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Hurst Exponent Estimates and Fractional Differencing in Price Models

Article Quant Q&A · Author: Nik
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Summary

The document considers whether a Hurst exponent different from one half justifies fractionally differencing stock prices before forecasting. It cautions that an estimated exponent is a statistic subject to sampling uncertainty: even data generated by a process with a true exponent of one half can produce estimates above or below that value.

Fractional differencing is presented as one possible modeling transformation, not an automatic consequence of detecting apparent memory. Differencing changes the data and may discard information, so the choice depends on the model and prediction objective. The response notes that models can use prices or log prices, ordinary differences, and fractionally differenced values, including as combined inputs. It offers no formal test, empirical comparison, or specific forecast results, so the answer frames the modeling options without establishing which works best for a given stock or sample.

Key ideas

  • A Hurst exponent estimate has sampling uncertainty, even when the underlying process has an exponent of one half.
  • An estimated value above or below one half alone does not establish exploitable memory.
  • Fractional differencing is a modeling choice that changes the information represented in the series.
  • Price levels, log levels, ordinary differences, and fractional differences can be considered based on the model and goal.

Tags

EquitiesStatisticsMean reversionTrend followingMachine learning
Full text
# Hurst Exponent and fractional differencing


# Hurst Exponent and fractional differencing












I have a quick question.

Currently I study daily OHLCVs of some stocks and find that many of them have the Hurst exponent not being equal to 0.5. I know that if it is less than 0.5 then it is a mean reverting series, if more than then it is a trending series. But my question is if $H$ $\neq$ $0.5$ then it is a series containing some memory, should I fractionally differentiate the original price series and work with the new, differentiated, series, for example, for the next day price movement prediction?

I worked through the formula of fractional differentiation and, indeed, found that when we use lags between 0 and 1, we assign some weights to previous prices so that some "info" about them can be included in current price.

I mean, it seems quite legit for me but I do not see much formal research going on using these principles while there are many stocks where the Hurst exponent indicates that they are not random walks (at least, in my sample). Are there any theoretical issues with the HE?

Would be really happy if you can guide on these questions and provide some sources to read!

Thank you!

## Answer by Stéphane (score 1, accepted)

https://quant.stackexchange.com/a/53020

The first thing you should ask concerns the amount of information contained in your estimator of the Hurst exponent.

Let's say you approach this problem from the perspective of frequentist statistics. No matter how you compute the Hurst exponent, the fact of the matter is that it is a statistic (i.e., a function of the data) and is thus subject to sampling uncertainty. Putting this in simple terms, suppose that you were to simulate several thousands of time series of prices using, say, a geometric brownian motion. If you computed your estimate for the Hurst exponent for each one of those simulated runs, the value would vary and even though in this case the exponent truly is 0.5 you would still find cases where it's higher than 0.5 and other cases where it is lower than 0.5, perhaps even much lower.

Now, what should you do with regards to modeling? It depends on your goals and on the model you will use. Just in case this never occured to you, any transformation of the original data you perform is part of your modeling choices. It's not innocuous to take a difference, whether fractional or integer and, yes, there are times when working directly with prices in levels or log levels is preferable. For example, if you're working with Neural Networks, nothing says you cannot target prices directly and use prices, differenced prices, and fractionally differenced prices as inputs. Likewise, if you think that taking a first difference is an overkill for prices (or, more typically, log prices), go ahead and work with a fractional difference.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.