how smooth is price discovery? evidence from cross-listed stock trading

Journal of International Money and Finance 32 (2013) 668–699

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Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

How smooth is price discovery? Evidence from cross-listedstock tradingq

Haiqiang Chen a,1, Paul Moon Sub Choi b,*, Yongmiao Hong a,c,2

aWang Yanan Institute for Studies in Economics, Xiamen University, Xiamen 361005, Chinab Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 120-750, Republic of KoreacDepartment of Economics, Cornell University, 424 Uris Hall, Ithaca, NY 14853, USA

JEL classification:C32G15G14

Keywords:Price discoveryInformation shareThreshold error correction modelSmooth transition error correction model

q Chen appreciates the financial support from tProvincial Key Laboratory of Statistical Sciencesresearch grant, and Kwanjeong Educational FounWarren B. Bailey, G. Andrew Karolyi, Maureen O’HaCheol Eun, Willoe Freeman, Stefan Frey, Jinho KimJung Soon Shin, Jungwon Suh, Yafeng Qin, Byung SaAssociation 2010 (Hong Kong), Securities and FinConference 2011 (Sydney, Australia), European FiConference 2012 (Rio de Janeiro, Brazil), and Amdiscussion and feedback. This paper previously circand empirics on cross-border stock trading.” Standrules apply and all errors are our own.

* Corresponding author. College of Business AdmSeoul 120-750, Republic of Korea. Tel.: þ82 2 3277

E-mail addresses: [email protected] (H. Chen)1 Tel.: þ86 592 218 8827; fax: þ86 592 218 7702 Tel.: þ1 607 255 5130; fax: þ1 607 255 2818.

0261-5606/$ – see front matter � 2012 Elsevier Lthttp://dx.doi.org/10.1016/j.jimonfin.2012.06.005

a b s t r a c t

The adjustment to parity can be nonlinear for a cross-listed pair:Convergence may be quicker when the price deviation issufficiently profitable. We propose a threshold error correctionmodel (ECM) to gauge the market-respective information shares ofCanadian listings traded on the Toronto Stock Exchange (TSX) andthe New York Stock Exchange (NYSE). Since dynamics mayalternatively be gradual, we further generalize the thresholdframework to a smooth transition ECM. The empirical implicationsare as follows: First, the TSX and the NYSE appear to haveintegrated over time. Second, parity-convergence accelerates upondiscounts on the cross-listings on the NYSE. Third, we find a largerfeedback from the NYSE if the price gap exceeds the threshold(required arbitrage return). Fourth, informed traders tend to

he Ministry of Education Key Laboratory of Econometrics and the Fujianat Xiamen University. Choi is grateful for the Ewha Womens Universitydation grants 03112BUS044 and 0422USD012. Special thanks are due tora. We would also like to thank Jinho Byun, Michael Chng, Joung Hwa Choi,, Yihui Lan, Gianluca Mattrocci, Lars Norden, Alexei Orlov, Kyung-Hee Park,m Yoo and the participants of INFINITI 2010 (Dublin, Ireland), Asian Financeancial Markets 2011 (Kaohsiung, Taiwan), Australian Finance and Bankingnancial Management Association 2012 (Barcelona, Spain), World Financeerican Economic Association 2013 (San Diego, California) for invaluableulated under the title: “Price discovery and threshold cointegration: Theoryard disclaimer rules apply and all errors are our own. Standard disclaimer

inistration, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu,3922; fax: þ82 2 3277 2776., [email protected] (P.M.S. Choi), [email protected] (Y. Hong).8.

d. All rights reserved.

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H. Chen et al. / Journal of International Money and Finance 32 (2013) 668–699 669

3 We illustrate our methods using cross-listingsmarkets, such as commodity futures, foreign exchaMenkhoff (2011).

4 See De Jong (2002), Harris et al. (2002), and H5 A group of multiple random-walk processes is

the processes. A time series is (weakly) stationary6 In Harris et al. (2002), the efficient price compo

Granger’s (1995) permanent-transitory decomposiprices where the normalized weights are markeexchange, the bigger its influence on setting thecoefficient vector, which can be conveniently obta

cluster on the NYSE upon discounts on the cross-listings. Fifth,information share and threshold are affected by the relative degreeof private information, market friction and liquidity measures,firm-level characteristics, and aggregate risks.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

We contribute to the literature by implementing nonlinear error correction mechanisms whenestimating the exchange-respective contribution to price discovery (information share) of cross-listings and their original listings.3 The existing methods assume linear convergence of price devia-tions to parity whereaswe hinge our premise on the reality that when premiums or discounts on cross-listings are profitably arbitrageable they disappear faster than otherwise, and that convergences maybe gradual and nonlinear.

Price discovery is the process by which information is priced in the market. When a security istraded in multiple markets, determining where and how price discovery occurs is important. Harriset al. (1995) and Hasbrouck (1995) examine the information share of the New York Stock Exchange(NYSE) for fragmented (multi-market traded) stocks on the NYSE and other U.S. exchanges, andconfirm the NYSE’s leadership role. As for international cross-listing, Bacidore and Sofianos (2002) andSolnik et al. (1996) suggest that most price discovery occurs in the home market where considerableinformation originates. Eun and Sabherwal (2003) report that U.S. host exchanges, to a lesser extentthan the Toronto Stock Exchange (TSX), determine the prices of Canadian cross-listings.

In the literature, there are two broad approaches to estimating the contribution of each market tothe price discovery of fragmented listings. Hasbrouck’s (1995) innovation variance approach extractsinformation shares by employing variance decomposition based on the vector moving averagerepresentation of an error correctionmodel (ECM). Harris et al.’s (1995, 2002) common factor approachemploys permanent-transitory decomposition of a cointegrated system to estimate the informationshare of each market. As Eun and Sabherwal (2003) show, Hasbrouck’s (1995) approach involves theCholesky factorization of the covariance matrix of innovations to prices on various exchanges, yieldingmultiple information shares. This may cause confounding effects on the identification of the venue ofprice discovery. Hasbrouck’s (2002) modification can be numerically onerous on implementation.4 Weexpand Harris et al.’s (1995, 2002) platform and complement Hasbrouck’s (1995) idea.

Harris et al. (1995) associate error correction dynamics with the price discovery of cross-listed pairswhich are cointegrated5 by the law of one price. The cointegrating vectors of the vector ECM (VECM)represent the long-run equilibrium (near-parity condition), while the error correction terms charac-terize the convergence mechanism. Through representation, the relative extent of the contributionmade by each market to the price discovery of fragmented stocks, using the estimates of adjustmentcoefficients, is assessed. Harris et al. (2002) buttress the method earlier formulated in Harris et al.(1995) by incorporating a microstructure model in which the price is assumed to be the sum of anefficient (permanent) price component and an error (transitory) term.6

. However, these methods can be applied to other informationally linkednges, bonds etc. See Liu and An (2011), Chen and Gau (2010), and Fricke and

asbrouck (2002) for further discussion.cointegrated if, by definition, there exists a stationary linear combination ofif the probability laws (of up to the second moments) are time-invariant.nent is unobservable and reflects the underlying fundamentals. Gonzalo andtion posits the permanent price as a linear combination of the observablet-respective information shares. The higher the normalized weight of anpermanent price. Normalized weights are orthogonal to the adjustmentined from an ECM.

H. Chen et al. / Journal of International Money and Finance 32 (2013) 668–699670

However, an implicit assumption made in Harris et al. (1995, 2002) is that adjustment to parity, thelong-run equilibrium, is continuous and linear. Various economic circumstances challenge suchrestrictions, particularly where transaction costs and policy intervention are present. According to theempirical findings of Gagnon and Karolyi (2010), impediments to arbitrage gravity are both explicit andimplicit. The former observable sources of market frictions are taxes,7 transactions costs, holding costs,and short-sale restrictions. Transaction costs of an arbitrage on a diverged cross-listed pair arecommissions to home and host exchanges, bid-ask spreads on the cross-listed pair and the foreignexchange rate, andprice impacts onboth cross-border listings.While their arbitrageportfolios are active,arbitrageurs incur holding costs which include lending fees (Barberis and Thaler, 2003), the unrealizedinterest on short-sales proceeds, the opportunity cost of capital, and firm-level risks (Pontiff, 2006).Although it does not persistently apply to the Canadian cross-listings on U.S. exchanges, temporary andregulatory bans of short sales on the home exchanges of specific American Depositary Receipts (ADRs)render arbitrageurs’ transactionsmore costly thereby increasing their required returns (Bris et al., 2007).

The hidden hurdles to swift parity-convergences of cross-listed pairs may stem from cross-borderdifferential adverse selection risks (Chen and Choi, 2012) and agency walls between the entrusted arbi-trageurs and their investor clients (Shleifer and Vishney,1997). Chen and Choi (2012) find that the originalCanadian listings on the TSX suffer more from asymmetric information than their cross-listings on theNYSE. As a result, an average Canadian cross-listed pair exhibits an adverse selection discount on the TSXand a relative premium8 on the NYSE by 30.6 basis points, based on the 10-min frequency US$-denominated prices of 56 cross-listed pairs through the sample period of 1998–2000. An uninformedarbitraguerwill be forced to closeherportfolio earlyat a substantial deviation in thepairuponsensing “flowtoxicity”. Also, an outsourced portfolio manager have less incentive to act less aggressively when investorssolely evaluate him by his returns in a “separation of brains and capital” (Shleifer and Vishney, 1997).

Given these intricacies of the trading environment with various transaction costs and obstacles,nonlinear parity-convergences of cross-listed pairs capture the market to a higher proximity. Therationale of nonlinear modeling is straightforward. A relatively small deviation of the price of a cross-listing from its parity-implied price is unarbitrageable if the dollar spread is insufficient to cover fees,commissions, liquidity shortfalls, and other related costs. In this case, the dollar premium or discountbehaves like a near-unit root process and will not converge to parity. Arbitrage forces will activate asthe spread widens beyond the threshold (outer regime), determined by transaction costs and associ-ated risk premiums of arbitrage. There may be another case that there is more than a single break-evenprice spread in a cross-listed pair: The transition from a profitable status to an unarbitrageable one canbe “smooth” due to multiple and overlapping regime-switching effects.

To date, we find a dearth of articles with nonlinear frameworks in the literature. Among thosefound, Rabinovitch et al. (2003) use a nonlinear threshold model to estimate the adjustment dynamicsof the return deviations for 20 Chilean and Argentine cross-listings. Koumkwa and Susmel (2005)suggest two nonlinear adjustment models: The exponential smooth transition autoregressive(ESTAR) and the logarithmic smooth transition autoregressive (LSTAR) to delineate the relativepremiums of Mexican ADRs. Chung et al. (2005) study the dynamic relationship between the prices ofthree Taiwanese ADRs and their underlying stocks using a threshold VECM.9 For nonparametric esti-mation of the convergence speed parameter of Gagnon and Karolyi (2004) to capture time-varyingnature, Chen et al. (2008) sampled the cross-listed pairs of Asia-Pacific firms and find that marketintegration is positively associated with parity-convergence speed. These articles are devoted to theasset pricing aspect of cross-listings, whereas our novel approaches focus on the price discovery ofmulti-market listings using nonlinear frameworks.

7 Although it does not apply to our sample period from 1998 to 2000, exchanges have begun imposing data-traffic taxes andcharges to the institutional arbitrageurs for their data-heavy, high frequency trades.

8 We define the “relative premium” as the percentage premium of a cross-listed stock traded on a foreign exchange againstthe home market share, adjusted by the exchange rate. The term “cross-listing premium” defined by Doidge et al. (2004) is theexcess value of foreign firms cross-listed in the U.S. relative to those not listed in the U.S, in terms of Tobin’s (1969) q ratio.

9 As a linear modeling precursor, see Kim et al. (2000) for vector autoregressive (VAR) and seemingly unrelated estimation(SURE) methods that analyze adjustments in ADR-implied prices.


We extend Harris et al.’s (1995, 2002) ECM to estimate the exchange-respective information sharesof Canadian cross-listed pairs traded on the NYSE and the TSX10 by first considering threshold coin-tegration according to Balke and Fomby (1997) and then generalizing it to a smooth transition version.Departing from linear modeling, our information share measures are derived from outer-regimeadjustment coefficients based on a two-regime threshold ECM, and average coefficient estimatesbased on a smooth transition ECM. The former method is intuitively appealing whereas the latteramendment risks less model misspecification.

Our alternative methods have many advantages. We theoretically depict and empirically analyzethe discrete dynamics of both abrupt and smooth parity-convergences, which are frequently observedin the market due to risk factors such as information asymmetry and market friction. A large deviationfrom parity, far beyond the threshold, is a very profitable arbitrage opportunity and is more likely toreflect information shocks than a small deviation from parity, which may reflect noise trading.11 Agradual and nonlinear conversion to parity is better detected with our smooth transition ECMframework. Thus, our methods capture the relative contributions to price discovery to a higher degreecompared with existing linear approaches in the literature which circumvent the time- and regime-contingent characteristics of information shocks.

In addition, we identify and explicate the factors that affect estimated information shares andthresholds. Specifically, the empirical implications are as follows: First, the TSX and the NYSE appear tohave integrated over time. Second, parity-convergence accelerates upon discounts on the cross-listingson the NYSE. Third, we find a larger feedback from the NYSE if the price gap exceeds the threshold(required arbitrage return). Fourth, informed traders tend to cluster on the NYSE upon discounts on thecross-listings. Fifth, information share and threshold are affected by the relative degree of privateinformation, market friction and liquidity measures, firm-level characteristics, and aggregate risks.

The remainder of this paper is organized as follows. Section 2 develops econometric models impliedfrom a price discovery model of cross-listings developed in Appendix A 12: The existing standard ECMand our threshold and smooth transition ECMs. Section 3 describes the data, defines the variables, andpresents statistical test and estimation results. Panel regression analyses are discussed in Section 4. Weconclude in Section 5.

2. Error correction models

The equilibrium model constructed in Appendix A captures the measures of market-wise contri-butions to price discovery which are related to the relative populations of respective market partici-pants (Eqs. (A.50) and (A.51)). However, it poses an empirical challenge since the “headcounts” areusually unknown. Fortunately, we can estimate the adjustment coefficients at and bt using the ECM (Eq.(A.48)) which only requires information of market prices. Another hurdle is that the adjustmentcoefficients at and bt are time-varying. In order to estimate the model, additional restrictions arenecessary to characterize the time paths of at and bt. In the following three subsections, by adoptingdifferent assumptions on arbitraguers, the equilibriummodel presented in Appendix A generates threedifferent version of ECMs: standard ECM, threshold ECM, and smooth transition ECM. These modelsprovide various estimates of the adjustment coefficients and, thus, information shares.

10 FollowingEun andSabherwal (2003) andChen andChoi (2012),we choose to studyCanadian stocks listed in theU.S. for severalreasons. First, Canadian equities are the largest groupof stocks cross-listed in theU.S. fromanysingle country. Second,manyof theseCanadian stocks trade actively on both the NYSE and the TSX which is essential for conducting intraday tests. Third, the tradinghours of theTSXcoincidewith thoseof theNYSE (9:30AM–4:00PM,EST). Finally, Canadian stocks trade in theU.S. as ordinary sharesdue to compatible accounting standards, whereas most other cross-listed shares are ADRs issued by U.S. custodian banks.11 A similar idea is illustrated by Gonzalo and Martinez (2006) in a model of the price discovery for stocks traded in a singlemarket. In their model, only new information which implies a profit greater than the transaction cost, measured by bid-askspread, is translated into the transaction price. In other words, the shocks that drive the efficient price component must bebig shocks to the transaction price. The transactions of uninformed agents cannot generate big inefficient changes in transactionprices, because informed traders will arbitrage the situation. Therefore, the shocks driving the pricing error component byuninformed traders must be small.12 We provide a price discovery model of cross-listings to illustrate the role of arbitrageurs in Appendix A. It is a general modelfrom which econometric models with differing assumptions are derived in Section 2.


2.1. Standard error correction model

We begin with a standard ECM, where the adjustment coefficients, at and bt, are assumed to beconstant in Eq. (A.48). All arbitrageurs are homogeneous and the market is perfectly competitive,i.e., there are neither transaction costs nor other market frictions. Under these assumptions, we havebAkt ¼ bA in Eqs. (A.46) and (A.47) for all t and for any arbitrageur k ¼ 1,2,.,N3, with bA > 0. It followsthat

at ¼ N3bAN2

bN2N1 þ N3bAN1 þ N3b

AN2

ha; (1)

bt ¼ N3bAN1

bN2N1 þ N3bAN1 þ N3b

AN2

hb; (2)

which are constant for all t, and

�Dp1tDp2t

�¼

��a; ab;�b

��p1t�1p2t�1

��0@ vt þ ~31t

vt þ ~32t

1A ¼

�a�b

�ðp2t�1 � p1t�1Þ �

0@ vt þ ~31t

vt þ ~32t

1A: (3)

Define the dollar premium on the cross-listing (p2t) on the NYSE against its original listing (p1t, US$-translated) on the TSX as

kthp2t � p1t : (4)

A standard ECM for the bivariate cointegrated system of the cross-listed pair (p1t,p2t) can bestructured as

Dp1t ¼ b10 þ a1kt�1 þXm1

j¼1b11Dp1t�j þ

Xm2

j¼1~b11Dp2t�j; (5)


j¼1b21Dp1t�j þ

Xm2

j¼1~b21Dp2t�j; (6)

where kt�1 gives the remaining cross-listing dollar premium or cointegrating residual. a1 and a2 are theadjustment coefficients of the TSX and the NYSE, respectively: They describe the degree of adjustmentto the deviation to restore the long-run equilibrium in each series. By the Granger representationtheorem (Engle and Granger, 1987; Engle and Yoo, 1987), if p1t and p2t are cointegrated, then at leastone of a1 and a2 must be nonzero. In other words, either p1t or p2t or both, will adjust fractionally torestore parity in the long run.

Harris et al. (1995, 2002) propose to use the ECM adjustment coefficients to estimate the relativeextent of exchange-respective contributions to the price discovery (information share) of shares whoseorder purchases are fragmented across multiple markets. For a Canadian company originally listed onthe TSX and cross-listed on the NYSE, the proportion of the adjustments that takes place on the TSX outof the total adjustments on both exchanges is the contribution share of the home exchange in settingthe long-run equilibrium price resulting from synchronous cross-border stock trading. In the extremecase where there is no feedback from the NYSE, such that a2 ¼ 0, then the NYSE makes no contributionto the price discovery of the cross-listed pair. Eun and Sabherwal (2003) further define the respectiveinformation shares of the TSX and the NYSE as

IS1hja1j

ja1j þ ja2jand IS2h

ja2jja1j þ ja2j

: (7)


Suppose p1t�1 < p2t�1 in the previous period (t� 1), then to reduce the gap between the two priceseither p1t increases or p2t decreases, or both. In this case one can conjecture that a1 is non-positive anda2 is non-negative. There are two other possibilities: (1) p1t�1 decreases but p2t�1 decreases more; or(2) p1t�1 increases but p2t�1 increases less.13 As Eun and Sabherwal (2003) determine that the lattertwo results are very unlikely, they are not considered in our study. A similar adjustment mechanismcan be designed to show that a1 is non-positive and a2 is non-negative for the symmetric situationwhen p1t�1 > p2t�1.

14 Therefore, we define the exchange-respective information shares of the TSX andthe NYSE as

IS1h�a1

�a1 þ a2and IS2h

a2�a1 þ a2

: (8)

2.2. Threshold error correction model

In reality, the market is imperfect due to various sources of market friction such as transaction costs,direct and indirect trading barriers, etc. We let the threshold (g) measure the sum of transaction costsand risk premiums required from arbitrageurs (required arbitrage return).15 Arbitrage opportunitiesexist when

kthp2t � p1t < �g or kt > g; (9)

which becomes jkt j>g. We still assume all arbitragers are homogeneous, such that they sharea common demand elasticity. Under these assumptions, for any arbitraguer k and for all t in Eqs. (A.46)and (A.47) we have

bAkt ¼�0 if jkt j � g

bA > 0 o:w:

�: (10)

Now, cointegration between p1t and p2t is dormant within a range of disequilibrium but the errorcorrection dynamics become active once the cross-listing dollar premium sufficiently diverges fromparity beyond the threshold. Balke and Fomby (1997) propose this regime-switching mechanism asthreshold cointegration, and the implied error correction dynamics can be characterized by a thresholdECM, given by

Dp1t ¼8<:b110 þ a11kt�1 þ

Pm1j¼1b11jDp1t�j þ

Pm2j¼1

~b11jDp2t�j; if jkt�1j � g

b120 þ a12kt�1 þPm1

j¼1b12jDp1t�j þPm2

j¼1~b12jDp2t�j; if jkt�1j > g

9=;; (11)

Dp2t ¼8<:b210 þ a21kit�1 þ


Pm2j¼1


b220 þ a22kit�1 þPm1



9=;: (12)

13 These odds may reflect an under-reaction to the information share of the market. When information incorporation takesmultiple periods, the price adjustment should persist unilaterally during all of them.14 This discussion is based on perfect arbitrage conditions, i.e., there are neither transaction costs nor arbitrage trading risks.However, as this is not necessarily true in practice these restrictions are not imposed in the estimation.15 The threshold as an “effective” break-even price spread depends on explicit and implicit sources of market frictions.According to Gagnon and Karolyi (2010), the observable, explicit sources of market frictions are taxes, transactions costs,holding costs, and short-sale restrictions. Transaction costs of an arbitrage on a diverged cross-listed pair are commissions tohome and host exchanges, bid-ask spreads on the cross-listed pair and the foreign exchange rate, and price impacts on bothcross-border listings. Arbitrageurs incur holding costs while their arbitrage portfolios are active which include lending fees(Barberis and Thaler, 2003), the unrealized interest on short-sales proceeds, the opportunity cost of capital, and firm-level risks(Pontiff, 2006). Temporary and regulatory bans of short sales on the home exchanges of certain ADRs render arbitrageurs’transactions more costly thereby increasing their required returns (Bris et al., 2007). The hidden hurdles to swift parity-convergences of cross-listed pairs may stem from cross-border differential adverse selection risks (Chen and Choi, 2012) andagency walls between the entrusted arbitrageurs and their investor clients (Shleifer and Vishney, 1997).


We allow the beta coefficients to change conditional on the regime since irrational traders in eachmarket, such as trend or contrarian traders, may look at both prices across the markets. When the pricespread is large in the outer regime ðjkt�1j>gÞ, traders may switch to another trading habit such that therelationship between the current price change and past price changes is different. In themiddle regimewhen jkt�1j � g, there are neither market forces nor arbitrageurs to sustain the cointegration of the pairof prices. Unless the pair shows a significant price gap exceeding the threshold minimum profit, theadjustment coefficients are zeroes (a11 ¼ a21 ¼ 0) and, thus, neither price (p1t or p2t) appropriatelyreflects risk. We define the information share, or the relative measure of contribution to pricediscovery, for respective markets using the outer regime coefficient estimates16 (a12 and a22):

IS1hja12j

ja12j þ a22and IS2h

a22ja12j þ a22

: (13)

2.3. Smooth transition error correction model

A common assumption for the standard and threshold ECMs is the homogeneity of arbitrageurs. Wenow relax this assumption to allow for heterogeneous arbitrageurs since they may require differingthresholds (gk’s) to establish their respective positions. For example, using a database of arbitragetransactions on the NYSE, Neal (1996) finds that larger mispricings are positively related to thefrequency of arbitrage porfolio construction. Moreover, fees paid by institutional investors depend onthe arrangement between the investors and their executing brokers. The opportunity costs faced bycapital-constrained arbitrageurs can be another reason for different threshold values: Investors withstricter capital constraints will skip small mispricings to wait for larger ones. Lastly, all traders are notalike in terms of risk aversion and/or risk perception. Overall, a unique break-even point for all arbi-traguers does not exist. Specifically, we assume in Eqs. (A.46) and (A.47) for all t and for any arbitrageurk ¼ 1,2,.,N3,

bAkt ¼�0 if jkt j � gkbAktð$Þ > 0 o:w:

�: (14)

The “aggregate” threshold is a smoothing function of price deviations such that

XN3

k¼1bAkt ¼ N3E

�bAkt

�¼ N3

(Z jkt�1 j

�NbAktdFðgÞ þ

Z N

jkt�1 jbAktdFðgÞ

)hbAðkt�1Þ; (15)

where F(g) is the cumulative density function of gk across all k. Further

at ¼ bAðjkt�1jÞN2

bN2N1 þ bAðjkt�1jÞN1 þ bAðjkt�1jÞN2

ha1ðkt�1Þ; (16)

bt ¼ bAðjkt jÞN1

bN2N1 þ bAðjkt jÞN1 þ bAðjkt jÞN2

ha2ðkt�1Þ: (17)

By inserting at and bt into Eq. (A.45), we obtain a smooth transition ECM written as:

Dp1t ¼ b10 þ a1ðjkt�1jÞkit�1 þXm1

j¼1b1jDp1t�j þ

Xm2

j¼1~b1jDp2t�j; (18)

16 Eun and Sabherwal (2003) estimate the adjustment coefficients in every period using a linear ECM following Harris et al.(1995).


Dp2t ¼ b20 þ a2ðjkt�1jÞkit�1 þXm1

j¼1b2jDp1t�j þ

Xm2

j¼1~b2jDp2t�j: (19)

The “average” information shares of respective markets can be defined as

IS1hjE½a1ðkt�1Þ�j

jE½a1ðkt�1Þ�j þ jE½a2ðkt�1Þ�j; (20)

IS2hjE½a2ðkt�1Þ�j

jE½a1ðkt�1Þ�j þ jE½a2ðkt�1Þ�j; (21)

where

E½a1ðkt�1Þ� ¼ 1T

XT

t¼1a1ðkt�1Þ; (22)

E½a2ðkt�1Þ� ¼ 1T

XT

t¼1a2ðkt�1Þ: (23)

In order to see whether informed traders will choose the market with a relative discount, we definethe “cross” information share of an exchangewhere a discount or premium on the cross-border tradingvenue. We have

IS11hjE½a1ðkt�1ÞIðkt�1 > 0Þ�j

jE½a1ðkt�1ÞIðkt�1 > 0Þ�j þ jE½a2ðkt�1ÞIðkt�1 > 0Þ�j; (24)

IS21hjE½a2ðkt�1ÞIðkt�1 > 0Þ�j

jE½a1ðkt�1ÞIðkt�1 > 0Þ�j þ jE½a2ðkt�1ÞIðkt�1 > 0Þ�j; (25)

IS12hjE½a1ðkt�1ÞIðkt�1 < 0Þ�j

jE½a1ðkt�1ÞIðkt�1 < 0Þ�j þ jE½a2ðkt�1ÞIðkt�1 < 0Þ�j; (26)

IS22hjE½a2ðkt�1ÞIðkt�1 < 0Þ�j

jE½a1ðkt�1ÞIðkt�1 < 0Þ�j þ jE½a2ðkt�1ÞIðkt�1 < 0Þ�j : (27)

For a cross-listed pair, IS11 and IS21 are the information shares of the TSX and the NYSE, respectively,when the TSX-listing trades at a discount (kt�1 > 0); IS12 and IS22 when the cross-listing postsa discount (kt�1 < 0).

3. Data, variables, and preliminary results

3.1. Data

In Table 1, 56 TSX-NYSE pairs are identified through the sample period, January 1, 1998 throughDecember 31, 2000. The fact that nearly half of these firms belong to the basic materials, oil and gasindustries speaks to the rich natural resource endowment in Canada. Specifically,13 corporations are inthe basic materials industry (23.2%), followed by 11 in oil and gas (19.6%), 8 in industrials (14.3%), 8 infinancials (14.3%), 4 in consumer services (7.1%), 3 in technology (5.4%), 3 in telecommunications (5.4%),3 in consumer goods (5.4%), 2 in healthcare (3.6%), and 1 in utilities (1.8%). On a daily basis, an average

Table 1Sample of Canadian firms co-listed on the TSX and the NYSE. Price and return observations are in firm-days for the sampleperiod: January 1, 1998, through December 31, 2000.

Company Industry TSX NYSE No. of obs.

Price (CAD) Return Price (US$) Return

Abitibi-Consolidated, Inc. Industrials 16.27 �0.001 11.01 �0.001 756Advantage Oil & Gas Ltd. Oil & Gas 2.26 �0.020 1.53 �0.004 268Agnico-Eagle Mines Limited Basic Materials 8.62 0.000 5.83 0.000 756Agrium Inc. Basic Materials 15.03 0.000 10.14 0.000 756Alcan Inc. Basic Materials 45.41 0.000 30.63 0.000 756Bank of Nova Scotia Financials 8.81 �0.001 5.95 �0.001 754Barrick Gold Corporation Basic Materials 26.97 0.000 18.17 �0.001 756BCE Inc. Telecommunications 70.36 0.000 47.92 0.000 756Biovail Corporation Health Care 67.89 0.000 46.77 0.000 756BMO Financial Group Financials 62.38 0.000 42.03 0.000 756Brookfield Properties

CorporationFinancials 17.85 0.008 13.16 0.001 426

Cameco Corporation Basic Materials 28.64 �0.001 19.36 �0.001 746Canadian Imperial

Bank of CommerceFinancials 55.03 0.000 37.18 0.000 735

Canadian NationalRailway Company

Industrials 65.88 �0.001 44.97 �0.001 756

Canadian PacificRailway Limited

Industrials 35.92 0.000 24.20 0.000 756

Canwest GlobalCommunications

Technology 19.81 �0.001 13.45 �0.001 721

Celestica Inc. Industrials 65.96 0.002 44.35 0.002 632CGI Group Inc. Technology 25.30 �0.002 16.67 �0.002 662Compton Petroleum

CorporationOil & Gas 38.44 0.001 25.97 �0.003 726

Corus Entertainment, Inc. Consumer Services 39.86 0.002 26.93 0.000 333Cott Corporation Consumer Goods 29.59 0.000 19.99 0.000 734Domtar Corporation Basic Materials 50.35 0.000 34.02 0.000 734Encana Corporation Oil & Gas 6.87 0.001 4.87 0.001 727Energy Metals Corporation Basic Materials 6.76 0.002 4.57 0.001 688Enerplus Resources Fund Oil & Gas 22.01 0.003 14.87 0.008 706Fairfax Financial Holdings

LimitedFinancials 52.00 0.000 35.13 0.000 742

Four Seasons Hotels Inc. Consumer Services 65.06 0.001 43.84 0.001 756Gildan Activewear Inc. Consumer Goods 31.71 �0.032 21.42 0.003 630Goldcorp Inc. Basic Materials 9.13 �0.001 6.58 0.000 260Intertape Polymer Group Inc. Industrials 31.97 �0.001 21.64 �0.001 737IPSCO Inc. Basic Materials 28.67 �0.002 19.52 �0.002 732Kinross Gold Corporation Basic Materials 3.15 �0.002 2.13 �0.003 755Magna International Inc. Consumer Goods 81.85 �0.001 55.15 �0.001 756Manulife Financial Corp. Financials 24.19 0.012 17.09 0.003 333MDS Inc. Health Care 43.05 0.013 29.09 �0.004 269Meridian Gold Inc. Basic Materials 7.68 0.001 5.16 0.001 756Nexen, Inc. Oil & Gas 36.28 0.000 23.65 0.001 34Nortel Networks Corporation Technology 98.89 �0.001 66.81 �0.001 756NOVA Chemicals Corporation Basic Materials 27.45 0.000 18.42 0.000 625Pengrowth Energy Trust Oil & Gas 17.29 0.000 11.68 0.003 735Petro-Canada Oil & Gas 23.53 0.001 15.86 0.001 754Potash Corporation of

Saskatchewan Inc.Basic Materials 90.68 0.000 61.20 0.000 756

Precision Drilling Trust Oil & Gas 33.84 0.001 22.76 0.001 756Quebecor World, Inc. Industrials 35.38 0.001 23.58 0.001 170RBC Financial Group Financials 70.97 �0.001 48.23 �0.001 756Rogers Communications Inc. Telecommunications 24.89 0.002 16.73 0.002 756Shaw Communications Inc. Consumer Services 42.80 0.000 27.54 0.000 631Stantec Inc. Industrials 43.06 0.001 29.09 �0.001 467Suncor Energy Inc. Oil & Gas 49.03 0.000 33.38 0.000 756Talisman Energy Inc. Oil & Gas 40.09 0.000 27.03 0.000 751


Table 1 (continued )

Company Industry TSX NYSE No. of obs.

Price (CAD) Return Price (US$) Return

TELUS Corporation Telecommunications 39.66 0.000 24.58 0.002 163The Thomson Corporation Industrials 17.09 0.001 11.54 0.000 735Tim Hortons Inc. Consumer Services 26.62 �0.001 17.98 �0.001 630Toronto-Dominion Bank Financials 48.29 0.000 33.19 �0.001 756TransAlta Corporation Utilities 4.82 0.000 3.26 �0.002 744TransCanada Corporation Oil & Gas 19.84 �0.001 13.47 �0.001 756

Mean 35.74 0.000 24.13 0.000 649Median 31.84 0.000 21.53 0.000 740Standard deviation 22.84 0.006 15.43 0.002 188


Canadian firm closed at US$24.13 on the NYSE and at CAD36.74 on the TSX with a higher returnvolatility on the home exchange (0.6% > 0.2%). The sample firms in this study were also analyzed byChen and Choi (2012) who argue that, based on 10-min frequency US$-translated prices, an averageNYSE listing earns a relative premium of 30.6 basis points against its TSX counterpart due to a higherhome-market adverse selection risk.

In order to estimate asymmetric information and other market friction measures, high-frequencydata are required for both the shares co-listed on the TSX and the NYSE, and the U.S.–Canadaexchange rate. Accordingly, the tick-by-tick trade and quote data for the TSX-listed Canadian stocks andthe Trade-And-Quote (TAQ) data of their cross-listings on the NYSE through the period are used. Theexchange rate intraday data was purchased from Olson & Associates. Following Eun and Sabherwal(2003) the mid-points of the U.S.–Canada exchange rate bid and ask quotes are updated everyminute, and the bid and ask quotes of the TSX-listed Canadian stocks are matched with theirconcurrent minutes’ exchange rate quotemid-points. Both the TSX and the NYSE trade from 9:30AM to4:00PM, EST, and the US$-denominated price deviations are calculated with simultaneously observedcross-border prices and minute-frequency exchange rates.17 As a result, on an average trading day, weobserve 39 data points for each pair. Our sample period covers 756 trading days, but not all stocks are inpairs throughout the sample period. We require that for each firm-year the prices be observed in thetwo markets for at least 6 consecutive months. Consequently, having dropped thinly traded stocks onboth exchanges, our final sample includes 40 cross-listed pairs and 104 firm-years.

3.2. Cointegration analysis

We first examine whether the pairs of times series on the TSX and the NYSE price series are unitroots We use the augmented Dickey and Fuller’s (1981) (ADF) test, which considers the lagged value ofthe first differences of the time series. If the test statistic is too large, thenwe reject the null hypothesisof the pair being a unit root and conclude that the time series is stationary. Consequently, the nullhypothesis was rejected for only 4 firm-year time series data, at a 5% significance level. Thus, weconclude that both price series in our sample are, overall, first-order integrated (I(1)), or unit units.

We subsequently examine, using Johansen’s (1991) test, cointegration between the two price series.The S&P TSXComposite and the S&P 500 indices (market indices of the TSX and the NYSE, respectively)are not included in the cointegrated system since Eun and Sabherwal (2003) find that the estimatedcoefficients of the two index series are statistically insignificant. Since we have two price series in eachregression equation, there is at most one cointegrating vector. We estimate the cointegrating vector((1,�b)T) for each cross-listed pair in each year. Our results show that most of the estimated

17 During the sample period, the time stamps of trades and quotes on the TSX and the NYSE are recorded per second. In orderto calculate the cross-border price deviations, the trade prices of both listings on the TSX and the NYSE are merged ina chronological order according to the time stamps. The trade prices of the TSX listings are translated to US$-denominatedvalues using the minute-frequency exchange rate. Any missing trade prices are extrapolated using their respective lastpreceding trade prices. As a result, each price deviation is the NYSE trade price less the US$-translated TSX trade price in thesimultaneous observation row, or the US$-denominated dollar premium on the NYSE cross-listing.

Table 2Estimated cointegrating vector. The prices of the sample TSX-NYSE cross-listed pairsare tested for cointegration per Johansen (1991). Since we have only two price seriesin each regression equation in the cointegrated system, there is at most one coin-tegrating vector. We estimate the normalized cointegrating vector (1,�b)T by eachfirm-year, where b is the cointegrating coefficient. Our results show that most of theestimated cointegrating vectors are (1,�1)T, which is of the expected valuesaccording to the law of one price. The t-statistics for the null hypothesis attests thatthe cointegrating vector equals (1,�1)T. The observations are in firm-years.

b t-Statistic

5 %-ile 0.9 �5.2525 %-ile 0.995 �1.29Median 0.999 0.2575 %-ile 1.002 0.9995 %-ile 1.011 2.94


cointegrating vectors are (1,�1)T, which are the expected values according to the law of one price. Table2 reports the summary statistics of the normalized estimation of the cointegrating vector for p1t andp2t. The t-statistics for the null hypothesis confirm that the cointegrating vector equals (1,�1)T.

In Table 2, we see that the median of the normalized estimates throughout the sample is (1,�1)Twhich confirms that the Canadian cross-listed pairs tend to follow the law of one price and are,therefore, cointegrated. Given the estimated cointegrating vector (1,�b)T, the estimated cross-listingdollar premium is ktyp2t � p1t . We, then, test kt’s for stationarity through the ADF test and find thatonly 3 out of 104 firm-years do not reject the null hypothesis of a unit root. Thus, we conclude that theTSX-NYSE cross-listed pairs are cointegrated.

3.3. Nonlinearity tests

The law of one price suggests that two market prices for the same stock should not drift far fromeach other. This relationship is confirmed by the cointegration analysis in Section 3.2. However, linearadjustment dynamics are not necessarily prescribed by market efficiency assumptions. Given variousmarket frictions, such as transactions costs and short-sale limitations, it is more likely that a nonlinearmodel, such as a threshold or smooth transition cointegration model, provides a better description ofthe convergence of two market prices. In this section, we conduct several nonlinearity tests todetermine the relationship of short-run adjustment dynamics to long-run parity equilibrium.

We estimate the symmetric bivariate threshold ECMmodel in Section 2.2 and apply the supremum-Lagrangian multiplier (supLM) test to check for nonlinearity. This test also has power to detect smoothtransition error correction dynamics as shown in Hansen and Seo (2002). We use Akaike’s (1974) andSchwarz’s (1978) Bayesian information criteria to choose the number of lags, and consistently choosea lag length of 1 (m1 ¼ m2 ¼ 1). The cointegrating vector is given as (1,�1)T, after performing cointe-gration tests. Themodel is estimated by themaximum likelihoodmethod described in Appendix B. Theestimation results are reported in Table 3.

Panel A of Table 3 exhibits the summary statistics of the threshold estimates and test statistics. The p-valuesare computedby theparametricbootstrapmethodsuggestedbyHansenandSeo (2002).Wefind thatthemeanandmedianof the supLM for thewhole sample are73.52 and31.05,whichexceeds the 95% criticalvalues of 24.78 and 23.10. Therefore, on average, we can reject the null hypothesis of no threshold effect.

To further confirm the test results, we apply a combined p-value test on all firm-years. Let pi be theasymptotic p-value of the supLM test for each individual stock-year i, for i ¼ 1, 2,.,N, where N is thetotal number of firm-years. We combine all p-values (pi’s) to construct the Z-test statistic proposed byChoi (2001), and have:

Zh1

2ffiffiffiffiN

pXN

i¼1f � 2lnðpiÞ � 2g; (28)

which is asymptotically standard normal under the null hypothesis of no threshold effect. In ouruntabulated case, the combined p-value test statistic Z is 33.41, significantly rejecting the null

Table 3Nonlinearity tests. We estimate the threshold (required cross-border arbitrage return) per our threshhold ECM frameworkfollowing Balke and Fomby (1997) and extended from Harris et al. (1995, 2002). The threshold effect (supremum Lagrangianmultiplier) test statistic is supLM and is estimated per Hansen and Seo (2002). In order to examine whether the threshold effecthappens on the coefficients of the error correction term or short-term dynamic term in the threshold ECM, WaldECM1 andWaldECM2 are theWald test statistics for the null hypotheses of “no threshold effect” on the error correction terms for the TSX andNYSE listings, respectively. WaldDC1 and WaldDC2 are the Wald test statistics for the null hypotheses of “no threshold effect” onthe short-term dynamic terms for the TSX and NYSE listings, respectively.

Threshold Outer regime, %-age supLM 95%-ile critical value. p-value

Panel A: Threshold estimates and supLM test statisticsMean 0.193 0.170 73.516 24.783 0.146St. Dev. 0.159 0.162 110.814 3.562 0.249

1%-ile 0.010 0.102 10.290 17.861 0.00010%-ile 0.059 0.103 14.142 21.113 0.00025%-ile 0.100 0.103 19.664 21.920 0.00050%-ile 0.157 0.108 31.054 23.101 0.00875%-ile 0.242 0.143 56.139 28.234 0.14290%-ile 0.319 0.279 242.699 29.078 0.60999%-ile 0.808 0.887 509.250 30.407 0.818

WaldECM1 p-Value WaldECM2 p-Value WaldDC1 p-Value WaldDC2 p-Value

Panel B: Wald statisticsMean 10.118 0.265 32.723 0.260 13.330 0.265 9.349 0.246St. Dev. 24.471 0.312 87.764 0.324 70.767 0.283 14.630 0.281

1%-ile 0.000 0.000 0.000 0.000 0.158 0.000 0.183 0.00010%-ile 0.099 0.000 0.054 0.000 0.796 0.002 1.162 0.00125%-ile 0.440 0.005 0.336 0.000 2.012 0.033 2.222 0.02850%-ile 2.637 0.104 3.193 0.074 5.676 0.167 5.554 0.10875%-ile 8.068 0.507 16.106 0.563 9.679 0.428 9.636 0.40090%-ile 15.839 0.755 75.568 0.818 13.810 0.723 14.601 0.75999%-ile 132.409 0.984 487.706 0.986 28.317 0.981 62.062 0.920


hypothesis with a 5% critical value at 1.96. Overall, we conclude that nonlinearity exists in the course ofparity convergence.18

Of interest is whether the threshold effect takes place on the coefficients of error correction terms oron the short-term dynamics terms in the threshold ECM of Eqs. (11) and (12). We separately test thethreshold effect on these coefficients. Panel B of Table 3 reports the test results. The first two columns(WaldECM1) report the Wald statistics for testing the null hypothesis on the error correction terms: H0:a11¼ a12. In otherwords,we testwhether the adjustment coefficients associatedwith the prices of TSX-listings (p1t’s) are different within and beyond the threshold. Likewise, the third and fourth columns(WaldECM2) test the samehypothesis for theNYSE-listings:H0: a21¼ a22. The last four columns (WaldDC1andWaldDC2) report theWald statistics for the null hypotheses of no threshold effect for the TSX and theNYSE, respectively: H0 : b111 ¼ b121;

~b111 ¼ ~b121; and H0 : b211 ¼ b221;~b211 ¼ ~b221: Rejecting these

null hypotheses means accepting the threshold effect occurs on the short-term dynamics terms in thethreshold ECM of the TSX and the NYSE, respectively. The untabulated, Z-test statistics (Choi, 2001) ofthe null hypotheses of error correction coefficients are 16.80 and 19.68, respectively, while those of theshort-term dynamics coefficients are 13.95 and 14.44, respectively. Thus, for both exchanges, weconclude that there exists nonlinearity in both the error correction and the short-term dynamics terms.

3.4. Estimation

3.4.1. Microstructure measuresUnlike the NYSE, which is a specialist-based auction exchange, the TSX is an electronic exchange,

which uses a central limit order book (CLOB) system, where orders are required to be posted in the

18 We also repeated the estimation and testing procedures when the cointegrating vector is estimated from the data ratherthan restricted to. The results are qualitatively equivalent and the conclusion remains unaffected.

Table 4Representative statistics of key independent variables. PIN is the probability of information-based trading per Easley et al. (1996).Spread is the relative quoted spread: bid-ask spread divided by the quoted mid point. Volume is the total daily trading volume interms of quantity. DollarVol is the total daily trading volume in terms of value.

Variable Exchange Mean Median St. Dev. Firm-years Hypotheses p-Value

PIN TSX 0.242 0.213 0.107 104 H0: PINTSX � PINNYSE 0.001NYSE 0.214 0.202 0.060 104 H1: PINTSX > PINNYSE

Spread TSX 0.015 0.007 0.025 104 H0: SpreadTSX � SpreadNYSE 0.000NYSE 0.022 0.015 0.022 104 H1: SpreadTSX < SpreadNYSE

Volume (�1000) TSX 576.458 272.687 937.816 104 H0: VolumeTSX � VolumeNYSE 0.000NYSE 276.955 59.472 847.500 104 H1: VolumeTSX > VolumeNYSE

Dollar Vol. (�$106) TSX 17.236 5.543 43.032 104 H0: DollarVolTSX � DollarVolNYSE 0.001NYSE 11.987 1.153 53.373 104 H1: DollarVolTSX > DollarVolNYSE


book to be valid.19 By studying decrements in the inside depth on one side of the quote that correspondto uncommon trade sizes (such as a trade of 1300 shares), matching trades with prevailing quotes witha 5-s lead (Lee and Ready, 1991) is reasonable. Given a mid-quotemade 5 s earlier, a trade is consideredbuyer-initiated if it is higher, and seller-initiated if lower.20 We construct datasets to estimate theprobability of informed trading (PIN) following Easley et al. (1996, 2002). The NYSE-resident specialistsare central to the theory of the PIN (Easley et al., 2001; Duarte and Young, 2009). There are officialmarket makers, known as registered traders, on the TSX whose function is akin to that of the NYSEspecialists. Thus, a comparison of trade informedness on the two exchanges by the PIN is deemedappropriate.21

The PINs for TSX and NYSE-listed Canadian stocks are estimated following Easley et al. (1996,1997a,b). Further, we adopt Easley et al.’s (2008) log-likelihood function specification for improvednumerical stability in PIN estimation. The bid-ask spreads are adjusted by the mid-quotes and, thus,measure the relative discrepancy between bid and ask quotes free from the exchange rate. FollowingEun and Sabherwal (2003), the mid-points of U.S.–Canada exchange rate bid and ask quotes areupdated every minute. The bid and ask quotes of the NYSE-listed Canadian stocks are matched withtheir concurrent minutes’ exchange rate quote mid-points.

Table 4 summarizes the exchange-wise estimates and cross-exchange difference tests (Wilcoxon,1945) of the PIN, relative quoted spread, daily trading volume, and daily trading dollar volume ofour sample cross-listed pairs. First, on average, the PIN on the TSX (0.242) exceeds that on the NYSE(0.214). Second, the relative quoted spread on the TSX (0.015) is narrower than that on the NYSE(0.022). Third, for a Canadian cross-listed pair, on average, it appears that the intensity of informedtrades tends to be heavier (a higher PIN) with a lower spread (competitive market making) on theTSX.22 The TSX dominates the NYSE in trading volume in terms of both quantity and value. Thestatistical significances of comparisons are well explained by near-zero p-values for all differencetests.23

19 We owe this comment to Daniel Weaver. See Eun and Sabherwal (2003) for a detailed institutional comparison between theTSX and the NYSE.20 See Schultz and Shive (2008) for trade misclassification of the TAQ on the NYSE, which becomes severe after 2000.21 We owe this comment to Lawrence Kryzanowski. See Fuller et al. (2008) for difficulties in estimating the PIN for Nasdaqtrades.22 In the cross-section, the PIN is positively correlated with bid-ask spread according to Easley et al. (2002). However, thenegative relationship between the two estimates shown in Table 4 is due to averaging and aggregation.23 Based on, unreported, 10-min frequency relative premiums of the cross-listed pairs traded throughout the sample period,the arithmetic mean, the median, and the standard deviation are 0.00306, 0.00004, and 0.03031, respectively. The averagerelative premium of 30.6 basis points with a 3.03% volatility is a statistically insignificant deviation from parity. This suggeststhe extent to which Toronto and New York are integrated.

Table 5Delta estimates of the NYSE. According to the smooth transition ECM, the Delta of a cross-listed pair is the convergence speedparameter which measures the reciprocal speed of converenge to parity. DeltaPrem and DeltaDisc are the convergence speedparameters given premiums and discounts on NYSE cross-listings, respectively.

Delta (d) DeltaPrem DeltaDisc

Panel A: Representative statisticsMean 0.669 0.688 0.652St. Dev. 0.105 0.133 0.123

1% 0.495 0.446 0.43410% 0.550 0.537 0.53225% 0.585 0.603 0.58050% 0.654 0.661 0.64175% 0.734 0.760 0.72290% 0.827 0.883 0.81499% 0.897 1.000 0.894

Delta (d) DeltaPrem DeltaDisc

Mean Median Mean Median Mean Median

Panel B: Annual mean estimates1998 0.709 0.709 0.729 0.724 0.701 0.7131999 0.653 0.653 0.668 0.644 0.641 0.6162000 0.650 0.650 0.674 0.616 0.619 0.620

Hypothesis Wilcoxon stat. p-Value

Panel C: Wilcoxon signed rank testH0: DeltaDisc � DeltaPrem 1312 6.446� 10�05

H1: DeltaDisc < DeltaPrem


3.4.2. Estimation of the threshold and convergence speed parameterIn our threshold ECM, the threshold (g), on average, measures the sum of transaction costs and risk

premiums of a cross-border arbitrage. The first column (Threshold) in Panel A of Table 3 reports thesummary statistics of the estimated thresholds, which range from 0.01 to 0.81, with a mean of 0.193:On average, when the cross-listing dollar premium/discount records more than �19.3 cents, respec-tively, arbitrageurs begin to take positions on both sides and drive the deviation back into the “no-arbitrage” band. The percentage of data points falling into the outer regime ranges from 10.24% to88.69%, with a mean of 17.0% (second column in Panel A of Table 3).

According to our smooth transition ECM, the convergence speed parameter (d), on average,measures the reciprocal speed of converenge to parity of a cross-listed pair (Section 2.3).24 The firstcolumn (Delta) in Panel A of Table 5 reports the summary statistics of the estimates of convergencespeed parameters with a mean of 0.669 and a median of 0.654. To see how the convergence speed isaffected by the price deviation, the second and third columns report the results for d when there isa premium (DeltaPrem) or discount (DeltaDisc) on the NYSE-listing. Panel B reports the trends of d andshows downward trends in both the mean and the median of d, which suggest that the NYSE and theTSX have integrated over time. In Panel C we apply the Wilcoxon signed rank test to test the nullhypothesis in Panel C: H0: DeltaDisc � DeltaPrem. The p-value is smaller than 0.01, thus, we can reject thenull hypothesis: The convergence between the two market prices accelerates when there is a relativediscount on the NYSE.25 A possible explanation is that arbitrageurs like to establish short positions onthe TSX since stocks are likely to be more liquid in the home market (Table 4).26

As an illustrative case, we take Abitibi-Consolidated, Inc. (TSX ticker: A; NYSE ticker: ABY) as anexample and report its estimation results, with the linear and threshold ECMs for the sample years of1998,1999, and 2000, in Panels A and B of Table 6, respectively. In Panel B, the linearity hypothesis of the

24 The smooth-transition model is estimated nonparametrically without any assumption on the functional form.25 Note that as d decreases, convergence accelerates.26 We cautiously view this interpretation as valid since the proportion of discounted cross-listings (p2 < p1) is 50.6% usingobservations at 10-min intervals throughout the sample period.

Table 6Estimation of error correction models for Abitibi-Consolidated, Inc.In Panel A, the linear ECM is given by


j¼1b1jDp1t�j þ

Xm2

j¼1~b1jDp2t�j;


j¼1b2jDp1t�j þ

Xm2

j¼1~b2jDp2t�j:

In Panel B, the threshold ECM is given by

Dp1t ¼8<:b110 þ a11kt�1 þ


Pm2j¼1


b120 þ a12kt�1 þPm1



9=;;

Dp2t ¼8<:b210 þ a21kit�1 þ


Pm2j¼1


b220 þ a22kit�1 þPm1



9=;:

The lag lengths of both models (m1 andm2) are both selected as 1 by Akaike’s (1974) and Schwarz’s (1978) Bayesian informationcriteria.

1998 1999 2000

Estimate St. error Estimate St. error Estimate St. error

Panel A: Linear ECMa1 �0.262 0.011 �0.251 0.013 �0.352 0.013b10 0.000 0.000 0.002 0.001 0.003 0.001b11 0.011 0.016 �0.040 0.014 �0.054 0.016~b11 0.058 0.016 0.048 0.016 0.061 0.020a2 0.072 0.013 0.130 0.016 0.079 0.012b20 �0.002 0.001 �0.001 0.001 �0.001 0.001b21 0.144 0.020 0.069 0.017 0.057 0.018~b21 �0.144 0.019 �0.096 0.019 �0.077 0.022

1998 1999 2000

Panel B: Threshold ECMThreshold (g)Estimate 0.077 0.093 0.085supremum-Wald 25.27 30.44 22.5295% Crit. value 22.83 23.79 23.01

1998 1999 2000


Coefficientsa11 �0.229 0.014 �0.289 0.014 �0.341 0.014b110 0.000 0.000 0.002 0.001 0.003 0.001b111 0.001 0.014 �0.024 0.014 �0.051 0.015~b111 0.023 0.015 0.028 0.016 0.037 0.020

a12 �0.259 0.018 �0.191 0.023 �0.340 0.025b120 �0.001 0.001 �0.001 0.002 0.002 0.002b121 0.035 0.045 �0.116 0.048 �0.064 0.038~b121 0.119 0.035 0.122 0.042 0.143 0.054

a21 0.067 0.016 0.058 0.015 0.059 0.014b210 �0.001 0.001 0.000 0.001 �0.001 0.001



1998 1999 2000


b211 0.130 0.019 0.089 0.017 0.060 0.017~b211 �0.171 0.019 �0.108 0.018 �0.088 0.022

a22 0.093 0.019 0.223 0.033 0.114 0.027b220 �0.004 0.002 �0.007 0.003 0.000 0.002b221 0.181 0.050 �0.044 0.054 0.047 0.048~b221 �0.090 0.045 �0.044 0.055 �0.030 0.056

Fig. 1. Adjustment coefficient functions of smooth transition ECM. The adjustment coefficient functions of a smooth transition ECMfor the cross-listed pair of Abitibi-Consolidated, Inc (TSX ticker: A; NYSE ticker: ABY) are plotted in the respective sample years of1998, 1999, and 2000.


ECM is tested using the supremum-Wald statistic suggested by Hansen (1996). According to Andrews(1991) and Hansen (1996), this test has the power to reject the null hypothesis for both threshold andsmooth nonlinear effects.27 The cross-listed pairs of Abiti-Consolidated, Inc. do not converge linearly asthe supremum-Wald statistics exceed their respective 95 percent critical values in 1998 and 1999. Therejection of the nonlinearity hypothesis in 2000 is shown in Fig. 1 which plots the adjustment coeffi-cient functions of the smooth transition ECM for A and ABY prices in the respective sample years of

27 Since the threshold ECM is discrete and the smooth transition ECM is continuous, these ECMs are not nested. Thus, thestandard likelihood ratio test is not applicable.

Table 7Information shares of the NYSE. The information share of the NYSE is a relative measure of the contribution made by the NYSE tothe price discovery of TSX-NYSE cross-listed pairs. IS is the linear information share following Harris et al.’s (1995, 2002)standard ECM. Per threshold ECM, ISIn and ISOut are the inner and outer-regime information shares of the NYSE. Per the smoothtransition ECM, ISST, ISPrem, and ISDisc are the information shares of NYSE using the whole sample, and given premiums anddiscounts on cross-listings, respectively.

Linear ECM Threshold ECM Smooth transition ECM

IS ISIn ISOut ISST ISPrem ISDisc

Panel A: Summary statisticsMean 0.430 0.362 0.435 0.374 0.386 0.379St. dev. 0.258 0.239 0.259 0.254 0.253 0.264

1%-ile 0.030 0.017 0.020 0.001 0.012 0.00310%-ile 0.087 0.073 0.106 0.059 0.067 0.06825%-ile 0.215 0.138 0.215 0.173 0.177 0.16150%-ile 0.416 0.358 0.418 0.352 0.369 0.36075%-ile 0.601 0.543 0.626 0.543 0.554 0.53690%-ile 0.816 0.669 0.804 0.707 0.739 0.79799%-ile 0.948 0.910 0.980 0.946 0.934 0.981

Linear ECM Threshold ECM Smooth transition ECM

IS ISIn ISOut ISST ISPrem ISDisc

Panel B: Annual mean estimates1998 0.393 0.367 0.386 0.386 0.413 0.3811999 0.484 0.368 0.514 0.382 0.378 0.4012000 0.410 0.352 0.442 0.357 0.350 0.374

Hypothesis Wilcoxon stat. p-value

Panel C: Wilcoxon signed rank test of smooth transition information shareH0: ISDisc � ISPrem 2877 0.036H1: ISDisc > ISPrem


1998,1999, and 2000. Of these, the adjustment coefficient curve of the cross-listing on theNYSE in 2000is nearly linear with a downward slope while the other graphs show nonlinear patterns.

3.4.3. Estimation of the information shareGiven a cross-listed pair, the information share of an exchange measures its contribution to price

discovery. We estimate information shares using the three ECMs described in Section 2. The firstcolumn in Panel A of Table 7 reports the estimated information share of the NYSE from the standard,linear ECM. With a shorter sample period (February–July, 1998) Eun and Sabherwal’s (2003) infor-mation share estimates of the NYSE range from 0.2% to 98.2%, with a mean of 38.1% in the cross-sectionof sample firms. They conclude that price discovery for most cross-listed pairs occurs in the TSX, butthat there is significant feedback from the NYSE. Our results, based on a longer sample period, areconsistent: The estimated linear information shares (IS) of the NYSE range from 3% to 94.5%, witha mean of 42.99%. There is no discernible trend over the sample period as the yearly average estimatesof the linear information share of the NYSE in 1998, 1999, and 2000 are 39.3%, 48.4%, and 41%,respectively (first column in Panel B of Table 7).

However, the linear ECM ignores nonlinearity in the course of parity-convergence, as shown inSection 3.3, and this suggests that the estimates from the linear ECM may be biased. Accordingly, weestimate the information shares via both the threshold and the smooth transition ECMs.

The second and third columns in Panel A of Table 7 report the results for the bivariate threshold ECMpresented in Section 2.2. The information share estimates of the NYSE in the inner regime (ISIn) rangefrom 1.7% to 91%, with an average of 36.2%, while in the outer regime (ISOut) they range from 2% to 98%,with an average of 43.5%.28 Thus the NYSE makes a larger contribution to price discovery in the outer

28 Remember that the range of the proportion of outer regime is from 10.2% to 88.7% with a mean of 17%.


regime. Arbitrageurs tend to engage in the market when price deviations are sizable, and their arbi-trage activities transfer information from the home market to the NYSE (Fremault, 1991).

The last three columns in Panel A of Table 7 report the results from the smooth transition ECM.There are three information shares: ISST, ISPrem, and ISDisc, which are the information shares defined onthe whole sample, sample with premiums, and sample with discounts on the NYSE cross-listings,respectively. The means for these three information shares are 37.4%, 38.6%, and 37.9%, respectively.In Panel C, we apply the Wilcoxon signed rank test to examine the null hypothesis H0:ISDisc � ISPrem.The p-value of the test is 0.036, which significantly rejects the null hypothesis. In other words, whenthe cross-listing trades with a discount on the NYSE, the information share of the NYSE is larger thanthat of the TSX, suggesting that, when the cross-listing is underpriced, informed traders choose to tradeon the NYSE rather than the home listing.

4. Main results

Our data and variables described and estimated in Section 3 are framed to support our claimsconcerning the dynamics of arbitrage across neither instantly efficient nor fully integrated cross-borderstockmarkets in Toronto and New York City. Some of the proposed and estimated explanatory variablesattempt to proxy for the conspicuous and the latent obstacles to seamless arbitrage transactions. If no-arbitrage price relations are achieved sufficiently quickly, then the conventional linear measurementmethodology of the contribution share of price discovery should not be outperformed by our nonlinear,threhold and smooth transition propositions. Practical and regulatory impediments to arbitrage will“delay”market efficiency. The transition tomarket efficiency is expected to be either abrubt, gradual, orboth. Using the obtained estimates and controls, we now empirically identify the possible factorsdetermining the scope of market influences on the price discovery of the sample cross-listed pairs andthe effective required returns to arbitrageurs.

4.1. Panel dataset construction

Based on the estimation results in Section 3.4, we construct a panel data for regression analyses ofthe estimates of information shares and thresholds with columns for dependent variables, explanatoryvariables, and control variables. Symbol is the NYSE ticker of a TSX-NYSE cross-listed pair. Year is theyear index of an estimated value.

� Dependent variables. IsOut and IsIn are the outer-regime and inner-regime information shares ofthe NYSE in a threshold ECM. IsLin is the information share estimate of the NYSE as shown in Harriset al. (1995, 2002). Threshold is the US$-denominated threshold estimate. IsSt, IsPrem, and IsDiscare the information shares of the NYSE in a smooth transition ECM for thewhole sample, thewholesample given premiums, and the whole sample given discounts on cross-listings, respectively.

� Key explanatory variables. The PIN captures the informativeness of a listing. We use estimates of thePINs of both listings of a cross-listed pair to proxy for their relative and average degree of efficientinformation. Since informed traders foster price discovery, the information share of an exchange isexpected to be larger relative to its cross-border counterpart, the higher the PIN of the listingcompared to that of its cross-listing. PinRatio is the ratio of the PIN of the NYSE over that of the TSX.PinAvg is the average PIN of the NYSE and the TSX. PinDiff is the PIN of the NYSE less that of theTSX. We also estimate the relative quoted spread measures on both exchanges to proxy for theirrespective degrees of market friction. The threshold (effective required return of cross-borderarbitrage) of a cross-listed pair is expected to be positively associated with a bid-ask spreadmeasure. SpreadRatio is the ratio of the relative quoted bid-ask spread of the NYSE over that of theTSX. SpreadAvg is the average relative quoted bid-ask spread of NYSE and the TSX. SpreadDiff is thedifference of the quoted bid-ask spread of the NYSE over that of the TSX.

� Control variables. UsVol is the average daily trading volume of the NYSE out of both the NYSE andthe TSX following Eun and Sabherwal (2003). VolAvg is the average of the log-transformations ofthe average daily trading volumemeasures of the NYSE and the TSX. VolDiff is the difference of thelog-transformation of the average daily trading volume of the NYSE over that of the TSX.

Table 8Panel regression results of threshold and linear ECM-implied information shares. The dependent variables of Panels A, B, and C are IsOut which is the outer-regime information share of theNYSE, a relative measure of contribution made by the NYSE to the price discovery of TSX-NYSE cross-listed pairs; IsIn which is the inner-regime information share of the NYSE; and IsLinwhich is the linear information share (Harris et al., 1995, 2002). Explanatory variables: PinRatio is the ratio of the PIN of the NYSE over that of the TSX. SpreadRatio is the ratio of the relativequoted bid-ask spread of the NYSE over that of the TSX. UsVol is the average daily trading volume of the NYSE out of both of the NYSE and the TSX following Eun and Sabherwal (2003).UsDollarVol is the average daily dollar trading volume of the NYSE out of both of the NYSE and the TSX. Control variables: Governance is the report on Business governance index of Canadianfirms published by Globe andMail (McFarland, 2002). Industry equals one if the cross-lister is a manufacturing firm, and zero otherwise. Size is the normalized averagemarket capitalizationon the TSX and the NYSE. VolatDiff is the difference of the U.S. and Canada’s market index return volatilities. Vix is the CBOE Volatility Index. The t-statistics of coefficient estimates aresuppressed for the lack of space. ***, **, and * stand for statistical significance based on two-sided tests at the 1%, 5%, and 10% level, respectively. The observations are in firm-years. All modelspecifications are controlled for fixed and year effects.

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10

Panel A: Outer-regime information sharesIntercept 0.651 *** 0.702 *** 0.632 *** 0.683 *** 0.262 *** 0.307 *** 0.206 *** 0.242 *** 0.008 * 0.011 *

PinRatio 0.127 ** 0.122 ** 0.133 ** 0.127 ** 0.151 *** 0.136 ** 0.179 *** 0.168 *** 0.096 ** 0.100 **

SpreadRatio 0.001 0.002 0.002 0.002 0.000 �0.001 �0.002 �0.002 0.000 0.000UsVol 0.386 *** 0.358 *** 0.414 *** 0.454 *** 0.104 *

UsDollarVol 0.300 *** 0.277 *** 0.282 *** 0.336 *** �0.026Industry �0.054 �0.050Governance �0.005 *** �0.005 *** �0.005 *** �0.005 ***

Size �0.390 ** �0.403 ** �0.353 ** �0.368 ** �0.443 ** �0.473 **

VolatDiff 19.048 *** 19.573 ***

Vix �0.018 �0.021

No. of Obs. 104 104 104 104 104 104 104 104 104 104Adjusted R2 0.277 0.252 0.273 0.249 0.207 0.154 0.203 0.144 0.240 0.200

Panel B: Inner-regime information sharesIntercept �0.027 �0.026 �0.027 �0.026 �0.020 �0.018 �0.022 * �0.021 �0.069 *** �0.068 ***

PinRatio �0.015 �0.016 �0.015 �0.016 �0.007 �0.008 0.008 0.007 0.033 0.032SpreadRatio 0.000 0.000 0.000 0.000 0.000 0.000 �0.001 �0.001 �0.066 *** �0.066 ***

UsVol 0.225 0.225 0.200 0.234 * 0.302UsDollarVol 0.222 0.222 0.213 0.247 * 0.324Industry 0.000 �0.001Governance 0.000 0.000 0.000 0.000Size �0.030 �0.036 �0.029 �0.035 �0.033 �0.036VolatDiff �35.897 *** �35.747 ***

Vix 0.054 0.054


Panel C: Linear information sharesIntercept 0.015 0.014 0.013 0.011 0.014 0.014 0.019 0.019 �0.069 *** �0.068 ***

PinRatio 0.049 0.055 0.049 0.055 0.052 0.057 0.065 0.071 0.033 0.032

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SpreadRatio 0.002 0.001 0.002 0.001 0.002 0.001 0.001 0.001 �0.066 *** �0.066 ***

UsVol �0.153 �0.151 �0.128 �0.034 0.302UsDollarVol �0.350 �0.348 �0.330 �0.189 0.324Industry �0.004 �0.005Governance 0.000 0.000 0.000 0.000Size �0.030 �0.045 �0.025 �0.040 �0.019 �0.034VolatDiff �35.897 *** �35.747 ***

Vix 0.054 0.054

No. of Obs. 104 104 104 104 104 104 104 104 104 104Adjusted R2 �0.014 0.006 �0.005 0.014 0.010 0.029 0.017 0.025 0.185 0.188

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UsDollarVol is the average daily dollar trading volume of the NYSE out of both the NYSE and theTSX. DollarVolAvg is the sum of log-transformations of the average daily dollar trading volumemeasures of the NYSE and the TSX. DollarVolDiff is the difference of the log-transformation of theaverage daily dollar trading volume of the NYSE over that of the TSX. Governance is the Report onBusiness index on the governance of Canadian firms published by the Globe and Mail (McFarland,2002). Industry equals one if the cross-lister is a manufacturing firm, and zero otherwise. Webelieve the governance risk of a Canadian firm is reflected in the threshold as a risk premium. Sizeis the normalized average market capitalization on the TSX and the NYSE. VolatAvg and VolatDiffare the average and difference of U.S. (S&P 500) and Canada’s (TSX 100) USD-denominated marketindex return volatilities, respectively, where Canada’s market return volatility includes theexchange rate volatility. Vix is the Chicago Board Options Exchage (CBOE) implied volatility of theS&P 500 stock index options.

4.2. Panel regression analyses

Figuerola-Ferretti and Gonzalo (2010) model and measure price discovery on the New YorkMercantile Exchange (NYMEX) and the International Petroleum Exchange (IPE). The two contractprices comove relatively closely, but transportation costs and grade differences pose difficulties indetermining arbitrage opportunities. They investigate two interesting questions: How does arbitrageensure adjustment to the long-run path given location and grade differences; andwhich of themarketsis the market leader, or the most important contributor to price discovery?

4.2.1. Regression of information shares per threshold cointegrationWe conduct regression analyses on the constructed panel data to identify the factors that affect the

relative extent of the contribution of the NYSE to price discovery. The estimated outer-regime infor-mation shares are regressed onto the panel of explanatory and control variables and reported in PanelA of Table 8, respectively. The contribution of the NYSE increases relative to that of the TSX as NYSE-based trades become more informative (a higher PIN). This is cross-border evidence that informedtrades contribute to fostering price discovery, in line with Chen and Choi (2012). Either in quantity orvalue, the higher the liquidity on the NYSE, the more it leads in price discovery. This is consistent withEun and Sabherwal’s (2003) findings where they estimate the information share of the NYSE by usingHarris et al.’s (1995, 2002) approach. They find that information share is directly related to the U.S.’sshare of total trading (UsVol), the proportion of informative trades on U.S. exchanges and the TSX(confirmed as proxied for by the PIN), and inversely related to the ratio of bid-ask spreads on U.S.exchanges and the TSX, which is not discernable.29 A better investor-protecting (Governance) andlarger (Size) Canadian firm tends to lead price setting on the TSX as seen inModels 1 through 4. Models9 and 10, controlling for the cross-border differential volatilities of market returns and foreignexchange rate (VolatDiff) and the market fear factor (Vix) in addition to the liquidity and friction oftrades on both exchanges, re-affirm the information story (PinRatio) of market contribution to pricediscovery shown in Models 1 through 8.

We conduct analoguous panel regressions for the inner-regime and linear information shares inPanels B and C of Table 8, respectively. Neither alternative measure of exchange-specific contribution toprice discovery has a higher explanatory power (adjusted R2) and economically and statisticallymeaningful implications. To this end, the outer-regime information shares (Table 8 Panel A) prove to benot only heuristically appealing but also economically reasonable and statistically robust.

4.2.2. Regression of the estimated thresholdFor each cross-listed pair, the threshold includes transactions costs consisting of bid-ask price

spreads on both exchanges and the foreign exchange rate, fixed costs, and liquidity shorfalls. Implicit

29 Hasbrouck (1995) finds a positive and significant correlation between contribution to price discovery made by the NYSEand its market share by trading volume using U.S. domestic data. Using the same data, Harris et al. (2002) find that informationshare increases when its bid-ask spread declines relative to the regional exchange.

Table 9Panel regression results of threshold values. The dependent variable is threshold which is the US$-denominated threshold estimate. Explanatory variables: PinDiff is the difference of the PINof the NYSE over that of the TSX. PinAvg is the average PIN of the NYSE and the TSX. SpreadDiff is the difference of the quoted bid-ask spread of the NYSE over that of the TSX. SpreadAvg isthe average relative quoted bid-ask spread of NYSE and the TSX. Control variables: VolAvg is the average of the log-transformations of the average daily trading volumemeasures of the NYSEand the TSX. VolDiff is the difference of the log-transformation of the average daily trading volume of the NYSE over that of the TSX. DollarVolAvg is the sum of log-transformations ofaverage daily dollar trading volume measures of the NYSE and the TSX. DollarVolDiff is the difference of the log-transformation of the average daily dollar trading volume of the NYSE overthat of the TSX. Governance is the Report on Business governance index of Canadian firms published by Globe and Mail (McFarland, 2002). Industry equals one if the cross-lister isa manufacturing firm, and zero otherwise. Size is the normalized average market capitalization on the TSX and the NYSE. VolatAvg and VolatDiff are the average and difference of U.S. andCanada’s market index return volatilities, respectively. Vix is the CBOE Volatility Index. The t-statistics of the coefficient estimates are suppresed for the lack of space. ***, **, and * stand forstatistical significance based on two-sided tests at the 1%, 5%, and 10% level, respectively. The observations are in firm-years. All model specifications are controlled for fixed and year effects.


Panel A: Regressions onto average measuresIntercept 1.275 2.488 * 1.085 2.591 * 0.373 1.852 0.625 1.742 * 0.029 0.029PinAvg �1.419 �2.152 �1.087 �2.082 �0.053 �0.945 �0.410 �1.131 �0.033 �0.014SpreadAvg 15.217 *** 11.387 * 15.419 *** 11.923 * 3.959 0.782 2.789 �0.214 �2.796 �3.070VolAvg 0.003 0.032 0.024 0.008 �0.010DollarVolAvg �0.066 �0.060 �0.067 �0.056 �0.003Industry 0.366 *** 0.370 ***

Governance �0.010 *** �0.011 *** �0.010 ** �0.010 **

Size 0.458 0.789 0.013 0.411 �0.290 0.126VolatAvg 15.062 ** 15.765 **

Vix �0.028 �0.027

No. of Obs. 104 104 104 104 104 104 104 104 104 104Adjusted R2 0.118 0.126 0.048 0.052 �0.034 �0.027 �0.029 �0.021 0.074 0.059


Panel B: Regressions onto difference measuresIntercept 1.031 *** 1.007 *** 1.278 *** 1.268 *** 0.567 *** 0.574 *** 0.589 *** 0.590 *** 0.041 * 0.029PinDiff �1.553 * �1.427 * �1.731 * �1.462 �1.206 �1.212 �1.067 �1.048 �0.076 �0.014SpreadDiff 10 461 *** 9.386 *** 10.091 *** 10.050 *** 9.299 *** 9.115 *** 7.959 *** 8.064 *** 1.929 ** 3.070 *

VolDiff �0.093 ** �0.051 �0.013 0.002 0.004DollarVolDiff �0.065 * �0.019 �0.011 0.004 �0.003Industry 0.495 *** 0.491 ***

Governance �0.013 *** �0.011 *** �0.011 *** �0.010 **

Size 0.194 0.192 �0.170 �0.132 �0.315 �0.318VolatDiff 21.082 *** 21.115 ***

Vix 0.038 * 0.038 *


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risk premiums, including those from information asymmetry andmacroeconomic uncertainty, can alsoaffect the determination of the threshold. Accordingly, Table 9 provides the results of panel regressionsof the estimated thresholds onto average measures (Panel A) and difference measures (Panel B) of theasymmetric information component (PIN). It also shows the inverse of market depth (spread),controlling for liquidity, either in quantity (UsVol) or value (UsDollarVol), firm-level idiosyncraticcharacteristics (Industry, Governance, and Size), and aggregate risks (VolatAvg, VolatDiff, and Vix).

As expected, our measure of market friction (relative quoted spread) significantly increases therequired dollar return of cross-border arbitrage as shown for 4 out of 10models using averagemeasures(Panel A) and 8 out of 10models using difference measures (Panel B). The better governed the firm is athome, the lower theminimum required profit, as allmodelswith theGovernance control variable show.Manufacturing firms (when Industry equals 1) tend to require larger relative premiums. Exposure toaggregate risks in market portfolio, foreign exchange, and market index options appear to buoy arbi-trage costs in Models 9, 10 (VolatAvg), 19 and 20 (VolatDiff and Vix). Overall, difference measures havea greater effect on the threshold level than averagemeasures do, as the adjustedR2’s of Panel B dominatethose of Panel A for all specifications. In sum, the effective break-even point (threshold) of cross-borderarbitrage appears to be affected by the relative degree of private information, market friction, andliquidity measures, idiosyncratic firm-level characteristics, and aggregate risks. These, much econom-ically appealing, empirical results lend support to the findings of Gagnon and Karolyi (2010).

4.2.3. Regression of information shares per smooth transition cointegrationSince the parity-convergence of a cross-listed pair can instead be gradual rather than abrupt, an

alternative measure of the contribution to price discovery is the smooth transition ECM-impliedinformation share proposed in Section 2.3 and estimated in Section 3.4.3. The information shares ofthe NYSE using the whole sample, and given premiums and discounts on cross-listings, are given byIsSt, IsPrem, and IsDisc, respectively. These information shares are dependent variables in the panelregressions onto the same group of explanatory and control variables shown in the regressions of thethreshold ECM-implied information shares (Table 8) whose respective results are shown in Panels A, B,and C of Table 10. We conduct separate regressions of information shares given premiums (Panel B)versus discounts (Panel C) on cross-listings since the NYSE is already shown to be dominant in thecontribution to price discovery (Table 7 Panel C).

In Panel A of Table 10, the contribution of the NYSE to the price discovery of a cross-listed pair,assuming gradual convergence to parity, appears to be more influential with a higher relative pop-ulation of informed traders (PinRatio), market friction (SpreadRatio), and liquidity in both quantity(UsVol) and value (UsDollarVol) on the NYSE vis-à-vis the TSX in Models 1 through 6, 9 and 10.Compared to the regression results of the outer-regime information share of the NYSE (Table 8 Panel A),the corporate governance (Governance) of the Canadian cross-lister does not appear to be effective indetermining the venue of price discovery when convergence is gradual (smooth transition cointegra-tion) rather than abrupt (threshold cointegration). Overall, the smooth transition ECM-implied infor-mation share appears to be explained by a similar basket of risk factors and controls as in the case of thethreshold ECM-implied outer-regime information share. The model fitness of both measures of infor-mation share is also deemed comparable: The adjusted R2’s of the smooth transition ECM regressionmodel (Table 10 Panel A) compared to those of the threshold ECM (Table 8 Panel A) are in the range of9.6%–29.7% versus 14.4%–27.7%, respectively. These results are in stark comparison to those of the linearECM-implied information share (Table 8 Panel C), both in terms of the statistical and economicsignificance of the risk factors and controls, and the model fitness. Thus, not only is there undeniablenonlinearity (Table 3) in the course of convergence to parity, but the contribution measures of pricediscovery are better explained when nonlinearity is assumed, in either a threshold regime switch ora smooth transition. Panels B and C of Table 10 report qualitatively similar regression outcomeswhetherwe use the information share given by premiums or discounts on cross-listings, respectively.30

30 Analogous to Tables 8–10, we also conduct panel regressions excluding 9 firm-years of cross-listed pairs that are notcointegrated per the Phillips and Perron (1988) test. The results are qualitatively unaffected other than the omission of a fewcontrol variables due to a lower dimension of the panel.

Table 10Panel regression results of smooth transition ECM information shares. The dependent variables of Panels A, B, and C are, per the smooth transition ECM, IsSt, IsPrem, and IsDisc which are theinformation shares of NYSE using thewhole sample, and given premiums and discounts on cross-listings, respectively. Explanatory variables: PinRatio is the ratio of the PIN of the NYSE overthat of the TSX. SpreadRatio is the ratio of the relative quoted bid-ask spread of the NYSE over that of the TSX. UsVol is the average daily trading volume of the NYSE out of both of the NYSEand the TSX following Eun and Sabherwal (2003). UsDollarVol is the average daily dollar trading volume of the NYSE out of both of the NYSE and the TSX. Control variables: Governance isthe Report on Business governance index of Canadian firms published by Globe and Mail (McFarland, 2002). Industry equals one if the cross-lister is a manufacturing firm, and zerootherwise. Size is the normalized average market capitalization on the TSX and the NYSE. VolatDiff is the difference of U.S. and Canada’s market index return volatilities. Vix is the CBOEVolatility Index. The t-statistics of the coefficient estimates are suppressed for the lack of space. ***, **, and * stand for statistical significance based on two-sided tests at the 1%, 5%, and 10%level, respectively. The observations are in firm-years. All model specifications are controlled for fixed and year effects.


Panel A: Information shares with whole sampleIntercept �0.121 �0.122 �0.060 �0.060 �0.066 �0.066 �0.025 �0.025 �0.004 �0.004PinRatio 0.120 ** 0.120 ** 0.119 *** 0.120 *** 0.124 *** 0.124 *** 0.064 0.064 0.073 * 0.073 *

SpreadRatio 0.096 *** 0.097 *** 0.098 *** 0.099 *** 0.099 *** 0.100 *** 0.055 * 0.056 * 0.051 0.052UsVol 0.726 *** 0.727 *** 0.735 *** 0.431 * 0.483 *

UsDollarVol 0.726 *** 0.727 *** 0.735 *** 0.448 * 0.499 *

Industry 0.016 0.016Governance 0.000 0.000 0.000 0.000Size 0.001 0.001 0.000 0.000 0.001 0.001VolatDiff �25.161 �25.081Vix �0.008 �0.008


Panel B: Information shares given premiums on cross-listingsIntercept �0.049 �0.050 �0.023 �0.023 �0.025 �0.025 �0.003 �0.003 �0.011 �0.011PinRatio 0.081 * 0.081 * 0.081 * 0.081 * 0.084 * 0.084 * 0.051 0.050 0.059 0.059SpreadRatio 0.117 *** 0.118 *** 0.118 *** 0.119 *** 0.118 *** 0.118 *** 0.067 ** 0.067 ** 0.064 ** 0.064 **

UsVol 0.431 0.432 0.443 * 0.120 0.176UsDollarVol 0.433 0.434 0.446 * 0.139 0.194Industry 0.007 0.007Governance 0.000 0.000 0.000 0.000Size 0.001 0.001 0.000 0.000 0.000 0.000VolatDiff �25.569 �25.685Vix 0.016 0.016


Panel C: Information shares given discounts on cross-listingsIntercept �0.165 �0.166 �0.082 �0.083 �0.091 �0.091 �0.043 * �0.043 * 0.014 0.014PinRatio 0.147 *** 0.147 *** 0.146 *** 0.147 *** 0.153 *** 0.154 *** 0.098 ** 0.099 ** 0.105 ** 0.106 **

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SpreadRatio 0.096 *** 0.096 *** 0.098 *** 0.099 *** 0.100 *** 0.100 *** 0.071 ** 0.072 ** 0.067 ** 0.068 **

UsVol 0.811 *** 0.812 *** 0.822 *** 0.706 *** 0.734 ***

UsDollarVol 0.811 *** 0.813 *** 0.822 *** 0.716 *** 0.743 ***

Industry 0.021 0.022Governance 0.000 0.000 0.000 0.000Size 0.001 0.001 0.000 0.000 0.001 0.001VolatDiff �15.034 �14.736Vix �0.042 �0.043


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5. Conclusion

For a pair of the original listing and its cross-listing, the adjustment to parity can be discontinuous:Convergence may be quicker when the relative premium is profitable. In other words, the dynamics ofcross-listed pairs fall into two regimes: within and beyond the threshold, e.g., transaction costs andassociated risk premiums of arbitrage. This paper extends Harris et al.’s (1995, 2002) ECM to estimatethe extent of the contribution to price discovery (information share) by considering threshold coin-tegration as shown in Balke and Fomby (1997). Alternatively, since convergence mayinstead be gradualand nonlinear, we further generalize the threshold framework to a smooth transition model.

According to our threshold and smooth transition ECMs, the information share and threshold areestimated and regressedwith the followingempirical implications: First, the TSXand theNYSE appear tohave integrated over time. Second, parity-convergence accelerates upon discounts on the cross-listingson the NYSE. Third, we find a larger feedback from the NYSE if the price gap exceeds the threshold(required arbitrage return). Fourth, informed traders tend to cluster on the NYSE upon discounts on thecross-listings. Fifth, information share and threshold are affected by the relative degree of privateinformation, market friction and liquidity measures, firm-level characteristics, and aggregate risks.

Lastly, a disclaimer. We do not account for exchange-rate market friction in our threshold ECMframework unlike Grammig et al. (2005). This is because of the stationarity of the U.S.–Canadaexchange rate (Issa et al., 2006). The synchronous trading environment of TSX-NYSE cross-listed pairsallows constructing a cointegration system without considering the exchange-rate bid-ask spreadwhich risks a sufficiently low margin of error (Eun and Sabherwal, 2003). However, including sucha source of randomness to models of the nonlinear dynamics of cross-listed stocks is of interest forfuture study.

Appendix A. Price discovery of cross-listingsWe extend Garbade and Silber’s (1983) model to develop an equilibrium framework to characterize

the interactive dynamics of a cross-listed pair simultaneously traded on two separate exchanges.Arbitrageurs linking the two markets are subject to market frictions, such as transaction fees, capitalconstraints, etc. We emphasize the role of arbitrageurs in the process of inter-market price discovery.

We first assume that there are two cross-border stock exchanges: the TSX and the NYSE. Weconveniently index the respective markets: i ¼ 1, 2. We further assume that there are N1 investors whoonly trade at home (TSX) and N2 who only abroad (NYSE), and N3 arbitrageurs who trade in andbetween both markets. We assume that choice of exchanges by the first two trader types (one-markettraders) is exogeneous and is due to various reasons such as distance, language, institutionalconstraints, transaction costs, etc.

We specify the behavior of one-market traders in market i. With an exponential utility function andnormally distributed random payoffs for the risky asset, the demand function is a linear function of themarket price pit, given by

Xijt ¼ Eijt � b�pit � mijt

�for j ¼ 1;2;.;Ni; (A.29)

where Ejti is the endowment (in terms of the shares of the risky asset) of trader j at the beginning of

period t; mijt is the reservation price at which the trader is willing to hold the endowment; and b (>0) isthe price elasticity of demand, assumed to be the same for all one-market traders for simplicity.31

We now consider the demand function of arbitrageurs. Arbitrageurs “buy low and sell high” in thetwo markets, thus their demand function only depends on the cross-border price deviation. Givenrespective prices p1t and p2t, arbitrageur k would submit a buy order in market 1 as

XA1kt ¼ bAktðp2t � p1tÞ for k ¼ 1;2;.;N3; (A.30)

31 The demand function of one-market traders is not bounded. This linear demand function with positive but finite demandelasticity reflects some regularity conditions, such as exponential utility and quadratic cost functions. A short-sales restrictionshould not affect the demand side. However, it may affect the equilibrium solution. We do not impose any constraints onborrowing or lending funds.


where bAkt > 0 is the demand elasticity and is assumed to be finite due to market friction, followingGarbade and Silber (1983). Also bAkt evolves over time as a result of time-varying transaction costs andtrading risks, such as exchange rate volatility and macroeconomic flucations etc. In Section 2, by givingdifferent structure of this time variation, we generate three econometric model for empirical studies.

We assume the abitrageurs hedge perfectly, thus,

XA2kt ¼ �XA

1kt ¼ bAktðp1t � p2tÞ; (A.31)

i.e. her short position in one market always equals the long position in the other market.32

In equilibrium, the two exchanges clear as total supply equals total demand, i.e.

XN1

j¼1E1jt ¼

XN1

j¼1X1jt þ

XN3

k¼1bAktðp2t � p1tÞ; (A.32)

XN2

j¼1E2jt ¼

XN2

j¼1X2jt þ

XN3

k¼1bAktðp1t � p2tÞ: (A.33)

Inserting the demand function (A.29) into Eqs. (A.32) and (A.33), we have

XN1

j¼1E1jt ¼

XN1

j¼1

nE1jt � b

�p1t � m1jt

�oþXN3

k¼1bAktðp2t � p1tÞ; (A.34)

XN2

j¼1E2jt ¼

XN2

j¼1

nE2jt � b

�p2t � m2jt

�oþXN3

k¼1bAktðp1t � p2tÞ: (A.35)

Solving these market clearing conditions for equilibrium prices of the cross-listed pair yields

p1t ¼�bN2 þ

PN3k¼1b

Akt

�N1m

1t þ

PN3k¼1b

AktN2m

2t�

bN2 þPN3

k¼1bAkt

�N1 þ

PN3k¼1b

AktN2

; (A.36)

p2t ¼�bN1 þ

PN3k¼1b

Akt

�N2m

2t þ

PN3k¼1b

AktN1m

1t�

bN1 þPN3

k¼1bAkt

�N2 þ

PN3k¼1b

AktN1

; (A.37)

where m1t ¼ ð1=N1ÞPN1

j¼1m1jt and m2t ¼ ð1=N2Þ

PN2j¼1m

2jt are the markets’ average reservation prices.

In order to derive dynamic price relationships, we specify an evolution mechanism of the reser-vation prices m1jt and m2jt , following Garbade and Silber (1983), as

mijt ¼ pit�1 þ vt þ 3ijt for i ¼ 1;2; j ¼ 1; 2;.;Ni: (A.38)

As the reservation price mijt reflects trader j’s belief on the expected return on the risky asset, Eq.(A.38) describes the updating process of her belief. Specifically, as market i clears at the end of periodt� 1with a partial equilibrium price pit�1, each trader is willing to hold her share of asset Eijt (whichwillbe her endowment in the subsequent period t), indicating that pit�1 was her reservation price after theclearing in period t � 1. At the beginning of period t, the trader updates her belief (or the new reser-vation price mijt), based on the past belief (or the reservation price pit�1). The new information, signalwhich consists of two components: vt, common to all investors in both markets, and an idiosyncraticcomponent 3ijt . We assume that vt and 3ijt are i.i.d normal random variables with a mean of zero anda constant variance, respectively.

32 We assume there is no financial constraint for the arbitrageurs.


In aggregate, the market reservation prices m1t and m2t can be expressed as

m1t ¼ 1N1

XN1

j¼1m1jt ¼ p1t�1 þ vt þ 1

N1

XN1

j¼131jt ; (A.39)

m2t ¼ 1N2

XN2

j¼1m2jt ¼ p2t�1 þ vt þ 1

N2

XN2

j¼132jt : (A.40)

Substituting m1t and m2t into the Eqs. (A.36) and (A.37), we have,

p1t ¼�bN2 þ

PN3k¼1b

Akt

�N1p1t�1 þ

PN3k¼1b

AktN2p2t�1�

bN2 þPN3

k¼1bAkt

�N1 þ

PN3k¼1b

AktN2

þ vt þ ~31t ; (A.41)

p2t ¼�bN1 þ bAN3

�N2p2t�1 þ

PN3k¼1b

AktN1p1t�1�

bN1 þPN3

k¼1bAkt

�N2 þ

PN3j¼1b

AktN1

þ vt þ ~32t ; (A.42)

where

~31t ¼�bN2 þ

PN3k¼1b

Akt

�PN1j¼1 31jt þ

�PN3k¼1b

Akt

��PN2j¼1 32jt

��bN2 þ

PN3k¼1b

Akt

�N1 þ

PN3k¼1b

AktN2

; (A.43)

~32t ¼�bN1 þ

PN3k¼1b

Akt

�PN2j¼1 32jt þ

�PN3k¼1b

Akt

��PN1j¼1 31jt

��bN1 þ

PN3k¼1b

Akt

�N2 þ

PN3k¼1b

AktN1

: (A.44)

An equivalent matrix representation is

�p1tp2t

�¼

�1� at ; atbt ;1� bt

��p1t�1p2t�1

��0@ vt þ ~31t

vt þ ~32t

1A; (A.45)

where

at ¼PN3

k¼1bAktN2

bN2N1 þPN3

k¼1bAktN1 þ

PN3k¼1b

AktN2

; (A.46)PN A

bt ¼3

k¼1bktN1

bN2N1 þPN3

k¼1bAktN1 þ

PN3k¼1b

AktN2

: (A.47)

We obtain a bivariate VECM by subtracting (p1t�1, p2t�1)T from both sides:

�Dp1tDp2t

�¼

��at ; atbt ;�bt

��p1t�1p2t�1

��0@ vt þ ~31t

vt þ ~32t

1A: (A.48)

This VECM describe the short-term dynamics toward the long run equilibrium, given the cointe-grating vector (1,�1)T. The short term adjustment coefficients at and bt for respective prices, p1t and p2t,


reflect their responses to deviations from the long-run equilibrium in the respective markets. We applythe permanent transitory decomposition (Gonzalo and Granger, 1995) to this VECM: The permanentcomponent is a linear combination of (p1t, p2t) formed by the scaled orthogonal vector of the adjust-ment coefficient vector (at, bt). Specifically, the permanent component is given by

ft ¼ btat þ bt

p1t þat

at þ btp2t ; (A.49)

where

btat þ bt

¼ N1

N1 þ N2; (A.50)

atat þ bt

¼ N2

N1 þ N2: (A.51)

The contribution share of each price to the permanent component is captured by bt=ðat þ btÞ andat=ðat þ btÞ. They reflect the respective information shares of markets 1 and 2 towards determining thelong-run equilibrium price. In other words, they are relative measures of market-specific contributionsto the price discovery of the cross-listed pair.

Define Dpthp2t � p1t as the dollar premium on the cross-listing against its original listing. It can beshown that

Dpt ¼ dtDpt�1 þ et ; (A.52)

where

dt ¼ 1� at � bt

¼ 1�PN3

k¼1bAktN2

bN2N1 þPN3

k¼1bAktN1 þ

PN3k¼1b

AktN2

�PN3

k¼1bAktN1

bN2N1 þPN3

k¼1bAktN1 þ

PN3k¼1b

AktN2

¼ bN2N1

bN2N1 þPN3

k¼1bAktN1 þ

PN3k¼1b

AktN2

: (A.53)

Following Garbade and Silber (1983), dt measures the reciprocal convergence speed of the twomarket prices to their long run equilibrium: The smaller dt is the faster is convergence.

Appendix B. Estimation and testing of parametersFor convenience, the firm indicator (i) is omitted in the following discussion. The threshold ECM

mentioned in Section 2.2 is represented as

Dxt ¼ AT1Xt�1d1tðgÞ þ AT

2Xt�1d2tðgÞ þ ut ; (B.54)

where Dxt ¼ ðp1t ; p2tÞ;Xt�1 ¼ ½1; kt�1;Dxt�1;Dxt�2;.Dxt�m�T ;d1tðgÞ ¼ 1ðjkit�1j � giÞ andd1tðgÞ ¼ 1ðjkt�1j>gÞ; AT

1 and AT2 are the parameters to be estimated; and g is the threshold parameter

to be estimated.The threshold VECM can be estimated using the maximum likelihood estimation (MLE) method

proposed by Hansen and Seo (2002). Assuming that the error term (ut) is i.i.d. Gaussian, the likelihoodfunction is

LnðA1;A2;S;gÞ ¼ �n2lnjSj � 1

2

Xnt¼1

utðA1;A2;gÞTS�1utðA1;A2;gÞ; (B.55)


where utðA1;A2;gÞ ¼ Dxt � AT1Xt�1d1tðgÞ � AT

2Xt�1d2tðgÞ: The covariance matrix (S) is an identitymatrix due to the i.i.d. Gaussian assumption of the error term. For a fixed c, A1 and A2 are estimated byan ordinary least squares (OLS) regression, thus

A1ðgÞ ¼�Xn

t¼1Xt�1X

Tt�1d1tðgÞ

��1Xn

t¼1Xt�1Dx

Tt d1tðgÞ; (B.56)

A2ðgÞ ¼�Xn

t¼1Xt�1X

Tt�1d2tðgÞ

��1Xn

t¼1Xt�1Dx

Tt d2tðgÞ; (B.57)

and then utðgÞ ¼ Dxt � AT1Xt�1d1tðgÞ � A

T2Xt�1d2tðgÞ: By substituting utðgÞ; the likelihood function

ðLnðA1;A2;S;gÞÞ becomes a univariate function of g:

LnðgÞ ¼ �n2

ln�1n

Xn

t¼1utðgÞutðgÞT

�� nðmþ 2Þ

2: (B.58)

Following Hansen (2000), the grid search method can be used to estimate gwithin a preset interval½g ;g�. Themle estimators for A1 and A2 can be obtained by inserting g. To further confirm the thresholdeffect, we test the following null hypothesis:

H0 : A1 ¼ A2 for any g˛hg ;g

i(B.59)

against

H1 : A1sA2 for some g˛hg ;g

i: (B.60)

We use the supremum-Lagrangianmultiplier (supLM) test (Hansen and Seo, 2002) to test the abovehypotheses. The supLM statistic is

LMðgÞ ¼ ðA1ðgÞ � A2ðgÞÞT ðV1ðgÞ þ V2ðgÞÞ�1ðA1ðgÞ � A2ðgÞÞ; (B.61)

where V1ðgÞ ¼ MjðgÞ�1UjðgÞMjðgÞ�1; MjðgÞ ¼ Imþ25PjðgÞTPjðgÞ;UjðgÞ ¼ GjðgÞTGjðgÞ; andPjðgÞ;GjðgÞ are the matrices of the stacked rows of Xt�1djtðgÞ and utðgÞ5Xt�1djtðgÞ, respectively.Define

supLM ¼ supg˛g ;g

LMðgÞ: (B.62)

A bootstrap method is used to generate the critical values since the asymptotic distribution is non-standard.

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how smooth is price discovery? evidence from cross-listed stock trading

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