:The Rescaled Range is calculated for a time series, X=X_1,X_2,\dots, X_n \, , as follows: • Calculate the
mean • : m=\frac{1}{n} \sum_{i=1}^{n} X_i \, • Create a mean adjusted series • : Y_t=X_{t}-m \text{ for } t=1,2, \dots ,n \, • Calculate the cumulative deviate series Z; • :Z_t= \sum_{i=1}^{t} Y_{i} \text{ 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 ) \text{ for } t=1,2, \dots, n \, • Create a
standard deviation series S; • :S_{t}= \sqrt{\frac{1}{t} \sum_{i=1}^{t}\left ( X_{i} - m(t) \right )^{2}} \text{ for } t=1,2, \dots ,n \, • :Where
m(t) is the mean for the time series values through time t X_1,X_2, \dots, X_t \, • Calculate the rescaled range series (R/S) • :\left ( R/S \right )_{t} = \frac{R_{t}}{S_{t}} \text{ for } t=1,2, \dots, n \, Lo (1991) advocates adjusting the standard deviation S for the expected increase in range R resulting from short-range
autocorrelation in the time series. This involves replacing S by \hat{S} , which is the square root of \hat{S}^2 = S^2 + 2 \sum_{j=1}^{q} \left( 1 - \frac{j}{q + 1} \right) C(j), where q is some maximum lag over which short-range autocorrelation might be substantial and C(j) is the sample
autocovariance at lag j. Using this adjusted rescaled range, he concludes that stock market return time series show no evidence of long-range memory. ==See also==