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L28 Hurst Exponent Estimation under ARFIMA

  • Fractionally Integrated \(\implies\) \(d\) can take fractional values

Hurst Exponent - An Index of Long-Range Dependence

  • \(H\) can be used as a measure of long-term memory of TS.
  • Developed in hydrology → optimum dam size (river Nile)
  • Used in finance, hydrology and physics (behavior of physics over time)

Construction

We need some ingredients…

\(y_{t}\) for \(t=1,2,\dots,n\) be a realization with mean \(\bar{y}_{n}\) and variance \(S_{n}^{2}\)

  • Mean adjusted partial sums defined as,
\[ Z_{t} = \sum_{j=1}^t y_{j} - t\bar{y}_{t}, \quad t=1,2,\dots,n \]

Example, \(Z_{1} = y_{1} - \bar{y}_{1}\), \(Z_{2} = y_{1} + y_{2} - \bar{y}_{t} - 2\bar{y}_{t}\) etc.

  • Adjusted range is defined as
\[ R_{n} = \max\{ Z_{t} \} - \min \{ Z_{t} \} \]
  • Rescaled adjusted range = \(\dfrac{R_{n}}{S_{n}}\)
  • \(E(\dfrac{R_{n}}{S_{n}}) \propto Cn^H\) as \(n\to \infty\)
  • If the realization is i.i.d., \(H = 0.5\)

For large \(n\),

$$
\log\left(\dfrac{R_{n}}{S_{n}}\right) = \alpha + H \times \log(n)
$$
- Can be viewed as \(\log\left(\dfrac{R_{n}}{S_{n}}\right)\) regressed on \(\log(n)\)
- This is how we can estimate, \(\alpha\) and \(H\)

Properties of \(H\)

  • \(H \in (0.5, 1):\) Long memory structure
  • \(H \in [1,\infty):\) infinite variance, non-stationary
  • \(H \in (0,0.5):\) Anti-persistence structure exists
  • \(H = 0.5:\) White noise structure exists (i.i.d.)

Application of \(H\) exponent

  • Calculate Hurst exponent, to gain insights into the nature of price movements.
\[ H \gt 0.5\ (H=0.7) \]
  • Stock price shows a persistent trend, trending market or momentum
  • Investors can use this persistence ka information to apply a trend-following strategies \(\implies\) price movements will persist
\[ H \approx 0.5\ (e.g., H = 0.5) \]
  • Stock price follows random walk \(\implies\) no observable trend. Equally likely to increase or decrease
\[ H \lt 0.5\ (e.\mathbf{g} H=0.3) \]
  • Stock price shows mean-reverting behavior.
  • After an increase, more likely to decrease and vice versa. (Stock overbought and oversold)