15.2 Hurst and Fractional Integration


15.2.1 Hurst (Hurst; 1951)

The Hurst constant $ H$ is an index of dependence and lies between 0 and 1 (Hurst; 1951). For $ 0 < H < 0.5$, the series is said to exhibit antipersistence. For $ 0.5 < H < 1$, the series is said to possess long-memory or persistence. For $ H = 0.5$, the series is said to be independent. Although the early work of Hurst was to address the problem of setting a level of discharge such that the reservoir would never overflow or fall below an undesirable level, recent applications have used the Hurst to analyse the fluctuations in financial markets.

In financial markets, $ H$ has been interpreted as an indicator of range of dependence, of irregularity and of nervousness (Hall, Härdle, Kleinow, and Schmidt; 1999). A higher $ H$ signals a less erratic or more regular behaviour; a lower $ H$ reveals a more nervous behaviour. For example, May (1999) has used the Hurst constant to generate buy-sell signals for financial time series. His strategy employs the $ H$ constant to gauge the stability of the time series. A large Hurst constant signals greater stability and persistence of uptrend, over at least short periods of time. Trade in the financial instruments is said to be subject to less nervousness and enjoys more stability. When $ H$ falls below a certain level, it signals that the market is nervous and a sell-signal is given.


15.2.2 Fractional Integration

A long-memory time series is fractionally integrated of degree $ d$, denoted by $ I(d)$, if $ d$ is related to the Hurst constant by the equality $ d = H - 0.5$. If $ d > 0.5 \quad (H > 1)$, the series is nonstationary. In case $ 0 < d < 0.5$, then the series is stationary. The non-integer parameter $ d$ is also known as the difference parameter. Notice that if a series is nonstationary, one can obtain a $ I(d)$ series with $ d$ in the range of $ (-0.5,
0.5)$ by differencing the original series until stationary is induced. When $ d = 0$, the series is an $ I(0)$ process and said to have no long-memory.