InteractionFrequency-fb2(1).pdf

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

)!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*

Log-

sp*

Log-

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 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

#a 13

#a 14 p t~1

sp*

b 11

b 12

b 13

t~1

sp*

t~1

" Dm .#

b 21

b 22

b 23

#a 15 p t~2

(1.a)

b 31

b 32

b 33

"Dummies#a 21 p t

#a 23

#a 24

t~1

#a 25

t~1

(1.b)

d 11

d 12

d 13

t~2

sp*

t~2

e 1t

e 2t

e 3t

"Dummies#a 31 p t

#a 32

#a 33

#a 34

(1.c)

d 21

d 22

d 23

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

, and n

where

p c Ç

!1)/

c 1

, sp*

"( sp

c È

!1)/

c 2

, and

"( n

c Ê

!1)/

c 3

. Notice that for

c 1

"c 2

"c 3

"1, and

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

.Itis

depends

on the variable and the currency. However, notice that the

1We avoided Full Information Maximum Likelihood estimation on the grounds of the strong non-normality of the residuals (see below).

p *

Log-

p *

Log-

where

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

, at least for

meters and their heteroskedasticity robust standard errors

are presented in Table 2.

"Dummies#a 12

sp*

#a 13

#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

(2.a)

sp*

"Dummies#a 21 p

#a 23

#a 24

sp*

t~1

#a 25

sp*

t~2

#a 26

sp*

t~3

(2.b)

"Dummies#a 31 p

#a 32

sp*

#a 34

t~1

#a 35

t~2

(2.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

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

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-

Table 2. Estimated coe¦cients and standard errors of the structural system (2.2)

DEM

i / j

9.146

0.012

0.210

!0.002

(5.611)

(1.656)

(3.678)

(!0.111)

0.012

0.000

0.398

0.108

0.079

(1.641)

(0.393)

(5.565)

(2.697)

(2.510)

!0.004

5.424

0.496

0.111

(!0.00)

(0.344)

(13.56)

(3.282)

JPY

aö

i / j

0.629

0.028

0.189

0.007

(5.340)

(2.189)

(4.137)

(0.227)

0.291

!0.007

0.296

0.095

0.088

(3.129)

(!0.881)

(5.597)

(2.162)

(2.683)

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

i / j

7.637

0.006

0.267

0.109

(7.213)

(2.809)

(4.897)

(3.019)

0.007

0.000

0.489

0.176

0.114

(1.651)

(1.650)

(9.243)

(4.126)

(3.770)

!3.237

38.196

1.051

!0.192

(!2.155)

(1.803)

(33.73)

(!5.692)

JPY

aö

i / j

0.483

0.011

0.303

0.085

(6.473)

(2.770)

(7.240)

(3.012)

0.153

0.002

0.369

0.173

0.147

(2.639)

(1.112)

(7.743)

(3.757)

(4.009)

!2.380

2.578

0.976

!0.233

(!2.876)

(2.908)

(28.81)

(!6.359)

Note: Heteroskedasticity robust t -statistics are in parentheses.

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