Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 073500106000000594

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Modeling Around-the-Clock Price Discovery for
Cross-Listed Stocks Using State Space Methods
Albert J Menkveld, Siem Jan Koopman & André Lucas
To cite this article: Albert J Menkveld, Siem Jan Koopman & André Lucas (2007) Modeling
Around-the-Clock Price Discovery for Cross-Listed Stocks Using State Space Methods, Journal
of Business & Economic Statistics, 25:2, 213-225, DOI: 10.1198/073500106000000594
To link to this article: http://dx.doi.org/10.1198/073500106000000594

Published online: 01 Jan 2012.

Submit your article to this journal

Article views: 178

View related articles

Citing articles: 1 View citing articles

Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]

Date: 12 January 2016, At: 23:03

Modeling Around-the-Clock Price Discovery for
Cross-Listed Stocks Using State Space Methods
Albert J. M ENKVELD
Department of Finance, Vrije Universiteit, Amsterdam, The Netherlands, NL-1081HV (ajmenkveld@feweb.vu.nl )

Siem Jan KOOPMAN
Department of Econometrics and Tinbergen Institute, Vrije Universiteit, Amsterdam, The Netherlands, NL-1081HV
(s.j.koopman@feweb.vu.nl )

André L UCAS

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

Department of Finance and Tinbergen Institute, Vrije Universiteit, Amsterdam, The Netherlands, NL-1081HV
(alucas@feweb.vu.nl )
U.S. trading in non-U.S. stocks has grown dramatically. Around the clock, these stocks trade in the home
market, in the U.S. market, and, potentially, in both markets simultaneously. We develop a general methodology based on a state space model to study 24-hour price discovery in a multiple-markets setting. As
opposed to the standard variance ratio approach, this model deals naturally with (1) simultaneous quotes
in an overlap, (2) missing observations in a nonoverlap, (3) noise due to transitory microstructure effects, and (4) contemporaneous correlation in returns due to market-wide factors. We apply our model to
Dutch stocks, cross-listed in the United States. Our findings suggest a minor role for the New York Stock
Exchange in price discovery for Dutch shares, in spite of its nontrivial and growing market share.
KEY WORDS: Efficient price; Financial markets; High-frequency data; Kalman filter; Unobserved
components time series models.

1. INTRODUCTION
In the last decade, international firms have increasingly
sought a U.S. listing, oftentimes achieved through cross-listing
their shares on either the New York Stock Exchange (NYSE)
or the NASDAQ. At the end of 2002, 467 non-U.S. firms were
listed on the NYSE and generated approximately 10% of total
volume that year (numbers are taken from the 2003 annual report). The NASDAQ lists even more non-U.S. firms. This trend
has prompted many academic studies. Most of them focus on
the benefits of cross-listing, such as reduced cost of capital and

enhanced liquidity of a firm’s stock (see Karolyi 2006 for a
survey).
A relatively unexplored question is how much U.S. trading
contributes to around-the-clock price discovery over and above
domestic trading. Reasoning could go both ways. On the one
hand, the home market, being closest to where the information
is produced, may be most important (see, e.g., Solnik 1996; Hau
2001; Bacidore and Sofianos 2002). On the other hand, U.S.
stock exchanges, being the largest and most liquid exchanges in
the world, may also play an important role in price discovery for
non-U.S. stocks, particularly now that their share in total U.S.
volume is rapidly increasing. Chan, Hameed, and Lau (2003),
for example, found that trading location matters irrespective of
business location for a group of companies that changed their
listings from Hong Kong to Singapore. The evidence for Canadian stocks and, if we focus on the overlapping period only,
for Dutch, German, and Spanish stocks shows that the NYSE
contributes significantly less to price discovery than the domestic exchange (see Eun and Sabherwal 2003; Hupperets and
Menkveld 2002; Grammig, Melvin, and Schlag 2005a; Pascual,
Pascual-Fuste, and Climent 2001, respectively). Grammig et al.
(2005b) documented, based on a cross-section of Canadian,

British, French, and German stocks, that the NYSE contribution
increases in its liquidity relative to that of the home market.
Our objective is to develop a general methodology to determine how informative trading is for around-the-clock price
discovery in a multiple-markets setting. It enables the analysis
of the questions raised for U.S. trading in non-U.S. stocks but
applies more generally to securities trading in multiple venues
and, possibly, multiple time zones. Examples include securities
trading on multiple trading platforms or through multiple broker/dealers, oftentimes referred to as fragmented trading (see
Stoll 2001a), London and Tokyo trading in U.S. treasury securities (see, e.g., Fleming and Lopez 2003), and foreign listings
on European exchanges (see Pagano, Randl, Roëll, and Zechner
2001).
The empirical literature on around-the-clock price discovery dates back to single-market studies comparing variance ratios of open-to-close and close-to-open returns. They generally
found that trading periods produce more information than nontrading periods (see Oldfield and Rogalski 1980; French and
Roll 1986; Harvey and Huang 1991; Jones, Kaul, and Lipson
1994). Chan, Fong, Kho, and Stulz (1996), for example, used
variance ratios to study U.S. trading in European stocks and
found that the morning-to-afternoon variance ratio is significantly larger than a benchmark ratio based on U.S. control
stocks. A natural extension of variance ratios to our multiplemarkets setting is to fix time points in the day, calculate the variance of intraday returns, take a cross-sectional average (across
stocks), and study the intraday pattern. This approach could fail

213

© 2007 American Statistical Association
Journal of Business & Economic Statistics
April 2007, Vol. 25, No. 2
DOI 10.1198/073500106000000594

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

214

for three reasons. First, in our setting, calculating returns involves arbitrary choices for prices in overlapping periods, as
we observe prices in multiple markets. Second, midquotes and
transaction prices are potentially noisy proxies for the unobserved efficient price as a result of microstructure effects, due
to, for example, market making or a nonzero tick size (see, e.g.,
Stoll 2001b). These transitory effects, or “noise,” are negligible for weekly, monthly, or annual returns, but not for intraday
returns. This is illustrated by studies that find 24-hour returns
based on opening prices to be, on average, up to 20% more
volatile than those based on closing prices (see Amihud and

Mendelson 1987; Stoll and Whaley 1990; Gerety and Mulherin
1994; Forster and George 1996). Transitory effects are potentially distorting as they artificially inflate price discovery within
a trading day. Third, Ronen (1997, p. 184) criticized “many of
the intradaily variance ratios used in this literature” because statistical inference ignores cross-sectional correlation in returns.
In this article we develop a methodology based on a state
space model to account for the three criticisms of the standard
variance ratio approach. Such a model arises naturally after
characterizing the (unobserved) efficient price process as a random walk (see, e.g., Campbell, Lo, and MacKinlay 1997). To
study around-the-clock price discovery, we endow this random
walk with deterministic, time-varying volatility. To account for
transitory price changes, we model the observed midquote as
the unobserved efficient price plus temporary deviations due
to “microstructure” effects and short-term market under- or
overreaction to information (see, e.g., Amihud and Mendelson
1987). In the overlap, both midquotes are functions of the same
unobserved efficient price plus idiosyncratic temporary deviations. To account for cross-correlation in returns, we model returns as the sum of a common and an idiosyncratic factor, where
the common factor captures, for example, macroeconomic information or portfolio-wide liquidity shocks. The model is estimated using maximum likelihood. The Kalman filter is used to
calculate the likelihood at each step in the optimization. A major advantage of the Kalman filter in our setting is that it deals
naturally with missing values in the nonoverlap trading periods. Moreover, it allows for decomposition of an observed price
change into a transitory and a permanent component based on

the entire sample (see Durbin and Koopman 2001).
For partially overlapping markets, the state space approach
compares favorably to alternative methodologies. In related
work, Hasbrouck (1995) proposed a vector error correction
model to measure “information shares” of exchanges for price
discovery during the period when both exchanges are open.
Although this approach accounts for transitory price changes,
it does not extend to our setting, because it cannot deal with
missing values in one of the markets in the nonoverlap. A generalized state space approach, on the other hand, does nest
the Hasbrouck model. We will come back to this issue when
discussing the model. Barclay and Hendershott (2003) used
weighted price contributions (WPCs) to study how informative
after-hours trading is. The WPC approach, however, does not
explicitly allow for transitory effects. In their pioneering study,
Barclay and Warner (1993, p. 300) developed WPCs and acknowledged that “it is not clear how any bias from ignoring
temporary price-change components could drive our results.”
The state space model is estimated based on a 1997–1998
sample of Dutch blue chips cross-listed in New York. The

Journal of Business & Economic Statistics, April 2007

United Kingdom excluded, Dutch stocks are the European
stocks that generate the most volume in New York. The dataset
is rich, because it includes all trades and quotes on both sides
of the Atlantic, as well as intraday quotes on the exchange rate
and intraday prices on the major Dutch index and the Standard
& Poor’s (S&P) 500.
The results demonstrate the empirical relevance of the model,
because the estimated variance pattern of the efficient price innovations differs significantly from the pattern based on the
standard variance ratio approach. Such an approach was pursued in earlier articles on British and Dutch cross-listed stocks
(see Werner and Kleidon 1996; Hupperets and Menkveld 2002,
respectively). The major difference is that the variance ratio
approach attributes most price discovery to the “Amsterdam
close,” whereas the state space approach considers the “New
York open” the most informative period. This difference is primarily due to significant temporary effects in Amsterdam and
New York midquotes, which are, implicitly, assumed to be absent in the variance ratio approach. Interestingly, transitory effects are insignificant for Amsterdam midquotes in the Amsterdam only trading period. We quantify price discovery consistent with existing literature and find that price discovery during
Amsterdam trading hours is three times stronger than that in the
New York only or the overnight period. This compares to, for
example, seven times stronger price discovery for NYSE stocks
during trading hours relative to the overnight period, as reported

in George and Hwang (2001). Our results survive a number
of robustness tests, including potential nonzero correlation between transient microstructure effects and efficient price innovations (see, e.g., Hasbrouck 1993; George and Hwang 2001).
The rest of the article is structured as follows. Section 2
presents and discusses a multivariate state space model for
midquotes of securities that are traded in different markets. Section 3 elaborates on trading Dutch securities in Amsterdam and
New York. Section 4 presents the model estimates and contains
robustness tests. Section 5 summarizes the main conclusions.
2. MODEL
The principles of the analysis in this article are based on an
unobserved “efficient” price and observed midquotes in multiple markets that trade the same security. State space models
are a natural tool in this setting as the efficient price can be
modeled as an unobservable state variable and the midquotes
as observations of this variable with measurement error to reflect transitory microstructure effects.
2.1 The Unobserved Efficient Price Model
We model the efficient price as a random walk with a deterministic linear trend to account for nonzero expected returns.
For the purpose of studying around-the-clock price discovery,
we consider T time points in the day. The efficient price process
is subject to deterministic, time-varying volatility depending on
the time of day. The efficient price innovation is decomposed
into a common factor and an idiosyncratic innovation. The common factor is associated with a macroeconomic or portfoliowide liquidity shock (see Subrahmanyam 1991; Caballe and

Krishnan 1994). The process for a multiple of N stock prices,

Menkveld, Koopman, and Lucas: Modeling Around-the-Clock Price Discovery

T intraday time points, and D trading days can, therefore, be
described as
α t,τ +1 = α t,τ + βξt,τ + ηt,τ ,

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

2
2
), ηt,τ ∼ N(0, ση,τ
C),
ξt,τ ∼ N(0, σξ,τ

(1)

for t = 1, . . . , D and τ = 1, . . . , T and with α t+1,1 = α t,T+1 .
The N × 1 state vector α t,τ contains the unobserved efficient

prices of N stocks at day t and time point τ . The scalar variable ξt,τ is the unobserved common factor and the N × 1 vector
β is fixed and contains the unknown common factor loadings.
The common factor is a zero-mean random variable with a deterministic intraday variance structure. The idiosyncratic disturbance vector ηt,τ is normally and independently distributed
2 C. The scaled variwith N × N diagonal variance matrix ση,τ
ance matrix C = diag(c1 , . . . , cN ) captures interstock volatility
differences. For ease of exposition, we ignore the time trend in
the efficient price process, but, in the implementation, we allow for such a trend by including an N × 1 parameter vector µτ
for the mean of ηt,τ . The common and idiosyncratic shocks are
mutually and serially uncorrelated at all time points.
Model (1) can also be represented using a single disturbance
term, that is,
α t,τ +1 = α t,τ + ζ t,τ ,
2
2
ζ t,τ ∼ N(0,  ζ ,τ ),  ζ ,τ = σξ,τ
ββ ′ + ση,τ
C,

(2)

where ζ t,τ = βξt,τ + ηt,τ . To ensure identification of the model,
we impose the sufficient parameter restrictions
N
1
2
βi = 1,
N
i=1

N
1

ci = 1,
N

(3)

i=1

1
2
2
+ ση,τ
,
= E(ζ ′t,τ ζ t,τ ) = σξ,τ
N

(4)

2 is the average variance of the total efficient price
where σE,τ
innovation from time point τ to τ + 1.

2.2 The Observation Model
Although we do not observe the efficient price, midquotes in
either or both markets at day t and time τ are the best proxies
because they do not suffer from the bid–ask bounce in transaction prices (see, e.g., Roll 1986). They are, nevertheless, noisy
because they suffer from transient microstructure effects, such
as rounding errors due to discrete price grids, temporary liquidity shocks, or inventory management by market makers. The
model for N midquotes observed during T intraday time points
and D days and for K different markets is specified as
pk,t,τ = α t,τ + ε k,t,τ ,

2
εk,t,τ ∼ N(0, σε,k,τ
IN ),

The transitory N × 1 error vector εk,t,τ is due to microstructure effects and other market irregularities. The observation error variances depend on the time of day and on the market, but
they are assumed to be equal across all stocks, an assumption
that will be relaxed at a later stage. The observation model (5)
implies a vector moving average process for price changes.
2.3 The Observation Model With Delayed
Price Reaction
The basic observation equation (5) is extended to allow for
market under- or overreaction to information, which cannot be
excluded ex ante in high-frequency analysis (see, e.g., Amihud
and Mendelson 1987). Both effects are rationalized in standard
microstructure theory. Market underreaction or positive serial
correlation is consistent with strategic trading by informed investors who split their order across time to maximize profit (see,
e.g., Kyle 1985). Market overreaction is consistent with liquidity suppliers who are compensated for their services through
price reversals (see, e.g., Stoll 1978). A natural way to account for these effects is to include the efficient price change
α t,τ − α t,τ −1 in the observation equation. We only consider one
lag in this article, but this approach generalizes to multiple lags.
In our application, the empirical autocorrelations show that one
lag suffices. We obtain
pk,t,τ = α t,τ + θ (α t,τ − α t,τ −1 ) + εk,t,τ
= α t,τ + θ ζ t,τ −1 + ε k,t,τ
= α t,τ + θ βξt,τ −1 + θ ηt,τ −1 + ε k,t,τ ,

where βi is the ith element of vector β. Although the choice of
parameter restrictions for identification is arbitrary, the current
2 and σ 2 to be interpreted as the average
restrictions allow σξ,τ
η,τ
(over N stocks) systematic and idiosyncratic variance, respectively. Around-the-clock price discovery in the sample is then
determined by
2
σE,τ

215

(5)

where the vector pk,t,τ contains midquotes for N stocks traded
at market k with k = 1, . . . , K, t = 1, . . . , D, and τ = 1, . . . , T.

(6)

where the scalar coefficient θ measures the price reaction to information. This specification, however, does not distinguish between, for example, underreaction to firm-specific or commonfactor information. Further, the coefficient θ may vary within
the day. To allow for both these generalizations, we consider
the specification
pk,t,τ = α t,τ + θξ,τ −1 βξt,τ −1 + θη,τ −1 ηt,τ −1 + ε k,t,τ ,

(7)

where θξ,τ and θη,τ are scalar coefficients for τ = 1, . . . , T
with θξ,0 ≡ θξ,T and θη,0 ≡ θη,T . The common-factor (firmspecific) efficient price innovation at time τ is premultiplied by
θξ,τ (θη,τ ) to indicate that midquotes underreact (negative θ ) or
overreact (positive θ ) to the innovation. The modeling framework allows us to determine whether these effects exist by testing the null hypothesis that θ is equal to 0. We use the Wald test
for this purpose.
The current specification of the state space model in state
equation (2) and observation equation (7) is easily generalized
to nest the Hasbrouck (1995) vector error correction model
for multimarket trading. We emphasize that both rely on cointegrated price processes; that is, the efficient price process is
in Hasbrouck’s terminology the “common trend.” The critical
advantage of our approach is its treatment of missing values in
one of the markets during the nonoverlap. Our current specification of the state space model, however, is slightly more restrictive than the Hasbrouck approach because we demand that
the (full) innovation over each period be incorporated in both
Amsterdam and New York prices. In the Hasbrouck approach,
one market could lag the other market completely, leading to

216

Journal of Business & Economic Statistics, April 2007

a 100% “information share” for the leading market. We could,
however, make the under- and overreaction parameters θ in the
article market and/or stock specific and potentially add more
lags. In that case, the Hasbrouck model is nested in ours. We
choose not to do this, because the Hasbrouck approach is particularly relevant for ultra-high-frequency data; current technology allows participants to view markets in real time and update the (efficient) price estimate almost instantaneously. It is
for this reason that Hasbrouck implemented his approach with
one-second intervals. In our approach, however, we study intervals as wide as a full hour, and we consider it, therefore,
reasonable that the innovation is incorporated into both prices
simultaneously.

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

2.4 State Space Representation
The standard state space model is formulated for a vector of
time series ys with a single time index s and is given by
ys = Zs δ s + ν s ,

δ s+1 = Ts δ s + Rs χ s , s = 1, . . . , M, (8)

where disturbances ν s ∼ N(0, Hs ) and χ s ∼ N(0, Qs ) are mutually and serially uncorrelated. The initial state vector δ 1 ∼
N(a, P) is uncorrelated with the disturbances. The mean vector a and variance matrix P are usually implied by the dynamic
process for δ s in (8). For example, when δ s represents a nonstationary process, the mean a is set to 0 whereas the variance
matrix P is scaled by a diffuse prior, for example, P = κI with
κ → ∞. In case some or all of the elements of the state vector
are driven by stationary processes, the corresponding values for
a and P are obtained from the unconditional means and variances, respectively. The system matrices or vectors Zs , Ts , Rs ,
Hs , and Qs are assumed fixed for s = 1, . . . , M. The elements
of the system matrices or vectors are known except for some
elements that are functions of a fixed parameter vector. This
general state space model is explored further in the textbook of
Durbin and Koopman (2001), among others.
The basic model (2) and (5) can be represented as a state
space model (8) by choosing
ys = (p′1,t,τ , . . . , p′K,t,τ )′ ,

δ s = α t,τ ,

s = (t − 1)T + τ,

with NK × 1 observation vector ys , N × 1 state vector δ s , and
M = TD. The state space disturbance vectors are specified as
ν s = (ε ′1,t,τ , . . . , ε′K,t,τ )′ ,

χ s = ζ t,τ .

The state space matrices are then given by
Zs = ι′K ⊗ IN ,
Hs =

Ts = Rs = IN ,

2
2
) ⊗ IN ,
, . . . , σε,K,τ
diag(σε,1,τ

Qs =  ζ ,τ ,

where ιK is the K × 1 vector of 1’s and IN is the N × N unity
matrix. We notice that Hs and Qs only vary within a trading day
t for t = 1, . . . , D.
The model of interest (2) and (7) can also be cast in state
space form. The state vector needs to be extended to include the
lagged disturbances ξt,τ −1 and ηt,τ −1 in the model. The (2N +
1) × 1 state vector δ s and the (N + 1) × 1 state disturbance
vector are then defined as
δ s = (α ′t,τ , η′t,τ −1 , ξt,τ −1 )′ ,

χ s = (η′t,τ , ξt,τ )′ ,

whereas the observation vector ys remains as defined. The state
space matrices are


IN 0 0
′
Zs = ιK ⊗ [IN , θη,τ IN , θξ,τ β],
Ts = 0 0 0 ,
0 0 0




IN β
σ2 C
0
Rs = IN 0 ,
Qs = η,τ
,
2
0
σ
ξ,τ
0 1
whereas Hs remains as defined.
The unknown parameters of the model are collected in the
2
2 , and σ 2
vector ψ , which contains the variances σε,k,τ
, σξ,τ
η,τ
for k = 1, . . . , K and τ = 1, . . . , T; the scalar coefficients βi and
ci for i = 1, . . . , N; and the price reaction coefficients θξ,τ and
θη,τ for τ = 1, . . . , T.
2.5 Estimation and Signal Extraction
The Kalman filter and associated algorithms can be used for
the estimation of the parameter vector ψ and the signal extraction of the unobserved state vector δ s (see Durbin and Koopman
2001, among others). The Kalman filter evaluates the conditional mean and variance of the state vector δ s given the past
observations Ys−1 = {y1 , . . . , ys−1 }, that is,
as|s−1 = E(δ s |Ys−1 ),

Ps|s−1 = var(δ s |Ys−1 ),
s = 1, . . . , M,

and we further have a1|0 = a and P1|0 = P, where a and P are
the mean and variance, respectively, of the initial state vector
δ 1 . The recursive equations are given by
as+1|s = Ts as|s−1 + Ks vs ,

′
Ps+1|s = Ts Ps|s−1 T′s + Rs Qs R′s − Ks F−1
s Ks ,

with one-step-ahead prediction error vector vs = ys − Zs as|s−1 ,
its variance matrix Fs = Zs Ps|s−1 Z′s + Hs , and Kalman gain matrix Ks = Ts Ps|s−1 Z′s F−1
s for s = 1, . . . , M. For a given parameter vector ψ , all system matrices and vectors have values and
the Kalman recursions can be carried out. The initializations of
the recursions at s = 1 are determined by the initial conditions
of the state space model as given by the mean vector a and the
variance matrix P. In case all of the elements of the state vector δ s follow stationary processes, a and P have proper values
that are based on unconditional means and variances of the stationary processes. However, the random walks α t,τ in (2), for
τ = 1, . . . , T, are part of the state vector and require initializations depending on the diffuse prior κ → ∞. The Kalman filter and associated algorithms can be adapted to deal explicitly
with κ; see Durbin and Koopman (2001, chap. 5) for more details. Finally, it can be shown that when the model is correctly
specified the standardized prediction errors are normally and
independently distributed with a unit variance.
An important feature of state space methods is their ability to
deal with missing values, which are paramount in our dataset,
because no observations are available on one of the exchanges
during the nonoverlap. When all elements in ys are missing, the
recursive equation, for example, reduces to
as+1|s = Ts as|s−1 ,

Ps+1|s = Ts Ps|s−1 T′s + Rs Qs R′s .

Menkveld, Koopman, and Lucas: Modeling Around-the-Clock Price Discovery

Forecasting is based on these equations as well, because we can
regard future observations as missing values in the dataset.
The parameter vector ψ, associated with the state space
model, is estimated by numerically maximizing the log-likelihood that can be evaluated by the Kalman filter as a result of the
prediction error decomposition. The log-likelihood function is
given by

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

l=−

M

M

s=1

s=1

1
′ −1
1

NKM
log |Fs | −
vs Fs vs .
log 2π −
2
2
2

For the application of around-the-clock price discovery, the observation vector ys is of a high dimension. It follows that the
variance matrix Fs is of a high dimension, which is inconvenient because it needs to be inverted and its determinant needs
to be computed for each s. Consequently, the computations are
relatively slow for a single log-likelihood evaluation. During
the process of log-likelihood maximization, the Kalman filter
is carried out repeatedly many times. A computationally more
efficient implementation of the Kalman filter for vector observations is based on updating ys element by element. This reduces the computational load considerably because inversions
of large matrices are no longer required; see Durbin and Koopman (2001, sec. 6.3) for more details and for computational
comparisons.
Signal extraction refers to the estimation of the unobserved
efficient price given all observations YM . The conditional mean
vector δˆ s = E(δ s |YM ) and conditional variance matrix Vs =
var(δ s |YM ) can be computed by a smoothing algorithm. Estimation and signal extraction were done in Ox using the SsfPack
state space routines (see Koopman, Shephard, and Doornik
1999).
3. DATA FROM THE AMSTERDAM AND
NEW YORK MARKETS
The volume of non-U.S. shares grew to approximately 10%
of total NYSE volume in 2002. European shares accounted
for most of this volume—approximately one third. Not surprisingly, U.K. shares accounted for most European volume,
followed by Dutch shares, which generated more volume than
French and German shares combined. The cross-listed Dutch
shares studied in this article are NY registered shares, as
opposed to the more common American depositary receipts
(ADRs). These are, however, not regarded as materially different in the eyes of investors, according to Citibank, one of
the key players in the depositary services industry. Most important is that both the NY registered share and the ADR can
be changed for the underlying common share at a small fee of
approximately 15 basis points.
In our sample, Dutch shares traded from the Amsterdam
open, 3:30 Eastern Standard Time (EST), to the New York
close, 16:00 EST, with a one-hour trade overlap, as depicted
in Figure 1. To study around-the-clock price discovery, the researcher can choose the intraday time points arbitrarily. The
only concern is that the time between these points is long
enough for the transitory (microstructure) effects to die out.
If this is not the case, the independence of the “noise” terms
εk,t,τ across time points might be violated. We choose to select

217

Figure 1. Time Line. This figure illustrates the time line for trading in Amsterdam and New York using Eastern Standard Time. The
economically interesting time points modeled in this article are indicated with arrows. Most are self-explanatory, except for the 8:00 time
point, which was introduced to pick up the potential effect of U.S.
macro-announcements and premarket open press releases of rival U.S.
firms.

six economically relevant time points inspired by the variance
patterns reported in earlier studies (Werner and Kleidon 1996;
Hupperets and Menkveld 2002). The first time point is 4:00,
which is half an hour after the Amsterdam open. We choose not
to take the actual open, because trading might not start directly,
creating a missing observation. Subsequent time points in the
Amsterdam trading period are 8:00, 9:00, and 10:00. These
are, purposefully, located around the economically interesting
event times 8:30 and 9:30, because at these times U.S. macroannouncements are published and the NYSE opens, respectively.
In the U.S. trading period, we further select 11:00 to incorporate
the Amsterdam close and 15:30 to study price discovery during
the remainder of the trading day. We choose to stay half an hour
ahead of the close to minimize disturbances due to last-minute
trading. The time between each of the points is at least one hour
for which we expect the transitory effects to die out. For the 30
Dow stocks, Hansen and Lunde (2006, p. 127) found that “the
noise may be ignored when intraday returns are sampled at relatively low frequencies, such as 20-minute sampling.”
The Amsterdam and the New York stock exchanges are both
continuous, consolidated auction markets in the terminology
proposed by Madhavan (2000). Both exchanges release trade
and quote information in real time. The main difference is that
the NYSE is a hybrid market, because orders can arrive at the
floor through both brokers and the electronic Superdot system.
The Amsterdam Stock Exchange is a pure electronic market
in which orders are routed to a consolidated limit order book
and are executed according to price–time priority. In our sample period, a market maker (“hoekman”) was assigned to each
book with the obligation to “smooth price discovery” by inserting limit orders at times of illiquidity. For the blue-chip stocks
we study, however, they rarely intervened. And, for our sample period, tick sizes were comparable across both exchanges:
U.S.$ 1/16 at the NYSE and NLG .1 (≈U.S.$ .05) at the Amsterdam Stock Exchange.
The dataset used in this study consists of trade and quote
data from Euronext–Amsterdam and the NYSE for July 1, 1997
through June 30, 1998. Seven Dutch blue-chip stocks crosslisted in New York have been selected for the current study: Aegon (AEG), Ahold (AHO), KLM, KPN, Philips (PHG), Royal
Dutch (RD), and Unilever (UN). These firms are multinationals in different industry groups and represent more than 50% of
the local index in terms of market capitalization. We use Dutch
guilder–U.S. dollar (NLG–USD) tick data from Olsen & Associates to convert all prices to USD, which is the benchmark
currency. In the robustness section, we show our results are robust to prices in NLG instead of USD.

218

Journal of Business & Economic Statistics, April 2007

Table 1. Summary Statistics for Trading in Amsterdam and New York

Variable

Stock
exchange

Share
AEG

AHO

KLM

KPN

PHG

RD

UN

1,360
1,214
1,076
799
40
106

1,284
753
929
686
37
66

1,005
376
642
390
32
90

1,412
808
1,118
743
25
38

730
859
600
914
20
44

581
493
438
533
18
19

26
49
89
1

28
32
20
6

25
35
53
1

18
15
77
24

15
15
139
72

14
13
57
20

Panel B: Trading statistics overlapping hour (15-minute averages)
Trade price
AMS
1,437
2,116
Volatility (bp2 )
NY
933
2,007
Midquote
AMS
1,038
1,708
NY
888
1,466
Volatility (bp2 )
Quoted
AMS
23
41
NY
61
120
Spread (bp)
Effective
AMS
20
28
NY
51
82
Spread (bp)
Volume
AMS
53
124
(1,000 shares)
NY
5
3

2,321
1,840
1,949
1,679
36
83
32
58
33
11

1,779
733
1,325
815
31
90
28
83
81
2

2,096
1,508
1,783
1,553
25
44
20
33
123
38

1,017
1,291
897
1,284
21
47
17
17
232
120

966
619
827
710
20
20
16
21
95
34

Panel A: Trading statistics full day (15-minute averages)
Trade price
AMS
922
Volatility (bp2 )
NY
336
Midquote
AMS
544
NY
274
Volatility (bp2 )
Quoted
AMS
23
NY
51
Spread (bp)

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

Effective
Spread (bp)
Volume
(1,000 shares)

AMS
NY
AMS
NY

18
19
34
3

NOTE: This table contains summary statistics for trading in Amsterdam and New York from July 1, 1997 through June 30, 1998. Panel A contains averages for the full trading day; panel B for
the overlapping hour. All variables are 15-minute averages. Trade Price Volatility is calculated as the variance of the 15-minute squared returns based on transaction prices and measured in basis
points. Midquote Volatility is calculated the same way, but based on midquotes. Quoted Spread is calculated as the time-weighted average of all prevailing quoted spreads in a 15-minute interval.
Effective Spread is calculated as the time-weighted average of twice the difference between the transaction price and the prevailing midquote. Both spreads are measured in basis points. Volume
is the 15-minute average number of shares traded.

Summary statistics for trading in the seven Dutch stocks are
shown in Table 1. They are very diverse as is apparent from
trade variable averages such as volatility, volume, and spread
(definitions are given in the table footnote). A closer look reveals that they are similar in two important ways. First, for
none of the stocks has New York been able to generate more
volume than Amsterdam. Second, quoted spreads are larger
in New York, up to almost 300%. This is most likely due to
the different market structure in New York, where many orders receive price improvement from the floor. The effective
spread, in this case, is a more appropriate measure, because it
is based on actual trades. Changing to this measure, we find
that differences shrink and for some stocks New York spreads
are lower. Although finding the most competitive exchange is
beyond the scope of this article, effective spread results show
that exchanges are very competitive, which is a promising result in view of the price discovery questions addressed in this
study. Comparing Amsterdam to New York based on statistics
for the overlapping hour yields a similar picture. The main difference is that average values for all variables are higher during
the overlap.
4. EMPIRICAL RESULTS
4.1 Variance Ratio Estimates
As a preliminary analysis, we follow the standard variance
ratio approach and calculate the variance pattern, based on
pooled intraday and overnight squared midquote returns. The
intraday returns are calculated based on the six identified time

points τi , where we arbitrarily choose the average midquote as a
proxy for the price during the overlap. In addition to the pooled
estimates, we also estimate variance patterns stock by stock to
study heterogeneity. We use a Newey–West approach to calculate standard errors to account for commonality in (squared)
returns (and, thus, deal with the Ronen 1997 critique) and potential nonzero autocovariances up to a one-week lead and lag
period.
For all estimates reported in this article outliers were removed for different reasons. First, in 1998 the change to daylight savings time in the Netherlands happened one week before it occurred in the United States. As a result, there was no
trading overlap from March 30 to April 3, 1998. This period
was removed from the sample because it was not representative. The week of nonoverlap could be used to test whether the
increased variance patterns are due to both markets being open
simultaneously or to more information being produced exogenously during those times of day. Exploratory results reveal the
former to be the case, but the period of one week is too small for
any statistical significance. Second, at the end of the trading day
on October 27, New York prices collapsed by 7%. They fully
recovered at the New York open the next day. This overnight
period was removed from the sample because it was a clear
temporary distortion. Third, on a Unilever quarterly announcement on May 1, 1998, the share price jumped by roughly 8%
on the Amsterdam open. This jump was removed because it
clearly was a one-time event and not representative for regular
around-the-clock price discovery on an “average day.”
Table 2 reports the variance estimates, which are scaled by
the number of hours between time points to create hourly equivalents. This facilitates easy comparison across time intervals.

Menkveld, Koopman, and Lucas: Modeling Around-the-Clock Price Discovery

Table 3. Intraday Return Autocorrelations

Table 2. Hourly Variance for Intraday and Overnight Returns
Time interval

Time intervals
Event
Start (EST)
End

Start
AMS
4:00
8:00

NY
preopen
8:00
9:00

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

Panel A: Pooled variance estimates
.36
.36
στ2 (×10,000)
(.03)
(.08)

219

NY
open
9:00
10:00

AMS
close
10:00
11:00

NY
only
11:00
15:30

Overnight
15:30
4:00

.36
(.03)

.53
(.05)

.15
(.01)

.12
(.01)

Panel B: Stock-specific variance estimates
AEG
.23
.30
.24
.30
.04
(.03)
(.08)
(.03)
(.05)
(.01)
AHO
.40
.69
.44
.55
.09
(.07)
(.28)
(.15)
(.11)
(.02)
KLM
.42
.39
.37
.49
.21
(.08)
(.15)
(.04)
(.07)
(.03)
KPN
.33
.25
.19
.37
.05
(.04)
(.06)
(.02)
(.07)
(.01)
PHG
.57
.40
.49
.67
.33
(.07)
(.06)
(.05)
(.08)
(.08)
RD
.26
.19
.33
.54
.23
(.04)
(.05)
(.05)
(.07)
(.03)
UN
.18
.15
.23
.36
.12
(.02)
(.02)
(.03)
(.05)
(.02)
Panel C: Pairwise variance ratio tests for pooled estimates
(Wald, χ 2 (1))
∗
∗
Start AMS
.00
.02
7.00
33.50
∗
NY preopen
.00
2.91
5.80
∗
∗
NY open
7.66
32.01
∗
AMS close
47.27
NY only

.10
(.02)
.14
(.03)
.13
(.02)
.09
(.01)
.15
(.02)
.07
(.01)
.07
(.01)

∗

48.24
∗
8.12
∗
46.62
∗
58.00
∗
4.58

NOTE: This table contains estimates of the midquote return variance στ2 for different intraday
time intervals based on July 1, 1997 through June 30, 1998. All stocks are included. Midquote
returns are first demeaned by subtracting the time-proportional average mean over the entire
sample and then scaled to correct for interstock volatility differences. Standard errors are based
on a Newey–West procedure to account for nonzero autocorrelation. We use the same procedure to also account for commonality across returns. Panel A presents the pooled variance
estimates. Panel B presents these estimates on a stock-by-stock basis. Panel C presents pairwise “variance ratio” tests for the pooled estimates of panel A. We test equality based on a Wald
test and the Newey–West covariance matrix that accounts for intraday correlation across returns
and autocorrelation up to a one-week lead and lag period.
∗
Significant at the 95% confidence level.

Panel A contains the pooled estimate and shows that for the
three intervals in the Amsterdam trading period, the average
variance is 3.6 × 10−5 , which corresponds to a standard deviation of 60 basis points per hour or an annualized volatility of
47%. Variance for the hour containing the Amsterdam close is
a significant 48% higher. In the terminology of the existing literature, we find that price discovery—the information flow per
unit of time—in this hour is a factor of 1.5 higher (see, e.g.,
French and Roll 1986; Jones et al. 1994; Ronen 1997; George
and Hwang 2001). Additionally, the Amsterdam nonoverlap is
a significant of factor 2.4 more informative than the NYSE
nonoverlap, which, in turn, is a significant factor of 1.3 more
informative than the overnight hours. Panel B reports the stockspecific patterns and illustrates the heterogeneity of the sample.
For six out of seven stocks, we find that the Amsterdam close
continues to be the most informative period. The result that the
New York only is more informative than the overnight, however, only holds for four stocks. Straightforward pooling, as is
often done in the literature, does not seem to be appropriate for
our sample. Also note that Royal Dutch (RD) as opposed to the
other stocks has a variance for the New York only period that
is roughly equal to that of the Amsterdam only period. This is
due to the proportionally higher trading volume of RD on the

4:00–8:00
8:00–9:00
9:00–10:00
10:00–11:00
11:00–15:30
15:30–4:00(+1)

Event

Lag 1

Lag 2

AMS only
NY preopen
NY open
AMS close
NY only
Overnight

−.077
.056
∗
−.125
∗
.251
−.050
∗
−.165

−.020
−.005
∗
−.170
.039
−.022

NOTE: This table presents the raw return autocorrelations up to the second lag of intraday
and overnight midquote returns. The midquote in the overlapping interval was arbitrarily fixed
at the average of the Amsterdam and New York midquotes. The autocorrelations are calculated
for the full sample period, from July 1, 1997, through June 30, 1998, and averaged across all
stocks. We explicitly account for commonality in returns when determining confidence intervals.
∗
Significant at the 95% confidence level.

NYSE. In the advocated state space approach, we control for
heterogeneity across stocks through a stock-specific variance
parameter (ci ), differential factor loadings on a common factor
(βi ), and, in a robustness check, stock-specific variance of the
microstructure effects. In the extreme case, we could abstain
from pooling altogether and estimate the model on a stock-bystock basis. In that case, however, it would no longer be possible to distinguish the market reaction to common versus stockspecific information. We, therefore, do not pursue this line of
research further.
To further motivate the state space model advocated in this
article, Table 3 reports the autocorrelations for intraday returns.
If measurement errors exist and if they are economically significant, we should find negative first-order autocorrelation. Most
of these autocorrelations are indeed negative and two of them
are significant. We find a significantly positive autocorrelation
for the period containing the Amsterdam close. Apparently,
markets underreact to information in the New York open, causing persistence in returns for the subsequent Amsterdam close
period. Higher order autocorrelations are insignificant, except
for the Amsterdam close period.
4.2 Estimation Results
We proceed by re-estimating the intraday variance pattern using the state space model advocated in this article. We test for
significance of parameters at a 95% level and omit the insignificant ones. Furthermore, we scale our (innovation) variance estimates by the number of hours between time points to create
hourly equivalents, which facilitates easy comparison across
time intervals. The results are shown in Tables 4 and 5.
Panel A of Table 4 presents the estimate of the variance pattern of the efficient price, which is plotted in Figure 2 along
with the variance pattern based on the direct variance ratio approach reported in Table 2. The state space model estimates differ in two important ways. First, most information is attributed
to the New York open, instead of the Amsterdam close period.
The reason is market underreaction to both common-factor and
firm-specific information (35% and 34%, respectively) in the
New York open period. In other words, the efficient price innovation in the New York open period is not yet fully revealed
in midquotes halfway through the overlapping period. We conjecture that this is due to informed investors who trade strategically and split their order over time to maximize profit. They
might have a preference for trading in the overlapping period

220

Journal of Business & Economic Statistics, April 2007

Table 4. Estimation Results for Efficient Price and Observation Models
Panel A: Variance efficient price innovation (×10,000, hourly)
Time intervals
Event
Start (EST)
End

σE2 ,τ
2
σξ,τ

Start
AMS
4:00
8:00

NY
preopen
8:00
9:00

NY
open
9:00
10:00

AMS
close
10:00
11:00

NY
only
11:00
15:30

Overnight
15:30
4:00

.35
(.02)
.13
(.01)

.33
(.02)
.18
(.02)

.38
(.03)
.15
(.02)

.15
(.01)

.09
(.01)
.05
(.01)
−.16
(.04)
.05
(.01)
.87
(.13)

.10
(.01)
.05
(.01)

.22
(.01)

.43
(.02)
.16
(.02)
−.35
(.04)
.27
(.01)
−.34
(.02)

θξ,τ
ση2,τ

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

θη,τ

.23
(.02)
−.30
(.08)

.05
(.00)

Panel B: Variance measurement error (×10,000)
Time points (EST)
Start (EST)

4:00

8:00

9:00

10:00

11:00

15:30

σε2,A,τ

.00
(.00)

.00
(.00)

.00
(.00)

.07
(.01)
.11
(.01)

.07
(.02)

.14
(.00)

σε2,NY,τ

NOTE: This table contains maximum likelihood estimates of the state space model (1) and (7)
based on intraday midquotes for July 1, 1997 through June 30, 1998. The observation vector
pk ,t ,τ contains the midquotes for all stocks traded in market k at day t and time point τ . Panel A
contains estimates of the efficient price innovation variance σE2,τ , the common price variance
2
σξ,τ
, the response to a common innovation θξ,τ , the firm-specific price variance ση2,τ , and the
2
+ ση2,τ . Panel B contains estiresponse to a firm-specific innovation θη,τ . Note that σE2,τ = σξ,τ
mates of measurement error variance in both markets, σε2,k ,τ with k ∈ {A, NY}. Standard errors
are in parentheses.

as they can hide their order flow by trading across two markets (see Kyle 1985 for informed trader behavior in a single
market and Chowdhry and Nanda 1991 and Menkveld 2005 for
a multiple-markets setting). This behavior could also explain
the market underreaction (30%) to firm-specific information in
the Amsterdam close period. Second, trading in New York after the Amsterdam close is not significantly more informative
than the overnight nontrading hours. The main reasons are that
the New York midquotes contain significant transitory effects
and that the New York market appears to overreact significantly
(87%) to firm-specific information. The overreaction is most
likely due to increased market-making activity by the NYSE
specialist after the home market, as a secondary pool of liquidity, disappears. Moulton and Wei (2005) studied NYSE trading
of European stocks and documented a significant increase (6%)
in specialist participation rates after the home markets close.
We also find that the market underreacts to common-factor information, but this effect is much smaller (16%) and, as we will
show later, is not robust.
We conduct formal tests of equality of the over-/underreaction coefficients for firm-specific and common-factor information. We cannot reject the hypothesis of an equal coefficient for
the New York open period [H0 : θξ,τ = θη,τ in model (7) for τ
corresponding to the New York open period]. The χ 2 (1) Wald
test for this hypothesis has a value of .025. For the New York
only period, however, this test has a value of 95.6, which clearly
rejects the null of equal coefficients. It is, therefore, important
that the model allow for a different market reaction to commonfactor relative to firm-specific information.

Figure 2. Raw Return Variance versus State Space Innovation Variance. The top figure illustrates the estimate of the intraday variance pattern
based on raw returns of all stocks, taking into account interstock volatility differences. It is presented on an hourly basis to enhance comparability.
The bottom figure represents the variance pattern based on the returns of the unobserved efficient price as estimated by the state space model.
The bars represent point estimates; the dotted lines 95% confidence intervals.

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:03 12 January 2016

Menkveld, Koopman, and Lucas: Modeling Around-the-Clock Price Discovery

221

Figure 3. Variance Decomposition Into Common and Idiosyncratic Components. The state space model specification allows for decomposition
of the efficient price returns into two components: a common component due to a common (market-wide or macro) factor driving the returns across
all stocks and an idiosyncratic component due to firm-specific returns. Hence, the variance can be decomposed by time of day. The efficient price
variance pattern, as depicted in Figure 2, is, therefore, the sum of two components: The top figure represents the common-factor component; the
bottom figure the stock-specific or idiosyncratic component. The bars represent point estimates; the dotted lines 95% confidence intervals.

To characterize around-the-clock price discovery in more detail, we further decompose information into firm-specific and
common-factor information by the time of day. Figure 3 illustrates this decomposition and leads to three important observations. First, the significantly larger innovations in the efficient
price during the overlap are due to increased firm-specific rather
than common-factor information. Apparently, a disproportionate amount of firm-specific information is produced through
trading in this period. This is consistent with our earlier conjecture on increased inform