What drives volatility persistence in the foreign exchange market?

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Journal of Financial Economics

Journal of Financial Economics 94 (2009) 192–213

0304-40

doi:10.1

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Frank D

Lennart

Richard

Joshua

Wright,

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journal homepage: www.elsevier.com/locate/jfec

What drives volatility persistence in the foreign exchange market?$

David Berger a, Alain Chaboud b, Erik Hjalmarsson b,�

a Department of Economics, Yale University, P.O. Box 208281, New Haven, CT 06520, USAb Division of International Finance, Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA

a r t i c l e i n f o

Article history:

Received 28 December 2007

Received in revised form

10 September 2008

Accepted 7 October 2008Available online 15 July 2009

JEL classification:

C22

F31

G12

Keywords:

Volatility persistence

High-frequency data

Realized volatility

Long memory

Exchange rates

5X/$ - see front matter Published by Elsevier

016/j.jfineco.2008.10.006

are extremely grateful to the referee, Tor

helping to improve the paper. Other helpfu

d by Federico Bandi, Tim Bollerslev, David Bo

iebold, Jon Faust, Joe Gagnon, Dale Hende

Hjalmarsson, Randi Hjalmarsson, Blake LeBa

Lyons, Michael Moore, Carol Osler, Andrew Pa

Rosenberg, Mark Seasholes, Clara Vega, Jon W

Par Osterholm, and seminar participants at t

the Federal Reserve Bank of New York, th

n Summer Meeting of the Econometric

ity, the European Summer Meeting of the E

pest, and the Central Bank Market Microstruc

st. The views in this paper are solely the re

and should not be interpreted as reflectin

f Governors of the Federal Reserve System or

ed with the Federal Reserve System.

responding author. Tel.: +1202 452 2426; fax:

ail address: [email protected] (E. Hjal

a b s t r a c t

We propose a new empirical specification of volatility that links volatility to the

information flow, measured as the order flow in the market, and to the price sensitivity

to that information. The time-varying market sensitivity to information is estimated

from high-frequency data, and movements in volatility can therefore be directly related

to movements in order flow and market sensitivity. Empirically, the model explains a

large share of the long-run variation in volatility. Importantly, the time variation in the

market’s sensitivity to information is at least as relevant in explaining the persistence of

volatility as the rate of information arrival itself. This may be evidence of a link between

changes over time in the aggregate behavior of market participants and the time-series

properties of realized volatility.

Published by Elsevier B.V.

B.V.

ben Andersen, for

l comments were

wman, Mark Carey,

rson, Olan Henry,

ron, Mico Loretan,

tton, Dagfinn Rime,

ongswan, Jonathan

he Federal Reserve

e IMF, the North

Society at Duke

conometric Society

ture Conference in

sponsibility of the

g the views of the

of any other person

+1202 263 4850.

marsson).

1. Introduction

The results of more than two decades of empiricalvolatility modeling all point towards the conclusion thatthe volatility of asset prices changes over time in a fairlypersistent manner. However, there is still no clearagreement on why this is the case. The most commonhypothesis is that the rate of information arrival affectingthe price of an asset is itself persistent, perhaps becauseeconomic data relevant to the price are released in aclustered fashion or because analysis relevant to the priceis produced in a clustered fashion. A less commonexplanation, which does not exclude the first one, is thatvariation over time in the sensitivity of market partici-pants to information may also help explain the pattern ofvolatility persistence. This paper presents a simple, directempirical test of the roles of both information arrival andmarket sensitivity in driving volatility persistence, using aunique high-frequency data set covering several years ofglobal interdealer foreign exchange trading.

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1 Liesenfeld (2001) finds a similar result using his mixture of

distributions model with a latent market sensitivity variable.

D. Berger et al. / Journal of Financial Economics 94 (2009) 192–213 193

Much of the previous research on factors drivingvolatility derives from studies of ‘mixture of distributions’models proposed by Clark (1973), Tauchen and Pitts(1983), and Andersen (1996), among others, wherevolatility and trading volume are jointly directed by theprocess of information arrival. Thus, in these models,persistence in the information arrival process generatespersistence in volatility and trading volume. In practice,however, the estimated persistence of volatility in suchmodels has typically been found to be well below that ofunivariate time-series models of volatility. Liesenfeld(2001), extending the previous mixture models, allowsfor volume and volatility to be driven not only by a latentinformation arrival process but also by an additionallatent process that governs the impact of information onprices. He shows that such a model, where the dynamicsof volatility are associated with both the rate of informa-tion arrival and the time-varying sensitivity to informa-tion, can capture much more of the persistence involatility.

Though our paper is inspired by the result of Liesenfeld(2001), it takes a very different approach from previouswork on the study of volatility persistence. In particular,we draw upon recent findings on the relationshipbetween returns and order flow. This relationship, whichhas been studied extensively in the foreignexchange market in the past few years (e.g., Evans andLyons, 2002, 2004), has been found to account for asubstantial share of observed movements in exchangerates. The main thrust of the current paper is to show thatthere are large and persistent variations over time in therelationship between returns and order flow, and to studyhow these variations, along with variations in the orderflow itself, are linked to the time-series behavior ofvolatility.

Evans and Lyons and other authors have argued thatthe contemporaneous explanatory power of order flow forexchange rate movements derives in great part from thefact that information relevant to the price is transmittedto the market through order flow, a point previouslydemonstrated by Hasbrouck (1991) for the stock market.To the extent that order flow represents information, ourempirical specification therefore allows us to decomposethe factors affecting volatility into two time-varyingcomponents: the flow of information itself and thesensitivity of the market to that information. It shouldbe stressed, however, that our results on the role ofmarket sensitivity in driving volatility do not depend uponthe interpretation of order flow as information, althoughwe focus on that interpretation.

The empirical analysis is conducted using a uniquedata set that represents a majority of global interdealertrading in the spot euro-dollar exchange rate at the one-minute frequency over a period of several years. Impor-tantly, the frequency and the content of the data permit usto estimate a time-varying market sensitivity parameterat the daily frequency. We can therefore treat the marketsensitivity as an observed variable, along with order flowand realized volatility. This allows us to analyze directlythe interactions of the time-series properties of thesethree variables, without relying on latent variables.

Focusing on the fact that volatility is a slow-movingand fairly persistent process that is well described by along-memory model, we frame the empirical analysis as atest of fractional cointegration. This provides a test of howclosely the long-run movements in volatility are linked tothe long-run swings in order flow and market sensitivity.This empirical setup offers several advantages. By ex-plicitly modeling the persistence in the data, it becomespossible to evaluate the degree to which the persistence inthe dependent variable is captured by movements in theright-hand side variables. In particular, the long-memoryframework provides a concise summary statistic of thedegree of persistence and thus easily enables comparisonbetween the persistence in the original data and in thecointegration residuals. In addition, by testing for (frac-tional) cointegration, we guard against the risk of spuriousregression, a possibility given the persistent nature of thedata. Finally, the estimation methods employed are robustto potential endogeneity in the regressors, which is clearlya concern in a standard ordinary least squares (OLS)analysis of these time series.

Overall, the empirical results are very strong. We showthat a very large share of the persistence in volatility canbe explained by variations in the market’s sensitivity toorder flow and variations in the order flow itself. Thus, ifone interprets order flow as information, the results showthat volatility persistence can be linked to both thepersistence of the market’s sensitivity to new informationand the persistence of the information flow itself.Furthermore, the empirical analysis clearly shows thatthe variations in market sensitivity play at least as large arole in driving volatility persistence as the variations inthe flow of information. This finding may point to animportant link between changes over time in theaggregate behavior of market participants and thetimes-series properties of volatility. From an econometricpoint of view, we observe that the measured persistencein the information flow is not large enough to capture thepersistence in volatility, whereas the persistence of thesensitivity parameter is very similar to that of volatility. Infact, our results show that the information flow appearsmore closely linked to the shorter-, or medium-term,behavior of volatility, whereas the market’s sensitivity toinformation is more closely related to the longer-runbehavior of volatility.1

The paper proceeds as follows. Section 2 introduces thehigh-frequency exchange rate data used in our analysis.Section 3 first derives our empirical specification, addres-sing its motivation, its empirical validity, and theconstructed variables that we use in our estimations. Itthen presents some preliminary OLS analysis of thecontemporaneous relationships between our constructedvariables. Section 4 outlines the fractional integration andcointegration methodology used in the remainder of thepaper. Section 5 presents our main estimation results, andSection 6 presents some additional extensions androbustness results. Section 7 studies the interaction

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D. Berger et al. / Journal of Financial Economics 94 (2009) 192–213194

between trading volume and our measure of marketsensitivity, ruling out a major role for volume in explain-ing our results. Section 8 concludes.

2. The data

We analyze high-frequency spot euro-dollar exchangerate data spanning January 1999 through December 2004.We have access not only to data on the exchange rateitself, but also to the volume of trade and the order flow.Our price data are available at the one-second frequency,from which we construct time series sampled at either theone-minute or the five-minute frequencies; the transactionvariables are available at the one-minute and five-minutefrequencies. The transactions data are proprietary andconfidential. The data were provided by Electronic BrokingServices (EBS), part of the ICAP Group, which operates anelectronic limit order book used by all large foreign exchangedealers across the globe to trade in a number of majorcurrency pairs. Over our sample period, EBS processed a clearmajority of the world’s interdealer transactions in spot euro-dollar, and the price on the EBS system was the referenceprice used by all dealers to generate derivatives prices andspot prices for their customers.2 Further details on the EBStrading system and the data can be found in Chaboud,Chernenko, Howorka, Iyer, Liu, and Wright (2004) and Berger,Chaboud, Chernenko, Howorka, Iyer, Liu, and Wright (2005).

The exchange rate data we use are the midpoint of thehighest bid and lowest ask quotes in the EBS limit-orderbook at the top of each time interval (one-minute or five-minute). These quotes are executable, not just indicative,and therefore represent a true price series. The tradingvolume data are the amounts traded per time interval,expressed in millions of the base currency (the euro).Order flow is measured as the net of buyer-initiatedtrading volume minus seller-initiated trading volume pertime interval. The direction of trade is based on actualtrading records: A trade is recorded as, for instance, buyer-initiated, if it is the result of a ‘hit’ on a posted ask quote.Order flow is also expressed in millions of base currency,with a positive number representing net buying pressureon the base currency.

We exclude all data collected from Friday 17:00 NewYork time to Sunday 17:00 New York time from oursample, as trading activity during these hours is minimaland not encouraged by the foreign exchange tradingcommunity.3 We also drop several holidays and days of

2 EBS does not publish trading volume data per currency pair. To give

a sense of an order of magnitude, average daily trading volume (the

dollar amount traded) on EBS in euro-dollar in 2003, for instance, was

well above that of the NYSE as a whole (average daily trading volume on

the NYSE in 2003 was about $40 billion). Traders on EBS can transact in

amounts ranging from 1 to 999 million of the base currency. In practice,

however, as large deals are routinely broken down, most transactions are

for amounts of one to five million, and the average trade size varies little

over time during our sample period. As a result, there is a very high

correlation between the trading volume in a given time period and the

number of transactions in that same period.3 In this market, by global convention, the value date changes at

17:00 New York time, which therefore represents the official cutoff

between two trading days.

unusually light volume near these holidays: December24–26, December 31–January 2, Good Friday, EasterMonday, Memorial Day, Labor Day, Thanksgiving and thefollowing day, and July 4 or the day on which it isobserved. Similar conventions have been used in otherresearch on foreign exchange markets, such as Andersen,Bollerslev, Diebold, and Vega (2003b). In addition, we alsoexclude September 22, 2000, the day of the coordinatedintervention operation in support of the euro by the G7,and September 11, 2001, and the three following days,which had very low market activity on EBS.4

For the analysis in this paper, we construct a sample atthe five-minute frequency over the entire 24-hour tradingday and a sample at the one-minute frequency that usesonly observations from the busiest trading hours of theday, obtained between 03:00 and 11:00 New York time,when the global foreign exchange market is most active.Fig. 1 presents a graph of average minute-by-minutetrading volume throughout the day in euro-dollar on theEBS system, indexed to the average one-minute tradingvolume in our entire sample.

3. Empirical specification and preliminary results

3.1. Motivation

The starting point for our analysis is the behavior ofintra-daily foreign exchange returns. Motivated by recentempirical findings, we consider the following contem-poraneous relationship between returns and order flow,

rt;i ¼ ltof t;i, (1)

where rt;i and of t;i are the returns and order flow in periodi on day t, respectively. Berger, Chaboud, Chernenko,Howorka, and Wright (2008) analyze this relationship forthe data used here and find a strong positive associationbetween returns and order flow. The empirical validity ofEq. (1) is discussed further below.

These lt coefficients can be, and have been, interpretedas the sensitivity of the price to the information thattraders receive through the trading process. A number ofresearchers, including Hasbrouck (1991), Payne (2003),and Evans and Lyons (2002, 2004), have argued that thewell-documented impact of order flow on prices in theforeign exchange market and other asset markets reflectsthe fact that order flow reveals to traders information thatis either private or just widely dispersed among economicagents, such as risk parameters or even early indicationsof changes in the pace of economic activity.5 Even if onedoes not fully accept that interpretation, the lt coeffi-

4 The coordinated foreign exchange intervention by the G-7 on

September 22, 2000, is the only instance of publicly announced

intervention in our sample. A few weeks later, in early November

2000, the ECB alone conducted much smaller intervention operations on

three separate days. We keep these three days in our sample, as these

interventions were not announced to the public at the time. As for the

September 11, 2001 date, one of EBS’s computer centers, located near the

World Trade Center in New York, was temporarily disabled by the

September 11 attack.5 The relationship in Eq. (1) is, of course, also the one derived in

Kyle’s (1985) model of continuous auctions.

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Table 1Results from regressing returns onto contemporaneous order flow.

This table reports the ordinary least squares (OLS) estimates from the

regression of returns on order flow. The regression is estimated using the

entire 1999–2004 sample of intra-daily observations, allowing for a non-

zero intercept and treating the slope coefficient l as identical for all t;

there are T ¼ 1;489 days in the sample. Panel A shows the results for the

full-day five-minute observations and Panel B shows the results for the

3–11am one-minute observations. The l’s should be interpreted as the

estimated exchange rate movement, in basis points, per billion euros of

order flow. The first column states the number of observations ðNÞ used

in each regression, and the following two columns give the estimates of

the intercept and the slope coefficient l, respectively; robust standard

errors are given in parentheses below the estimates. The last column

shows the R2’s of the regressions.

N Intercept l R2

Panel A: Full-day, five-minute

428,832 �0.114 52.457 0.475

(0.004) (0.223)

Panel B: 3–11am, one-minute

714,720 �0.038 48.696 0.417

(0.002) (0.162)

Fig. 1. The average minute-by-minute volume of trade over the day, standardized to have a mean of 100, calculated from the full one minute sample from

1999 to 2004. Each point on the graph is the average standardized volume, for a given minute of the day, across all days in the sample.

7 Using the Elliott and Muller (2006) dqLL test for parameter

constancy, applied to daily returns and order flow, the null of parameter


cients unquestionably reflect how traders adjust the pricein reaction to order flow, perhaps reflecting factors such asthe traders’ willingness or ability to hold inventory orchanges in their appetite for risk.

By squaring and summing up each side in Eq. (1) overall daily intervals, the following equation for daily realizedvariance, RVt , is obtained:

RVt �XK

i¼1

r2t;i ¼ l2

t

XK

i¼1

of 2t;i. (2)

Define OFð2Þt �PK

i¼1of 2t;i and write

RVt ¼ l2t OFð2Þt . (3)

That is, daily variance is a function of the aggregate dailysquared order flow and the squared sensitivity of the priceto order flow.6

The usefulness of this derivation, of course, hinges onthe validity of the original Eq. (1), or its empiricalcounterpart,

rt;i ¼ ltof t;i þ �t;i. (4)

Table 1 shows the results from estimating Eq. (4) with afixed slope coefficient l for the entire sample, allowing fora non-zero intercept. The results are promising, with an R2

of 47% in the full-day five-minute sample and an R2 of 42%in the 3–11am one-minute sample, which must beconsidered highly successful for any asset return. Theseresults are also close to what Evans and Lyons (2002)report in their study of the impact of daily order flow ondaily exchange rate returns; aggregating the data usedhere to a daily frequency yields an impressive R2 of 45% inthe regression of daily returns on order flow.

6 Suppressing the constant in Eq. (1) will not have an impact on the

derivations, since any (cross-) terms involving this constant will be of

negligible size. That is, as in the arguments for the consistency of

realized variance as an estimator of the quadratic variation, the drift

term is of an order of magnitude smaller than the diffusion term.

Of course, the parameter lt may not be fixed over time,and it is thus likely that an even better fit of Eq. (4) can beobtained by estimating lt separately for each day.7 A

constancy is rejected at the 5% level but not at the 1% level. It is not clear,

however, how much power the test has against an alternative where the

parameter changes are not entirely persistent, but merely possess long-

memory, as suggested by the subsequent empirical analysis. When thedqLL test is applied to the five-minute or one-minute data, using the full

six-year sample, the null hypothesis is strongly rejected at the 1% level

for both samples.

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Table 2Summary of daily results from regressing returns onto contemporaneous order flow separately for each day.

This table reports a summary of the daily estimates of the slope coefficients lt in daily regressions of returns on order flow. The estimates of lt are

obtained day-by-day using intra-daily data, for each day in the 1999–2004 sample; there are T ¼ 1;489 days in the sample. That is, for each day in the

sample, a separate lt is estimated, based only on intra-daily observations for that day and allowing for a non-zero intercept. Panel A shows the results for

the full-day five-minute observations and Panel B shows the results for the 3–11am one-minute observations. Summary statistics for the resulting

estimates of lt and the R2’s for the regressions are reported. The l’s should be interpreted as the estimated exchange rate movement, in basis points, per

billion euros of order flow. The first two columns report the mean and standard deviations of the daily estimates and R2’s. The remaining columns give the

percentiles of the empirical distributions of the daily estimates and R2’s.

Mean Std.dev. 1% 5% 10% 25% 50% 75% 90% 95% 99%


lt55.147 15.901 26.820 34.642 38.202 44.320 53.306 63.312 73.887 83.875 106.858

R2 0.511 0.096 0.157 0.339 0.403 0.467 0.525 0.573 0.614 0.633 0.674


lt50.573 14.219 26.949 32.325 35.250 40.534 48.291 58.060 68.226 76.724 95.441

R2 0.463 0.076 0.186 0.340 0.379 0.430 0.471 0.509 0.542 0.562 0.588

Fig. 2. Plots of the logged daily data. The left-hand side panels show the daily data generated from the five-minute full-day intra-daily observations,

whereas the right-hand side panels show the daily data generated from the 3–11am one-minute frequency intra-daily data. The top two graphs show

realized volatilities, the middle graphs show the market sensitivity parameters ðltÞ, and the bottom two graphs show the integrated squared order flows.


summary of the results from such daily regressions isshown in Table 2, and the logged daily lt ’s are plotted inFig. 2. The mean of the daily estimates is a bit larger than

the overall estimates reported in Table 1, whereas themedian daily estimates are, in fact, very close to those inTable 1. The R2’s are also somewhat larger with a median

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of 52.5% for the full-day sample estimates at the five-minute frequency. The percentiles of the daily R2’s shownin Table 2 provide additional support for a strongrelationship between returns and order flow at highfrequencies; for instance, the lower 5% quantile for thedaily regressions run on the full-day sample at a five-minute frequency is above 33%.

In summary, the results reported in Tables 1 and 2show that, at high frequencies, returns and order flowtend to move in a consistent direction to a large degree,thus giving support to Eq. (4).8 It is also evident from Fig. 2that this relationship is not constant across time. Theestimation of the lt ’s in Eq. (4) at a daily frequency allowsus to replace the unobserved lt ’s in Eq. (3) with these‘realized’ daily lt ’s, in the same manner as we use realizedvariance instead of the true unobserved integratedvariance.

An implicit assumption in the above derivations is thatalthough the lt ’s change across days, they are assumed tobe constant within the day. This is a simplifying assump-tion as, in practice, there is also an intra-daily pattern inthe relationship between returns and order flow. However,in order to keep the exposition as simple and transparentas possible, we treat the lt ’s as constant within the day inthe main empirical analysis. We then show in Section 6that allowing for an intra-daily pattern does not signifi-cantly change any of the results or conclusions, and evenmarginally strengthens some of the empirical results. Thebaseline specification with the market sensitivity fixedwithin each day can be interpreted as a simplified modelwhere only the ‘average’ change in market sensitivity fromday to day is controlled for. Since the results are verysimilar when intra-daily effects are added to the model,this suggests that it is the changes in this averagecoefficient that are most important to our results.9

3.2. Empirical specification

The empirical specification that we test is the followingempirical log version of Eq. (3),

logðRVtÞ ¼ aþ 2bl logðltÞ þ bOF logðOFð2Þt Þ þ ut , (5)

8 Another diagnostic of the relationship between returns and order

flow is given by the fraction of time that the signs of returns and order

flow coincide. In the five-minute data, this happens 71% of the time. For

order flows close to zero, it is not likely that there is a strong ‘pull’ on

returns in either direction, and thus, if one focuses only on the periods

with order flow larger than the 75th percentile and smaller than the

25th percentile of the empirical distribution of order flow, this fraction

increases to 85%. In the daily data, where some of the high-frequency

noise in returns is removed, the corresponding numbers are 72% and

88%, respectively. Thus, there is a fairly strong sign correspondence

between returns and order flow, both at extremely high frequencies and

at lower daily frequencies.9 In a separate Appendix, which is available on the Journal’s Web

page, some additional robustness results are also presented. In particular,

we analyze whether there is any predictive ability of lagged order flow

for current returns; i.e., whether lags of order flow should be included in

Eq. (4). Some (fairly weak) evidence of predictability is found, but the

effects are small enough that, again, none of the results presented in the

main empirical analysis change in any significant way.

where log of realized volatility (variance) is now afunction of the log of the market sensitivity parametersand the integrated squared order flow. This provides aconvenient specification since it allows for tests ofwhether it is the variations in both the lt ’s and theintegrated squared order flow that help explain realizedvolatility or whether it is the variation in only one of themthat primarily influences volatility. Furthermore, we canalso test whether bl ¼ bOF ¼ 1 as implied by Eq. (3); thereare no natural restrictions on a (see footnote 10). In theremainder of the paper, we will refer only to the log-transformed variables unless otherwise noted, and we willrefer to logðRVtÞ as the (log) realized volatility.

As shown below, realized volatility is fairly persistent andwell characterized as a long-memory or fractionally inte-grated process. In order for Eq. (5) to provide a meaningfuleconometric relationship, some of the persistence in realizedvolatility must therefore be explained by the lt ’s and theintegrated squared order flow; i.e. (fractional) cointegrationmust exist. Otherwise, the error term will have the samepersistence as the original data and Eq. (5) will make littlesense from an econometric point of view. It is important tonote, however, that the strong results from estimating Eq. (4)do not necessarily imply that such a relationship holds in Eq.(5). For instance, consider the case when the true model ofreturns is given by Eq. (4). The instantaneous variance ofreturns, assuming that lt is fixed over short intervals and thatthe covariance term is zero, is now given bys2

r;t ¼ l2t s2

of ;t þ s2�;t , in obvious notation. In order for all of

the persistence in the variance of returns, s2r;t , to be described

by l2t s2

of ;t, it follows that s2�;t must be non-persistent. That is,

all of the persistence in the variance of returns must comefrom the part of returns that is captured by the order flowrelationship.10Simply estimating Eq. (4) does not revealwhether this is the case. Essentially, Eq. (4) is focused onthe first moment of the data, whereas the analysis of Eq. (5) isstrongly linked to the persistence in the second moment.

As always in regression analysis, an important concernis the possible effect of any potential missing factors in Eq.(5), and the empirical analysis is specifically designed toaddress such concerns. First, since we focus on the long-run properties of volatility and test for fractional coin-tegration, as described in detail below, the exclusion ofany transient factors should not bias the analysis. Second,by testing for fractional cointegration and finding thatvirtually all of the persistence in volatility is captured bythe variables included in Eq. (5), it can be ruled out thatthe residuals contain important missing persistent factors.Finally, in going from Eqs. (1) to (5), there are strongrestrictions on the regression coefficients, in the form ofbl ¼ bOF ¼ 1. If these restrictions hold in addition tofractional cointegration, the evidence must be consideredvery strong that the lt ’s and integrated squared order flowcapture the long-run behavior in volatility and that theydo so in a ‘structural,’ or economically relevant, sense.

10 This discussion also illustrates why there are no simple restric-

tions on the value of a in Eq. (5). Ignoring the log-transformation, a will

reflect the average value of the volatility due to the variation in returns

that is not related to order flow (i.e., s2�;t). In general, this value will be

unknown and not subject to any informative restrictions.

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Table 3Summary statistics of the daily log-transformed data.

The mean, standard deviation, skewness, and kurtosis, as well as the

minimum and maximum values, over the full 1999–2004 sample period,

are shown for each variable. The summary statistics are given for both

daily samples used in the analysis; i.e., the daily sample constructed

from full-day five-minute observations and the daily sample constructed

from 3–11am one-minute observations. Each day there are 288 intra-

daily full-day five-minute observations and 480 3–11am one-minute

observations from which the daily series are constructed. There are T ¼

1;487 observations in the daily samples (see footnote 13 for details).

logðRVtÞ represents the daily log-transformed realized volatility, defined

as the sum of squared intra-daily returns. logðltÞ represents the daily log-

transformed realized market sensitivity coefficients, defined as the OLS

estimates of the coefficient on order flow in daily regressions of returns

on contemporaneous order flow, using intra-daily observations.

logðOFð2Þt Þ represents the daily log-transformed integrated squared order

flow, defined as the sum of the squared intra-daily order flow

observations. logðVtÞ represents the daily log-transformed volume of

trade, where the original volume is expressed in millions of the base

currency (the euro). In the full-day five-minute sample, the daily volume

of trade represents the volume for the full day. In the 3–11am one-

minute sample, the daily volume of trade represents the volume in the

period 3–11am each day. The first moment of the volume of trade is

proprietary and cannot be displayed.


That is, validity of the coefficient restrictions providesevidence that Eq. (5) not only constitutes a statisticallyvalid relationship, but also an economically relevant one.

As for the first stage estimation in Eq. (4), if order flowis correlated with the residuals, then the OLS estimates ofthe lt ’s will be biased. If this bias is time-varying (andpersistent), some of the observed variation in theestimated daily lt ’s may simply be a result of changes inthe bias term. In this case, though Eq. (5) may still providea valid (economic) relationship for volatility, one could notnecessarily interpret the estimated bl as solely represent-ing the changes in volatility due to market sensitivity.Such a correlation might arise, for instance, if there is anunobserved risk-premium in the returns, which is corre-lated with the order flow. However, also in this case, thefractional cointegration tests and coefficient restrictionsin Eq. (5) are likely to raise warning signals of misspeci-fication.11Thus, although one can never completely guardagainst model misspecification, the procedures we use foranalyzing Eq. (5) go a long way towards addressingomitted variable concerns both in the actual volatilityEq. (5) as well as in the first stage Eq. (4).

Variable Mean Std.dev. Skewness Kurtosis Min Max


logðRVtÞ �0.917 0.523 0.185 3.871 �2.697 1.873

logðltÞ 3.974 0.279 �0.367 7.921 1.416 5.008

logðOFð2Þt Þ13.502 0.471 �0.911 5.393 10.957 14.761

logðVtÞ 0.333 �1.207 6.687


logðRVtÞ �1.634 0.565 0.167 3.549 �3.447 1.086

logðltÞ 3.890 0.265 0.174 3.338 2.678 4.837

logðOFð2Þt Þ12.823 0.459 �1.046 5.801 10.332 14.203

logðVtÞ 0.352 �1.215 6.515

3.3. Construction of the daily data

The daily data used in estimating Eq. (5) are con-structed in a manner identical to that described in thederivations above. That is, from the five minute exchangerate data, we construct continuously compounded returns(log differences), where rt;i is the return on day t ininterval i. There are 288 five-minute intervals each day;interval 1 is the time period from 17:00 to 17:05 (NewYork time), since, by convention, each trading day in theglobal foreign exchange market begins and ends at 17:00.We then calculate realized variance on day t asRVt ¼

P288i¼1 r2

i;t .12 Similarly, squared integrated order flow

is created as OFð2Þt �P288

i¼1 of 2t;i where of t;i is the five-

11 Suppose Eq. (1) is replaced with rt;i ¼ ltof t;i þ xt;i, where xt;i is a

risk-premium. It follows that RVt ¼ l2t OFð2Þt þ 2ltOFXt þ Xð2Þt , where

Xð2Þt �PK

i¼1x2t;i and OFXt �

PKi¼1of t;ixt;i . The OLS estimate of lt in the

fitted regression (4) will satisfy lt ¼ lt þ OFXt=OFð2Þt þ opð1Þ and in terms

of lt , RVt ¼ l2

t OFð2Þt � OFX2t =OFð2Þt þ Xð2Þt � l

2

t OFð2Þt þ Bt . Ignoring the ef-

fects of the log-transformation, the fractional cointegration analysis of

Eq. (5) should detect the additional term Bt , if it is persistent. In the

special case that Bt is persistent and cointegrated with l2

t and/or OFð2Þt ,

the fractional cointegration analysis might not detect its presence, but

the coefficient restrictions on bl and bOF are, on the other hand, not

likely to hold. Finally, if the bias terms are not persistent, they should not

be relevant to the long-run analysis performed in the paper. The log-

transformation and the non-linear interactions between the variables

make it difficult to formally analyze the effects of the bias, but it still

seems likely that the cointegration analysis of Eq. (5) will have some

power to detect this issue.12 Realized variance, calculated from high-frequency intra-daily

observations, has by now become the most established way of

measuring daily variance in a model-free fashion. The correlation

between the daily realized variance, obtained from five-minute returns,

and the daily squared returns is around 0.3, which reflects the fact that

although the daily squared return is an unbiased estimate of daily

variance, it is extremely noisy and imprecise; this is the main point of the

seminal work by Andersen and Bollerslev (1998), who argue that daily

squared returns will typically not be a very useful measure of ex post

daily variance.

minute order flow in interval i on day t. Daily variablesbased on the one-minute frequency intra-daily data forthe busiest trading hours between 3am and 11am areconstructed in an analogous manner. The lt variables areobtained from the daily OLS estimates of Eq. (4), allowingfor a non-zero intercept, as described above. There areT ¼ 1;489 daily observations in the data.13 Tables 3 and 4show summary statistics of the data, including the dailyvolume of trade, as well as the correlations between thevariables. The log-data are graphed in Fig. 2.

3.4. Preliminary OLS analysis

OLS analysis of Eq. (5) suffers from several potentialpitfalls. First, there is no reason to believe that the right-hand side variables in Eq. (5) are exogenous in the

13 In each of the two daily samples thus obtained, there are two days

for which the estimated lt is negative, although not statistically

significantly different from zero. Interestingly, these are days when U.S.

monthly payroll data were released, with large jumps in the exchange

rate instantaneously associated with the data releases. Since negative

values of lt make little economic sense and prevent a log-transforma-

tion, we drop these rare days from our sample. The actual number of

days used in each sample is thus T ¼ 1;487.

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Table 4Correlation matrices for the daily data.

Each panel shows the correlation structure between the variables in

the daily samples for the full 1999–2004 sample period. The summary

statistics are given for both daily samples used in the analysis; i.e., the

daily sample constructed from full-day five-minute observations and the

daily sample constructed from 3–11am one-minute observations. Each

day there are 288 intra-daily full-day five-minute observations and 480

3–11am one-minute observations from which the daily series are

constructed. There are T ¼ 1;487 observations in the daily samples

(see footnote 13 for details). logðRVtÞ represents the daily log-trans-

formed realized volatility, defined as the sum of squared intra-daily

returns. logðltÞ represents the daily log-transformed realized market

sensitivity coefficients, defined as the OLS estimates of the coefficient on

order flow in daily regressions of returns on contemporaneous order

flow, using intra-daily observations. logðOFð2Þt Þ represents the daily log-

transformed integrated squared order flow, defined as the sum of the

squared intra-daily order flow observations. logðVtÞ represents the daily

log-transformed volume of trade, where the original volume is expressed

in millions of the base currency (the euro). In the full-day five-minute

sample, the daily volume of trade represents the volume for the full day.

In the 3–11am one-minute sample, the daily volume of trade represents

the volume in the period 3–11am each day.

Variable logðRVtÞ logðltÞ logðOFð2Þt ÞlogðVtÞ


logðRVtÞ 1.000

logðltÞ 0.564 1.000

logðOFð2Þt Þ0.433 �0.398 1.000

logðVtÞ 0.528 �0.232 0.882 1.000


logðRVtÞ 1.000

logðltÞ 0.602 1.000

logðOFð2Þt Þ0.467 �0.321 1.000

logðVtÞ 0.595 �0.136 0.933 1.000

Table 5Results from OLS estimation of the volatility equation.

This table reports the results from ordinary least squares (OLS)

estimation of the following model (Eq. (5)),

logðRVtÞ ¼ aþ 2bl logðltÞ þ bOF logðOFð2Þt Þ þ ut ,

where logðRVtÞ, logðlt Þ, and logðOFð2Þt Þ, are the log-realized volatility, the

log-market sensitivity parameter, and the log-integrated squared order

flow, on day t, respectively. Daily data from 1999 to 2004 are used in the

estimation, and the standard errors are given in parentheses below the

estimates. Panel A shows the results for the daily sample constructed

from full-day five-minute observations and Panel B shows the results for

the daily sample constructed from 3–11am one-minute observations.

The first row in each panel shows the results from the unrestricted

estimation, whereas rows two and three in each panel shows the results

from the restricted estimation, with bOF ¼ 0 and bl ¼ 0, respectively. The

last column shows the R2. T ¼ 1;487 daily observations are used in the

estimation.

a bl bOF R2


�19.154 0.822 0.867 0.831

(0.223) (0.011) (0.013)

�5.127 0.530 0.318

(0.160) (0.020)

�7.410 0.481 0.187

(0.351) (0.026)


�20.206 0.894 0.905 0.849

(0.214) (0.011) (0.013)

�6.636 0.642 0.363

(0.172) (0.022)

�9.003 0.574 0.218

(0.362) (0.028)


econometric sense, and OLS estimation may thereforeresult in biased estimates. Second, and as importantly,since the variables are all fairly persistent, there is the riskof spurious regression results, in the sense that large t-statistics could be found even though there is no actualrelationship between the variables (e.g., Granger andNewbold, 1974; Phillips, 1986; Tsay and Chung, 2000).Finally, lt is a generated regressor and thus subject tomeasurement error, which may lead to additional bias inthe OLS estimates. However, it is still instructive toconsider the OLS estimates of Eq. (5) and compare themto the results obtained from regression analyses thatexplicitly take into account the persistence and potentialendogeneity in the data.

Table 5 shows the results from the OLS estimation ofEq. (5). Based on these results, Eq. (5) appears to provide areasonably good fit to the data, with a large R2 (around80%) and highly significant t-statistics for all parameters.However, the coefficient estimates deviate significantlyfrom their theoretical values of bl ¼ bOF ¼ 1, a result duemostly to bias in the OLS estimation as shown below.Table 5 also reports the results from estimating Eq. (5)when either bl or bOF are restricted to equal zero, that is,the results from regressing realized volatility onto eitherjust the lt ’s or just the squared order flow. Judging by themuch lower R2’s in these restricted estimations, it seems

that both of the regressors in Eq. (5) help explain themovements of volatility over time. By themselves, the lt ’sappear to explain more of the variation in volatility thanthe squared order flow, but neither variable does a verygood job alone.

In the next section, we outline econometric methodsthat take into account the persistent nature of the data, aswell as measurement errors, and provide explicit tests ofthe validity of Eq. (5).

4. Econometric methodology

4.1. Fractional integration and cointegration

The high degree of serial correlation in volatility, evenat long lags, is a well established empirical regularity. Oneof the most common models for capturing this ‘long-memory’ property is the so-called fractionally integratedmodel, often referred to simply as a long-memory model.A process xt is said to be fractionally integrated withmemory parameter d if ð1� LÞdxt ¼ vt , where L is theusual lag operator satisfying Lxt ¼ xt�1, and vt is a (short-memory) stationary process. This specification generalizesthe concept of an integrated process with d ¼ 1 to allowfor fractional values of d. For jdjo1=2, the process xt isstationary, although it will only slowly return to its long-

ARTICLE IN PRESS


run unconditional mean, and for jdj � 1=2, xt is a non-stationary process.

Similarly, the concept of cointegration can also beexpanded to fractional cointegration. If Zt is a vectorprocess of fractionally integrated variables, each withmemory parameter d, and there exists a non-zero linearcombination of the elements in Zt with memory duod,then Zt is said to be fractionally cointegrated.

The subsequent econometric analysis in this paper isbased around these concepts of fractional integration andcointegration. We show that the variables in Eq. (5)possess long-memory and then test whether Eq. (5) is afractional cointegrating relationship. This approach allowsus to directly assess how much of the persistence involatility can be attributed to the persistence in theexplanatory variables by estimating the parameter du

from the fractional cointegrating residuals. The test forcointegration also provides an explicit test againstspuriousness; the issue of endogenous regressors, as wellas measurement errors, will be addressed by relying onfrequency domain methods as described below.

4.2. Estimation of d

Univariate estimation of the long-memory parameter d

for each variable has been well analyzed and a number ofdifferent procedures have been proposed in the literature.We primarily use a recent estimator developed byShimotsu and Phillips (2004) and Shimotsu (2004), whichthey refer to as the exact local Whittle (ELW) estimator. TheELW estimator is more efficient than the commonly usedlog-periodogram regression estimator (Geweke and Por-ter-Hudak, 1983; Robinson, 1995a) and unlike the stan-dard local Whittle estimator (Kunsch, 1987; Robinson,1995b), it is consistent and asymptotically normallydistributed for any value of d. The ELW estimator relieson a frequency domain representation of the data anduses only the first m frequencies closest to the origin,which eliminates the effects of the short-run dynamics ofthe data.

If the variable under consideration is a combination ofa long-memory process and an excessively noisy short-memory process, perhaps due to measurement error,standard estimators of d such as those mentioned abovemay be downward biased in small samples, although theyare still consistent. We therefore also consider a modifiedversion of the local Whittle estimator proposed byHurvich and Ray (2003) and Hurvich, Moulines, andSoulier (2005), which is designed to reduce this potentialbias. This estimator will be referred to as the local Whittle

with noise (LWN) estimator; it also relies on a frequencydomain representation, using only the first m frequenciesclosest to the origin. The LWN estimator does suffer fromone significant drawback, however. It is of primaryinterest here to test whether d ¼ 0 for the residuals inEq. (5), but the LWN estimator is only consistent for d40.As d approaches zero, the LWN estimator becomesincreasingly imprecise. In the empirical analysis, the mainfocus will therefore be on the ELW estimator, and the LWN

estimator will be used as a robustness check. In practice,the two estimators deliver very similar results.

4.3. Cointegration estimation and testing

It is well known that in a standard cointegrationframework with unit-root regressors and stationaryerrors, the standard OLS estimates of the cointegrationvector are still consistent when the regressors areendogenous. Briefly speaking, this holds because thestrength of the signal in the non-stationary regressors isof an order of magnitude stronger than the biasing effectresulting from the endogeneity. A similar argument can bemade in the stationary fractionally cointegrated case withdo1=2, which seems to be the relevant case for thepresent study, but only when focusing explicitly on thelong-run movements in the data. That is, for low-frequency variations, the signal from the long-memoryregressors dominates the biasing endogeneity effect.

The most convenient way of extracting the long-runmovements in the data is by transforming the data intothe frequency domain, where the frequencies close to zerorepresent the long-run. A least squares estimator thatrelies only on observations in a narrow band of frequen-cies is referred to as a narrow band least squares (NBLS)estimator. Robinson (1994) shows that in the presence offractional cointegration, with do1=2, the NBLS estimatoraround the zero frequency does yield consistent estimatesof the cointegrating vector. The NBLS estimator is alsoconsistent in the non-stationary case of d � 1=2, as long asthere is fractional cointegration. In the stationary case,under fairly general assumptions, the NBLS estimator isasymptotically normally distributed (Christensen andNielsen, 2006).

NBLS estimation also addresses concerns over mea-surement errors. The lt ’s are generated regressors andtherefore subject to measurement errors, which generallylead to bias in standard OLS estimation because themeasurement errors induce a correlation between theregressors and the error term. However, as just outlined,when fractional cointegration exists and the estimation isperformed using NBLS, correlation between the regressorand the error term still allows for consistent estimation,with correct standard errors.

The NBLS estimator is a function of the number offrequencies close to zero used in the estimation; we callthat number the bandwidth parameter m0. Similarly, welabel as m1 the bandwidth parameter used in theestimation of du for the NBLS residuals. The asymptoticdistribution of the estimator for du will be the same as ifthe true cointegration errors were used, provided that m0

and m1 satisfy m0=m1 ! 0 as T !1, in addition to theusual restrictions (Nielsen and Frederiksen, 2006).

To sum up, we use a three-stage approach to test forfractional cointegration. In the first stage, the memoryparameters for the original data, and in particular for theleft-hand side variable, are estimated. In the second stage,the cointegrating vector, or regression coefficients, areestimated using NBLS, and the residuals are obtained. Inthe final stage, the memory parameter for the residuals, du

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15 Thus, it cannot be ruled out statistically that some of the variables

are non-stationary. The NBLS estimator will remain consistent for non-

stationary data, although it will no longer be asymptotically normally

distributed. Since most estimates point towards stationarity or near

stationarity, however, inference based on the assumption of a stationary

fractional cointegration relationship still seems the most suitable. One

would also expect that, for small deviations from stationarity, i.e., for d

greater than but close to 0.5, the estimators are close to normally


is estimated and compared to the memory estimate forthe dependent variable.

5. Empirical results

At the end of Section 3, we presented some preliminarysupport for Eq. (5) based on OLS analysis. In this section,we use the econometric tools described above to showthat the initial conclusions from the OLS estimation can infact be substantially strengthened. We find strong evi-dence of a fractional cointegrating relationship betweenrealized volatility, the integrated squared order flow, andthe market sensitivity. Indeed, in many cases, we cannotreject the null hypothesis that all of the long-memory involatility is explained by these two covariates. This isespecially true for the sample based only on data for thebusiest trading hours, sampled at the one-minute fre-quency. In this case, the null hypothesis cannot be rejectedfor any of the bandwidths that are used. In addition, therestrictions on the regression coefficients are generallywell supported by the data.

The empirical results are shown in Table 6. Panel Ashows the results for the samples based on intra-dailydata sampled at the five-minute frequency, using all hoursof the day. Panel B shows the corresponding results whenonly intra-daily data between the hours of 03:00 and11:00, sampled at the one-minute frequency, are used. Inboth panels, the estimates of d for each of the daily log-transformed variables are shown, along with the esti-mates of the cointegrating vector, and the correspondingestimates of the long-memory parameter, du, in theresiduals. Since there is little guidance on how to choosethe bandwidth parameters in practice, all estimates arecalculated for a number of different bandwidths.

5.1. Estimates of d

The top of both panels in Table 6 shows the estimatedlong-memory parameters for each data series. Threedifferent bandwidths are considered, m ¼ ½T0:5

� ¼ 38,m ¼ ½T0:6

� ¼ 80, and m ¼ ½T0:7� ¼ 166, where ½�� indicates

the integer part of a real number.14 Results for both theELW estimator and the LWN estimator are shown, wherethe latter explicitly attempts to control for noise andmeasurement error in the variables.

Starting with the ELW estimator, for the largerbandwidths, m ¼ ½T0:6

� and m ¼ ½T0:7�, the estimates of d

for realized volatility are all between 0.4 and 0.5, similarto those found in other studies (e.g., Andersen, Bollerslev,Diebold, and Labys, 2003a; Bollerslev and Wright, 2000).The estimates for the lt ’s are similar to those for realizedvolatility, but generally somewhat larger. The estimatesfor the integrated squared order flow are smaller and areall in the region ð0:3;0:4Þ. The LWN estimates are generallysimilar, although for m ¼ ½T0:6

�, the estimates of d for

14 The bandwidths used here for the estimation of d, and below in

the NBLS estimation, are all of the form m ¼ ½Tg�, for g ¼ 0:3;0:4;0:5;0:6,

and 0.7. The highest frequencies included in the spectral analysis for

these bandwidths correspond approximately to time-domain horizons of

185, 80, 40, 20, and 10 days, respectively.

realized volatility are substantially larger when the biascorrected estimator is used.

For some bandwidths, and when using the LWNestimator in particular, the point estimates of d for bothrealized volatility and the realized lt ’s are greater than0.5, and thus in the non-stationary region. This is incontrast to the commonly held belief that realizedvolatility is a stationary process with do0:5, althoughBandi and Perron (2006) also find similar results for stock-return volatility. Of course, for realized volatility, for allbandwidths considered here, a 95% confidence interval ford would always include values greater than 0.5.15 Ingeneral, some caution should be applied when interpret-ing measures of long-run properties such as the memoryparameter d. Although the data series each contain almost1,500 observations, the temporal span is only six years.One can therefore not rule out that there are still smallsample biases in the estimates of d, particularly since thevariables are subject to imperfect measurement, and oneshould not over-interpret any observed differences be-tween the estimates for the different variables, or theabsolute magnitude of particular estimates.

In any case, there is strong evidence of significant long-memory in all variables, and the estimates indicate thatthe memory in both realized volatility and marketsensitivity are quite similar whereas the squared orderflow appears to have somewhat less memory. Thesubsequent fractional cointegration analysis does notrequire identical memory in all of the variables; it issufficient that one of the right-hand side variables has thesame memory as the left-hand side variable.

5.2. Cointegration estimates

The bottom parts of the panels in Table 6 show theresults from the fractional cointegration analysis with aset of different bandwidths. As a comparison to thenarrow band estimates, the second to last row in eachpanel gives the results from a full bandwidth or,equivalently, OLS estimation; these results are thusidentical to those shown in Table 5, except for theadditional estimates of du, based on the OLS residuals,which are also shown.16

The results for the sample based on intra-daily five-minute returns from all hours of the day are shown inPanel A of Table 6. It is immediately obvious that there is a

distributed asymptotically. Most importantly, the estimator of du for the

regression residuals will have the same asymptotic distribution regard-

less of whether the data are stationary or not.16 Clearly, for the OLS estimator, which is equivalent to the frequency

domain least squares estimator with all the frequencies included (i.e.,

m0 ¼ T), the condition that m0=m1 ! 0 is not satisfied. The results are

still likely to be indicative, however.

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Table 6Results from fractional cointegration analysis of the volatility equation.

This table reports the estimates of the long-memory parameters in the data as well as the narrow band least squares (NBLS) regression estimates, and estimates of the long-memory parameter for the

residuals, in the following regression equation (Eq. (5)),


where logðRVtÞ, logðltÞ, and logðOFð2Þt Þ, are the log-realized volatility, the log-market sensitivity parameter, and the log-integrated squared order flow, on day t, respectively. Daily data from 1999 to 2004 are used

in the estimation. Standard errors are given in parentheses below the estimates and estimates of the intercept are not shown. Panel A shows the results for the daily sample constructed from full-day five-minute

observations and Panel B shows the results for the daily sample constructed from 3–11am one-minute observations. The first part of each panel shows the exact local Whittle (ELW) and local Whittle with noise

(LWN) estimates of d for the log of realized volatility ðRVtÞ, the log of the market sensitivity parameter ðltÞ, and the log of the squared integrated orderflow ðOFð2Þt Þ. The estimates for three different bandwidths, m,

are reported. The second part of each panel reports the NBLS estimates of the above regression equation (Eq. (5)), for different bandwidth choices. The second to last row in these panels corresponds to ordinary

least squares (OLS) estimation, and the last row imposes the null hypothesis bOF ¼ bl ¼ 1. The columns labeled du , with the bandwidth ðm1Þ listed below, give the ELW estimates of the long-memory parameter

for the residuals of the respective regression and the columns labeled dLWN

u give the corresponding LWN estimates. The bandwidths that are used are all integer parts of fractional powers of the sample size T.

Observe that ½T0:3� ¼ 8, ½T0:4

� ¼ 18, ½T0:5� ¼ 38, ½T0:6

� ¼ 80, ½T0:7� ¼ 166, and T ¼ 1;487.


ELW long-memory estimates, d LWN long-memory estimates, dLWN

Bandwidth logðRVt Þ logðltÞ logðOFð2Þt ÞlogðRVtÞ logðltÞ logðOFð2Þt Þ

m ¼ ½T0:5� 0.559 0.604 0.329 0.623 0.598 0.400

(0.081) (0.081) (0.081) (0.106) (0.108) (0.142)

m ¼ ½T0:6� 0.445 0.476 0.363 0.720 0.623 0.393

(0.056) (0.056) (0.056) (0.067) (0.073) (0.099)

m ¼ ½T0:7� 0.456 0.488 0.309 0.469 0.499 0.394

(0.039) (0.039) (0.039) (0.061) (0.058) (0.069)

Cointegration analysis

Bandwidth bl bOF du du du dLWN

u dLWN

u dLWN

u

m1 ¼ ½T0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7� m1 ¼ ½T

0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� 0.949 0.958 0.173 0.085 0.086 0.169 0.084 0.086

(0.039) (0.088) (0.081) (0.056) (0.039) (0.281) (0.363) (0.244)

m0 ¼ ½T0:4� 0.955 0.967 0.082 0.084 0.080 0.084

(0.032) (0.058) (0.056) (0.039) (0.375) (0.252)

m0 ¼ ½T0:5� 0.953 0.936 0.088 0.087

(0.026) (0.045) (0.039) (0.242)

m0 ¼ T (OLS) 0.822 0.867 0.216 0.235

(0.011) (0.013) (0.039) (0.637)

bl ¼ bOF ¼ 1 imposed 1 1 0.185 0.088 0.085 0.181 0.085 0.084

(0.081) (0.056) (0.039) (0.265) (0.357) (0.250)

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ELW long-memory estimates, d LWN long-memory estimates, dLWN

Bandwidth logðRVt Þ logðltÞ logðOFð2Þt ÞlogðRVtÞ logðltÞ logðOFð2Þt Þ

m ¼ ½T0:5� 0.548 0.586 0.303 0.569 0.581 0.393

(0.081) (0.081) (0.081) (0.112) (0.110) (0.144)

m ¼ ½T0:6� 0.429 0.552 0.328 0.704 0.514 0.379

(0.056) (0.056) (0.056) (0.068) (0.082) (0.102)

m ¼ ½T0:7� 0.441 0.487 0.317 0.432 0.516 0.351

(0.039) (0.039) (0.039) (0.064) (0.057) (0.075)

Cointegration analysis

Bandwidth bl bOF du du du dLWN

u dLWN

u dLWN

u

m1 ¼ ½T0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7� m1 ¼ ½T

0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� 0.957 0.885 0.116 �0.010 0.025 0.116 0.010 0.022

(0.028) (0.060) (0.081) (0.056) (0.039) (0.392) (2.823) (0.908)

m0 ¼ ½T0:4� 0.965 0.899 �0.011 0.021 0.010 0.019

(0.027) (0.049) (0.056) (0.039) (2.823) (1.054)

m0 ¼ ½T0:5� 0.967 0.886 0.023 0.021

(0.023) (0.037) (0.039) (0.944)

m0 ¼ T(OLS) 0.894 0.905 0.101 0.102

(0.011) (0.013) (0.039) (0.210)

bl ¼ bOF ¼ 1 imposed 1 1 0.173 0.034 0.042 0.176 0.040 0.042

(0.081) (0.056) (0.039) (0.271) (0.722) (0.480)

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Table 7Results from fully modified narrow band least squares estimation of the

volatility equation.

This table reports the fully modified narrow band least squares

(FMNBLS) regression estimates, as well as estimates of the long-memory

parameter for the residuals, for the following equation (Eq. (5)),


where logðRVtÞ, logðltÞ, and logðOFð2Þt Þ, are the log-realized volatility, the


flow, on day t, respectively. Daily data from 1999 to 2004 are used in the

estimation. Standard errors are given in parentheses below the estimates

and estimates of the intercept are not shown. Panel A shows the results

for the daily sample constructed from full-day five-minute observations

and Panel B shows the results for the daily sample constructed from

3–11am one-minute observations. Each panel reports the FMNBLS

estimates of the above regression equation (Eq. (5)), for different

bandwidth choices (see footnote 17 for a brief description of the role

of each bandwidth). The column labeled du gives the exact local Whittle

(ELW) estimates of the long-memory parameter for the residuals of the

respective regression, using bandwidth m1. The bandwidths that are

used are all integer parts of fractional powers of the sample size T.

Observe that ½T0:4� ¼ 18, ½T0:5

� ¼ 38, ½T0:6� ¼ 80, ½T0:7

� ¼ 166, and

T ¼ 1;487.


Bandwidths bl bOF du

m0 ¼ ½T0:4�; m1 ¼ ½T

0:6� 0.970 1.019 0.075

m2 ¼ ½T0:6�; m3 ¼ m0

(0.032) (0.059) (0.056)

m0 ¼ ½T0:4�; m1 ¼ ½T

0:7� 0.983 1.046 0.081

m2 ¼ ½T0:7�; m3 ¼ m0

(0.033) (0.061) (0.039)

m0 ¼ ½T0:5�; m1 ¼ ½T

0:7� 0.984 0.988 0.081

m2 ¼ ½T0:7�; m3 ¼ m0

(0.027) (0.046) (0.039)


Bandwidths bl bOF du

m0 ¼ ½T0:4�; m1 ¼ ½T

0:6� 0.970 0.913 �0.010

m2 ¼ ½T0:6�; m3 ¼ m0

(0.027) (0.049) (0.056)

m0 ¼ ½T0:4�; m1 ¼ ½T

0:7� 0.982 0.937 0.022

m2 ¼ ½T0:7�; m3 ¼ m0

(0.028) (0.050) (0.039)

m0 ¼ ½T0:5�; m1 ¼ ½T

0:7� 0.991 0.915 0.027

m2 ¼ ½T0:7�; m3 ¼ m0

(0.023) (0.038) (0.039)


fairly large difference between the standard OLS estimatesand the narrow band estimates. The NBLS estimates of bland bOF are much closer to unity, as the model wouldpredict: based on the standard errors, we typically cannotreject the null hypothesis that bl ¼ bOF ¼ 1. The differ-ences across bandwidths are fairly small and do notchange the overall outcome of the estimates. This issomewhat striking, given that the smallest bandwidthused for the regression estimates, m0 ¼ ½T

0:3� ¼ 8, in fact

only contains the first eight frequencies; this is a reflectionof how much of the signal in a persistent process isconcentrated in the first few frequencies. The OLSestimates show, however, that the introduction of higherfrequencies will eventually bias the results downward.

The estimates of du, the long-memory parameter forthe residuals in Eq. (5), are shown in both panels of Table6. Results for both the ELW and LWN estimators areprovided. The point estimates from these two proceduresare very close but the standard errors for the LWNestimates are often very large, as discussed previously.Given this, the discussion below refers exclusively to theELW estimates.

Focusing again on Panel A, it is evident that thememory in the residuals, ut , is substantially less than thememory in the original realized volatility. Although formost bandwidths it is possible to reject the null hypoth-esis that du ¼ 0, the evidence of fractional cointegration isvery strong, with estimates of du typically much smallerthan the estimates of d for realized volatility. Only for thesmallest bandwidth is the estimate of du, about 0.17,substantially larger than zero in absolute terms. However,for that bandwidth, the (ELW) estimate of d for realizedvolatility is equal to 0.559, suggesting a memory reduc-tion of almost 0.4. When using the OLS residuals, theestimate of du, 0.216, is much larger than the correspond-ing estimates from the narrow band residuals, which areequal to about 0.08, with the same bandwidth used for theestimation of du.

There is thus very strong evidence that Eq. (5) shouldbe seen as a fractional cointegrating relationship. It cannotbe ruled out that there is still a small long-memorycomponent in the residuals, but it is evident that theamount of persistence in the residuals is much lower thanin the realized volatility.

The results for the one-minute sample shown in PanelB of Table 6 are also generally in line with those justdiscussed for the full-day five-minute sample. For theone-minute sample, the cointegration results are evenstronger, and we cannot reject the hypothesis of du ¼ 0 forany bandwidths; the point estimates of du are alsotypically very close to zero. The estimates of bOF aresomewhat smaller than those for the full-day five-minutedata shown in Panel A, and on the borderline of beingsignificantly different from one.

Although the narrow band estimates are generallysupportive of the coefficient restrictions, it is alsointeresting to impose the null hypothesis of bl ¼ bOF ¼ 1and estimate the long-memory in the resulting residuals.The results from this exercise are shown in the bottomrows of both panels in Table 6. The estimates of du in thefive-minute sample are virtually identical to those

obtained when the narrow band estimation was used toestimate bl and bOF . In the one-minute sample, the onlynoticeable difference is seen for the smallest bandwidth,where the ELW estimate of du is now marginallysignificantly different from zero, although still small inabsolute terms and no larger than the estimates in thefive-minute sample. The evidence of fractional cointegra-tion thus remains very strong when the coefficientrestrictions are in place, further validating Eq. (5).

To gather additional direct evidence on the validity ofthe coefficient restrictions, we also consider an improve-ment of the NBLS estimator. Nielsen and Frederiksen(2006) show how to modify the NBLS estimator and

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Fig. 3. Plots of the demeaned daily log-realized volatility and the corresponding regression residuals. The top two panels show plots of the demeaned log-

realized volatility, constructed from the full-day five-minute and 3–11am one-minute intra-daily data, respectively. The lower panels show the

corresponding regression residuals from the narrow band least squares (NBLS) estimation of Eq. (5), using bandwidth m0 ¼ ½T0:4� ¼ 18; i.e., the NBLS

residuals from regressing logged realized volatility on the logged market sensitivity coefficients and the logged integrated squared order flow.


achieve estimates that are more closely centered aroundthe true parameter values; they label the resultingestimator fully modified NBLS (FMNBLS). The FMNBLSestimator improves upon the NBLS estimator when thereis correlation between the regressors and the residuals,and it should be even better equipped than the NBLSestimator for dealing with measurement error. Theestimation results are shown in Table 7 for some differentbandwidth combinations, along with the ELW estimates ofthe memory in the fitted residuals.17 The FMNBLSestimates are noticeably closer to unity than the NBLSestimates, both in the five-minute and one-minute

17 The FMNBLS estimator relies on a preliminary NBLS estimation,

using bandwidth m0, as well as estimates of d and du , based on the NBLS

residuals, using bandwidth m1; the ELW estimator is used to calculate

the values of d and du . The correction term used in the FMNBLS estimator

is calculated using a bandwidth m2 and the actual FMNBLS estimates are

obtained using a bandwidth m3, which is set equal to m0. The estimate of

du in the FMNBLS residuals is calculated using the bandwidth m1. As

outlined in Nielsen and Frederiksen (2006), there are some restrictions

on the choice of each of these bandwidths, and the bandwidths used in

Table 7 satisfy those restrictions.

samples. Only the estimates of bOF in the one-minutesample deviate to any significant extent from one,although the FMNBLS estimates in this case are still animprovement over the NBLS estimates. The estimates of du

are virtually identical to those in Table 6 for the samebandwidths (du is calculated using the bandwidth m1 inTable 7). The results in Table 7 thus provide additionalevidence that the theoretical restrictions on bl and bOF aregenerally consistent with the data.

Finally, some additional graphical evidence of frac-tional cointegration is shown in Figs. 3 and 4. Fig. 3 showsplots of realized volatility and the corresponding NBLSregression residuals. The difference between the originaldata and the residuals is striking, and the graphs clearlyshow the large reduction in persistent behavior in theresiduals.18 A similar case is made in Fig. 4, which plotsthe auto-correlograms for realized volatility and the NBLSresiduals. Again, there is an obvious and remarkabledifference in the autocorrelation of realized volatility and

18 Many of the largest remaining residuals in the lower panels (the

occasional spikes) occur on days with important macroannouncements.

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Fig. 4. Auto-correlograms of daily log-realized volatility and the corresponding regression residuals. The top two panels show the auto-correlograms of

log-realized volatility, constructed from the full-day five-minute and 3–11am one-minute intra-daily data, respectively. The lower panels show the auto-

correlograms for the corresponding regression residuals from the narrow band least squares (NBLS) estimation of Eq. (5), using bandwidth

m0 ¼ ½T0:4� ¼ 18; i.e., the NBLS residuals from regressing logged realized volatility on the logged market sensitivity coefficients and the logged integrated

squared order flow.


the regression residuals. In summary, the results pre-sented in this section provide strong evidence that thelong-run time-series behavior of exchange rate volatilitycan be explained to a very large degree by movements inthe associated order flow and in the market’s sensitivity tothat order flow.

5.3. Order flow or market sensitivity: Which influence

dominates?

The results in the previous section give strong supportto the ability of market sensitivity and integrated squaredorder flow to jointly explain the persistence in volatility.Although the estimated slope coefficients for both of thesevariables are highly significant and close to their theore-tical values, it cannot be ruled out that the fractionalcointegration result is primarily driven by one of thesevariables. We test this possibility here by using the samefractional cointegration tests as above on each of the twoexplanatory variables separately. The results are shown inTable 8. Starting with the results for lt, it is clear that theestimated slope coefficient bl is smaller than in the

specification with two regressors, but still of a somewhatsimilar magnitude. The estimates of du, the memoryparameter for the residuals, are now substantially larger,however, with values around 0.3 (for brevity, only the ELWestimates are displayed in Table 8, since the LWNestimates were all very similar). The estimates of du arestill smaller than the estimates of d for the originalrealized volatility data, which are typically around 0.45,but it is evident that the lt ’s explain less of the persistencein volatility by themselves than they do jointly with thesquared order flow.

The results for the integrated squared order flow,however, show no evidence that this variable, by itself, canexplain much, or even any, of the persistence in volatility:the estimates of du are very similar to the estimates of d inrealized volatility. Given the apparent lack of fractionalcointegration in this regression, the estimates are likely tobe spurious. This may explain why the slope coefficient isnow negative for the NBLS estimates, and is also related tothe large variation observed across bandwidths. The OLSestimates are, in fact, the only ones that appear somewhatsimilar to the results found for the model with bothexplanatory variables included. However, the lack of any

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Table 8Results from restricted narrow band estimation of the volatility equation.

This table reports the restricted narrow band least squares (NBLS) regression estimates, as well as estimates of the long-memory parameter for the

residuals, for the following equation (Eq. (5)),


with either the coefficient on the market sensitivity ðblÞ or the coefficient on integrated squared order flow ðbOF Þ restricted to be equal to zero. logðRVtÞ,

logðltÞ, and logðOFð2Þt Þ, are the log-realized volatility, the log-market sensitivity parameter, and the log-integrated squared order flow, on day t, respectively.

Daily data from 1999 to 2004 are used in the estimation. Standard errors are given in parentheses below the estimates and estimates of the intercept are

not shown. Panel A shows the results for the daily sample constructed from full-day five-minute observations and Panel B shows the results for the daily

sample constructed from 3–11am one-minute observations. The first column shows the bandwidths used in the estimation, where the last row in each

panel corresponds to OLS estimation. The left-hand side of each panel shows the results for bOF ¼ 0 and the right-hand side shows the results for bl ¼ 0.

The columns labeled du , with the bandwidth ðm1Þ listed below, give the exact local Whittle (ELW) estimates of the long-memory parameter for the

residuals of the respective regressions. The bandwidths that are used are all integer parts of fractional powers of the sample size T. Observe that ½T0:3� ¼ 8,

½T0:4� ¼ 18, ½T0:5

� ¼ 38, ½T0:6� ¼ 80, ½T0:7

� ¼ 166, and T ¼ 1;487.


Bandwidth bl du du du bOF du du du

m1 ¼ ½T0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7� m1 ¼ ½T

0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� 0.739 0.343 0.336 0.311 �0.162 0.553 0.432 0.435

(0.132) (0.081) (0.056) (0.039) (0.658) (0.081) (0.056) (0.039)

m0 ¼ ½T0:4� 0.740 0.336 0.311 0.203 0.461 0.484

(0.115) (0.056) (0.039) (0.381) (0.056) (0.039)

m0 ¼ ½T0:5� 0.721 0.313 0.238 0.489

(0.084) (0.039) (0.240) (0.039)

m0 ¼ T ðOLSÞ 0.530 0.351 0.481 0.522

(0.200) (0.039) (0.026) (0.039)


Bandwidth bl du du du bOF du du du

m1 ¼ ½T0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7� m1 ¼ ½T

0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� 0.766 0.297 0.271 0.301 �0.099 0.552 0.418 0.429

(0.130) (0.081) (0.056) (0.039) (0.648) (0.081) (0.056) (0.039)

m0 ¼ ½T0:4� 0.766 0.271 0.301 0.189 0.450 0.464

(0.110) (0.056) (0.039) (0.380) (0.056) (0.039)

m0 ¼ ½T0:5� 0.749 0.300 0.242 0.470

(0.083) (0.039) (0.238) (0.039)

m0 ¼ T ðOLSÞ 0.642 0.319 0.574 0.500

(0.022) (0.039) (0.028) (0.039)

19 These results are entirely consistent with Eq. (5) providing a valid

cointegrating relationship between RVt , lt , and OFð2Þt , when RVt and lt

have memory 0.45 and OFð2Þt memory 0.3. In this case, regressing RVt on

lt alone is valid from an econometric viewpoint since the cointegration

residual has less memory (0.3) than the original data (0.45) and there is

thus a fractional cointegrating relationship with memory 0.3 in the

residuals. Regressing RVt on OFð2Þt alone, on the other hand, is a spurious

regression in the sense that the residual lt now have the same memory

as the original data, and no relationship can be established.


evidence of cointegration in this case, and the subsequentspurious nature of the regression, makes it difficult tointerpret the coefficient estimates. Still, all of the evidencewe have uncovered strongly suggests that the role ofmarket sensitivity in explaining volatility persistence is atleast as large as that of the rate of information arrival.

The failure of the integrated squared order flow tocapture a meaningful share of the persistence in volatilityby itself is not very surprising, given the estimates of thelong-memory parameters shown in Table 6. The pointestimates clearly indicate that the persistence in volatilityis likely to be greater than that in the squared order flow.Hence, there is simply not enough persistence in thesquared order flow to explain the long-run movements involatility. Market sensitivity, on the other hand, is found tohave a degree of persistence very similar to that ofvolatility. Taken together, the results in Tables 6–8 suggestthe possibility that market sensitivity primarily explainsthe most persistent behavior in volatility, captured by thereduction in memory from 0.45 to 0.3, whereas the

integrated squared order flow captures the somewhat lesspersistent behavior represented by the remaining mem-ory of about 0.3, which is not explained by marketsensitivity.19

6. Intra-daily patterns in the return-order flowrelationship

The analysis in this paper focuses on the variationacross days in the market sensitivity coefficients, lt . As

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Fig. 5. Intra-daily pattern of market sensitivity. The graph plots the estimates of the slope coefficients from regressing returns on order flow separately for

each bi-hourly slice of the day; the dashed lines show the robust 95% confidence interval. That is, each point in the graph represents the slope coefficient

estimate in the regression of returns on order flow, using data at the five-minute frequency from all days in the sample but only from the given two-hour

slot of the day. Data from 1999 to 2004 are used in the estimation. The coefficient estimates should be interpreted as the estimated exchange rate

movement, in basis points, per billion euros of order flow.

20 These derivations assume that the intra-daily pattern is invariant

across days. In the separate Appendix available on the Journal’s Web

page, we provide evidence that this is indeed a valid assumption.


shown in Berger, Chaboud, Chernenko, Howorka, andWright (2008), there is also evidence of an intra-dailypattern in the relationship between returns and orderflow in foreign exchange markets. To the extent that thedaily lt ’s capture the average daily price sensitivity, theprevious results are likely to hold also in the presence ofan intra-daily pattern in the relationship between returnsand order flow. Nevertheless, in the current section, weshow how intra-daily effects can be explicitly accountedfor.

Consider first the empirical evidence of an intra-dailypattern in the return-order flow relationship. For thispurpose, each day is split up into 12 two-hour periods andthe following regression is fitted,

rt;i ¼X12

h¼1

ðahDhðiÞ þ ghDhðiÞof t;iÞ þ �t;i, (6)

where DhðiÞ is a dummy variable that is equal to one if theobservation i is in the hth two-hour segment of the day,and zero otherwise. That is, for each of the two-hourperiods of the day, returns are regressed on order flowusing observations from all days in the sample, so that 12intra-daily price sensitivity coefficients are obtained; inthe 3–11am one-minute sample, there are obviously onlyfour bi-hourly time slots per day. Using the full-day five-minute data, the estimates of the intra-daily coefficientsare plotted in Fig. 5, along with a 95% confidence interval,based on robust standard errors. There is a fairlypronounced intra-daily pattern in the coefficients, withthe lowest price impact occurring between 3am and 11amNew York time, which, as discussed previously, are thebusiest trading hours in the euro-dollar market.

The relationship between volatility and the marketsensitivity coefficients can easily be modified to explicitlyaccount for these intra-daily effects. In particular, thebasic relationship in Eq. (1) can be extended to incorpo-rate intra-daily effects, such that on day t the return-order

flow relationship is given by,

rt;i ¼ ltgðiÞof t;i, (7)

where gðiÞ captures the intra-daily pattern and is short-hand for gðiÞ ¼

PHh¼1ghDhðiÞ with H equal to 12 in the full-

day data and four in the 3–11am data. Eq. (7) thusgeneralizes Eq. (1) to allow for a multiplicative interactionbetween the daily effect, lt , and the intra-daily effects, gh.Thus, on a given day, the intra-daily pattern displayed inFig. 5 is scaled up or down by the overall daily effect, lt . Asbefore, if one squares each side of Eq. (7) and sums up overall daily intervals, it follows that

RVt ¼ l2t

XK

i¼1

ðgðiÞof t;iÞ2� l2

t OFð2ÞW ;t , (8)

where OFð2ÞW ;t �PK

i¼1ðgðiÞof t;iÞ2 is the daily integrated

squared order flow weighted by the intra-daily effects.Eq. (8) generalizes Eq. (3) to allow for intra-daily effects,by weighting the (squared) order flow differently, depend-ing on the time of the day.20

The coefficients in Eq. (7) are estimated jointly withnon-linear least squares using the entire six years of data,sampled either at the one-minute or five-minute fre-quency; daily as well as bi-hourly intercepts are allowedfor in the estimation. Tests of fractional cointegration inthe resulting empirical log-version of Eq. (8) deliver verysimilar results to those reported in Table 6; a limited set ofempirical results are shown in Table 9. As might beexpected—given the pattern seen in Fig. 5—the resultsfrom the fractional cointegration analysis when control-ling for intra-daily effects in the 3–11am one-minute dataare virtually identical to previous results. In the full-dayfive-minute sample, the estimate of bOF is now substan-

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Table 9Results from fractional cointegration analysis with intra-daily effects.

This table reports the narrow band least squares (NBLS) regression

results, as well as estimates of the long-memory parameter for the

residuals, for the following equation (the empirical log-version of Eq. (8))

logðRVtÞ ¼ aþ 2bl logðltÞ þ bOF logðOFð2ÞW ;tÞ þ ut ,

where logðRVtÞ, logðlt Þ, and logðOFð2ÞW ;tÞ, are the log-realized volatility, the


flow weighted by intra-daily effects, on day t, respectively. In this

specification, the daily lt ’s are estimated from Eq. (7), jointly with the

intra-daily effects, using intra-daily observation for the entire

1999–2004 sample. Daily data from 1999 to 2004 are used in the

estimation of the above equation, and standard errors are given in

parentheses below the estimates; estimates of the intercept are not

shown. Panel A shows the results for the daily sample constructed from

full-day five-minute observations and Panel B shows the results for the

daily sample constructed from 3–11am one-minute observations. The

regression estimates are obtained using NBLS with bandwidth

m0 ¼ ½T0:4� ¼ 18. The two columns labeled du give the exact local

Whittle (ELW) estimates of the long-memory parameter for the residuals

of the regressions, using bandwidths m1 ¼ ½T0:6� ¼ 80 and

m1 ¼ ½T0:7� ¼ 166, respectively. There are T ¼ 1;487 daily observations

in each sample.

bl bOF du du

m1 ¼ ½T0:6� m1 ¼ ½T

0:7�


0.956 1.004 0.098 .099

(0.032) (0.055) (0.056) (0.039)


0.963 0.898 0.006 0.043

(0.028) (0.051) (0.056) (0.039)


tially closer to unity than before, and there is thusevidence that controlling for the intra-daily pattern leadsto a slightly better specified model.

21 In the stock-market literature, it is common to use the number of

transactions, rather than the volume of trade, as a measure of market

activity. As mentioned in footnote 2, in the electronic interdealer foreign

exchange market, the majority of trades is for amounts between one and

five million euros, as large orders are routinely broken up before

execution. The average trade size shows little variation over time during

our sample period, and the number of transactions and the trading

volume show a high degree of correlation.22 Let bt and st be the amounts bought and sold in period t. The

volume is then defined as volt ¼ bt þ st and the order flow is defined as

of t ¼ bt � st . Suppose that bt and st are bivariate normally distributed

with, respectively, means mb and ms , variances s2b and s2

s , and covariance

sbs . If mb � ms is close to zero, E½jof t j� �ffiffiffiffiffiffiffiffiffi2=p

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

b þ s2s � 2sbs

q, which is a

function solely of the variance parameters and not the mean parameters.

7. Trading volume and market sensitivity

The measure of market sensitivity that we haveanalyzed, the magnitude of the relationship betweenorder flow and movements in prices, can clearly also beviewed as a measure of market liquidity. It is thereforepossible that variations over time in that measure couldbe mechanically related to variations in trading volume,explaining many of our results. Intuitively, one may expectthat, for instance, a larger amount of order flow is neededto move prices on a day with a high trading volume,implying that days with high trading volume should beassociated with low market sensitivity. In this section, weexplicitly consider the role of trading volume in ourempirical analysis.

Throughout this analysis, the volume of trade repre-sents the total amount transacted over a given timeperiod, i.e., the sum of buy and sell orders, expressed in

millions of the base currency (the euro).21 At the highestof frequencies (tick by tick), absolute (squared) order flowand (squared) trading volume are, by definition, equal. Asthe sampling frequency decreases, the two series divergequickly, and intervals with high trading volume couldhave an order flow of, for instance, zero. At the relativelyhigh frequencies that we consider in this paper, squaredorder flow integrated over a day and daily trading volume(not squared) are still very highly correlated, as seen inTable 4.

Trading volume tends to trend upward over time.Given the correlation between volume and (absolute)order flow at high frequencies, this naturally raises thequestion of the long-run behavior of the magnitude oforder flow over time. Since the mean of order flow is closeto zero, the expected absolute value of order flow willprimarily be a function of the variance in volume, or moreprecisely the variance in the buy and sell amounts.22

Therefore, if the standard deviation of volume increaseswith its mean, the absolute magnitude of the order flowwill also increase with the volume. On the other hand, ifthe mean volume increases but the variance remainsconstant, the magnitude of the order flow will beunaffected. In practice, it seems likely that the magnitudeof order flow increases over time with the overall volumeof trade, but not necessarily one-for-one.

Fig. 6 shows standardized moving averages of the dailyvolume and absolute order flow. As is seen, there is a smallupward trend in volume, and there is some evidence thatthe absolute order flow increases with volume, althoughthe evidence is far from conclusive. The graph also showsthat the lt ’s tend to decrease over the sample, hinting atthe possible negative relationship between volume andmarket sensitivity that was mentioned in the beginning ofthis section. It is also useful to compare averages ofvolume and absolute order flow from the first and the lastyears of the sample. From 1999 to 2004, the averageannual volume increased by 48% and the average absoluteorder flow by 9%. In addition, the annual average of thedaily lt ’s decreased by 17% from 1999 to 2004.

If (absolute) order flow and volume do trend upwardjointly over time, and the sensitivity to a given amount oforder flow decreases over time as a result, this suggeststhat the relationship between returns and order flowcould perhaps be more properly captured as a relationshipnot between returns and the actual size of the order flow,but between returns and the relative size of the order flow

ARTICLE IN PRESS

Fig. 6. Plots of the 40-day moving averages of daily volume, daily absolute order flow, and the daily market sensitivity parameters (lt) from 1999 to 2004.

The moving averages are standardized to all start at unity and the market sensitivity coefficients, lt , are obtained from the full-day five-minute intra-daily

returns. The series are shown in levels and not logs.

Table 10Estimates of the long-memory parameter d for the volume of trade.

This table reports the exact local Whittle (ELW) and local Whittle with

noise (LWN) estimates of the long-memory parameter in logged daily

trading volume; the standard errors are given in parentheses below the

estimates. Daily data from 1999 to 2004 are used in the estimation, with

T ¼ 1;487 observations. The columns on the left show the results for the

daily volume over the full-day and the columns on the right show the

results for the daily volume over the period 3–11am. The estimates for

three different bandwidths, m ¼ ½T0:5� ¼ 38, m ¼ ½T0:6

� ¼ 80, and

m ¼ ½T0:7� ¼ 166, are reported.

Bandwidth Full-day, five-minute 3–11am, one-minute

ELW

estimates

LWN

estimates

ELW

estimates

LWN

estimates

m ¼ ½T0:5� 0.379 0.474 0.364 0.467

(0.081) (0.126) (0.081) (0.127)

m ¼ ½T0:6� 0.414 0.459 0.379 0.428

(0.056) (0.089) (0.056) (0.093)

m ¼ ½T0:7� 0.388 0.418 0.366 0.399

(0.039) (0.066) (0.039) (0.068)


as compared to the overall volume in the market. In orderto test this empirically, we estimate the followingregression,

rt ¼ aþ lof tVyt þ �t , (9)

where rt , of t , and Vt are the daily returns, order flow, andvolume of trade, respectively. Eq. (9) captures the notionthat the price impact of trades is determined not by theirabsolute size, but by their size relative to the overallvolume of trade in the market. This intuition is clearestwhen y ¼ �1, although any negative value of y isconsistent with the basic notion that, all else equal, alarger amount of order flow is needed to move prices on aday with a high trading volume. Estimating the value of yand testing whether it is significantly different from zeroprovides a test of whether volume is a significantdeterminant of the price sensitivity to order flow, andthus also whether some of the observed variation in thedaily lt ’s can be explained by shifts in volume. To avoidthe above-mentioned high correlation between high-frequency volume and order flow, we restrict the analysisto daily data. In addition to using the daily volume Vt , wealso consider two specifications where Vt is replaced by a100- or 250-day trailing average of volume, in order to getat the possibility that order flow is ‘compared’ by marketparticipants to a long-run average of volume, rather thanthe current volume.

Eq. (9) is estimated with non-linear least squares.When the non-averaged daily volume is used, theestimate of l is equal to 40.44, which is very similar towhat is found in a daily linear regression of returns onorder flow, although the (robust) standard error is muchlarger (15.97). The estimate of y is equal to �0:013 with astandard error of 0.101, and is thus not significantlydifferent from zero; using a long-run trailing average ofthe daily volume gives qualitatively identical results. Thisestimation thus does not provide any evidence that

changes in volume affect the price impact of order flowand, in consequence, no evidence that the observedvariation in the sensitivity coefficients is driven bychanges in the volume of trade.

To further analyze the potential role of volume inexplaining the variation in the lt ’s, we also directlyestimate the relationship between the market sensitivitycoefficients and the daily volume of trade. As discussed atlength in previous sections, standard OLS analysis is likelyto deliver spurious results due to both the potentialendogeneity of the regressor and the long-memory natureof the data. We therefore again focus on the low-frequency movements and perform tests for fractionalcointegration. As seen in Table 10, the daily log-volume

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Table 11Results from regressing market sensitivity onto trading volume.

This table reports the narrow band least squares (NBLS) regression

estimates, bV , from regressing the log of market sensitivity, logðltÞ, onto

the log of the volume of trade, logðVt Þ. Daily data from 1999 to 2004 are

used in the estimation. Standard errors are given in parentheses below

the estimates and estimates of the intercept are not shown. Panel A

shows the results for the daily sample constructed from full-day five-

minute observations and Panel B shows the results for the daily sample

constructed from 3–11am one-minute observations. The first column

shows the bandwidths used in the estimation, where the last row in each

panel corresponds to ordinary least squares (OLS) estimation. The

columns labeled du , with the bandwidth ðm1Þ listed below, give the exact

local Whittle (ELW) estimates of the long-memory parameter for the

residuals of the respective regression. The bandwidths that are used are

all integer parts of fractional powers of the sample size T. Observe that

½T0:3� ¼ 8, ½T0:4

� ¼ 18, ½T0:5� ¼ 38, ½T0:6

� ¼ 80, ½T0:7� ¼ 166, and T ¼ 1;487.


Bandwidth bV du du du

m1 ¼ ½T0:5� m1 ¼ ½T

0:6� m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� �0.438 0.563 0.456 0.483

(0.412) (0.081) (0.056) (0.039)

m0 ¼ ½T0:4� �0.273 0.465 0.492

(0.237) (0.056) (0.039)

m0 ¼ ½T0:5� �0.258 0.492

(0.154) (0.039)

m0 ¼ T (OLS) �0.194 0.494

(0.021) (0.039)


Bandwidth bV du m1 ¼ ½T0:5� du m1 ¼ ½T

0:6� du m1 ¼ ½T

0:7�

m0 ¼ ½T0:3� �0.236 0.566 0.484 0.484

(0.419) (0.081) (0.056) (0.039)

m0 ¼ ½T0:4� �0.154 0.487 0.488

(0.240) (0.056) (0.039)

m0 ¼ ½T0:5� �0.158 0.487

(0.154) (0.039)

m0 ¼ T (OLS) �0.099 0.488

(0.019) (0.039)

23 The slight time trend seen in volume in Fig. 6 is most likely better

modeled as a linear time trend than as a part of the long-memory

process that governs the stochastic trend in volume. Estimates of the

long-memory parameter in volume may therefore more accurately be

obtained by first linearly detrending the volume; the resulting estimates

of d are around 0.3 in the detrended data. However, in terms of relating

the lt ’s to the trading volume over time, it makes more sense to focus on

the relationship between the lt ’s and the non-detrended volume, since

part of what we wish to understand is whether an increase in volume

leads to a decrease in the market sensitivity coefficients. Results not

presented here, however, also show that the finding of no significant

relationship (in a fractional cointegration sense) holds when relating the

detrended volume to the lt ’s.24 We do not focus directly on the relationship between trading

volume and volatility in this paper, but our findings indirectly provide a

possible explanation for the mixed results found in the ‘mixture of

distributions’ literature. Trading volume and order flow are highly

correlated at high frequencies, and we have shown that the role of order

flow in explaining long swings in volatility is not very large. On the other

hand, there is a very low observed correlation between trading volume

and market sensitivity. Given that market sensitivity seems to play an

important role in driving volatility, our results also help explain the fairly

weak explanatory power of volume for volatility that has been observed.


possesses long-memory with estimates of d close to 0.4,which is somewhat lower than the memory estimates forlt presented earlier but still large enough to justify ananalysis of fractional cointegration between the twovariables. The results from regressing the daily marketsensitivity coefficients, lt , onto the daily volume of tradeare shown in Table 11. The table shows the NBLS estimatesof the slope coefficient for volume, along with theestimates of the long-memory parameter in the fittedresiduals; the estimates of the long-memory parameterare similar for the ELW and LWN estimators, and only theELW results are shown. Overall, the results are notsupportive of an inverse relationship betweenmarket sensitivity and trading volume. There is noevidence of fractional cointegration, as seen from theestimates of the long-memory parameter in the regressionresiduals, which are very close to the estimates of thememory in lt shown in Table 6. The regression is thus

apparently spurious and inference on the slope coefficientnot valid.23

Overall, there is thus essentially no formal support forthe conjecture that volume and market sensitivity co-move negatively, as a simple liquidity story could suggest.The changes in market sensitivity that we documenttherefore appear to occur for reasons other than changesin the ‘physical’ depth of the market, as measured by theoverall volume of trade.

However, some caution should be applied wheninterpreting these results. In particular, we view theformal empirical findings in this section as strongevidence against the notion that changes in volumeexplain any significant portion of the variation in marketsensitivity at the frequencies relevant for our analysis. Thatis, the analysis performed in this paper is designed touncover long-run dynamics and co-movement in afractional (co-) integration sense. These ‘long-run’ testsare inevitably restricted by the sample length availableand effectively test relationships at frequencies of a fewmonths (see footnote 14). In contrast, it is quite possiblethat any significant volume effects occur over much longertime horizons, as the size of the market substantiallychanges, and cannot be captured by the variation overthe sample period available. Indeed, to the extent that thevolume of trade trends upwards over time, and themagnitude of order flow does as well, it seems likely thatthe price impact of a given amount of order flow willdecrease in the long-run, as it becomes relatively smallercompared to the overall amount transacted in the market.It is clear from the empirical analysis, however, that overthe time horizons considered here, the dynamics ofvolatility and market sensitivity uncovered in this paperare not driven by a simple volume effect.24

8. Conclusion

We have shown that movements in the market’ssensitivity to information, jointly with movements in therate of information arrival, can, to a very large degree,

ARTICLE IN PRESS


explain the long-run dynamics of realized exchange ratevolatility. Importantly, our analysis reveals that variationsin market sensitivity play at least as large a role in drivingvolatility persistence as do variations in the flow ofinformation reaching the market through the tradingprocess. This may be evidence of a link between changesover time in the aggregate behavior of market participantsand the time-series properties of realized volatility.

The analysis in this paper builds upon ideas containedin the ‘mixture of distributions’ literature, in particular inLiesenfeld (2001). However, our volatility model is basedon relationships described in the market microstructureliterature, and its estimation relies on the availability of along time-series of price and trading data at very highfrequency. As such, focusing on the micro and high-frequency aspects of the market has helped us to betterexplain movements in price volatility at frequencies moretypical of the macro level.

The fact that time-varying market sensitivity plays animportant role in driving volatility fits well within theframework of several theoretical and empirical models ofasset pricing and trading. However, these models providediffering explanations as to what may actually causechanges in market sensitivity over time. Some modelsfocus more on the content of the information reaching themarket, some on changes in the traders’ attitude towardsrisk, and some on the impact of changes in the composi-tion or practices of market participants.

On the information side, in Kyle’s (1985) tradingmodel, for instance, the response of the price to orderflow depends on the amount of private information thatinformed traders have, relative to the expected amount ofnoise trading in the market. Within Kyle’s framework, onemay expect that, for example, times of heighteneduncertainty about economic conditions, such as inflexionpoints in economic activity or in monetary policy, couldbe associated with a relative information advantage forthe informed traders, thus resulting in persistent periodsof higher market sensitivity.

In the models of Evans and Lyons (2002) and Killeen,Lyons, and Moore (2006), which are aimed directly atunderstanding the impact of order flow in the foreignexchange market, the market sensitivity is assumed to bean increasing function of the absolute risk aversionparameter. This suggests the more drastic explanationthat variations over time in market sensitivity may reflectchanges in deeper, more fundamental parameters amongmarket participants, including changes in risk aversion. Inthe same vein, McQueen and Vorkink (2004) offer a modelwhere shocks to investors’ wealth temporarily affect boththeir attitude towards risk and the amount of attentionthey pay to news, resulting in variations in theirsensitivity to information.

Focusing more on the operational practices of marketparticipants, Garleanu and Pedersen (2007), for instance,argue that standard risk management practices, such asvalue at risk, can impact market liquidity in a persistentway. In their model, heightened concerns by marketparticipants, due to either fundamental or behavioralfactors, lead to more stringent risk management practicesand to a decrease in market liquidity. A feedback effect,

with the resulting higher volatility leading to furthertightening in risk management, could then cause persis-tent periods of reduced liquidity and of increasedvolatility. Such a mechanism is also potentially consistentwith our observed swings in market sensitivity.

Our present work provides direct evidence of the roleof market sensitivity in driving variations over time inrealized volatility in the foreign exchange market. Ex-tending our work to other markets and studying whetherthe pattern of variation in market sensitivity is specific tocertain assets or markets, or if there is a commoncomponent, should provide additional evidence on thefactors driving the variation in market sensitivity andvolatility.

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What drives volatility persistence in the foreign exchange market?

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