Newsgroups: bit.listserv.frac-l
Date:         Mon, 9 Nov 1992 10:50:14 EST
From:         McClelland_R%PSB%BLS@NIH3PLUS.BITNET
Subject:      Hurst exponent & other methods

|Date: 11-9-92  10:42am
|From: McClelland_R:PSB:BLS
|  To: FRAC-L@GITVM1:BITNET:NIH
|  cc: xum@nihcu:bitnet:nih
|Subj: Hurst exponent & other methods
|In-Reply-To: Message from FRAC-L@GITVM1:BITNET of 11-9-92
------------------------------------------------------------------

>Dear Frac-lers:
>I fully agree that we should start some new discussions
>on this list rather than only ask and answer questions like
>how to leave the list or from where to get some files.

>I want to start some work on time series analysis and I am
>quite interested in questions about estimating the best time lag
>and enbedding dimension.
>In one of the last e-mails the HURST-Exponent of time series
>was mentioned. I have never heard about that. Could one of you
>explain to me how  this exponent is defined?
>
>                         With best regards,
>
>                                           Andy J. Novak
>                                          Vienna/ Austria

Andy:

The Hurst exponent H measures the bias from estimating the data
generating process (dgp) using a pure random walk.

Define M(t,n) as the mean at time t looking forward n periods.

Define the accumulated departures from the mean over the n periods as

Y(t,i) = sumj=t+1, i, X(j) - M(t,n)  for t < i < t+n

the range of Y(t,i) over the n periods as

R(t,n) = maxt < i < t+n, Y(t,i)

and the standard deviation of X(j) over the n periods as

S(t,n) = (1/n)*sumj=t+1, t+n, [X(j) - M(t,n)]^2^.5 .

Define the rescaled range as R(t,n)/S(t,n).

Finally, define the average of the rescaled range over all t as
R(n)/S(n).

Then R(n)/S(n) will be proportional to n^H, where H is  the Hurst
exponent. If the dgp is a random walk, then H=0.5. H > 0.5 indicates
positive persistence.

There are several ways to estimate the Hurst exponent, such as the F
Hurst and the G Hurst. Wallis and Matalas (1970) describe these methods.
A more recent method is described in Lo (1991) (which I have not read)
and Ambrose, Ancel and Griffiths (1992) (from which I cribbed the
description of the Hurst exponent.)

There are several other tools to investigate time series, such as
dimension, Kolmogorov entropy and the BDS statistic. All three of these
can be estimated using the correlation integral, which Craig Hiemstra
of Loyola@Baltimore
has called "the vegomatic of statistics". Ruelle (1989) has a concise
description of dimension, entropy and fractals, etc. Brock, Hsieh and
Lebaron (1991) is an entire book devoted to the BDS. Casdagli (1992) has
some suggestions on selecting the embedding dimension by using a
dimension estimate. Finally, my paper with Jaditz and Johnson describes
dimension, entropy and the BDS in the context of the correlation
integral.

I hope you find some of this useful.

  |\/\/\/|  Rob McClelland
  |      |  Bureau of Labor Statistics      Voice (202)606-6573 ext. 607
  |      |  2 Massachusetts Ave, NE         Fax (202) 606-7080
  | (o)(o)  Room 3105                       BITNET XUM@NIHCU
  C      _) Washington, DC 20212
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Ambrose, B Ancel E and M Griffiths "The fractal structure of
real estate investment trust returns: the search for evidence
of market segmentation and nonlinear dependency" manuscript, U of
Wisconsin-Milwaukee School of Business Administration 1991

Brock,W Hsieh, D & B Lebaron *A test of Nonlinear Dynamics, Chaos and
Instability* 1991 MIT press Cambridge MA

Jaditz, T Johnson D and R McClelland "Testing Nonlinear Dynamics" BLS
Working Paper 216 1991

Lo, Andrew "Long-term memory in stock market prices" MIT working paper
3014-89-EFA

Ruelle, D *Chaotic Evolution and Strange Attractors* 1989 Cambridge
University Press, Cambridge

Wallis, JR and Matalas, NC "Small sample properties of H and K
estimators of the Hurst coefficient h" *Water Resources Research* v 6
1970 1583-1594