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Applied Economics
, 1996, 28, 377
Ð
386
The interaction between the frequency of
market quotations, spread and volatility
in the foreign exchange market
ANTONIS A. DEMOS and CHARLES A. E. GOODHART
Department of Economics
, º
niversity of Reading
,
P
.
O
.
Box 218
, ¼
hiteknights
,
Reading
RG62AA
, º
K
and
Department of Economics
, ¸
ondon School of Economics
,
Financial
Markets Group
,
Houghton St
, ¸
ondon
¼
C2A 2AE
, º
K
There is an empirical relationship between volatility, average spread, and number of
quotations in the foreign exchange spot market. The estimation procedure involves
two steps. In the Þrst one the optimal functional form between these variables is
determined through a maximization procedure of the unrestricted VAR, involving the
Box
Ð
Cox transformation. The second step uses the two-stage least squares method to
estimate the transformed variables in a simultaneous equation system framework. The
results indicate that the number of quotations successfully approximates activity in
the spot market. Furthermore, the number of quotations and temporal dummies
reduce signiÞcantly the conditional heteroskedasticity e¤ect. We also discuss informa-
tion aspects of the model as well as its implications for Þnancial informational
theories. Inter- and intra-day patterns of the three variables are also revealed.
I. INTRODUCTION
laborious, but could still be worth attempting at a later
stage.
Another way is to follow previous studies of mixture of
distributions [see, for example,
Harris (1987),
Gallant
et al
.
(1989) and (1990)
, and
Laux and Ng (1991)
] and use volume
as a proxy for the number of information events. However,
Jones, Kaul and Lipson (1991)
show that volume is a noisy
and imperfect proxy for information arrival, and that the
number
of transactions is a better variable in a model with
a Þxed number of traders. However, there are no volume
data available in the forex market [see, for example
Good-
hart and Demos (1990)
]. Instead the frequency of quote
arrivals over ReutersÕ screens is used as the proxy for market
activity. This may capture the e¤ect of market activity on
volatility, up to the extent that news is reßected in changes
in current market activity.
The next question is whether it is permissible and appro-
priate to examine the
contemporaneous
interaction between
quote arrival and volatility, or only to relate volatility to
quote arrival using information available at
t
!1 and
earlier. The previous literature indicates that this decision is
important. The results using information on market activ-
ity, whether quote frequency or volume, at
t
!1 and earlier
suggest that such data has no signiÞcant ability to predict
volatility, given past data on volatility, [for example,
Jones,
It is common in the literature for variations in the arrival of
ÔnewsÕ in Þnancial markets to be measured directly from the
data on the volatility of prices/returns. [See, for example,
Engle and Ng (1991)]
. In one sense this approach assumes
what needs to be tested, i.e. that ÔnewsÕ drives volatility.
Moreover, the ARCH e¤ects commonly found in such
Þnancial series, [see
Bollerslev
et al
. (1992)]
, may well rep-
resent some combination of the autoregressive character-
istics of ÔnewsÕ arrival, i.e. the bunching of ÔnewsÕ, and of
ÔpureÕ market volatility. Given the theoretical results on
the mixtures-of-distributions hypothesis by
Clark (1973),
Tauchen and Pitts (1983),
and
Andersen (1991)
among
others, when time is measured in calendar time, the condi-
tional variance of returns will be an increasing function of
the actual number of information arrivals [see
Bollerslev
and Domowitz (1991)]
.
A number of questions follow. The Þrst is what indicator
of information arrival to use. One possibility would be to try
to exploit the data available over the ÔnewsÕ pages on the
electronic screens, for example, Reuters AAMM page of
ÔnewsÕ of interest to market dealers [see
Goodhart (1990),
Goodhart
et al
. (1991)]
. The construction of any such index
would undoubtedly be somewhat subjective, and extremely
0003
Ð
6846
(
1996 Chapman & Hall
377
378
A
.
A
.
Demos and C
.
A
.
E
.
Goodhart
Kaul and Lipson (1991),
Lamoureux and Lastrapes (1990),
Bollerslev and Domowitz (1991)]
. On the other hand,
Lamoureux and Lastrapes (1990)
and
Laux and Ng (1991)
Þnd that the use of
contemporaneous
data on market activity
virtually removes all persistence in the conditional variance
in their series, being daily stock returns and intra-day cur-
rency future returns respectively.
Bollerslev and Domowitz
(1991)
doubt the validity of using contemporaneous data on
the grounds of simultaneity and that the traders informa-
tion set does not include contemporaneous data on market
activity. Simultaneity is dealt with by using a simultaneous
equation system estimation procedure. With respect to the
second objection, market tradersÕ way of life is watching the
screen, so they will be virtually instantaneously aware of
a change in the speed of ßow of new quotes. Furthermore, it
is argued that the entry of a quote on the screen must
have both temporal and causal priority over volatility
developments, since the latter can only be estimated
once
decisions to enter a new quote have been taken
and executed. Hence the hypothesis is that, in this ultra-
high frequency data set, the ÔcausalÕ linkages will be
found to be stronger from quote frequency to volatility
when both are taken over the same short time interval, than
vice versa.
Here we examine international patterns of intra-day trad-
ing activity and some properties of the time series of returns
for the Deutschemark/Dollar and Yen/Dollar exchange
rates in the foreign exchange market through the interbank
trade. The purpose is to provide some information useful in
the further development of the microstructure of trading
models and to compare the empirical results with previous
ones and theoretical models already in existence.
The results in Bollerslev and Domowitz (1991) are ex-
tended in two di¤erent ways. First, certain arguments are
outlined (in Section III) explaining why quote frequency
data might be better entered in log, rather than in numer-
ical, form, and we search for the best Þtting transformation
of the data using the Box
Ð
Cox transformation. Second, in
Goodhart and Demos (1990),
we argue that there are certain
predictable temporal regularities in the foreign exchange
market (for example, the regular release of economic data at
certain pre-announced times, the passage of the market
through the time zones punctuated by market openings and
lunch breaks (especially in Tokyo)). Consequently temporal
weekly, daily and half-hourly dummies are added to all
equations. As will be shown in Section III, these two cha-
nges do make a di¤erence to the results. The conditioning of
the variables of interest on such temporal dummies allows
us to distinguish between public and private information,
something of great importance to informational theories of
market micro-structure (see, for example,
Admati and
Pßeiderer (1988),
Son (1991),
etc.).
Although the emphasis here is on the relationship be-
tween quote frequency and volatility, since it is a less-re-
searched area, we examine the three-fold interrelationships
between quote frequency, volatility and bid-ask spreads.
The positive relationship between volatility and the spread
is well-known in the literature [see, for example,
Ho
and Stoll (1983)
and
Berkman (1991)
]. We suggested
earlier that the absence of any signiÞcant ability of
prior quote frequency to predict volatility implied that
volatility may have incorporated both the contempor-
aneous evidence from quote arrivals and other sources of
information. If so, we would not expect quote arrivals, either
contemporaneous or lagged, to inßuence spreads, given
volatility.
Where, however, one might Þnd some relationship be-
tween spreads and quote frequency would be among the
constant temporal dummy variables. Whereas some sources
of news are continuously unfolding, the market has a pat-
tern of openings, lunch breaks, and closes, which might
inßuence both quote frequency and spreads, independently
of the pattern of price/return volatility. The work of
OldÞeld
and Rogalski (1980),
Wood, McInish and Ord (1985),
French and Roll (1986),
and
Harris (1986)
among others
have stimulated considerable interest in documenting the
pattern of stock market returns and their variances around
the clock.
Admati and Pßeiderer (1988)
, and
Foster and
Viswanathan (1990)
o¤er some theoretical explanations for
some of these empirical Þndings. Here we aim to extend this
work by looking also at the temporal patterns of quote
frequency and spreads. We examine the relationship be-
tween the sets of temporal dummy variables in Section IV.
We conclude in Section V.
II. THE DATA SET
)), and the number of new
quotations within this interval are recorded. In a few instan-
ces there were too few observations in a half-hour to calcu-
late a meaningful estimate of volatility. In such cases we
substituted the values for the lowest calculable observed
volatility, and the accompanying spread, in a half-hour of
that week. This resulted in around potentially 2500 half-
hourly observations. In fact, 5 out of the 12 weeks were
chosen for analysis, avoiding any weeks with public hol-
idays in the main country participants. The results are
robust to this choice.
At this point we should review some pitfalls associated
with the approximation of market activity by the number of
quotations. Market participants have claimed that during
very busy periods traders may be too occupied in dealing
through their telephones to update their screens immediate-
ly (see
Goodhart and Demos (1990))
.
Per contra
, when the
market is dull some market participants may enter new
t
)!ln(
e
t~1
The continuously quoted data are divided into discrete
segments in the following way. The 24-hour weekday is
divided into 48 half-hour intervals and the average spread,
standard deviation of the percentage Þrst di¤erence of the
rates quoted (ln(
e
Interaction between quotations
,
spread
,
and volatility in FOREX
379
Table 1.
Quasi log
-
likelihood values as a function of the BoxÐCox exponent
DEM
JPY
Log-
sp*
t
Log-
Log-
sp*
t
Log-
c
likelihood
likelihood
likelihood
c
likelihood
likelihood
likelihood
1.0
!1304.8
!1675.5
!5395.5
1.0
!1699.8
!1736.9
!5202.1
0.5
!1053.3
!1532.9
2
5170.2
0.5
!1386.8
!1706.4
!4894.5
0.3
!1012.7
!1489.6
!5228.3
0.4
!1353.6
!1703.9
2
4882.1
0.2
2
1008.6
!1470.4
!5311.9
0.3
!1330.3
!1702.3
!4894.1
0.1
!1016.9
!1452.6
!5438.0
0.2
!1316.8
2
1701.9
!4934.2
0.0
!1040.9
!1436.2
!5607.8
0.1
2
1312.9
!1702.7
!5005.8
!0.5
!1429.9
!1375.0
!6990.1
0.0
!1314.9
!1703.9
!5110.8
!1.0
!2255.8
2
1350.2
!8867.4
!2.0
!4525.9
!1385.2
!13 130.0
Note: Bold indicates the optimum
c
.
quotes to generate some business. However, in general
the temporal pattern of the markets may di¤er from the
temporal pattern of the ÔnewsÕ generation process. Markets
often close almost entirely, for example, at weekends and
over the Tokyo lunch hour, or become very busy, while
some ÔnewsÕ is continuously occurring. Although we would
expect more ÔnewsÕ always to be associated with a higher
frequency of quotes, as long as some markets are in opera-
tion, the functional form of this relationship, for example,
linear, log-linear, etc., remains unknown.
The functional form of the relationship between these
variables needs careful consideration. There is no apparent
reason why the average spread, volatility, and number of
quotations should be linearly related, rather than, say, log-
linearly. On theoretical grounds both functional relation-
ships would have the same characteristics as discussed in
Sections I and II. Hence, we left the data to decide on this by
using the following procedure.
We Þrst transformed the three variables using the
Box
Ð
Cox transformation. The reduced form of the SES is
a restricted Vector Autoregression (VAR) of order 2; we
estimated the unrestricted form for each currency for di¤er-
ent values of the Box
Ð
Cox exponent, i.e. the following
VAR(2) was estimated for di¤erent values of
III. ESTIMATION AND RESULTS
c
1
,
c
2
, and
c
3
The following Simultaneous Equation System (SES) is to be
estimated:
(the exponents):
p
t
"Dummies#a
12
sp
#a
13
n
#a
14
p
t~1
*
t
sp*
t
n*
t
p
b
11
b
12
b
13
*
t~1
sp*
t~1
n*
t~1
t
t
"
Dm
.#
b
21
b
22
b
23
#a
15
p
t~2
(1.a)
b
31
b
32
b
33
sp
"Dummies#a
21
p
t
#a
23
n
t
#a
24
sp
t~1
#a
25
sp
t~1
(1.b)
d
11
d
12
d
13
*
t~2
sp*
t~2
n*
t~2
e
1t
e
2t
e
3t
n
"Dummies#a
31
p
t
#a
32
sp
#a
33
n
#a
34
n
(1.c)
#
d
21
d
22
d
23
#
t
t
t~1
t~2
d
31
d
32
d
33
are the standard deviation of the
percentage change of an exchange rate, the average
spread, and the number of quotations within the
t
th half-
hour interval, and the system is separately estimated
for the two currencies under interest, i.e. the Deutschemark
and Japanese Yen, against the US dollar. As Þnancial
time series su¤er from conditional heteroskedasticity
e¤ects, we include lagged dependent variables in
Equations
1.a to 1.c.
Moreover this helps in the identiÞcation of
the system. The estimation method is two-stage least
squares.
p
t
,
sp
t
, and
n
t
where
p
*
t
"(
p
c
Ç
t
!1)/
c
1
,
sp*
t
"(
sp
c
È
!1)/
c
2
, and
n*
t
"(
n
c
Ê
!1)/
c
3
. Notice that for
c
1
"c
2
"c
3
"1, and
t
c
1
"c
2
"c
3
"0 we have the linear and log-linear forms,
respectively.
In
Table 1
we present the values of the quasi log-likeli-
hood function for the transformed variables, for di¤erent,
but common across the three variables, values of
c
.Itis
depends
on the variable and the currency. However, notice that the
c
1
1We avoided Full Information Maximum Likelihood estimation on the grounds of the strong non-normality of the residuals (see below).
p
*
t
Log-
n*
t
p
*
t
Log-
n*
t
p
t
p
where
t
immediately apparent that the optimal value of
380
A
.
A
.
Demos and C
.
A
.
E
.
Goodhart
values between 1
and !2 for the Deutschemark, and 1 and 0 for the Yen.
What we are doing here in e¤ect is a grid search of the
pseudo-likelihood function with respect to the
c
, at least for
c
meters and their heteroskedasticity robust standard errors
are presented in
Table 2.
p
*
t
"Dummies#a
12
sp*
t
#a
13
n*
t
#a
14
p
*
t~1
#a
15
p
*
t~2
parameter.
Although we chose the steps of the grid to be 0.05, in
Table 1
only some representative values of the log-likelihood func-
tion are reported, for two reasons. First, the likelihood
function is not very ßat around the optimum, with the
possible exception of the Yen average spread equation, and
second, because of space considerations.
The optimal
c
(2.a)
sp*
t
"Dummies#a
21
p
*
t
#a
23
n*
t
#a
24
sp*
t~1
#a
25
sp*
t~2
#a
26
sp*
t~3
(2.b)
n*
t
"Dummies#a
31
p
*
t
#a
32
sp*
t
#a
34
n*
t~1
#a
35
n*
t~2
(2.c)
c
values for the Deutschemark are
c
1
"0.2,
Some important points emerge from this table. First, the
results are quite robust across the two currencies, although
the functional form of the variable is di¤erent. Second,
notice that in the volatility equation (
Equation 2.a)
the
average spread and the number of quotations have a strong
positive e¤ect on volatility. These positive relationships
of spread-volatility and volatility-activity are well-
documented facts in the literature.
Ho and Stoll (1983),
Berkman (1991),
as well as the probit model of
Hausman,
Lo and MacKinley (1991)
of trade by trade stock market
data document the Þrst relationship, whereas
Lamoureux
and Lastrapes (1990)
and
Laux and Ng (1991)
support the
second. The second relationship also supports the model of
Brock and Kleidon (1990)
where the link between variations
in demand and the variability of prices is through variations
in the bid and ask prices.
In the average spread equation (
Equation 2.b)
the number
of observations is insigniÞcant. This justiÞes our earlier
hypothesis that volatility has incorporated both the con-
temporaneous evidence from quote arrivals and other
sources of information and consequently quote arrivals do
not inßuence spread, given volatility.
c
2
"!1,
c
3
"0.5, and for the Yen
c
1
"0.1,
c
2
"0.2, and
"0.4. We did a second grid search but this time we kept
one of the
c
s constant at its optimum value, say
c
1
, and
,
around their optimal, using a step length of 0.01. For both
currencies the optimum values of
c
Õs,
c
2
and
c
3
Õs stayed as above. Hence,
it seems that neither the linear nor the log-linear functional
forms are the best approximations to the data generating
process functionals. However, from
Table 1
it is apparent
that the log-linear form is a better approximation than the
linear one, with the possible exception of the number of
quotations for the Deutschemark.
Diagnostic tests on this simultaneous system are reported
in Appendix A. In particular, the
Wu (1973)
and
Hausman
(1978)
F
tests for exogeneity of the three variables, with one
exception, are rejected. However, the tests for the omission
of relevant lagged variables could not reject, at least for the
spread equation (see Appendix A), so we included one more
lag in this equation.
Consequently, we estimated the following SES by two-
stage least squares. The estimates of the structural para-
c
Table 2.
Estimated coe¦cients and standard errors of the structural system (2.2)
DEM
aL
ij
i
/
j
1
2
3
4
5
6
1
9.146
0.012
0.210
!0.002
(5.611)
(1.656)
(3.678)
(!0.111)
2
0.012
0.000
0.398
0.108
0.079
(1.641)
(0.393)
(5.565)
(2.697)
(2.510)
3
!0.004
5.424
0.496
0.111
(!0.00)
(0.344)
(13.56)
(3.282)
JPY
aö
ij
i
/
j
1
2
3
4
5
6
1
0.629
0.028
0.189
0.007
(5.340)
(2.189)
(4.137)
(0.227)
2
0.291
!0.007
0.296
0.095
0.088
(3.129)
(!0.881)
(5.597)
(2.162)
(2.683)
3
1.022
!0.805
0.457
0.038
(1.091)
(!0.781)
(11.58)
(1.217)
Note: Heteroskedasticity robust
t
-statistics are in parentheses.
log-likelihood function appears to be unimodal, with
respect to the parameter
c
3
varying simultaneously the values of the other
Interaction between quotations
,
spread
,
and volatility in FOREX
381
In the number of quotations equation (
Equation 2.c)
volatility and average spread are highly insigniÞcant. This
implies that there may be some kind of ÔcausationÕ from the
number of quotations to volatility and some kind of feed-
back relationship between volatility and average spread.
However, the number of observations is not weakly
exogenous to the system as the variance covariance matrix
of the residuals is not diagonal. In fact, the correlation
matrix of the residuals of the system (
Equation 2.a to 2.c)
is
presented in
Table 4.
Hence, we conclude that, apart from the residual e¤ects,
volatility and average spread are simultaneously deter-
mined and there may be a feedback rule between number of
quotations and volatility. However, the number of quota-
tions a¤ects the average spread process through volatility
only. This relationship is stronger for the Yen than for the
Deutschemark.
Furthermore, notice that the second lagged volatility in
Equation 2.a
is insigniÞcant, and the coe¦cient estimate of
the Þrst lag has a very low value (around 0.2 for both
currencies), which implies a very weak autoregressive condi-
tional heteroskedasticity e¤ect. However, this is not the case
when average spread and number of observations are ex-
cluded from this equation. In such a case the OLS estimates
of the Þrst and second lag volatility, of the regression of
volatility on Dummies and 2 lagged volatilities, equal 0.322
(6.079), and 0.070 (1.746) for the Mark and 0.319 (7.237), and
0.0717 (2.206) for the Yen (the robust
t
-statistics are in
parentheses). This implies that these two variables take out
a considerable amount of the conditional heteroskedasticity
e¤ect observed in exchange rate time series. This points out
to the fact that heteroskedasticity type e¤ects, which cap-
tured by ARCH or GARCH type models in a univariate
setups, are mainly due to missing variables in the econo-
metricianÕs information set.
Moreover, the addition of our dummy variables further
reduces the second order ARCH type e¤ect in the series. If
the SES (
Equations 2.a to 2.c)
is estimated without the
dummy variables the results exhibited in
Table 3
are
obtained.
Now the Þrst lag estimated coe¦cient takes a consider-
ably higher value than in the case where dummy variables
are included, and the second lag coe¦cient becomes signiÞ-
cant. Notice also that now in the number of quotations
equation volatility has a strong negative e¤ect, something
which is also documented in
Bollerslev and Domowitz
(1991)
, where the dummy variables are excluded from their
model.
To conclude this section we can say that the simultaneity
and the inclusion of dummy variables capture a consider-
able part of heteroskedasticity type e¤ect, observed ex-
change rate markets. This in e¤ect is due to unobservable
news reßected either in the bid-ask spread or in the dummy
variables which are responsible for changes in tradersÕ de-
sired inventory positions with the result of changing
spreads, according with the theories of
OÕHara and OldÞeld
(1986)
and
Amihud and Mendelson (1980).
These changes in
spread can explain a considerable part of volatility move-
ments, and consequently decreasing the heteroskedasticity
type e¤ects.
IV. TEMPORAL HALF-HOURLY EFFECTS
The temporal dummies capture events (publicly announced
news releases, market openings and closings) whose timing,
Table 3.
Estimated coe¦cients and standard errors of the structural system (2.2) without dummy
variables
DEM
aL
ij
i
/
j
1
2
3
4
5
6
1
7.637
0.006
0.267
0.109
(7.213)
(2.809)
(4.897)
(3.019)
2
0.007
0.000
0.489
0.176
0.114
(1.651)
(1.650)
(9.243)
(4.126)
(3.770)
3
!3.237
38.196
1.051
!0.192
(!2.155)
(1.803)
(33.73)
(!5.692)
JPY
aö
i
/
j
1
2
3
4
5
6
1
0.483
0.011
0.303
0.085
(6.473)
(2.770)
(7.240)
(3.012)
2
0.153
0.002
0.369
0.173
0.147
(2.639)
(1.112)
(7.743)
(3.757)
(4.009)
3
!2.380
2.578
0.976
!0.233
(!2.876)
(2.908)
(28.81)
(!6.359)
Note: Heteroskedasticity robust
t
-statistics are in parentheses.
ij
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