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Basic Models of Complex Systems: Crux of the Duplex Method plus Case Study of the Dow Stock Index

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Abstract and Figures

The world is full of complex and baffling systems. For instance the stock market confounds all manner of investors, be they amateurs or professionals. On the upside, though, a duplex framework can portray flighty systems with the utmost of simplicity, clarity and efficacy. An example concerns the monthly waves of the Dow stock index.
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Basic Models of Complex Systems
Crux of the Duplex Method
plus Case Study of
the Dow Stock Index
Steven Kim
MintKit Institute Kenwave Research
Keywords:
Duplex, Complex Systems, Stocks, Markets, Dow, Index, Models, R, Program, Code,
Seasonal, Waves, Efficiency, EMH, Random Walk, Myths, Finance, Economics,
Statistics
© 2020 MintKit.com and Kenwave.com
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Short Summary
The world is full of complex and baffling systems. For instance the stock market
confounds all manner of investors, be they amateurs or professionals. On the upside, a
duplex framework can portray flighty systems with the utmost of simplicity, clarity and
efficacy. An example concerns the monthly waves of the Dow stock index.
Mid Summary
The world abounds with complex and puzzling systems. For instance, the stock market
fuddles all manner of investors ranging from newcomers to veterans. According to the
reigning doctrine of financial economics, every market is an efficient system where the
current price always reflects all the information available to investors. In this sleek
environment, no one can discern any cues for predicting the market in a credible way.
On the upside, though, a duplex framework can debunk the myths of the stock market in
decisive ways. An example involves the seasonal waves behind the monthly moves of
the Dow stock index. The benefits of the wavy model include the ease of acquiring the
information required, the leanness of the dataset employed, the ubiquity of the software
deployed, the universality of the experimental setup, and the strength of the conclusions
at high levels of statistical significance.
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Long Summary
The world around us is chockful of complex and baffling systems. For example, the
stock market confounds all manner of investors ranging from part-time amateurs to full-
time professionals. According to the reigning doctrine of financial economics, every
market is an efficient system where the current price always reflects all the information
available to investors. In this sleek environment, no one can detect useful clues for
predicting the market with any measure of consistency.
On the bright side, though, a duplex framework can portray dicey systems with the
utmost of simplicity, clarity and efficacy. The merits of the binomial scheme show up, for
instance, in debunking the myths of the stock market. An example involves the seasonal
patterns behind the monthly moves of the Dow Jones Industrial Average.
The case study presented here spotlights the simplicity of the cyclic model from a
conceptual slant as well as a pragmatic stance. Other drawcards include the ease of
acquiring the information required, the leanness of the dataset employed, the ubiquity of
the software deployed, the universality of the experimental setup, and the strength of
the conclusions at high levels of statistical significance.
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* * *
Ruling Myths and Trampled Facts
We live in a world full of complex and chaotic systems. A good example concerns the
stock market that stymies all manner of investors ranging from casual amateurs to
gung-ho professionals.
According to the Efficient Market Hypothesis, the current price always reflects the
totality of information available to the investing public (Fama, 1965). As a byproduct, no
one can detect any clues for predicting the market in a trusty fashion.
Instead, the market is deemed to move in an utterly erratic way. In particular, a popular
myth known as the Random Walk shuffle contends that the price level shifts with equal
likelihood and to similar extent in either direction, whether to the upside or downside.
At first glance, the image of pure randomness does ring true in practice. For instance,
the average investor is unable to beat the market averages such as the Dow Jones
index. While the lack of success may seem like a letdown, the truth is even worse. In
actuality, the participants in the aggregate lag comfortably behind the benchmarks of the
bourse.
If we look more closely, the lousy performance of the actors springs mostly from their
frantic efforts to beat the competition. Amid the frenzy, the brutes of greed and fear prod
the antsy players into making impulsive moves that are not only groundless and futile
but actually counterproductive and harmful to their cause.
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On the bright side, though, the market displays a smattering of patterns that can be
exploited by a sober person. An example concerns the seasonal cycle behind the
monthly moves of the Dow benchmark.
To fathom the elusive waves in a stringent fashion, we turn to the duplex method of
modeling shifty systems. The sturdy framework makes use of the binomial test: the
simplest and strongest, as well as safest and surest, way to profile chancy events
regardless of the domain.
To this end, we first transform the conceptual models of the stock market into a trio of
precise templates. The formal blueprints are then converted into R code: the top choice
of programming language and software platform for statistical assays. The trenchant
results serve to debunk the fable of efficiency and confirm the existence of hardy
patterns in the marketplace.
In short, the benefits of the seasonal model lie in simplicity and potency in sundry forms.
The drawcards include the ease of acquiring the information required, the leanness of
the dataset employed, the ubiquity of the software deployed, the universality of the
experimental setup, and the strength of the conclusions at high levels of statistical
significance.
Method
To underscore the gulf between the theory and reality, we will take a minimalist
approach. For starters, we use only a minute fraction of the wealth of information freely
available to all comers at the most popular portal among the investing public. Moreover,
the quantitative analysis employs only the simplest technique in hypothesis testing. In
addition, the code required for the study invokes a tiny subset of the core functions
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within the R system: the top choice of programming language and software platform for
data science.
Precise Models
According to the Efficient doctrine, every asset displays equal odds of rising or falling at
any stage. In that case, the current price always reflects the best guesstimate of the
price level going forward. We will refer to this parody of the market as the Idle Model.
We now take a step back and look at the big picture of the stock market over the long
range. Anyone can see at once that the price level has a way of climbing higher as the
decades go by. The global trend shows up, for instance, in the ascent of the Dow Jones
Industrial Average since its inception in 1896. The index is also known by the shorthand
of DJIA or simply DJI.
In view of the uprise over the long range, a variation on the theme of ideal markets
asserts that the upward tilt poses the best guess going forward. For instance, the
optimal forecast of the price level a month hence is the current price plus a suitable
increment of the gradual uptrend. We will refer to the trending version of the Efficient
mantra as the Drift Model.
Choice of Dataset
The Dow Jones Industrial Average is the most popular benchmark of the stock market
amongst the general public. For instance, the DJI is the yardstick of choice when the
mass media round the world report on the movements of the U.S. bourse on the
evening news.
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As we noted earlier, the DJI in its modern form evolved from a prototype dating back to
the late 1800s. Given the long history, the use of a lengthy dataset may seem like a
good idea. That might in fact be the case if the target system happens to be stable and
displays a fixed set of properties.
But that’s not the case for us here. For one thing, the action in the stock market stems in
part from internal outcrops such as the rollout of novel products. A case in point is an
option contract or an inverse fund that may be used to offset the risk of a market crash.
A second driver involves the changing habits of the participants within the financial ring.
An example concerns the swelling throng of fund managers along with their penchant
for window dressing toward the end of the year; namely, the switch-out of downcast
assets with high-flying stocks in an effort to mask the crummy performance in their
annual reports.
A third facet concerns the growing role of the stock market for the mass of humanity. An
exemplar involves the multitude of entrepreneurs and employees who pour a bigger
fraction of their earnings into the bourse around the end of the year.
More generally, the lot of the stock market is tied to the fortunes of the real economy
especially over the medium term and the long range. Moreover, the economy itself has
a way of evolving with the passage of time. An example lies in the dwindling role of the
agrarian sector as well as the industrial branch over the past century.
In this fluid milieu, we ought to balance the richness of a lengthy dataset against the
pertinence of a recent sample. We will use the data from Yahoo Finance, the most
popular portal among the investing public (Yahoo, 2020). At the time of research, the
records available at the site for the Dow index stretched back to 29 January 1985. In
that case, a suitable dataset for our study involves the closing values of the benchmark
at the end of each month starting with the onset of 1985.
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Since we want to distill the seasonal cycle from the historical record, we will sample the
data on a monthly basis until January 2020. In this way, we can calculate the forward
returns over precisely 35 years beginning in 1985.
Check for Null Returns
The financial forum finds itself in a constant state of flux as every asset swells and
swoons as time goes by. Despite the ceaseless heaving, though, the price level could
settle at the same point from one period to the next.
On the other hand, our dataset does not contain any repeated values in direct
succession. As a result, the monthly return is never zero. So we can ignore the prospect
of a null value in assigning each move into one of two disjoint buckets; namely, to the
upside or downside.
Profiling the Data
As we saw earlier, the Dow index grows over time despite a series of sizable upswings
and downstrokes along the way. Moreover, our dataset in particular also displays these
traits as shown in Figure 1.
On the whole, the benchmark appears to burgeon at an exponential rate over the entire
stretch. That makes sense since the yardstick reflects a simple average of the stock
prices for a rolling roster of 30 hulking companies in the prime of their lives. At a
minimum, then, the value of the ensemble should keep pace with the march of inflation
that tends to advance by a couple of percent or more per year.
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Fig. 1. Monthly level of DJI from January 1985 to January 2020.
Fig. 2. Log to base 10 of DJI by month over the entire timespan.
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We now profile the overall trend by plotting the logarithm of the raw readings over time.
In Fig. 2, the vertical axis denotes the log to base 10 of the closing prices. The graphic
suggests that the overall trend is more or less steady. In that case, the original variable
displays an exponential streak.
An exponential rate of growth reflects a steady gain in relative terms; in other words, a
fixed percentage on a periodic basis. Fig. 3 shows the relative shift from one month to
the next throughout the timeline.
Fig. 3. Forward monthly return for DJI from 1985 to 2019.
The next step involves a quantitative assay. As a lead-in, we have just reckoned the
fractional changes in price on a monthly basis over the entire stretch. At each stage, we
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pinned down the relative gain—also called the forward return—over the next period as a
percentage of the current price.
From the graphic, we can see that the returns vary a great deal. Despite the outspread,
though, the scattering pattern appears to be more or less stable especially after the first
hundred cases or so.
Crafting the Models
Our next step is to convert the verbal descriptions of the stock market into formal
models. Two of the templates represent the Efficient bromide. Meanwhile the third
model depicts the seasonal cycle of monthly returns throughout the year.
We begin with the Idle mockup which declares that the market rises or falls with equal
likelihood and to similar extent. Since the expected moves to the upside and downside
are perfectly balanced, the best forecast of the monthly return at any stage is fixed at
zero percent.
On the other hand, the setup differs somewhat for the Drift Model. The latter send-up
does rely in a small way on the actual behavior of the market; namely, the uptrend over
the long range. For the dataset at hand, the geometric mean of the monthly gains
amounted to some 0.7382 percent. Roughly speaking, then, the Dow index was wont to
rise by nearly three-quarters of a percent on average each month.
The third and last model captures the seasonal patterns in the stock market. We will
refer to this portrait of monthly moves as the Sway Model.
Fig. 4 shows the mean return by month throughout the year. We can see that, more
often than not, the average move had a positive bent. In particular, the biggest gains
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occurred in March, June, October and November. Moreover, the market on average
hardly moved at all during May. On the downside, the worst performance cropped up in
July and August.
Meanwhile, Fig. 5 offers a richer view of the dataset. Each boxplot depicts the forward
returns for a particular month. For a given column, the thick bar within the rectangle
represents the median level rather than the mean value for the pertinent period.
Furthermore, each circle denotes an outlier.
The boxplot for each month reflects a great deal of dispersion compared to the waver of
median values throughout the year. The largest number of outliers appear in the fifth,
ninth and eleventh thumbnails. More precisely, each of May, September and November
features three extreme values.
Fig. 4. Mean return for each month of the year.
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Fig. 5. Boxplot of forward monthly returns.
Despite the broad scatter of readings along with the clutter, we can discern some
differences in the average values. For instance, the median return for October appears
to surpass those of September and November. Meanwhile, an example on the flip side
applies to August whose average gain falls below those of July and September. On the
whole, the relative placement of the median values resembles the configuration of mean
returns that we saw in the previous chart.
Choice of Average for the Sway Model
We could use the chain of mean values or the median levels to serve as the backbone
of the Sway Model. To this end, we first check which arrangement leads to higher
accuracy; or more precisely, less variation in the absolute values of the errors against
the entire set of forward returns.
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We find that the string of median values does a better job of modeling the monthly gains
on average than the chain of mean returns. So we adopt the median array as the
marrow of the Sway Model. That is, the median value of the forward returns for each
month will serve as the implicit forecast for that period.
Graphic Matchup of Results
We now compare the three models for profiling the seasonal patterns. The output in the
form of Fig. 6 presents a boxplot of the errors for each model.
Fig. 6. Boxplots of errors by model.
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Compared to the full scatter of errors, the average values of the residuals hardly varied
at all. On the other hand, a graphic display can be misleading. For this reason, we turn
to several forms of quantitative analysis.
Precise Peg of Ups and Downs
According to the Efficient chorus along with the Random Walk burlesque, the moves to
the upside or downside occur with equal likelihood. In that case, the null hypothesis
posits the chance of upturns to equal 0.5. In the absence of further information or
supposition, the alternative premise entails a two-tailed test in which the critical regions
for rejection lie on the low side as well as the high end.
In the current study, the number of successes—that is, the turnups in price—came out
to 266 out of 420 trials. The ratio amounted to a little over 0.633 which exceeded the
anticipated value of one-half.
More to the point, the assay rendered a p-value of 5.108x10–8. The latter value fell well
below the usual threshold of 0.05 that serves as the upper bound for rejecting the null
hypothesis. As a result, we could eject the premise that the movements of the market
were thoroughly random. That is, the Idle Model contradicted the data at a high level of
statistical significance.
In the case of the Drift motif, the best forecast at each stage reflects the fixed uptrend
over the long range. Other than that, the sloping template resembles the idling motif. In
particular, the odds of leading or lagging the trendline are deemed to be equal. For this
reason, the null hypothesis presumes the fraction of positive residuals to be one-half.
In actuality, the number of upturns equaled 230 out of 420 cases thus yielding a quotient
of some 0.548. The latter fraction strayed by a moderate amount from the expected
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value of 0.5. Moreover, the p-value of 0.05691 fell just above the threshold of 0.05 for
rejecting the null hypothesis. As a result, the Drift mockup lodged within the realm of
credibility.
In brief, the market displayed a small tendency to rise rather than fall in relation to the
prolonged uptrend. Despite the imbalance, a binomial test of directional moves showed
the Drift motif to be more or less compatible with the data.
We now proceed to the Sway Model. The number of successes, or miscues to the
upside, equaled 204 out of 420 cases. The resulting quotient came out to 0.486.
Moreover, the p-value of 0.5915 proved the null hypothesis to be wholly credible. In
other words, the Sway Model pinned down the midpoint of the movements in the
following sense: the implicit forecast was about as likely to overrun as to undershoot the
actual return going forward.
We have just assessed the pricing models in terms of the fraction of correct calls. In
particular, we proved that the Idle gospel runs afoul of the reality. Moreover, the Drift
template was somewhat plausible from a statistical stance. Best of all, the Sway Model
turned in trustworthy results; the seasonal template bisected the prospects so well that
the fraction of positive errors came much closer to the expected value of 0.5.
To recap, we examined the fitness of the candidate models based on the frequency of
errors to the upside versus downside regardless of the extent of the flubs. Each
template was checked separately to determine whether or not it conforms to the
historical record. A different way to appraise the models is to compare their performance
against each other.
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Global Odds of Orientation
The trio of pricing models may be pitted against each other in a pairwise fashion. More
precisely, one template outranks another if its tally of miscues undercuts its rival’s.
According to the null hypothesis, the fraction of wins for any model against any other
should equal one-half. Even so, we may have good reason to believe that a given
model ought to outplay another; in that case, the alternative conjecture posits the
fraction of wins to surpass one-half.
As an initial step, the Drift template was compared against the Idle idol. The null
hypothesis presumed no difference in efficacy thus entailing a win ratio of 0.5.
In the previous section, we saw that the Drift makeshift was wont to outshine the Idle
mockup. In that case, a fitting premise for the alternative thesis avers that the former
should trounce the latter with a probability in excess of one-half.
The Drift Model in fact bested the Idle canard in 244 cases out of 420 trials, thus scoring
a win ratio of around 58.1%. Moreover the p-value of 0.0005272 proved the advantage
of the winner to be highly significant.
The results of the previous section also suggest that the Sway Model should outstrip the
Idle version. For this reason, the null hypothesis assumed no difference in accuracy
while the alternate premise favored a fraction in excess of 0.5. A rundown of the data
showed the Sway template to outclass the Idle motif in 244 cases thus amounting to a
hit rate of 58.1%. Moreover, the p-value of 0.0005272 marked a stout level of statistical
significance.
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The next checkup involved a similar assay for the Sway framework against the Drift
mockup. The former beat the latter in 55.2% of the cases and yielded a p-value of
0.01788. In this way, the cyclic template outran its trending rival at a respectable level of
statistical clout.
To sum up, the Drift makeshift outranked the Idle icon in terms of the odds of rising and
falling. Better yet, the Sway template outflanked both of the Efficient takeoffs with
remarkable consistency. The advantage of the seasonal framework prevailed at ample
levels of statistical significance throughout the timespan covered by the study.
As we saw earlier, the directional moves of the market were by themselves enough to
expunge the dogma of perfect efficiency. Moreover, this section has rendered similar
results based on the performance of the pricing models against each other.
From a larger stance, these types of appraisals may also be performed on smaller
portions of the annual cycle. An example involves a marked tendency of the market to
falter during the summer then advance in the autumn.
Odds of Intra-year Waver
In the previous two sections, the monthly moves of the stock market were canvassed
across the entire timeline without paying heed to the miscues by month over the year.
On the other hand, we may perform similar assays of the relative performance of a
couple of months within the year.
For a specific example, we return to Fig. 5 which presents the boxplot of returns for
each month of the year. From this chart, we infer that the median value in the tenth
boxplot is higher than that of the eighth thumbnail. In other words, the median return for
October outranks that of August.
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These two months in fact turned in the best and worst scores on average by way of
median returns. As a result, August and October comprise the top candidates for
confirming the gap in performance between different portions of the year.
Two Samples Without Coupling
For the first test of intra-year movements, we examine the two months independently.
That is, we check whether each month displays a notable tendency to rise or fall
regardless of the behavior of the other month during the same year.
The null hypothesis presumes that August and October display equal odds of rising or
falling. For a change of pace, we consider an inquisitor who is so entranced by the
Efficient scripture as to deny the mass of evidence thus far. The fiend asserts instead
that neither month flaunts any advantage over the other. In that case, the alternate
hypothesis entails a two-tailed test.
We now check the difference in performance by way of a binomial test. For starters, we
find that October climbed higher in 25 out of 35 years thus notching a win ratio of
71.4%. The corresponding value for August was 45.7%. The mismatch between the
foregoing rates displayed a p-value of 0.003298. In this way, October prevailed against
August at a staunch level of significance.
In this assay, we examined the independent prospect of each month rising or falling
regardless of the year of occurrence. The checkup took the form of a test for
proportions.
As an aside, we note that Fisher’s exact test would have been an apt method if we were
checking for the consistency of directional moves. In the latter case, the query would
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take the following form: Is the market prone to rise in October when August does
likewise, and vice versa?
On the other hand, our concern here is more pragmatic: whether October tends to beat
August rather than move in the same direction. Hence the decision to employ a test for
proportions.
From a different slant, we may also check the ratio of paired returns on a yearly basis.
In other words, we want to determine whether October was likely to outshine August in
the same year.
Paired Samples by Year
The latest workout calls for a binomial test of matched pairs. A rundown of the data
showed October to eclipse August in 25 out of 35 years. The upshot was a win ratio of
71.4%.
A test of paired samples differs somewhat from a checkup of unpaired groups as
described in the previous subsection. In the latest grilling, the p-value for the paired test
came out to 0.008337. While the turnout was not as stringent as the prior result for the
unpaired samples, it was highly compelling even so.
In the larger scheme of things, the ratio of wins notched by a given month against
another may well differ in the case of a paired sample as opposed to an unpaired batch.
Naturally, the distinction also applies to the ensuing levels of statistical significance.
In the foregoing contest of October versus August, the peppy month prevailed in 25
cases on its own; and exactly as often in a pairwise matchup against the droopy month.
Although the number of wins for October happened to equal 25 in both assays, there
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was no preordained reason for the scores to be identical. In keeping with the distinct
themes and objectives, the significance levels did in fact differ between the tests for the
paired sample versus the unpaired grouping.
To bring up a second factor, the checkup of independent samples involved a two-tailed
test under the premise of perfect efficiency. By contrast, the paired grouping was
consigned to the strictness of a one-tailed test as a way to acknowledge the domain
knowledge uncovered thus far. Both types of appraisal yielded hefty levels of statistical
import in confirming the vantage of October over August.
From a larger stance, other combinations of assumptions may be put to the test in the
future. Given the strength of the conclusions to date, however, it seems safe to surmise
that the Efficient dogma will in general be annulled with ample assurance in other
cogent studies to be conducted down the line.
Special Features of the Assay
The foregoing forms of appraisal differ from the conventional method for vetting the
disparity of a couple of groups. More precisely, the usual practice relies on a rough-and-
ready scheme based on the t-test along with its iffy assumptions. All too often, though,
the constraints behind the latter probe are invalid in the real world. Even so, the disdain
for inconvenient facts is spotlighted by the mass of studies trained on the financial forum
including the stock market.
To be precise, a t-test of unpaired samples stands on a couple of shaky premises: the
data from both groups must be normally distributed as well as display the same
variance. In a similar way, a t-test for a matched sample assumes that the difference
between each pair of observations conforms to a Gaussian ogive.
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By contrast to the usual spree of hand waving and windy denial, the procedure followed
here has invoked the binomial test. For the current study, the only requirement to speak
of concerns the independence of movements.
From a different angle, the latter condition forms the pith of the Efficient doctrine. For
this reason, the premise of independence is not merely admissible but in fact obligatory
for an assay designed to evaluate the orthodox canon.
In short, the duplex framework does not need extraneous crutches. To cite a
counterexample that runs riot in the literature, the t-test draws on a false image of the
stock market; namely, the hegemony of the normal ogive.
On the bright side, though, the binomial test upholds a sound alternative along with an
honest approach to modeling. In this formula, the lack of arbitrary constraints plays a
vital role in the leanness, elegance and robustness of the procedure.
To wrap up, an appreciable gap in performance arose between certain months of the
year. In particular, October was prone to outpace August of the same year at a
persuasive level of statistical clout.
In the previous subsection, an independent sampling of the divergent months also
confirmed the lopsided odds of rising versus falling when the moves were not paired up
by year. In these ways, the existence of a noteworthy gap between a couple of months
was by itself enough to scrub the whitewash of efficiency at an ample level of
assurance.
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Conclusion
Duplex models can portray complex systems with the utmost of simplicity, clarity and
efficacy in areas ranging from the dearth of premises to the strength of conclusions. The
advantage of the binomial scheme shows up, for instance, in purging the dreck of myths
and misconceptions in the financial forum. A showcase involves the existence of
seasonal patterns behind the monthly moves of the Dow stock index.
According to the dominant creed of financial economics, the real and financial markets
are so nimble as to absorb every scrap of information with startling speed and react at
once with perfect rationality. A direct outgrowth of the Efficient voodoo is the Random
Walk spoof that pictures the market as an erratic system immune to prediction except
perhaps as an occasional fluke. Another offshoot lies in the mantra of buying and
holding a dicey asset forever as an unbeatable strategy for investment.
Unfortunately, the charade of efficiency flouts the facts of life in plenteous ways. To
underscore the gulf between the illusion and reality, the current study has employed
only a minute fraction of the wealth of resources available to the general public.
If we look at the big picture, the duplex framework showcased here is the plainest way
to model shifty systems regardless of their complexity. The crux of the methodology lies
in the binomial test: the most basic tool in statistics for checking presumptions and
drawing conclusions. The two-tone method offers a medley of benefits ranging from the
clarity of concepts and ease of usage to the soundness of inference and sureness of
conclusions.
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Further Resources
A useful resource for investors of all stripes, ranging from newcomers to veterans, lies in
a Web site maintained by MintKit Institute (mintkit.com). The portal presents a wealth of
articles, books and other materials dealing with the challenges and opportunities that
await earnest investors bent on sound growth in a global marketplace.
For instance, an article at the gateway provides a compact survey of the fables and
bungles that hamper the bulk of amateurs as well as professionals in the financial
arena. The write-up is titled, “Myths versus Mistakes” (MintKit, 2020).
The duplex framework for modeling complex systems is described in detail in an ebook.
A summary of the guidebook is given in a blog post at Kenwave (2020). The same blurb
provides links to sites where the full publication is available as a PDF file.
Supplementary Materials
The statistical assays reported here were encoded in the R language. The resulting
program has been uploaded to GitHub, an open repository of virtual resources on the
Internet. The R script as well as the raw data are available at the following locale:
“github.com/mintkitcom”.
The program resides in the Code folder as a document in txt format under the title of
DjiSeasonMonthPeg. Meanwhile the dataset abides in the Data node as a file in csv
form under the rubric of DjiMonth1985-2020.
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References
Fama, E. The Behavior of Stock Market Prices. J. of Business, v. 38(1), Jan. 1965: 34–
105. http://doi.org/10.1086/294743.
Kenwave. Duplex Models of Complex Systems. 2020.
https://w.kenwave.com/2020/08/duplex-models-of-complex-systems.html.
MintKit. Myths versus Mistakes: Riot of Passive Muffs and Active Goofs in Financial
Markets. http://www.mintkit.com/myths-vs-mistakes.
Yahoo Finance. Dow Jones Industrial Average (^DJI) – Historical Data.
https://finance.yahoo.com/quote/%5EDJI/history?p=%5EDJI.
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ResearchGate has not been able to resolve any citations for this publication.
Duplex Models of Complex Systems
  • Kenwave
Kenwave. Duplex Models of Complex Systems. 2020.
Myths versus Mistakes: Riot of Passive Muffs and Active Goofs in Financial Markets
  • Mintkit
MintKit. Myths versus Mistakes: Riot of Passive Muffs and Active Goofs in Financial Markets. http://www.mintkit.com/myths-vs-mistakes.