2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
Do “Random Series” Exist?
Lewis L Smith
Consulting Economist in Energy and Advanced Technologies
mmbtupr@aol.com
787-640-3931
Carolina, Puerto Rico, USA
June 22-24, 2008
Oxford, UK
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2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
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Do “Random Series” Exist ?
Lewis L. Smith mmbtupr@aol.com
ABSTRACT
The existence and nature of randomness is not an arcane matter, best left to specialists. It is fundamental,
not only in helping to define the nature of the universe but also in more mundane matters. Examples are
making money in the stock market, improving the accuracy of economic forecasts and saving lives in
hospitals, especially by the correct interpretation of diagnostic printouts.
The whole question is still open. Some say that not only quantum mechanics but also mathematics, is
random in at least one fundamental aspect. Others believe that someday even most “random”
phenomena will be explained. Most economists side with the latter group. However, there are lots of
random-looking systems and data around. So to get about their business, economists have sought to
tame this randomness with intellectual patches.
Unfortunately, the patches have started to come apart, and a great many questions are now open to
discussion. What do we mean by random? Does a single distribution really fit all possible random
outcomes? Are the data generating processes of securities markets and their outputs chaotic, random or
what? If random, how many and what kind of distributions do we need to characterize price behavior in
a given market? What do we do about extreme values (outliers)? And more generally, what do we do
about nonrandom systems which generate runs of data which look and/or test random? And vice versa?
To deal with these problems, this paper proposes to separate the characterization of processes from the
characterization of output. In place of random processes, we define “independent-event generators”, and
in place of random series, we set forth a research program to define “operationally random runs”.
INTRODUCTION
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At first glance, this subject would seem to be an arcane matter, best left to specialists such as
econometricians, mathematicians, meteorologists, philosophers, physicists and statisticians. But it is not.
The question of randomness is fundamental, not only in helping to define the nature of the universe but
also in more mundane matters. Examples are making money in the stock market, improving the
accuracy of economic forecasts and saving lives in hospitals, especially by the correct interpretation of
diagnostic printouts.
Since well before the time of Leibnitz (1646-1716), scholars have been arguing over whether the
universe is completely explainable or whether some of its behavior is “random” and in what sense. On
one hand, there is a thread running from Godel (accidentally) to Chaitin and others which ends up saying
that not only quantum mechanics but also mathematics, is random in at least one fundamental aspect. On
the other, there is a contrary thread running through Einstein to Wolfram, who tries to explain too much
with cellular automata. In brief, as Einstein would put it, Does God “play dice” with the world? And if
so, when and how? (Calude 1997, Casti 2000, Chaitin 2005, Einstein 1926, Wolfram 2002.)
Economists’ hearts are with Wolfram. Indeed a strong bias towards order pervades “mainstream”
economics. For example, economies and markets are alleged to spend most of their time at or near
equilibrium, after the manner of self-righting life boats or stationary steam boilers, preferably one of the
simpler designs invented in the 19th century. All this raises suspicions of “physics envy” and even stirs
calls for a new paradigm. (Mas-Colell 1995, Pesaran 2006, Smith 2006.)
RANDOMNESS AND ECONOMICS
However, the goal of explaining almost everything is far off, and present reality is full of systems whose
data-generating processes and outputs sometimes look and/or test random, whether or not they really
are. Economics in particular is heavily dependent on concepts of randomness. For example:
[1]
The parameters used by econometrics to define economic relationships are stochastic.
That is, they are “best” estimates of supposedly random variables, each made with a specified margin of
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error. Moreover, their validity depends in large part, on the “quality” of these random aspects. (Kennedy
1998.)
[2]
Many analysts still believe that some very important time series are generated largely or
exclusively by random processes. These include the prices of securities whose markets are the largest in
the world and among those with the lowest transaction costs. (Mandelbrot 2004.) Randomness also
shows up in the models of agent-based computational economics. (Arthur 2006, Hommes 2006,
Voorhees 2006, Wolfram 2002.)
[3]
Randomness plays multiple roles in the applications to economics of the new discipline
of complexity, which studies what such systems as biological species, cardiovascular systems,
economies, markets and human societies have in common. For example, with too many definitions of
complexity “on the table”, it is operationally convenient to define a complex system as one whose
processes generate at least one “track” (time series) whose successive line segments (regions) exhibit at
least four modes of behavior, one of which must be — random! (Smith 2002.)
So to get about their business, economists have made a virtue of an unpleasant reality by attempting to
tame randomness with a number of asymptotic truths, bald-faced assertions, fundamental assumptions
and/or working hypotheses. For example:
[1]
The essence of randomness is the spatial and/or time wise independence of the numerical
output of a system. This independence could be due to process design, to pure chance or to a hidden
order so arcane that we cannot perceive it. But whatever the case, we are unable to predict the next
outcome on the basis of any of the previous ones or any combination thereof. In brief, the process has
absolutely no memory. (Beltrami 1999)
[2]
1
An implicit corollary of the foregoing is that each and every run of data generated by any
such process is random, in a heuristic sense at least. (Pandharipande 2002.)
1
Some would argue that properly speaking, randomness is a property of the generating process, not of its outputs. But in
practice, this distinction is frequently ignored, so we will do likewise.
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[3]
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Conversely a run which looks and/or tests random has (hopefully) been generated by a
random process.
[4]
All random outcomes follow the Gaussian distribution or, thanks to the magical powers
of the Central-Limit Theorem, approach it asymptotically as n increases, as if economic events had
something in common with the heights of French soldiers in the 18th Century! Unfortunately we are
seldom told how far away we are from this state of bliss in any particular instance.
DISARRAY IN THE RANDOMNESS COMMUNITY
Over time, the first of the preceding items has acquired competition and the other three have turned out
to be wrong on too many occasions. In particular:
[1]
The definitions of randomness currently “on the table” use a variety of incompatible
concepts, such as the nature of the generating process, the characteristics of one or more of its tracks, a
hybrid of these two or the length of the shortest computer program which reproduces a given track. As a
result, the “randomness” of given process or track depends on the definition used. Indeed there is no all
purpose “index” of randomness. (Pandharipande 2002.)
[2]
More often than one expects, some genuinely random processes generate runs which do
not look and/or test random (in the popular sense) while non-random processes sometimes generate
random ones.
[3]
Since Pareto’s pioneering work with income distributions (Pareto 1896-97), there is
increasing recognition that “departures from normal” among probability distributions are in fact, quite
normal! For example, the heights of a collection of human beings at any one point in time follows a
Gaussian Distribution; the expected lives of wooden utility poles, a Poisson Distribution; and the speed
of the wind at a given height and location for a given period, a Weibull Distribution. In the cases of
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econometric parameters, residual errors and test statistics, we find a bewildering variety of distributions,
many of which are not even asymptotically Gaussian. (Razzak 2007.)
[4]
In securities markets, it quite possible for the distribution of outcomes in the/ “tails” of a
price series to be different from that which characterizes the bulk of the sample and sometimes, from
each other. (Sornette 2003.)
[5]
Popular concepts of randomness do not tell us how to distinguish errors from other kinds
of extreme values. Nor do they give us adequate guidance on “noise”, which need not be Gaussian.
We now examine these matters in detail from the point of view of “operating types”, such as business
economists, organizational managers and strategic planners. We begin with what were long thought to
be the “easy cases”, ones in which both the processes and their tracks were held to be random.
SECURITIES MARKETS
For several decades, it was “gospel truth” in academic and financial circles that the processes of
securities markets were not only random but generated outputs which tested random. In particular, the
log differences of any and all prices for any period in almost all such markets were alleged to faithfully
obey the Gaussian distribution, with nary a chaotic thrust, off-the-wall event or statistical outlier to
disturb one’s sleep. (Lo 1999.)
In this enchanted world, happenings such as those of 10/29, 10/77, 10/87 and O3/00 had nothing to do
with the dynamics or structure of a particular market or with the distribution of returns. Such
occurrences were genuine, bona fide outliers, rare aberrations caused by factors which were external,
unusual and unique to each occasion. Examples are the gross errors attributed to the Federal Reserve
System’s Board of Governors in 1929 or those never found “sufficient causes” of the 10/87 debacle,
whose absence is a sure signature of chaotic, not random behavior. (Peters 2002, Sornette 2003, Jan
2003.)
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Moreover, this illusion protected brokers. If their clients suffered heavy losses, it was not the broker’s
fault. And no one could make money by “playing the market”, as opposed to investing for the long haul.
Those who did were just lucky and would eventually “go broke”, as did the famous speculator Jesse
Livermore. (LeFevre 1923.)
Thanks to the dogged work of Mandelbrot, Sornette, Taleb and other skeptics, researchers began to have
doubts and so today, there are many open questions. Are markets random, chaotic or what ? What is the
relevant output — prices, price changes or run length? Should we use one, two or three distributions to
describe output, and which ones? And what do we do about extreme values?
The resolution of these disputes is not an easy matter. There are many candidate distributions, and many
of these “shade into” one or more of the others. Worse yet, as Sornette puts it, there is no “silver bullet”
test by which one can tell if the tail of a particular distribution follows a power law. (Bartoluzzi 2004;
Chakraborti 2007); Espinosa-Méndez 2007; Guastelo 2005; LeBaron 2005; Lokipii, 2006; Motulsky,
2006; Pan, 2003; Papadimitrio, 2002; Perron 2002; Ramsey 1994; Scalas 2006; Seo 2007; Silverberg,
2004; Sornette all, Taleb 2007.)
Of course, one may sometimes obtain a tight fit to historic data with a “custom-made” distribution
obtained by the use of copula functions. But over fitting is a risk, the methodology is complicated and
requires both judgment calls and the use of a Monte Carlo experiment, which plunges us into murky
world of pseudo-random number generators of which, more later. (Giacomini 2006)
Even if one discovers the “correct” distribution to characterize a particular output, it may not provide an
adequate measure of market risk. Moreover, if the generating process is unstable, one may be forced to
measure the additional risk by the risky procedure of Lyapunov exponents. (Bask 2007, Das 2002a.)
TECHNICAL ANALYSIS
To add insult to injury, those who devoutly believe that price formation in securities markets is a purely
random process must face up to the existence on Wall Street of an empirical art known as “technical
analysis”, which most academics either ignore or classify as “irrational behavior”. (Menkhoff 2007.)
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Some of its practitioners not only make money, but have the effrontery to publish their income-tax
returns on the Internet!
These persons, called “technicians”, see a veritable plethora of regularities in price charts: candlesticks,
cycles, double bottoms, double tops, Elliot waves, Fibonacci ratios, flags, Gann cycles, Gartley patterns,
heads, pennants, pitchforks, shoulders, trend lines, triangles, wedges et cetera. (Bulkowski 2006;
Griffoen 2003; Maiani 2002; Penn 2007.)
Nevertheless, the financial success of a minority of technicians and the discovery of 38,500 patterns by
Bulkowski does not signify “the death of randomness” on Wall Street. To begin with, all technicians
recognize that any price or price index may slip into “trading range” from time to time and that any
pattern may fail on occasion.2 Furthermore, they frankly admit that many patterns are not reliable
enough to trade on, that some are too hard to program and that still others have become too popular to
give a trader “an edge”. So if academic “hard liners” wish, they may still look down on technicians as if
they were priests of ancient Rome examining the entrails of sacred geese, but with the mantra — “past is
prologue”. (Blackman 2003; Bonabeau 2003; Bulkowski 2005; Connors 2004, Eisenstad 2001, Katz
2006, Mandelbrot 2004, Menkhoff 2007, Yamanaka 2007.)
The attempts of some technicians to explain market behavior entirely as a sequence of non-random
cycles have also come a cropper, except in the hands of the most experienced practitioners. These
include not only such traditional techniques as Dow Theory, Elliot Waves and Gann Curves, but more
modern ones such as the Bartlett algorithm, assorted Fourier transforms, high-resolution spectral
analysis and multiple-signal classification algorithms. (Ehlers 2007; Hetherington, 2002; Penn 2007,
Ramsey 1994, Rouse 2007 and Teseo 2002.)
Fortunately the techniques of wavelets, familiar to engineers and natural scientists, holds promise both
for economics and for finance and even more so, the related one of wave-form dictionaries. However,
these techniques require judgment calls between a rich variety of options and (sometimes) a knowledge
2
Dukas (2006) finds that for 80 years of the Dow Jones Industrial Average, the index was in a bullish phase (upward
trending) 45% of the time, a bearish phase (downward trending) 24% and in “a trading range”, 31%.
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of programming, so careful study and considerable experience are a prerequisite to their successful use.
(Schleicher 2002, Ramsey 1994, 2002.)
EXTREME VALUES
In characterizing the output of any process by a probability distribution, one must deal with extreme
values. Are they part and parcel of some “fat-tailed distribution” which fully describes the data at hand,
do have their own “tail” distribution or are they truly exogenous. That is, are they anomalous, separate
events, each one “coming off its own wall” perhaps, such as catastrophic events which lie beyond any
power-law distribution? And what about errors in measuring or recording? How do we separate “the
sheep from the goats” and obtain a “true” representation of the data-generating process? (Chambers
2007, Maheu 2007, Sornette all, Taleb 2007.)
Decisions relating to extreme values are of particular significance whenever one uses estimators in
which second or higher moments play important role. Unfortunately there is no consensus as to how to
answer these questions in the case of nonlinear regressions, frequently the most appropriate ones for
financial time series. The classification of a specific event as an outlier in such cases is not only difficult,
but it involves at least one subjective judgment and at least one tradeoff between two desirable
characteristics of parameter estimates. (Le Baron 2005; Moltusky 2006; Perron 2001 and 2002, White
2005.)
RANDOM PROCESSES WITH NON-RANDOM OUTPUTS
Paradoxically many processes known with absolute certainty to be random, frequently generate runs of
data which do not look and/or test random by any of the popular criteria. Coin flippers and roulette
wheels are obvious examples. Their outputs, especially the cumulative versions, may quite accidentally
manifest one or more autocorrelations, including pseudo cycles, pseudo trends and unusual runs, any
one of which may end abruptly when one least expects it! Needless to say, such an event can cause
great distress to those not familiar with the behavior of such processes, especially if they have placed a
bet on the next outcome!
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For example, given n tosses of a fair coin, each possible sequence has exactly the same probability of
occurring, whether it is “orderly”, like HHHHHH… and HTHTHT… or well scrambled, by far the most
common case. However, it is very human to forget that random events often exhibit patterns. So upon
seeing HHHHHH… in a short run, almost all lay persons and some researchers will “cry foul”. (Stewart
1998.)
First-order Markov chains with a unit root, such as those used to represent the cumulative results of coin
flipping, provide some of the worst examples. The longer the run, the more likely it is that the track will
exhibit long, strong “trends” and make increasingly lengthy “excursions” away from its expected mean.
In fact, the variance of such a series is unbounded and grows with time.
For example, consider the
“Record of 10,000 coin tosses”. (Mandelbrot 2004, p. 32). This cumulative track takes some 800 tosses
to cross the 50% line for the first time and 3,000 tosses to cross it for the second. Even after 10,000
tosses, the track has not yet made a third crossing, although it is getting close!
Of course, given enough realizations, the famous “law of averages” will always reassert itself from time
to time. But frequently it fails within the shorter runs which are all that casino gamblers and “players” in
securities markets can afford. So the latter usually run out of money long before things “even up”.
(Leibfarth 2006.) In sum, drawdown usually trumps regression to the mean, a truth apparently forgotten
by the principals of both Amaranth and Long-Term Capital. (Peters 2002.)
DETERMINISTIC PROCESSES WITH RANDOM OUTPUTS
Conversely many largely or purely deterministic systems can generate long runs which look and/or test
random by popular criteria. A classic example is the chaotic system which begins as a tropical storm off
the west coast of Africa, wanders erratically to within few hundred miles of Barbados and then suddenly
turns into a hurricane.
Worse yet, some purely deterministic systems are intentionally designed to generate nothing but runs of
data which test random by standard tests, although in the very long run, they will repeat themselves.
Prominent among them are the computer algorithms known as “Pseudo-random Number Generators”
(PRNG’s). Unfortunately, none of these provides an “all weather” standard of randomness. Each has its
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own deficiencies and idiosyncrasies, so one must chose among them with care. (Bauke 2004; Hellekalek
2004; OmniNerd 2007 and P Lab all.)
Another example is provided by the constant π, well known to ancient Greek geometers and modern
students of trigonometry. Although the true value has an infinite number of digits, Leibnitz worked out
an exact formula for the first n digits, a formula which can be easily calculated by a short computer
program. If one generates enough of these digits, one can select a long run which tests random and use
it for evaluating still other series, in lieu of the output of a PRNG. This example also violates all
definitions of randomness based on the length of the generating computer program. (Chaitin 2005 and
Shalizi 2003.)
In summary, nearly all processes of human confection, whether deterministic, random or some of each,
are sooner or later going to generate some output which looks or tests as if it came from somewhere
else.
ILL-DEFINED PROCESSES WITH AMBIGUOUS OUTPUTS
In many cases, we face a double challenge. Our understanding of a process limited, and the behavior of
it output is ambiguous. There are several ways by which we may tempted to increase our knowledge of
the situation.
If the possible classifications of a given track can be reduced to two, a comparison of the series may
settle the issue. Unfortunately, large sample sizes and sophisticated mathematical techniques are often
required, without any guarantee of success. (Das 2002b, Geweke 2006.)
For example, in their respective fields, biologists and medical researchers have great difficulty in
distinguishing between time series generated by the processes of deterministic systems and those
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generated by random ones, although the difference may be critical for diagnosis and treatment of
patients. (Das 2005.)
For example, “…in many cases, it may be impossible to distinguish between a process which follows a
deterministic (algorithm) subject to (random) ‘noise’ and a random process (which has been) subjected
to selection or reinforcement”. (Vorhees 2006.)
Alternatively we may try to construct a forecasting function from the data at hand, analyze an existing
model of the process or construct our own model of the process. The first of these endeavors is fraught
with peril; the other two are usually a waste of time, at least as far as randomness is concerned.
For example, large amounts of data are often required for the construction of a good forecasting
function, such as Sornette’s heroic “Formula 19”, which requires eight years of stock-market prices to
be properly estimated. (Sornette 2003, p. 236.) Moreover, the longer the time series, the greater the risk
that the researcher will be “ambushed” by phenomena such as changing seasonality, cycles, extreme
values, fractional integration, gaps, improbable runs, log periodicity, long memory, near unit roots,
nonlinearity,
non-stationary
noise,
nonstandard
distributions,
omitted
variables,
power-law
accelerations, regime switching, stationarity problems, stochastic trends, structural breaks, thick tails,
thresholds, time-varying parameters, time-varying volatility and volatility clusters. Any one may affect
parameters, variables or both, and many are hard to detect and/or distinguish from each other.
(Bartolozzi 2004, Bassleer 2006, Bernard 2008, Byrne 2006, Chakraborti 2007, Diebold 2006, Dolado
2005), Dow 2006, Harvey 1994 and 1997, Hjalmarsson 2007, Harvey 1997, Heathcote 2005, Johansen
2007, Komunjer 2007, Koop 2007, Koopmand 2006, Kuswanto 2008, Maheu 2007, Noriega 2006,
Perron 2005: Pindyck 1999, Royal 2003, Sornette all, Teräsvirta 2005 and Zietz 2001.)
Moreover, it is not enough to classify whole time series or segments thereof. Operating types need to
know when the mode of behavior is going to change. Unfortunately there is the “70% barrier”, in both
econometrics and technical analysis. It is extremely difficult to call more than 70% of the turning points
in any given time series. (Cobly 2006, Gopalakrishnan 2007, Peters 2002, Pring 2004, Taleb 2007 and
Zellner 2002.)
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As for existing models, they have probably been designed for another purpose and therefore are unlikely
to be of much help in detecting randomness. (Karagedikli 2007; Zellner 2002.) Building a model “from
scratch” is a complex, difficult and resource-consuming activity, as well established by a vast literature.
(Favero 2007, Gilboa 2007, Phillips 2003 and 2007, and Zellner 2002 and 2004.) More to the point,
given the propensity of dynamic systems to imitate each other’s outputs from time to time, it is unlikely
to enlighten us very much.
Speaking of the travails of central-bank economists, Razzak (2007) neatly summarizes the modeling
problem : “… we will not be able to find out what the (data-generating process) is exactly. We are also
not able to precisely pin down the nature of the trend”.
WHERE TO TURN?
At the present time, the only way to be sure of obtaining a genuinely random process that generates
random outputs on every occasion, is have recourse to certain physical processes in nature. An example
is the “Quantum Random Bit Generator”, a a fast, non-deterministic, widely distributed, random-bit
generator developed and operated by a group of computer scientists at the Ruder Bokovic Institute in
Zagreb, Croatia. (QRBG Service 2007.) It relies on the intrinsic randomness of photonic emission in
semiconductors, a quantum physical process, and on subsequent detection of these emissions by
photoelectric effect
TOWARDS SOME OPERATIONAL DEFINITIONS
It seems futile to persist in the search for a concept which embraces all kinds of random processes and
their outputs. We suggest that the term “random process” be dropped ant that processes which generate
independent events be called just that, “independent-event generators”. We further suggest that ther term
“random series” be dropped and that line segments which test random (by criteria yet to be determined)
be called “operationally random runs”.
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At first glance, we have a surfeit of riches with regard to tests. For example, the three volumes of Walsh
(1962-68) run to 1,982 pages! In particular, there are a large number of diagnostic tests used to evaluate
the residuals of functions estimated by econometric techniques, tests which can be applied to line
segments in general. (Kennedy 1998.)
Unfortunately, there is no all purpose test for randomness. Moreover, tests for randomness and related
concepts suffer not only from problems of definitions but from most of the problems associated with the
construction of forecasting functions. In addition, they are plagued by other problems, such as
asymptotic distributions and degrees of power which vary with the alternatives to be compared and/or
the characteristics of the data. Also runs which test random may be stationary or non-stationary. White
noise is always random and stationary, but not all runs which test random or stationary are white noise.
(Lo 1999, Kennedy 1999, Patterson 2000.)
The most popular measures of correlation measure only linear relationships. In any case, the
corresponding indices represent average conditions along the entire distance between dates, when
conditions may be changing in that distance. Such fluctuations in the degree of correlation may well
reduce the value of the calculated index and cause one to miss those conditions under which a particular
correlation is significant. This problem is of course, made particularly acute wherever structural breaks
occur. (Bergsma 2006, Phillips 2004.) And so on.
Under the circumstances, we propose that a small number of researchers attempt to define an
“operationally random run”, in the spirit of parallel processing. Each one would take a different stock
market and (in the USA at least) emphasize a moving window of 201 trading days, since 200 days is a
popular and statistically significant period for traders. At a minimum, the data in the period would be
subject to the following calculations and tests, every time the window advanced a day:
(1)
Tests of autocorrelation of the absolute values of the price changes.
(2)
A sign test.
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(3)
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The largest uninterrupted gain and largest uninterrupted loss, based on trading days.
(Sornette 2003, pp. 51+.)
(4)
The distribution of these gains and losses. These last
two items may also serve as rough alerts of the approach of a chaotic mode.
In addition,
(5)
The low-volatility index (VIX) should be downloaded for
each trading day from a convenient stock-price service, and compared with its moving average for
eleven trading days ending. When VIX is 5% below its 10 period moving average, stock markets tend to
be in a “dead money zone”. (Connors 2004; Ruffy 2004.)
The researchers would have their own private discussion group on the Internet.
CONCLUSION
Yes, random series do exist. They are not artifacts of our ignorance. However, they do not always
appear where we expect to find them, nor do their outputs always look like what we expect to see.
POSTSCRIPT
We can learn much from peering at the mathematical “entrails” produced by the manipulation of data
but in the end, we cannot avoid the hard tasks of “collecting the dots” across space, time and disciplines,
connecting them and then marshalling evidence, all as best we can. We can never “tell the story”
without the numbers, the numbers seldom if ever, “tell the whole story”!
In the final analysis, we come back to what Professor Manoranjan Dutta of Rutgers University told
Puerto Rico’s first econometrics class one summer day in 1966, “There is no substitute for knowledge of
the problem!”
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REFERENCES
Arthur, W. Brian (2006) : Out-of-Equilibrium Economics and Agent-Based Modeling, in Judd, K. and
Tesfatsion, L. (editors) : Handbook of Computational Economics (vol. 2). Elsevier/North Holland,
Amsterdam, Netherlands.
Bartolozzi, M., Leinweber, D. B.; Thomas, A.W. (2004) : Self-organized criticality and stock market
dynamics : an empirical study. Dept. of Physics, U. of Adelaide, Australia.
Bask, Mikael (2007) : Measuring potential market risk. Discussion paper 20-2007, Bank of Finland,
Helsinki, Finland.
Bassler, Kevin E.; Gunaratne, Gemunu H. and McCauley, Josephy L. (Jul 2006) : Markov processes,
Hurst exponents and nonlinear diffusion equations. Physics Dept., University of Houston TX.
Bauke, Heiko and Mertens, Stephan (2004) : Pseudo-random coins show more heads than tails. Journal
of Statistical Physics 114, pp. 1149-1169.
Beltrami, Edward (1999) : What is Random? Springer-Verlag, New York NY
Bergsma, Wicher (July 2006) : A new correlation coefficient and associated tests of independence.
London School of Economics and Political Science.
Bernard, Jean-Thomas; Khalaf, Lynda; Kichian, Maral and McMahon, Sebastienne (Feb 2008) : Oil
prices : heavy tails, mean reversion and the convenience yield. Bank of Canada and Ministry of Finance,
Quebec, Quebec, Canada.
Blackman, Matthew (May 2003) : Most of what you need to know about technical analysis. Technical
Analysis of Stocks and Commodities, pp. 44-49.
Bonabeau, Eric (2003) : Don’t trust your gut, reprint R0305J. Harvard Business Review. (reprint)
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2008 Oxford Business &Economics Conference Program
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18
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
19
Geweke, John F. ; Horowitz, Joel L. and Pesaran, M. Hashem (Nov 2006) : Econometrics : a bird’s eye
view. Discussion paper 2458. Intitute for the Study of Labor, Bonn, Germany.
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
20
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June 22-24, 2008
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2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
21
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
22
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
23
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June 22-24, 2008
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2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
24
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
25
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June 22-24, 2008
Oxford, UK
2008 Oxford Business &Economics Conference Program
ISBN : 978-0-9742114-7-3
26
Teräsvirta, Timo (May 2005) : Forecasting economic variables with nonlinear models. Stockholm
School of Economics, Stockholm, Sweden.
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June 22-24, 2008
Oxford, UK