PRICE DISCOVERY IN INTERNATIONAL EQUITY
TRADING:
Evidence from High-Frequency NYSE and XETRA Data
by
Joachim Grammiga, Michael Melvinb, and Christian Schlagc
a,c
b
University of Frankfurt
Arizona State University
TWO QUESTIONS
1. Where does price discovery occur for firms traded
internationally?
2. Given an exchange rate shock, how do stock prices adjust in
the home market and the derivative market?
TOPIC ORDER
1. Institutional Details
2. Equilibrium Relationships
3. Framework for Analysis
4. Data and Estimation Results
5. Conclusions
INSTITUTIONAL DETAILS
American Depositary Receipts (ADRs)
*Deutsche Telekom (DT)
*SAP
Global Registered Shares (GRSs)
*DaimlerChrysler (DCX)
XETRA and NYSE
A SIMPLE MODEL OF INTERNATIONAL EQUITY
TRADING
E t  E t  1  ut
Pth  Pth1  vt
Ptu  E t  1  P h  w t
t 1
Et  Pth  Ptu  Et 1  ut  Pth1  vt  Et 1  Pth1  wt  ut  vt  wt
Expect cointegration among the 3 variables
Two stock prices will differ by the exchange rate plus random
terms
Have 1 cointegrating vector and 2 common trends:
*exchange rate innovations
*home market price
GRANGER REPRESENTATION THEOREM
a) If the level of prices and the exchange rate can be
represented by a nonstationary vector autoregression like:
Pt     1 Pt  1   2 Pt  2  ...   q Pt  q   t
b) and P has the Wold representation:
( 1  L )Pt     ( L ) t
{note: Wold’s decomposition for any zero-mean cov.-stationary process
yt can be written as a function of the past white-noise errors  one would
make in forecasting yt as a linear function of lagged yt}
c) and there is a cointegrating vector A’ such that
Zt=A’Pt
is stationary where
A' ( 1 )  0
then there exists a vector error correction (VEC) model:
Pt    BZ t  1   1 Pt  1   2 Pt  2  ...  Pt  q  1   t
Johansen’s Cointegration Estimation Method
1) Estimate VAR for Pt
Pt     1Pt  1   2 Pt  2  ...   q  1Pt  q  1  ut
2) Estimate lagged P regression on lagged P
Pt  1  b  1Pt  1   2 Pt  2  ...  q  1Pt  q  1  v t
3) Calculate Var-Cov matrices of residuals
T
 uu  ( 1 / T )  ut ut '
t 1
T
 vv  ( 1 / T )  vt vt '
t 1
T
 uv  ( 1 / T )  ut vt '
t 1
 vu   'uv
1  1
find eigenvalues of  vv
vu uu uv ordered 1  2 ...  n
then max. of log likelihood function s.t. constraint of h
cointegrating relations is:
h
L  ( Tn / 2 ) log( 2 )  ( Tn / 2 )  ( T / 2 ) log |  uu | ( T / 2 )  log( 1  i )
i 1
then conduct likelihood ratio test for number of cointegrating
relations
4) Calculate parameter estimates
a1,a2,...,ah are eigenvectors associated with the h largest
eigenvalues
any cointegrating vector A can be written as
A  b1 a1  b2 a 2  ...  bhah
normalize so that A'  vv A  1
Then typically normalize so that first element of A equals 1
INFORMATION SHARES
What is contribution of innovations in each asset price to
“price discovery”?
“Information Share” = proportion of innovation variance in
asset i explained by innovations in asset j
Since each asset price is I(1) and cov-stationary, our 3 equation
system can be expressed as a Vector Moving Average (VMA):
Pt   ( L ) t
where  is a polynomial in the lag operator and  is a zeromean vector of random disturbances with covariance matrix

Granger Representation Theorem included cointegration
restriction that A' ( 1 )  0 where  ( 1 ) is the matrix
polynomial evaluated at L=1
Note on VMA
E t  u t
Pth  v t
Ptu  aE t  1  bPth 1  w t  aut  1  bv t  1  w t  w t  1
define  te  ut ,  th  v t ,  tu  w t  aut  1  bv t  1
so Ptu   tu  w t  1   tu   u  a e  b h
t 1
t 2
t 2
and the system can be written as:
Pt   ( L ) t
or
 E 
 t  1
 P h    0
 t   2
 P u  aL
 t 
0
1
bL2
 e
  t 
0   th 
 
( 1  L )  u 
 t 
0
1 0 0
so  ( 1 )  0 1 0 


a b 0 
We let data choose actual (1)
The restriction A' ( 1 )  0 means that the matrix operator
(1) is singular
We find the elements numerically by dynamic simulation of the
estimated VEC (Hamilton, 1994)
*dynamic multipliers associated with unit shock to innovations
in each variable
Our priors on (1):
With cointegrating vector of essentially A’=[1 1 –1] and general
system of
 E 
 e 



 t   11
12
13   t 
 P h   
 22  23   th 
21
t



 
u





 P   31
32
33   tu 
t


 
We expect:
* 12
  13  0
* 22   32
* 23   33
Intuition:
*E unaffected by stock price innovations
*Ph innovations have symmetric effect on XETRA & NYSE
*Pu innovations have symmetric effect on XETRA & NYSE
*E innovations may have different effect on XETRA & NYSE
Once (1) is known we can infer the information shares
impact of innovation in price j on price i is
 ij  j
Total variance associated with innovations in i is ii diagonal
element of  '
Since  has off-diagonal elements due to contemporaneous
correlation in innovations, to find info. share need a
triangularization of 
*Cholesky factorization yields matrix C such that CC’=
Information shares found as:
S ij  ([C ] ij ) 2 /( ' )ii
this is total innovation variance associated with i explained by
innovations in j
* ([C ] ij ) 2 is the product of i’th row of  matrix and j’th
column of C matrix
* ( ' )ii is diagonal element associated with total variance
of i innovations
*normalization ensures that info. shares sum to 1 for each asset
DATA
XETRA and NYSE prices on DCX, DT, and SAP
*NYSE TAQ data
*Deutsche Börse A.G. proprietary quotes
USD/EUR from Reuters indicative quoting screen
1 August – 31 October, 1999
Overlap of trading hours
*14:30-16:00 GMT until 19 Sept.
*14:30-16:30 GMT from 20 Sept.
Table 1 summary stats
*SAP trades at 12 to 1 ratio
*mean USD/EUR was 1.0607
Figure 1 plots of logs of variables (avg. of bid & ask)
*level of USD/EUR ranges from 1.0355 to 1.0889
*XETRA & NYSE prices track closely together
Figure 2 plots quoting intensity
*synergies between two markets?
*DT is uniquely “German”
Informal evidence of Frankfurt as “primary” market with New
York as “derivative” market
ESTIMATION
Sample at 10 second intervals
*tradeoff contemporaneous correlation against
“microstructure effects”
*temporal aggregation can dissolve 1-way causality that
appears at higher frequency
*higher frequencies may yield nonsynchronous quoting, bidask bounce or other “noise”
*pick 10 seconds by examining partial correlations across
residuals of VEC models estimated using different frequency
data
ADF tests revealed unit roots in log of each price
Johansen cointegration tests support 1 cointegrating vector
Lag length chosen by Schwarz Info. Criterion (SIC)
(note: SIC = -2(log L)/T + n(log T)/T)
*start at 18 lags (3 hours) and estimate VEC for every lag
length down to 1
*SIC identifies 3 lags for SAP and DT and 4 lags for DCX
Table 2 VEC estimates
*Johansen stats identify 1 cointegrating vector
* cointegrating vectors are all about [1 1 –1] for [E Ph Pu]
VECs dynamically simulated to find long-run multipliers of
unit shock to innovations
(note: set all innovations to zero, then set first observation =1
for one variable and simulate; repeat for other variables)
Table 3 (1) matrices meet priors
*E unaffected by innovations to stock prices
*long-run impact of shock to 1 stock price is symmetric for
itself and the other stock price
Table 3 surprise (or at least diffuse priors)?
*long-run impact of shock to Ph greater than long-run impact
of shock to Pu
*E shocks have greater impact on Pu than Ph
Cholesky factorization of  yields upper bound on info. share
for variable ordered first and lower bound on info. share for
variable ordered last
*common problem in VARs with impulse response or variance
decompositions
Swanson & Granger propose using the “Directed-AcyclicGraph” (DAG) approach to choose ordering:
1) specify relationship expected due to prior beliefs based on
theory
2) estimate model and calculate all partial correlations among
residuals of individual equations
3) construct hypothesis tests of null that some partial
correlations equal zero in accord with step 1)
4) if supported by partial correlations, order variables
accordingly
DAG for exogenous E that may affect both Ph and Pu while Pu
is also affected by Ph :
 Et 

 Pth 

 Ptu

ut
vt
wt
DAG portrays recursive contemporaneous correlation
DAG implied system is:
E t  u t
Pth   h E t  v t
Ptu   ue E t   uh Pth  w t
Following Swanson & Granger, the DAG implies that the
partial correlation  ( E t , Ptu | Pth )  0
*estimate by regressing residuals of individual VEC equations
on each other
Use order of E, Ph, Pu in calculating information shares
reported in Table 4
Table 4 info. shares
*E unaffected by stock price innovations
*E info share of about 0 for Ph and more than 5% for Pu
* Ph info share on Ph ranges from 99% for DT to 79% for SAP
* Ph info share on Pu ranges from 94% for DT to 76% for SAP
* Pu info share on Ph ranges from 20% for SAP to 1% for DT
* Pu info share on Pu ranges from 19% for SAP to 1% for DT
DT appears to be more of a domestic firm while DCX and SAP
are multinationals with more of a role for international price
discovery
*U.S. revenue as share of total revenue supports this
DT, 1%
DCX and SAP about 50%
Why bigger role for NYSE in price discovery for SAP?
*”new economy” stock?
Table 5 additional check for robustness of results over
alternative orderings
*tight bounds so ordering of variables relatively unimportant
CONCLUSIONS
Where does price discovery occur for internationally-traded
firms?
*largely in home market
*home-market info share larger for purely domestic firm than
multinational
How do international stock prices adjust to an exchange rate
shock?
*home market price appears to be independent of exchange
rate
*foreign market (NYSE) price contains all of the adjustment
Table 1
Descriptive Statistics for Firms and Markets
Summary statistics are reported for three firms: DaimlerChrysler (DCX), Deutsche Telekom (DT),
and SAP. Trading in the U.S. occurs on the NYSE and trading in Germany occurs on the XETRA
system. The data are for the period of trading overlap each day: 14:30-16:00 (or 16:30 from
September 20, 1999) over the period from August 1 to October 31, 1999. XETRA prices are quoted
in euro and NYSE prices are quoted in dollars (the mean bid quote for the exchange rate over the
sample period was 1.0607 dollars per euro). DCX and DT shares in the U.S. trade at a 1 to 1 ratio
against German shares. SAP shares in the U.S. trade at a 12 to 1 ratio against the German shares.
Avg. bid
price
-----------
Avg. ask
price
-----------
Avg. daily no.
of quotes
---------------
Avg. daily
Avg. daily
trading volume turnover
---------------------------
XETRA
69.72
69.79
731
794,523
55,478,987
NYSE
73.80
73.96
215
203,909
15,102,534
XETRA
40.60
40.67
593
1,024,785
41,579,511
NYSE
42.97
43.14
151
572,902
28,491,202
XETRA
402.00
402.70
563
92,917
37,657,635
NYSE
35.77
35.91
162
368,992
13,388,669
DCX
DT
SAP
Table 2
VEC Estimation Results
Vector error correction models are estimated for each firm. The form of equation is:
Pt    BZ t 1   1 Pt 1   2 Pt 2  ...  Pt q 1   t
where Pt contains the change in the logs of the exchange rate, the XETRA price, and the NYSE price; Zt-1
is the lagged log-levels of each variable in the cointegrating equation estimated by the Johansen method.
The table reports the cointegating vector that applies to Z. , B, and  are coefficients to be estimated. The
estimated values of these coefficients are reported below. The bootstrap standard errors are in parentheses.
At the bottom of each table is the number of observations for that firm (each day, the time of the first
observation is defined by the first quote), the log likelihood associated with the estimated system, and
summary statistics associated with the Johansen test for the order of cointegration, where h = number of
cointegrating relations, LR = likelihood ratio statistic, krit. 5 %: critical values of LR statistics taken
from Hamilton (1994), pp. 767-768. In each case, the results support 1 cointegrating vector.
Table 2a: DCX Estimation results
Cointegrating Eq.
pFX
1.00000 (0.00000)
pXETRA
1.01520 (0.01762)
pNYSE
-1.01512 (0.01737)
Error Correction
pFX
pXETRA
pNYSE
zt
-0.00032 (0.00039)
-0.00453 (0.00474)
0.01472 (0.00927)
pFX(-1)
-0.42990 (0.00502)
0.00589 (0.00277)
-0.03192 (0.00482)
pXETRA(-1)
0.00378 (0.00254)
-0.04856 (0.00276)
0.00646 (0.00437)
pNYSE(-1)
0.00226 (0.00265)
0.00582 (0.00072)
-0.01045 (0.00933)
-0.20214 (0.00526)
0.00034 (0.00972)
-0.01650 (0.00541)
pXETRA(-2)
0.00215 (0.00275)
0.02318 (0.00528)
0.04392 (0.00483)
pNYSE(-2)
-0.00167 (0.00252)
0.01474 (0.00479)
-0.00401 (0.00073)
pFX(-3)
-0.05865 (0.00537)
-0.00026 (0.00946)
-0.01150 (0.00957)
pXETRA(-3)
0.00012 (0.00278)
0.02015 (0.00499)
0.05482 (0.00434)
pNYSE(-3)
0.00550 (0.00274)
0.01023 (0.00490)
-0.01399 (0.00542)
pFX(-4)
-0.02920 (0.00474)
0.01358 (0.00927)
-0.02043 (0.01015)
pXETRA(-4)
-0.00186 (0.00277)
0.01666 (0.00482)
0.05287 (0.00521)
pNYSE(-4)
-0.00169 (0.00276)
0.01001 (0.00437)
-0.00660 (0.00502)
pFX(-2)
Number of obs.
38704
Log Likelihood
812177.0
Johansen’s trace statistic
H0 HA
LR
krit. 5%
h=0 h=3 437.51 24.31
h=1 h=3 4.33 12.53
h=2 h=3 0.08 3.84
Johansen’s max. Eigenvalue
statistic
H0 HA
LR
krit.5%
h=0 h=1 433.18 17.89
h=1 h=2
4.25 11.44
h=2 h=3
0.08 3.84
Table 2b: DT Estimation results
Cointegrating Eq.
pFX
1.00000 (0.00000)
pXETRA
1.01516 (0.01622)
pNYSE
-1.01507 (0.01596)
Error Correction
pFX
pXETRA
pNYSE
zt
-0.00018 (0.00031)
-0.00213 (0.00088)
0.02304 (0.0087)
pFX(-1)
-0.42928 (0.00472)
0.00725 (0.01437)
-0.00893 (0.01281)
pXETRA(-1)
-0.00146 (0.00202)
-0.05629 (0.00538)
0.01970 (0.00537)
pNYSE(-1)
-0.00216 (0.00205)
0.01031 (0.00534)
-0.09353 (0.00481)
pFX(-2)
-0.19805 (0.00535)
0.00060 (0.01492)
0.00803 (0.01544)
pXETRA(-2)
0.00316 (0.00198)
-0.00242 (0.00510)
0.04307 (0.00515)
pNYSE(-2)
-0.00113 (0.00178)
0.00476 (0.00473)
-0.00292 (0.00499)
pFX(-3)
-0.04761 (0.00519)
-0.01701 (0.01426)
-0.00323 (0.01425)
pXETRA(-3)
-0.00229 (0.00194)
-0.00682 (0.00464)
0.03118 (0.00514)
pNYSE(-3)
0.00356 (0.00196)
0.00507 (0.00474)
0.00545 (0.00474)
Number of obs.
38300
Log Likelihood
772688.2
Johansen’s trace statistic
H0 HA
h=0 h=3
h=1 h=3
h=2 h=3
LR krit.5 %
684.36 24.31
7.07 12.53
0.20 3.84
Johansen’s max. Eigenvalue
statistic
H0 HA
h=0 h=1
h=1 h=2
h=2 h=3
LR
677.29
6.86
0.20
krit.5 %
17.89
11.44
3.84
Table 2c: SAP Estimation results
Cointegrating Eq.
pFX
1.00000 (0.00000)
pXETRA
1.00538 (0.02273)
pNYSE
-1.00523 (0.02249)
Error Correction
pFX
pXETRA
pNYSE
zt
-0.00007 (0.00030)
-0.00633 (0.00073)
0.01516 (0.00098)
pFX(-1)
-0.42807 (0.00542)
0.00859 (0.01207)
0.00762 (0.01544)
pXETRA(-1)
-0.00370 (0.00211)
-0.06548 (0.00504)
-0.01802 (0.00601)
pNYSE(-1)
-0.00381 (0.00175)
0.01045 (0.00440)
0.02253 (0.00530)
pFX(-2)
-0.19745 (0.00560)
0.01124 (0.01337)
-0.02241 (0.01560)
pXETRA(-2)
0.00057 (0.00214)
-0.01023 (0.00537)
0.01774 (0.00619)
pNYSE(-2)
-0.00086 (0.00178)
0.01420 (0.00422)
0.00620 (0.00518)
pFX(-3)
-0.04530 (0.00570)
0.01316 (0.01253)
-0.04529 (0.01301)
pXETRA(-3)
0.00207 (0.00217)
-0.00842 (0.00549)
0.01163 (0.00652)
pNYSE(-3)
-0.00076 (0.00184)
0.01652 (0.00435)
-0.02144 (0.00548)
Number of obs.
38754
Log Likelihood
785231.2
Johansen’s trace statistic
H0 HA
h=0 h=3
h=1 h=3
h=2 h=3
Johansen’s max. Eigenvalue
statistic
H0 HA
LR
krit.5%
h=0 h=1 433.20 17.89
h=1 h=2
2.12 11.44
h=2 h=3
0.01 3.84
LR krit.5%
435.32 24.31
2.12 12.53
0.01 3.84
Table 3
Vector Moving Average Coefficients
The matrices below are estimates of the (1) matrix associated with the vector moving average (VMA)
models:
 Et   11  12  13   te 
 h 
 h 
Pt    21  22  23   t 
Pt u   31  32  33   tu 


 
The reported coefficients indicate that the exchange rate appears to be unaffected by innovations in the
stock prices. The long-run impact of a shock to the home-market stock price appears to be similar for both
XETRA and NYSE prices. Similarly, the long-run impact of a shock to the U.S. stock price appears to be
the same for both XETRA and NYSE prices. The long-run impact of a shock to the home-market price is
larger than the impact of a shock to the U.S. price in all cases. Shocks to the exchange rate appear to have
a larger impact on the NYSE price than the XETRA price.
DCX
 0.576 (0.010)  0.005 (0.011) 0.011 (0.012) 
 0.132 (0.026) 0.822 (0.031) 0.250 (0.033)
 0.435 (0.028)
0.818 (0.032) 0.261 (0.033) 

DT
 0.594 (0.006)  0.004 (0.007) 0.004 (0.007)
 0.046 (0.024) 0.879 (0.030) 0.081 (0.032) 
 0.539 (0.027)
0.875 (0.030) 0.085 (0.032) 

SAP
 0.596 (0.007)  0.005 (0.008) 0.001 (0.008) 
 0.149 (0.022) 0.689 (0.024) 0.287 (0.026)
 0.444 (0.024)
0.685 (0.023) 0.288 (0.025) 

Table 4
Information Shares of the Exchange Rate, Home-Market Price, and U.S. Price in
Price Discovery of Internationally-Traded Equities
The information shares are the proportion of the variance in the value of asset i that can be attributed to
innovations in the price of asset j. The estimates are drawn from a VEC model involving the dollar/euro
exchange rate, the home-market (XETRA) price, and the U.S. (NYSE) price. That particular order of the
three variables is utilized in the triangularization of the covariance matrix. Elements of each row may not
sum exactly to 1 due to rounding to 3 decimal places.
DCX
Exchange Rate
XETRA Price
NYSE Price
DT
Exchange Rate
XETRA Price
NYSE Price
SAP
Exchange Rate
XETRA Price
NYSE Price
Exchange Rate
Innovation
XETRA
Innovation
NYSE
Innovation
0.999 (0.005)
0.007 (0.003)
0.073 (0.007)
0.000 (0.002)
0.906 (0.028)
0.838 (0.023)
0.001 (0.003)
0.087 (0.026)
0.089 (0.026)
0.999 (0.005)
0.000 (0.000)
0.049 (0.005)
0.000 (0.002)
0.991 (0.007)
0.942 (0.008)
0.000 (0.002)
0.009 (0.007)
0.009 (0.007)
1.000 (0.003)
0.006 (0.002)
0.059 (0.006)
0.000 (0.002)
0.798 (0.041)
0.752 (0.035)
0.000 (0.002)
0.196 (0.040)
0.189 (0.038)
Table 5
Bounds for Information Shares
Permuting the order of the variables in the Cholesky decomposition of the covariance matrix allows the
computation of the upper and lower bounds on information shares. The variable going first in the order has
its share maximized and the variable listed last has its share minimized. The table gives the upper and
lower bounds for each innovation pair. Only a single value is reported when the upper and lower bounds
round to the same number at 3 decimal places.
DCX
Exchange Rate
XETRA Price
NYSE Price
DT
Exchange Rate
XETRA Price
NYSE Price
SAP
Exchange Rate
XETRA Price
NYSE Price
Exchange Rate
Innovation
XETRA
Innovation
NYSE
Innovation
0.998
0.007
0.081-0.072
0.000
0.906-0.901
0.838-0.833
0.003-0.001
0.093-0.086
0.097-0.089
0.999
0.000
0.050-0.049
0.000
0.991-0.988
0.941-0.938
0.000
0.012-0.009
0.012-0.009
1.000
0.007-0.006
0.058-0.057
0.000
0.797-0.794
0.758-0.750
0.000
0.199-0.196
0.191-0.189
Figure 1
Time-Series Plots of the XETRA and NYSE Stock Prices
and the Dollar/Euro Exchange Rate
The data plotted in the figures shows the stock prices in Frankfurt trading (XETRA) and New York trading
(NYSE) for 3 firms, DaimlerChrysler (DCX), Deutsche Telekom (DT), and SAP. In addition, the
dollar/euro exchange rate is plotted. The sample period is August 1, 1999 to October 31, 1999.
Figure 2
Intra-daily Quoting Intensities
The figures show the average number of quotes per second for each 5-minute interval over the XETRA and
NYSE trading day for the period August 1 – September 19, 1999 when XETRA closed at 16:00 GMT.