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Consider a time series $X: S \to \mathbb{R}$, where $S := \{\nu, 2\nu, 3\nu, \ldots T\}$, and $T$ is a multiple of $\nu > 0$. For each $\tau \in (0, \tau_{\max}] \cap S$ and $q \in \mathbb{N}$, define $$K_q(\tau) := \frac{\langle \vert X(t + \tau) - X(t) \vert^q \rangle}{\langle \vert X(t) \vert^q \rangle}$$ where the $\langle \cdot \rangle$ operator denotes averaging over all $t \in S$. Then the generalized Hurst exponent $H(q)$ satisfies $$K_q(\tau) \sim \left( \frac{\tau}{\nu} \right)^{qH(q)} $$ For fixed $q$, we can obtain $H(q)$ by linear regression over all values of $\tau \in (0, \tau_{\max}] \cap S$. My question is what value do we specify for $\tau_{\max}$? The function $K_q(\tau)$ follows a predictable curve until a certain value of $\tau$ (which I assume to be the true value of $\tau_{\max}$), and behaves unpredictably thereafter. For a single time series we could use trial and error to identify an appropriate value for $\tau_{\max}$, but this is obviously not the best approach. $\tau_{\max}$ clearly depends on the length of the time series, and also appears to depend on the variability of $X$. Is there a formula, or maybe some kind of heuristic, that we can use to identify an appropriate value for $\tau_{\max}$?

Thanks.

user3294195
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1 Answers1

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Having read Hurst's papers as well as Mandelbrot's on H(q), the quick answer is that there is no protocol for deriving τmax. Based on how the rescaled range is estimated, my approach has been to choose random draws from the full range of possible combinations across the windows of time. This produces estimates of H(q) consistent with expectations for, e.g., variability in stock prices.

Mike Hunter
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  • Thanks. So if I understand correctly, you draw values of $\tau$ in the range $(0, T)$, and suppose your sample is $\{\tau_1, \ldots, \tau_n\}$, you compute $\{K_q(\tau_1), \ldots, K_q(\tau_n)\}$, and estimate $H(q)$ by linear regression on this sample? – user3294195 May 25 '16 at 02:19
  • Yes, exactly. The trick is to randomly draw contiguous τ. – Mike Hunter May 25 '16 at 10:59
  • The issue I see here is that for large values of $\tau$ ($\gg \tau_{\max}$), several datapoints in the middle of the time series would be excluded when computing $K_q(\tau)$. – user3294195 May 25 '16 at 15:40
  • How would that happen if the bands or windows are randomly selected based on draws from a uniform distribution? – Mike Hunter May 25 '16 at 15:45
  • Based on what I stated in the first comment, say you have a time series of length 10 (increments of 1), and you draw $\tau = \{1, 3, 5, 7\}$. When $\tau = 7$, you can only compute $X(8) - X(1), X(9) - X(2), X(10) - X(3)$, thereby excluding points $X(4), \ldots, X(7)$. – user3294195 May 25 '16 at 15:51
  • Ah! My suggestion is that you randomly draw bootstrapped *bands* of time, not individual points, where the "bands" consist of ranges within the observed series. For example and using your data: randomly draw values from *X(1) to X(5), X(3) to X(7), X(2) to X(9), X(5) to X(6)*, and so on, estimating the parameters (std dev, etc.) for each "band" to create the inputs for the regression. This is my reading of the rescaled range approach to estimating H(q). As noted, this method produces results consistent with expectations for things like stock prices. – Mike Hunter May 25 '16 at 16:05
  • Right, so you were talking about rescaled range analysis. This is the Generalized Hurst exponent, a slightly different method. – user3294195 May 25 '16 at 16:37
  • I wouldn't get too hung up on the theoretical nuances extending to the derivation of these metrics -- they aren't that different in their underlying approach -- a Google search shows papers using a rolling window for estimating the *generalized Hurst exponent*. Similarly, many papers have noted the equivalence between, e.g., Mandelbrot's *D* and H(q). – Mike Hunter May 25 '16 at 16:58