StrategicCross-TradingintheU.S.Stock Market*

STRATEGIC CROSS-TRADING IN THE U.S. STOCK MARKET233function of the number of sophisticated market participants and the dispersion of their beliefs.2 All proofs are in the Appendix.2.1 THE BASIC SETTING m EðvjSvm Þ P0 ¼ ðSvm P0 Þ,ð1Þwhere varð m Þ ¼ v is nonsingular. It then follows that any twovectors m and k have a joint MND and covð m , k Þ c ¼ , a symmetric positive definite (SPD) matrix. Therefore, Eð k jSvm Þ ¼ m and can beinterpreted as the correlation between any two information endowments mand k : The lower (higher) is , the more (less) heterogeneous—that is, theless (more) correlated and, of course, precise—is speculators’ private information about v.2Holden and Subrahmanyam (1992), Foster and Viswanathan (1996), Back, Cao, andWillard (2000), and Pasquariello and Vega (2007) develop one-asset extensions of Kyle(1985) to link market liquidity to the trading activity of heterogeneously informedtraders. Pasquariello (2007) characterizes the circumstances under which such activitymay magnify equilibrium price comovement in Caballé and Krishnan’s (1994) multitrader, multi-asset generalization of Kyle (1985).Downloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015The model consists of a two-period economy in which N risky assets areexchanged. Trading occurs only at the end of the first period (t ¼ 1). At theend of the second period (t ¼ 2), the payoffs of the risky assets, an N 1multivariate normally distributed (MND) random vector v with mean P0and nonsingular covariance matrix v , are realized. The economy ispopulated by three types of risk neutral traders: a discrete number M ofinformed traders (labeled speculators), liquidity traders, and perfectly competitive MMs. All traders know the structure of the economy and thedecision process leading to order flow and prices.At t ¼ 0, there is neither information asymmetry about v nor trading.Sometime between t ¼ 0 and t ¼ 1, each speculator m receives a privateand noisy signal of v, Svm . We assume that each vector Svm is drawn froma MND with mean P0 and covariance matrix s and that, for any twospeculators m and k, covðv, Svm Þ ¼ covðv, Svk Þ ¼ covðSvm , Svk Þ ¼ v . Wefurther parametrize the degree of diversity among speculators’ private information by imposing that s ¼ 1 v and 2 ð0, 1Þ, as in Pasquariello andVega (2009). These assumptions imply that speculator m’s information advantage about v at t ¼ 1, before trading with the MMs, is given by

234P. PASQUARIELLO AND C. VEGA2.1.a. EquilibriumConsistent with Caballé and Krishnan (1994), we define a Bayesian Nashequilibrium of this economy as a set of M þ 1 vector functionsX1 ð Þ, . . . , XM ð Þ, and P1 ð Þ such that the following two conditions hold:(1) Profit maximization: Xm ðSvm Þ ¼ arg max Eð m jSvm Þ;(2) Semi-strong market efficiency: P1 ð!1 Þ ¼ Eðvj!1 Þ.Proposition 1 characterizes the unique linear equilibrium for this economy.Proposition 1There exists a unique linear equilibrium given by the price functionP1 ¼ P0 þ !1ð2Þand by each speculator m’s demand strategy1 1 m ,2 þ ðM 1Þ ð3ÞpffiffiffiffiffiffiffiffiM 1 2 ¼½2 þ ðM 1Þ z vð4ÞXm ¼whereis an SPD matrix.The optimal trading strategy of each speculator depends on the privateinformation she receives about v ( m ) as well as on the depth of the market( 1 ). Speculators are imperfectly competitive. Hence, albeit risk neutral,theyexploittheir information advantage in each market cautiously ( Xm ðnÞ 1) to avoid dissipating their informational advantage withtheir trades, as in the single-asset setting of Kyle (1985). For the same purpose, speculators also trade strategically across assets@Xm ðjÞ@ m ðnÞ6¼ 0 .Downloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015At t ¼ 1, both speculators and liquidity traders move first and submit theirorders to the MMs before the price vector P1 has been set. We define thevector of market orders of speculator m to be Xm. Thus, her profit is given by m ðXm , P1 Þ ¼ X0m ðv P1 Þ. Liquidity traders generate a vector of randomdemands z, MND with mean 0 (a zero vector) and nonsingular covariancematrix z . For simplicity, we impose that noise trading z has identicalvariance and is independent across assets ( z ¼ z2 I) and from any otherrandom vector. MMs do not receivePM any information, but observe the netorder flow for each asset !1 ¼ m¼1 Xm þ z and set the market-clearingprices P1 ¼ P1 ð!1 Þ.

STRATEGIC CROSS-TRADING IN THE U.S. STOCK MARKET235Intuitively, the MMs know the structure of the economy (the covariancematrix v ). Hence, unless all securities’ terminal payoffs are fundamentallyunrelated (i.e., unless v is diagonal), they rationally use the order flow foreach asset to learn about the liquidationvalues of other assets when settingthe market-clearing price vector P1@P1 ðnÞ@!1 ðjÞ6¼ 0 . Speculators are aware of this2.1.b. Testable ImplicationsProposition 1 generates unambiguous predictions on direct ( ðn, nÞ) andcross-price ( ðn, jÞ) impact. In the model of Section 2.1, speculators arerisk neutral, financially unconstrained, and formulate “fundamentally correct” inference from their private signals (@S@ vmm ððjnÞÞ ¼ 0 if v ðn, jÞ ¼ 0), whilenoise trading is uncorrelated across assets ( z is diagonal). Hence, neithercorrelated information shocks (King and Wadhwani, 1990; Chan, 1993),correlated liquidity shocks (Bernhardt and Taub, 2008; see also Section5.2), nor portfolio rebalancing (Kodres and Pritsker, 2002) drive theircross-trading decisions. Nonetheless, Proposition 1 implies that if theunderlying economy is fundamentally interconnected—a nondiagonal v —the equilibrium market liquidity matrix of Equation (4) is alsonondiagonal: Order flow in one security has a contemporaneous impacton the equilibrium prices of many securities ( ðn, jÞ 6¼ 0)—even thosewhose terminal values are unrelated to that security’s payoff ( v ðn, jÞ ¼ 0).Such an impact reflects both (1) speculators’ strategic trading activity toaffect the MMs’ inference from the observed order flow; and (2) theMMs’ attempt to learn from it about the traded assets’ payoffs v as wellas to be compensated for the losses they anticipate from it by their expectedprofits from noise trading.Remark 1.If the economy is fundamentally interconnected there exist cross-price impact,even among fundamentally unrelated assets.The number of speculators (M) and the correlation among their privateinformation ( ) affect both direct and cross-price impact. The intensity ofcompetition among speculators influences their ability to attenuate the informativeness of the order flow in each security. More numerous speculatorsDownloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015learning process, labeled cross-inference. Thus, they strategically place theirtrades in many assets—rather than independently trading in each asset—tolimit the amount of information divulged by their market orders. As a resultof this effort, labeled strategic cross-trading, Equations (2) and (3) representa noisy rational expectations equilibrium.

236P. PASQUARIELLO AND C. VEGACorollary 1.Under most parametrizations, direct and absolute cross-price impact aredecreasing in the number of speculators and increasing in the heterogeneityof their information.To gain further insight into these results, we construct a simple numericalexample along the lines of Pasquariello (2007). We assume that there arethree assets in the economy (N ¼ 3), that their liquidation values are relatedto each other by way of the following parametrization of v :232 0:5 0 v ¼ 4 0:5 1:5 0:5 5,ð5Þ0 0:5 2and that z2 ¼ 1. According to Equation (5), assets 1 and 3 are fundamentallyunrelated (i.e., cov½vð1Þ, vð3Þ ¼ 0) yet both exposed to asset 2(cov½vð1Þ, vð2Þ 0 and cov½vð2Þ, vð3Þ 0). We then focus on asset 1 and3For instance, in the limit, if M speculators were homogeneously informed—thatzffi 1 2 ðv P0 Þ—it can be shownis, if ¼ 1 such that s ¼ v , Svm ¼ v, and Xm ¼ p ffiffiffi M v thatthefinitedifference jMXm j ¼ ðM þ 1ÞXm ðatM þ 1Þ MXm ðatMÞ ffiffi ¼ p zffiffi M þ 1 M v 1 2 ðv P0 Þ 0. 1 2 4Xm jz ffi In particular, @j@ ¼ 2p ffiffiffiffiffi v ðSvm P0 Þ 0.M Downloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015trade more aggressively—that is, their aggregate amount of trading ishigher—in every asset because competition among them precludes any collusive trading strategy.3 This behavior reduces the intensity of adverse selection for the MMs in each market, thus leading to lower direct and absolute(i.e., unsigned) cross-price impact (lower ðn, nÞ and ðn, jÞ ).The heterogeneity of speculators’ signals moderates their tradingaggressiveness. When information is less correlated (smaller ), each speculator has some monopoly power on her signal vector, because at least part ofit is known exclusively to her. Hence, they trade more cautiously—that is,their absolute amount of trading is lower—in each asset to reveal less of theirown information advantage m .4 This “quasi-monopolistic” behavior makesthe MMs more vulnerable to adverse selection. However, the smaller is thelower is the precision of each speculator’s private signal of v (as s ¼ 1 v ),hence the less severe is adverse selection for the MMs in all assets. In thepresence of many, thus less cautious, speculators (i.e., for nontrivial M, ascommon to most financialmarkets),the former effect dominates the latter and both ðn, nÞ and ðn, jÞ increase for lower . Corollary 1 summarizesthese empirical implications of our model.

STRATEGIC CROSS-TRADING IN THE U.S. STOCK MARKET237Λ(1,1)(a) 0.40.35lambda(1,1)0.30.250.20.15M 100M (b)Λ(1,3)0M 500-0.0005lambda(1,3)-0.001-0.0015-0.002M .550.650.750.850.95rhoFigure 1. Three-asset economy: measures of liquidity. This figure plots measures of direct( ð1, 1Þ) and cross-price impact ( ð1, 3Þ) for the three-asset economy (N ¼ 3) parametrizedin Section 2.1.b as a function of the degree of information heterogeneity among speculators( ) in the presence of M of them. Specifically, we plot ð1; 1Þ (a), and ð1, 3Þ (b), as afunction of 2 ½0:05, 0:95 when v is given by Equation (5), z2 ¼ 1, and M ¼ 100 (dottedline) or M ¼ 500 (solid line).plot its equilibrium direct ( ð1, 1Þ) and cross-price impact from asset 3( ð1, 3Þ) as a function of the private signal correlation parameter —inFigures 1a and 1b, respectively—for M ¼ 100 (dotted line) and M ¼ 500(solid line).As a result of speculators’ strategic cross-trading and MMs’ cross-inference, order flow in asset 3 impacts the equilibrium price of asset 1, althoughtheir terminal payoffs are unrelated: ð1, 3Þ 6¼ 0 in Figure 1b althoughDownloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 20150.05

238P. PASQUARIELLO AND C. VEGAcov½vð1Þ, vð3Þ ¼ 0. For instance, ceteris paribus, a negative private information shock to asset 1 alone (i.e., to m ð1Þ alone) prompts imperfectlycompetitive speculators, moving first and aware of MMs’ potential cross-m ð1Þinference, not only to sell asset 1 @X@ m ð1Þ 0, as expected but also to buy @Xm ð3Þm ð2Þasset 2 @X 0andtosellasset3 0. The latter two trades are@ m ð1Þ@ m ð1ÞDownloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015to minimize the dissipation of private information and profits that wouldoccur if speculators sold asset 1 alone. Intuitively, selling asset 1 alone leadsthe MMs to a steeper downward revision of its price (via ð1, 1Þ 0), henceto lower expected profits for speculators. In Kyle (1985), this possibilityinduces speculators to more cautious selling of asset 1. In the economy ofEquation (5), speculators can also engage in strategic trading in other assets.Speculators’ purchases of asset 2 raise the possibility that a positive shock tothe common portion of the payoffs of both assets 1 and 2 may have occurred(as cov½vð1Þ, vð2Þ 0). Thus, these purchases may attenuate the MMs’downward revision of the price of asset 1, yet at the cost of a potentialupward revision of the price of asset 2. Speculators’ sales of asset 3 raisethe possibility that a negative shock to the payoffs of both assets 2 and 3 mayhave occurred (as cov½vð2Þ, vð3Þ 0). Thus, these sales may attenuate theMMs’ potential upward revision of the price of asset 2, yet at the cost ofa potential downward revision of the price of asset 3. Sales of asset 3 alsoraise the possibility that the accompanying purchases of asset 2 may stemfrom positive common information about v(1) and vð2Þ (rather than aboutv(2) and vð3Þ). Thus, these sales may attenuate the MMs’ downward revisionof the price of asset 1 as well.Aware of this potential strategic cross-trading activity, the MMs makethe equilibrium price of asset 1 (P1 ð1Þ) sensitive to observed order flownot only in asset 1 ( ð1, 1Þ 0) but also in assets 2 ( ð1, 2Þ 0) and 3( ð1, 3Þ 0). Intuitively, a positive ð1, 2Þ allows the MMs to account forpotentially positive common information about vð1Þ and v(2) from purchases in asset 2 (as cov½vð1Þ, vð2Þ 0) when setting P1 ð1Þ. Besides, anegative ð1, 3Þ allows the MMs to account for further potentiallypositive common information about v(1) and v(2) from purchases inasset 2—in light of potentially negative common information about v(2)and v(3) from sales of asset 3 (as cov½vð2Þ, vð3Þ 0)—when setting P1 ð1Þ.Figure 1 also shows that a smaller number of speculators (lower M) orgreater information heterogeneity among them (lower ) intensify suchtrading activity, and thus worsening MMs’ adverse selection problemsand increasing both direct ( ð1, 1Þ) and absolute cross-price impact( ð1, 3Þ) for asset 1.

STRATEGIC CROSS-TRADING IN THE U.S. STOCK MARKET2393. Data DescriptionWe test the implications of the model of Section 2 using a comprehensivesample of U.S. stock market transaction-level data, and firm-levelcharacteristics.3.1 U.S. STOCK MARKET DATADownloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015We use intraday, transaction-level TAQ data (trades and quotes) duringregular market hours (9:30 a.m. to 4 p.m. ET) for all domestic commonstocks (CRSP share code 10 or 11) listed on the NYSE and the NASDAQbetween January 1 1993 and June 30 2004 (2,889 trading days). We excludeReal Estate Investment Trusts, closed-end funds, foreign stocks, andAmerican Depository Receipts since their trading characteristics mightdiffer from those of ordinary equities (Chordia and Subrahmanyam, 2004;Boehmer and Wu, 2008). Firm-level accounting information (e.g., quarterlyearnings-per-share (EPS)) is from the COMPUSTAT database. MergingTAQ, CRSP, and COMPUSTAT data yields a sample of 3,773 firms(unique identifiers) over our sample period.We filter the TAQ data by deleting a small number of trades and quotesrepresenting possible data error (e.g., negative prices or quoted depths) orwith unusual characteristics (as listed in Bessembinder, 1999, footnote 5).Consistent with the vast literature using TAQ data (Hasbrouck, 2007), wethen sign intraday trades using the Lee and Ready (1991) algorithm: (1) If atransaction occurs above (below) the prevailing quote mid-point, we label ita purchase (sale); (2) if a transaction occurs at the quote mid-point, we labelit a purchase (sale) if the sign of the last price change is positive (negative).Assigning the direction of trades via the Hasbrouck (1988, 1991) algorithmleads to qualitatively and quantitatively similar inference. As inBessembinder (2003a), we do not allow for a five-second lag between TAQreports and compare exchange quotes from NYSE (NASDAQ) exclusivelywith NYSE (NASDAQ) transaction prices—that is, we only consider orderflow taking place in the listing exchange—since off-exchange quotations(e.g., from regional stock exchanges) rarely improve on the exchangequote (Blume and Goldstein, 1997).Our model, a multi-asset extension of Kyle (1985), conjectures a relationship between a firm’s stock price changes and both its own and other firms’net order flow. Chordia and Subrahmanyam (2004, p. 486) observe that “theKyle setting is more naturally applicable in the context of signed order imbalances over a time interval, as opposed to trade-by-trade data, since thetheory is not one of sequential trades by individual traders.” Jones, Kaul,

240P. PASQUARIELLO AND C. VEGAWe divide the buy–sell imbalance by the total number of trades inEquation (6) to eliminate the impact of total trading activity (Chordia andSubrahmanyam, 2004). In unreported analysis, we find our inference to berobust to defining order imbalance as the net scaled dollar trading volume(Jones, Kaul, and Lipson, 1994) or to using alternative normalizations of thebuy–sell imbalance (e.g., by scaling it by the number of shares outstanding ora moving average of the total number of trades over the trailing year).3.2 INFORMATION VARIABLESAccording to the model of Section 2, the magnitude of equilibrium crossprice impact among traded assets depends on the extent of market-wideinformation heterogeneity among speculators ( ) and on the number ofspeculators in the economy (M).We use professional forecasts of either individual firm’s long-term EPSgrowth or U.S. macroeconomic announcements to proxy for the beliefs ofsophisticated market participants about U.S. stocks’ fundamentals. Thestandard deviation across professional forecasts is a commonly usedmeasure of aggregate and stock-level information heterogeneity (Diether,Malloy, and Scherbina, 2002; Kallberg and Pasquariello, 2008).We obtain our first proxy for by using the unadjusted I/B/E/S SummaryHistory database of analyst forecasts of the long-term growth of individualstocks’ EPS. Long-term growth forecasts are less likely to be biased by firms’potential “earnings guidance” (Yu, 2011) and normalization for cross-firmcomparability (Qu, Starks, and Yan, 2004). The inference that follows isrobust to use fiscal-year EPS forecasts. We define the diversity of opinionabout the long-term prospects of each firm i in the TAQ/CRSP/COMPUSTAT sample in each month m between January 1993 and June2004 as the standard deviation across multiple (i.e., two or more) analystforecasts of that firm’s long-term EPS growth (when available),Downloaded from http://rof.oxfordjournals.org/ at University of Michigan on March 2, 2015and Lipson (1994) an

(even unrelated) assets—in the U.S. stock market. Our investigation of the trading activity in New York Stock Exchange (NYSE) and National Association of Securities Dealers Automated Quotation System (NASDAQ) stocks between 1993 and 2004 reveals that, consistent with our model, (1) daily order imbalance in one industry or random stock