# Valen Lab

This is a documentation page for Long Memory and the Nile: Herodotus, Hurst and H blog post.

### Given Dataset of Nile overflow data (https://bit.ly/2rPABAH)

** The original data consists of average monthly flow measurements from January 1869 to December 1984. Which I couldn't get access to the full dataset. For this exercise, I used Jan 1869 to December 1874.

From this mini data,

From the website: Long Memory and the Nile: Herodotus, Hurst and H. We can see that there is a similarity between variation of flow at Dongola Station by Month and Monthly Flow of Nile River Dongola Station.

### Limitation:

** The original data consists of average monthly flow measurements from January 1869 to December 1984. Which I couldn't get access to the full dataset. For this exercise, I used Jan 1869 to December 1874.

#### Hurst Exponent

The goal of the Hurst Exponent is to provide us with a scalar value that will help us to identify (within the limits of statistical estimation) whether a series is mean reverting, random walking or trending.

The idea behind the Hurst Exponent calculation is that we can use the variance of a log price series to assess the rate of diffusive behavior. For an arbitrary time lag ττ, the variance is given by:

Since we are comparing the rate of diffusion to that of a Geometric Brownian Motion, we can use the fact that at large ττ we have that the variance is proportional to ττ in the case of a GBM:

The key insight is that if any autocorrelations exist (i.e. any sequential price movements possess non-zero correlation) then the above relationship is not valid. Instead, it can be modified to include an exponent value "2H2H", which gives us the Hurst Exponent value HH:

A time series can then be characterized in the following manner:

**H<0.5H<0.5**- The time series is mean reverting**H=0.5H=0.5**- The time series is a Geometric Brownian Motion**H>0.5H>0.5**- The time series is trending

In addition to characterization of the time series the Hurst Exponent also describes the extent to which a series behaves in the manner categorized. For instance, a value of HH near 0 is a highly mean reverting series, while for HH near 1 the series is strongly trending.

The code below represents the hurst exponent and random changes value. Due to its small sample data size. I wasn't able to produce a hurst exponent. Instead, I've substitute with a random series.

Any questions or concerns, you can email me here: steven.yoo@nyu.edu

Cheers,

Steven

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