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# Hurst exponent mean reversion investing

Hopefully we will draw some conclusions around if, when and how we might apply the very attractive theory of Hurst in a manner that is practical to systematic traders. In this post, we perform the analysis in Python, which is something of a departure from tradition for Robot Wealth. We are currently building our skills in both Python and the Microsoft. Introduction to Hurst The Hurst exponent, H, measures the long-term memory of a time series, characterising it as either mean-reverting, trending or a random walk.

Smaller and larger values of H indicate stronger mean-reversion and trending, respectively. As the algorithm shows, calculation of Hurst is related to the autocorrelations of the time series. Autocorrelation also known as serial correlation refers to the correlation between a time series and lagged values of itself. In particular, Hurst is related to the rate at which these autocorrelations decrease as the lag increases. We know that we get different values of Hurst depending on which lags we use in its calculation.

So which lags should we focus on? After trying a few different ranges, we found that using lags resulted in a Hurst exponent of 0. We can draw a significant conclusion from these results: that the lags used to calculate Hurst have a much greater impact on the calculation than the particular segment of the time series analysed for SPY, anyway. Further, when we look at the entire time series, we see that SPY is moderately mean-reverting for shorter lags. Lags up to about 20 are most mean reverting, and then H increases as we increase the lags used in its calculation.

If we continue to increase the lags used to calculate H up to the range , we find that H indicates a moderately trending time series. We also find that for moderate lag values, H tends to approach 0. This leads to the conclusion that SPY is neither absolutely mean reverting nor absolutely trending. Instead, Hurst indicates that it is moderately mean reverting over short time periods and tends to exhibit momentum over the longer term.

I find this to be a pleasing result, one that is in line with the results of Jagadeesh and Titman , who investigated momentum and found that multiple-month relative returns predict future returns. The result is also in line with what we tend to see over shorter time horizons in equities markets. See general information about how to correct material in RePEc.

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