MARKET MICROSTRUCTURE INVARIANCE: EMPIRICAL HYPOTHESES. ANNA A. OBIZHAEVA New Economic School, Moscow , Russia

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1 Econometrca, Vol. 84, No. 4 (July, 2016), MARKET MICROSTRUCTURE INVARIANCE: EMPIRICAL HYPOTHESES ALBERT S. KYLE Robert H. Smth School of Busness, Unversty of Maryland, College Park, MD 20742, U.S.A. ANNA A. OBIZHAEVA New Economc School, Moscow , Russa The copyrght to ths Artcle s held by the Econometrc Socety. It may be downloaded, prnted and reproduced only for educatonal or research purposes, ncludng use n course packs. No downloadng or copyng may be done for any commercal purpose wthout the explct permsson of the Econometrc Socety. For such commercal purposes contact the Offce of the Econometrc Socety (contact nformaton may be found at the webste or n the back cover of Econometrca). Ths statement must be ncluded on all copes of ths Artcle that are made avalable electroncally or n any other format.

2 Econometrca, Vol. 84, No. 4 (July, 2016), MARKET MICROSTRUCTURE INVARIANCE: EMPIRICAL HYPOTHESES BY ALBERT S. KYLE AND ANNA A. OBIZHAEVA 1 Usng the ntuton that fnancal markets transfer rsks n busness tme, market mcrostructure nvarance s defned as the hypotheses that the dstrbutons of rsk transfers ( bets ) and transacton costs are constant across assets when measured per unt of busness tme. The nvarance hypotheses mply that bet sze and transacton costs have specfc, emprcally testable relatonshps to observable dollar volume and volatlty. Portfolo transtons can be vewed as natural experments for measurng transacton costs, and ndvdual orders can be treated as proxes for bets. Emprcal tests based on a data set of 400,000+ portfolo transton orders support the nvarance hypotheses. The constants calbrated from structural estmaton mply specfc predctons for the arrval rate of bets ( market velocty ), the dstrbuton of bet szes, and transacton costs. KEYWORDS: Market mcrostructure, lqudty, bd-ask spread, market mpact, transacton costs, order sze, nvarance, structural estmaton. 0. INTRODUCTION THIS PAPER PROPOSES AND TESTS TWO EMPIRICAL HYPOTHESES that we call market mcrostructure nvarance. When portfolo managers trade fnancal assets, they can be modeled as playng tradng games n whch rsks are transferred. Market mcrostructure nvarance begns wth the ntuton that these rsk transfers, whch we call bets, take place n busness tme. The rate at whch busness tme passes market velocty s the rate at whch new bets arrve nto the market. For actvely traded assets, busness tme passes quckly; for nactvely traded assets, busness tme passes slowly. Market mcrostructure characterstcs such as bet sze, market mpact, and bd-spreads vary across assets and across tme. Market mcrostructure nvarance hypotheszes 1 We are grateful to Elena Asparouhova, Peter Bossaerts, Xaver Gabax, Lawrence Glosten, Larry Harrs, Pankaj Jan, Mark Loewensten, Natale Popovc, Sergey N. Smrnov, Georgos Skoulaks, Vsh Vswanathan, and Wenyuan Xu for helpful comments. Obzhaeva s also grateful to the Paul Woolley Center at the London School of Economcs for ts hosptalty as well as Smon Myrgren, Sébasten Page, and especally Mark Krtzman for ther help. Kyle has worked as a consultant for varous companes, exchanges, and government agences. He s a non-executve drector of a U.S.-based asset management company. Ths paper supersedes a prevous manuscrpt, Market Mcrostructure Invarance: Theory and Emprcal Tests (October 17, 2014), whch also contans a theoretcal model whch s now descrbed n Kyle and Obzhaeva (2016b). The prevous paper was a revsed verson of an earler manuscrpt (June 7, 2013) whch combned and superseded two earler papers: the theoretcal paper Market Mcrostructure Invarants: Theory and Implcatons of Calbraton (December 12, 2011) and the emprcal paper Market Mcrostructure Invarants: Emprcal Evdence From Portfolo Transtons (December 12, 2011). These two papers superseded an older combned manuscrpt Market Mcrostructure Invarants (May 8, 2011) The Econometrc Socety DOI: /ECTA10486

3 1346 A. S. KYLE AND A. A. OBIZHAEVA that these mcrostructure characterstcs become constants mcrostructure nvarants when vewed n busness tme. Secton 1 formulates two nvarance prncples as emprcal hypotheses, conjectured to apply for all assets and across tme. Invarance of Bets: The dstrbuton of the dollar rsk transferred by a bet s the same when the dollar rsk s measured n unts of busness tme. Invarance of Transactons Costs: The expected dollar transacton cost of executng a bet s the same functon of the sze of the bet when the bet s sze s measured as the dollar rsk t transfers n unts of busness tme. When busness tme s converted to calendar tme, these nvarance hypotheses mply specfc emprcal restrctons relatng market mcrostructure characterstcs to volume and volatlty. The mplcatons of the frst nvarance hypothess can be descrbed usng the concept of tradng actvty, defned as the product of dollar volume and returns volatlty. Invarance mples that the number of bets per calendar day s proportonal to the two-thrds power of tradng actvty. Average bet sze, expressed as a fracton of tradng volume, s nversely proportonal to the twothrds power of tradng actvty; otherwse, the shape of the dstrbuton of bet sze s the same across assets and tme. The mplcatons of the second nvarance hypothess can be descrbed usng a measure of llqudty defned as the cube root of the rato of returns varance to dollar volume. Invarance mples that the percentage bd-ask spread s proportonal to ths measure of llqudty. Percentage transacton costs are proportonal to the product of ths asset-specfc llqudty measure and some nvarant functon of bet sze, scaled by volume n busness tme to convert bet sze nto nvarant dollar rsk transfer. Invarance does not restrct the shape of ths functon; t can be consstent wth ether lnear or square-root models of prce mpact. Secton 2 shows how nvarance can be used to mpose testable restrctons on transacton-cost models descrbed n the theoretcal market mcrostructure. For example, the model of Kyle (1985) mples that market depth s proportonal to the standard devaton of order mbalances. Order mbalances are not drectly observable n transactons data. By mposng restrctons on the sze and number of bets whch determne the composton of the order flow nvarance shows how to nfer the standard devaton of order mbalances from volume and volatlty and thereby make correct emprcal predctons. Secton 3 descrbes the portfolo transtons data used to test nvarance relatonshps concernng bet szes and transacton costs. The data set conssts of more than 400,000 portfolo transton orders executed over the perod 2001 through 2005 by a leadng vendor of portfolo transton servces. In portfolo transtons, nsttutonal fund sponsors hre a thrd party to execute the orders necessary to transfer funds from legacy portfolo managers to new managers n order to replace fund managers, change asset allocatons, or accommodate cash nflows and outflows. Portfolo transtons provde a good natural experment for dentfyng bets and measurng transacton costs.

4 MARKET MICROSTRUCTURE INVARIANCE 1347 Secton 4 examnes whether bet szes are consstent wth the nvarance-ofbets hypothess under the dentfyng assumpton that portfolo transton orders are proportonal to bets. When scaled as suggested by nvarance, the dstrbutons of portfolo transton orders are ndeed smlar across volume and volatlty groups. Regresson analyss also confrms ths fndng. Moreover, ths dstrbuton s well-descrbed by a log-normal wth estmated log-varance of 2 53 (Fgure 2). The bmodal dstrbuton of sgned order sze (obtaned by multplyng the sze of sell orders by 1) has much more kurtoss than the normal dstrbuton often assumed for analytcal convenence n the theoretcal lterature. The fat tals of the estmated log-normal dstrbuton suggest that very large bets represent a sgnfcant fracton of tradng volume and an even more sgnfcant fracton of returns varance. Kyle and Obzhaeva (2016a) nvestgated the dea that executon of large stock market bets may trgger stock market crashes. Secton 5 uses mplementaton shortfall to examne whether transacton costs are consstent wth the nvarance hypotheses. Even though our statstcal tests usually reject the nvarance hypothess, the results are economcally close to those mpled by nvarance. Consstent wth nvarance, transactoncost functons can be closely approxmated by the product of an asset-specfc llqudty measure (proportonal to the cube root of the rato of returns varance to dollar volume) and an nvarant functon of bet sze (Fgure 4). Invarance tself does not mpose a partcular form on the transacton-cost functon. Emprcally, both a lnear model and a square-root model explan transacton costs well. A square-root model explans transacton costs for orders n the 90th to 99th percentles better than a lnear model; a lnear model explans transacton costs for the largest 1% of orders slghtly better than the squareroot model. Quoted bd-ask spreads are also consstent wth the predctons of nvarance. Secton 6 calbrates several deep parameters and shows how to extrapolate them to obtan estmates for the dstrbuton of bet sze, the number of bets, and transacton-cost functons. Gven values of a tny number of proportonalty constants, the nvarance relatonshps allow mcroscopc features of the market for a fnancal asset, such as number of bets and ther sze, to be nferred from macroscopc market characterstcs, such as dollar volume and returns volatlty. The potental benefts of nvarance hypotheses for emprcal market mcrostructure are enormous. In the area of transacton-cost measurement, for example, controlled experments are costly and natural experments, such as portfolo transtons, are rare; even well-specfed tests of transacton-cost models tend to have low statstcal power. Market mcrostructure nvarance defnes parsmonous structural relatonshps leadng to precse predctons about how varous mcrostructure characterstcs, ncludng transacton costs, vary across tme and assets wth dfferent dollar volume and returns volatlty. These predctons can be tested wth structural estmates of a handful of parameters, poolng data from many dfferent assets.

5 1348 A. S. KYLE AND A. A. OBIZHAEVA Due to market frctons, we do not expect the emprcal nvarance hypotheses to hold exactly across all assets and all tmes. The predctons of nvarance may hold most closely when tck sze s small, market makers are compettve, and transacton fees and taxes are mnmal. If not, the nvarance hypotheses provde a benchmark from whch the mportance of these frctons can be compared across markets. Ths paper focuses on market mcrostructure nvarance as two emprcal hypotheses. Kyle and Obzhaeva (2016b) developed an equlbrum structural model n whch these hypotheses are endogenous mplcatons of a dynamc equlbrum model of nformed tradng. The model derves nvarance relatonshps under the assumpton that the effort requred to generate one dscrete bet does not vary across assets and tme. The dea of usng nvarance prncples n fnance and economcs, at least mplctly, s not new. The theory of Modglan and Mller (1958) s an example of an nvarance prncple. The dea of measurng tradng n fnancal markets n busness tme or transacton tme s not new ether. The tme-change lterature has a long hstory, begnnng wth Mandelbrot and Taylor (1967), who lnked busness tme to transactons, and Clark (1973), who lnked busness tme to volume. Allas (1956, 1966) are other early examples of models wth tme deformaton. More recent papers nclude Hasbrouck (1999), Ané and Geman (2000), Dufour and Engle (2000), Plerou, Gopkrshnan, Amaral, Gabax, and Stanley (2000), andderman (2002). Some of these papers are based on the dea that returns volatlty s constant n transacton tme. Ths s dfferent from the nvarance hypothess that the dollar rsks transferred by bets have the same probablty dstrbuton n bet tme. 1. MARKET MICROSTRUCTURE INVARIANCE AS EMPIRICAL HYPOTHESES Market mcrostructure characterstcs such as order sze, order arrval rate, prce mpact, and bd-ask spread vary across assets and across tme. We defne market mcrostructure nvarance as the emprcal hypotheses that these varatons almost dsappear when these characterstcs are examned at an asset-specfc busness-tme scale whch measures the rate at whch rsk transfers take place. Although the dscusson below s mostly based on cross-sectonal mplcatons of nvarance for equty markets for ndvdual stocks, we beleve that nvarance hypotheses generalze to markets for commodtes, bonds, currences, and aggregate ndces such as exchange-traded funds and stock ndex futures contracts. We also beleve that nvarance hypotheses generalze to tme seres. We wll thus use subscrpts j for assets and t for tme perods n what follows. For smplcty, we assume that a bet transfers only dosyncratc rsk about a sngle asset, not market rsk. Modelng both dosyncratc and market rsks smultaneously takes us beyond the scope of ths paper.

6 MARKET MICROSTRUCTURE INVARIANCE 1349 Bets and Busness Tme In the market for an ndvdual asset, nsttutonal asset managers buy and sell shares to mplement bets. Our concept of a bet s new. We thnk of a bet as a decson to acqure a long-term poston of a specfc sze, dstrbuted approxmately ndependently from other such decsons. Intermedares wth short-term tradng strateges market makers, hgh frequency traders, and other arbtragers clear markets by takng the other sde of bets placed by long-term traders. Bets can be dffcult for researchers to observe. Bets are nether orders nor trades nor prnts; bets are portfolo decsons whch mplement tradng deas; they are smlar to meta-orders. Consder an asset manager who places one bet by purchasng 100,000 shares of IBM stock. The bet mght be mplemented by placng orders over several days, and each of the orders mght be shredded nto many small trades. To mplement a bet, the trader mght place a sequence of orders to purchase 20,000 shares of stock per day for fve days n a row. Each of these orders mght be broken nto smaller peces for executon. For example, on day one, there may be trades of 2,000, 3,000, 5,000, and 10,000 shares executed at dfferent prces. Each of these smaller trades may show up n the Trade and Quote (TAQ) database as multple prnts. Snce the varous ndvdual orders, trades, and prnts are postvely correlated because they mplement a common bet, t would not be approprate to thnk of them as ndependent ncrements n the ntended order flow; they are peces of bets, not bets. To recover the sze of the orgnal bet, all trades whch mplement the bet must be added together. Thus, ndvdual bets are almost mpossble to reconstruct from publcly dssemnated records of tme-stamped prces and quanttes such as those contaned n TAQ data. Bets result from new deas, whch can be shared. If an analyst s recommendaton to buy a stock s followed by buy orders from multple customers, all of these orders are part of the same bet. For example, f an analyst ssues a buy recommendaton to ten dfferent customers and each of the customers quckly places executable orders to buy 10,000 shares, t mght be approprate to thnk of the ten orders as one bet for 100,000 shares. Snce the ten purchases are all based on the same nformaton, the ten ndvdual orders lack statstcal ndependence. Conceptually, t s ths ndependence property of bets that allows us to lnk ther arrval rate to the speed of busness tme. To fx deas, assume that bets arrve randomly. Let γ jt denote the expected arrvalrateofbetsnassetj at tme t; γ jt s measured n bets per calendar day. Suppose that a bet arrves at tme t. Let Q jt denote a random varable whose probablty dstrbuton represents the sgned sze of ths bet; Qjt s measured n shares (postve for buys, negatve for sells) wth E{ Q jt }=0. The expected

7 1350 A. S. KYLE AND A. A. OBIZHAEVA bet arrval rate γ jt measures market velocty, the rate at whch busness tme passes for a partcular asset. 2 The varables Q jt and γ jt are usually dffcult to observe. The nvarance hypotheses help to lnk these varables to volume and volatlty, whch are easer to observe. To set up ths lnk, t s frst useful to make two dentfyng assumptons. Strctly speakng, these assumptons are not necessary for developng the ntuton for nvarance. Instead, they help to defne ssues for future emprcal work. Frst, let V jt denote tradng volume, measured n shares per day. It conssts of bet volume reflectng the arrval of bets and ntermedaton volume reflectng trades of ntermedares. Assume that, on average, each unt of bet volume results n ζ jt unts of total volume, mplyng one unt of bet volume leads to ζ jt 1 unts of ntermedaton volume. If all trades are bets and there are no ntermedares, then ζ jt = 1, snce each unt of tradng volume matches a buybet wth a sell-bet. If a monopolstc specalst ntermedates all bets wthout nvolvement of other ntermedares, then ζ jt = 2. If each bet s ntermedated by dfferent market makers, each of whom lays off nventory by tradng wth other market makers, then ζ jt = 3. If postons are passed around among multple ntermedares, then ζ jt 4. Defne expected bet volume V jt as the share volume from bets, V jt := γ jt E{ Q jt }. In terms of bet volume V jt and the volume multpler ζ jt,tradng volume s equal to V jt = ζ jt /2 V jt, where dvdng by two mples that a buy-bet matched to a sell-bet s counted as one unt of volume, not two unts. Bet volume V jt and tradng volume V jt therefore satsfy the relatonshp (1) V jt := γ jt E { Q jt } = 2 V jt ζ jt Whle bet volume V jt s not drectly observed, the second equalty n equaton (1) shows how t can be nferred from tradng volume V jt f the volume multpler ζ jt s known. In what follows, we make the dentfyng assumpton, consstent wth Occam s razor, that ζ jt s constant across assets and tme; thus, for some constant ζ, we assume ζ jt = ζ for all j and t. Second, defne returns volatlty σ jt as the percentage standard devaton of an asset s daly returns. Some prce fluctuatons result from the market mpact of bets whle others result from release of nformaton drectly wthout tradng, such as overnght news announcements. Let ψ 2 jt denote the fracton of returns varance σ 2 jt resultng from bet-related order mbalances. Defne bet volatlty σ jt as the standard devaton of returns resultng from the market mpact 2 Over long perods of tme, the nventores of ntermedares are unlkely to grow n an unbounded manner; ths requres bets to have small negatve autocorrelaton. Also, both the bet arrval rate and the dstrbuton of bet sze change over longer perods of tme as the level of tradng actvty n an asset ncreases or decreases.

8 MARKET MICROSTRUCTURE INVARIANCE 1351 of bets, not news announcements. Bet volatlty σ jt and returns volatlty σ jt satsfy (2) σ jt = ψ jt σ jt Whle bet volatlty s not drectly observed, t can be nferred from returns volatlty σ jt usng the volatlty multpler ψ jt.letp jt denote the prce of the asset; then dollar bet volatlty s P jt σ jt = ψ jt P jt σ jt. To smplfy the emprcal analyss below, we make the dentfyng assumpton that ψ jt s the same across assets and tme; thus, for some constant ψ, we assume ψ jt = ψ for all j and t. To llustrate, f ζ = 2andψ = 0 80, then for all assets and tme perods, bets are ntermedated by a monopolst market maker, and bets generate 64 percent of returns varance, whereas the remanng 36 percent of returns varance comes from news announcements. The assumptons that ζ jt and ψ jt are constants are mportant for testng the predctons of market mcrostructure nvarance emprcally. These assumptons can be tested emprcally. If ζ jt and ψ jt are correlated wth V jt and σ jt, emprcal estmates of parameters predcted by nvarance may be based. An nterestng alternatve approach, whch takes us beyond the scope of ths paper, s to examne these correlatons emprcally and then to make necessary adjustments n tests of our nvarance hypotheses. The assumptons that ζ jt and ψ jt are constants are not mportant for understandng market mcrostructure nvarance theoretcally. To understand nvarance theoretcally, t suffces to assume ζ = 2andψ = 1, n whch case V jt = V jt and σ jt = σ jt, so that the dstncton between varables wth and wthout bars can be gnored. Invarance of Bets We call our frst nvarance hypothess nvarance of bets. Snce busness tme s lnked to the expected arrval of bets γ jt, returns volatlty n one unt of busness tme 1/γ jt s equal to σ jt γ 1/2 jt. A bet of dollar sze P jt Q jt generates a standard devaton of dollar mark-to-market gans or losses equal to P jt Q jt σ jt γ 1/2 jt n one unt of busness tme. The sgned standard devaton P jt Q jt σ jt γ 1/2 jt measures both the drecton and the dollar sze of the rsk transfer resultng from the bet. The sze of the bet can be measured as the dollar amount of rsk t transfers per unt of busness tme, whch we denote Ĩ jt and defne by (3) Ĩ jt := P jt Q jt σ jt γ 1/2 jt Snce the prmary functon of fnancal markets s to transfer rsks, t s economcally more meanngful to measure the sze of bets n terms of the dollar

9 1352 A. S. KYLE AND A. A. OBIZHAEVA rsks they transfer rather than the dollar value or number of shares transacted. Indeed, transactons can be large n terms of shares traded but small n terms of dollar amounts transacted, as n markets for low-prced stocks. Transactons can also be large n dollar terms but small n terms of rsks transferred, as n the market for U.S. Treasury blls wth low returns volatlty. Furthermore, as dscussed next, we beleve that emprcal regulartes n bet sze and transacton costs become more apparent when returns volatlty s examned n busnesstme unts. The varable Ĩ jt n equaton (3) s a good canddate for measurng the economc content of bets because t s mmune to splts and changes n leverage. Indeed, a stock splt whch changes the number of shares does not change the dollar sze of a bet P jt Q jt, returns volatlty σ jt, or the number of bets γ jt n equaton (3). For example, a two-for-one stock splt should theoretcally double the share volume of bets, but reduce by one-half the dollar value of each share, wthout affectng dollar sze of bets or returns volatlty. Also, a change n leverage does not change dollar volatlty P jt σ jt, contract sze of a bet Qjt, or the number of bets γ jt n equaton (3). For example, f a company levers up ts equty by payng a debt-fnanced cash dvdend equal to ffty percent of the value of the equty, then the volatlty of the remanng equty, ex-dvdend, should double, whle the prce should halve, thus keepng dollar volatlty constant. Ths s consstent wth the ntuton that each share of leveraged stock stll represents the same pro rata share of frm rsk as a share of un-leveraged stock. HYPOTHESIS Invarance of Bets: The dstrbuton of the dollar rsk transferred by a bet n unts of busness tme s the same across asset j and tme t, n the sense that there exsts a random varable Ĩ such that for any j and t, (4) Ĩ jt d = Ĩ that s, the dstrbuton of rsk transfers Ĩ jt s a market mcrostructure nvarant. Ths hypothess mples that the dstrbuton of bet szes s such that for any asset j and tme t, bets n the same percentle transfer rsks of the same sze n busness tme. It does not say that volatlty n busness tme s constant. Consder the followng numercal example. Suppose that a 99th percentle bet n stock A s for $10 mllon (e.g., 100,000 shares at $100 per share) whle a 99th percentle bet n stock B s for $1 mllon (e.g., 100,000 shares at $10 per share). The dollar szes of these bets dffer by a factor of 10. Snce both bets occupy the same percentle n the bet-sze dstrbuton for ther respectve stocks, the nvarance of bets mples that the realzed value of Ĩ jt s the same n both cases. Even though stock A may be more actvely traded than stock B, ts returns volatlty per unt of busness tme must be lower by a factor of 10

10 MARKET MICROSTRUCTURE INVARIANCE 1353 for the nvarance hypothess to hold. For example, f the stocks have the same volatlty n calendar tme, say two percent per day, then the arrval rate of bets for stock A must be 100 = 10 2 tmes greater than stock B. We next derve the mplcatons of ths nvarance hypothess for observable calendar-tme measures of volume and volatlty. By analogy wth the defnton of Ĩ jt, defne tradng actvty W jt as the product of expected dollar tradng volume P jt V jt and calendar returns volatlty σ jt : (5) W jt := σ jt P jt V jt Tradng actvty measures the aggregate dollar rsk transferred by all bets durng one calendar day. Smlarly, defne bet actvty W jt as the product of dollar bet volume P jt V jt and bet volatlty σ jt, that s, W jt := σ jt P jt V jt. 3 Gven values of the volume multpler ζ jt and the volatlty multpler ψ jt, more-easlyobserved tradng actvty W jt can be converted nto less-easly-observed bet actvty W jt usng the relatonshp W jt = W jt 2ψ jt /ζ jt. Bet actvty W jt can be expressed as the product of an nvarant constant and a power of unobservable market velocty γ jt : (6) W jt = σ jt P jt γ jt E { Q jt } = γ 3/2 jt E { Ĩ jt } = γ 3/2 jt E { Ĩ } In equaton (6), the frst equalty follows from the defnton of W jt and equaton (1), the second equalty follows from equaton (3), and the thrd equalty follows from the nvarance of bets (4). Invarance of bets therefore makes t possble to nfer market velocty γ jt from the level of bet actvty W jt,upto some dollar proportonalty constant E{ Ĩ }, whch accordng the nvarance of bets does not vary across assets j or tmes t. Defne ι := (E{ Ĩ }) 1/3 ;snceĩ has an nvarant probablty dstrbuton, ι s a constant. Equaton (6) makes t possble to express the unobservable bet arrval rate γ jt and the expected sze of bets E{ Q jt } n terms of the observable varables P jt, V jt,andσ jt (and W jt ): (7) γ jt = W 2/3 jt ι 2 E { Q jt } = W 1/3 1 jt ι 2 P jt σ jt The shape of the entre dstrbuton of bet sze Q jt can be obtaned by pluggng γ jt from equaton (7) nto equaton (3). Traders often measure the sze of orders as a fracton of average daly volume. Smlarly, expressng bet sze Q jt 3 In prncple, we could dstngush between P jt and P jt based on adjustments for transacton fees, fee rebates, taxes, and tck sze effects. To keep matters smple, we gnore these ssues and effectvely assume P jt = P jt.

11 1354 A. S. KYLE AND A. A. OBIZHAEVA as a fracton of expected bet volume V jt yelds the followng predcton for the dstrbuton of bet szes: (8) Q jt V jt d = W 2/3 jt Ĩ ι Equatons (7) and(8) summarze the emprcal mplcatons of nvarance for the dstrbuton of bet sze Q jt and the arrval rate of bets γ jt. We test these mplcatons n Secton 4. Equaton (7) descrbes the mpled composton of order flow. The specfc exponents 2/3 and1/3 are very mportant. Equaton (7) mples that f W jt ncreases by one percent, then the arrval rate of bets γ jt ncreases by 2/3 ofone percent and the dstrbuton of bet sze Q jt shfts upwards by 1/3 ofonepercent. The exponents 1/3 and2/3 have smple ntuton. For example, suppose the expected arrval rate of bets γ jt speeds up by a factor of 4, but volatlty n calendar tme σ jt does not change. Then volatlty per unt of busness tme σ γ 1/2 jt decreases by a factor of 2. The nvarance hypothess (4) therefore requres bet sze Q jt to ncrease by a factor of 2 to keep the dstrbuton of Ĩ jt nvarant. The resultng ncrease n volume by a factor of 8 = 4 3/2 can be decomposed nto an ncrease n the number of bets by a factor of 8 2/3 = 4andan ncrease n the sze of bets by a factor of 8 1/3 = 2. As bet actvty ncreases, the number of bets ncreases twce as fast as ther sze. Ths specfc relatonshp between the number and sze of bets les at the very heart of nvarance. By suggestng a partcular composton of order flow, nvarance lnks observable volume to unobservable order mbalances, whch n turn has further mplcatons for transacton costs. Indeed, our next hypothess about transacton costs reles on the order flow havng ths specfc composton. Invarance of Transacton Costs We call our second nvarance hypothess nvarance of transacton costs. The rsk transferred per unt of busness tme by a bet of Q jt shares s measured by Ĩ jt = P jt Qjt σ jt γ 1/2 jt.letc B jt (Ĩ jt ) denote the expected dollar cost of executng ths bet. HYPOTHESIS Invarance of Transacton Costs: The dollar expected transacton cost of executng a bet s the same functon of the sze of the bet when ts sze s measured as the dollar rsk t transfers n unts of busness tme, n the sense that there exsts a functon C B (I) such that for any j and t, (9) C B jt (I) = C B (I)

12 MARKET MICROSTRUCTURE INVARIANCE 1355 that s, the dollar transacton-cost functon C B jt (I) s a market mcrostructure nvarant. Ths hypothess mples that the dollar cost of executng bets of the same percentle s the same across tme and assets. Defne the uncondtonal expected dollar cost C B jt := E{C B jt (Ĩ jt )}. Invarance of transacton costs mples C B jt := E{C B (Ĩ jt )}, and nvarance of bets further mples C B jt = C B for some constant C B. The hypothess says that the dollar costs, not the percentage costs, are the same across tme or across assets for orders n the same percentles. As n the prevous example, suppose that a 99th percentle bet n stock A s for $10 mllon whle a 99th percentle bet n stock B s for $1 mllon (e.g., 100,000 shares at $10 per share). Whle the dollar szes of these bets are dfferent, ther correspondng measures of dollar rsk transfers are the same. Even though the bet n stock A has 10 tmes the dollar value of the bet n stock B, nvarance of transacton costs mples that the expected cost of executng both bets must be the same n dollars because both bets transfer the same amount of rsk per stock-specfc unt of busness tme. Traders typcally measure transacton costs n bass ponts, not dollars. Invarance of transacton costs mples that the percentage transacton cost for stock B must be 10 tmes greater than for stock A. As dscussed next, nvarance of transacton costs places strong, emprcal testable restrctons on transacton-cost models. Consder the mplcatons for the percentage costs of executng bets. Let C jt (Q) denote the asset-specfc expected cost of executng a bet of Q shares, expressed as a fracton of the notonal value of the bet P jt Q : (10) C jt (Q) := C B jt(i) P jt Q where I P jt Q σ jt γ 1/2 jt The notatons Q and I are two ways to refer to the same bet. The quantty I rescales the bet from share unts nto dollar unts so that bets become comparable across assets and tme. Usng equaton (3), ths percentage cost functon can be expressed as the product of two factors: (11) C jt (Q) = C B jt E { P jt Q jt } CB jt(i)/ C B jt I /E { Ĩ jt } We next dscuss these two factors separately n more detal. The frst factor on the rght sde of equaton (11), denoted 1/L jt, s the assetspecfc lqudty measure defned by (12) 1 C B jt := L jt E { P jt Q jt }

13 1356 A. S. KYLE AND A. A. OBIZHAEVA It measures the dollar-volume-weghted expected percentage cost of executng a bet. For an asset manager who places many bets n the same asset, ths measure ntutvely expresses the expected transacton cost as a fracton of the dollar value traded. For example, f an asset manager executes $10 mllon at a cost of 20 bass ponts, $5 mllon at a cost of 10 bass ponts, and $100 mllon at a cost of 80 bass ponts, then the mpled approxmaton for ths measure of llqudty 1/L jt s equal to 72 bass ponts (= ( )/115). The second factor on the rght sde of equaton (11), denoted f(i), s the nvarant average cost functon defned as (13) f(i):= C B jt(i)/ C B jt I /E { Ĩ jt } = C B(I)/ C B I /E { Ĩ } The second equalty follows drectly from the nvarance hypotheses. Thus, the two nvarance hypotheses mply that the functon f(i), whch does not requre subscrpts j and t, descrbes the shape of transacton-cost functons n a manner that does not vary across assets or across tme. Intutvely, the functon f(i) s the rato of C B (I) to I when both are expressed as multples of the means of C B (Ĩ) and Ĩ, respectvely. For example, f I 1 denotes a bet that s equal to an average unsgned bet of sze E{ Ĩ } and ts dollar cost s 1 20 tmes hgher than the average dollar cost C B, then f(i 1 ) = If I 2 denotes a bet that s 5 tmes greater than an average unsgned bet of sze E{ Ĩ } and ts dollar cost s 10 tmes greater than the average dollar cost C B, then f(i 2 ) = 10/5 = 2. To summarze, we obtan the followng mportant decomposton of transacton-cost functons: THEOREM Decomposton of Transacton-Cost Functons: The percentage transacton cost C jt (Q) of executng a bet of Q shares n asset j at tme t s equal to the product of the asset-specfc llqudty measure 1/L jt and an nvarant transacton-cost functon f(i), (14) C jt (Q) = 1 f(i) where I P jt Q L jt σ jt γ 1/2 jt The decomposton (14) represents the strong restrcton whch nvarance hypotheses place on transacton-cost models. The percentage transacton-cost functon C jt (Q) vares sgnfcantly across assets and tme. When ts argument Q s converted nto an equvalent rsk-transfer I and ts value s scaled wth the asset-specfc llqudty ndex 1/L jt, then ths functon turns nto an nvarant functon f(i).aswedscussnext,1/l jt s proportonal to returns volatlty n busness tme.

14 MARKET MICROSTRUCTURE INVARIANCE 1357 An Illqudty Measure It can be shown that the asset-specfc llqudty measure 1/L jt,defnedn equaton (12), s proportonal to bet-nduced returns volatlty n busness tme. Furthermore, t can be also convenently expressed n terms of observable volume and volatlty. THEOREM Illqudty Index: The llqudty ndex 1/L jt for asset j and tme t satsfes (15) 1 = C B L jt E { Ĩ } σ jt = ι 2 γ 1/2 CB jt σ jt = W 1/3 jt ( ζψ 2 2 ) 1/3 ( σ 2 ) 1/3 ι 2 jt CB P jt V jt The frst equalty says that 1/L jt s proportonal to returns volatlty n busness tme σ jt γ 1/2 jt wth nvarant proportonalty factor C B /E{ Ĩ }. The second equalty expresses 1/L jt n terms of bet actvty W jt and bet volatlty σ jt.the thrd equalty expresses 1/L jt as the cube root of the asset-specfc rato of observable returns varance σ 2 to observable dollar volume P jt jt V jt, multpled by an nvarant proportonalty factor. In equaton (15), the frst equalty follows from the defnton of I jt n equaton (3). The second equalty can be proved usng equaton (7)forγ jt. The thrd equalty follows from the defnton of W jt n equaton (6), from equatons (1) and (2), and the dentfyng assumpton ζ jt = ζ and ψ jt = ψ. The rght sde of equaton (15) s the asset-specfc llqudty ndex mpled by nvarance. Snce the volume multpler ζ and the volatlty multpler ψ are assumed not to vary across assets, the quantty L jt [P jt V jt /σ 2 jt ]1/3 becomes a smple ndex of lqudty based on observable volume and volatlty, wth a proportonalty constant that does not vary wth j and t. The dea that lqudty s related to dollar volume per unt of returns varance P jt V jt /σ 2 jt s ntutve. Traders beleve that transacton costs are hgh n markets wth low dollar volume and hgh volatlty. The cube root s necessary to make the llqudty measure behave properly when leverage changes. If a stock s levered up by a factor of 2, then P jt halves and σ 2 jt ncreases by a factor of 4. Wthout the cube root, the rato of returns varance to dollar volume therefore ncreases by a factor of 8. If dollar rsk transfers do not change, then the dollar transacton cost should not change ether. Ths requres percentage transacton costs to double snce the dollar sze of bets P jt Q jt halves whle the share sze Q jt remans the same. Takng a cube root changes the factor of 8 to 2, so that percentage transacton costs double as requred. As dscussed n the next secton, the lqudty measure L jt s an ntutve and practcal alternatve to other measures of lqudty, such as Amhud (2002)and Stambough and Pastor (2003). Its value can easly be calbrated from prce and volume data provded by the Center for Research n Securty Prces (CRSP).

15 1358 A. S. KYLE AND A. A. OBIZHAEVA The lqudty measure L jt n equaton (15) s also smlar to the defnton of market temperature χ = σ jt γ 1/2 jt n Derman (2002); substtutng for γ jt from equaton (7), we obtan χ = ι [P jt V jt ] 1/3 [ σ jt ] 4/3 L jt σ 2. jt Whle 1/L jt s defned as a measure of tradng llqudty, t may also be a good measure of fundng llqudty as well. A reasonable measure of fundng lqudty s the recprocal of a repo harcut that suffcently protects a credtor from losses f the credtor sells the collateral due to default by the borrower. Such a harcut should be proportonal to the volatlty of the asset s return over the horzon durng whch defaulted collateral would be lqudated. As suggested by nvarance, ths horzon should be proportonal to busness tme 1/γ jt, makng volatlty over the lqudaton horzon proportonal to σ jt γ 1/2 jt, whch as we showed earler s tself proportonal to 1/L jt. Thus, nvarance suggests that both tradng lqudty and fundng lqudty are proportonal to L jt. Transacton-Cost Models When bet sze s measured as a fracton of bet volume Q/ V jt, the cost functon C jt (Q) can be also expressed convenently n terms of bet actvty W jt and the nvarant constants ι := (E{ Ĩ }) 1/3 and C B as (16) C jt (Q) = σ jt W 1/3 jt ( W 2/3 ι 2 jt CB f ι Q ) V jt Ths can be proved usng equaton (14), the defnton of 1/L jt n equaton (12), the nvarance of bets (7), the nvarance of transacton costs (9), the defnton of Ĩ jt n equaton (3), and the defnton of γ jt n equaton (7). Ths s the transacton-costs model mpled by nvarance. We test ths specfcaton emprcally n Secton 5. The two nvarance hypotheses do not mply a specfc functonal form for functon f(i) n the transacton-cost model (16). In what follows, we focus on two specfc functonal forms as benchmarks: lnear prce-mpact costs and square-root prce-mpact costs. Both are specal cases of a more general power functon specfcaton for f(i). The lnear prce-mpact functon s consstent wth prce-mpact models based on adverse selecton, such as Kyle (1985).The square-root prce-mpact functon s consstent wth emprcal fndngs n the econophyscs lterature, such as Gabax, Gopkrshnan, Plerou, and Stanley (2006), although these results are based on sngle trades rather than bets. Some papers, ncludng Almgren, Thum, Hauptmann, and Hong (2005), fnd an exponent closer to 0 60 than to the square-root exponent For both functonal forms, we also nclude a proportonal bd-ask spread cost component. For the lnear model, express f(ĩ) as the sum of a bd-ask spread component and a lnear prce-mpact cost component, f(ĩ) := (ι 2 CB ) 1 κ 0 + (ι C B ) 1

16 MARKET MICROSTRUCTURE INVARIANCE 1359 κ I Ĩ, where nvarance mples that the bd-ask spread cost parameter κ 0, the market mpact cost parameter κ I, and the constants ι and C B do not vary across assets. Snce the specfc coeffcents ι 2 CB and ι C B n the specfcaton for f(ĩ) are chosen to cancel out n a nce way, equaton (14) mples that the cost functon C jt (Q) has the smple form (17) ( C jt (Q) = σ jt κ 0 W 1/3 jt + κ I W 1/3 jt Q ) V jt When bet szes are measured as a fracton of expected bet volume and transacton costs are measured n bass ponts and further scaled n unts of bet volatlty σ jt,equaton(17) says that bd-ask spread costs are proportonal W 1/3 jt and market mpact costs are proportonal to W 1/3 jt for a gven fracton of bet volume. For the square-root model, express f(ĩ) as the sum of a bd-ask spread component and a square-root functon of Ĩ, obtanng f(ĩ) := (ι 2 CB ) 1 κ 0 + (ι 3/2 CB ) 1 κ I Ĩ 1/2, where nvarance mples that κ 0, κ I, ι, and C B do not vary across assets. The proportonal cost functon C jt (Q) from (14) s then gven by (18) ( C jt (Q) = σ jt κ 0 W 1/3 jt + κ I Q V jt 1/2) When transacton costs are measured n unts of bet volatlty σ jt,bd-ask spread costs reman proportonal to W 1/3 jt, but the square-root model mples that the bet actvty coeffcent W 1/3 jt cancels out of the prce-mpact term. Indeed, the square root s the only functon for whch nvarance leads to the emprcal predcton that mpact costs (measured n unts of returns volatlty) depend only on bet sze as a fracton of bet volume Q jt / V jt and not on any other asset characterstcs. If there are no bd-ask spread costs so that κ 0 = 0, then the square-root model mples the parsmonous transacton-cost functon C jt (Q) = σ jt κ I [ Q / V jt ] 1/2. Torre (1997) proposed a square-root model lke specfcaton (18) basedon emprcal regulartes observed by Loeb (1983). Practtoners sometmes refer to t as the Barra model. Grnold and Kahn (1999) used an nventory rsk model to derve a square-root prce-mpact formula. Gabax et al. (2006) formalzed ths approach under the assumptons that orders are executed as a constant fracton of volume and lqudty provders have mean-varance utlty functons lnear n expected wealth and ts standard devaton (not varance). To formalze our predctons about the bd-ask spread, let s jt denote the dollar bd-ask spread. As shown earler for ntercepts n equaton (17) or(18), the

17 1360 A. S. KYLE AND A. A. OBIZHAEVA nvarance hypotheses mply that the percentage spread s jt /P jt s proportonal to σ jt W 1/3 jt : (19) s jt = κ 0 σ jt W 1/3 jt P jt For example, holdng volatlty constant, ncreasng tradng actvty by a factor of 8 reduces the percentage bd-ask spread by a factor of 2. Equatons (15) and (19) above mply that the percentage bd-ask spread s proportonal to both the llqudty measure 1/L jt and to volatlty n busness tme σ jt γ 1/2 jt, wth nvarant constants of proportonalty. Numercal Example The followng numercal example llustrates the nvarance hypotheses. Suppose a stock has daly volume of $40 mllon and daly returns volatlty of 2%. Suppose there are approxmately 100 bets per day and the mean sze of a bet s $400,000. A daly volatlty of 2% mples a standard devaton of mark-tomarket dollar gans and losses equal to $8,000 per calendar day (2% tmes $400,000) for the mean bet. Snce busness tme passes at the rate bets arrve nto the market, 100 bets per day mples about 1 bet every 4 mnutes; the busness clock therefore tcks once every 4 mnutes. Over a 4-mnute perod, the standard devaton of returns s 20 bass ponts (200/ 100). Thus, the mean bet has a standard devaton of rsk transfer of $800 per unt of busness tme. Invarance mples that the specfc number $800 s constant across stocks and across tme. For example, f the arrval rate of bets ncreases by a factor of 4, then the busness clock tcks 4 tmes faster or once every mnute, and the standard devaton of returns per tck on that clock s reduced from 20 bass ponts to 10 bass ponts (200/ 400), keepng daly volatlty constant. For the standard devaton of mark-to-market dollar gans and losses on the mean bet to reman constant at $800, nvarance mples that the dollar sze of the mean bet must ncrease by a factor of 2 from $400,000 to $800,000. Thus, holdng volatlty constant, the bet arrval rate ncreases by a factor of 4 and the sze of bets ncreases by a factor of 2. Ths mples that daly volume ncreases by a factor of 8 from $40 mllon to $320 mllon. Holdng volatlty constant, the bet arrval rate ncreases by a factor proportonal to the 2/3 power of the factor by whch dollar volume changes, and the sze of bets ncreases by the 1/3 power of the factor by whch dollar volume changes. Invarance of transacton costs further says that the dollar cost of executng a bet of a gven sze percentle s the same across dfferent stocks. The dollar costs of executng the mean bets s therefore the same constant across assets and across tme; suppose t s equal to $2,000. Then, doublng the dollar sze of the mean bet from $400,000 n the frst stock to $800,000 n the second stock decreases the cost of $2,000, measured n bass ponts, by a factor of 2 from 50

18 MARKET MICROSTRUCTURE INVARIANCE 1361 bass ponts to 25 bass ponts. The percentage transacton cost of executng a bet s therefore nversely proportonal to the 1/3 power of the factor by whch dollar volume changes, holdng volatlty constant. Smlar arguments are vald for bets n dfferent percentles of the bet-sze dstrbuton. Suppose, for example, that the standard devaton of mark-tomarket gans or losses on a 99th percentle bet s $10,000 per tck of busness tme. Snce the standard devaton of returns per tck of busness tme s equal to 20 bass ponts n the frst stock and 10 bass ponts n the second stock, the sze of such a 99th percentle bet s equal to $5 mllon n the frst stock and $10 mllon n the second stock; the 99th percentle bets dffer n dollar sze by a factor of 2. Suppose the dollar costs of executng all 99th percentle bets s equal to $50,000. Ths corresponds to a percentage cost of 100 bass ponts n the frst stock and 50 bass ponts n the second stock. The percentage cost for the second stock s lower by a factor of 2; ths dfference s nversely proportonal to the 1/3 power of the factor by whch dollar volume changes, holdng volatlty constant. Smlarly, the percentage bd-ask spread of the second stock wll be lower by a factor of 2 than the percentage bd-ask spread of the frst stock. Dscusson Our nvarance hypotheses have essental propertes whch potentally allow them to be extended to more general settngs. Frst, nvarance relatonshps are consstent wth rrelevance of the unts n whch tme s measured. The values of I, C B jt (I), f(i),and1/l jt and therefore the economc content of the predctons of nvarance reman the same regardless of whether researchers measure γ jt, V jt, σ jt,and W jt usng daly weekly, monthly, or annual unts of tme. Ths s unlke some other models, such as ARCH and GARCH. Second, nvarance relatonshps are based on the mplct assumpton that bets are executed at an endogenously determned natural speed that trades off the benefts of faster executon aganst hgher transacton costs. Invarance does not rule out the possblty that unusually fast executon of a bet would lead to executon costs hgher than the costs mpled by the functons C B jt (I) and f(i). For example, t s possble to consder more general nvarant cost functons C B jt (I T/γ jt ) and f(i T/γ jt ) that depend not only on the sze of bets but also on executon horzons T converted from unts of calendar tme nto unts of busness tme T/γ jt. Thrd, the values of nvarants Ĩ and C B (I) are measured n dollars. Although not consdered n the current paper, nvarance relatonshps can also be appled to an nternatonal context n whch markets have dfferent currences or dfferent real exchange rates; they can also be appled across perods of tme when the prce level s changng sgnfcantly. Invarance s consstent wth the dea that these nomnal values Ĩ and C B (I) should be equal to the nomnal cost

19 1362 A. S. KYLE AND A. A. OBIZHAEVA of fnancal servces calculated from the productvty-adjusted wages of fnance professonals n the local currency of the gven market durng the gven tme perod. Snce wages are measured n dollars per day and productvty s measured n bets per day, the rato of wages to productvty s measured n dollars per bet, exactly the same unts as Ĩ and C B (I). Lke fundamental constants n physcs, dvdng the nvarants Ĩ and C B (I) by the rato of wages to productvty makes them dmensonless. Fourth, t s possble to develop an equlbrum market mcrostructure model that endogenously generates nvarance hypotheses. Several modelng assumptons are essental. Tradng volume results from bets placed by traders. Bets nduce prce volatlty, and long-term prce mpact of bets s lnear n ts sze. The effort requred to generate one bet s the same across assets and tme. There are no barrers of entry nto securtes tradng. Then, f for some reason proft opportuntes ncrease, more traders enter the market, the market becomes more lqud, prces become more accurate, profts per trader decrease, and traders scale up szes of ther bets n order to break even and to contnue coverng the costs of generatng tradng deas. Tradng volume ncreases due to both an ncrease n the number of bets and an ncrease n ther szes. The order flow has a specfc 2/3 1/3 composton because bet szes must be kept nversely proportonal to returns volatlty per unt of busness tme for the profts of traders to reman constant across assets and tme. Kyle and Obzhaeva (2016b) dscussed a model along these lnes. We model market mcrostructure usng nvarance n a manner smlar to the way modern physcsts model turbulence. Kolmogorov (1941a) derved hs two-thrds law (or fve-thrds law ) for the energy dstrbuton n a turbulent flud based on dmensonal analyss and scalng. 4 Ouranalysssalsosmlar n sprt to nferrng the sze and number of molecules n a mole of gas from measurable large-scale physcal quanttes. 2. MICROSTRUCTURE INVARIANCE IN THE CONTEXT OF THE MARKET MICROSTRUCTURE LITERATURE Market mcrostructure nvarance bulds a brdge from theoretcal models of market mcrostructure to emprcal tests of those models. Theoretcal mcrostructure models usually suggest measures of lqudty based on the dea that order mbalances move prces. By scalng busness tme to be proportonal to the rate at whch bets arrve, market mcrostructure nvarance mposes cross-sectonal or tme-seres restrctons whch make t easer to mplement lqudty measures based on order mbalances. 4 We thank an anonymous referee and Sergey N. Smrnov for pontng out the connecton to Kolmogorov s model of turbulence.

20 MARKET MICROSTRUCTURE INVARIANCE 1363 Many theoretcal models use game theory to model tradng. These models typcally make specfc assumptons about the rsk averson of traders, the consstency of belefs across traders, the flow of publc and prvate nformaton whch nformed traders use to trade, the flow of orders from lqudty traders, and aucton mechansms n the context of whch market makers compete to take the other sdes of trades. Some models emphasze adverse selecton, such as Treynor (1995), Kyle (1985), Glosten and Mlgrom (1985), andback and Baruch (2004); some models emphasze nventory dynamcs, such Grossman and Mller (1988) and Campbell and Kyle (1993); some models emphasze both, such as Grossman and Stgltz (1980) and Wang (1993). Whle these theoretcal models are all based on the dea that order mbalances move prces (wth partcular parameters dependng on specfcs of each model), t s dffcult to nfer precse emprcal mplcatons from these models. Theoretcal models usually provde nether a unfed framework for mappng the theoretcal concept of an order mbalance nto ts emprcal measurements nor precse predctons concernng how prce mpact vares across dfferent assets. Instead, researchers have taken an approach based on ad hoc emprcal ntuton. For example, prce changes can be regressed on mperfect emprcal proxes for order mbalances for example, the dfference between uptck and downtck volume, popularzed by Lee and Ready (1991) to obtan market mpact coeffcents, whch can then be related to stock characterstcs such as market captalzaton, tradng volume, and volatlty. Breen, Hodrck, and Korajczyk (2002) s an example of ths approach. A volumnous emprcal lterature descrbes how the rate at whch orders arrve n calendar tme, the dollar sze of orders, the market mpact costs, and bd-ask spread costs vary across dfferent assets. For example, Brennan and Subrahmanyam (1998) estmated order sze as a functon of varous stock characterstcs. Hasbrouck (2007) and Holden and Jacobsen (2014) provded surveys of ths emprcal lterature. In contrast to the exstng lterature, mcrostructure nvarance generates precse, emprcally testable predctons about how the sze of bets, arrval rate of bets, market mpact costs, and bd-ask spread costs vary across assets wth dfferent levels of tradng actvty. These predctons are consstent wth ntuton shared by many models. The undentfed parameters n theoretcal models show up as nvarant constants (e.g., E{ Ĩ } and C B ), whch can be calbrated from data. In ths sense, mcrostructure nvarance s a modelng prncple applcable to dfferent models, not a model tself. It complements theoretcal models by makng t easer to test them emprcally.

21 1364 A. S. KYLE AND A. A. OBIZHAEVA Applyng Invarance to the Model of Kyle (1985) We use the contnuous-tme model of Kyle (1985) as an example to dscuss how nvarance helps to map the theoretcal predctons of a model of order mbalances to data on volume and volatlty. 5 In ths model, the market depth formula λ = σ V /σ U measures market depth (n unts of dollars per share-squared) as the rato of the standard devaton of asset prce changes σ V (measured n dollars per share per square root of tme) to the standard devaton n order mbalances σ U (measured n shares per square root of tme). Ths formula asserts that prce fluctuatons result from the lnear mpact of order mbalances. The market depth formula tself does not depend on specfc assumptons about nteractons among market makers, nformed traders, and nose traders. An emprcal mplementaton of the market mpact formula λ = σ V /σ U should not be consdered a test of the specfc assumptons of the model of Kyle (1985), such as the exstence of a monopolstc nformed trader who trades smoothly and patently n a context where less patent lqudty traders trade more aggressvely and market makers set asset prces effcently. Instead, emprcal mplementaton of the formula λ = σ V /σ U attempts the more general task of measurng a market mpact coeffcent λ based on the assumpton that prce fluctuatons result from the lnear mpact of order-flow nnovatons, a property shared by many models. Measurng the numerator σ V s much more straghtforward than measurng the denomnator σ U. The value of σ V s easly nferred from the asset prce and returns volatlty. We have σ V = σ jt P jt. Measurng the denomnator σ U s dffcult because the connecton between observed tradng volume and order mbalances s not straghtforward. Intutvely, σ U should be related to tradng volume n some way. The contnuoustme model provdes no help concernng what ths relatonshp s; n the Brownan moton model of Kyle (1985), tradng volume s nfnte. Wthout some other approach for measurng σ U, the model s not testable. We can thnk of Brownan moton as an approxmaton to order mbalances resultng from dscrete, random, zero-mean decsons by traders to change asset holdngs. We call these decsons bets. Snce bets are ndependently dstrbuted, the standard devaton of order mbalances s gven by σ U = γ 1/2 jt [E{ Q 2 jt }]1/2. Ths approach s also consstent wth the sprt of other models, such as Glosten and Mlgrom (1985) and Back and Baruch (2004). 5 To smplfy ths dscusson, we omt the dstncton between varables wth and wthout a bar by assumng ζ = 2, ψ = 1, and therefore σ jt = σ jt and V jt = V jt.

22 MARKET MICROSTRUCTURE INVARIANCE 1365 The formulas for the numerator σ V and denomnator σ U mply that the prce mpact of a bet of X shares, expressed as a fracton of the value of a share P jt, s gven by (20) λ jt P 2 jt (X P jt ) = σ V σ U X P jt = σ jt γ 1/2 jt [ E { Q2 jt X }] 1/2 Ths formula reflects smple ntuton. A one standard devaton bet X = [E{ Q 2 jt }]1/2 has a prce mpact σ jt γ 1/2 jt equal to one standard devaton of returns volatlty σ jt measured over a tme nterval 1/γ jt equal to the expected tme nterval between bet arrvals. Emprcal tests of ths formula requre assumptons about how Q jt and γ jt vary wth volume and volatlty so that the standard devaton of order mbalances can be calculated. The nvarance of bets provdes the requred assumptons. Usng equatons (7) and(8) to determne how γ jt and moments of Q jt vary wth observable volume and volatlty, t follows that the prce-mpact cost of an order of dollar sze X P jt, as a fracton of the value traded, s (21) λ jt P 2 jt (X P jt ) = σ jt P jt γ 1/2 jt [ { }] 2/3 E Ĩ = [ {Ĩ2 }] 1/2 E (E Q ) 2 1/2 (X P jt ) jt σ jt W 1/3 jt (X P jt ) P jt V jt The percentage prce mpact s proportonal to W 1/3 jt σ jt /(P jt V jt ),whchtself s proportonal to the squared llqudty measure 1/L 2 jt. Invarance of bets makes the proportonalty factor [E{Ĩ 2 }] 1/2 [E{ Ĩ }] 2/3 nvarant. Thus, mplementaton of the market mpact formula (21) requres calbraton of only one proportonalty constant for all assets and all tme perods. By applyng the nvarance-of-bets hypothess to the model of Kyle (1985), wehaveobtaneda lnear verson of nvarance of transactons costs consstent wth equaton (17) wth no bd-ask spread term. Note that ths constant does not depend on the unts of tme n whch varables are measured, because Ĩ s measured n unts of dollars. As an alternatve to nvarance, the formula λ = σ V /σ U can be mplemented emprcally by mposng dfferent assumptons concernng the connecton between σ U and tradng volume. For example, we can assume that the expected arrval rate of bets s some unknown constant, the same for all assets and tme perods; ths wll further mply that σ U s proportonal to volume V jt and the

23 1366 A. S. KYLE AND A. A. OBIZHAEVA llqudty measure n equaton (20) s proportonal to σ jt /(P jt V jt ).Basedon ths assumpton, we obtan (22) λ jt P 2 jt (X P jt ) := P jt γ 1/2 jt σ jt (E Q 2 jt ) 1/2 (X P jt ) σ jt P jt V jt (X P jt ) Ths emprcal mplementaton of the transacton-costs formula can be, loosely speakng, thought of as the llqudty rato n Amhud (2002). Indeed, Amhud s llqudty rato s the tme-seres average of the daly ratos of the absolute value of percentage returns to dollar volume. To the extent that dollar volume s relatvely stable across tme and returns are drawn from the same dstrbuton, ths llqudty rato s effectvely proportonal to σ jt /(P jt V jt ).Although ths s a logcally consstent way to connect theory wth emprcal mplementaton, t s unrealstc to assume that the most actvely traded and least actvely traded assets have the same number of bets per day; emprcal ntuton suggests that assets wth hgh levels of tradng actvty have more bets per day than assets wth low levels of tradng actvty. We are aware of no emprcal studes whch clam that the number of orders or bets n dfferent assets s the same. Thus, the assumpton that the standard devaton of order mbalances s proportonal to volume seems to be unrealstc. The same ssue can be addressed by thnkng about tme unts. Unlke our llqudty measure 1/L jt = ι 2 CB [P jt V jt /σ 2 jt ] 1/3, the Amhud rato σ jt / (P jt V jt ) has tme unts. Indeed, σ 2 and P jt jt V jt n our llqudty measure have the same tme unts, but σ jt and P jt V jt n the Amhud rato do not; the Amhud rato thus depends on the tme horzon over whch volume and volatlty are measured. Snce the left sde of equaton (22) does not have tme unts, then to keep the left sde consstent wth the rght sde, the proportonalty constant n that equaton must change when tme unts are changed. Furthermore, f the nvarance-mpled market mpact formula (21) s ndeed correct, then Amhud s market mpact formula (22) theoretcally mples a dfferent proportonalty constant for every stock. Ths problem can be fxed that s, the same proportonalty coeffcent can be obtaned for every stock usng Amhud s approach f data for each stock are sampled at a dfferent stock-specfc frequency approprate to the stock s level of tradng actvty. Invarance mples that the approprate samplng frequency should be approxmately proportonal to 1/γ jt, whch s proportonal to W 2/3 jt. Illqudty ratos calculated usng data sampled at the same calendar tme frequences, as mplemented n many emprcal studes, mplctly rely on the unrealstc assumpton that the standard devaton of order mbalances s proportonal to tradng volume. By contrast, our llqudty measure 1/L jt does not depend on tme unts, and therefore t does not matter over what tme horzons ts components are measured. Even f dfferent horzons are used by researchers for dfferent assets, ts value wll be the same.

24 MARKET MICROSTRUCTURE INVARIANCE DATA Portfolo Transtons Data We test the emprcal mplcatons of market mcrostructure nvarance usng a propretary data set of portfolo transtons from a leadng vendor of portfolo transton servces. 6 Durng the evaluaton perod, ths portfolo transton vendor supervsed more than 30 percent of outsourced U.S. portfolo transtons. The sample ncludes 2,552 portfolo transtons executed over the perod for U.S. clents. A portfolo transton may nvolve orders for hundreds of ndvdual stocks. Each order s a stock-transton par potentally executed over multple days usng a combnaton of nternal crosses, external crosses, and open-market transactons. The portfolo transtons data set contans felds dentfyng the portfolo transton; ts startng and endng dates; the stock traded; the trade date; the number of shares traded; a buy or sell ndcator; the average executon prce; the pre-transton benchmark prce (closng prce the day before the transton trades began); commssons; SEC fees; and a tradng venue ndcator dstngushng among nternal crossng networks, external crossng networks, open market transactons, and n-knd transfers. When old legacy and new target portfolos overlap, postons are transferred from the legacy to the new portfolo as n-knd transfers. For example, f the legacy portfolo holds 10,000 shares of IBM stock and the new portfolo holds 4,000 shares of IBM, then 4,000 shares are transferred n-knd and the balance of 6,000 shares s sold. The n-knd transfers do not ncur transacton costs and have no effect on our emprcal analyss. The 6,000 shares sold consttute one portfolo transton order, even f the 6,000 shares are sold over multple days. We augment the portfolo transtons data wth stock prce, returns, and volume data from CRSP. Only common stocks (CRSP share codes of 10 and 11) lsted on the New York Stock Exchange (NYSE), the Amercan Stock Exchange (Amex), and NASDAQ n the perod from January 2001 through December 2005 are ncluded n the sample. ADRs, REITs, and closed-end funds are excluded. Also excluded are stocks wth mssng CRSP nformaton necessary to construct varables used for emprcal tests, transton orders n hghprced Berkshre Hathaway class A shares, and transton observatons whch appeared to contan typographcal errors and obvous naccuraces. Snce t s unclear from the data whether adjustments for dvdends and stock splts are made n a consstent manner across all transtons, observatons wth nonzero payouts durng the frst week followng the startng date of portfolo transtons were excluded from statstcal tests. 6 The non-dsclosure agreement does not allow revealng the name of the vendor or makng the data descrbng ndvdual customer trades publc. Research valdatng the nvarance hypotheses, ncludng research usng publc data sources, s descrbed at the end of the Supplemental Materal (Kyle and Obzhaeva (2016c)).

25 1368 A. S. KYLE AND A. A. OBIZHAEVA After exclusons, there are 439,765 observatons (orders), ncludng 201,401 buy orders and 238,364 sell orders. CRSP Data: Prces, Volume, and Volatlty For each of the transton-stock observatons ( = ), we collect data on the stock s pre-transton prce, expected volume, and expected volatlty. The prce, denoted P, s the closng prce for the stock the evenng before the frst trade s made n any of the stocks n the portfolo transton. A proxy for expected daly tradng volume, denoted V (n shares), s the average daly tradng volume for the stock n the prevous full pre-transton calendar month. The expected volatlty of daly returns, denoted σ for order, s calculated usng past daly returns n two dfferent ways. Frst, for each stock j and each calendar month m, we estmate the monthly standard devaton of returns σ j m as the square root of the sum of squared daly returns for the full calendar month m (wthout de-meanng or adjustng for autocorrelaton). We defne σ = σ j m /N 1/2 m,wherej corresponds to the stock traded n order, m s the prevous full calendar month precedng order,and N m s the number of CRSP tradng days n month m. Second, to reduce effects from the postve skewness of the standard devaton estmates, we estmate for each stock j a thrd-order movng average process for the changes n ln[σ j m ] for all months m over the entre perod Specfcally, lettng L denote the lag operator, we estmate (1 L) ln[σ j m ]=Θ j 0 + (1 Θ j 1 L Θ j 2 L 2 Θ j 3 L 3 )u j m. Lettng y j m denote the estmate of ln[σ j m ] and Vˆ j the varance of the predcton error, we alternatvely defne the condtonal forecast for the volatlty of daly returns by σ = exp(y j m + Vˆ j /2)/N 1/2 m,wherem s the current full calendar month for order. These volatlty estmates can be thought of as nstrumental varables for true expected volatlty. Whle below we report results usng the second defnton of σ based on the log-arima model, these results reman quanttatvely smlar when we use the frst defnton of σ based on smple hstorcal volatlty durng the precedng full calendar month. Except to the extent that the ARIMA model uses n-sample data to estmate model parameters, we use the pre-transton varables known to the market before portfolo transton trades are executed n order to avod any spurous effects from usng contemporaneous varables. Descrptve Statstcs Table I reports descrptve statstcs for traded stocks n panel A and for ndvdual transton orders n panel B. The frst column reports statstcs for all

26 TABLE I DESCRIPTIVE STATISTICS a All Panel A: Propertes of Stocks Med(V P) Med(σ) Med(Sprd) Med(Turn) Panel B: Propertes of Portfolo Transtons Orders Avg(X/V ) Med(X/V ) Avg(X/Cap) Med(X/Cap) Avg C(X) Avg Comm Avg SEC fee #Obs 439,765 71,000 68,689 41,238 49,000 28,073 29,330 29,778 34,409 40,640 47,608 #Stks 2,583 1, a Table reports the characterstcs of stocks and transton orders. Panel A shows the medan average daly dollar volume (n $ mllon), the medan daly volatlty (percent), the medan percentage spread (n bass ponts), the medan monthly turnover rate (n percent). Panel B shows the average and medan order sze (n percent of daly volume and n bass ponts of market captalzaton) as well as average mplementaton shortfall (n bass ponts), the average commsson (n bass ponts), and the average SEC fee for sell orders (n percent per 10 bass ponts). The thresholds of ten volume groups correspond to 30th, 50th, 60th, 70th, 75th, 80th, 85th, 90th, and 95th percentles of dollar volume for common stocks lsted on the NYSE. Group 1 (Group 10) contans orders n stocks wth lowest (hghest) dollar tradng volume. The sample ranges from January 2001 to December MARKET MICROSTRUCTURE INVARIANCE 1369

27 1370 A. S. KYLE AND A. A. OBIZHAEVA stocks n aggregate; the remanng ten columns report statstcs for stocks n ten dollar-volume groups. Instead of dvdng the stocks nto ten decles wth the same number of stocks n each decle, volume break ponts are set at the 30th, 50th, 60th, 70th, 75th, 80th, 85th, 90th, and 95th percentles of tradng volume for the unverse of stocks lsted on the NYSE wth CRSP share codes of 10 and 11. Group 1 contans stocks n the bottom 30th percentle of dollar tradng volume. Group 10 approxmately corresponds to the unverse of S&P 100 stocks. The top fve groups approxmately cover the unverse of S&P 500 stocks. Narrower percentle bands for the more actve stocks make t possble to focus on the stocks whch are most mportant economcally. For each month, the thresholds are recalculated and the stocks are reshuffled across bns. Panel A of Table I reports descrptve statstcs for traded stocks. For the entre sample, the medan daly volume s $18.72 mllon, rangng from $1.13 mllon for the lowest volume group to $ mllon for the hghest volume group. The medan volatlty s 1.93 percent per day, rangng from 1.76 percent n the hghest-volume decle to 2.16 n the lowest-volume decle. Snce there s so much more cross-sectonal varaton n dollar volume than n volatlty across stocks, the varaton n tradng actvty across stocks s related mostly to varaton n dollar volume. Tradng actvty dffers by a factor of 150 between stocks n the lowest group and stocks n the hghest group, and ths varaton creates statstcal power helpful n determnng how transacton costs and order szes vary wth tradng actvty. The medan quoted bd-ask spread, obtaned from the transton data set, s bass ponts; ts mean s bass ponts. From lowest-volume group to hghest-volume group, the medan spread declnes monotoncally from to 4.83 bass ponts, by a factor of A back-of-the-envelope calculaton based on nvarance suggests that spreads should decrease approxmately by a factor of 150 1/ from lowest- to hghest-volume group. The dfference between 5 31 and 8 48 s partally explaned by dfferences n returns volatlty across the volume groups and warrants further nvestgaton. The monotonc declne of almost one order of magntude s potentally large enough to generate sgnfcant statstcal power n estmates of a bd-ask spread component of transacton costs based on mplementaton shortfall. Panel B of Table I reports propertes of portfolo transton order szes. The average order sze s 4 20% of average daly volume, declnng monotoncally across the ten volume groups from 16 23% n the smallest group to 0 49% n the largest group, by a factor of The medan order s 0 57% of average daly volume, also declnng monotoncally from 3 33% n the smallest group to 0 14% n the largest group, by a factor of The nvarance hypothess mples that order szes should declne by a factor of approxmately 150 2/ , a value whch matches the data closely. The medans are much smaller than the means, ndcatng that dstrbutons of order szes are skewed to the rght. We show below that the dstrbuton of order szes closely fts a log-normal.

28 MARKET MICROSTRUCTURE INVARIANCE 1371 The average tradng cost (estmated based on mplementaton shortfall, as explaned below) s bass ponts per order, rangng from bass ponts n the lowest-volume group to 6 16 bass ponts n the hghestvolume group. Invarance suggests that these costs should fall by a factor of 150 1/3 5 31, somewhat smaller than the actual declne. The cost estmates exclude commssons and SEC fees. 7 One portfolo transton typcally contans orders for dozens or hundreds of stocks. It typcally takes several days to execute all of the orders. About 60% of orders are executed durng the frst day of a portfolo transton. Snce transton managers often operate under a cash-n-advance constrant usng proceeds from sellng stocks n a legacy portfolo to acqure stocks n a target portfolo sell orders tend to be executed slghtly faster than buy orders (1 72 days versus 1 85 days). In terms of dollar volume, about 41%, 23%, 15%, 7%, and 5% of dollar volume s executed on the frst day through the ffth days, respectvely. The two longest transtons n the sample were executed over 18 and 19 busness days. The tme frame for a portfolo transton s usually set before ts actual mplementaton begns. 4. EMPIRICAL TESTS BASED ON ORDER SIZES Market mcrostructure nvarance predcts that the dstrbuton of W 2/3 jt Q jt / V jt does not vary across stocks or tme (see equaton (8)). We test these predctons usng data on portfolo transton orders, makng the dentfyng assumpton that portfolo transton orders are proportonal to bets. Portfolo Transtons and Bets Snce bets are statstcally ndependent ntended orders, bets can be conceptually dffcult for researchers to observe. Consder, for example, a trader who makes a decson on Monday to make one bet to buy 100,000 shares of stock, then mplements the bet by purchasng 20,000 shares on Monday and 80,000 shares on Thursday. To an econometrcan, ths one bet for 100,000 shares may be dffcult to dstngush from two bets for 20,000 shares and 80,000 shares, respectvely. In the context of a portfolo transton, dentfyng a bet s easer because the sze of the order for 100,000 shares s known and recorded on Monday, even f the order s executed over several subsequent days. Portfolo transton orders may not have a sze dstrbuton matchng precsely the sze dstrbuton of typcal bets. Transton orders may be smaller 7 The SEC fee represents a cost of about 0 29 bass ponts, whch does not vary much across volume groups. The average commsson s 7 43 bass ponts, declnng monotoncally by a factor of 7 30 from bass ponts for the lowest group to 2 68 bass ponts for the hghest group. Snce commssons may be negotated for the entre transton, the allocaton of commsson costs to ndvdual stocks s an accountng exercse wth lttle economc meanng.

29 1372 A. S. KYLE AND A. A. OBIZHAEVA than bets f transtons tend to lqudate a porton of an asset manager s postons or larger than bets f transtons lqudate the sum of bets made by the asset manager n the past. When both target and legacy portfolos hold long postons n the same stock, the portfolo transton order may represent the dfference between two bets. Let X denote the unsgned number of shares transacted n portfolo transton order, = The quantty X sums shares traded over multple days, excludng n-knd transfers. We make the dentfyng assumpton that, for some constant δ whch does not vary across stocks wth dfferent characterstcs such as volatlty and tradng actvty, the dstrbuton of scaled portfolo transton orders δ X s the same as the dstrbuton of unsgned bets n the same stock at the same tme, denoted Q. Ifδ = 1, the dstrbuton of portfolo transton orders matches the dstrbuton of bets. If the scalng constant δ were correlated wth volatlty or tradng actvty, parameter estmates mght be based. The Emprcal Hypotheses of Invarance and Log-Normalty for the Sze Dstrbuton of Portfolo Transton Orders Let W := V P σ and W := V P σ denote tradng actvty and bet actvty, respectvely, for the stock n transton order. Under the dentfyng assumpton that portfolo transton orders are proportonal to bets, nvarance of bets mples nvarance of portfolo transton orders. Specfcally, replacng Q jt wth X n equaton (8) mples that the dstrbuton of W 2/3 X / V does not vary wth stock characterstcs such as volume, volatlty, stock prce, or market captalzaton. To facltate ntutve nterpretaton of parameter estmates, we scale observatons by a hypothetcal benchmark stock wth prce P of $40 per share, daly volume V of one mllon shares, and volatlty σ of 2% per day, mplyng W = Ths benchmark stock would belong to the bottom tercle of S&P 500 (volume group 7 n Table I). Combnng nvarance of portfolo transton orders wth equatons (1) and (2) to convert the bet actvty varables W and V nto tradng actvty varables W and V and takng logs, nvarance mples the emprcally testable relatonshp (23) [( W ln W ) 2/3 X V ] = ln[ q]+ ε Under the dentfyng assumptons that the volume multpler ζ, the volatlty multpler ψ, and the deflator δ do not vary across observatons, ln[ q] s an nvarant constant ln[ q]=e{ln[ Q /V ]} ln[δ] and ε s a zero-mean error

30 MARKET MICROSTRUCTURE INVARIANCE 1373 wth the same nvarant dstrbuton as ln[ Ĩ ] E{ln[ Ĩ ]}. 8 Adjustment by W n equaton (23) scales each observaton on the left sde so that t has the same nvarant dstrbuton as the log of a hypothetcal portfolo transton order n the benchmark stock, expressed as a fracton of ts expected daly volume. We wll also examne the stronger log-normalty hypothess not mpled by mcrostructure nvarance that the dstrbuton of unsgned order szes adjusted for tradng actvty [W /W ] 2/3 X /V has a log-normal dstrbuton, that s, ε n equaton (23) has a normal dstrbuton. The log-normalty hypothess mples that the rght sde of equaton (23) s characterzed by two nvarant constants, the mean ln[ q] and the varance of ε. Next, we mplement several tests to examne ths hypothess. The Graphcal Relatonshp Between Order Szes and Tradng Actvty One way to examne the nvarance hypothess s to plot the log of order sze as a fracton of average daly volume ln[x /V ] aganst the log of scaled tradng actvty ln[w /W ].Fgure1 presents a cloud of ponts for all 400,000+ portfolo transton orders. The lne ln[x /V ]= /3 ln[w /W ] s also FIGURE 1. Order sze and tradng actvty. The fgure plots ln[x /V ] on the vertcal axs aganst ln[w /W ] on the horzontal axs, where X s portfolo transton order sze n shares, V s average daly volume n shares, and W = P V σ s tradng actvty. The ftted lne s ln[x /V ]= /3 ln[w /W ], where the ntercept s estmated from an OLS regresson wth the slope fxed at 2/3. There are 400,000+ data ponts from January 2001 to December More generally, ln[ q]:=e{ln[ Q /V ]} 1/3 ln[ζ /ζ ] 2/3 ln[ψ /ψ ] ln[δ ].