Informational relatedness among the broad indices

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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


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


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

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