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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
High Frequency Data and Volatility
in Foreign Exchange Rates
by
Bin Zhou
MIT
Sloan School Working Paper 3485-92
September 1992
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
September 1992
M.I.T.
NOV
LIBRARIP?
1
3 1992
High Frequency Data and Volatility
in Foreign Exchange Rates
Bin Zhou
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
September 1992
Abstract: Exchange
rates, like
substantial heteroscedasticity.
many
other financial time series, display
This poses obstacles
in detecting
trend and
changes. Understanding volatility becomes extremely important in studying
financial time series. Unfortunately, estimating volatility
data, such as daily, weekly, or monthly observations,
is
from low frequency
very
difficult.
Re-
cent availability of ultra-high frequency observations, such as tick-by-tick
data, to large financial institutions creates a
volatile
time
series. Tick-by-tick
new
possibility for analysis of
data provides us a near continuous observa-
much
tion of the process that gives us potential to study volatility in
However, high frequency data has extremely high negative
correlation in
its
return.
In this paper,
and US Dollar (DM/$) exchange
we use
tick-by-tick
rates to explore this
first
detail.
order auto-
Deutsche Mark
new type
of data.
propose a model to explain the negative autocorrelation and a volatility
timator using the high frequency data.
DM/$
We
es-
Daily and hourly volatility of the
exchange rates are estimated and the behaviors of the
volatility are
discussed.
Key Words:
Financial time series; tick-by-tick data; heteroscedasticity;
Introduction
1
There
is
considerable literature analyzing the behavior of exchange rates.
However, structural exchange rate modeling has not been very successful.
studying monthly data, Meese and RogofF (1983a,b) have shown that
By
a
random walk model
at least as well as
fits
more complicated structural
models.
Empirical study (Hsieh 1988) has shown that daily returns are approxi-
mately symmetric and leptokurtic
are
weak but not independent and
heavy
tion of the
(i.e.,
heavy
tailed).
is
[8]
([6],
simple.
The argument
[4]).
The amount
is
for
mean and
is
is
explana-
indepen-
variance change
changing variance of returns
of "information flow" that cause
not constant over time. Hence there
of the price changes
One
the hypothesis that data
is
dently distributed as a normal distribution whose
over time
autocorrelations
identically distributed (iid).
tailed distribution
and
The
change
in prices
no reason to beheve that the variance
constant over time.
Clark (1973) and
many
others
(Mandelbrot and Taylor 1969, Praetz 1972) have argued that observed
turns
come from a mixture
of
normal distributions.
Xt denotes the daily return of the
given information
cjt is all
number
the
If
random
the
re-
variable
price, the conditional distribution of ,Y(
is:
XtliJt
where
is
-^NifiJiu,))
the information available at time
of transactions (Mandelbrot
(1)
t.
The quantity
ujt
could be
and Taylor 1969), or trading volume
(Clark 1973).
One parametrization
ied by
of this conditional heteroscedasticity was
first
stud-
Engle (1982) where:
p
fiut)=ao + J2a,{Xt-,-fi)\
ao>Oa,>0,
i
=
l,...,p
(2)
«=i
It is
called the autoregressive conditional heteroscedasticity
since the heteroscedasticity
Because the
in financial
ARCH
represented in an autoregressiv'e fashion.
ARCH model exhibits the conditional heteroscedasticity present
time series and
used to analyze
is
(ARCH) model
many
is
mathematically easy to manipulate,
financial time series. Previous studies
model provides a
close
it
has been
found that the
approximation to many financial time
series.
Since then
many
other parametrizations, such as the generalized autoregres-
sive conditional heteroscedasticity
(GARCH) model
(BoUerslev 1986), have
been proposed. They captured some characteristics of the
volatility such as
But there also exists problems. The parameters need to
these models use historical data. The volatility estimates
volatility clustering.
be estimated. All
are often lagged.
The
data has opened up new possibility
availability of high-frequency
estimating the
volatility.
in
Tick-by-tick data provides us with a near contin-
uous observation of the process.
It
gives us potential to study volatility in
great detail. Understanding volatility
is
the key issue in the conditional het-
model
(1) and any other financial time series model. This
paper explores tick-by-tick data and uses the data to estimate and study the
eroscedasticity
volatility.
High-Frequency Data
2
Because of
growing computer power, gathering financial data
fast
than ever. Data
began
is
easier
no longer recorded daily or weekly. Many large institutions
to collect so called tick-by-tick
exchange rate
in
the early eighties.
Different from stock market, the foreign exchange market has
graphical location, and no "business-hour" limitations.
tiated
is
and traded over the telephone.
volume are not known
to the public.
The
no geo-
deals are nego-
The transaction prices and trading
The exchange rates used for most re-
search are the quotes from large data suppliers such as the Reuters, Telerate,
or Knight Ridder.
pliers.
Any market maker can submit new
The quotes then
are conveyed to data subscriber's screens.
suppliers cover the market information worldwide
day.
The quotes
quotes to the data sup-
The data
and twenty-four hours a
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 which transactions are being conducted.
some participants
It is
to manipulate indicative prices occasionally
possible for
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,
it
match the true
is
generally
felt
that these indicative prices would closely
prices experienced in the market.
A
reader
who
is
unfamil-
Table
PUB.
1:
A Sample
of Tick-by-tick
Exchange Rates
Table
n
2:
Summary
Statistics of Tick-by-tick
Returns
Figure
1:
Original Quotes
From the Reuters and Telerate
adu
Figure
2000
2:
Validated Quotes
4000
sooo
ddu
BOOO
Figure
3:
Sample Kurtosis of Return
of n-ticks
I
600
400
200
ddu
unstable and then decreases.
The
last stage,
decreasing kurtosis,
is
due to
increasing negative autocorrelation in data (see Figure 4).
Although we expect a
slightly negative first lag autocorrelation as
ported in other exchange rate literature, a
in tick-by-tick return
—47%
it
re-
negative autocorrelation
was surprising. To be cautious, we also calculated the
The
auto-
correlation coefficients are —0.4718, —0.4691, —0.4665 and —0.4632.
The
autocorrelations in four subgroups according to four quarters.
negative autocorrelation
is
consistent.
assumed for low frequency data. Our
any fundamental difference between the high and low
not follow a Brownian motion as
question
is if
there
is
Obviously, high frequency data does
it is
frequency data. After further studying the data, we find that the difference
in
high and low frequency data
is
level of noises.
The
noise
is
neghgible in
low frequency data, but becomes very significant in high frequency data.
noise
may come from many
different sources. For
example,
it
The
could be round
As we know that round
off Brownian motion introduces negative autocorrelation in its return. There
are also updating noises in quotes. To be visible in the market, traders keep
off error.
All financial
data
is
quoted
in finite digits.
Figure
First-order Autocorrelations of Return of n-ticks
4:
k
^,.^M*#**^
I
400
200
600
updating their quotes. The new update
previous quotes even the market
is
800
1000
often sHghtly different from the
Typographical errors are another
is still.
source of noise. Summarizing these arguments, we assume following process
for the
exchange
rate:
=
S{t)
where S{t)
is
dit)
+
+
B{Tit))
logarithm of the exchange rate, B{-)
motion, both
d{-)
and
6^
the Brownian motion B{.).
is
the standard Brownian
are assumed deterministic functions, r(-) has
t(-)
and
positive increments,
(3)
tt
is
the
The
mean
noise,
random
zero
Cj,
is
noise independent to
combination of several sources
which were mentioned above. This extra noise causes most of the negative
autocorrelation in high frequency data.
=
Let X{s,t)
S{t)
—
X{s,
where
Zt
fi{s, t)
=
is
S{s), the return in interval
t)
=
fiis, t)
a standard normal
d{t)
—
d{s).
The
Var(X(3, 0)
+
ais, t)Zt
random
^'(5,
t)
+
variable, a^{s,t)
8
r]\t)
+
Then
+ tt-e,
variance of the return
=
[s,t].
(4)
=
r(f)
is:
t]\s)
+
2c{s,
t)
—
t{s)
and
where
=
r/^(0
and
Var(e()
like
cT^{s,t) diminishes.
sample
random walk.
The return, X{s,t),
a
high-frequency data, the noise
-47%
\s — t\, noise
When |5 — <|
is
for the
DM/$
is
—
\s
t\
increcises,
becomes negligible and
decreases to near zero,
the difference of two noises.
order autocorrelation of such series
first
When
Cov(e,,et).
For large
cr^{s,t) increases as well.
X{s,t) behaves
=
c(s,t)
is
about —.5.
When we
The
study
no longer negHgible. An autocorrelation of
exchange rate indicates that
level of noises
is
very high
in tick-by-tick data.
There are several
difficulties
is
lack of information about r(<) which
volatility. (7^{t
—
=
6,t)
T{t)
—
r(<
—
or simply 6- volatility. Next section,
volatility for
3
analyzing the process
difficulties in
6)
we
call
(3).
it
One
of the
the cumulative
called the ^-increment of volatility
is
we devote our
attention to estimate the
any increment.
Volatility Estimation
we concentrate on estimating the volatility of a given time
T{n) — t(0). The function T{t) can be estimated increment by in-
In this section
[0,n],
crement.
We
first
derive an optimal estimator of the volatility based on
assumption of constant variance and zero mean. Then we generalize the
sults to general process (3).
re-
Proofs of the theorems are listed in appendix
I.
Theorem
from
1
Assume
that {S{t),t
0, l,...,n}
is
a series of observations
the process
S{t)
where
=
et,t
=
distribution
l,..,n, are
and
T[t)
=
=
B{T{t))
+
e,
(5)
independent and identically distributed with normal
a^t
+ b.
Let Xt
=
S{t)
— S{t — I). Then
the
maximum
likelihood estimator of a^ is
a'MLE = {l/n)±{X! + 2X,X,_,-^,)^-^^-^
where
P
=
^
ana
p
=
—
(6)
MLE
This
An
close.
^k^
However, we noticed that p and p' are very
unbiased estimator can be obtained by eliminating the factors
not unbiased.
is
and 4:
a'u
= {Un)Ei^f +
2X,X,.,).
(7)
1
Theorem
Under
2
Theorem
the assumptions of
I,
mean and variance
the
of the estimator {!) are:
Ea^u
=
^^
Var(<7\)
=
{\/n)a'i6
(8)
and
From
we
(9),
we apply estimator
cLSsume that n
is
multiple of
Theorem
— S{i — k),i =
The variance
is
is
r?
(l/n)a''(6^
minimized
...,
n where we
(A7,,+2.Y,,X._,,,).
(10)
given in following theorem:
3 Under the assumptions of theorem
=
k, 2k,
i=k,2k,...n
unbiased and the variance
Var(a2f;,fc)
X down
S{i)
Let
E
= "
is
k.
Since aggregation increases the variance
=
(7) to Xi^k
(t2 u,k
Estimator (10)
rj'^/cr^.
(9)
be optimized by properly
find that variance of a'^u can
adjusting the variance ratio
(7^,
+ ie^ + S^^ + ^'/n\
at k
=
+
[-Jj^]
IG-^
or k
1
n^
+ S/-) + V/"'
=
[-j^^]
+
1,
where
(H)
[x]
rounds
to the next integer.
Since the noise
ej
is
independent, the variance of the estimator can be
further reduced by averaging the estimator (7) at different start points:
<^'
=
E(A'.^ +
^
kn
,^,
10
2X,^k.kX,.2k.k)
(12)
In fact,
can be proved that:
it
+ 8-j-^) + ^'/n^
16^
K O
Kff
< -a\6k +
Var(a2)
ft
The above
three theorems assumed
However the estimator can
that
we have observations
Then the
volatility r{<„)
—
^(^0, in)
also
{5(<,),?
r(<o) can
4
—'2k^—'2k
=
+
l,...,n}
from process
(3).
be estimated by:
2X,_,,,X„2k,k)
(14)
.=1
Assume Cov(ti,ti-k)=0 for
E\/(<0,<n)
in
= ] f^iXlk +
^
Theorem
=
and constant variances.
a more general case. Suppose
noises
iid
be used
(13)
alii.
Then
T{t^)-T{to)
k-l
+
^(i/^')[a2(f,_i_;t,<._fc)-cr2(<„_,+i, <„_,)]
i=l
1=0
n
+{\lk)Y,[^i\U,U_,)
+
2ti{ti,U_k)ti{ti_k^,_2k)]
(15)
:=0
where
r]^{t)
=
Var{et).
Only assumption we made in this theorem is the uncorrelated noises.
can be any increasing function and tt is not necessary stationary. Since
T{t)
n
J2Wit„U_k) +
2fi{t„ t,_k)Kt,-k, t,.2k)]
<
3 max{|/.(i„
U-k)\Mn) -
d{0)]
i=k
the last term in (15)
smooth. Therefore,
if
is
negligible in high frequency data
for large n, the
no jumps occurred
in
estimator (14)
is
if
the drift d{t)
approximately unbiased
time interval [^o^n]- This estimator
be updated when new data becomes available.
the volatility T{t) dynamically.
11
It
is
is
also easy to
will allow us to
estimate
Estimating Volatility of Exchange Rates
4
In this section,
rates.
we apply the
The data
volatility
estimator (14) to
DM/$
exchange
has one discontinuity which was in the week of August 13
when the data base was shutdown due to a power outage accident in lower
Manhattan area. The return of that week is set to be zero and so does the
volatility.
we estimated the variance ratio rj^/a^ to
be approximate 6. Minimizing upper bond of variance (13), we have k =
6. Using all available tick-by-tick data, we estimate the volatility of DM/$
exchange rate in entire 1990. The estimate is .010349. To verify this estimate,
we compare it to the estimate under the Brownian motion assumption. If the
data had no noise and followed a Brownian motion, the quadratic variation
To choose the parameters
k,
{n/k)
Qk=
Yl x{Uk-k.tikf
i=k
would be a standard estimator of the
is
used on data with noise,
it
volatility.
When
the quadratic variation
overestimates the volatility.
along with the sample frequency.
The expectation
The
bias decreases
of the quadratic variation
is:
{n/k)
1=1
{n/k)
i=l
where
c{t,f.,t,k-k)
=
When
Cov(e(,^, e(_^_^).
e('s
are uncorrelated
and
/I's
are
negligible,
{n/k)
EQ, =^r(^„)-r(<o) +
2
^tjI
(17)
1=1
which decreases as k increases.
We
axes have a logarithmic scale.
When
variation
is
is
k
=
Qk
against k in Figure
the frequency
about the same as our estimate except
caused by small sample
When
plotted
I,
size.
is
is
12
Both
low, the quadratic
for a high variation,
In high frequency, the bias
the quadratic variation
5.
is
which
tremendous.
about thirteen times the
size of
our
Figure
3
Quadratic Variations Using n-tick Returns
5:
.
s
>
tidu
estimate. Therefore, from (17), the total variance of the noise
about
is
six
times the total volatility which confirmed our early estimate of the ratio.
To estimate
daily volatihties,
we need
a day since the foreign exchange market
market.
We
is
time.
GMT
is
is
much
less.
is
as a
7:00pm
is
more than
7,000.
Quotes
in
For small n, volatility estimate from (14)
could be negative. In such a case, we
The
GMT
This twenty-four hour period covers most activities of the
weekends or hoHday
The
Mean Time (GMT)
9:00am Tokyo time and 24:00
world market. Average weekday daily ticks
volatility.
and the end of
a twenty four hour international
choose 24 hours from 0:00 Greenwich
day because that 0:00
New York
to define the start
let
the estimate to zero to avoid negative
daily volatility estimates of
DM/$
are plotted in Figure
6.
and Aubank
gust 2, 3, 1990. On January 4 and 5, the German central
surprisingly
intervened the foreign exchange market and pushed the dollar lower. On
January 30, a wide market followed a rumor that Mr. Gorbachev was consix largest volatilities
were on January
4,
sidering resigning as secretary of the former Soviet
5,
30, July 12
Communist
Party.
July 12, the dollar tumbled because possible lower interest rate by the
13
On
US
Figure
6:
Daily Volatility Estimates of 1990
DM/$
9
a:
•r
a
iiiiiJliiiiltl
1990
J990
Mu
On August
Federal Reserve.
IMO
1990
1990
1990
990
990
ha
;990
hi
Auf
Scy
SI>T
Dec
Mt/
and
2
3,
1991
Ju
the Dollar had another wild ride as
the news of Iraq's invasion of Kuwait spread around the world. However, the
large volatility does not always have large price change.
for
above
six
respectively.
The
daily change
days are -0.0537, 0.0162, 0.0212, -0.0178, 0.0058 and -0.0074
On August
2
and
the exchange rate only changed 58 and 74
3,
points which are about the average.
When we
study low frequency data
like daily price, noise
becomes
negli-
and the price change, Xi, approximately normal distributed with mean
zero and variance erf, or rescaled return ¥{ = X,/ai is a standard normal
gible
random
variable. Therefore
we can
the normality of rescaled return
on Saturdays), we plot
return Yi in Figure
Table
3.
The 95%
7.
Y,.
test
our model assumption
(3)
by testing
Excluding zero volatility estimates
Q-Q normal plots
The basic statistics
for
both return
A",
Xi and
Yi
of both
confidence intervals of the
moments
(all
and rescaled
are shown in
of Yi are given in
the parentheses. Table 3 also shows the Kolmogorov-Smirnov goodness-of-fit
test for normality.
Comparing the
conclude that
much
Yi
is
statistics in
closer to having a
14
column
A',
and column
normal distribution.
It
Yi,
we
indicates
Figure
Q-Q
7:
ReKiled Daily Returns
Daily Returni
-2-10
QntnolM
«f
2
1
Stindud Monnil
Table
3:
Plot of Daily Returns
-2-10
Q^ittdu
of
1
Srindard Konnftl
Basic Statistics of Daily Returns
2
Table
4:
Average Daily Volatility
.
Figure
4 soE'O:
8:
Average Daily
Volatilities
-r
Figure
3
soE'Oe
.
3
00E-08
..
2
SOE'Ot
.
2
ooE-oe
1
50E-09
9:
Average Hourly
Volatilities
.
8
8
8
8
8
8
S
17
8
8
8
8
S
S
8
Figure
10:
Q-Q
Plot of Hourly Returns
Conclusion
5
High frequency data can be described as a Brownian motion with a
This noise brings a strong
data.
The
first
noise.
lag negative autocorrelation in high frequency
autocorrelation decreases as frequency decreases since the role of
the noise reduces. High frequency data can be used to estimate the volatility
in high
frequency in reasonable precision.
Different
from other
volatility
estimator, our volatility estimator mainly uses the data within the period
we
are interested instead of historical data. This allows us to capture the market
volatility quickly
without delay. The estimate
is
nearly unbiased
when
price
has no jump. Since trading volume and volatility are highly correlated, this
is more important in exchange market than in
unknown trading volume.
Besides of many other applications of the volatility, we are more interested
type of volatility estimation
other market because of
in
modeling and forecasting the time
series. Volatility
the return to address the heteroscadesticity
can analyze the data
in volatility scale in
eliminate the heteroscadesticity.
We
like
19
ARCH
model.
We
also
stead of calendar time scale to
will discuss
future papers.
the
can be used to rescale
more
of these issues in our
^
Appendix
•
Proof of Theorems
I:
Proof of Theorem
1:
Under assumptions
of the theorem,
=
Xt
is
where
p
random
a normal
=
77^
variable with
the variance of
is
—Tj^/v^.
The
Notice that Xt\Xt-i
L{s\
+
e,
mean
-
c,_i
zero and variance v^
Xt has the
£(.
likelihood function of Xt
=
L{s\p;Xu...,Xr,)
variance s^
aZt
=
i'^(l
p;
X,
,
is
first
=
(7^
+ 2r/^,
lag autocorrelation
is
/(A'„|X„_i)/(X„_i|X„_2).../(.Yi|Xo)
also a
normal variable with mean pXt-i and
—
p^),
we have
...,
X„)
=
(27r.^)-"/^exp(-
t ^^^^^^).
—
(=1
The
log-likelihood function
i{s\p;X,,...,X„\Xo)
The
is
= -^M2;r.^)-X: ^^'7f-^^'
partial derivative of the likelihood function with respect to p
di
dp
^^ iXt-pXt-,)Xt-i
h
Setting the derivative equal to zero and solve for
«
P
Similarly, for 5^,
_
-
Y^^-\
r-n
/),
XjXt-i
v-2
we have
d^
^
n
^ {Xt-pXt-,Y ^
20
we have
is
or
s
1
^
= -j2{Xt-px,_,f
2
"«=i
^
Substituting p hy p
m
(=1
(=1
t=l
above formula, we have
"
(=1
(=1
= -t-'^ii^-PP')
where
It is
easy to show that
a'
Therefore, the
= v'{l+2p) = s'{l+2p)/{l-p')
maximum
=
•
Proof of Theorem
hkelihood estimator of a^
i|:,.v?..v,A-,..A,lL^
2:
= (l/n)f;(Var(A^) + 2EX,X,_i)
E{l/n)f2iXf + 2X,Xt-,)
1
1
= {l/n)±{a' +
1
and
Vara\ =
is
(l/n)2Var X:(A?
+ 2A^X,_i)
1
21
2rj^-2r,^)
= a\
1
+2aZt-xtt
-
2aZt-itt-x
n
a^
Proof of Theorem
Since VarX,,i
—
Ws.v{ka^u,,)
2e(e(_2
+
2e,_ift_2)
+
a^
3:
ka^
(9) implies that
,
= {k/n}{ka')\6 + 16-^ +
Var(aV) =
8-^) + ^Vin/kf
(l/n)(<72)2(6A:
+ IG^ + sf^) + V/n^
ka*
CT''
•
variance reaches
Proof of Theorem
EVito,tr.)
=
minimum when
4:
(\/k)J2[a'it,_,J,)
+fl'^{t„t,_k)
=
-
n^
or
The
e^
n^
n
•
-
[r{tn)
-
+
+
r]'{U)-ri'{U_k)
2^l{t„t,_k)fi{t,_k,t,_2k)]
T{to)]
k-1
1
=1
k
+ {Uk)j:iv'{t,-k)-v'itn-.)]
1=0
n
1=0
22
tl]
Appendix
II:
Validation
Program
There are many reasons to have outliers in original data set shown in
Figure 1. Most quotes are typed in by human, there are unavoidable keying
errors.
Most
outliers
we found
are this type of errors. Outlier also could be
caused by large bid and ask spread. In such case, at least one of bid or ask
and becomes an
Following program
price does not reflect true market price
electronic error also
makes
outliers.
outlier. Occasionally,
is
designed to remove
above three types of outhers:
A
i)
ii)
iii)
quote
is
considered as an outlier and removed from the time series
a rate
is
more than
bid and ask spread
a rate
is
5 or less than
is
1,
more than 50
above or below than
its
if
or
points, or
neighbor prices more than a certain
threshold.
(iii),
we
of the data.
If
For
carry out two regressions using ten bid prices on each side
the current bid price
both regressions,
it
is
is
higher or lower than c-point from
considered as an outlier, where c
is
range from 15 to
30 points dependent on the variances of the neighbor points.
Whenever an
is detected, we go back ten steps and repeat above procedure.
About 0.73% of data have been removed by this validation program.
The first ten removed data due to (iii) are listed in Table 6 with possible
outlier
explanation of error.
23
Table
6:
The
First
Removed Data
Ten Outliers Removed From 1990
DM/$
[6]
Friedman, Daniel and Stoddard Vandersteel (1982), "Short-run fluctuation in foreign exchange rates."
[7]
Goodhart, C.A.E. and
Money and
[11]
Karatzas, loannis and Steven E. Shreve (1988), Brownian Motion and
New
York.
Mandelbrot, B. and H. Taylor (1969), "On the distribution of stock price
differences." Operations Research, 15, 1057-1062.
Meese, R. A. and K. Rogoff (1983a), "Empirical exchange rate models
of the seventies: do they
[12]
fi-
rates: 1974-1983." J. of Inter. Econ., 24, 129-145.
Stochastic Calculus, Springer- Verlag,
[10]
in
Finance, 10, 23-52.
Hsieh, David A. (1988), "The statistical properties of daily foreign ex-
change
[9]
"Every minute counts
L. Figliuoli (1991),
nacial markets." J. of Inter.
[8]
Econ., 13, 171-186.
J. Intern.
fit
out of sample?."
J.
of Inter. Econ., 14, 3-24.
Meese, R. A. and K. Rogoff (1983b), "The out of sample failure of
empirical exchange rate models:
in J. Frenkel, ed..
sampling error or misspecification?."
Exchange Rates And International Microeconomics,
Chicago: University of Chicago Press.
[13]
Praetz, P.D. (1972), "The distribution of share price changes." Journal
of Business, 45, 49-55.
[14]
Taylar, Stephen (1988), Modeling financial time series,
Sons.
25
John Willey
&
V928
122
Date Due
"Wr 2 7
1991
7\9^
OCT.?-
Lib-26-67
MIT LIBRARIES
.DUPL
3 9080 02879 8350