The Hurst constant is an index of dependence and lies between 0 and 1
(Hurst; 1951).
For
, the series is said to exhibit antipersistence. For
, the
series is said to possess long-memory or persistence. For
, 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, has been interpreted as an indicator of range of dependence, of
irregularity and of nervousness (Hall, Härdle, Kleinow, and Schmidt; 1999). A higher
signals a less erratic or more regular behaviour; a lower
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
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
falls below a certain level, it signals that the market is nervous and a sell-signal is
given.
A long-memory time series is fractionally integrated of degree ,
denoted by
, if
is related to the Hurst constant by the equality
. If
, the
series is nonstationary. In case
, then the series is stationary. The
non-integer parameter
is also known as the difference parameter. Notice that if a
series is nonstationary, one can obtain a
series with
in the range of
by differencing the original series until stationary is induced. When
, the series is
an
process and said to have no long-memory.