Online Appendix Supplemental Material for Market Microstructure Invariance: Empirical Hypotheses

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1 Onlne Appendx Supplemental Materal for Market Mcrostructure Invarance: Emprcal Hypotheses Albert S. Kyle Unversty of Maryland Anna A. Obzhaeva New Economc School

2 Table A.1: Quantle Estmates of Order Sze. p1 p5 p25 p50 p75 p95 p99 ln [ q (0.008) (0.006) (0.004) (0.003) (0.004) (0.006) (0.009) α (0.005) (0.003) (0.002) (0.002) (0.002) (0.003) (0.005) Pseudo R Q /V δ #Obs 439, , , , , , ,765 Table presents the estmates ln[ q and α 0 for the quantle regresson: [ X ln = ln [ q [ W +α 0 ln V W + ǫ. Each observaton corresponds to transton order wth order sze X, pretranston prce P, expected daly volume V, expected daly volatlty σ, tradng actvty W. The parameter q s the measure of order sze such that for δ = 1, Q /V δ measures the medan bet sze for the benchmark stock, n bass ponts of average daly volume. The benchmark stock has daly volatlty of 2%, share prce of $40, and daly volume of one mllon shares. The standard errors are shown n parentheses. The sample ranges from January 2001 to December

3 Table A.2: OLS Estmates for Order Sze: Model Calbraton. NYSE NASDAQ All Buy Sell Buy Sell Restrcted Specfcaton: α 0 = 2/3, b 1 = b 2 = b 3 = b 4 = 0 ln [ q (0.019) (0.023) (0.019) (0.042) (0.039) Q /V δ MSE R Unrestrcted Specfcaton Wth 5 Degrees of Freedom: α 0 = 2/3. ln [ q (0.019) (0.026) (0.019) (0.051) (0.047) b (0.040) (0.050) (0.043) (0.094) (0.087) b (0.019) (0.021) (0.017) (0.049) (0.040) b (0.010) (0.012) (0.009) (0.026) (0.021) b (0.015) (0.020) (0.017) (0.040) (0.032) R #Obs 439, , ,377 69,871 87,987 Table presents the estmates and the mean squared error (MSE) for the regresson [ X ln = ln [ q [ W [ σ [ P [ V [ ν +α 0 ln V W +b 1 ln +b 2 ln +b 3 ln b 4 ln + ǫ. 1/12 wth α 0 restrcted to be 2/3 as predcted by nvarance and b 1 = b 2 = b 3 = 0. Each observaton corresponds to transton order wth order sze X, pretranston prce P, expected daly volume V, expected daly volatlty σ, tradng actvty W, and monthly turnover rate ν. The parameter q s the measure of order sze such that for δ = 1, Q /V δ measures the medan bet sze for the benchmark stock, n bass ponts of average daly volume. The benchmark stock has daly volatlty of 2%, share prce of $40, and daly volumeof onemllon shares. TheR 2 s arereportedforrestrcted specfcaton wth α 0 = 2/3,b 1 = b 2 = b 3 = b 4 = 0 as well as for unrestrcted specfcaton wth coeffcents ln[ q and b 1,b 2,b 3,b 4 allowed to vary freely. The standard errors are clustered at weekly levels for 17 ndustres and shown n parentheses. The sample ranges from January 2001 to December

4 Table A.3: Transacton-Cost Estmates n Regresson wth Lnear Impact. NYSE NASDAQ All Buy Sell Buy Sell β mkt (0.013) (0.016) (0.016) (0.037) (0.036) κ (0.890) (1.600) (1.154) (2.147) (1.501) α (0.020) (0.048) (0.029) (0.051) (0.031) κ I (0.252) (0.460) (0.346) (0.663) (0.765) α (0.028) (0.038) (0.041) (0.056) (0.058) R #Obs 439, , ,377 69,871 87,987 Table presents the estmates for β mkt,α 1,κ 0,α 2, and κ I n the regresson: I BS, S (0.02) σ mkt, (0.02) [ = β mkt R +I BS, κ W [ σ 0 α1+ibs, κ W [ φi W I α2 W 0.01 (1) where z = 1 and φi /0.01 = X /(0.01V ) (W /W ) 2/3. S s mplementaton shortfall. R mkt, s the value-weght market returnfor thefrstday of transton. The tradng actvty W s the product of expected volatlty σ, pre-transton prce P, and expected volume V. The scalng constant W = (0.02)(40)(10 6 ) s the tradng actvty for the benchmark stock wth volatlty of 2% per day, prce $40 per share, and tradng volume of one mllon shares per day. X s the number of shares n the order. The parameter κ I 104 s the market mpact cost of executng a trade of one percent of daly volume n the benchmark stock; and κ s the effectve spread cost; both are measured n bass ponts. The standard errors are clustered at weekly levels for 17 ndustres and shown n parentheses. The sample ranges from January 2001 to December z+ ǫ. 3

5 Table A.4: Transacton Costs: Model Calbraton. NYSE NASDAQ All Buy Sell Buy Sell Lnear Model: z = 1, β 1 = β 2 = β 3 = β 4 = β 5 = β 6 = β 7 = β 8 = 0. β mkt (0.0135) (0.0159) (0.0158) (0.0371) (0.0365) κ (0.5776) (1.1215) (0.7943) (1.5627) (0.7811) κ I (0.1903) (0.3700) (0.2650) (0.3953) (0.3267) R Square-Root Model: z = 1/2, β 1 = β 2 = β 3 = β 4 = β 5 = β 6 = β 7 = β 8 = 0. β mkt (0.0134) (0.0158) (0.0159) (0.0365) (0.0364) κ (0.7035) (1.2779) (0.9264) (2.0554) (0.8244) κ I (0.7416) (1.2177) (1.2979) (1.4564) (1.2069) R Unrestrcted Specfcaton Wth 12 Degrees of Freedom. β mkt (0.013) (0.016) (0.015) (0.036) (0.036) κ (0.675) (0.124) (0.556) (1.698) (1.148) β (0.147) (0.890) (0.392) (0.489) (0.131) β (0.072) (1.230) (0.231) (0.127) (0.109) β (0.159) (0.754) (0.296) (0.238) (0.155) β (0.173) (0.620) (0.490) (0.313) (0.175) κ I (1.307) (2.471) (1.804) (3.340) (2.033) z (0.041) (0.039) (0.042) (0.094) (0.083) β (0.135) (0.192) (0.229) (0.252) (0.242) β * (0.061) (0.113) (0.100) (0.120) (0.113) β (0.037) (0.050) (0.052) (0.099) (0.100) β (0.067) (0.086) (0.101) (0.143) (0.153) R #Obs 439, , ,377 69,871 87,987

6 Table presents the estmates for the regresson: I BS, S (0.02) = β mkt R mkt, (0.02) [ I BS, κ W 0 σ σ +I BS, κ I W 1/3 [ φi z [ W 1/3 σ β 5 P β 6 V β 7 ν β W (0.02)(40)(10 6 )(1/12) + ǫ. σ β 1 P β 2 V β 3 ν β 4 (0.02)(40)(10 6 )(1/12) + where φi /0.01 = X /(0.01V ) (W /W ) 2/3. S s mplementaton shortfall. R mkt, s the value-weght market return for the frst day of transton. The tradng actvty W s the product of expected volatlty σ, pre-transton prce P, and expected volume V. The scalng constant W = (0.02)(40)(10 6 ) s the tradng actvty for the benchmark stock wth volatlty of 2% per day, prce $40 per share, and tradng volume of one mllon shares per day. X s the number of shares n the order. The parameter κ I 104 s the market mpact cost of executng a trade of one percent of daly volume n the benchmark stock; and κ s the effectve spread cost; both are measured n bass ponts. The R 2 s are reported for restrcted specfcaton as well as for unrestrcted specfcaton wth twelve coeffcents β mkt, z, κ I,κ 0,β 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8 allowed to vary freely. The standard errors are clustered at weekly levels for 17 ndustres and shown n parentheses. The sample ranges from January 2001 to December

7 Table A.5: Transacton-Cost Estmates n Regresson wth Quoted Spread. NYSE NASDAQ All Buy Sell Buy Sell β mkt (0.013) (0.016) (0.015) (0.036) (0.037) κ I (0.261) (0.504) (0.366) (0.700) (0.749) α (0.029) (0.036) (0.039) (0.053) (0.060) β S (0.053) (0.110) (0.094) (0.127) (0.073) R #Obs 436, , ,600 69,218 86,731 Table presents the estmates for β mkt,κ I,α 2, and β S n the regresson: I BS, S (0.02) σ mkt, (0.02) = β mkt R +I BS, β S 1 σ 2 s (0.02) [ φi [ W +I BS, κ I α2+ ǫ P σ 0.01 W. [ where nvarant I = X (0.01)V W 2/3. W Each observaton corresponds to order. I BS, s a buy/sell ndcator, S s mplementaton shortfall, R mkt, s the value-weght market return for the frst day of transton. The term (0.02)/σ adjusts for heteroscedastcty. The tradng actvty W s the product of expected volatlty σ, pre-transton prce P, and expected volume V. The scalng constant W = (0.02)(40)(10 6 ) s the tradng actvty for the benchmark stock wth volatlty of 2% per day, prce $40 per share, and tradng volume of one mllon shares per day. X s the number of shares n the order. The parameter κ I 104 s the market mpact cost of executng a trade of one percent of daly volume n the benchmark stock, measured n bass ponts. s /P s the quoted percentage spread. The standard errors are clustered at weekly levels for 17 ndustres and shown n parentheses. The sample ranges from January 2001 to December

8 Current Research on Invarance Hypotheses The nvarance hypotheses can be examned usng many dfferent datasets. Here s a summary of our current research usng datasets avalable to academc researchers: The Ancerno dataset ncludes more orders than the dataset of portfolo transtons used n ths paper. The dataset groups trades nto meta-orders whch approxmate our concept of a bet. Prelmnary research by Albert S. Kyle and Kngsley Fong fnds that proxes for bets n Ancerno data have sze patterns consstent wth the nvarance hypotheses. The Ancerno data s also desgned to facltate measurement of transactons costs. Ths dataset s therefore approprate for valdatng our emprcal results for both bet sze and transacton costs. Andersen et al. (2015) examne the varaton n trade frequency, contract volume, and volatlty n the S&P 500 E-mn futures contract across mnutes of the 24-hour day. The results conform to predcted nvarance relatonshps. Kyle and Obzhaeva (2016) compare extrapolatons of the lnear market-mpact estmates n ths paper to publcly documented quanttes sold n fve stock market crashes. The prce declnes n the 1987 crash and the 2008 lqudaton of Jerome Kervel s rogue trades at Socete Generale, whch occurred over tme frames smlar to large portfolo transtons, were close the the predcted declnes. Transtory prce declnes were larger than predcted n the two flash crashes, when sales occurred unusually rapdly, and smaller n the 1929 crash, when sales were stretched out over weeks and months. Whle consstent wth the nvarance hypotheses, these results also suggest that the speed of executon nfluences temporary market mpact. Kyle, Obzhaeva and Tuzun (2016) examne the hypothess that the sze of prnts of stock market trades n Trade and Quotatons (TAQ) data are proportonal to the sze of bets. Ths hypothess holds up well durng the 1990s, consstent wth the nterpretaton that bets were negotated and prnted as block trades. The hypothess breaks down after 2001, when trade sze collapsed toward the round-lot mnmum sze of 100 shares for many trades. The changes after 2001 may be the result of the reducton n the mnmum tck sze to one cent and the ncreased use of electronc order-shreddng algorthms. Kyle et al. (2012) examne whether the monthly frequency of Thomson Reuters news artcles s proportonal to the 2/3 power of tradng actvty for ndvdual stocks. The estmated exponent s close to the predcted value of 2/3 pror to a strategc decson by Thomson Reuters to ncrease the number of news artcles about less actvely traded stocks, after whch the coeffcent changes n a manner consstent wth ther strategc decson. 7

9 Usng a dfferent propretary dataset, Bae et al. (2014) examne the number of tmes ndvdual tradng accounts swtch between buyng and sellng ndvdual stocks. Consstent wth the predctons of nvarance, t s shown that the number of swtchng ponts s proportonal to the 2/3 power of tradng actvty. References Andersen, Torben G., Oleg Bondarenko, Albert S. Kyle, and Anna A. Obzhaeva Intraday Tradng Invarance n the E-mn S&P 500 Futures Market. Workng Paper, avalable at Bae, Kyounghun, Albert S. Kyle, Eun Jung Lee, and Anna A. Obzhaeva An Invarance Relatonshp n the Number of Buy-Sell Swtchng Ponts. Workng Paper, avalable at Kyle, Albert S., and Anna A. Obzhaeva Large Bets and Stock Market Crashes. Workng Paper, avalable at Kyle, Albert S., Anna A. Obzhaeva, and Tugkan Tuzun Mcrostructure Invarance n U.S. Stock Market Trades. Workng Paper, avalable wth older ttle at Kyle, Albert S., Anna A. Obzhaeva, Ntsh R. Snha, and Tugkan Tuzun News Artcles and the Invarance Hypothess. Workng Paper, avalable at 8

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