Hurst exponent explained: Long-term memory in time series (Excel) 

Thanks so much for the comment and glad you enjoyed the video. Hurst exponent can be definitely related to half-life, and together they can augment one's perspective on the persistence or antipersistence of a time series.

Hi, thank you for the kind words! I’m glad you enjoyed the videos :)

Hi, and glad you liked the video! Mandelbrot's multifractal model, if I recall correctly, can be considered as a generalisation of Hurst exponent (processes that can be described by Hurst exponent Mandelbrot himself calls "self-affine" and considers them a special case of broader self-similar processes). I might make a video on that in the distant future!

@@NEDLeducation that would be awesome thanks mate

@@NEDLeducation I would love to see a presentation on that as well. I'm trying to work through Mandelbrot (Mis-Behavior of Markets) and applying it in Python, but it's been challenging for me. thank you

Hi Hen, and glad you liked the video! As for your comment, the drift in the calculations is eliminated, as the rescaled range and the Hurst exponent are calculated based on demeaned returns. which takes care of the drift. Hope it helps!

@@NEDLeducation Demeeaned, does that mean detrended? Anyway if drift is eliminated, can you translate that back into the prices you have used and get the series without drift?

Hi JaVale, and glad to see you are successfully applying the tests to your data! As for your question, the Hurst exponent in your case is higher than 0.5, and significantly so (p-value is lower than 5%). That would imply the data is persistent in the long-term (there are long-term trends rather than reversals). As such, the returns are not independent (the market is inefficient). For trading strategies, that would mean that long-term momentum strategies are likely to be profitable. Hope it helps!

By the way, the video on approximate entropy is in the pipeline and to be released in the next couple of weeks, so stay tuned :)

Cheers buddy, thanks a lot!!

opa, br aq tb so por curiosidade, pq ta procurando sobre o coeficiente d hurst>

Hi Muhammad, and glad you liked the video! Will definitely consider addressing this at some point in the future!

Hi Noah, and many thanks for your subscription! Yes, you can apply Hurst at any frequency.

Hi Facu, and glad you liked the video! As for your question, if you use hourly data, you can afford to go for longer intervals back in time (for example, 2048 hours, or even 4096 hours, which would be around a year or two years, respectively in terms of trading hours). However, keeping it at 1024 is reasonable as well so feel free to keep this specification for hourly data, especially if your investing horizons are not very long.

Hi Kemal, and glad you liked the video! The Hurst exponent can be considered a special case of broader multifractal model developed by Mandelbrot, among others. A general case is quite tricky computationally, but I am looking forward to implement it at some point.

Hi Chintan, and glad you liked the video! Generally, Hurst exponent is calculated for samples of size equal to a power of 2, and you need enough data points to be able to estimate the Hurst exponent precisely enough. The smallest sample that still performs decently is (from my experience) 256.

Hi Shiney, and happy you are enjoying the channel! As for your question, it is preferable that the sample size is a power of 2 as it allows for the same length of all subsamples. This, in turn, leads to our estimate of Hurst to be the most precise.

Hi Jansen, and glad you liked the video! In principle, you can workaround the MAXIFS or MINIFS function by building a simple MAX or MIN function on top of an IF function, just as I have done later in the video to calculate standard deviations. Hope it helps!

@@NEDLeducation could you post the respective command? thx...

Hi, and glad you liked the video! As for your question, the fractal dimension can be calculated as two minus the Hurst exponent. For a more efficient way of calculating the Hurst exponent, you can have a look at this video that codes the procedure in Python: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-v0sivj2wGcA.html

@@NEDLeducation Think you! Would you please recommend some books to me?

Hi, and thanks for the great question! This formula works as we implement an IF function on top of the whole array of returns that only takes those that belong to the subsample we need.

@@NEDLeducation sorry but it doesnt work for me. Maybe its different for open office

@@NEDLeducation would it be trouble for you to send the file somewhere ?? Maybe, just in case you have the time

Hi Debditya, and thanks for the question! Hurst exponent requires estimation to be undertaken on a power of 2 subsample, so you generally select the most recent 1024 (256, 512, 2048, etc.) trading days/candles to get the value. The selection of the optimal period is where finance becomes more of an art than a science, but you can always see whether the results are consistent in different specifications. I do address it in greater detail in my Python tutorial on Hurst exponents: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-v0sivj2wGcA.html

Hi, and glad you have enjoyed the video! Thanks for the excellent question. DFA can be viewed as a generalisation of the Hurst exponent, so if you want to use Hurst as presented in this video (or, in a more efficient way using Python, here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-v0sivj2wGcA.html), you can refer to it as a particular case of DFA. DFA and, for example, multi-fractal models of asset returns that are associated with Mandelbrot, relax the assumption that the scaling process is linear (that Hurst exponent assumes) and thus are much harder to explicitly test. Hope this makes sense :)

Hi Adam, and glad you liked the video! A time-varying Hurst exponent is quite a computationally intensive feat, so I might well do a Python tutorial on that in the future. The logic would be to either estimate it in subsamples of a power of 2 observations each, or move in a rolling window of a power of 2 observations. While Excel can do the former, it would be painful to accommodate the latter there.

​@@NEDLeducation Actually, I am working on my dissertation, and I am considering whether I need to choose the time-varying Hurst Exponent to compare the changes of market efficiency with the influence before and after a special policy. I know that the time-varying Hurst Exponent needs to be smoothed, but this is only my shallow understanding. Indeed, it is quite difficult. Anyway, thanks for your reply and material!

@@NEDLeducation that will be awesome

Hi Rodrigo, and glad you are enjoying the channel! Little discrepancies in caclulation results might be a result of methodology, for example, how many subsamples do you select (here I stop at 256), or which method do you use to calculate the Hurst exponent. I fit the slope using OLS, sometimes it is also done using maximum likelihood. The difference should not be substantial though. Hope it helps!

@@NEDLeducation I'm greatful!! thanks again, and please, keep making videos, your work really helps a lot to all

Hi, and glad to see you have applied the technique to your own dataset! Your Hurst exponent in 0.67, and it is significantly higher than the null hypothesis of 0.5 (as p-value is very low), meaning that there is long-term persistence in your data. Hope it helps!

Hi Ibrahim, and thanks for the question! It is preferable to have a power of 2 as a sample size. For your application, you can look at the most recent 2048 observations (2^11). Hope it helps!

Hi Jack, and thanks for the question! Yes, you can estimate it based on a rolling window of 1024 trading days, for example, and see how it evolves through time. Alternatively, the data can be divided into samples of 256 (a power of two closest to a full trading year), and then Hurst can be estimated on an annual basis. Hope it helps!

NEDL thanks for getting back to me so quickly. If I wanted to do that, would I have to change the structure of the spreadsheet used in the video? Forgive me for the ignorance, but it seems like the sheet produces only one hurst exponent value, as the averages and index numbers are fixed. Ideally I’d like to track H over a rolling basis and then present that on a line chart. I hope my question makes sense and thanks!

Hi again, and thanks for the follow-up! Yes, this spreadsheet is designed to estimate one Hurst exponent value. For a dynamic way of generating one for rolling windows, Excel is perhaps not the best software (as you need to calculate a new subsample breakdown and estimate a new regression for a box plot for each observation) and a more efficient solution would be to code it in Python. If you want to discuss it further, you can email me onto savvashanaev@yandex.ru. Hope it helps!

@@jackhaywood4331 Python . The secret is Python :-)

Hi Guilherme, and glad you liked the video! The easiest approach that comes to mind is to continuously apply the Hurst exponent to the most recent 1024 (or any other power of two) observations and see how results change with time. Overall, there is nothing particularly special about powers of 2, it could well have been powers of 3, or even any sumbsample sizes that divide the total sample size, and here it is just crucial that your sample size is not a prime :) Powers of 2 are conventionally chosen as they give a reasonably high level of statistical power by generating the highest number of evenly spread points on a log-log plot which leads to a precise slope estimation (and that is what we are ultimately looking for in Hurst). Hope it helps!

@@NEDLeducation sure helped! Thanks for the video! Gret content as always, proud to be a patreon!

Привет, и спасибо за комментарий! На самых первых видео есть русские субтитры, есть планы в будущем сделать их и для самых популярных более новых видео. Всегда стараюсь говорить на более-менее понятном английском, ну и разумеется альфа и бета на любом языке альфа и бета :)