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Hi, RTs,
The following is the conclusion of an international study of grains and
oilseeds price distribution among statisticians in universities. Since
the site contains some mathematics, I think it will better for me to
quote the conclusion. The site is at:
http://homepages.luc.edu/~tmallia/agr.htm I sent this site before but
my e-mail system received an error message. Please don't be annoyed if
you receive it twice.
Do you agree with the result?
"Market Efficiency remains the central theory of financial economics.
During the past twenty years this theory has been refined analytically,
mathematically and statistically. Numerous sophisticated statistical
tests have been employed to test this theory and a voluminous
bibliography exists that confirms it. However, a very large literature
has also been accumulated rejecting market efficiency and the random
walk hypothesis.
Notice, however, that it is not enough to reject randomness and market
efficiency. In order to make scientific
progress we must specify alternatives to randomness. In this paper we
offer an answer to the question: if asset
returns do not follow random walk what are they? Using daily
agricultural futures data for about ten years our
answer is: returns are fractal.
What does it mean to say that returns are fractal. The concept can be
given a rigorous mathematical definition
presented in Falconer (1990). Intuitively, returns are fractal if they
are characterized by properties such as: fine structure, local and
global irregularities , self similarity and noninteger dimension. Such
fractal processes generalize the well known random walks and martingales
of financial economics.
To support our claim that agricultural futures returns are fractal we
offer three pieces of statistical evidence. First, we conduct tests and
reject the hypothesis that returns are normally distributed. We then
estimate the four parameters of the Pareto-Levy stable distribution.
This distribution generalizes the special case of the normal
distribution. Using certain mathematical facts we conclude that the
estimates of the four parameters are consistent with the conjecture that
the stochastic process generating the returns is fractal.
The second set of tests uses the classical rescaled range analysis by
computing the Hurst exponent. The third test is an extension of the
second using a recent modification poposed by Lo (1991). Using both
these tests, we once again find evidence that returns are fractal.
What are the implications of our findings? Suppose that all financial
returns (not only the agricultural futures studied in this paper during
the decade 1981-1991) were fractal. This would imply that financial
returns behave in ways much more general than random walks. Put
differently, random walks are only a very special case of general
fractal processes. Technically, this means that while a fractal process
may have a Hurst exponent that ranges theoretically over an infinite
set, the random walk is only one special case when the Hurst exponent
receives the value 0.5. This means that market efficiency is a special
theory and not a general theory; it holds sometimes but not always. In
other words the empirical evidence that efficiency holds in some cases
and does not hold in others is now consisitent with the evidence that
returns are fractal. Obviously, much more research is needed to confirm
or reject the fractal behavior of returns for nonagricultural futures.
"
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