Informational relatedness
among
the broad indices
:
The context of extreme return clusters
B
y
Anh Phuong Nguyen
*
Economics Department
New Mexico State University
Las Cruces, New Mexico 88003
T. Harikumar
Finance Department
New Mexico State University
Las Cruces, New Mexico 88003
Jayashree Harikumar
Physical Sciences Laboratory
New Mexico State University
Las Cruces, New Mexico 88003
_________________________
*
Corresponding author: email:
phuonga@nmsu.edu
. We thank the panel of reviewers in the following
conferences: 9
th
International Business Research Conference
, Melbourne, Australia 2008,
Sponso
red by
World Business Institute,
Business Economics Institute Conference, L
as Vegas Nevada, USA
, and
International Conference on Business, Economics and Information Technology, Nagoya 2009,
sponsored
by
University of Guam, School of Business and Public Administration (Guam)
,
Penn State Altoona,
Division of Business and Engineeri
ng (Pennsylvania)
,
Nagoya University Graduate School of Economics,
Economic Research Center and KITANKAI and DAIKO FOUNDATION (Japan)
.
2
Informational relatedness
among
the broad indices
:
The context of extreme return clusters
Abstract
Financial mark
et participants are always faced with having to make decisions
based on a continual flow of information. The aggregation of these decisions is
reflected in broad indices such as DJIA, S&P 500 and NASDAQ. This research
studies clusters of extreme returns (i
.e, returns exceeding the 90
th
percentile) in
each of the above indices over a period of 1960

2008. We modify the algorithm
advanced by Laurini (2004) to detect these informationally independent extreme
return clusters. We back test the algorithm and find
that it accurately detects the
era of terrorism and the subprime crisis. Using this algorithm we detect and
extreme return clusters find that the information that drives DJIA and S&P 500
are similar and is likely to be a superset of what drives NASDAQ. The
se indices
show relatively asynchronous behavior as they differ in terms of when they enter
a cluster and when they leave the cluster. We suggest a returns based and a
volatility based strategy to benefit from cross

index predictions of entry to and
exit f
rom a cluster.
3
Informational relatedness
among
the broad indices
:
T
he context of extreme return clusters
1.
Introduction
In recent years, we have witnessed an increasing level of integration in the global
financial markets. A hint of a r
ecession in the US markets affects
returns on
the Nikkei,
Hang Seng,
DAX, FTSE 100 and many more. More recently, we have witnessed the
connectedness among markets since the sub

prime crisis started unfolding in the US
financial system. The failures of lar
ge investment banking houses, the failures of giant
mortgage companies, the foreclosures of properties, the credit crunch have not only
caused a climate of gloom in the US markets but has sent reverberations around the
global markets in Japan, Asia and Eur
ope.
In contrast, the
markets around the world
also
r
eact
to positive information such as bailouts
,
ta
keovers of failing institutions, Federal
Rese
rve cutting interest rates, etc, leading to investor euphoria.
Such
global
information
also
affects
domestic
indices
such as Dow Jones
Industrial Average (DJIA), Standard and Poor’s 500 (S&P 500) and National Association
of Securities Dealers and Automated Quotes (
NASDAQ
).
1
This high correlation indicates
that some underlying information affects all three indice
s is similar manner.
While,
i
nvestors typically chase high re
turns for a given level of risk,
the correlation coefficient
does not provide them with
timely
information on whether the
prevailing
investing
climate is
conducive for achieving their goals
.
We
realize that i
nvestors would b
enefit
greatly if they discern timely
informa
tion that can drive retur
ns beyond a certain
high
1
We find that the correlation between pairs of returns for these three indices is highly significant at the 1%
level.
4
threshold.
Consequently
, we characterize the type of information that potentially yield
such extreme returns
and thus provide inves
tors with relevant information
.
The unit of study in this paper is an
extreme return cluster
that is driven by an
information set that is independent of information related to other clusters.
The beginning
of an extreme return cluster is marked by investo
r’s reaction to that information
set and
the end is determined by
this information
being fully processed and digested. W
e
implement
the
algorithm in
Laurini (2004) to determine the ending of a extreme return
cluster
and
along the same lines develop a modi
fied version of
Laurini
’s
algorithm to
determine the beginning of a
n
extreme return cluster.
This
research examines DJIA, S&P
500 and
NASDAQ
by
partition
ing
the time

series of index return
s
into clusters of
ex
treme returns.
2
We find several of interestin
g results
on
the way the three indices react to
information that drives return beyond
a
threshold. We find that at least 90% chance
that
S&P 500 and DJIA have overlapping clusters
. The cha
nces are lower at 75% for
NASDAQ
. Next, we align clusters that are c
ommon across indices and find that DJIA
and S&P 500 cluster are normally distributed from 0% to 100% of overlap days. The
distribution
of cluster in
NASDAQ
is biased towards a higher percentage overlap days. A
pairwise analysis of common cluster across ind
ices suggests that DIJA and S&P 500
clusters are likely to be driven by the same information and such information is superset
of information that drives clusters in
NASDAQ
. Next, we turn to examine the presence of
lead/lag in entering and/or leaving of clu
sters. For instance, we find that if
an investor
observes
that
NASDAQ
is in cluster, he or she can expect DJIA to enter cluster in about
2
Galbraith and Zernov (2006) examine the conditional volatility and find that the Nasdaq
and S&P 500
indices exhibit very similar values. However, when they focused on extreme dependence they find that
Nasdaq revealed more extreme dependence than S&P 500.
5
12 days. For the ending of the cluster, we find there is roughly 12 to 14 days lag from the
time DJIA and/
or S
&P 500 an
d
NASDAQ
following suit. We employ a return strategy to
predict lead/lag at the starting of a cluster and a volatility strategy for the ending.
The rest of the paper is laid out as follows.
Section 2 contains the motivation for
this research. Section 3
reviews the related literature in this area. We discuss a variety of
cluster detection algorithms and argue that the algorithm advanced by Laurini (2004) is
the most appropriate for our study. We apply the algorithm to index returns from DJIA,
S&P 500 a
nd
NASDAQ
. Section 4 contains the specifications of the GARCH model we
employ to estimate conditional volatilities. We then present the details of the algorithm
that detects the starting point and the ending point of an extreme return cluster.
These
algor
ithms use the time

series of returns and conditional volatility and mark starting and
ending points of informationally independent extreme return clusters
.
Section 5 shows
how the algorithm detects real

life informationally independent clusters such as the
period of terrorism (
9/2000 to 8/2003) and the advent of the
sub

prime crisis (
7/2007
until the present date).
Section 6 contains the results of applying this algorithm to the
time

series of returns and volatilities for DJIA, S&P 500 and
NASDAQ
.
Section
7
contains our
concluding re
marks
.
2.
Motivation
As obvious it might seem, we ask w
hat
independent information is.
Consider the
sub

prime credit crisis that began around January 2007. We are continuing to experience
the impact
of related information till th
e time this article is being written
. This type of
information is quite distinct and even independent
of the period subsequent to (say) the
beginning of the
period of terrorism and Iraq war around September 2001 and March
6
2003
.
In this sense, the sub

prim
e era contains in
formation that is independent of
the
information
during the
period of terrorism
. These periods characterize informationally
independent
return
clusters.
The data allows us to back

test the detection technique,
since we know the calendar t
ime
during which
these independent information sets were
prevalent.
With a remarkable degree of accuracy, we show that the methodology we
implement detects the calendar dates associated with the
terrorism
period and the sub

prime crisis period.
Although
a
return
cluster
is caused by
its associated information set, does its
character change depending on the type
of
independent information?
It is conceivable
that subsets of independent information sets share the same characteristics. For example,
consider
the following pieces of information: Fed rate cuts, Japan’s intention to buy
Merrill Lynch, Congress passing the bill to infuse $250 billion
into the banking sector,
etc. These pieces of information are a subset of the broader information set related to t
he
sub

prime crisis. However, t
hese are pieces of information that are independent of each
other and yield extreme positive reaction by investors
.
We realize
the importance of
understanding and delineating
information
that yields extreme returns. By usin
g the
detection technique mentioned above, we find it possible to detect and characterize such
information by choosing a
parameter
triplet including a return threshold, a volatility
threshold and a length of period denoting related information.
We recogn
ize
that
a certain set of parameters could provide the characteristics of
ext
reme information,
and once extreme information clusters are detected for each index,
we go
further and
ask
whether this information is the same (or partially overlapping)
7
across t
he three indices
.
This is important for an investor to know as it is sufficient to
trade in one index if the same information set affects others,
as well.
It is likely that t
he information set that drives a
n extreme return
cluster in one
index is either s
ame or a subset of information that affects another index
. If this is the
case, this situation will pose some very interesting questions; a) Do the indices react to
new information synchronously i.e., do they enter an extreme return cluster at the same
ti
me? b) Can a typical investor observe the index return and realize whether the market
has begun processing new information that might potentially yield an extreme
return?
c)
If an investor realizes that an index is in a common extreme return cluster, can h
e or she
determine how soon another index will follow suit? d) Can an investor benefit from a
situation of synchronous and/or asynchronous index reaction to new information? T
hese
questions have not yet been addressed in prior research. We find these qu
estions to be
very interesting and motivat
e us
to pursue this line of research.
3
. Literature
review
Researchers in this area have taken a variety of approaches to identify
informationally independent clusters.
The character of each cluster depends on th
e way
information
is defined.
Volatility c
lusters based on
volatile and non

volatile
periods
Hovsepian, Anselmo and Mazumdar (2005)
classify clusters based on rel
atively
volatile and
non

volatile periods.
They
implement a three

step approach to detect
volatility clusters. In the first step, they consider a time

series data of currency returns
and fit a GARCH model to compute conditional volatilities. Next, they classify the
volatility series into relatively volatile and relatively non

volatile segments
based on the
8
difference in the changes in the conditional variance as compared to the overall variance
in a time

series segment using a χ
2
test. This information on partitions of relative
volatility periods is the basis of a machine learning technique cal
led support vector
machines (SVM), which then forms the basis of cluster detection.
3
There are many limitations with this approach. First, using 10 observations to
designate a volatile period seems arbitrary and restrictive. Volatility clusters can range
from a few weeks to several months. Secondly, the paper does not explain whether the
learning by SVM is similar to what investors would learn in actual markets.
Additionally, if time segments are artificially kept the same length to be acceptable inputs
to SVM, it is not necessary that individual investors should be doing the same.
Since the
volatility clusters are determined somewhat arbitrarily, it does not ensure that the
information set associated with a cluster is independent of another.
Further, wh
ile this
method detects v
olatility clusters, it is not suited for our study that aims to detect extreme
return clusters
.
The restrictive nature of the technique
also
does not allow us to study how
long it takes the market to digest the information.
Extrem
e return c
lusters based on
blocks
and runs of
returns
Leadbetter (1983) proposes a method
to detect extreme returns cluster. This
method
divides the sample of
T
observations
into
k
blocks of size
r
, where
r
=
T
/
k
and
computes the extremal index
equal to the ratio of the number of blocks showing at
least one value in excess of a threshold to the total number of ‘exceedences’
of the
threshold on the sample. The
greater
the cluster size, the lower the value of
an
d the
3
The length of the relatively volatile period is not the same as the non

volatile perio
ds. Since the SVM
methodology processes only time segments of equal length, the data is pre

processed using a periodogram
which returns the power spectrum density estimate of each segment.
9
more the dependence in the sample. O’Brien (1974) develops
a
runs
method
of defining
extreme return
clusters by using a given number of observations below a threshold to
separate clusters. The method requires specification of a de

clustering paramet
er
r
as the
number of consecutive observations below a threshold required to define separation.
In the case of blocks estimation, these clusters are specified
a priori
by the block
size parameter. In practice, it is more than likely for sequences of indep
endent
information t
o permeate through the market in
varying duration. A fixed grid of blocks
does not correspond well with the pattern of randomly occurring, random length periods
of clustering typically observed in financial data.
As a result, we do not
implement the
block method of detecting
extreme return
clusters.
On the other hand, the runs method only focuses on return patterns and does not
consider movements in volatility. When new information enters the market, we observe
an impact on price level
s and changes in prices.
Mandelbrot (1963) observes that clusters
occur when large changes tend to be followed by large changes, of either sign, and small
changes tend to be followed by small changes.
4
Hence, we do not use the runs method by
itself.
Extre
me return c
lusters based on returns and volatility
We use the
two

threshold
detection method advanced by Laurini (200
3, 2004) and
define an independent extreme return
cluster in terms of movements in returns
and
volatility. We find this approach appealing
as it is motivated by the empirical findings
4
Owyang (2001) finds that volatility appears in clusters directl
y after changes in inflation. He also finds
that periods of high (low) mean level of inflation correspondingly has a high (low) variance of inflation.
Lux and Marchesi (2000) attempt to provide a reason for cluster formation based on a model with chartist
s
and fundamentalists. An outbreak of volatility is shown to occur if the fraction of agents using chartist
techniques surpasses a certain threshold.
10
exhibited by financial returns. For instance, when a cluster of extreme observations is
ended, the volatility drops down to a suitable level. Th
is marks the end of a period of
influence of a certain type of inde
pendent information
and causes a
cluster
to terminate.
This method of cluster detection
lends itself to certain modifications to be
able to address
all the questions we raise in the earlier section
.
4. Data, Model and Parameter Estimation
Estimation of
conditional volatilities
In order to implement the two

threshold method,
we need a time series of returns
and a corresponding time

series of volatility
for each index
.
It is well known that
financial returns series exhibit statistically significant serial
dependence in return
volatility. This phenomenon was first modeled by Engle (1982) as an autoregressive and
conditional heteroskedasticty (ARCH
) process with lagged residuals and generalized by
Bollerslev (1986) to estimate conditional volatilities as a f
unction of lagged residuals and
lagged volatilities
(GARCH)
.
5
We consider DJIA, S&P 500
and
NASDAQ
indices in our analy
sis of
independent
clusters.
In the case of DJIA and S&P 500 we use the time

series returns from 1960 to
2008. For
NASDAQ
, we use returns
for the period
1980 to 2008.
We estimate
t
he
conditional volatility usin
g GARCH (
1, 1
) process using
t
h
e following system of
equations
6
7
;
5
Johnston and Scott (1999), Hsieh (1988), Kugler and Lenz (1990), McCurdy and Morgan (1988) and
Taylor
(1986) support an autoregressive and conditional heteroskedasticity (ARCH) and a GARCH type
process. The study by Fujihara and Park (1990) finds that three out of the five currencies they study
support the ARCH model. These effects are important and have
been incorporated in option pricing models
(Duan (1995), Heston and Nandi (2000)).
6
For a thorough review of ARCH/GARCH modeling in finance see Bollerslev, Chou and Kroner (1992).
7
One might find this to be a limitation of our study. However, consider the study by Hansen and Lunde
(2001) where they use the DM

USD exchange rate data and compare the forecasting ability of 330
GARCH

type models. They find that none of the models beat
GARCH (1,1) in its’ forecasting ability.
11
where
,
X
t
denotes index returns,
μ denotes the average returns,
σ
2
t
denotes the conditional
volatility at time
t
,
ε
t
~
N
(0,σ
2
t
),
α
> 0,
β
j
≥ 0, and
λ
i
≥
0
.
8
The results of this procedure
are presented in Table 1.
A casual glance at the coefficients in Table 1 indicates that it
would no
t be proper to use one estimate for the entire period from 1960

2008.
[
Insert Table 1 here]
Endpoint
detection algorithm
Laurini’s (2004) examines
extremal index
and estimate
s
this parameter
to
equal the
ratio of the number of exceed
ances of extreme values to the number of the independ
ent
extreme return clusters
. While emphasizing the fundamental importance of
indentifying
independent
extreme return
clusters, Laurini suggests that estimating
is equivalent
to
id
entifying
such
independent clusters. He proposed the two threshold method to
calculate
. This method is shown to yield stable range of values for
thus indicating
the presence of
independent clusters. Laurin
i showed that the two thresholds approach
performs better for stochastic volatility and ARCH (1) processes. Using the Monte Carlo
experiment, Laurini (2004) found realistic the hypothesis of normality of the two
threshold estimator thus emphasizing the va
lidation of this new metho
d for indentifying
independent extreme return
clusters.
T
he two

threshold method of det
ecting clusters is as follows.
L
et {
X
t
} denote a return
series and {σ
t
} denote the conditional volatility series. Let
u
denote the threshold
for the
8
As we study daily returns for long periods, it is likely that the parameter estimates might experience
changes or shifts over time. To address this problem, we arbitrarily partition the data into 10 year period
s
and estimate the above set of equations for each sub

period.
12
{
X
t
} process and
c
denote the threshold for the {σ
t
} process and
m
denote the length of a
run.
The following
conditions
yield the
cutoff point
t
*
that define
s
the endpoint of
an
extreme return
cluster
:
1.
I
f
X
1
>
u,
Max (
X
2
, …,
X
m
) ≤
u
, and
Min (σ
2
,…, σ
m

1
) >
c
,
then
t
*
= m .
2.
I
f
X
1
>
u
,
Max (
X
2
, …,
X
T
) ≤
u
,
Min (σ
2
,…, σ
T

1
) >
c,
and σ
T
≤
c
,
then
t
*
= T
where
9
We illustrate these conditions in
F
igure (
1
).
[Insert Figure (1
) here]
T
he end
point of an informational cluster is determined
by the behavior of
index returns
and
its volatility of an
index
.
Condition 1 is depicted in the first set of returns and
volatility figures and condition 2 is illustrated in the second set.
For each condit
ion, the
X

axis denotes time in days and the Y

axis denotes returns (upper figure) and volatility
(lower figure).
The observations on returns and volatility movements prior to day 1, in
each of the figu
res, represent the in

cluster status of an index be
fore it’s ending.
Condition 1
depicts
the case where
t
he returns stay lower than the threshold
u
period of
m
days
and marks the end of the cluster
even though volatility does not d
ecrease
.
The
endpoint is determined by the runs method of returns and th
e pattern of volatility does n
ot
signal the end of a cluster.
In c
ondition 2,
the drop in returns is accompanied by a drop in
volatility
prior to
m
days,
thus indicating a definite change in the informational regime.
In this case,
the endpoint point is
marked as
the date on which the volatility dropped
9
We implement this algorithm using MATLAB and will furnish the code upon request. See Laurini and
Tawn (2003) for the choice of
u
,
c
, and
mi
.
13
below the threshold
c
, at any time T
< m = 50.
The above algorithm
yields
endpoints
of
clusters that help partition the time

series of
returns into
informationally independent
clusters.
Detecting th
e endpoint is not sufficient to determine if investors process
information synchronously. I
f
we can identify both the starting point and the ending of an
extr
eme return cluster, we can compare the overlap across indices and analyze their
synchronous
and le
ad/lag
behavior.
L
aurini (2004)
det
ects independent clusters
on the
assumption that the pool of extreme information in one cluster must be able to drive the
returns to an extreme level
and/
or generate investor activity at a certain level of volatility
for
a brief period
.
Based on this assumption, t
he investor
activity
p
rior to the occurrence
of the first ext
reme return event
can potentially trigger the start of a cluster
.
W
e
use this
idea
to detect the starting point of an independent extreme information c
luster.
Startpoint detection
algorithm
According to the endpoint algorithm in Laurini (2004), t
he period after the
endpoint
of an extreme return cluster
marks the beginning of a new cluster.
I
n practice
,
however, this need not define a
starting point
a
s the
investor activity
during this period
may not generate
an
extreme return event.
W
e
find it
appropriate to adapt the reasoning
in Laurini’s endpoint algorithm to detect the starting point. Hence, once an endpoint is
detected and
a cluster ends,
we sugg
est that
the starting
point of a
new
cluster is
marked
by
the occurrence of the first extreme return event
resulting from an increased level of
investor
activity
in response to new and independent information.
T
he following
algorit
hm
shows that a
starting
point
is detected
by
observing the returns and volatility
path leading
up
to the
time the returns
first cross the threshold. More specifically, The
14
following conditions yield the cutoff point
t
* that defines the starting point of an extreme
return cluster
:
1.
I
f
Max
(
X
1
,
X
2
, …,
X
m

1
) ≤
u,
Min
(σ
1
, σ
2
, …, σ
m

1
) >
c
,
and
X
m
>
u,
then
t
*
= m .
2. I
f
Max
(
X
T

1
,
X
T
,
X
T+1
, …,
X
m

1
) ≤
u,
σ
T

1
≤
c ,
and
Min
(σ
T
, σ
T+1
, …, σ
m

1
) >
c,
and
X
m
>
u,
then
t
*
= T
where
Th
ese conditions are depicted in Figure (2
).
[Insert figure (
2
) here]
The first set of returns and volatility figures pertain to condition 1 and the second set
depicts condition 2.
For each condition, the X

axis denotes time in days an
d the Y

axis
denotes returns (upper figure) and volatility (lower figure).
The value of
m
in the x

axis
denotes the first time returns exceed the threshold
u
after the previous cluster ended.
The
return and volatility observations to the right side of
m
denote the in

cluster status of the
index. In the figures for c
ondition 1
, we find that the returns are less than the threshold
u
for
m
prior days and the volatility is greater than its threshold
c
. We interpret this
situation as a case where investors p
rocess information and react to such information
causing volatility to be greater than
c
and pushing returns beyond the threshold value of
u
,
thus
resulting in an extreme value.
The figure
s for condition 2 show
that
the starting
point is indicated by the
instant volatility first exceeds the threshold
c
and remains above
the threshold till
X
m
>
u
for the first time on date
m
. Again, it is the timing of the
volatility shift that is presumed to cause an extreme event to occur. These two conditions,
15
give some
idea about the starting point of an extreme value cluster.
In what follows, we
illustrate the role of the parameters
u
and
c
in the cluster detection and relate the choice
of these parameters to empirical phenomena.
Choice of the parameters u, c and m in
determining independent clusters.
Recall that
the return
s
threshold
is denoted as
u
, the volatility threshold
is denoted
as
c
, and
m
is a parameter
that denotes the length of period of related information
.
The
value of
u
determines
a pool of
return
clus
ters
with returns exceeding
u
.
A
high
value of
u
focus
es
on
information that generates extreme returns
.
A
low
value of
u
tends to allow
returns driven by
information that yields
extreme and n
on

extreme
returns
.
Thus, t
he
choice of
u
determines
clusters ba
sed on the return reaction and the associated
information that causes it.
The
level of
u
determines
the type of information
set,
and the
characteristics of such
information, in turn, determine
the p
arameter
s
m
and
c
.
The parameter
m
denotes the
maximum n
um
ber of days that extreme
information is related
. Laurini (2004) suggest
s
that the value of m equal to 50 days is reasonable.
This implies
that any two extreme
return events within a cluster are
driven by related information
provided that
these events
are
separated by fewer than 50
days
. Additionally,
during these days, the related
information within a cluster surrounding the extreme event sustains an appropriate
threshold
of volatility
(
c
).
The volatility process indicates investor activity in the market
.
Clusters that include
less

extreme return values (or, more from the middle of the distribution) are more likely
to be associated with lower levels of volatility, due to (possibly) a limited surprise
element in its associated information. In contrast,
the
informatio
n that results in extreme
16
returns
could
be expected to
have
an
aggressive effect on the investor’s behavior.
Thus,
if one is interested in characterizing pools of clusters containing extreme return values, it
would be reasonable to consider a h
igh level of volatility
as
is likely to induce such
clusters.
W
e argue that the
value of
c
needs to be
chosen in combination with
the value of
u
.
For example, a
high value of
u
(say 90%)
is likely to be
consistent with a high value of
c
(say 75%) where
su
ch clusters include
extreme
return
observations
.
5. An Illustration of the algorithm
We consider
an eight year
period
from
2000 to the present date
to illustrate how
the
algorithms
detect informationally independent clusters.
While there are many cluste
rs
in t
his period
, there are
two distinct
ly long
clusters
;
A
period
of terrorism
(
9/2000 to
8/2003
)
and the period of
s
ub
prime crisis (
7/2007 until the present date
). It is easy to see
that the information content that affects the index returns during the
Iraq war is
completely different and
independent
of the inf
ormation content during the sub
prime
crisis period. Additionally, each of these informationally independent clusters contain
s
clusters of extreme values that are driven by independently arriving
pieces of information
that continue to be related to the overarching cluster.
Recall, that the choice of parameter values for the algorithm depends on the type
of cluster one intends to study.
We
are interested in
studying
the type of independent
informa
tion that
results in extreme
returns
. Hence, we choose the parameter
u
to be 90%.
W
e define
m
to be 50 days
based on Laurini (2004).
Finally,
the value of
c
is critical in
determining the fraction of distribution to include in the cluster and the associat
ed
underlying information related to the cluster.
I
f w
e choose
a low value of
c
(say 25% )
relative to
u
(75% or 90%)
,
the algorithm
detect
s
clusters that include coarse information
17
that relates to a larger fraction of the return distribution (i.e., contai
ns extreme and non

extreme values).
We term these clusters as indicating an ‘
Era
’ of information that
does
not delineate extreme return events.
Consider fig (
3
) below.
[Insert
f
igure (
3
)
here
]
Figure (
3a
) depicts clusters based on the triplet
{75%, 25%,
50}
and fig (
3b
)
presents clusters based on {90%, 25%, 50}.
Although, these sets of parameters attempt to
detect different types of extreme return cluster
s, the algorithm picks the same
Era
of
information for the value of
c
= 25%.
For instance,
using the
DJIA index,
the algorithm
picked a terrorism era from September 18, 2000 to August 28, 2003 and the
S
ubprime
crisis
era
from July 11, 2007 until the present date.
There are many extreme return observations that are embedded in the each of the
eras charact
erized by their respective independent information.
To extract these extreme
return clusters, we
choose the triplet {90%, 75%, 50}. As mentioned earlier, the choice
of 75% as a
reasonable
threshold for volatility allows us to
isolate
a subset of
investor
activity that has the potential to
make
returns
exceed
the 90% threshold. Figure (
3c
)
shows that the
extreme return clusters not only belong to their respective era but have
their own
associated
independent
information.
For example,
visually
there are
9
extreme
return observations
(
with
more than
3
days)
embedded in the terrorism era and
4
extreme
event observations during the subprime era.
10
Focusing on the subprime era
,
we
study
two
extreme return
cluster
s
;
cluster
D
(
Januar
y 14, 2008 to February 19, 2
008)
and
cluster
E
(
March 11, 2008 to April 11,
2008
)
as depicted in figure (4)
.
To
better u
nderstand the information
drives
extreme
returns
in
cluster
D
, w
e
examined
business
news
such as CNN Money, Reuters,
10
The actual number of cluster, however, is 17 a
nd 7 respectively.
18
Marketwatch and Bloomberg
news
and find that
t
he
rate cut by federal reserve (from
4.25% to 3%)
on 1/30/2008
,
good news from J.P. Morgan about housing and Warren
Buffet’s $800bn plan to bail out
bond issuers suffering from the subprime crisis
(reported
on 2/13/2008)
are
some of the pieces of
(
subprim
e related
)
news that caused
the extreme
return to exceed the threshold.
Cluster E
is associated with a fed rate cut (from 3% to
2.25%)
on 3/18/2008, positive news and optimism surrounding the government role in
Freddie Mac and Fannie Mae
on 3/21/2008 and b
etter than
anticipated performance of
financials.
The
above illustration shows that
the cluster detection algorithm is
ab
l
e
to detect
extreme return
clusters
that have relevance
in
the
context of the real

world
.
In what
follows, we use this illustration a
s a motivation and study extreme returns clusters
in an
abstract manner without having to relate them to calendar dates.
6. Results
Th
e
aim of this study is to understand how the three broad indices are informat
ionally
related to each other.
We present t
he results in three subsections.
Initially, we
examine
the extent of overlap
between
clusters in each index
and present the results in the section
below titled
overlapping clusters
.
The greater the overlap in days between two clusters,
the more likely that
it is being driven by the same information set.
In this context, we
provide results on synchronous clusters where the clusters
common to two or more
indices
begin
on the same day
or
end on the same day.
The study of overlap
ping clusters
,
however, does n
ot
provide an understanding of
whether
the
investors trading in the shares
of
one
index
react to information sooner than
another index
. It is also interesting to know
if
the indices that lead also leave the clu
ster earlier thereby giving an idea about the
19
amount of time taken to digest an information set.
We present results for this part
in the
section below titled
‘
Lead/Lag in extreme clusters across indices
’.
Finally, w
ith these
results, we investigate whether an investor can
benefit from knowing that an
index is in
cluster and being able to use that information as signal about the movement in the other
two indices.
Overlapping Clusters
We employ the
cluster detection
algorithm
s
described earlier along
with triplet
{90%,
75%, 50
days}
and
compute
the starting and ending points of
extreme value
clusters characteriz
ed by independent information. We assume that
the return cluster is
significant
if an index remains in a cluster for
at least
10 day
s. That is, it takes
at least 10
days for
investors t
o
fully process the information
. Moreover, our research
examines the
impact of information across indices and
unlike a 2 day or a
3 day cluster where the
information set is a short burst, a 10 day interval
may be driven by information common
to
other indic
es also.
However, once we identify a cluster with at least 10 days duration,
we
compare this cluster with others around the same calendar time even if they have
fewer than 10 days duration. In this sense, we capture the impact of the
index containing
the
dominant cluster on
other indices.
The re
su
lts are presented in Tables (
2
),
(
3
)
and (
4
)
.
We identify 87 extreme value clusters in DJIA, 86 clusters in S&P 500 from 1960

2008
and 44 clusters in Nasdaq from 1980

2008.
Table (
2
)
shows
at least 90% chance
of overlap of cluster
s
belonging to
DJIA
and
S
&P 500. In contrast, the chance of overlap of cluster
s
between
NASDAQ
and DJIA is
[Insert Table 2 here]
20
77.3% and that with S&P 500 is 75%
.
Next we investigate if these overlap
clusters are
driven by the same
information.
Even though it seems that these indices may be driven by different information
sets, we present evidence that
indicates the presence of common clusters across all three
indices.
Table (
3
) indicates that
d
uring the period 1980

08
there are
54
extreme
return
[Insert Table 3 here]
clusters in DJIA and 52 in S&P 500 and 44 in
NASDAQ
.
Also,
there are
28
clus
ters
common to all the indices
.
We examine the
se 28
common clusters
and for each
index
we
compute
the percent of days in
the cluster that
ove
rlap
with the other two indices. For
instance, 7 clusters in DJIA has a 100% overlap in days
jointly
with the other two indices.
In the case of NASDAQ, 13 out of the 28 clusters overlap 100% with DJIA and S&P
500.
The table presents the frequency tabulat
ion of the number of clusters that
correspond to days overlap in percentage.
While the
frequencies in the table are only indicative of the extent of overlap, it
would be interesting to know if the distribution of actual percentage overlap days
for a
clust
er in a given
index is
skewed in any manner
compared to a benchmark distribution
describing the null hypothesis.
A priori
,
we
expect a lower probability of
occurrence for
extreme observations
of
the
variable percentage
overlap days and a greater probabilit
y
with
±
1
σ
f
rom
the mean.
Hence, we assume the
Normal distribution as the
benchmark
distribution and use the Kolmogorov

Smirnov (KS) test of goodness of fit. We perform a
one

tail test to see if the actual distributions are skewed in the direction of a g
reater than
‘normal’ overlap. Specifically, Let F(x) denote the cumulative distribution of the
standardized percentage overlap days and let G(x) denote the cumulative distribution of a
21
standard normal variable.
11
If the standardized percentage overlap days
is biased to
higher values we expect to observe G(x) > F(x). Hence, the null hypothesis is H0: G(x) =
F(x) and the alternate hypothesis is G(x) > F(x).
Based on the one tail KS

test, we find that the distribution for DJIA and S&P 500
are not different f
rom normal but the one for
NASDAQ
is biased m
ore toward higher
percentages. This result indicates
that
the information set that drives
NASDAQ
is
possibly a subset of the information that drives the other two indices
. However, to better
understand the relat
ionship between the information related to those indices, we find
necessary to study each pair of indices.
Table (
4)
presents
the pair

wise analysis of
clusters
common to all three indices.
[Insert Table 4 here]
For each pair of indices (A and B) compar
ed, we compute the percentage of days in A
that overlap with B and
vice versa
. A higher percentage of overlap between clusters
indicates that the two indices react to common information set during that cluster period.
A
10
0% overlap occurs when the start
ing and ending dates of a cluster belonging to one
index is contained
in the cluster belonging to another index.
We find that 38 out of
the 79
common clusters
in DJIA
have a 100% overlap with
common
clusters in S&P 500 and 46
out of 79 clusters in S&P 500
have a 100% overlap with
common clusters in DJIA
.
As
these are informationally independent clusters, the above observation
implies that
both
the indices are driven by the same information set.
Thus,
the higher the percentage of
overlap days the more like
ly
is
common
information set that drives the two indices.
Table (4
) reports the results from the KS

test
. The test does not reject the null
hypothesis in the case of
percentage overlap of DJIA
and S&P 500
clusters in comparison
11
Standardized percentage overlap days = (actual percentage overlap days
–
mean)/std deviation.
22
with
common clusters in
NAS
DAQ
. The test rejects the null
hypothesis
for all other
pairs of
comparisons.
The percentage of days overlap in the common clusters between
DJIA and S&P 500 exhibit a significant skew towards the higher percentage. This
suggests that these two indices ar
e driven by very similar information sets. When we
examine the clusters in NASDAQ, we find a significant portion of them exhibit a high
percent overlap with the clusters in DJIA and S&P 500. However, the KS

test indicates
that
the portion of
DJIA and S&P 5
00
common
clusters
that overlap with NASDAQ
clusters is not significantly different from what the normal benchmark distribution
suggests.
Th
ese results imply
that the information set that drives DJIA and/or S&P 500
may not always pertain to stocks in
NASDA
Q
and that it is
very
likely for
DJIA and/or
S&P 500 is in a
n
informational cluster when
NASDAQ
is observed to be in a cluster
.
More generally, the information set that drives
NASDAQ
index seems to be a subset of
the information set that drives DJIA and S&
P 500.
Thus,
as mentioned in Table 1,
whenever DJIA (S&P 500) is in cluster, with 77.3% (75%)
probability
NASDAQ
is also
in
a cluster driven by the same i
nformation.
Similarly, whenever
DJIA (
S&P 500) is in
cluster, with
probability
90.8% (91.8%) S&P 50
0 (DJIA) is in a cluster
driven by the
same information.
Collectively, these results characterize the simultaneity
of reaction to information
within these broad indices. The above results do not provide information on the
synchronous behavior of
cluster
s
.
This information would help
investors
take positions to
profit from possible movements in returns and/or volatility. The next two sub

sections
present results that relate to these issues.
23
Lead/Lag in extreme clusters across indices
The earlier section r
eports that with a high probability the
independent extreme
return
clusters in the
three indices are
driven by the same information.
12
The earlier
result, however, does not pr
ovide us with any guidance on the
lead/lag pattern across
indices or whether inves
tors could realize extreme return information and digest them
synchronously across the indices.
This question is investigated below.
Table (
5
) contains results relating to lead

lag in clusters.
Panel A reports
[Insert Table
5
here]
comparative data on th
e synchronous behavior in the three indices.
The left s
ide table in
Panel A shows that
59.49%
of the 79 common clusters between DJIA
and S&P
, begin on
the same day
, 44.12% of the
34
common clusters between DJIA and
NASDAQ
,
begin on
the same day
and 41.94%
of the
31
common
clus
ters between S&P 500 and
NASDAQ
,
begin on the same day.
These numbers indicate that
the trading in DJIA and S&P 500, to
a large extent,
realize the same information around the same time
. The same cannot be
claimed
for these indices an
d
NASDAQ
as less than 50% of the clusters
have a
synchronous starting point.
The ending point results are given in the right side of Panel A. We find that
22.78% of the common clusters between DJIA and S&P 500 end on the same day.
DJIA
and S&P 500 indic
es have a higher percentage of clusters that start on the same day and a
lower percentage of common clusters that end on the same day. This implies that these
indices seem to react to information more synchronously but vary in the processing times.
There a
re very few clusters in
NASDAQ
that end on the same day as either D
JIA or S&P
12
Recall, that DJIA and S&P 500 are likely driven by the same information set We find that this
information set is likely to be a superset o
f the information set that drives Nasdaq.
24
500. This suggests that investors trading in DJIA and S&P 500 stocks react to similar
information
b
ut not
NASDAQ
.
The significant difference in the
synchronous
starting
and
end
ing point
s imply
that
t
he investors are active in recognizing the starting o
f
extreme return information but
process information differently across indices
.
Panel B reports results on the lead

lag activity in these indices. The left side of
Panel B contain
s results for asynchronous
starting
dates
and the right side contains results
for asynchronous ending
dates
of clusters.
We find that
21
.52%
(18.99%)
of common
clusters
in S&P 500
(
DJIA)
enter a cluster earlier than
DJIA
(S&P 500).
In contrast,
35.29% (29.
03%) of the clusters in
NASDAQ
enter a cluster earlier than DJIA (S&P
500).
A high percentage of the
NASDAQ
clusters that start earlier are followed by S&P
500 and DJIA.
Additionally
,
as the information in DJIA and S&P 500 is likely to contain
informatio
n not related to stocks in
NASDAQ
, we interpret the numbers 20
.59% and
29.03% as overestimates
and conclude that a smaller fraction of the clusters starting early
in DJIA and S&P may be informationally related.
This overestimate emphasizes the
observation
that
NASDAQ
seems to lead both indices
.
Overall, a high percentage of the
NASDAQ
clusters that start earlier are followed by S&P 500 and DJIA.
NASDAQ
seems
to lead both in
dices when entering the cluster
.
S&P seems to enter a cluster
first
in
comparison
t
o DJIA.
As for exiting a cluster,
S&P seems to end first compare to DJIA
and NASDAQ
trails
both
DJIA and S&P 500
,
equally
.
The right side of
Panel C contains results for asynchronous ending dates
.
Typically, a cluster ends when
a series of low returns i
s followed by
a drop in volatility
below the threshold. We find that 31.65%
of DJIA clusters end earlier than S&P 500
. In
contrast, 45.57% of S&P 500 clusters end before DJIA.
Both DJIA and S&P 500 have
25
roughly 38% of their clusters that end earlier than
NASDAQ
.
The
percentages for
NASDAQ are
overestimated for reasons discussed earlier.
The left

side of Panel C
shows that the average
number of days
per cluster that
DJIA leads S&P 500 or
vice versa
is
about 6.50 days.
NASDAQ
leads
DJIA by 11.75
days and
S&P
500 by 7.56 days.
Again, as mentioned earlier the underlined percentages
are estimates that might contain different
information and hence we cannot
say precisely
whether
DJIA
and S&P leads
NASDAQ
.
We know from the previous tables that
NASDAQ seems to l
ead DJIA and S&P
.
Thus, by knowing the number of days lead we
can obtain an idea if an investor has sufficient nu
mber of days to take a pos
ition to profit
from the lag in another
index. For instance, if an investor observes that
NASDAQ
is in
cluster, he or
she can expect DJIA to enter a cluster in about 11.75 days.
In the case of
end lead days, there is
no
significant difference in the pattern of endings for S&P and
DJIA. However,
there is a lag of roughly 12 to 14 days from the time DJIA and/or S&P
500 e
n
ds and
NASDAQ
following suit. It appears that the traders in
NASDAQ
stocks
take a longer time to fully digest the information, thus taking more time to end
.
Overall, DJIA and S&P 500 seem
to lead one another by approximately the same
time in both ending
and starting points.
In comparison with S&P 500, it takes a longe
r
time for DJIA to follow NASDAQ
in the start.
NASDAQ
take
s
approximately the same
time to follow the other two indices in the end.
NASDAQ
take
s
fewer days to lead others
in starting point a
nd more days to follow the other two indices in ending points.
This
confirms the
previous finding about
the subset information of NASDAQ
clusters.
26
Return and Volatility Strategies
The lead/lag analysis presented in the p
revious section is interesting
from an
academic standpoint. However, from an investor’s view, it is more important to know if
an index is in a cluster and whether it gives any signal
about the behavior of another
index.
Specifically
, if an investor knows that an index is
either
in clu
ster or out of a
cluster, he or she could
draw inference
s
about
the cluster

based information of
another
index.
In what follows, we characterize the opportunities that an investor faces
and show
the use of
a
returns strategy
(used
when an index enters a
c
luster) and
volatility strategy
(used when an index leaves a cluster).
Returns based strategy
Based on
the modified Laurini’
s algorithm,
an investor can realize that an index is
in cluster when the volatility
first
exceeds a threshold value
(
c
=75%)
and
th
e
n
returns
exceed its threshold value
(
u
=90%)
.
13
It is the occurrence of an extreme return that marks
the index as being in

cluster
, thus enabling the investor to predict the first extreme return
of another index using the asynchronous in

cluster status.
It
is for the above reason that
we use
returns
based strategy in the context of in

cluster trades.
As an example, on
September 29, 1980,
an invest
or could observe that
DJIA,
S&P 500 and
NASDAQ
synchronous
ly started entering
a cluster by exhibiting an
incr
ease in volatility beyond the threshold.
On
e
day later, investors
have full information
that S&P 500 and
NASDAQ
are in
a cluster by observing that their
return
s
had
exceeded
the
threshold
.
However, on this day
although
DJIA
has not provided complete
infor
mation of its in

cluster status, the investors could infer from
the in

cluster status of
13
Empirically, only condition 2 triggers a starting point as indicated by the instant volatility first exceeds
the threshold
c
and remains above the threshold till
X
m
>
u
for the first time on date
m
. In practice,
values of
c
and
u
can be estimated using historical distributions of GARCH volatilities and returns.
27
S&P 500 and
NASDAQ
and
predict an extreme return movement
for DJIA
in
a
few
days
.
Although not reported, we find from analyzing the
common cluster relevant to the
exa
mple
that DJIA indeed follows investor prediction and sends a complete signal of
being in

cluster within 4 days of the
signal from
other two indices.
The investor has an
opportunity to align his or her portfolio based on this
return
prediction.
In general,
it
takes roughly
seven to ten days for one index to follow the other
signal of being in

cluster
.
14
Based on our analysi
s, we find that 37.9% of the common clusters between
DJIA and S&P 500
lend themselves to the returns strategy. As for the other
pairs o
f
indices, between 38.2% and
70.6% of common clusters between DJIA and
NASDAQ
and
between 25.8% to 58.1% of common clusters between S&P 500 and
NASDAQ
afford the
investors the same opportunity.
Volatility based strategy
The end of a cluster is marked by a
drop in returns followed by a drop in
volatility.
15
When an index leads another index during its exit from a cluster, it sends a
conclusive signal that the volatility of the lead index has declined below the threshold
level
c
.
Thus, if an investor observes
that an index is the first to leave a cluster, he or she
can use this as a signal to predict a drop in volatility in another in
dex. Continuing with
the same example as in the returns strategy,
DJIA leads the pack in exiting the same
cluster on
October 9,
1980. An investor can observe DJIA and imply that
NASDAQ
and
S&P 500
will also be exiting that cluster and experience a drop in volatility in a few days.
Our analysis of clusters indicates that
NASDAQ
exits within 2 days and S&P 500 exits
14
We have not reported these results in the paper and will furnish them upon request.
15
Just as in footnote 13, empirically we find that only condition 2 o
f Laurini’s algorithm triggers the end of
a cluster.
28
within 7 days
of DJIA’s exiting a cluster.
T
he investor has an opportunity to align his or
her portfolio based on
volatility
prediction
.
From Table (5
), Panel C we find that
the lag in
S&P 500 (DJIA)
index after other
leaves a cluster is
about
6
days
(
7 days).
In com
parison, when either S&P 500 or
DJIA
leaves a cluster, it takes
about two weeks
for
NASDAQ
to follow suit.
Additionally, we
find that 77.2% of the common clusters between DJIA and S&P 500 lend themselves to
the volatility strategy. As for the other pair
s of indices, between 38.2% and 97.1% of
common clusters between DJIA and
NASDAQ
and between
38.7
% to
93.5
% of common
clusters between S&P 500 and
NASDAQ
afford the investors the same opportunity.
7. Conclusions
With the help of modern technology and tel
ecommunications, the global and
domestic financial markets are closely connected with each other in an informational
sense. The information related to the markets in one country is transmitted almost
instantly to the financial markets in other countries a
round the world. Such information
affects even indices within domestic markets,
al
beit
in different degrees.
This research
provides a novel way of analyzing
information
that
affects the domestic indices such as
DJIA, S&P 500 and
NASDAQ
by characterizing
clusters of positive extreme returns
.
We partition the return space to detect clusters by using two algorithms that
depend on returns and volatility movements. Our results indicate that very similar
information drives DJIA and S&P. We
find it
more likel
y that only a subset of this
information affects the
NASDAQ
index.
We find that these patterns translate into the
extent of overlap in days between common clusters across indices and also in the patterns
of lead/lag while entering or leaving a cluster. Ou
r analysis shows that if an investor
29
observes that
NASDAQ
is in cluster, he or she can expects DJIA to enter cluster in about
12 days. In the case of a cluster ending, we find there is roughly 12 to 14 days lag from
the time DJIA and/or S&P 500 and
NASDAQ
following suit.
We employ a return strategy to predict lead/lag at the starting of a cluster and a
volatility strategy for the ending.
It is the occurrence of an extreme return that marks the
index as being in

cluster, thus enabling the investor to pred
ict the first extreme return of
another index using the asynchronous in

cluster status. Additionally, if an investor
observes that an index is the first to leave a cluster, he or she can use this as a signal to
predict a drop in volatility in another inde
x. We show that the detection methodology
described in this research would provide investors with ample opportunity to predict
cross

index movement in returns and/or volatility and proactively engage in ret
urns and
volatility strategies. Addi
tionally, inv
estors can take positions in index options to take
advantage of cross index return and/or volatility movements.
The methodology in this paper is well suited to partition information sets that
yield common extreme return clusters across indices.
It is very
plausible to search for
common information
that yields common return clusters between an
index and individual
stock returns. This approach attempts to get at the notion of systematic risk for extremes
of the return distributions.
In a companion paper, we
employ Latent Semantic Analysis to screen volumes of
news articles prior to cluster beginnings to
determine the associated keywords. These
keywords can then be tested out of sample to ascertain its’ ability to predict if
investor
reaction
during a volatil
e period is followed by an extreme return event. We leave these
questions for future research.
30
Figure 1
: This figure illustrates the two (mutually exclusive) conditions
in Laurini
(2004)
that mark the end
of an informationally independent extreme return
cluster.
Condition 1 is described in the first set of return and volatility figures and condition 2 is
illustrated in the second set of figures below.
31
Figure 2
: This figure illustrates the two (mutually exclusive) conditions in Modified
Laurini’s a
lgorithm that mark the beginning of an informationally independent extreme
return cluster. Condition 1 is described in the first set of return and volatility figures and
condition 2 is illustrated in the second set of figures below. Note that
m
denotes the
first
time return exceed the threshold
u
after the end of a previous cluster.
32
Figure 3
:
These figures illustrate role of the return and volatility thresholds in detecting
independent information extreme return clusters. Figures3a and 3b illustrate
s the return
threshold sensitivity for the triplets {75%, 25% 50} and {90%, 25%, 50}, respectively.
Figures3c and 3d illustrates the volatility threshold sensitivity for the triplets {90%, 75%
50} and {90%, 25%, 50}, respectively
33
Figure 4:
This
figure uses the sub

prime crisis era to detect extreme return clusters using
the triplet {90%, 75%, 50}. Figure 4a presents the extreme return clusters and Figure 4b
presents the corresponding conditional volatilities. While these extreme return clusters
w
ere embedded in Figure 3d, the volatility threshold of 75% brings it is in focus here.
34
Table 1
: The parameter estimates of a GARCH (1,1) model are presented below for the
three indices DJIA, S&P500 and
NASDAQ
for different sub

periods. The table
also
presents the initial volatility (Std Dev) of the GARCH process. This daily volatility is
estimated by using the return data for the immediately preceding 30 days from the start of
each sub

period.
DJIA
________________________________________________
______________________
1960

70
1970

80
1980

90
1990

00
2000

08
______________________________________________________________________
Intercept
0.0000659
0.000185
0.000470
0.000565

0.000020
ARCH(0)
2.05047x10

6
0.4599x10

7
4.8367x10

6
8.089x10

7
1.026
1x10

6
ARCH(1)
0.1393
0.0586
0.0918
0.0512
0.0759
GARCH(1)
0.8135
0.9296
0.8686
0.9393
0.9179
Std Dev
0.004781
0.007574
0.006029
0.005143
0.007783
_______________________________________________________________________
S&P 500
__________________
_____________________________________________________
1960

70
1970

80
1980

90
1990

00
2000

08
_______________________________________________________________________
Intercept
0.000173
0.0000630
0.000469
0.000564

000090
ARCH(0)
01.601x10

6
7.5201x10

7
4.946x10

6
5.4987x10

7
9.8951x10

7
ARCH(1)
0.1771
0.0629
0.0940
0.0524
0.0709
GARCH(1)
0.7909
0.9260
0.8610
0.9416
0.9233
Std Dev
0.003924
0.006978
0.005909
0.0006252
0.00748
______________________________________________________________________
_
NASDAQ
_______________________________________________
1980

90
1990

00
2000

08
_______________________________________________
Intercept
0.000436
0.000867

0.000285
ARCH(0)
3.6132x10

6
3.8592x10

6
8.5965x10

7
ARCH(1)
0.1672
0.1282
0.0573
GARCH(
1)
0.7807
0.8423
0.9405
Std Dev
0.00435
0.005882
0.013529
______________________________________________
35
Table 2
:
This table contains
percentages of
extreme return
clusters that are common
across indices.
For instance, 90.8% of DJIA clusters are common
to S&P 500 and 91.8%
clusters in S&P 500 are common with DJIA.
___________________________________________
DJIA
S&P 500
NASDAQ
___________________________________________
DJIA

90.8%
63.9%
S&P 500
91.8%

63.5%
NASDAQ
77.3%
75%

___________________________________________
36
Table 3
:
This table examines the extreme return clusters that are
simultaneously
common to S&P 500,
DJIA
and
NASDAQ
. Since data for
NASDAQ
is available only
from 1980, we restr
ict our analysis to the other two indices to the same period.
______________________
Panel A
Total Number of Clusters
______________________
DJIA
54
S&P 500
52
NASDAQ
44
______________________
Panel B
_______________________
_______________________________________
Days Overlap (%)
1
DJIA and
S&P 5
00 and
NASDAQ
DJIA
S&P 500
NASDAQ
______________________________________________________________
100
7
4
13
99

90
1
0
3
89

80
2
2
1
79

70
2
6
1
69

60
4
5
2
59

50
2
3
0
49

40
0
0
2
39

30
3
3
2
29

20
6
4
0
19

10
1
1
3
9
–
0
0
0
1
Total common clusters
28
28
28
K

S Test
2
Do not
Reject
Do not
Reject
Reject
p

value
0.1967
0.3958
0.0040
___________________________________________________
1.
Overlap days (%) is
the percentage of days in
index
A that overlap with B and
vice versa
.
2. KS

1

tail test (Null): The distribution of overlap days for an index is normally distri
buted.
KS

1

tail test (Alt) : The distribution of overlap days for an index is biased towards
higher percentages of the overlap days.
37
Table 4:
This table presents a pair

wise analysis of the overlap days in common extreme
re
turn clusters in S&P 500,
DJIA
and
NASDAQ
. We use daily data during 1960

2008 for
DJIA and S&P 500 and
1960

08 and
1980

2008 for
NASDAQ
.
Panel A
________________________________________
Total Number of Clusters
___________________________
_
______
DJIA
87
S&P 500
86
NASDAQ
44
________________________________________
Panel B
____________________________________________________________________________________
__
Days
Overlap (%)
DJIA

S&P 500
DJIA
–
NASDAQ
S
&P

NASDAQ
DJIA
S&P 500
DJIA
NASDAQ
S&P
NASDAQ
____________________________________________________________________________________
__
100
38
46
11
17
1
0
16
99

90
14
8
1
4
1
2
89

80
11
2
2
1
1
1
79

70
4
9
3
2
5
2
69

60
2
3
4
1
4
2
59

50
2
5
2
1
0
0
49

40
2
1
0
2
0
2
39

30
2
3
4
2
4
1
29

20
3
0
5
0
5
0
19

10
0
2
2
3
2
3
9
–
0
1
0
0
1
0
2
Total
79
79
34
34
31
31
KS Test
Reject
Reject
Do Not
Reject
Do not
Reject
p

value
9.0450x10

5
6.3216x10

8
0.0816
0.0015
0.1185
0.0023
___________________________________________________
_____________________
1.
Overlap days (%) is
the percentage of days in
index
A that overlap with B and
vice versa
.
2. KS

1

tail test (Null): The distribution of overlap days for an index is normally distributed.
KS

1

tail test (Alt) : The distribution of overlap days for an index is biased towards
higher percentages of the overlap days.
38
Table 5
:
This table
aligns clusters based on starting dates and ending dates and presents
results for synchronous behavior in Panel A, Asynchronous behavior in Panel B and
provides additional statistics on the average number of days an index leads a start of a
cl
uster and the average number of days an index leads in leaving a cluster.
PANEL A
Synchronous starting point
Synchronous ending point
_________________________
______
________________________
______
DJIA and S&P 500
59.49%
DJIA and S&P
500
22.78%
DJIA and
NASDAQ
44.12%
DJIA and
NASDAQ
2.94%
S&P 500 and
NASDAQ
41.94%
S&P 500 and
NASDAQ
6.45%
_________________________
______
_________________________
_____
PANEL B
Asynchronous starting point
1
Asy
nchronous ending point
______________________________
__
_________________________
__________
DJIA
S&P 500
NASDAQ
DJIA
S&P500
NASDAQ
DJIA
18.99%
20.59%
DJIA
31.65% 38.24%
S&P 500
21.52%
29.03%
S&P
500
45.57%
38.71%
NASDAQ
35.29%
29.03%
NASDAQ
58.82% 54.84%
_____
___________________________
___________________________________
P
ANEL C
Average start
lead
days per cluster
Average end
lead
days per cluster
____
________________
___________
____________________________________
DJIA
S&P 500
NASDAQ
DJIA
S&P 500
NASDAQ
DJIA
6.40
17.00
DJIA
7.88
14.00
S&P 500
6.58
5.11
S&P 500
6.08
12.18
NASDAQ
11.75
7.56
N
ASDAQ
22.75
26.35
_____
__________________________
____
________________________________
1 The underlined figures represent an overestimate as the information that drives DJIA
and S&P 500 is
a superset of
NASDAQ.
39
References
T. Bollerslev, 1987, ‘
Con
ditionally Heteroskedastic Time Series Model for Speculative
Prices and Rates of Return’,
The Review of Economics and Statistics
, Vol. 69, No.
3 (Aug., 1987), pp. 542

547
Bollerslev, T., R. Y. Chou and K.F. Kroner 1992, ‘ARCH Modeling in Finance: A
Review of the Theory and Empirical Evidence,’
Journal of Econometrics,
52, 5

59.
J. Duan, 1995, ‘The GARCH Option Pricing Model,’
Mathematical Finance
, 5(1),
January, 13

32.
R.F.
Engle,
1982, ‘
Autoregressive Conditional Heteroskedasticity with Estimates
of the
Variance of U.K.
Inflation.’
Econometrica.
50: 987

1008.
Fujihara, R and K. Park (1990), “The Probability Distribution of Futures Prices in the
Foreign Exchange Market: A Comparison of Candidate Processes,”
Journal of
Futures Markets
, 623

641.
Jo
hn G. Galbraith & Serguei Zernov, 2006. "
Extreme Dependence In The Nasdaq And
S&P Composite Indexes
," Departmental Working Papers 2006

14, McGill
University, Department of Economics.
D. Guillaume, M. Dacorogna, R. Dav´e, U. M¨uller, R. Olsen, and O. Pic
tet,
From the birds eye view to the microscope: A survey of new stylized facts of the
intraday foreign exchange markets,
Finance and Stochastics
, 1 (1997), pp. 95
–
131
F. Laurini, J.A. Tawn (2003), ‘New Estimators for the Extremal Index and
Other Cluster
Characteristics,’
Extremes
,
Vol 6, 3, 189

211.
F. Laurini,
2004,
‘
Clusters of Extreme Observations and Extremal Index Estimate in
Garch Processes,
”
Studies in Nonlinear Dynamics & Econometrics
, Vol 8, Issue
2,
1

21.
Leadbetter, M.R. (1983) Extremes an
d Local Dependence in Stationary Sequences.
Z
eitschrift
fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 64, 291

306.
Hansen, Peter Reinhard and Asger Lunde 2001, ‘A Forecast Comparison of Volatility
Models: Does Anything Beat a GARCH(1,1)?’ WP No.
01

01, Department of
Economics, Brown University.
H
eston, Steven
and Saikat Nandi 2000, ‘
A Closed

For
m GARCH Option Valuation
Model,’
Journal of Financial Review
, Fall, 13(3), 585

625.
40
K. Hovsepian, P. Anselmo, and S. Mazumdar, “Detection and prediction
of relative
Clustered
volatility in financial markets,” in
Fourth International Conference on
ComputationalIntelligence in Economics and Finance
(
CIEF 2005
), Salt Lake
City, 2005.
D. Hsieh, 1988, ‘The Statistical Properties of Daily Exchange Rates: 1974

1993,’
Journal of International Economics
, 129

145.
Johnston, K. and E. Scott 1999, ‘
The Statistical Distribution of Foreign Exchange Rates:
De
pendent vs. Independent Models,’
Journal of Financial and Strategic
Decisions
, 12, 2, 39

49.
A. Kirman, Ants,
rationality, and
recruitment,
Quarterly Journal of Economics
,
108
(1993), pp. 137
–
156.
A. Kirman and G. Teyssiere, Microeconomic models for long

memory in the
volatility of financial time series,
Studies in nonlinear dynamics and
econometrics, 5 (2002),
pp. 281
–
302.
Kluger, P. and C. Len
z 1990, ‘
Chaos, ARCH and the Foreign Exchange Market:
Empi
rical Results from Weekly Data,’
Unpublished Manuscript,
Volswirtschaftliches Institut, Zurich.
B. LeBaron, Agent

based computational finance : suggested reading
s and early
research,
Journal of Economic Dynamics and Control
, 24 (2000), pp. 679
–
702.
32. , Evolution and time horizons in an agent

based stock market,
Macroeconomic
Dynamics
, 5 (2001), pp. 225
–
254.
T. Lux and M. Marchesi,
Volatility clustering in finan
cial markets : a micro simulation
of interacting agents,
International Journal of Theoretical and Applied
Finance
, 3 (2000), pp. 675
–
702.
B.B. Mandelbrot (1963) The variation of certain speculative prices,
Journal of Business
,
XXXVI, 392

417.
McCurdy,
T. H. and I. G. Morgan (1988), “Testing the martingale Hypothesis in
Deutsche Mark Futures with Models Specifying the Form of Heteroscedasticity,”
Journal of Applied Econometrics,
3, 187

202.
O’Brien, G.L. (1974) The Maximum Term of a Uniformly Mixing Seq
uence.
41
Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete 30, 57

63.
M.T.
Owyang
2001, Persistence, Excess Volatility, and Volatility Clusters in Inflation,
The Federal Reserve Bank of St. Louis
,
Nov

Dec, 41

51.
Taylor, S (1986),
Modeli
ng Financial Time Series
. New York, Wiley.
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο