Calculates the Hurst exponent (a measure of persistence or long memory) for time series with more than 96 observations.
Syntax
Hurst(X, Alpha, Return_type)
X is the input data sample (a onedimensional array of cells (e.g. rows or columns)).
Alpha is the statistical significance of the test (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.
Return_type is a number that determines the type of return value: 1 (or missing)=Empirical, 2=(AnisLloyd/Peters) Corrected, 3=Theoretical, 4=Upper Limit, 5=Lower Limit.
RETURN_TYPE  NUMBER RETURNED 

1 or omitted  Empirical Hurst exponent 
2  Corrected Hurst exponent (AnisLloyd/Peters) 
3  Theoretical Hurst exponent (AnisLloyd/Peters) 
4  Upper limit of empirical confidence interval 
5  Lower limit of empirical confidence interval 
Remarks
 The input data series must have at least 96 nonmissing values.Otherwise, Hurst function returns #VALUE.
 The input data series may include missing values (e.g. #N/A, #VALUE!, #NUM!, empty cell), but they will not be included in the calculations.
 The Hurst exponent, $h$, is defined in terms of the Rescaled Range as follows:
$$E \left [ \frac{R(n)}{S(n)} \right ]=Cn^H \qquad \mathrm{as} \ n \to \infty $$
Where:
 $\left [ \frac{R(n)}{S(n)} \right ]$ is the Rescaled Range.
 $E \left [x \right ]$ is the expected value.
 $n$ is the time of the last observation (e.g. it corresponds to $X_n$ in the input time series data.)
 $h$ is a constant.
 The Hurst exponent is a measure of autocorrelation (persistence and long memory).
 A value of $0 \lt H \lt 0.5$ indicates a time series with negative autocorrelation (e.g. a decrease between values will probably be followed by another decrease),
 A value of $0.5 \lt H \lt 1$ indicates a time series with positive autocorrelation (e.g. an increase between values will probably be followed by another increase),
 A value of $H=0.5$ indicates a "true random walk," where it is equally likely that a decrease or increase will follow from any particular value (e.g. the time series has no memory of previous values)
 The Hurst exponent's namesake, Harold Edwin Hurst (18801978), was a British hydrologist who researched reservoir capacity along the Nile river.
 The Rescaled Range is calculated for a time series, $X=X_1,X_2,\dots, X_n$, as follows:
 Calculate the mean:
$$m=\dfrac{1}{n} \sum_{i=1}^{n} X_i$$  Create a mean adjusted series:
$$Y_t=X_{t}m \qquad \mathrm{for}\ t=1,2, \dots ,n$$  Calculate the cumulative deviate series Z:
$$Z_t= \sum_{i=1}^{t} Y_{i} \qquad \mathrm{for}\ t=1,2, \dots ,n$$  Create a range series R:
$$ R_t = max\left (Z_1, Z_2, \dots, Z_t \right ) min\left (Z_1, Z_2, \dots, Z_t \right ) \qquad \mathrm{for}\ t=1,2, \dots, n$$  Create a standard deviation series R:
$$S_{t}= \sqrt{\dfrac{1}{t} \sum_{i=1}^{t}\left ( X_{i}  u \right )^{2}}\qquad \mathrm{for} \ t=1,2, \dots ,n$$
Where:
$h$ is the mean for the time series values $X_1,X_2, \dots, X_t$  Calculate the rescaled range series (R/S):
$$\left ( R/S \right )_{t} = \frac{R_{t}}{S_{t}} \qquad \mathrm{for}\ t=1,2, \dots, n$$
 Calculate the mean:
 The Hurst Exponent is estimated by fitting the powerlaw $E[R(n)/S(n)]=C\times n^H$ to the data. This is done by taking the logarithm of both sides, and fitting a straight line. The slope of the line gives H (i.e. Hurst Exponent Estimate).
 The method above is known to produce a biased estimate of the powerlaw exponent, and for small data set, there is a deviation from 0.5 slope (i.e. whitenoise). AnisLIoyd estimated the whitenoise theoretical value of the R/S statistics to be as follow: $${\displaystyle \operatorname {E} [R(n)/S(n)]={\begin{cases}{\frac {\Gamma ({\frac {n1}{2}})}{{\sqrt {\pi }}\Gamma ({\frac {n}{2}})}}\sum \limits _{i=1}^{n1}{\sqrt {\frac {ni}{i}}},&{\text{for }}n\leq 340\\{\frac {1}{\sqrt {n{\frac {\pi }{2}}}}}\sum \limits _{i=1}^{n1}{\sqrt {\frac {ni}{i}}},&{\text{for }}n>340\end{cases}}}$$ Where $\Gamma$ is the Euler Gamma Function
 No asymptotic distribution theory has been derived for most of the Hurst exponent estimators so far, but an approximate functional forms for AnisLIoyd corrected R/S estimate is available. For 95% confidence interval, the functional form for confidence interval limits are expressed as follow: $$LL=0.5  exp(4.21 7.33\times ln(ln(M)))$$ $$UL=0.5 + exp(4.77 3.10\times ln(ln(M)))$$ Where $M = log_2(N)$
 Finally, The AnisLloyd corrected R/S Hurst exponent is calculated as 0.5 plus the slope of ${\displaystyle R(n)/S(n)\operatorname {E} [R(n)/S(n)]}$
Examples
Example 1:


Formula  Description (Result)  

=HURST($B$2:$B$30,0.05,1)  Empirical Hurst exponent (0.583)  
=HURST($B$2:$B$30,0.05,2)  Corrected Hurst exponent (0.492) 
References
 [1] A.A.Anis, E.H.Lloyd (1976) The expected value of the adjusted rescaled Hurst range of independent normal summands, Biometrica 63, 283298.
 [2] H.E.Hurst (1951) Longterm storage capacity of reservoirs, Transactions of the American Society of Civil Engineers 116, 770808.
 [3] E.E.Peters (1994) Fractal Market Analysis, Wiley.
 [4] R.Weron (2002) Estimating long range dependence: finite sample properties and confidence intervals, Physica A 312, 285299.
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740
0 Comments