Applied Economics, 1996, 28, 377 386
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, ºniversity of Reading, P.O. Box 218, źhiteknights, Reading
RG6 2AA, ºK and Department of Economics, ¸ondon School of Economics, Financial
Markets Group, Houghton St, ¸ondon źC2A 2AE, ºK
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 first one the optimal functional form between these variables is
determined through a maximization procedure of the unrestricted VAR, involving the
Box 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 significantly the conditional heteroskedasticity effect. We also discuss informa-
tion aspects of the model as well as its implications for financial 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.
It is common in the literature for variations in the arrival of Another way is to follow previous studies of mixture of
news in financial markets to be measured directly from the distributions [see, for example, Harris (1987), Gallant et al.
data on the volatility of prices/returns. [See, for example, (1989) and (1990), and Laux and Ng (1991)] and use volume
Engle and Ng (1991)]. In one sense this approach assumes as a proxy for the number of information events. However,
what needs to be tested, i.e. that news drives volatility. Jones, Kaul and Lipson (1991) show that volume is a noisy
Moreover, the ARCH effects commonly found in such and imperfect proxy for information arrival, and that the
financial series, [see Bollerslev et al. (1992)], may well rep- number of transactions is a better variable in a model with
resent some combination of the autoregressive character- a fixed number of traders. However, there are no volume
istics of news arrival, i.e. the bunching of news , and of data available in the forex market [see, for example Good-
pure market volatility. Given the theoretical results on hart and Demos (1990)]. Instead the frequency of quote
the mixtures-of-distributions hypothesis by Clark (1973), arrivals over Reuters screens is used as the proxy for market
Tauchen and Pitts (1983), and Andersen (1991) among activity. This may capture the effect of market activity on
others, when time is measured in calendar time, the condi- volatility, up to the extent that news is reflected in changes
tional variance of returns will be an increasing function of in current market activity.
the actual number of information arrivals [see Bollerslev The next question is whether it is permissible and appro-
and Domowitz (1991)]. priate to examine the contemporaneous interaction between
A number of questions follow. The first is what indicator quote arrival and volatility, or only to relate volatility to
of information arrival to use. One possibility would be to try quote arrival using information available at t!1 and
to exploit the data available over the news pages on the earlier. The previous literature indicates that this decision is
electronic screens, for example, Reuters AAMM page of important. The results using information on market activ-
news of interest to market dealers [see Goodhart (1990), ity, whether quote frequency or volume, at t!1 and earlier
Goodhart et al. (1991)]. The construction of any such index suggest that such data has no significant ability to predict
would undoubtedly be somewhat subjective, and extremely volatility, given past data on volatility, [for example, Jones,
0003 6846 1996 Chapman & Hall 377
378 A. A. Demos and C. A. E. Goodhart
Kaul and Lipson (1991), Lamoureux and Lastrapes (1990), between quote frequency, volatility and bid-ask spreads.
Bollerslev and Domowitz (1991)]. On the other hand, The positive relationship between volatility and the spread
Lamoureux and Lastrapes (1990) and Laux and Ng (1991) is well-known in the literature [see, for example, Ho
find that the use of contemporaneous data on market activity and Stoll (1983) and Berkman (1991)]. We suggested
virtually removes all persistence in the conditional variance earlier that the absence of any significant ability of
in their series, being daily stock returns and intra-day cur- prior quote frequency to predict volatility implied that
rency future returns respectively. Bollerslev and Domowitz volatility may have incorporated both the contempor-
(1991) doubt the validity of using contemporaneous data on aneous evidence from quote arrivals and other sources of
the grounds of simultaneity and that the traders informa- information. If so, we would not expect quote arrivals, either
tion set does not include contemporaneous data on market contemporaneous or lagged, to influence spreads, given
activity. Simultaneity is dealt with by using a simultaneous volatility.
equation system estimation procedure. With respect to the Where, however, one might find some relationship be-
second objection, market traders way of life is watching the tween spreads and quote frequency would be among the
screen, so they will be virtually instantaneously aware of constant temporal dummy variables. Whereas some sources
a change in the speed of flow of new quotes. Furthermore, it of news are continuously unfolding, the market has a pat-
is argued that the entry of a quote on the screen must tern of openings, lunch breaks, and closes, which might
have both temporal and causal priority over volatility influence both quote frequency and spreads, independently
developments, since the latter can only be estimated of the pattern of price/return volatility. The work of Oldfield
once decisions to enter a new quote have been taken and Rogalski (1980), Wood, McInish and Ord (1985),
and executed. Hence the hypothesis is that, in this ultra- French and Roll (1986), and Harris (1986) among others
high frequency data set, the causal linkages will be have stimulated considerable interest in documenting the
found to be stronger from quote frequency to volatility pattern of stock market returns and their variances around
when both are taken over the same short time interval, than the clock. Admati and Pfleiderer (1988), and Foster and
vice versa. Viswanathan (1990) offer some theoretical explanations for
Here we examine international patterns of intra-day trad- some of these empirical findings. Here we aim to extend this
ing activity and some properties of the time series of returns work by looking also at the temporal patterns of quote
for the Deutschemark/Dollar and Yen/Dollar exchange frequency and spreads. We examine the relationship be-
rates in the foreign exchange market through the interbank tween the sets of temporal dummy variables in Section IV.
trade. The purpose is to provide some information useful in We conclude in Section V.
the further development of the microstructure of trading
models and to compare the empirical results with previous
ones and theoretical models already in existence. II. THE DATA SET
The results in Bollerslev and Domowitz (1991) are ex-
tended in two different ways. First, certain arguments are The continuously quoted data are divided into discrete
outlined (in Section III) explaining why quote frequency segments in the following way. The 24-hour weekday is
data might be better entered in log, rather than in numer- divided into 48 half-hour intervals and the average spread,
ical, form, and we search for the best fitting transformation standard deviation of the percentage first difference of the
of the data using the Box Cox transformation. Second, in rates quoted (ln(e )!ln(e )), and the number of new
Goodhart and Demos (1990), we argue that there are certain quotations within this interval are recorded. In a few instan-
predictable temporal regularities in the foreign exchange ces there were too few observations in a half-hour to calcu-
market (for example, the regular release of economic data at late a meaningful estimate of volatility. In such cases we
certain pre-announced times, the passage of the market substituted the values for the lowest calculable observed
through the time zones punctuated by market openings and volatility, and the accompanying spread, in a half-hour of
lunch breaks (especially in Tokyo)). Consequently temporal that week. This resulted in around potentially 2500 half-
weekly, daily and half-hourly dummies are added to all hourly observations. In fact, 5 out of the 12 weeks were
equations. As will be shown in Section III, these two cha- chosen for analysis, avoiding any weeks with public hol-
nges do make a difference to the results. The conditioning of idays in the main country participants. The results are
the variables of interest on such temporal dummies allows robust to this choice.
us to distinguish between public and private information, At this point we should review some pitfalls associated
something of great importance to informational theories of with the approximation of market activity by the number of
market micro-structure (see, for example, Admati and quotations. Market participants have claimed that during
Pfleiderer (1988), Son (1991), etc.). very busy periods traders may be too occupied in dealing
Although the emphasis here is on the relationship be- through their telephones to update their screens immediate-
tween quote frequency and volatility, since it is a less-re- ly (see Goodhart and Demos (1990)). Per contra, when the
searched area, we examine the three-fold interrelationships market is dull some market participants may enter new
Interaction between quotations, spread, and volatility in FOREX 379
Table 1. Quasi log-likelihood values as a function of the Box Cox exponent
DEM JPY
* sp* n* * sp* n*
Log- Log- Log- Log- Log- Log-
likelihood likelihood likelihood 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 5170.2 0.5 !1386.8 !1706.4 !4894.5
0.3 !1012.7 !1489.6 !5228.3 0.4 !1353.6 !1703.9 4882.1
0.2 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 1701.9 !4934.2
0.0 !1040.9 !1436.2 !5607.8 0.1 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 1350.2 !8867.4
!2.0 !4525.9 !1385.2 !13 130.0
Note: Bold indicates the optimum .
quotes to generate some business. However, in general The functional form of the relationship between these
the temporal pattern of the markets may differ from the variables needs careful consideration. There is no apparent
temporal pattern of the news generation process. Markets reason why the average spread, volatility, and number of
often close almost entirely, for example, at weekends and quotations should be linearly related, rather than, say, log-
over the Tokyo lunch hour, or become very busy, while linearly. On theoretical grounds both functional relation-
some news is continuously occurring. Although we would ships would have the same characteristics as discussed in
expect more news always to be associated with a higher Sections I and II. Hence, we left the data to decide on this by
frequency of quotes, as long as some markets are in opera- using the following procedure.
tion, the functional form of this relationship, for example, We first transformed the three variables using the
linear, log-linear, etc., remains unknown. Box 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 differ-
ent values of the Box Cox exponent, i.e. the following
III. ESTIMATION AND RESULTS
VAR(2) was estimated for different values of , , and
(the exponents):
The following Simultaneous Equation System (SES) is to be
estimated:
* *
"Dummies# sp # n #
sp* "Dm.# sp*
# (1.a)
n* n*
sp "Dummies# # n # sp
# sp (1.b)
*
# sp* #
n "Dummies# # sp # n # n (1.c)
n*
where , sp , and n are the standard deviation of the
percentage change of an exchange rate, the average
spread, and the number of quotations within the tth half- where *"( !1)/ , sp*"(sp !1)/ , and
hour interval, and the system is separately estimated n*"(n !1)/ . Notice that for " " "1, and
for the two currencies under interest, i.e. the Deutschemark " " "0 we have the linear and log-linear forms,
and Japanese Yen, against the US dollar. As financial respectively.
time series suffer from conditional heteroskedasticity In Table 1 we present the values of the quasi log-likeli-
effects, we include lagged dependent variables in Equations hood function for the transformed variables, for different,
1.a to 1.c. Moreover this helps in the identification of but common across the three variables, values of . It is
the system. The estimation method is two-stage least immediately apparent that the optimal value of depends
squares. on the variable and the currency. However, notice that the
We avoided Full Information Maximum Likelihood estimation on the grounds of the strong non-normality of the residuals (see below).
380 A. A. Demos and C. A. E. Goodhart
log-likelihood function appears to be unimodal, with meters and their heteroskedasticity robust standard errors
respect to the parameter , at least for values between 1 are presented in Table 2.
and !2 for the Deutschemark, and 1 and 0 for the Yen.
*"Dummies# sp*# n*# *
What we are doing here in effect is a grid search of the
# *
pseudo-likelihood function with respect to the parameter. (2.a)
Although we chose the steps of the grid to be 0.05, in Table 1
sp*"Dummies# *# n*# sp*
only some representative values of the log-likelihood func-
# sp* # sp* (2.b)
tion are reported, for two reasons. First, the likelihood
function is not very flat around the optimum, with the
n*"Dummies# *# sp*# n*
possible exception of the Yen average spread equation, and
# n* (2.c)
second, because of space considerations.
The optimal values for the Deutschemark are "0.2, Some important points emerge from this table. First, the
"!1, "0.5, and for the Yen "0.1, "0.2, and results are quite robust across the two currencies, although
"0.4. We did a second grid search but this time we kept the functional form of the variable is different. Second,
one of the s constant at its optimum value, say , and notice that in the volatility equation (Equation 2.a) the
varying simultaneously the values of the other s, and , average spread and the number of quotations have a strong
around their optimal, using a step length of 0.01. For both positive effect on volatility. These positive relationships
currencies the optimum values of s stayed as above. Hence, of spread-volatility and volatility-activity are well-
it seems that neither the linear nor the log-linear functional documented facts in the literature. Ho and Stoll (1983),
forms are the best approximations to the data generating Berkman (1991), as well as the probit model of Hausman,
process functionals. However, from Table 1 it is apparent Lo and MacKinley (1991) of trade by trade stock market
that the log-linear form is a better approximation than the data document the first relationship, whereas Lamoureux
linear one, with the possible exception of the number of and Lastrapes (1990) and Laux and Ng (1991) support the
quotations for the Deutschemark. second. The second relationship also supports the model of
Diagnostic tests on this simultaneous system are reported Brock and Kleidon (1990) where the link between variations
in Appendix A. In particular, the Wu (1973) and Hausman in demand and the variability of prices is through variations
(1978) F tests for exogeneity of the three variables, with one in the bid and ask prices.
exception, are rejected. However, the tests for the omission In the average spread equation (Equation 2.b) the number
of relevant lagged variables could not reject, at least for the of observations is insignificant. This justifies our earlier
spread equation (see Appendix A), so we included one more hypothesis that volatility has incorporated both the con-
lag in this equation. temporaneous evidence from quote arrivals and other
Consequently, we estimated the following SES by two- sources of information and consequently quote arrivals do
stage least squares. The estimates of the structural para- not influence spread, given volatility.
Table 2. Estimated coefficients and standard errors of the structural system (2.2)
DEM
L
i/j 123 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
Ć
i/j 123 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.
Interaction between quotations, spread, and volatility in FOREX 381
In the number of quotations equation (Equation 2.c) setups, are mainly due to missing variables in the econo-
volatility and average spread are highly insignificant. This metrician s information set.
implies that there may be some kind of causation from the Moreover, the addition of our dummy variables further
number of quotations to volatility and some kind of feed- reduces the second order ARCH type effect in the series. If
back relationship between volatility and average spread. the SES (Equations 2.a to 2.c) is estimated without the
However, the number of observations is not weakly dummy variables the results exhibited in Table 3 are
exogenous to the system as the variance covariance matrix obtained.
of the residuals is not diagonal. In fact, the correlation Now the first lag estimated coefficient takes a consider-
matrix of the residuals of the system (Equation 2.a to 2.c) is ably higher value than in the case where dummy variables
presented in Table 4. are included, and the second lag coefficient becomes signifi-
Hence, we conclude that, apart from the residual effects, cant. Notice also that now in the number of quotations
volatility and average spread are simultaneously deter- equation volatility has a strong negative effect, something
mined and there may be a feedback rule between number of which is also documented in Bollerslev and Domowitz
quotations and volatility. However, the number of quota- (1991), where the dummy variables are excluded from their
tions affects the average spread process through volatility model.
only. This relationship is stronger for the Yen than for the To conclude this section we can say that the simultaneity
Deutschemark. and the inclusion of dummy variables capture a consider-
Furthermore, notice that the second lagged volatility in able part of heteroskedasticity type effect, observed ex-
Equation 2.a is insignificant, and the coefficient estimate of change rate markets. This in effect is due to unobservable
the first lag has a very low value (around 0.2 for both news reflected either in the bid-ask spread or in the dummy
currencies), which implies a very weak autoregressive condi- variables which are responsible for changes in traders de-
tional heteroskedasticity effect. However, this is not the case sired inventory positions with the result of changing
when average spread and number of observations are ex- spreads, according with the theories of O Hara and Oldfield
cluded from this equation. In such a case the OLS estimates (1986) and Amihud and Mendelson (1980). These changes in
of the first and second lag volatility, of the regression of spread can explain a considerable part of volatility move-
volatility on Dummies and 2 lagged volatilities, equal 0.322 ments, and consequently decreasing the heteroskedasticity
(6.079), and 0.070 (1.746) for the Mark and 0.319 (7.237), and type effects.
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 IV. TEMPORAL HALF-HOURLY EFFECTS
effect observed in exchange rate time series. This points out
to the fact that heteroskedasticity type effects, which cap- The temporal dummies capture events (publicly announced
tured by ARCH or GARCH type models in a univariate news releases, market openings and closings) whose timing,
Table 3. Estimated coefficients and standard errors of the structural system (2.2) without dummy
variables
DEM
L
i/j 12 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
Ć
i/j 12 34 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.
382 A. A. Demos and C. A. E. Goodhart
though not generally their exact scale, is known in advance. Japanese economy is announced either early in the Japanese
Public new related to macroeconomic variables is simulta- morning, i.e. around 1:00 BST, or in the late Japanese
neously announced to all traders, at a time known in ad- afternoon, i.e. 6:00 BST. The same time period is character-
vance since the scheduled time of all economic related news ized by high spread and screen activity. However, it appears
is predetermined, and reported on another part of the that Japanese economic-related news has no effect on the
Reuters system, the FXNB page. The stochastic element in volatility of the JPY currency. Although in line with the
such cases is the actual announcement, not the timing of it. results of Ito and Rolley (1987), this remains peculiar. Fur-
In general, the majority of the US announcements are thermore, the same is true for the Deutschemark in relation
around 13:30 hours British Summer Time (BST), and the to German economic announcements, which are mostly
German ones around 10:00 hours BST. Consequently, the released either around 9:30 or 14:00 BST. Hence, it seems
relationship between the dummy variables and the charac- that only US economic news affects the variability of DEM
teristics of interest to us in the market predominantly reflect and JPY exchange rates.
response of these variables to publicly known events. Per There is a further curiosity in the half-hourly dummies
contra, the relationship between these variables, after condi- which is worth mentioning. During the Tokyo lunch time
tioning on such temporal constants, will primarily reflect break (4:00 5:00 BST) there is a dramatic decrease of vola-
private information to a somewhat greater extent. tility coupled with an increase in spread and a decrease in
Notice that the constant represents the last half hour of the number of quotations in the first half-hour period (be-
the last Friday in the sample. During this half hour all the tween 4:00 4:30 BST), followed by an increase in volatility
main markets are closed and only a few traders, if any at all, coupled with a decrease in spread which cannot be ex-
input quotations. Therefore, the constant in the estimation plained by public information theories. Perhaps traders who
reflects, on average, the smallest number of observations in come back early from lunch take wild positions to make
the sample, but not necessarily the lowest level of volatility their early return worthwhile. On the other hand this vola-
or the smallest average spread. Let us now concentrate on tility increase could be a statistical artefact due to the small
these dummy effects. number of quotations during that period; that is, a few
The estimated dummy coefficients, for both currencies observations out of equilibrium level can have a dramatic
and per equation, are not presented here because of space increase in the sample variance of the rate.
considerations. Let us consider the half hour dummies first. The increase of average spread during the beginning of
In graphs 1a to 3b in Figure 1 the values of the estimated the Tokyo (4:00 BST) lunch hour for both currencies could
dummy coefficients for both currencies are presented. They be attributed to that traders during the lunch hour widening
reveal an interesting feature. In the last part of the day BST their spreads to protect themselves from any unexpected
time, from about the closing time of the European ex- news, whereas when they return to their desks the average
changes and until the closing time of the New York ex- spread returns to normal.
change, volatility is unusually high. Notice that this takes For both markets 7:00 BST seems to be an unusually high
place in both currency markets. spread period. This coincides with the opening of the Euro-
During this period there are few, or no, economic (or pean market and the closing of the Asian one; possibly
other public) announcements from Europe or Asia (consid- European traders want to protect themselves from potential
ering only Japan). Most US economic announcements are superior information that their Asian counterparts could
made before the opening of the New York Stock Exchange, possess. However, this is less marked in the JPY market.
at 13.30 BST. There is a small spike at the relevant half hour This opposes the Admati and Pfleiderer (1988) model, where
(27), but this remains quite small compared with the higher spread is lowest at the beginning of the trading day, due to
volatilities apparent later on in the US market day. liquidity considerations, and in line with the Foster and
Hence, it seems that public news is not the explanation of Viswanathan (1990) model where spread is highest at the
this volatility increase. Furthermore, this increase seems start of the day. Another high spread period for the DEM
even more difficult to explain in the light of the Admati and market is around 14:00 BST, shortly after the release of US
Pfleiderer (1988) theory. During this period we certainly macroeconomic news. It is also the common time for coor-
have a reduction in the number of traders in the market, as dinated interventions to occur [see Goodhart and Hesse
only the New York exchange is in operation, so this increase (1992)]. As at the same time there is some small increase in
can hardly be attributed to an increase in the number of the volatility of the market the spread increase can be
liquidity traders. attributed to the traders, fear of central bank interventions.
There is then an apparent decrease in volatility for both The busiest period of the day in terms of the number of
currencies, during the early morning period between 1:30 quotations, measured by the half-hourly dummies, is the
and 3:30 (BST). Most of the economic-related news for the return in activity after the Tokyo lunch-break and around
See Table 5 is Demos and Goodhart (1992).
Interaction between quotations, spread, and volatility in FOREX 383
Fig. 1. Graphs of volatility, average spread, and number of quotations equations
5:30 6:00 BST, whereas the least busy is the Tokyo lunch tions) falls steadily as the US markets grind to a halt, before
hour for both currencies. After the burst of activity in the Australia opens the new day.
post Tokyo lunch-break, activity declines until there is The increased spread during periods of high market acti-
a smaller secondary peak when New York opens, between vity in both markets is best explained by the model of
13.30 and 14.30 BST, (27 29 on our graphs), before London Subrahmanyan (1989), where more trading by informed
(Europe) closes. Thereafter activity (the number of quota- risk-averse traders brings about lower liquidity and higher
384 A. A. Demos and C. A. E. Goodhart
Table 4. Correlation matrix of the residuals for Equations 2.a 2.c
DEM JPY
(2.a) (2.b) (2.c) (2.a) (2.b) (2.c)
(2.a) 1 1
(2.b) !0.267 1 !0.502 1
(2.c) 0.158 0.023 1 !0.074 0.185 1
costs. Furthermore, the higher spread towards the end of the number of transactions in the spot FOREX market, for
trading day, observed in the Deutschemark market but not which data are unavailable. This is in line with studies in
in the Japanese Yen market, is predicted by the dealer stock market volume and volatility data [see Gallant, Rossi,
market model of Son (1991), where risk-averse traders avoid and Tauchen (1990), and Lamoureux and Lastrapes (1990)].
trading close to the end of their day to avoid overnight It turns out that informational theories can only partially
inventory holdings. explain the facts documented here. Although, high trading
There are few signs of any significant pattern in volatility and volatility at the opening of markets can be explained
between the days of the week, except for some indications of along the lines of the Admati and Pfleiderer (1988) theory,
higher volatility in the Yen on Thursdays, and also positive the different behaviour of the two currencies in different
but insignificantly so for DEM. The average spread was, markets at the same (and different) time periods points
however, significantly higher on Fridays than earlier in the towards the need to take into account local and currency-
week, with some tendency for it to be lowest on Thursdays specific behaviour. The same can be said for the models of
and Wednesdays. This is roughly the inverse to the daily Foster and Viswanathan (1990), Subrahmanyan (1989), and
pattern for the frequency of quote arrivals (activity), which is Son (1991).
lowest on Friday, and tends to peak in mid-week, Tuesday An important result of this paper is that the inclusion of
and Wednesday. half-hourly dummies, and taking account of simultaneity
The weekly dummies during the period showed a pattern between volatility, average spread, and number of quota-
of steadily increasing market activity from week to week. tions, considerably reduces the GARCH type effects in the
The final week (Week 5) was not only extremely active, but conditional variance of these two exchange rates. What
exhibited a marked and highly significant increase in spread remains of such GARCH effects can then probably be
size. Volatility also increased in the final week, but the attributed to private information and the uncertainty asso-
increase was much less significant. ciated with it.
Finally, having fitted weekly, daily and half-hour dum-
mies, we can identify inter- and intra-day patterns of acti-
V. CONCLUSIONS vity, volatility and average spread. Some of these, for
example, the impact of the Tokyo lunch hour, we have
We have assessed the behaviour of the spot foreign ex- previously documented. Others are already well known in
change market quotations in terms of volatility, average markets, for example, the rise in spreads and decline in
spread, and the number of quotations within half-hour activity on Fridays. But we were surprised by the finding of
intervals, as well as certain informational aspects of these the continuing high volatility, in both currencies, through-
processes. It seems that a log-linear relationship among out the period of US market opening, despite steadily falling
these three processes is a considerably better approximation activity, which we had expected. Much of the public in-
to the true data generating process functional form, than the formation on economic news in the US is released at, or
linear one; however, it is by far worse than the functional before, the market opening, so exactly what keeps volatility
form presented here. so high during the afternoons in the US is a mystery to us.
A new variable was introduced: the number of observa-
tions within a specific time interval. This variable plays an
important role in the determination of volatility and aver- ACKNOWLEDGEMENTS
age spread, either directly or through the error terms. The
contemporaneous correlation of the number of quotations We wish to thank Seth Greenblatt, Steve Satchell,
and volatility leads us to hypothesize that the former pro- Enrique Sentana, and especially Ron Smith for helpful com-
cess could be a proxy for the volume of trade, or for the ments. Financial support from the Financial Markets
Strictly speaking, however, the Admati and Pfleiderer (1988) model applies to individual traders and to markets with well-defined opening
and closing times.
Interaction between quotations, spread, and volatility in FOREX 385
Group and the Economic and Social Research Council is Harvey, C. and Huang, R. (1990) Inter and Intraday Volatility in
Foreign Currency Futures Market, mimeo, Duke University.
gratefully acknowledged. All remaining mistakes are ours.
Harris, L. (1987) Transactions Data Tests of the Mixture of Distri-
butions Hypothesis, Journal of Quantitative and Financial
Analysis, 22, 127 42.
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APPENDIX A
Goodhart, C. A. E. and Hesse, T. (1992) Central Bank FOREX
Intervention Assessed in Continuous Time, Financial Mar-
For the optimal s obtained, from the procedure described
kets Group Discussion Paper No. 123, London School of
Economics. above, we tested for omission of relevant lags [see Spanos
386 A. A. Demos and C. A. E. Goodhart
(1986)], specifically two more, in the VAR formulation. The with the Deutschemark we decided to stay with this speci-
F statistics per currency and variable were the following: fication.
2.25, 5.03, and 1.43 for the Deutschemark and 1.88, 4.271, The Jarque-Bera (1980) normality tests on the VAR resid-
and 3.81 for the Yen (F(6,R) "2.64). For 10-order serial uals stand at 2445.0, 696.6, and 185.3 for the Mark and
%
correlation of the residuals, the F statistics were 2.08, 2.52, 777.3, 529.6, and 125.9 for the Yen, implying a massive
and 1.13 and 1.70, 2.82, and 1.34 for the Deutschemark and rejection of the null hypothesis. Furthermore, the one-sided
Yen respectively (F(10,R) "2.32). It seems that at least Lagrange Multiplier test for ARCH type effects [see Demos
%
for the spread equation having only two lags does not and Sentana (1991)] again massively rejects the null of
capture the systematic dynamics. Hence, in the VAR formu- conditional homoskedasticity. Notice that in the normality
lation one more lag is added. test using linear of log-linear form the statistics had, more or
The F-statistics for two more lags, this time, are: 1.25, less, two to three times the values reported above. A ques-
0.98, and 1.65, and 1.47, 2.60, and 3.04, for the Mark and tion arises immediately on the validity of the distributions,
Yen respectively. However, the 10-order serial correlation mainly of the various statistics that are used. However,
F-statistics are highly significant for both currencies. This is provided that the usual regularity conditions hold, that is,
probably due to overfitting in the volatility and number of the existence of higher moments for the distribution of the
quotes equations. Consequently, we re-estimated the VAR errors, the usual arguments for the asymptotic validity of
imposing zero coefficients to the third lag of volatility and the tests apply.
number of quotations. The 10-order serial correlation statis- The exogeneity Wu (1973) Hausman (1978) F statistics
tics now are: 1.54, 1.38, and 1.23, and 1.62, 2.31, and 1.66 for are 5.51, 4.10, and 5.95, and 4.60, 2.75, 5.80 for the Mark and
the two currencies, suggesting that indeed overfitting was Yen respectively. Hence with the exception of the average
the cause of spurious serial correlation. The omission of two spread in Yen the exogeneity of the other variables is rejec-
more lags, in the systematic dynamics of the VAR are now ted. The Basmann (1974) test for the overidentified restric-
1.57, 0.86, and 2.13 for the Deutschemark and 1.49, 2.22, and tions does not reject the null hypothesis as it stands at 1.57,
3.89 for the Yen. Although the systematic dynamics for the 2.19, and 1.52 for the Mark and 1.95, 0.56, and 0.93 for the
number of quotations, for the Yen only, indicates that Yen. This is an indication that the specification of the
more lags are needed, and provided that this is not the case system is correct (see Spanos (1986)).
Notice that even in small samples it is not clear if the two-stage least square estimator over or underestimates the normal probability [see
Knight (1986)].
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