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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
High Frequency Data and
Volatility
in Foreign Excliange Rates
by
Bin Zhou
MIT
Sloan School Working Paper 3485-92
Revised January 14, 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
High Frequency Data and Volatility
in Foreign Exchange Rates
by
Bin Zhou
MIT
Sloan School Working Paper 3485-92
Revised January 14, 1993
MAR
2 5 1993
High Frequency Data and Volatility
in Foreign Exchange Rates
Bin Zhou
'
Management
Sloan School of
Massachusetts Institute of Technology
Cambridge,
MA
02139
September 1992
Abstract:
Exchange
rates, like
eroscedasticity.
many
other financial time series, display substantial het-
This poses obstacles
detecting trends and changes.
in
derstanding volatility becomes extremely important
series.
ity of ultra-high
monthly observations,
very difficult.
is
The
new
possibility for the analysis of volatile time
This article uses tick-by-tick Deutsche Mark and
exchange rates to explore
this
new type
of data.
high frequency data have extremely high negative
in their return.
recent availabil-
frequency observations, such as tick-by-tick data, to large
financial institutions creates a
ity
studying financial time
Unfortunately, estimating volatility from low frequency data, such as
daily, weekly, or
series.
in
Un-
A model
US
Dollar
(DM/$)
Unlike low frequency data,
first
order autocorrelation
explaining the negative autocorrelation and volatil-
estimators using the high frequency data are proposed. Daily and hourly
volatility of the
DM/$
exchange rates are estimated and the behaviors of the
volatility are discussed.
KEY WORDS:
Financial time series; tick-by-tick data; heteroscedastic-
ity.
'The author thanks Professor David Donolio for his advice and helpful comments. The
author also thanks Morgan Guaranty Trust Company for providing the data for this research.
INTRODUCTION
1
There
How-
considerable literature analyzing the behavior of exchange rates.
is
exchange rate modeling has not been very successful. By
studying monthly data, Meese and Rogoff (19S3a,b) have shown that a random walk model fits at least as well as more complicated structural models.
ever, structural
Empirical studies such as those by Hsieh (19SS) and Diebold (1988) have
shown that
heavy
daily returns are approximately symmetric and leptokurtic
tailed).
The
(i.e.,
autocorrelations are weak but not independent and identi-
One explanation
cally distributed (i.i.d.).
for the
heavy tailed distribution
is
the hypothesis that data are independently distributed as a normal distribu-
whose mean and variance change over time (Friedman and Vandersteel
1982, Hsieh 19S8 and Diebold 1988). Since market volatility depends information flow and the amount of "'information flow" is not constant over time.
tion
There
is
no reason to believe that the variance of the price changes
many
stant over time. Clark (1973) and
is
con-
others (Mandelbrot and Taylor 1969,
Praetz 1972) have argued that observed returns come from a mixture of nor-
mal
distributions.
If
random
the
variable
price, the conditional distribution of
A',
denotes the daily return of the
A( given information
is:
A-,k~A^(//./(^-<))
where
the
a,'(
the information available at time
is all
number
of transactions
(1)
t.
The quantity
cj(
could be
(Mandelbrot and Taylor 1969), or trading volume
(Clark 1973).
One parametrization
of this conditional heteroscedasticity
by Engle (1982). Engle approximates the
volatility
was
first
studied
by
p
f(^'t)
It
is
=
Oo
+ ^a,{.\\_,-
p.)\
ao>OQ,
>0,
i
=
called the autoregressive conditional heteroscedasticity
since the heteroscedasticity
Because the
in financial
time series and
many
many
is
(2)
(ARCH) model
an autoregressive fashion.
mathematically easy to manipulate,
close
it
has been
Previous studies found that the
approximation of many financial time
series.
other parametrizations, such as the generalized autoregres-
sive conditional heteroscedasticity
been proposed.
in
financial time series.
model provides a
Since then
represented
p
ARCH model exhibits the conditional heteroscedasticity present
used to analyze
ARCH
is
l
(GARCH) model
They capture some
(BoUerslev 1986), have
characteristics of the volatility such as
1
volatility clustering.
need to
However,
be estimated. The
other parametric models, the parameters
like
volatility estimates
ARCH, G.A.RCH
from
models
are often lagged because the historical data are used.
The
Tick- by- tick data provide us with a near continuous
in estimating volatility.
observation of the process.
Understanding
tail.
ticity
data has opened up new possibilities
availability of high-frequency
model
( 1 )
gives us the chance to study volatility in great de-
It
volatility
and any other
tick-by-tick data
is
the key issue
the conditional heteroscedas-
in
financial time series models. This
and uses the data
to estimate
and study
paper explores
volatility.
HIGH-FREQUENCY DATA
2
Because of fast growing computer power, gathering financial data
ever.
Data are no longer recorded
began to
exchange rate
In contrast to stock markets, foreign
ical location,
Many
daily or weekly.
collect so called tick-by-tick
in
is
easier than
large institutions
the early eighties.
exchange markets have no geograph-
and no "business-hour" limitations. Traders negotiate deals and
make exchanges over
known to the
are not
The transaction prices and trading volume
The exchange rates used for most research are the
the telephone.
public.
quotes from large data suppliers such as Reuters, Telerate, or Knight Ridder.
Any market- maker
can submit new quotes to the data suppliers.
are then conveyed to data subscribers' screens.
The data
The quotes
suppliers cover the
market information worldwide and twenty-four hours a day. The quotes are
intended to be used by market participants as a general indication of where
exchange rates stands, but does not necessarily represent the actual rate
at
some participants
to
which transactions are being conducted.
possible for
It is
manipulate indicative prices occasionally and create a favorable market movement. However, since a bank's reputation and credibility as a market-maker
emerges from favorable relations with other market participants,
felt
that these indicative prices closely
market.'^
Goodhart and
Figliuoli (1991) studied
rates (the closing tick of a minute)
ries
match the true
it is
generally
prices experienced in the
minute-by-minute exchange
from Reuters.
They found
that the se-
exhibited (time varying) leptokurtosis, unit roots, and first-order negative
correlation.
The paper
used only three days' data from Reuters.
In this study, tick-by-tick
data
for
data are provided by Morgan Guaranty Trust
-Reader who
to read
is
unfamiliar witii this type of data
Goodhart and
Figliuoli's (1991)
paper
The
Morgan). They
the entire year of 1990 are used.
Company
in foreign
for details.
(J. P.
exchange markets may want
Table
PUB.
1:
A Sample
of Tick-by-tick
Exchange Rates
Figure
1:
Original Quotes
From
the Reuters and Telerate.
Qckt
Figure
2:
X'alidated Quotes.
Table
n
2:
Summary
.Statistics of
Tick-bv-tick Returns
ICCC
2SC
Figure
Figure
4:
3:
Sample Kuitoses
of
Return of Different Frequencies.
First-order Autocorrelations of Return of Different Frequencies.
To be
also updating noises in quotes.
There are
The new update
keep updating their quotes.
the previous quotes even
if
visible in the market, traders
often slightly different from
is
the market price remains the same. Typographical
Summarizing these arguments,
the exchange rate:
errors are another source of noise.
the following process for
=
S(t)
where S{t)
d{t)
+
+
B(T(t))
nian motion, both
d{-)
and
(.3)
the standard Brow-
is
assumed deterministic functions,
r(-) are
assume
t,
the logarithm of the exchange rate, B{-)
is
I
r(-)
has
e, is the mean zero random noise independent to the
noise, f(, is a combination of several sources that
The
Brownian motion B(.).
were mentioned above. No distribution assumptions have been made for this
positive increments, and
noise.
It
could be a very general stochastic process. This extra noise causes
most of the negative autocorrelation
=
Let X{s,t)
-
S{t)
the return
5(.'')'
A'(.s.n =//(.s./)
where
Zi
li{g,t.)
=
d(t)
—
The
d(s).
—
r]f
=
and c[s,t)
V'ar(et)
increases as well. For large
does the noise. X{s,
\s
—
a'{.,J)
+
e,
-63
(4)
variable, a^(s,t)
+
//;
+
=
r(/)
—
t{s)
and
is:
i]l
- 2c{sJ)
—
=
Cov(ts.tt).
1\.
the variances of noises
When
|^
<|
increases,
become
a^{sj)
negligible, so
random walk. When \s — t\ decreases
The return. X{sJ). is the difference of two
beha\es just
t)
a(.../)Z,
\-ariance of the return
Var(.V(.s. /))
where
+
Then
in interval [sj].
random
a standard normal
is
high frequency data.
in
to near zero, a--{sj) diminishes.
like
a
The sample first order autocorrelation of such series is about -50%.
When we study high-frequency data, the noise is no longer negligible. An
autocorrelation of -47% for the D.M/S exchange rate indicates that the level
noises.
of noises
very high
is
There are several
culties
a^{t
—
is
in tick-by-tick
difficulties in
data.
analyzing the process
lack of information about T(t). which
=
6,t)
6-volatility.
—
T(t)
In the
T(t
—
6)
is
next section,
I
call
(.3).
One
of the
the cumulative
diffi-
volatility.
called the (^-increment of volatility or simply
I
will
devote
my
attention to estimating the
volatility increment.
VOLATILITY ESTIMATION
3
In this section,
[a,b], T{b)
—
1
r(fl).
concentrate
will
The
011
estimating the volatility of a given time
function -(/) can be estimated increment by increment.
I
first
derive an optimal estimator of the volatility based on the assumption of
constant variance and zero mean.
tions
For simplicity,
on the noise component. A more generalized
result will
of the section. Proofs of the theorems are listed in
Theorem
from
Assume
1
that
{S{t,).i
0.1,.. .,/i}
be given
in later
appendix A.
a series of observations
is
the process
S(t)
where
=
add normal assump-
also
I
=
e(_,i
T{t)
=
+
B(T(t))
e,
(.5)
independent and identically distributed with normal
l,..,n, are
and
distribution
=
+
a^t
likelihood estimator of a-
h.
Let
A',
=
S[t^)
-
S{t,_i).
Then
the
maximum
is
where
p
This
MLE
for large n,
I
=
and
-
is not unbiased.
However, noticed that p and p' are very close
can have an unbiased estimator by eliminating the factors (1 —
/>p')/(l-p^) and
/,//.':
^\.
Theorem
p
Under
2
= (l/,OX:(A7 +
2A-,A-._i).
assumptions of Theorem
the
1,
(7)
mean and variance
the
of
the estimator (7) are:
Ea\,
=
a^
\^r[a\;)
=
(l/„)a^(6+16-4 + S-^) +
(8)
and
where rf
From
is
I?
n-*
V/n^
(9)
variance oftt,.
(9),
I
find that variance of
the variance ratio
rj^/cr^.
estimator (7) to A',,fc
is multiple of /-. Let
=
Estimator (10)
is
can be optimized by properly adjusting
Since aggregation increases the variance
S{i)
'^"'^'.^
cr^^^
-
= -
S[i
E
-
k).i
-
(A?.,
k/2k
+
unbiased and the variance
n where
2AaA-,_,,,).
is
given
in
I
cr^,
I
apply
assume that n
(10)
following theorem:
Theorem
Under
3
The variance
where
[x]
is
assumptions of Theorem
the
minimized
rounds x down
Since the noise
t(
=
at k
1
3a-)] or k
[(2;/^)/(
=
[(27?^)/(
3a-)]
+
1,
to the next integer.
indepeiiclent. tlie \ariance of the estimator can be
is
further reduced by averaging the estinmtor (7) using overlapping returns:
^(A7,, +
a2
ku
In fact,
it
1=
2A-,,,A-,_,,,)
can be proved that:
^4
2
Varia')
<-a'{6k+
/;
The
abo\'p three theorems
However,
if
I
relax
all
+ 3-^) + iij'/n'
16^
knk^a'^
assumed
noises
i.i.d.
these assumptions,
I
can
from process
(3).
The
have a nearly unbiased
still
estimate the volatility r(/„)
—
V(i,.tn)
=
,
-2X,,,X,_,,,)
1=1
for all
=
1
+(i/^')i:'['/L-'/L,]
n
1=1)
=
\'aiiti
i
.
Then
Y.^i/k)[a'(t, _,_,,!,_,) -a'(/„_, + ,,/n-J]
1
ri^{t)
I
r(tn)-T(to)
+
where
— 2A' +
).
1
n]
propose to
.
= \Y.{Xl, +
Assume Cov(e,^.Lt^_J=()
EV(tQjn)
—2^',
Under minimal assumptions,
r(/o) by:
A'
4
=
variances of the returns are not necessarily constant.
noises can be nonstationary.
Theorem
(13)
and constant variances.
estimator. Suppose that wr ha\'e observations {5(i,),«
The
(12)
1
(14)
The only assumption
I
have made
in this
theorem
is
that of uncorrelated
noises. Since
n
'^[fi-{t,J,_k)+Mtnt,-k-)^'(t,-k,U-2k)]
<
:3max{|/i(^,/,_;.)|}[fi(n.) -f/(0)]
i=fc
the last term
smooth
in
(15)
in interval
is
negligible in high frequency data
[^o-^n]-
proximately unbiased
estimator
is
if
also easy to
if
the drift d(t)
Therefore, for large n, the estimator (14)
no jumps occurred
the time interval
in
update when new data become
is
is
ap-
[/o,<n].
This
will
allow
available.
It
us to estimate the volatility r(/) dynamically.
ESTIMATING VOLATILITY OF
EXCHANGE RATES
4
In this section,
frequencies.
I
about the exchange rates
DM/$
of
these estimates,
I
want
exchange rates
to evaluate
my
return for that week
To choose the parameter
approximately
all
(i.
in
data base was shutdown due to a power outage
13, the
Manhattan. The
at different
assumptions
Again, 1990
as well as the volatility estimator.
exchange rates are used. The data have one discontinuity:
August
Using
DM/S
estin^ate the volatilities of
By examining
k,
is
set at zero, as
is
.Minimizing the upper bond of variance (13),
available tick-b\'-tick data,
I
first
in lower
the volatility estimate.
estimated the variance ratio
I
the week
r/^/cr^
I
to be
= 6.
DM/$
have k
estimate the volatility of the
in entire 1990. The estimate is .010349. To verify this estimate,
compared it to the estimate under the Brownian motion assumption. If the
data had no noise and followed a Brownian motion, the quadratic variation
exchange rate
I
i
would be a standard estimator
is
used on data with noise,
it
quency decreases, so does the
=k
of the volatility.
When
the quadratic variation
overestimates the volatility.
bias.
The expectation
.As
the sample
fre-
of the quadratic variation
is:
(n/k)
1=1
(n/k)
=
r{tn)
-
T{to)
+ Y,
[lL
+
'/u-_.
i=\
10
-
c{t,kJ^k-k]
+
^l''{t,kJ,k-k)]{lQ)
1000
Figure
o:
where c{t,t;Ja-k] =
Quadratic Variations Using n-tick Returns
VVlien
Cov((.,,^., e(,,_,
are uncorrelated and
Ct's
f.i's
are
negligible,
in/k)
17)
(=1
which decreases as k increases.
have a logarithmic
is
scale.
When
plotted
I
the frequency
about the same as our estimate e.xcept
sample
size.
from
is
(17), the total variance of the noise
is
which confirmed our early estimate of the
To estimate
daily volatilities,
I
I
about
GMT
is
is
1,
the
a twenty-four hour international
Mean Time (GMT)
9:00am Tokyo time and 24:00
fewer ticks on weekends and holidays.
estimate from (14) could be negati\'e.
to zero to avoid negative volatility.
=
six times the total volatility
GMT
If
The
11
such
is
is
as a
in
day
7:00pm New
activities of the
market. There are average more than 7,000 ticks per day
many
A-
our estimate. Therefore,
York time. This twenty-four hour period covers most
are
to a small
ratio.
choose 24 hours from 0:00 Greenwich
because that 0:00
Both axes
need to define the start and the end of
a day since the foreign exchange market
market.
size of
due
When
tremendous.
is
about thirteen times the
5.
low, the c[uadratic variation
is
for a high variation
At a high frequency, the bias
quadratic variation
against k in Figure
Qk
world
weekdays. There
For small n, the volatility
the case,
I
let
the estimate go
daily volatility estimates of
DM/S
are
ilijiiiiiiili
lllllll iii
1990
;J90
Ftb
Ju\
1990
1990
1990
1990
1990
1990
1990
Mtr
Apr
M«^
Jun
hi
Aa(
S&p
Figure
plotted in Figure
A good
High
12
in
;?90
Nov
'S
D*c
DM/$
be able to catch market turbulences.
volatility estimates should indicate
volatility estimates,
volatility estimates
released.
1990
6.
volatility estimator should
and .August
1990
Oct
Daily Volatility Estimates of 1990
6:
examining the daily
six
III
2.
3.
On January
above .0001.
On
1990.
4
and
5,
all
the
unusual activities
I
in
There are on .January
these six days, there
German
the market. In
found such correlation. There are
central
is
30,
July
significant
news
4,
5,
bank surprisingly intervened
the foreign exchange market and pushed the dollar lower.
On January
30, a
wild market followed a rumor that Mr. Gorbachev was considering resigning
as secretary of the former Soviet
tumbled because possible lower
August 2 and
of
3,
Communist
interest rate by the
On July 12, the dollar
US Federal Reserve. On
the Dollar had another wild ride as the news of Iraq's invasion
Kuwait spread around the world. However,
mean a
Party.
large change in price.
The
large volatility does not always
daily changes for
above
six
0.0537, 0.0162, 0.0212. -0.017S. 0.0058 and -0.0074 respectively.
and
3,
On August
The
2
price changes prior to .\ugust 2 were not large either. Therefore
ARCH
model is used, these activities will be missed.
Another way to check the assumption of our model for exchange
an
-
the exchange rate only changed 58 and 74 points that are about the
average.
if
days were
and the accuracy of our
returns.
When
volatility estimates
is
rates (3)
to test normality of scaled daily
a model (3) provides a good approximation and the volatility
12
Daily
Returm
-1
Rescaled Daily Returm
-1
1
Quftnolej
Figure
Table
Q-Q
7:
|-'l(jt
o1
Stuidtrd Norn^4l
of Daily Returns
Basic Statistics of Daily Returns
3:
M ean
V,
-ar.
Skew.
Kurt.
KS-Test
X
320
-3.1e-4
3.62e-5
-.429
6.25
1.275(p = .00)
Y=X/(j
SE
320
-6.0e-2
1.17e+0
-.329
3.20
0.7SS(p=.13)
(.000)
(.0')3)
.137)
(.274)
estimate
is
accurate, the rescaled daily retiu'n
standard normal random variable. Noise
is
normal plots
for
statistics of
both
both return
A',
and
\\
are
shown
we
see that
V,
(all
Table
3.
Y, in
The
is
If
we compare the
iiuich closer to
studying daily
Figure
7.
plot
I
The
Q-Q
basic
errors are
statistics in
column
A',
having a normal distribution.
good and the process
frequency observations of the exchange rates.
daily volatility estimates can be used in
many
other ways.
ample, we can use them to check calendar effects on daily
average daily
am
The standard
indicates that the volatilit\- estimates are reasonably
(3) well describes the high
I
on Saturdays),
Table 3 also shows the Kolmogorov-Smirnov
goodness-of-fit test for normality.
Vj,
in
should be close to a
negligible since
and rescaled return
.V,
also given in the parentheses.
and column
— XJa^
Y]
returns here. Excluding zero volatility estimates
It
1
Qutnales
Sttndtrd NonDftl
o(
volatilities are calculated.
The
volatilities.
results are listed in
For ex-
Seven
Table 4 as
Tabl<>
I:
.\\'erage Daily Volatility
.
•
.
3
50E-06
.
•
3
00E06
.
2
50E-06
•
1
50E-06
.
5
OOE-07
.
.
Scaled Hourly
Hourly Returns
-2
of
Qat&dlu
Sttndtid Normtl
Figure
Table
5:
2
2
2
Quannlec
10:
Remrni
Q-Q
of
Studwd NoRTv&l
Plot of Hourly Returns
Basic Statistics of Hourly Returns
Mean
\'ar.
Skew.
Kurt.
KS-Test
X
4377
-4.3e-5
2.2Se-6
-.255
8.32
3.S47(p=.00)
Y=X/cT
SE
4377
-6.4e-3
').13e-l
.038
2.95
1.604(p=.00)
(.014)
(.020)
(.037)
(.074)
16
High
reduces.
frequeiic\' data
frequency with reasonable^
ran he used to estimate the volatility
])rc(isioii.
In
in
high
contrast to other volatility estima-
tors, our volatility estimator mainly uses the data within the period
we are
interested in instead ot historical data. This allows us to capture the market
volatility cjuickly
without d(May. The estimate
have no jumps. The
Q-Q normal
nearl\'
is
unbiased when prices
plot of rescaled daily or hourly returns by
the volatility estimates shows an almost straight
line,
which other
volatility
estimators can not achie\e.
we consider volatility as a stochastic process, we can have the same discussions as we did in this paper by condition on \-olatiIity. .'\fter we obtained
the volatility estimates, wc can fiuther sludy the \'olatility process itself. For
e.xample, we can stud\' the (list libutioii of the daily volatility, the dynamic
If
structure of the daily process. Oui' study indicates that daily volatility can be
appro.ximated by a lognormal distribution.
ARIMA
techniques on the logarithm of volatility for modeling and forecasting
the volatility.
divergence
are
much
in
Unlike estimating
MLE
easier.
.ARCH
coefficients,
where we often run into
iterations, estimating .ARIM.A. coefficients for volatilities
The
volatility estimates
volatility forecasting
can
l)e iis(>d
and the sample variance
to
Therefore, we can apply regular
image that the daily
ot i.i.d.
in
many other
normal
volatility
is
also impro\-ed.
ways. Since the sample
mean
are independent,
easy
\'arial)les
estimator
is
little
and market price movement.
17
it
is
dependent on daily
turn. This property enables us to do research on relations
volatility
Besides, these
re-
between the market
APPENDIX
PROOF OF THEOREMS
A:
Proof of Thtonin I:
Under assumptions of the theorem,
is
7/^
a normal
is
random
the variance of
likelihood function of
A',
L{s\p-X,
= v\l
-/9-),
I
=
first
lag autocorrelation p
\'ariable
mean
with
"
VJ =
+ 27/^, where
= —if I v^ The
.
/3A,_i
and variance
(A,-/.A,_,)\
(27r.^)-"/-exp(-^
~^
1=1
log-likelihood function
is
Vj = -^log(2;r.^)-f:'-'^'-^-^-
^{sKp:X,
The
(7^
have
L[s\p-X,
The
=
/(.V„|A„_,)/(A„_,|A'„_,).../(A,|A'o)
normal
also a
is
zero and \'ariance r"
is
V„)
Notice that A,|A,_i
^2
has the
A',
e,,.
mean
with
\-ariable
^-
partial derivative of the likelihood function with respect to p
^^
dp
-P-V,-i)-V,-i
(-V,
^f^
,^
t^
Setting the derivative equal to zero and solve for
I
I
have
z:l,-v-v,-i
._
Similarly, for .s^
p.
have
d(
n
ds'
2s'
(A,-/)A,_,) 2
"
,
^
^
(
=
1
or
= ^±(X.-pX,_,f
1
/i
n
n
"
1=1
!=1
1=1
18
=
is
Substituting p by p
above formula.
in
"
I
have
1=1
1=1
-i:x^i-pp)
=
where
"
"^
It is
T.U
-V;
•
easy to sliow that
a-'
=
maximum
Therefore, the
^'
r-(l
+->) =
lii<elilioocl
+-2/))/(l -/)-)
.s^(l
estimator of a^
-•n
= -E-V(i + '"
»e---'^--^/^
is
1
[y-'p']
D-V;.2.V,.V,.,|-<'-'^^''
" :t[
Proof of Thtoretn
p
[^
-
p-)
2:
E(1/»)XI(A7 +
-2.V,.V,_,)
=
(l//OE(V«r(-^'')
+
-EA',.V,_,
1=1
1=1
n
1
=
1
and
Var(a\0
=
(
l//7)'Var X^(A';
+
2.V,A',_i)
1=1
+2(TZ,_,e,,
=
-
2CTZ,_,t,,_,
-
2e(,e(,_,
-a'(6 + 16-4 + 8-^ + —;/'.
)
n
Proof of Thforem
a*^
(7'
/i-
S:
19
+
2e,,_,e(,_J
+
e]^
-
el
Since Var(.V,.^)
=
ha'.
V^r{k-a\;,)
(^))
iinpli(\s
that
= -{ka'fiG + IG-^ +
S-^)
2
+
or
a-
;/
The
ka''
/?•
variance reaches minimmii when
Proof of Theorem
.{:
n
1=1
=
[n/J-^(/o)l
k-\
II
APPENDIX
There are many reasons
B:
VALIDATION
PROGRAM
to have outliers in the original data set
shown
in
Most cjuotes are typed in by humans, there are unavoidable keying
Most outliers I found are these types of errors. Outlier also could
errors.
be caused by large bid and ask spreads. In such a case, at least one of bid
or ask prices does not reflect the true market price and becomes an outlier.
Occasionally, electronic errors also mak(^ outliers. The following program is
Figure
1.
designed to remove above three t\pes of outliers:
A
i)
ii)
cjuote
is
considered as an outlier and remo\ed from the time series
a rate
is
more than
bid and ask spread
5 or less than
is
1.
more than 50
20
or
points, or
if
iii)
a
rate
above or below
is
its
iieiglibor
prices
lj\'
more than a certain
threshold.
For
(iii),
the data.
If
regressions,
I
carry out two regressions using ten bid prices on each side of
the current bid [nice
it
is
I
higher or lower than c-point from both
considered an outlier, where
dependent on the
detected.
is
c
is
range from 15 to 30 points
\-ariances of the neighbor points.
Whenever an
go back ten steps and icpeat abo\-e procedure.
21
outlier
is
References
[I]
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Richard T. and
market
volatility
Tim
exchange
in
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Bollerslev (19S9),
rates."
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[2]
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conditional
het-
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[3]
Calderon-Rossel. Jorge and Moshe Ben-IIorim (1982). "The behavior of
"
./.
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./.
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[61
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./.
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Meese. R. A. and
1\.
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RogofT
(
(
1983h).
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pirical
University of Chicago Press.
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[14]
Royston,
J.
P.
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and
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Royston.
.J.
P. (19821)),
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Shapiro, S.S. and M.B. W'ilk
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23
\
5939
53
.Jolin
VVilley
k
Date Due
Lib-26-67
MIT IIBRARIES
3
TOfiO
DUPL
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3