my results have narrowed the gap when compared to those presented in the original paper. I think I am messing up on the log transformation of R/S. I will look at the MATLAB code and see… right now I think I have the exponent estimated at around .59.
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December 20th, 2007 on 11:49 am
It seems to me you could accomplish the same thing by measuring serial autocorrelation directly. Why all the work if it might not be needed?
December 20th, 2007 on 12:30 pm
Bill - thanks for stopping by!
Your question is a very straightforward response to something that was running through my head. Essentially, serial autocorrelation and the Hurst exponent test for the same thing - is x(t) impacted by x(t-1), or more specifically, is the error term distributed in a Gaussian fashion or no? So, to that end, there really is no point in handling the equations.
The only difference I may have found comes in terms of the bounds. To start, 0 to 1 is easier to interpret than the rules of the DW (my default test as of now) when on the fly. Also, I know with DW, the bounds are not reliable if the dependent variable y(t) is dependent upon a lag y(t-n).
However, for application, I think using serial autocorrelation would be better as a conditional than the Hurst exponent. Part of that, I would think, is due to the notion that the Hurst exponent is just a transformation of the decay in the serial autocorrelation equation - see Equation 3 on the Bear Cave link.
Thanks for the thought provoking question - as I’ll add in the soon to be added About section, I’m rusty on stats and calculus (corporate finance does that), so I appreciate the questions.
December 20th, 2007 on 12:54 pm
Sorry for the double-dip.
Here’s what I’m talking about. Assign the first two data points each a 0. For each data point thereafter, assign a 1 if the sign of the change from the previous data point is the same as the change from the next previous to previous was (i.e., a two-data-point trend gets 1 point, a three-data-point trend gets 2 points, etc.). Average the points over the time period.
The comparable or target average for a random walk is (odds of positive)^2 + (odds of negative change)^2. For a truly random series this will be 0.5. If a series has an upward bias (most 1-day periods in the stock market are positive) then 0.527^2+0.473^2 = 0.5015. A high average is serial correlation. A low average is mean reversion. Perform binomial approximation of the normal to ascertain statistical significance, keep in mind how many degrees of freedom we’re playing with. I got a 53.4% average for the S&P from 1950 to today, which is 7.89 standard deviations from what I’d expect if there wasn’t a serial autocorrelation bias, and this is a positive test.
Perform over different timeframes, because a series may look random on a daily or weekly basis, but be very trendy on a quarterly basis. Perform over different time PERIODS, because it can look pretty different. 2006 looks almost mean-reverting on a daily basis, actually disproved the null at 0.05% significance, but considering it was the second-lowest year out of 48 tested, I would probably hold to a higher standard than 0.05%. Maybe 0.01%.
December 20th, 2007 on 2:43 pm
Excellent Bill - that is much simpler. I appreciate the feedback and see the logic.