Prechter posted the following response on the Elliott Wave International website:2
“At some point Ralph Elliott’s “principle” and my cartoon simulations both
use recursive interpolation in which each part is a reduced-scale version of the whole.”
That’s our point. It is more accurate to say, though, that Elliott did not use recursive
interpolation but rather reported it in the actual stock market. Mandelbrot condescendingly
uses quotes and lower case for “principle,” but the model has a name, which he
knows full well, and the idea is grand enough fully to deserve it.
“The idea is ancient…”
Please. The idea of recursive interpolation (for art and so forth) may be ancient, but the use of it
to describe and model the stock market is not. It originated with Elliott.
“…but his use and mine stand in absolute contrast.”
That they do.
“Elliott drew a certain nonrandom ‘wave’ that he claimed ‘really forecasts’ every
real-world market…”
The Wave Principle is indeed substantially non-random. Its specificity is what makes it useful. In
probabilistic terms, and with varying degrees of success, it can indeed be employed to forecast most
real-world markets.
“…however, this simplistic wave was first stretched, squeezed or otherwise adjusted…”
In this comment we find both error and irony. Elliott did not model the market as a certain simple
picture and then stretch, squeeze or adjust it. He was describing reality, and reality itself is diverse.
This accusation is akin to saying that someone who sketched pictures of a dozen types of trees and then
proposed a composite model to depict tree-ness “started with a simplistic tree model and then
stretched, squeezed and otherwise adjusted it.” It’s backwards.
There is irony in Mandelbrot’s depiction because in fact it describes precisely what he himself
is doing in creating his “multifractal model.” He begins with a simple fractal generator
and then adjusts it by stretching it on the vertical axis and squeezing it
on the horizontal axis, all quite on an ad hoc basis, as you can clearly see in Figure 1, taken directly
from his article.
“…by hand.”
Was Elliott supposed to have used a computer? In 1934?
“In contrast, fractal or multifractal models must follow firm mathematical rules that allow quantitative
developments throughout, as mine do.”
Must they? It is important to distinguish between a model that we can manipulate precisely
and a model of reality. The ability to model with precision requires mathematical rigidity, but a model
of reality requires a commitment to truth, which may preclude mathematical rigidity, at least in the
model’s formative stages when knowledge is limited. This result seems particularly applicable
to living systems and even more especially to human behavior. A model with firm mathematical rules that
purported to predict your friend Charlie would be absurd, and one that allowed for Charlie to do anything would
be useless, yet we get both of these things from Mandelbrot’s model with respect to financial
markets. One could certainly build a useful model of Charlie’s past and potential behavior, but
it would have neither that basis nor that outcome. Consider the question of whether trees follow “firm
mathematical rules.” Perhaps they do, but no currently known model using such rules would be useful
for forecasting the final outcome of any particular tree. On the other hand, a robust model based on
detailed empirics — using mathematical rigidities and probability statements — might help
you anticipate much of it. Elliott’s model is of this type.
The above discussion is in no way a concession to Mandelbrot’s claim that only his model can
be described mathematically. Elliott’s model can be quantified with firm mathematical
rules. That is why we were able to create our EWAVES computer program3 to label data series according
to Elliott’s rules and guidelines. I have no illusions, though, that the resulting program has
captured every possible nuance of market behavior, perfectly assigned its ranges of probability, eliminated
all impossible patterns or completely encapsulated the robust Wave Principle model, about which there
is surely more to learn.
Neither is this discussion a concession to Mandelbrot’s claim to having modeled the stock market
in the first place. Although he claims that his model “follow(s) firm mathematical rules,” it
doesn’t follow any useful rules of form but rather allows for infinite manipulation.
Can you imagine the dismissive response to R.N. Elliott had he claimed that markets can do anything
at all as long as they fluctuate up and down? What kind of model is that, and of what practical value
are such “firm mathematical rules”? Do you want a model that admits random “quantitative
developments throughout” if that’s not what happens in reality? Random walk theorists
can describe their model with mathematical precision, too, but it is not therefore relevant to the stock
market. Elliott’s model and Mandelbrot’s model both allow for an infinite variety of quantitative
developments, except that Elliott’s model limits aspects of the variety that may occur,
making it useful in real-world forecasting. Mandelbrot’s model eliminates no potential market
event (except that prices cannot move backwards in time!). The incorporation of infinite possibilities
for price behavior in Mandelbrot’s purported model in fact makes it no different from a random
walk and thus, for practical purposes, no model at all.
“In any event, the random or nonrandom cartoons themselves are of no interest…”
In his 1997 book, Mandelbrot presents drawings like those in Figures 2 and 4 in the preceding chapter,
as well as a few others, that utilize a specific pattern (such as 3-up, 3-down, using certain lengths)
in order to display the idea of multifractal self-affinity. He calls these illustrations “cartoons,” an
apt word in this case, perhaps, because with respect to the actual patterns of financial market behavior
that Elliott elucidated, they represent either arbitrary constructs or potentialities, not actualities.
As we can see by Mandelbrot’s inclusionary use of the term above, he means to include Elliott’s
drawings among such “cartoons,” which may be why he chose such a diminutive word. Elliott’s
illustrations, however, are no more cartoons than a detailed painting by a realist. Elliott drew aspects
of reality, not abstract constructs. For this reason, Elliott’s nonrandom cartoons are
of interest to thousands of people. Indeed, they are fascinating both in their reliable reflection of
reality4 and in their theoretical implications. Mandelbrot’s random cartoons are
of far less interest. (Having seen Paree, I would extend that judgment to his entire model.)
If Elliott’s depictions are of no interest to Mandelbrot, that’s fine. He has placed this
fact on the record to his great detriment if and when the Wave Principle model is vindicated. Yet shouldn’t
Mandelbrot be thrilled to discover that someone who set out to depict actual stock market behavior recorded
a structure that his model accommodates? Apparently, the problem with his recognizing this feat is that
Elliott did it decades ago. Had someone come up with this depiction ten years into the future, Mandelbrot
could claim that his model, however vaguely, anticipated it. Given the true chronology, admitting Elliott’s
Wave Principle into the academic discussion places Mandelbrot in the position of being a latecomer who
is either giving Elliott’s model some added justification or proposing a broader derivative umbrella
under which to consider market fractality, which may have theoretical value (though still no practical
value) only if Elliott’s model is disproved. That must be why to him, the Wave Principle is “of
no interest.”
If Mandelbrot were to undertake a theoretical modeling of the automobile, he could perhaps take credit
for having invented “carness.” The mere fact that people have already built cars
should have no bearing on his claim. He could dismiss them as “non-random cartoons of no interest.”
We should further note that Mandelbrot never created any cartoons — or used that term — until after he
saw R.N. Elliott’s illustrations depicting the Wave Principle. Mandelbrot’s cartoons and
his related commentary may have some value apart from the Wave Principle, but their timing and presentation
appear designed at least partly with the hope of enveloping Elliott’s model within the scope of
his work while simultaneously relegating it to near insignificance. When Elliott’s specific observations
become accepted in the scientific community, this strategy, if that’s what it is, will be revealed
as inadequate to the task and turn the tables on the implied relative significance between the contributions
to financial market modeling.
“…they serve only to introduce the subtle quantitative properties and tools of my model
of price variation — fractional Brownian motion in multifractal time.”
That’s what Mandelbrot’s cartoons do. Elliott’s model does not depict Brownian motion;
it depicts stock market motion. They are very different categories.
“The rules of this model are not recursive...”
Mandelbrot’s model may not be recursive, but his fractal generator is; that’s what Figures
2 and 4 imply, and that’s what he himself says: “...my cartoon simulations...use recursive
interpolation....” If there is no recursion, then what is the point of the illustrations other
than to muscle in on Elliott’s territory? Elliott’s model is not rigidly recursive;
there is great variety within the Wave Principle. Finally, the stock market is robustly recursive,
so a model that is not — like Mandelbrot’s — does not capture the essence
of the stock market.
“...but fully specified mathematically and can be adjusted to fit the historical financial data.”
To repeat, of what value is a model that is “fully specified mathematically” but whose result
is no different from saying, “The stock market can do anything”? Elliott’s model describes
what aspects of stock market movement are certain and which are not, and therein lies much of
its value.
