VoV(%Spread)” for percent spread cost and a ne · 2019-08-29 · We test the monthly and daily versions of these new and existing liquidity proxies against liquidity benchmarks

Are Volatility Over Volume Liquidity Proxies Useful For Global Or US Research?*

Kingsley Y. L. Fong** Craig W. Holden** Ondrej Tobek**

University of New South Wales Indiana University University of Cambridge

March 2018

Abstract

We examine a general class of volatility over volume liquidity proxies as computed from low frequency

(daily) data. We start from the Kyle and Obizhaeva (2016) hypothesis of transaction cost invariance to

identify a new volatility over volume liquidity proxy “VoV(%Spread)” for percent spread cost and a new

volatility over volume liquidity proxy “VoV(λ)” for the slope of the transaction cost function “λ”. We test

the monthly and daily versions of these new and existing liquidity proxies against liquidity benchmarks as

estimated from high frequency (intraday) data on both a global and US basis. We find that both the monthly

and daily versions of VoV(λ) dominate the equivalent versions of Amihud and other cost-per-dollar-volume

proxies on both a global and US basis. We also find that both the monthly and daily versions of

VoV(%Spread) dominate the equivalent versions of other percent-cost proxies for US studies that cover

pre-1993 years. In a case study, we find that our new VoV liquidity proxies yield different research

inferences than the best previous liquidity proxies from the prior literature. The success of our invariance-

based liquidity proxies across exchanges and over time supports the prediction of Kyle and Obizhaeva of a

specific functional form for transaction costs across exchanges and over time.

JEL classification: C15, G12, G20. Keywords: Liquidity, transaction costs, effective spread, price impact.

* We are thank seminar participants at the 2017 SFS Asia-Pacific Cavalcade conference and the 2017

Financial Research Network conference. We are solely responsible for any errors.

** Kingsley Y.L. Fong email: [email protected], Craig W. Holden emai: cholden@ indiana.edu, and

Ondrej Tobek email: [email protected]. Correspondence address: Craig W. Holden, Kelley School of

Business, Indiana University, 1309 E. Tenth St., Bloomington, IN 47405. Phone: 1-812-855-3383.

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1. Introduction

We examine a general class of volatility over volume (VoV) liquidity proxies as computed from

low frequency (daily) data. This builds on a recent literature that proposes the use of low-frequency liquidity

proxies. The idea is that low-frequency liquidity proxies can do a good job of capturing liquidity

benchmarks as computed from high-frequency (intraday) data and at the same time provide enormous

computational savings compared to the use of high-frequency data. Fong, Holden, and Trzcinka (FHT)

(2017) show that these computational savings for global research are of the order of 1,000X to 10,000X

depending on the specific decade being analyzed.1

Kyle and Obizhaeva (2016) develop the empirical hypothesis of “market microstructure

invariance.” This includes the sub-hypothesis of “transaction cost invariance,” which means that transaction

costs per unit of business time have a constant distribution across all assets and over all time periods. We

start from the specific functional forms of their hypothesis to identify two new liquidity proxies: (1)

“VoV(%Spread)” which proxies for the percent effective spread and (2) “VoV(λ)” which proxies for the

slope of the transaction cost function (typically called λ).

Liquidity proxies are important because liquidity is connected in fundamental ways to asset pricing,

corporate finance, and market microstructure. A large literature on global and US research relies on monthly

liquidity proxies2 and daily liquidity proxies.3

1 FHT compute the ratio of high-frequency data points to low-frequency data points in their sample. Due to the

exponential increase in high-frequency data, this ratio rose from 42X in 1996 to 962X in 2007 to 12,616X in 2014. 2 Monthly liquidity proxies are used in Jain (2005), Lesmond (2005), Stahel (2005), Bailey, Karolyi, and Salva,

(2006), Levine and Schmukler (2006), Bekaert, Harvey, and Lundbland (2007), LaFond, Lang, and Skaife (2007),

Chan, Jain, and Xia (2008), Henkel (2008), Henkel, Jain, and Lundblad (2008), DeNicolo and Ivaschenko (2009),

Griffin, Kelly, and Nardari (2010), Hearn, Piesse, and Strange (2010), Clark (2011), Griffin, Hirschey, and Kelly

(2011), Lee (2011), Lang, Lins, and Maffett (2012), Liang and Wei (2012), Asness, Moskowitz, and Pedersen

(2013), Marshall, Nguyen, and Visaltanachoti (2013), and Bekaert, Harvey, Lundblad, Siegel (2014), Hearn (2014),

Karnaukh, Renaldo and Soderlind (2015), Orlov (2016), and Massa, Mataigne, Vermaelen, and Xu (2017). 3 Daily liquidity proxies are used in Bhattacharya, Daouk, Jorgenson, and Kehr (2000), Attig, Gadhoum, and Lang

(2003), Gomez-Puig (2006), Gersl and Komarkov (2009), Erten and Okay (2012), Karolyi, Lee, and van Dijk

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Using a dataset very similar to FHT, we test the top-three monthly and daily versions of low-

frequency liquidity proxies considered by FHT plus several new proxies not considered there. We test all

of these proxies against the corresponding monthly and daily versions of high-frequency liquidity

benchmarks on a global basis and in the US. We test the performance of monthly and daily liquidity proxies

on five dimensions of performance that are crucial for asset pricing, corporate finance and market

microstructure applications: (1) higher average cross-sectional correlation with the benchmarks, (2) higher

average Spearman’s cross-sectional correlation with the benchmarks, (3) higher portfolio time-series

correlation of proxy first differences with benchmark first differences, (4) higher individual stock time-

series correlation of proxy first differences with benchmark first differences, and (5) lower average root

mean squared error (RMSE) with the benchmarks.

For global research, we find that monthly VoV(λ) dominates monthly Amihud and five other

monthly cost-per-dollar-volume proxies by a statistically and economically significant amount. We also

find that daily VoV(λ) dominates daily Amihud and three other daily cost-per-dollar-volume proxies by a

statistically and economically significant amount. For US studies covering 1993-present, we find that

monthly (daily) VoV(%Spread) ties monthly (daily) Closing Percent Quoted Spread. For US studies

including pre-1993 years, when Closing Percent Quoted Spread is not available, we find that monthly

(daily) VoV(%Spread) dominates all other monthly (daily) percent-cost proxies tested by a statistically and

economically significant amount.

Specifically, on a global basis we find that monthly (daily) VoV(λ) has the highest average cross-

sectional correlation, highest average Spearman’s cross-sectional correlation, highest portfolio time-series

correlation of first differences, and highest individual stock time-series correlation of first differences with

monthly (daily) high-frequency benchmark lambda for both time-periods analyzed. The gain in

performance is economically meaningful. Specifically, monthly VoV(λ) has an average correlation across

(2012), Beber and Pagano (2013), Lee, Tseng, and Yang, (2014), Hanselaar, Stulz, and van Dijk (2016), Lee,

Sapriza, and Wu (2016), and Boubakri, Saad, and Samet (2017).

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the eight contests (i.e., four correlation measures X two time-periods) of 0.562 versus 0.464 for monthly

Amihud. Daily VoV(λ) has an average correlation across the eight contests of 0.534 versus 0.444 for daily

Amihud. We find that the dominant performance of monthly VoV() is robust by time-period, by size

quintiles, by volatility quintiles, net of controlling for size and volatility, and by exchange. We find that

none of the monthly (daily) cost-per-dollar-volume proxies captures the level of monthly (daily) lambda.

Qualitatively similar results hold for the US.

For US studies including pre-1993 years (for which data is available back to 1926), we find that

monthly (daily) VoV(%Spread) has the highest average cross-sectional correlation, highest average

Spearman’s cross-sectional correlation, highest portfolio time-series correlation of first differences, and

highest individual stock time-series correlation of first differences with monthly (daily) high-frequency

benchmark percent effective spread. The gain in performance is economically meaningful. Specifically,

monthly VoV(%Spread) has an average correlation across the four contests (i.e., four correlation measures)

of 0.705 versus 0.593 for the second-best monthly percent-cost proxy. Daily VoV(%Spread) has an average

correlation across the four contests of 0.584 versus 0.206 for the second-best daily percent-cost proxy. We

find that monthly (daily) VoV(%Spread) does the best job of capturing the level of monthly (daily) percent

effective spread.

Do our new VoV liquidity proxies yield different research inferences than the best previous

liquidity proxies from the prior literature? To address this question, we perform a case study that compares

daily VoV() and daily Amihud when analyzing the pattern of S&P 1500 illiquidity before, during, and

after the August 2007 “Quant Meltdown” event. We find that the daily VoV() proxy yields the inference

that S&P 1500 illiquidity is significantly higher than the base period in all three periods (before, during,

and after). By contrast, the daily Amihud proxy yields the inference that S&P 1500 illiquidity in not

significantly different than the base in all three periods. We conclude that our new VoV liquidity proxies

matter.

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We contribute to two lines of literature. First, contribute to the liquidity proxy literature by

developing four new liquidity proxies (monthly and daily VoV(%Spread) and monthly and daily VoV())

and by comprehensively testing them against many alternatives on both a global and US basis. Compared

with FHT, we analyze their top monthly and daily percent-cost proxies, but also analyze several that they

do not: VoV(%Spread), Gibbs, and Close-High-Low. Further, we analyze their top monthly and daily cost-

per-dollar-volume proxies, but also analyze several that they do not: the VoV(), Gibbs Impact, and Close-

High-Low Impact. Despite the expanded competition, all four of our new proxies prove to be the best for

one or more purposes.

Second, the success of our invariance-based liquidity proxies across exchanges and over time

supports the prediction of Kyle and Obizhaeva of a specific functional form for transaction costs across

exchanges and over time. Furthermore, we find there is very limited performance gain and there is risk of

overfitting the data when we allow the data to determine the optimal parameter in the general class of

volatility over volume proxies from the predictions of the transaction cost invariance hypothesis.

The paper is organized as follows. Section 2 explains the volatility over volume liquidity proxies.

Section 3 describes the high-frequency liquidity benchmarks. Section 4 reviews the low-frequency proxies

form the existing literature. Section 5 describes the data. Sections 6 presents global performance results.

Section 7 presents US performance results. Section 8 performs a case study of whether the new liquidity

proxies yield different research inferences than the best previous liquidity proxies. Section 9 concludes.

Appendix A derives VoV(%Spread) and VoV() from transaction cost invariance formulas in Kyle and

Obizhaeva. Appendix B summarizes the formulas of low-frequency proxies from the existing literature. An

online supplemental appendix provides additional robustness checks by exchange.

2. Volatility Over Volume Liquidity Proxies

We examine a general class of volatility over volume liquidity proxies in which a function of

volatility is in the numerator and a function of volume is in the denominator. Let be the standard

deviation of daily returns and V be average daily dollar volume in real dollars (i.e., inflation-adjusted). We

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define volatility over volume (VoV) class of liquidity proxies as liquidity proxies that have the following

form

b

c

aVoV

V

, (1)

where a, b, and c are strictly positive constants.

The key idea of Kyle and Obizhaeva is that markets for different assets run at different speeds,

which they call “business time.” Business time is the speed with which “bets” arrive at the market. A bet is

the total amount of risk transfer that is desired to be traded by a given trader. Thus, their key hypothesis of

“market microstructure invariance” is that all markets look the same when measured per unit of business

time. Specifically, this hypothesis has two sub-hypotheses. The first is “bet invariance,” which means that

bets per unit of business time have a constant distribution across all assets and over all time periods. The

second is “transaction cost invariance,” which means that transaction costs per unit of business time have

a constant distribution across all assets and over all time periods.

Transaction cost invariance supports alternative functional forms. Kyle and Obizhaeva consider

three alternative forms: (1) a constant percent quoted spread cost, (2) linear price impact costs, and (3)

square-root price impact costs. The appendix shows how the constant percent quoted spread cost case leads

to the following volatility over volume percent-cost proxy VoV(%Spread)

2

3

1

3

%VoV Spreada

V

, (2)

where a is a constant, 2

,3

b and 1

3c . In our empirical implementation, we set

2,

3b

1

3c , and

convert V to real dollars4 and then find the value of a that matches the mean value of monthly

4 We convert to real dollars as follows: Real Dollar Volume = Nominal Dollar Volume / CPI. For CPI we use the “Consumer Price Index for All Urban Consumers: All Items (CPIAUCNS)” available from the online FRED database provide by the Federal Reserve Bank of St Louis and normalize it to 1.0 on January 1, 2000 (i.e, we divide by the value of CPIAUCNS on 1/1/2000).

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VoV(%Spread) with monthly percent effective spread in our US sample. We obtain parameter a = 8.0 and

use this parameter value for both the monthly and daily versions and when analyzing both the global and

US samples.5

The appendix also shows how the square-root price impact cost case leads to the following volatility

over volume cost-per-dollar-volume proxy VoV()

1

2

VoVa

V

, (3)

where a is a constant, 1,b and 1

2c . In our empirical implementation, we set 1,b and

1

2c and then

find the value of a that matches the mean value of VoV() with lambda in our global sample. We obtain

parameter a = 1 and use this parameter value for both the monthly and daily versions and when analyzing

both the global and US samples.6

The volatility over volume class of liquidity proxies encompasses a wide range of special cases.

When 1,a b c it bears a general resemblance to the well-known Amihud liquidity proxy, which is

given by

t

t

rAverage

V

, (4)

where tr is the daily return on day t , tV is the dollar volume on day t , and the average is computed over

all positive volume days. The numerator tr of Amihud is similar to numerator of the VoV class of

liquidity proxies with 1b , in that both are measures of volatility.

An important difference is that the VoV class of liquidity proxies is the ratio of two statistics,

whereas the Amihud liquidity proxy is the average of daily ratios which may include noisy values when

5 a = 8.0 is very close to 7.62 estimated in Kyle and Obizhaeva for fixed costs component of their linear version of cost function. 6 a = 1.0 is again very close to 1.2 estimated in Kyle and Obizhaeva for variable costs component of their square root version of cost function.

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outlier low-volume days occur. This point is easily illustrated by a simple numerical example. Suppose that

the returns and dollar volume for a given stock are as shown in Table 1. On days 1-4, the dollar volume is

between $200,000 and $300,000, but on day 5a it was just $5,000, whereas in the alternative case on day

5b it is $250,000. The column labeled “Amihud ratio” is the daily ratio of absolute return over dollar

volume. Notice that the low volume on day 5a causes the Amihud ratio on day 5a to be much larger than

the other days. The Amihud liquidity proxy when computed over days 1 to 4 and 5a (and multiplied by

1,000,000) is 1.69. The analogous Amihud liquidity proxy computed over days 1 to 4 and 5b is 0.12. The

former is 14.2 times larger than the later. Thus, Amihud yields very noisy results when outlier low-volume

days occur. By contrast, VoV(λ) computed over days 1 to 4 and 5a is 76.52 and 68.33 when computed over

days 1 to 4 and 5b. It is robust to the outlier realization of low-volume days.

3. High-Frequency Liquidity Benchmarks

We test our new volatility over volume liquidity proxies and other liquidity proxies from the

existing literature against high-frequency liquidity benchmarks. Specifically, we following Fong, Holden,

and Trzcinka (2017) in distinguishing between the percent-cost benchmarks vs. cost-per-dollar-volume

benchmarks. A percent-cost benchmark measures the percentage cost of doing a small trade. By contrast, a

cost-per-dollar-volume benchmark measures the marginal transaction cost per US dollar of volume.

Our percent-cost benchmark is percent effective spread. The percent effective spread of a given

stock on the trade is defined as

Percent Effective Spreadk = 2Dk (ln(Pk) - ln(Mk)), (1)

where Dk is an indicator variable that equals +1 if the thk trade is a buy and -1 if the thk trade is a sell, Pk

is the price of the thk trade and Mk is the midpoint of the consolidated best bid and offer prevailing

immediately prior to the time of the thk trade, where the exact amount of time prior depends on the time-

stamp of each exchange. Next, we aggregate over either a day or month, which in either case we refer to as

period i. A stock’s Percent Effective Spreadi for period i is the volume-weighted average of Percent

Effective Spreadk computed over all trades in period i.

thk

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Our cost-per-dollar-volume benchmark is the slope of the price function, which is typically called

. We follow Goyenko, Holden, and Trzcinka (2009) and Hasbrouck (2009) in calculating as the slope

coefficient of the regression

n n nr S u , (5)

as estimated over five minute intervals. For the thn five-minute interval, nr is the stock return, nS =

kn knkSign v v

is the signed square-root of volume measured in US dollars, knv is the signed volume

measured in US dollars of the thk trade in the thn five-minute interval, and nu is the error term.7

4. Low-Frequency Proxies from the Existing Literature

We analyze five monthly percent-cost low-frequency proxies from the existing literature. This

includes what FHT finds to be the top-three performing monthly percent-cost proxies: “Closing Percent

Quoted Spread” from Chung and Zhang (2014), “High-Low” from Corwin and Schultz (2012), and “FHT”

from Fong, Holden, and Trzcinka (2017). We also include two leading percent-cost proxies that were not

analyzed by FHT. FHT explain that they did not include the “Gibbs” proxy from Hasbrouck (2004) because

it is very computationally-intensive. “Close-High-Low” from Abdi and Ranaldo (2017) was developed after

the FHT paper.

We analyze six monthly cost-per-dollar-volume low-frequency proxies from the existing literature.

This includes ‘‘Amihud’’from Amihud (2002) plus five proxies based on the “Extended Amihud Class” of

Goyenko, Holden, and Trzcinka (2009), Section 5.2. These five proxies are found by dividing five of the

monthly percent-cost proxies mentioned above by the average US dollar value of daily volume.

Specifically, these five proxies are: Gibbs Impact, FHT Impact, High-Low Impact, Closing-High-Low

Impact, and Closing Percent Quoted Spread Impact.

7 Noss et al. (2017) run similar benchmarking exercise in FOREX markets and found similar results as in our study. They benchmark the proxies to steepness of order book throughout the day which can be computed without any measurement error and does not depend on functional specification. This adds further robustness to our findings.

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Most of the monthly liquidity proxies cannot be computed on a daily basis, because they require

multiple daily observations to compute the proportion of zero return days, to run a regression, or compute

a maximum likelihood estimation. Therefore, we analyze three daily percent-cost low-frequency proxies

from the prior literature: “High-Low,” “Closing-High-Low,” and “Closing Percent Quoted Spread.”

Similarly, we analyze four daily cost-per-dollar-volume low-frequency proxies from the prior literature:

“High-Low Impact,” “Closing-High-Low Impact,” “Closing Percent Quoted Spread Impact,” and

“Amihud.”

5. Data

We obtain intraday trades, intraday quotes, and securities identifiers for non-US stocks from the

Thomson Reuters Tick History (TRTH) database, which is provided by the Securities Industry Research

Center of Asia-Pacific. We obtain other non-US data on returns, size, securities, individual security

information, and daily exchange rates from Datastream. We obtain intraday trade and quote data for US

stocks from the Trade and Quote database provided by the New York Stock Exchange. We obtain returns

and size data from the Center for Research in Security Prices (CRSP) and Compustat.

Our earlier sample covers 42 exchanges around the world from 1996 to 2007. It includes the leading

exchange by volume in 36 countries and three exchanges in China (the Hong Kong Stock Exchange,

Shanghai Stock Exchange, and Shenzhen Stock Exchange) and three exchanges in the US (the New York

Stock Exchange, American Stock Exchange, and NASDAQ). ). Given the large number of stocks and large

amount of data in the US market, we select a random sample of 400 firms out of the universe of all eligible

US firms in 1996, replace any firms that are ineligible in 1997 with randomly drawn firms out of the

universe of all eligible US firms in 1997, and so on rolling forward to 2007. Following the methodology of

Hasbrouck (2009), a stock must meet five criteria to be eligible: (1) it has to be a common stock, (2) it has

to be present on the first and last TAQ master file for the year, (3) it has to have the NYSE, AMEX or

NASDAQ as the primary listing exchange, (4) it does not change primary exchange, ticker symbol or cusip

over the year, and (5) has to be listed in CRSP.

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Our later sample cover the same 42 exchanges from 2008-2014. We select a random sample of 30

stocks per exchange – stratified by size tercile – out of all eligible firms on a given exchange in 2008. Our

stratification is to randomly select 10 large stocks, 10 medium stocks, and 10 small stocks from each

exchange. We replace any firms that are ineligible in 2009 with a randomly drawn firm from the same

exchange and the same size tercile, and repeat this process rolling forward year-by-year to 2014.

We impose several filters in order to have reliable and consistent proxy estimates. First, we require

that a stock have at least five positive-volume days and 11 non-zero return days in the month. The daily

sample is based on the stock-days contained within the set of stock-months that have at least five positive-

volume days and 11 non-zero return days. Second, for Datastream we follow the recommendation of Ince

and Porter (2006) to remove any stock-month with extreme return reversal. Finally, we winsorize the

sample at both ends at 1% level. That is, we set the values of proxies that are above the 99th percentile value

equal to the 99th percentile value and set the values of proxies below 1st percentile value equal to the 1st

percentile value. Our resulting sample is slightly larger than the FHT dataset.

For non-US exchanges, we convert the local currency value of volume into US dollars equivalent

using the daily (average over the month) exchange rate for daily (monthly) proxies. For VoV(%Spread), we

adjust dollar volume for inflation by dividing by the US CPI, which in turn has been normalized to equal

1.0 on 1/1/2000. To illustrate the importance of adjusting dollar volume for inflation over long periods of

time, Figure 1 plots the VoV(%Spread) when dollar volume is adjusted for inflation, the VoV(%Spread)

when dollar volume is not adjusted for inflation, and Close-High-Low from 1926 to 2015. We compare to

the Close-High-Low proxy, because Abdi and Ranaldo (2017) show it to be the best percent-cost proxy

from the existing literature for US research when including pre-1993 years. We find that the inflation-

adjusted version of VoV(%Spread) is very close to the Close-High-Low proxy over the entire 90 year span,

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whereas the unadjusted version of VoV(%Spread) is more than 2X off from 1926-1950 due to backwards-

in-time cumulative effect of inflation raised to the 1/3 power in the denominator of VoV(%Spread).8

For VoV(λ), we do not adjust dollar volume for inflation (i.e., we use nominal dollar volume). Our

reason for not doing so is because the RHS of the benchmark regression in equation (5) uses nominal dollar

volume and all of the competing cost-per-dollar-volume proxies use nominal dollar volume. In practical

terms, none of the VoV(λ) performance results are materially affected by this issue.

For the daily VoV proxies, we use the Parkinson (1980) volatility formula. In the case of a one-day

estimator (T=1), the formula becomes: 8 ln H L , where H is the daily high price and L is the

daily low price.9

Our earlier, high-frequency sample has 8.0 billion trades and 17.7 billion quotes. We compute the

corresponding benchmarks and proxies for 24,419 firms in 1,500,611 stock-months. Our later, high-

frequency sample has 1.8 billion trades and 14.7 billion quotes. We compute the corresponding benchmarks

and proxies for 3,087 firms in 84,789 stock-months.

Table 2 provides some descriptive statistics for the global and US samples of both monthly and

daily percent-cost proxies and benchmarks and both monthly and daily cost-per-dollar-volume proxies and

benchmarks. For the global sample, descriptive statistics are provided for both the earlier sample (1996-

2007) and the later sample (2008-2014). For the US sample, descriptive statistics are provided for the

combined sample period (1996-2014). Panel A shows that the means of all of the monthly percent-cost

proxies are in the same ballpark as the mean of percent effective spread and that this is true for both time

periods. Panel B shows that the means of all of the daily percent-cost proxies are also in the same ballpark

8 Another adjustment that we make to VoV(%Spread) is that we multiply the volume from Datastream in India by 10. The total volume for Indian exchanges in the Datastream is roughly 10 times lower than the value in the World Federation of Exchanges. 9 Following Corwin and Schultz (2012), we use the last valid high and low prices for a given stock for days where high is equal to low. We define valid high and low prices as prices on days where the stock was traded and high was not equal to low.

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as the mean of daily percent effective spread for both time periods. Panel C shows that the same is true in

the US sample at both monthly and daily frequencies.

Panel D shows that the medians of all of the cost-per-dollar-volume proxies are far different than

the median of lambda. Panel E shows that medians of the daily versions of all of the cost-per-dollar-volume

proxies are far different than the median of lambda, except for the median of VoV() in the earlier sample,

which is reasonably close to the median of lambda in that period. Panel F shows that in the US sample the

medians of all of the cost-per-dollar-volume proxies are far different from the median of lambda at both

monthly and daily frequencies. In summary, the monthly and daily percent-cost proxies are roughly on the

same level as the benchmark, but the monthly and daily cost-per-dollar-volume proxies get the level wrong.

6. Global Performance Results

6.1 Monthly Cost-Per-Dollar-Volume Proxies

Table 3 shows the global performance of monthly cost-per-dollar-volume proxies compared to the

monthly cost-per-dollar-volume benchmark, lambda. Five panels report five performance dimensions

(average cross-sectional correlation, average Spearman’s cross-sectional correlation, portfolio time-series

correlation of first differences, individual stock time-series correlation of first differences, and average

RMSE) and a sixth panel provides a summary statistic across all of the correlation results. Monthly Amihud

is by far the most widely used cost-per-dollar-volume proxy from the existing literature. As a rough

indicator of its enormous influence, we note that Amihud (2002) has more than 5,000 Google scholar

citations. Thus, in the analysis below we often highlight the relative performance of our new cost-per-dollar

proxy, monthly VoV(), and monthly Amihud.

Panel A reports the average cross-sectional correlation of seven monthly cost-per-dollar-volume

proxies compared to monthly lambda for both time-periods. The average cross-sectional correlations are

computed in the spirit of Fama and MacBeth (1973) by calculating the cross-sectional correlation each

month across all firms on a given exchange and then calculating the average correlation value over all

exchange-months to get the global average. For the entire paper, we adopt the convention of placing a solid

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box around the highest correlation in the row and a dashed box around any correlations that are statistically

indistinguishable from the highest correlation in the row at the 5% level.10 The idea is to determine the best

proxy on a given performance dimension and the full set proxies that are statistically indistinguishable from

the best. If a correlation is bold-faced, this means that it is statistically different from zero at the 5% level.11

We find that monthly VoV() dominates all six other monthly cost-per-dollar-volume proxies in

both time-periods. It has the highest correlations in both time-periods and is statistically significantly higher

than the correlations of any other proxy in both rows. There is economically large difference in performance

between monthly VoV() and the other monthly proxies.

Figure 2 shows the global average cross-sectional correlations between monthly cost-per-dollar-

volume proxies and monthly lambda over time. We see that monthly VoV()’s cross-section correlation is

higher than monthly Amihud’s cross-sectional correlation in every single month over the 19-year sample

period. In most months the gap in performance is large, which makes it an economically significantly

improvement as well.

Returning to Table 3, Panel B reports the average cross-sectional Spearman (rank-order) correlation

of monthly cost-per-dollar-volume proxies compared to monthly lambda. In the earlier period, monthly

VoV() ties with monthly Gibbs Impact for the highest Spearman correlation and is statistically

indistinguishable from the monthly Close-High-Low Impact’s Spearman correlation. In the later period,

monthly VoV() has the highest Spearman correlation and statistically distinguishable from the other

proxies, but the size of the difference is modest.

10 We test if two average cross-sectional correlations on the same row are statistically distinguishable by t-tests on the time-series of correlations in the spirit of Fama and MacBeth. Specifically, we calculate the cross-sectional correlation of each proxy for each month and then regress the correlations of one proxy on the correlations of another proxy. We assume that the time-series correlations of each proxy are i.i.d. over time, and test if the regression intercept is zero and the slope is one. Standard errors are adjusted for autocorrelation with a Newey-West correction using four lags. 11 We test if correlations are significantly different from zero using the following t-test 2 1⁄ , where r is the correlation coefficient and n is the degrees of freedom. See Swinscow (1997), chapter 11.

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Panel C reports the portfolio time-series correlation of the first differences of monthly cost-per-

dollar-volume proxies compared to the first differences of monthly lambda. That is, we compute the

equally-weighted portfolio of each monthly cost-per-dollar volume proxy across all stocks in each month

and compute the equally-weighted portfolio of monthly lambda across all stocks in each month. Then we

compute the correlation of the proxy portfolio time-series first differences and the lambda portfolio time-

series first differences. This performance criteria may be an especially useful as a measure of fitness for

liquidity risk research. Monthly VoV() has the highest portfolio time-series correlation in both periods.

The monthly Close-High-Low, Closing Percent Quoted Spread, and Amihud correlations are statistically

indistinguishable from monthly VoV() in the later period, where we test whether portfolio time-series

correlations are statistically different from each other using a Fisher’s Z-test. There is economically large

difference in performance between monthly VoV() and monthly Amihud proxy in both periods, but the

difference is especially large in the later period.

Panel D reports the individual stock time-series correlation of the first differences of monthly cost-

per-dollar-volume proxies compared to the first differences of monthly lambda. We find that monthly

VoV() has the highest individual stock time-series correlation of the first differences in both periods and

is statistically significantly higher than the correlations of any other monthly proxy in both rows. There is

economically large difference in performance between monthly VoV() and monthly Amihud in both

periods.

Panel E reports the simple average of the correlations on the eight contests above (i.e., four panels

above X two time-periods per panel). This simple summary statistic allows us to get an overview of each

proxy’s correlation performance. We find that monthly VoV() has an average correlation of 0.562, which

compares to 0.464 for monthly Amihud. Thus, Monthly VoV() provides a large and economically

meaningful gain in correlation performance over monthly Amihud.

Figure 3 plots the global values of monthly VoV(), monthly Amihud, and monthly lambda by

quantiles (i.e., cutoffs of equal probability) of monthly lambda. Note that the y-axis is on a log-scale. Both

15

monthly VoV() and monthly Amihud get the rank ordering correct. That is, they both yield higher values

when monthly lambda is higher and lower values when it is lower. Monthly VoV() is close the actual

values of monthly lambda for high quantiles, but up to two orders of magnitude off for low quantiles.

Conversely, monthly Amihud is close to monthly lambda for low quantiles, but up to 1.5 orders of

magnitude off for high quantiles. Interestingly, monthly Amihud is much more noisy than monthly VoV().

Considering the illustration above that Amihud is sensitive to low-volume outliers, one would expect this

kind of noisy disturbances in Amihud.

Returning to Table 3, Panel F reports the average root mean squared error (RMSE) of monthly cost-

per-dollar-volume proxies compared monthly lambda and then divided by the median of monthly lambda.

The RMSE is calculated every month for given exchange and then averaged over all sample months. The

average RMSE is useful for determining if a particular liquidity proxy does a good job of capturing the

level of a liquidity benchmark, not just that it is correlated with a liquidity benchmark. Capturing the level

is important for many research purposes, such as computing the returns to a trading strategy net of a

correctly-scale proxy for transaction costs. In the case of RMSE comparisons, the solid box denoting the

best performance is placed around the lowest average RMSE in the row and the dashed box is placed around

the average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row. We

test if average RMSEs are statistically indistinguishable from each other using paired t-test. Boldfaced

RMSE indicated that the ability of a proxy to predict the benchmark is statistically greater than zero at the

5% level.12

We find that monthly VoV() has the lowest ratio of average RMSE over median of monthly lambda

at 74 in the earlier period and 93 in the later period. The rest of the proxies have even larger ratios. This

means that the average error is nearly two orders of magnitude larger than the median of lambda itself.

Thus, none of the monthly liquidity proxies does a good job of capturing the level of monthly lambda.

12 Throughout the paper, we use the U-statistic of Theil (1996) to test if RMSEs are statistically significant. Under Theil’s convention if U2 = 1, then the proxy has no ability to predict the benchmark. We test if U2 is significantly less than 1 based on a F distribution, where the number of degrees of freedom is the sample size.

16

To summarize this sub-section, the global performance of monthly VoV() strongly dominates

monthly Amihud and the rest of the monthly cost-per-dollar-volume liquidity proxies. In Panels A-D, it has

the highest correlation with monthly lambda in all eight contests. These contests span cross-sectional and

time-series; regular and Spearman; and portfolio and individual stock correlations of first differences.

Monthly VoV()’s correlation is statistically significantly higher than the correlation of monthly Amihud

in all eight cases. Its average correlation across all eight contests provides a large and economically

meaningful gain in correlation performance. Further, monthly VoV() is much more robust (i.e. less noisy)

than monthly Amihud. However, none of the monthly proxies captures the level of monthly lambda.

6.2 Daily Cost-Per-Dollar-Volume Proxies

Table 4 shows the global performance of daily cost-per-dollar-volume proxies compared to daily

lambda. It reports the same five performance dimensions (average cross-sectional correlation, average

Spearman’s cross-sectional correlation, portfolio time-series correlation of first differences, individual

stock time-series correlation of first differences, and average RMSE) and provides a summary statistic

across all of the correlation results.

We find that daily VoV() has the highest correlation with daily lambda in all eight contests (panels

A-D X 2 time-periods per panel). In seven of the eight contests, its correlation is statistically significantly

higher than the correlations of all other cost-per-dollar-volume proxies and one case it is insignificantly

different than daily High-Low Impact. The gap between the performance of daily VoV() and the

performance of the second-place finisher on the row is moderate-to-large in all cases. Panel E reports that

daily VoV()’s average correlation across all eight contests is 0.534 compared to 0.444 for daily Amihud,

which is a large and economically meaningful gain in correlation performance.

Panel F reports the average RMSE divided by the median of daily lambda. The average error is

one-to-two orders of magnitude larger than the median of daily lambda. Thus, none of the daily liquidity

proxies does a good job of capturing the level of daily lambda.

6.3. Monthly Percent-Cost Proxies

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Table 5 shows the global performance of monthly percent-cost proxies compared to the monthly

percent-cost benchmark, percent effective spread. It reports the same five performance dimensions and

provides a summary statistic across all of the correlation results.

We find that monthly Closing Percent Quoted Spread has the highest correlation in six of the eight

contests (specifically in both time-periods of Panels A, B, and D) and its correlation is statistically

significantly higher than that of any other monthly percent-cost proxy in those six cases. Monthly

VoV(%Spread) has the second highest correlation in those six cases and there is a large gap to the third

highest correlation in those six cases. Monthly VoV(%Spread) and monthly Closing Percent Quoted Spread

have the highest portfolio time-series correlation or are indistinguishable from the highest correlation in

both time-periods in Panel C. These correlation results are summarized in Panel E, which shows monthly

Closing Percent Quoted Spread with the highest average correlation at 0.784, monthly VoV(%Spread) is a

close second at 0.757, and there is a large gap to the third highest at 0.615.

Figure 4 shows the global average cross-sectional correlations between monthly percent-cost

proxies and monthly percent effective spread over time. We see that monthly Closing Percent Quoted

Spread has the highest cross-section correlation in all months (except for two) over the 19-year sample

period. Monthly VoV(%Spread) is a close second in all months (except for two) over 19 years. Then there

is a large drop in performance to the rest of the monthly percent-cost proxies.

Returning to Table 5, Panel F reports the average RMSEs. Monthly Close-High-Low and monthly

High-Low have the lowest average RMSE for the earlier and later periods, respectively. All of the average

RMSEs are statistically significant for all of the monthly percent-cost proxies, which in marked contrast to

analogous results for monthly cost-per-dollar-volume proxies. Essentially, all of the monthly percent-cost

proxies get the level right. The gap in performance between the VoV(%Spread) average RMSEs and the

lowest average RMSEs in a row is relatively small.

Figure 5 plots the global values of monthly VoV(%Spread), monthly Closing Percent Quoted

Spread, and monthly percent effective spread by quantiles (i.e., cutoffs of equal probability) of monthly

percent effective spread. Note that the y-axis is now on a linear-scale. Both monthly VoV(%Spread) and

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monthly Closing Percent Quoted Spread are very close to monthly percent effective spread across-the-

board. Monthly VoV(%Spread) does slight better with high quantiles and monthly Closing Percent Quoted

Spread does slightly better with low quantiles, but both get the level right.

To summarize Table 5, monthly VoV(%Spread) performs very well and comes in a close second.

However, monthly Closing Percent Quoted Spread is the best monthly percent-cost proxy for global

research. This confirms the prior monthly recommendation of FHT, but does so over an expanded set of

test proxies.

6.4 Daily Percent-Cost Proxies

Table 6 shows the global performance of daily percent-cost proxies compared to daily percent-cost

benchmark, percent effective spread. It reports the same five performance dimensions and provides a

summary statistic across all of the correlation results.

We find that daily Closing Percent Closing Spread has the highest correlation with daily percent

effective spread in four of the eight contests in Panels A-D and is statistically significantly higher than any

other daily percent-cost proxy in those four cases. VoV(%Spread) has the highest correlation in the other

four contests and is statistically significantly higher than the other daily percent-cost proxy in those four

cases. Panel E reports that daily Closing Percent Quoted Spread’s average correlation across all eight

contests is 0.673 compared to 0.652 for daily VoV(%Spread), which is a small gap in performance. The

third place proxy finishes far behind with an average correlation across all eight contests of 0.302.

Panel F reports that daily VoV(%Spread) and daily Closing Percent Quoted Spread have the lowest

average RMSE in the earlier and later periods, respectively. All of the daily percent-cost proxies do a good

job of capturing the level, except for the case of VoV(%Spread) in the later period when it is statistically

insignificant.

To summarize Table 6, daily VoV(%Spread) performs very well, but comes in a close second. Daily

Closing Percent Quoted Spread is the best daily percent-cost proxy for global research, which confirms the

prior daily recommendation of FHT, but does so over an expanded set of test proxies.

6.5 Monthly Robustness Checks

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In this subsection we check the robustness of our monthly results. Table 7 reports the global

performance of monthly cost-per-dollar-volume proxies with monthly lambda by size and volatility

quintiles and after controlling for size and volatility. Panels A and B report average cross-sectional

correlations broken out by size quintiles and by volatility quintiles, respectively. We find that monthly

VoV() has the highest average cross-sectional correlations for all five size quintiles and for all five

volatility quintiles. In all cases, it is significantly higher than any other monthly cost-per-dollar-volume

proxy and there is an economically meaningful gain in correlation performance.

Panels C and D report individual stock time-series correlations of first differences by size quintiles

and by volatility quintiles, respectively. We find that monthly VoV() has the highest individual stock time-

series correlations of first differences for all five size quintiles and for all five volatility quintiles. In all

cases, it is significantly higher than any other monthly cost-per-dollar-volume proxy and there is an

economically meaningful gain in correlation performance.

Panel E reports pooled partial correlations after controlling for log size and volatility. The

correlations are computed over the pooled sample (i.e., across all stocks and across all months). The partial

correlations are computed by regressing each monthly cost-per-dollar-volume proxy on monthly log size

and volatility. Next, monthly lambda is regressed on monthly log size and volatility. Finally, the residuals

from both regressions are collected and correlated with each other to yield the partial correlations. We find

that monthly VoV() has the highest pooled partial correlation and it statistically significantly higher than

any other monthly cost-per-dollar-volume proxy. Panel F reports pooled partial Spearman’s correlations

after controlling for log size and volatility. We find that monthly Gibbs Impact has the highest pooled

Spearman’s partial correlation and that monthly VoV() is statistically indistinguishable from it.

In summary, the dominant performance of monthly VoV() is robust by size and volatility

subsamples. Further, it is robust net of controlling for size and volatility.

Table 8 reports the global performance of monthly percent-cost proxies with monthly percent

effective spread by size and volatility quintiles and after controlling for size and volatility. Panels A and B

20

report average cross-sectional correlations by size quintiles and by volatility quintiles, respectively. We

find that monthly Closing Percent Quoted Spread has the highest average cross-sectional correlations for

all five size quintiles and for all five volatility quintiles. We also find that monthly VoV(%Spread) has the

second highest correlation in all cases and there is a large gap to the third highest correlation.

Panels C and D report individual stock time-series correlations of first differences by size quintiles

and by volatility quintiles, respectively. We find that monthly Closing Percent Quoted Spread has the

highest individual stock time-series correlations of first differences for all five size quintiles and for all five

volatility quintiles. Again we also find that monthly VoV(%Spread) has the second highest correlation in

all cases and there is a large gap to the third highest correlation.

Panel E reports pooled partial correlations after controlling for log size and volatility. We find that

monthly Closing Percent Quoted Spread has the highest pooled partial correlation and it statistically

significantly higher than any other monthly cost-per-dollar-volume proxy. Panel F reports pooled partial

Spearman’s correlations after controlling for log size and volatility. We find that monthly Closing Percent

Quoted Spread has the highest pooled Spearman’s partial correlation. We find that monthly VoV(%Spread)

has the second highest pooled partial correlation in both Panel E and F and there is a large gap to the third

highest correlation.

In summary, the first-place performance of monthly Closing Percent Quoted Spread and the

second-place performance of monthly VoV(%Spread) are robust by size and volatility subsamples. Further,

they are robust net of controlling for size and volatility.

6.6 Monthly General Volatility Over Volume Class Proxies

Now we turn to the general VoV class as given by the equation b ca V . The new volume over

volatility proxies that we have tested so far have fixed parameters for a, b, and c. Specifically, the

parameters for VoV(%Spread) are 8 , 2 3,a CPI b and 1 3c and the parameters for VoV(λ) are

1, 1,a b and 1 2c . Now we ask, how much additional global performance do we obtain if we

21

optimally select the a, b, and c parameters from the set of non-negative real numbers so as to maximize

pooled correlation?

Table 9 provides the answer. It shows the global performance of monthly general VoV class proxies

under the parameters that maximize pooled correlation and top monthly liquidity proxies compared to

monthly liquidity benchmarks. Performance is defined as the pooled correlation (i.e., combined time-series

and cross-sectional) over all global stock-months in the earlier or later periods.

Panel A is for monthly cost-per-dollar-volume proxies compared to monthly lambda. We see the

optimal values of a, b, c, and ρ for each time period. In the earlier period, the general VoV class pooled

correlation is 0.780 versus 0.777 for VoV(λ). In the later period, the general VoV class yields 0.425 versus

0.405 for VoV(λ). In both cases, gain in performance is very small and comes with the risk of overfitting

the data in-sample.

Panel B is for monthly percent-cost proxies compared to monthly percent effective spread. We see

the optimal values of a, b, c, and ρ for each time period. In the earlier period, the general VoV class yields

0.836 versus 0.817 for VoV(%Spread) and in the later period, the general VoV class yields 0.849 versus

0.826 for VoV(%Spread). Again, both periods yield very small gains in performance and come with the risk

of overfitting.

In summary, there is very little to gain by going with the general VoV class and the overfitted

performance in-sample may not hold up out-of-sample. Therefore, our recommendation is to stay with the

VoV(%Spread) and VoV(λ) proxies which are grounded by the transaction cost invariance hypothesis, rather

than the general VoV class of proxies.

7. US Performance Results

7.1 Cost-Per-Dollar-Volume Proxies

Table 10 shows the US performance of monthly (daily) cost-per-dollar-volume proxies compared

to the monthly (daily) cost-per-dollar-volume benchmark, lambda. It reports the same five performance

dimensions and provides a summary statistic across all monthly (daily) correlation results.

22

We find that monthly (daily) VoV(λ) has the highest correlation in two (four) of the four contests.

Monthly FHT Impact has the highest correlation in one case and monthly Close-High-Low Impact has the

highest correlation in another case, but monthly VoV(λ) is insignificantly different in one of these cases.

Panel E summarizes that monthly VoV(λ) has an average correlation of 0.584, which is a small gain over

the 0.569 correlation of the second place finisher, but is a large gain over the 0.433 correlation of monthly

Amihud. Daily VoV(λ) has an average correlation of 0.546, which is a large gain over 0.401 of the second

place finisher.

Panel F reports that the average error is more than an order of magnitude larger than the median of

monthly (daily) lambda. Thus, none of the monthly (daily) liquidity proxies does a good job of capturing

the level of monthly (daily) lambda.

In summary, US performance of monthly and daily VoV() mirrors their strong global performance.

Specifically, the US performance of monthly VoV() strongly dominates monthly Amihud and the US

performance of daily VoV() strongly dominates all daily cost-per-dollar-volume proxies. None of the

monthly (daily) cost-per-dollar-volume proxies capture the level of monthly (daily) lambda.

7.2 Percent-Cost Proxies

There is an important difference in the structure of US stock data versus global stock data. US stock

trade data (including closing trade price, daily volume, daily high, and daily low) is available from 1925 to

the present, but US stock quote data (including closing bid and closing ask) is only continuously available

across the whole US market from 1993 to the present.13 In other words, there is a period of 68 years in

which US trade data is available to construct percent-cost proxies, but US quote data is partially available

at best. This essentially rules out the use of Closing Percent Quoted Spread for studies that cover pre-1993

years. By contrast, a large sample of global stock quote data is available for most exchanges from 1996 to

the present.

13 Specifically, CRSP provides closing bid and ask prices for NYSE/AMEX stocks form 1926-1941 and 1993-present. It provides the same for NASDAQ Global Market and Global Select Market stocks form 1982-present and for NASDAQ Capital Market stocks from 1992-present.

23

This corresponds roughly to the time period that global stock trade data is available across a wide

cross-section of exchanges worldwide. Therefore, Closing Percent Quoted Spread works as a percent-cost

proxy for global samples and it is candidate proxy for US studies that cover 1993 to the present, but we

need to find alternatives to it for US studies that cover pre-1993 years.

Table 11 shows the US performance of monthly (daily) percent-cost proxies compared to the

monthly (daily) percent-cost benchmark, percent effective spread. It reports the same five performance

dimensions and provides a summary statistic across all monthly (daily) correlation results.

We find that monthly (daily) VoV(%Spread) has the highest correlation in two (four) of the four

contests and is statistically significantly higher than any other monthly (daily) percent-cost proxy in two

(three) of those cases. Closing Percent Quoted Spread has the highest correlation in the remaining two

(zero) contests. Panel E shows that monthly Closing Percent Quoted Spread has an average correlation of

0.719, which is a tiny gain over the 0.705 correlation of monthly VoV(%Spread) and which in turn in a

large gain over the 0.593 correlation of the third place finisher. Daily VoV(%Spread) has an average

correlation of 0.584, which is a moderate gain over the 0.516 correlation of daily Closing Percent Quoted

Spread and which is a large gain over the 0.206 correlation of the third place finisher. Panel F reports that

monthly (daily) VoV(%Spread) has the lowest average RMSE and is statistically significantly lower than

any other monthly (daily) percent-cost proxy.

In summary, for US studies that cover 1993-present, monthly (daily) VoV(%Spread) and monthly

(daily) Closing Percent Quoted Spread are very closely matched. Both perform well and either one could

be used with confidence. For US studies that cover pre-1993 years, monthly (daily) VoV(%Spread) strongly

dominates any other monthly (daily) percent-cost proxy. Monthly (daily) VoV(%Spread) will provide large

performance gains over the alternatives.

8. Do VoV Liquidity Proxies Affect Research Inferences? The Case Of The Quant Meltdown

Do our new VoV liquidity proxies matter? That is, do they yield different research inferences than

the best previous liquidity proxies from the prior literature? To find out, we perform a case study that

24

compares daily VoV() and daily Amihud when analyzing the pattern of illiquidity before, during, and after

the August 2007 “Quant Meltdown” event.

Khandani and Lo (2007, 2011) note that over the four-day period of August 6-9, 2007 some of the

most successful equity hedge funds in history reported record losses.14 Specifically, these loss

announcements were made by quantitatively-managed, market-neutral hedge funds, giving rise to the

popular term for the event: the “Quant Meltdown.” Investors reacted to this news by withdrawing large

amounts of capital from these funds, which caused them to liquidate large amounts of their invested

portfolios. These forced liquidations pushed down the price of the securities being sold, which in turn

imposed loses on other similarly-invested hedge funds. Which in turn lead investors to pull money out of

those hedge funds as well, which forced them to liquidate some of their holdings. Khandani and Lo propose

the “Unwind Hypothesis” to explain this sequence of events around the Quant Meltdown announcements.

They find that some of the downward market impacts happened before the announcements, because some

of the quantitatively-managed hedge funds had quietly began to unwind their positions in late July and early

August as they realized the magnitude of the losses that they were facing.

For their empirical analysis, Khandani and Lo use the high-frequency Trade and Quote (TAQ)

dataset. They use this high-frequency data to compute a cost-per-dollar-volume “price impact” measure for

the S&P 1500 stock portfolio. They find that “the pattern of price impact displayed in Fig. 9 documents a

substantial drop in liquidity in the days leading up to August 6, 2007.” They further note that after the quant

meltdown announcements, the resulting large-scale liquidations caused further price impacts as the “market

makers burned by the turn of events in the week of August 6, reduced their market making capital in the

14 Khandani and Lo provide the following examples of major loss announcements. The Wall Street Journal reported on August 10, 2007, that ‘‘After the close of trading, Renaissance Technologies Corp., a hedge-fund company with one of the best records in recent years, told investors that a key fund has lost 8.7% so far in August and is down 7.4% in 2007. Another big fund company, Highbridge Capital Management, told investors its Highbridge Statistical Opportunities Fund was down 18% as of the 8th of the month, and was down 16% for the year. The $1.8 billion publicly traded Highbridge Statistical Market Neutral Fund was down 5.2% for the month as of Wednesday … Tykhe Capital, LLC – a New York- based quantitative, or computer-driven, hedge-fund firm that manages about $1.8 billion – has suffered losses of about 20% in its largest hedge fund so far this month…’’ (see Zuckerman, Hagerty, and Gauthier-Villars, 2007), and on August14, The Wall Street Journal reported that the Goldman Sachs Global Equity Opportunities Fund ‘‘…lost more than 30% of its value last week…’’ (Sender, Kelly, and Zuckerman, 2007).

25

following days and in turn caused the price impacts to rise substantially starting on August 10 and remain

high for the following week.” They conclude that their evidence supports the Unwind Hypothesis.

We examine the same three periods that were before, during, and after the quant meltdown

announcements. Specifically, we follow the exact dates mentioned by Khandani and Lo (see above) as

when they found higher price impact. The three periods are: before (July 30 – August 3), during (August 6

– August 9), and after (August 10 – August 17).

We analyze the S&P 1500 stocks. For each stock, we compute their cost-per-dollar-volume

illiquidity using both the daily VoV() and daily Amihud proxies. Following Khandani and Lo, we compute

the portfolio’s overall illiquidity on an equally-weighted basis.

Figure 6 shows the percent change in S&P 1500 illiquidity from July 2 to August 31, 2007 relative

to the average value over the base period. The base period are the two weeks of July 2 – July 13, 2007,

which is well in advance of the before period. The solid red circles are based on daily VoV() and open

blue squares are based on daily Amihud. The black vertical lines demark the before, during, and after

periods. First, it is immediately clear that daily Amihud is much more volatile than daily VoV(). For

example, during the base period the standard deviation of daily Amihud is 5.7 times larger than the standard

deviation of daily VoV(). Secondly, starting from the base period daily VoV() rises in the before period,

rises again in the during period, and rises to a peak in the after period. By contrast, daily Amihud declines

slightly in the before period, continues to be negative in the during period, and rises moderately in the after

period. That is, daily VoV() matches the S&P 1500 illiquidity pattern described by Khandani and Lo,

whereas daily Amihud does not match.

Table 12 reports the percent change in S&P 1500 illiquidity before, during, and after the quant

meltdown relative to the average value over the base period. Compared to the base period, VoV() rises

52.0% in the before period, rises again to 88.4% in the during period, and rises to a peak 118.9% in the

after period. All three values are significant at either the 5% or 1% levels. By contrast, daily Amihud

declines slightly to -5.8% in the before period, continues to be negative at -2.1% in the during period, and

26

rises moderately to 24.7% in the after period. None of the three values are significant. The insignificance

in the after period is driven both by the lower level of daily Amihud and by the higher volatility of the

Amihud standard error.

In summary, the two proxies yield different research inferences. The daily VoV() proxy yields the

inference that S&P 500 illiquidity is significantly higher than the base in all three periods. By contrast, the

daily Amihud proxy yields the inference that S&P 500 liquidity in not significantly different than the base

in all three periods. We conclude that our new VoV liquidity proxies do matter.

9. Conclusion

We examine a general class of volatility over volume liquidity proxies as computed from low

frequency (daily) data. We start from the Kyle and Obizhaeva (2016) hypothesis of transaction cost

invariance to identify a new volatility over volume liquidity proxy “VoV(%Spread)” for percent spread cost

and a new volatility over volume liquidity proxy “VoV(λ)” for the slope of the transaction cost function “λ”.

We test the monthly and daily versions of these new and existing liquidity proxies against liquidity

benchmarks as estimated from high frequency (intraday) data on both a global and US basis. We find that

both the monthly and daily versions of VoV(λ) dominate the equivalent versions of Amihud and other cost-

per-dollar-volume proxies on both a global and US basis. We also find that both the monthly and daily

versions of VoV(%Spread) dominate the equivalent versions of other percent-cost proxies for US studies

that cover pre-1993 years. In a case study, we find that our new VoV liquidity proxies yield different

research inferences than the best previous liquidity proxies from the prior literature. The success of our

invariance-based liquidity proxies across exchanges and over time supports the prediction of Kyle and

Obizhaeva of a specific functional form for transaction costs across exchanges and over time.

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32

Appendix A: Liquidity Measure Derivations

VoV(%Spread) Derivation

Transaction cost invariance supports alternative functional forms. One of those cases is a constant

percent quoted spread cost as given by their equation (19) for the percent quoted spread

1

30 ,jt

jt jtjt

sW

P

(15)

where jts is the dollar quoted spread of asset j on date t , jtP is the price of asset j on date t , 0 is a

constant, and jt is bet volatility, which is the standard deviation of returns generated by bet price impact.

jtW is bet activity the of asset j on date t , which is defined to be the product of price ,jtP bet volume

,jtV and bet volatility jt , as given by

.jt jt jt jtW P V (16)

Bet volume jtV is assumed to be related to trading volume jtV as follows

2.jt jt

jt

V V

(17)

where jt is a volume multiplier. Bet volatility jt is assumed to be related to returns volatility jt as

follows

jt jt jt (18)

where jt is the scale factor.

Substituting (16), (17), and (18) into (15), we obtain

1

3

0

1 2 1 2 13 3 3 3 3

0

2

2 ,

jtjt jt jt jt jt jt

jt jt

jt jt jt jt

sP V

P

V

(19)

33

where the second equality follows by just rearranging terms. Equation (19) is our VoV(%Spread) proxy and

can be written in volatility over volume form as

%

bjt

cjt

VoV Spreads a

P V

, (20)

where 1 2 1

3 3 302 ,jt jta

2

,3

b and 1

3c .

VoV(λ) Derivation

Alternatively, transaction cost invariance supports a linear expression for price impact jt jtC Q of

asset j on date t . Substituting 0 0 into Kyle and Obizhaeva equation (18), we obtain

1

2

,jtjt jt I jt

jt

QC Q

V (21)

where I is a constant and jtQ is unsigned number of shares in a bet. Substituting equations (17) and (18)

into (21), we obtain

1

2

.2

jtjt jt I jt jt

jtjt

QC Q

V

(22)

Define jtD as the sign of the net order imbalance in asset j on date t . Multiplying both sides of equation

(22) by jtD and multiplying the RHS numerator and denominator by 1

2jtP , we obtain

1

21

1 122 2

.

2

I jt jtjt jt jt jt jt jt

jt jtjt

D C Q D P Q

P V

(23)

34

Equation (23) is essentially a regression of signed price impact returns on the signed square root of dollar

volume. The RHS coefficient of this regression, which is shown in curly brackets, is our VoV(λ) price

impact proxy. It can be written in the volatility over volume form as

b

cVoV

a

V

, (24)

where we can map our notation to the Kyle and Obizhaeva notation as follows jt and jt jtV P V and

where ,2

I jt jta

1,b and

1

2c .

Appendix B: Low-Frequency Proxies from the Existing Literature.

The Gibbs sampler is an estimation process with three steps for each iterative sweep. First, given the

sample of prices P from all days in the time interval, starting values for the buy/sell indicator ,Q a prior

for the effective half-spread c , and a prior for the variance of public information innovations 2u ,

estimate c using a Bayesian regression that is restricted to the positive domain. Second, given , Q,P the

prior for 2u , and the updated estimate of c , estimate the residuals and make a new draw of 2

u from an

inverted gamma distribution. Third, given ,P the updated estimate of c , and the new draw of 2u , make

new draws of and Q V . You run 1,000 sweeps of the sampler, discard the first 200 as burn-in, and then

take the mean of the c values in the remaining 800 sweeps as the final estimate of c .

1 1+2 ,

2

zFHT N

where is the return standard deviation, z is the frequency of zero returns, and

1N is the inverse function of the cumulative normal distribution.

2 1;

1

t

t

eHigh Low Average

e

where 2

,3 2 2 3 2 2

t t tt

t is the sum over two

days of the squared daily log(high/low), and t is the squared log(High/Low) where the High (Low)

value is over two days.

35

411

max ,0 ,N

t t t tN tCHL c c

where tc is the close log-price on day t and the mid-range

/ 2t t tl h is the average of the daily high and low log-prices on day t.

/ 2

t t

t t

Closing Ask Closing BidClosing Percent Quoted Spread Average

Closing Ask Closing Bid

Listed below are five cost-per-dollar-volume proxies for month i (or day i) based on the “Extended

Amihud” class of proxies as defined in Goyenko, Holden, and Trzcinka (2009), Section 5.2:

Gibbs Impacti = Gibbsi / (Average Daily US Dollar Value of Local Volume) i.

FHT Impacti = FHTi / (Average Daily US Dollar Value of Local Volume) i.

High-Low Impacti = High-Lowi / (Average Daily US Dollar Value of Local Volume) i.

Close-High-Low Impacti = Close-High-Lowi / (Average Daily US Dollar Value of Local Volume) i.

Closing Percent Quoted Spread Impacti = Closing Percent Quoted Spreadi / (Average Daily US Dollar

Value of Local Volume) i.

t

t

rAmihud Average

Dollar Volume

, where the average is computed over positive volume days

only and where tr is the stock return on day t and tDollar Volume is the US Dollar value of volume on

day t.

36

Figure 1. Monthly VoV(%Spread) adjusted for inflation vs. unadjusted compared to close-high-low in the

US over time.

Figure 2. Global average cross-sectional correlations between monthly cost-per-dollar-volume proxies

and monthly lambda over time.

37

Figure 3. The global values of monthly VoV(λ) and monthly Amihud compared to monthly lambda by the

quantile of monthly lambda.

Figure 4. Global average cross-sectional correlations between monthly percent-cost proxies and monthly

percent effective spread over time.

38

Figure 5. The global values of monthly VoV(%Spread) and monthly closing percent quoted spread

compared to monthly percent effective spread by quantile of monthly percent effective spread.

Figure 6. Using VoV() and Amihud to compute the percent change in S&P 1500 illiquidity before,

during, and after the August 6-9, 2007 “quant meltdown.”

‐100.0%

‐50.0%

0.0%

50.0%

100.0%

150.0%

200.0%

7/2/07 7/9/07 7/16/07 7/23/07 7/30/07 8/6/07 8/13/07 8/20/07 8/27/07 9/3/07

Percent Chan

ge in S&P 1500 Illiquidity

(Base Period: July 2 ‐July 13, 2007)

VoV(λ)Amihud

After = Aug 10 ‐ Aug 17

During = Aug 6 ‐ Aug 9

Before = July 30‐ Aug 3

39

Table 1.

Numerical Illustration

Day Return

Dollar

Volume

Amihud

ratio

1 0.050 $300,000 0.00000017

2 ‐0.030 $250,000 0.00000012

3 0.020 $200,000 0.00000010

4 ‐0.010 $210,000 0.00000005

5a 0.040 $5,000 0.00000800

5b 0.040 $250,000 0.00000016

Std Dev of

Return

Average

Volume

VoV(λ)

* 1,000,000

Amihud

* 1,000,000

Days 1‐4 & 5a 0.034 $193,000 76.52 1.69

Days 1‐4 & 5b 0.034 $242,000 68.33 0.12

Ratio 1.1 14.2

40

Table 2

Descriptive Statistics for the Global and US Samples

Percent‐

Cost

Benchmark

Gibbs FHT

High‐

Low

Close‐

High‐Low

Closing %

Quo Sprd

VoV

(%Spread)

% Effective

Spread

Stock‐

Months Stocks

Panel A: Mean of the Global Monthly Percent‐Cost Proxies and Benchmarks

1996‐2007 0.024 0.014 0.013 0.015 0.021 0.020 0.017 1,500,611 24,419

2008‐2014 0.021 0.011 0.013 0.015 0.018 0.019 0.015 85,897 1,807

Panel B: Mean of the Global Daily Percent‐Cost Proxies and Benchmarks

1996‐2007 0.011 0.014 0.017 0.015 0.015 28,042,581 23,308

2008‐2014 0.012 0.014 0.016 0.018 0.014 1,687,131 1,840

Panel C: Mean of the US Percent‐Cost Proxies and Benchmarks 1996‐2014

Monthly 0.028 0.010 0.017 0.019 0.023 0.016 0.020 60,537 962

Daily 0.013 0.015 0.013 0.011 0.011 793,057 467

Cost‐Per‐$‐

Volume

Benchmark

Gibbs

Impact

FHT

Impact

High‐

Low

Impact

Close‐

High‐Low

Impact

Closing %

Quo Sprd

Impact Amihud VoV() Lambda

Panel D: Median of the Global Monthly Cost‐Per‐Dollar‐Volume Proxies and Benchmarks

1996‐2007 0.060 0.015 0.028 0.036 0.032 0.099 47.253 26.013

2008‐2014 0.050 0.009 0.026 0.031 0.030 0.084 41.152 9.190

Panel E: Median of the Global Daily Cost‐Per‐Dollar‐Volume Proxies and Benchmarks

1996‐2007 0.007 0.001 0.007 0.016 22.966 19.310

2008‐2014 0.004 0.000 0.004 0.008 17.434 6.246

Panel F: Median of the US Cost‐Per‐Dollar‐Volume Proxies and Benchmarks 1996‐2014

Monthly 0.030 0.002 0.017 0.020 0.016 0.049 36.763 8.581

Daily 0.002 0.000 0.001 0.004 12.721 2.406

Percent‐Cost Proxies

Cost‐Per‐Dollar‐Volume Proxies

The monthly (daily) high‐frequency liquidity benchmarks, percent effective spread and lambda, are

calculated from every trade and corresponding BBO quote in the SIRCA Thomson Reuters Tick

History database for a sample stock‐month (stock‐day). The monthly (daily) percent‐cost proxies

and cost‐per‐dollar‐volume proxies are calculated from daily stock price data for a sample stock‐

month (stock‐day). The earlier (later) global sample spans 42 exchanges around the world from

1996‐2007 (2008‐2014). All stock‐months are required to have at least five positive‐volume days and

eleven non‐zero return days. The medians of all of the cost‐per‐dollar‐volume proxies and

benchmark have been multiplied by 1,000.

41

Table 3

Gibbs

Impact

FHT

Impact

High‐Low

Impact

Close‐

High‐Low

Impact

Closing %

Quo Sprd

Impact Amihud VoV()

1996‐2007 0.671 0.605 0.644 0.661 0.685 0.588 0.755

2008‐2014 0.424 0.377 0.426 0.417 0.406 0.404 0.471

1996‐2007 0.490 0.330 0.481 0.486 0.467 0.468 0.490

2008‐2014 0.538 0.389 0.533 0.532 0.522 0.525 0.551

1996‐2007 0.781 0.805 0.760 0.764 0.797 0.854 0.937

2008‐2014 0.232 0.240 0.272 0.293 0.265 0.305 0.510

1996‐2007 0.363 0.218 0.348 0.348 0.365 0.304 0.406

2008‐2014 0.322 0.175 0.321 0.319 0.319 0.262 0.375

Panel E: Average Correlation of the Eight Contests Above

Summary Stat. 0.478 0.392 0.473 0.478 0.478 0.464 0.562

1996‐2007 102 102 104 103 112 95 74

2008‐2014 103 104 104 104 102 100 93

The Global Performance of Monthly Cost‐Per‐Dollar‐Volume Proxies Compared to Monthly

Lambda

Panel D: Individual Stock Time‐Series Correlation of First Differences of Monthly Cost‐Per‐Dollar‐

Volume Proxies Compared to the Monthly Lambda

Panel F: Average Root Mean Squared Error (RMSE) of Monthly Cost‐Per‐Dollar‐Volume Proxies

Compared to Monthly Lambda / Median of Monthly Lambda

The high‐frequency cost‐per‐dollar‐volume benchmark, monthly lambda, is calculated from

every trade and corresponding BBO quote in the SIRCA Thomson Reuters Tick History database

for a sample stock‐month. The monthly cost‐per‐dollar‐volume proxies are calculated from daily

stock price data for a sample stock‐month. The earlier (later) sample spans 42 exchanges around

the world from 1996‐2007 (2008‐2014). All stock‐months are required to have at least five positive‐

volume days and eleven non‐zero return days. A solid box means the highest correlation or the

lowest average root mean squared error (RMSE) in the row. Dashed boxes mean correlations that

are statistically indistinguishable from the highest correlation or average RMSEs that are

statistically indistinguishable from the lowest average RMSE in the row at the 5% level. Bold‐

faced numbers are statistically different from zero or proxies have predictive power that is

significant at the 5% level.

Panel A: Average Cross‐Sectional Correlation of Monthly Cost‐Per‐Dollar‐Volume Proxies

Compared to Monthly Lambda

Panel B: Average Spearman's Cross‐Sectional Correlation of Monthly Cost‐Per‐Dollar‐Volume

Proxies Compared to Monthly Lambda

Panel C: Portfolio Time‐Series Correlation of First Differences of Monthly Cost‐Per‐Dollar‐

Volume Proxies Compared to Monthly Lambda

42

Table 4

High‐Low

Impact

Close‐

High‐Low

Impact

Closing %

Quo Sprd

Impact Amihud VoV()

1996‐2007 0.481 0.421 0.528 0.530 0.638

2008‐2014 0.347 0.304 0.364 0.372 0.429

1996‐2007 0.259 0.117 0.339 0.366 0.440

2008‐2014 0.360 0.165 0.467 0.430 0.541

1996‐2007 0.761 0.797 0.721 0.775 0.827

2008‐2014 0.834 0.811 0.809 0.826 0.851

1996‐2007 0.057 0.071 0.110 0.116 0.258

2008‐2014 0.089 0.062 0.136 0.133 0.291


Summary Stat. 0.399 0.344 0.434 0.444 0.534

1996‐2007 51 51 46 51 67

2008‐2014 113 113 108 113 103

Panel D: Individual Stock Time‐Series Correlation of First Differences of Daily Cost‐Per‐Dollar‐

Volume Proxies Compared to the Daily Lambda

Panel F: Average Root Mean Squared Error (RMSE) of Daily Cost‐Per‐Dollar‐Volume Proxies

Compared to Daily Lambda / Median of Daily Lambda

The high‐frequency cost‐per‐dollar‐volume benchmark, daily lambda, is calculated from every

trade and corresponding BBO quote in the SIRCA Thomson Reuters Tick History database for a

sample stock‐day. The daily cost‐per‐dollar‐volume proxies are calculated from daily stock price

data for a sample stock‐day. The earlier (later) sample spans 42 exchanges around the world

from 1996‐2007 (2008‐2014). All stock‐days are contained within stock‐months with at least five

positive‐volume days and eleven non‐zero return days. A solid box means the highest

correlation or the lowest average root mean squared error (RMSE) in the row. Dashed boxes

mean correlations that are statistically indistinguishable from the highest correlation or average

RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the 5%

level. Bold‐faced numbers are statistically different from zero or proxies have predictive power

that is significant at the 5% level.

The Global Performance of Daily Cost‐Per‐Dollar‐Volume Proxies Compared to Daily Lambda

Panel A: Average Cross‐Sectional Correlation of Daily Cost‐Per‐Dollar‐Volume Proxies

Compared to Daily Lambda

Panel B: Average Spearman's Cross‐Sectional Correlation of Daily Cost‐Per‐Dollar‐Volume

Proxies Compared to Daily Lambda

Panel C: Portfolio Time‐Series Correlation of First Differences of Daily Cost‐Per‐Dollar‐Volume

Proxies Compared to the Daily Lambda

43

Table 5

Gibbs FHT High‐Low

Close‐

High‐Low

Closing %

Quo Sprd VoV(%Spread)

1996‐2007 0.679 0.688 0.727 0.749 0.864 0.820

2008‐2014 0.654 0.710 0.749 0.687 0.857 0.828

1996‐2007 0.548 0.672 0.567 0.613 0.899 0.849

2008‐2014 0.520 0.675 0.586 0.637 0.922 0.911

1996‐2007 0.640 0.725 0.855 0.873 0.892 0.923

2008‐2014 0.796 0.802 0.954 0.858 0.952 0.947

1996‐2007 0.209 0.150 0.237 0.245 0.468 0.392

2008‐2014 0.195 0.120 0.246 0.248 0.419 0.382


Summary Stat. 0.530 0.568 0.615 0.614 0.784 0.757

1996‐2007 0.019 0.019 0.017 0.015 0.016 0.018

2008‐2014 0.016 0.015 0.013 0.015 0.013 0.014

The Global Performance of Monthly Percent‐Cost Proxies Compared to Monthly Percent

Effective Spread

Panel D: Individual Stock Time‐Series Correlation of First Differences of Monthly Percent‐Cost

Proxies Compared to Monthly Percent Effective Spread

Panel F: Average Root Mean Squared Error (RMSE) of Monthly Percent‐Cost Proxies Compared to

Monthly Percent Effective Spread

The high‐frequency percent‐cost benchmark, monthly percent effective spread, is calculated

from every trade and corresponding BBO quote in the SIRCA Thomson Reuters Tick History

database for a sample stock‐month. The monthly percent‐cost proxies are calculated from daily

stock price data for a sample stock‐month. The earlier (later) sample spans 42 exchanges around

the world from 1996‐2007 (2008‐2014). All stock‐months are required to have at least five positive‐

volume days and eleven non‐zero return days. A solid box means the highest correlation or the

lowest average root mean squared error (RMSE) in the row. Dashed boxes mean correlations that

are statistically indistinguishable from the highest correlation or average RMSEs that are

statistically indistinguishable from the lowest average RMSE in the row at the 5% level. Bold‐

faced numbers are statistically different from zero or proxies have predictive power that is

significant at the 5% level.

Panel A: Average Cross‐Sectional Correlation of Monthly Percent‐Cost Proxies Compared to

Monthly Percent Effective Spread

Panel B: Average Spearman's Cross‐Sectional Correlation of Monthly Percent‐Cost Proxies

Compared to Monthly Percent Effective Spread

Panel C: Portfolio Time‐Series Correlation of First Differences of Monthly Percent‐Cost Proxies

Compared to Monthly Percent Effective Spread

44

Table 6

High‐Low

Close‐

High‐Low

Closing %


1996‐2007 0.395 0.401 0.765 0.660

2008‐2014 0.529 0.476 0.790 0.716

1996‐2007 0.214 0.196 0.800 0.737

2008‐2014 0.272 0.235 0.824 0.853

1996‐2007 0.673 0.589 0.917 0.952

2008‐2014 0.359 0.316 0.834 0.760

1996‐2007 ‐0.013 0.088 0.233 0.271

2008‐2014 ‐0.013 0.091 0.222 0.266


Summary Stat. 0.302 0.299 0.673 0.652

1996‐2007 0.020 0.024 0.018 0.016

2008‐2014 0.019 0.023 0.017 0.026

Panel D: Individual Stock Time‐Series Correlation of First Differences of Daily Percent‐Cost

Proxies Compared to Daily Percent Effective Spread

Panel F: Average Root Mean Squared Error (RMSE) of Daily Percent‐Cost Proxies Compared to

Daily Percent Effective Spread

The high‐frequency percent‐cost benchmark, daily percent effective spread, is calculated from


for a sample stock‐day. The daily percent‐cost proxies are calculated from daily stock price data

for a sample stock‐day. The earlier (later) sample spans 42 exchanges around the world from

1996‐2007 (2008‐2014). All stock days are contained within stock‐months with at least five

positive‐volume days and eleven non‐zero return days. A solid box means the highest

correlation or the lowest average root mean squared error (RMSE) in the row. Dashed boxes

mean correlations that are statistically indistinguishable from the highest correlation or average

RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the 5%

level. Bold‐faced numbers are statistically different from zero or proxies have predictive power

that is significant at the 5% level.

The Global Performance of Daily Percent‐Cost Proxies Compared to Daily Percent Effective

Spread

Panel A: Average Cross‐Sectional Correlation of Daily Percent‐Cost Proxies Compared to Daily

Percent Effective Spread

Panel B: Average Spearman's Cross‐Sectional Correlation of Daily Percent‐Cost Proxies

Compared to Daily Percent Effective Spread

Panel C: Portfolio Time‐Series Correlation of First Differences of Daily Percent‐Cost Proxies

Compared to Daily Percent Effective Spread

45

Table 7

Gibbs

Impact

FHT

Impact

High‐Low

Impact

Close‐

High‐Low

Impact

Closing %

Quo Sprd

Impact Amihud VoV()

Largest quintile 0.244 0.224 0.235 0.222 0.236 0.215 0.344

Quintile 2 0.349 0.333 0.344 0.334 0.331 0.285 0.373

Quintile 3 0.290 0.249 0.277 0.281 0.275 0.235 0.326

Quintile 4 0.291 0.265 0.279 0.291 0.289 0.227 0.313

Smallest quintile 0.569 0.503 0.552 0.560 0.572 0.505 0.625


Quintile 2 0.488 0.412 0.477 0.475 0.472 0.392 0.523

Quintile 3 0.557 0.441 0.543 0.544 0.539 0.450 0.592

Quintile 4 0.581 0.468 0.565 0.566 0.573 0.526 0.649



Quintile 2 0.432 0.221 0.401 0.406 0.417 0.386 0.487

Quintile 3 0.367 0.233 0.340 0.340 0.369 0.301 0.414

Quintile 4 0.334 0.268 0.308 0.312 0.340 0.231 0.372



Quintile 2 0.428 0.218 0.409 0.403 0.414 0.388 0.476

Quintile 3 0.391 0.226 0.382 0.377 0.399 0.340 0.429

Quintile 4 0.349 0.260 0.336 0.342 0.356 0.251 0.388


0.635 0.593 0.591 0.616 0.636 0.528 0.739

0.278 0.066 0.254 0.268 0.266 0.231 0.273

Panel E: Pooled Partial Correlations after Controlling for Log Size and Volatility

Panel F: Pooled Partial Spearman's Correlations after Controlling for Log Size and Volatility

Panel C: Individual Stock Time‐Series Correlations of First Differences by Size Quintiles

Panel D: Individual Stock Time‐Series Correlations of First Differences by Volatility Quintiles

The Global Performance of Monthly Cost‐Per‐Dollar‐Volume Proxies with Monthly Lambda by

Size and Volatility Quintiles and After Controlling for Size and Volatility

Panel A: Average Cross‐sectional Correlations by Size Quintiles

Panel B: Average Cross‐sectional Correlations by Volatility Quintiles

The monthly high‐frequency liquidity benchmark, lambda, is calculated from every trade and

corresponding BBO quote in the SIRCA Thomson Reuters Tick History database for a sample

stock‐month. The monthly cost‐per‐dollar‐volume proxies are calculated from daily stock price

data for a sample stock‐month. The full global sample spans 42 exchanges around the world

from 1996‐2014. All stock‐months are required to have at least five positive‐volume days and

eleven non‐zero return days.

46

Table 8


Close‐

High‐Low

Closing %

Quo Sprd

VoV

(%Spread)

Largest quintile 0.200 0.435 0.266 0.263 0.635 0.552

Quintile 2 0.176 0.474 0.229 0.279 0.680 0.540

Quintile 3 0.265 0.520 0.361 0.383 0.724 0.564

Quintile 4 0.405 0.553 0.509 0.482 0.776 0.594

Smallest quintile 0.662 0.638 0.700 0.720 0.828 0.768


Quintile 2 0.159 0.502 0.381 0.394 0.712 0.658

Quintile 3 0.193 0.496 0.428 0.431 0.746 0.675

Quintile 4 0.230 0.482 0.481 0.466 0.768 0.702



Quintile 2 0.203 0.110 0.212 0.216 0.454 0.386

Quintile 3 0.208 0.154 0.240 0.243 0.506 0.419

Quintile 4 0.212 0.203 0.265 0.282 0.543 0.446



Quintile 2 0.198 0.106 0.207 0.210 0.435 0.369

Quintile 3 0.185 0.130 0.215 0.216 0.474 0.387

Quintile 4 0.201 0.189 0.253 0.267 0.515 0.423


0.376 0.461 0.512 0.514 0.758 0.657

0.082 0.461 0.149 0.223 0.824 0.620

Panel D: Individual Stock Time‐Series Correlations of First Differences by Volatility Quintiles

Panel F: Pooled Partial Spearman's Correlations after Controlling for Log Size and Volatility

Panel E: Pooled Partial Correlations after Controlling for Log Size and Volatility

The monthly high‐frequency liquidity benchmark, percent effective spread, is calculated from


for a sample stock‐month. The monthly percent‐cost proxies are calculated from daily stock price

data for a sample stock‐month. The full global sample spans 42 exchanges around the world from

1996‐2014. All stock‐months are required to have at least five positive‐volume days and eleven

non‐zero return days.

The Global Performance of Monthly Percent‐Cost Proxies with Monthly Percent Effective Spread

by Size and Volatility Quintiles and After Controlling for Size and Volatility

Panel A: Average Cross‐sectional Correlations by Size Quintiles

Panel B: Average Cross‐sectional Correlations by Volatility Quintiles

Panel C: Individual Stock Time‐Series Correlations of First Differences by Size Quintiles

47

Table 9

a b c VoV() Amihud

1996‐2007 0.829 0.512 1.220 0.780 0.777 0.678

(0.001) (0.001) (0.004) (0.001) (0.001) (0.001)

2008‐2014 0.526 0.485 0.222 0.425 0.405 0.308

(0.014) (0.003) (0.010) (0.003) (0.003) (0.003)

a b c VoV(%Spread)

Closing %

Quo Sprd

1996‐2007 0.640 0.238 2.400 0.836 0.817 0.863

(0.001) (0.000) (0.004) (0.000) (0.000) (0.000)

2008‐2014 0.723 0.229 2.700 0.849 0.826 0.849

(0.003) (0.001) (0.024) (0.002) (0.002) (0.002)

The Global Performance of Monthly General Volatility Over Volume Class Proxies Under Optimal

Parameters And Top Monthly Liquidity Proxies Compared to Monthly Liquidity Benchmarks

The high‐frequency benchmarks, monthly percent effective spread and monthly lambda, are

calculated from every trade and corresponding BBO quote in the SIRCA Thomson Reuters Tick

History database for a sample stock‐month. The monthly general volatility over volume class

proxies and top liquidity proxies are calculated from daily stock price data for a sample stock‐

month. The earlier (later) sample spans 42 exchanges around the world from 1996‐2007 (2008‐

2014). All stock‐months are required to have at least five positive‐volume days and eleven non‐

zero return days. The general volatility over volume model specification is a*( ^b)/(V^c) . The

model parameters are estimated by maximizing pooled correlation with the liquidity benchmark

using non‐linear least squares over all stock‐months. Standard errors are reported in the

parenthesis.

General Volatility Over Volume Class Proxy Under Top Percent‐Cost Proxies

Panel B: Pooled Correlation of a Monthly General Volatility Over Volume Class Proxy and Top

Monthly Percent‐Cost Proxies Compared to Monthly Percent Effective Spread

General Volatility Over Volume Class Proxy Under

Parameters That Maximize Pooled Correlation

Top Cost‐Per‐Dollar‐

Volume Proxies

Panel A: Pooled Correlation of a Monthly General Volatility Over Volume Class Proxy and Top

Monthly Cost‐Per‐Dollar‐Volume Proxies Compared to Monthly Lambda

48

Table 10

Gibbs

Impact

FHT

Impact

High‐Low

Impact

Close‐

High‐Low

Impact

Closing %

Quo Sprd

Impact Amihud VoV()

US Monthly 0.579 0.435 0.603 0.606 0.580 0.489 0.663

US Daily 0.476 0.424 0.536 0.556 0.694

US Monthly 0.770 0.774 0.778 0.773 0.760 0.763 0.788

US Daily 0.445 0.224 0.600 0.563 0.656

US Monthly 0.528 0.557 0.517 0.490 0.536 0.204 0.551

US Daily 0.250 0.269 0.352 0.364 0.530

US Monthly 0.354 0.201 0.379 0.384 0.378 0.276 0.333

US Daily 0.064 0.076 0.098 0.123 0.304

Panel E: Average Correlation of the Contests Above

US Monthly 0.558 0.492 0.569 0.563 0.564 0.433 0.584

US Daily 0.309 0.248 0.396 0.401 0.546

US Monthly 18.1 18.3 18.3 18.2 18.1 17.7 23.3

US Daily 13 13 13.1 13 26.6

Panel F: Average Root Mean Squared Error (RMSE) of Monthly (Daily) Cost‐Per‐Dollar‐Volume

Proxies Compared to Monthly (Daily) Lambda / Median of Monthly (Daily) Lambda

The US Performance of Monthly (Daily) Cost‐Per‐Dollar‐Volume Proxies Compared to Monthly

(Daily) Lambda from 1996‐2014

The high‐frequency percent‐cost benchmark, monthly (daily) lambda, is calculated from every

trade and corresponding BBO quote in the NYSE Trade and Quote database for a sample stock‐

month (stock‐day). The monthly (daily) cost‐per‐dollar‐volume proxies are calculated from CRSP

daily stock price data for a sample stock‐month (stock‐day). The US sample spans the NYSE,

AMEX, and NASDAQ exchanges from 1996‐2014. All stock‐months (stock‐days) are required to

have (are contained within stock‐months with) at least five positive‐volume days and eleven non‐

zero return days. A solid box means the highest correlation or the lowest average root mean

squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically

indistinguishable from the highest correlation or average RMSEs that are statistically

indistinguishable from the lowest average RMSE in the row at the 5% level. Bold‐faced numbers

are statistically different from zero or proxies have predictive power that is significant at the 5%

level.

Panel A: Average Cross‐Sectional Correlation of Monthly (Daily) Cost‐Per‐Dollar‐Volume Proxies

Compared to Monthly (Daily) Lambda

Panel B: Average Spearman's Cross‐Sectional Correlation of Monthly (Daily) Cost‐Per‐Dollar‐

Volume Proxies Compared to Monthly (Daily) Lambda

Panel C: Portfolio Time‐Series Correlation of First Differences of Monthly (Daily) Cost‐Per‐Dollar‐

Volume Proxies Compared to Monthly (Daily) Lambda

Panel D: Individual Stock Time‐Series Correlation of First Differences of Monthly (Daily) Cost‐Per‐

Dollar‐Volume Proxies Compared to the Monthly (Daily) Lambda

49

Table 11


Close‐

High‐Low

Closing %


US Monthly 0.607 0.590 0.660 0.711 0.861 0.871

US Daily 0.358 0.361 0.742 0.767

US Monthly 0.630 0.604 0.666 0.720 0.909 0.938

US Daily 0.327 0.252 0.786 0.897

US Monthly 0.661 0.659 0.727 0.740 0.822 0.749

US Daily 0.066 0.148 0.396 0.424

US Monthly 0.181 0.055 0.179 0.202 0.282 0.263

US Daily ‐0.001 0.063 0.141 0.249

Panel E: Average Correlation of the Contests Above

US Monthly 0.520 0.477 0.558 0.593 0.719 0.705

US Daily 0.188 0.206 0.516 0.584

US Monthly 0.020 0.018 0.015 0.014 0.015 0.012

US Daily 0.018 0.024 0.014 0.012

Panel F: Average Root Mean Squared Error (RMSE) of Monthly (Daily) Percent‐Cost Proxies

Compared to Monthly (Daily) Percent Effective Spread

The US Performance of Monthly (Daily) Percent‐Cost Proxies Compared to Monthly (Daily)

Percent Effective Spread from 1996‐2014

The high‐frequency percent‐cost benchmark, monthly (daily) percent effective spread, is

calculated from every trade and corresponding BBO quote in the NYSE Trade and Quote database

for a sample stock‐month (stock‐day). The monthly (daily) percent‐cost proxies are calculated

from CRSP daily stock price data for a sample stock‐month (stock‐day). The US sample spans the

NYSE, AMEX, and NASDAQ exchanges from 1996‐2014. All stock‐months (stock‐days) are required

to have (are contained within stock‐months with) at least five positive‐volume days and eleven

non‐zero return days. A solid box means the highest correlation or the lowest average root mean

squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically

indistinguishable from the highest correlation or average RMSEs that are statistically

indistinguishable from the lowest average RMSE in the row at the 5% level. Bold‐faced numbers

are statistically different from zero or proxies have predictive power that is significant at the 5%

level.

Panel A: Average Cross‐Sectional Correlation of Monthly (Daily) Percent‐Cost Proxies Compared

to Monthly (Daily) Percent Effective Spread

Panel B: Average Spearman's Cross‐Sectional Correlation of Monthly (Daily) Percent‐Cost Proxies

Compared to Monthly (Daily) Percent Effective Spread

Panel C: Portfolio Time‐Series Correlation of First Differences of Monthly (Daily) Percent‐Cost

Proxies Compared to Monthly (Daily) Percent Effective Spread

Panel D: Individual Stock Time‐Series Correlation of First Differences of Monthly (Daily) Percent‐

Cost Proxies Compared to Monthly (Daily) Percent Effective Spread

50

Table 12

VoV() Amihud

Before (July 30 ‐ Aug 3) 52.0% ‐5.8%

2.99** ‐0.06

During (Aug 6 ‐ Aug 9) 88.4% ‐2.1%

5.69*** ‐0.02

After (Aug 10 ‐ Aug 17) 118.9% 24.7%

6.25*** 0.23

Do VoV() and Amihud yield different inferences about the change in S&P 1500 illiquidity

before, during, and after the August 6‐9, 2007 "quant meltdown"?

Both VoV() and Amihud are used to measure the percent change in S&P 1500 illiquidity before

(July 30 ‐ August 3), during (August 6 ‐ August 9), and after (August 10 ‐ August 17) the August 6‐9,

2007 "quant meltdown." The base period is July 2 ‐ July 13, 2007. CRSP daily stock data provides

the inputs to both cost‐per‐dollar‐volume proxies. The t‐statistic is shown below each estimate.

*, **, and *** means significant at the 10%, 5%, and 1% levels, respectively.

Online Appendix for Are Volatility Over Volume Liquidity

Proxies Useful For Global Or US Research?

Table IA.1The Performance of Monthly Cost-Per-Dollar-Volume Proxies Compared to Monthly λ by Exchange

The high-frequency cost-per-dollar-volume benchmark, monthly lambda, is calculated from every trade and corresponding BBOquote in the SIRCA Thomson Reuters Tick History database for a sample stock-month. The monthly cost-per-dollar-volumeproxies are calculated from daily stock price data for a sample stock-month. The earlier (later) sample spans 42 exchangesaround the world from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is significant at the 5%level.

Average Cross-Sectional Correlation FD of Portfolio Time-Series Correlation

HLImpact

CHLImpact

Closing %QS

ImpactAmihud VoV(λ)

HLImpact

CHLImpact

Closing %QS


Argentina Buenos Ar. 0.729 0.757 0.758 0.701 0.834 0.429 0.518 0.706 0.569 0.732

Australia Australian 0.232 0.233 0.244 0.153 0.296 0.160 0.156 0.154 0.017 0.183

Austria Vienna 0.670 0.676 0.672 0.673 0.842 0.579 0.604 0.507 0.445 0.712

Belgium Brussels 0.642 0.648 0.648 0.606 0.747 0.380 0.405 0.418 0.312 0.686

Brazil Sao Paulo 0.572 0.566 0.549 0.514 0.602 0.193 0.200 0.166 0.079 0.183

Canada Toronto 0.754 0.775 0.721 0.680 0.858 0.554 0.574 0.509 0.308 0.688

Chile Santiago 0.362 0.291 0.417 0.377 0.405 0.368 0.242 0.373 0.588 0.509

China Hong Kong 0.543 0.551 0.552 0.424 0.645 0.370 0.435 0.307 0.299 0.431

China Shanghai 0.755 0.740 0.697 0.730 0.773 0.688 0.727 0.522 0.724 0.837

China Shenzhen 0.809 0.793 0.667 0.794 0.856 0.763 0.790 0.684 0.694 0.843

Denmark Copenhag. 0.413 0.420 0.440 0.388 0.487 0.331 0.311 0.334 0.348 0.482

France Paris 0.689 0.678 0.686 0.637 0.781 0.465 0.436 0.499 0.458 0.662

Finland Helsinki 0.351 0.354 0.371 0.336 0.402 0.398 0.350 0.329 0.350 0.465

Germany Frankfurt 0.498 0.507 0.534 0.459 0.638 0.755 0.720 0.673 0.472 0.752

Greece Athens 0.661 0.650 0.664 0.623 0.709 0.185 0.099 0.298 0.125 0.359

India Bombay 0.548 0.557 0.572 0.474 0.643 0.427 0.442 0.470 0.503 0.771

Indonesia Jakarta 0.386 0.391 0.396 0.289 0.430 0.319 0.325 0.327 0.262 0.386

Ireland Irish 0.505 0.526 0.525 0.509 0.596 0.176 0.237 0.178 0.100 0.263

Israel Tel Aviv 0.708 0.702 0.719 0.679 0.787 0.702 0.653 0.604 0.658 0.725

Italy Milan 0.538 0.536 0.536 0.485 0.619 0.593 0.508 0.419 0.424 0.179

Japan Tokyo 0.741 0.758 0.731 0.731 0.869 0.631 0.736 0.641 0.843 0.906

Malaysia Kuala Lum. 0.707 0.727 0.724 0.591 0.794 0.585 0.686 0.657 0.037 0.834

Mexico Mexican 0.755 0.762 0.750 0.635 0.817 0.506 0.562 0.652 0.299 0.703

Netherlands AEX 0.715 0.724 0.716 0.670 0.842 0.444 0.484 0.483 0.427 0.680

New Zeland New Zea. 0.491 0.505 0.511 0.444 0.544 0.338 0.343 0.356 0.096 0.328

Norway Oslo 0.424 0.434 0.432 0.393 0.469 0.537 0.532 0.500 0.446 0.538

Philippines Phillipine 0.397 0.406 0.398 0.334 0.460 0.148 0.156 0.143 0.255 0.254

Poland Warsaw 0.553 0.560 0.558 0.477 0.636 -0.252 -0.305 -0.363 -0.073 0.025

Portugal Lisbon 0.756 0.756 0.748 0.712 0.853 0.481 0.482 0.572 0.192 0.626

Singapore Singapore 0.686 0.691 0.688 0.557 0.754 0.516 0.562 0.545 0.446 0.724

South Africa Johannes. 0.624 0.630 0.648 0.415 0.699 0.491 0.517 0.567 0.179 0.593

South Korea Korea 0.802 0.778 0.765 0.715 0.896 0.676 0.707 0.735 0.736 0.812

Spain Barcelona 0.791 0.776 0.762 0.771 0.833 0.522 0.481 0.648 0.579 0.681

Sweden Stockholm 0.481 0.489 0.482 0.432 0.540 0.552 0.485 0.510 0.516 0.580

Switzerland SWX Swiss 0.513 0.522 0.525 0.466 0.592 0.533 0.568 0.559 0.419 0.717

Taiwan Taiwan 0.868 0.854 0.817 0.828 0.925 0.787 0.676 0.660 0.628 0.816

Thailand Thailand 0.327 0.336 0.319 0.262 0.382 0.159 0.255 0.216 0.417 0.422

Turkey Istanbul 0.746 0.743 0.694 0.687 0.775 0.761 0.852 0.731 0.600 0.892

UK London 0.737 0.719 0.753 0.613 0.829 0.508 0.563 0.673 0.403 0.700

US New York 0.753 0.746 0.686 0.719 0.810 0.496 0.498 0.396 0.535 0.474

US American 0.564 0.558 0.539 0.508 0.586 0.390 0.251 0.328 -0.364 0.335

US NASDAQ 0.659 0.674 0.635 0.512 0.725 0.633 0.668 0.614 0.232 0.625

US All US 0.603 0.606 0.580 0.488 0.663 0.517 0.490 0.536 0.204 0.551

Developed 0.411 0.424 0.409 0.339 0.489 0.324 0.353 0.318 0.146 0.561

Developing 0.556 0.564 0.573 0.512 0.654 0.769 0.775 0.777 0.805 0.931

Global 0.564 0.571 0.582 0.520 0.650 0.733 0.741 0.780 0.791 0.906

2

Table IA.2The Performance of Monthly Cost-Per-Dollar-Volume Proxies Compared to Monthly λ by Exchange

The high-frequency cost-per-dollar-volume benchmark, monthly lambda, is calculated from every trade and corresponding BBOquote in the SIRCA Thomson Reuters Tick History database for a sample stock-month. The monthly cost-per-dollar-volumeproxies are calculated from daily stock price data for a sample stock-month. The earlier (later) sample spans 42 exchangesaround the world from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is significant at the 5%level.

FD of Individual Stock Time-Series Correlation Average Root Mean Squared Error

HLImpact

CHLImpact

Closing %QS


HLImpact

CHLImpact

Closing %QS



Australia Australian 0.175 0.185 0.203 0.130 0.257 11 11 8.99 13 50

Austria Vienna 0.145 0.198 0.137 0.211 0.473 4.63 4.61 4.39 4.48 9.35

Belgium Brussels 0.296 0.303 0.309 0.265 0.372 4.70 4.68 4.66 4.56 12

Brazil Sao Paulo 0.316 0.271 0.301 0.182 0.255 46 46 46 45 43

Canada Toronto 0.440 0.448 0.444 0.360 0.494 7.11 7.07 3.55 6.76 15

Chile Santiago 0.097 0.080 0.151 0.036 0.184 3.42 3.42 3.38 3.42 3.46

China Hong Kong 0.192 0.197 0.190 0.133 0.336 10 10 10 10 13

China Shanghai 0.624 0.629 0.585 0.613 0.677 0.96 0.96 0.96 0.96 0.37

China Shenzhen 0.631 0.633 0.579 0.592 0.693 1.06 1.06 1.06 1.06 0.34

Denmark Copenhag. 0.177 0.173 0.188 0.139 0.214 64 64 62 62 70

France Paris 0.349 0.363 0.341 0.318 0.451 6.68 6.62 6.85 6.70 15

Finland Helsinki 0.056 0.059 0.073 0.049 0.082 3.73 3.73 3.71 3.96 34


Greece Athens 0.480 0.460 0.502 0.370 0.326 4.63 4.57 5.16 4.31 15

India Bombay 0.315 0.324 0.347 0.242 0.367 4.49 4.48 4.56 4.19 3.16

Indonesia Jakarta 0.115 0.110 0.127 0.074 0.138 509 509 541 499 472

Ireland Irish 0.113 0.132 0.158 0.087 0.206 18 18 18 17 49

Israel Tel Aviv 0.427 0.431 0.469 0.327 0.424 3.99 3.99 3.91 3.91 2.30

Italy Milan 0.265 0.246 0.232 0.165 0.183 6.81 6.78 6.68 6.66 29

Japan Tokyo 0.392 0.420 0.440 0.432 0.501 3.00 3.00 2.67 3.00 2.33


Mexico Mexican 0.258 0.275 0.310 0.208 0.336 9.43 9.42 9.37 8.60 5.05

Netherlands AEX 0.279 0.283 0.270 0.256 0.393 10 10 9.97 9.74 18

New Zeland New Zea. 0.193 0.213 0.241 0.119 0.227 8.23 8.23 8.22 8.25 21

Norway Oslo 0.164 0.182 0.182 0.103 0.200 1830 1830 1830 1800 2270

Philippines Phillipine 0.077 0.092 0.087 0.084 0.142 203 203 192 201 208

Poland Warsaw 0.374 0.381 0.391 0.228 0.402 18 17 15 18 19

Portugal Lisbon 0.388 0.372 0.352 0.259 0.470 8.47 8.39 8.26 8.10 24

Singapore Singapore 0.255 0.270 0.271 0.211 0.344 5.74 5.72 5.76 6.21 19

South Africa Johannes. 0.250 0.247 0.289 0.119 0.315 15 15 15 16 16

South Korea Korea 0.572 0.568 0.578 0.435 0.582 2.56 2.34 2.36 3.53 225

Spain Barcelona 0.409 0.381 0.408 0.312 0.328 5.13 5.13 5.11 5.08 12



Taiwan Taiwan 0.491 0.295 0.517 0.594 0.597 3.02 3.04 2.64 2.72 48

Thailand Thailand 0.014 0.017 0.004 0.027 0.050 46 46 20 45 42

Turkey Istanbul 0.662 0.619 0.701 0.544 0.500 1.91 1.91 1.91 1.90 2.56

UK London 0.273 0.258 0.286 0.232 0.317 15 15 15 15 7.92

US New York 0.281 0.269 0.262 0.263 0.220 6.67 6.66 6.61 6.63 12

US American 0.274 0.304 0.288 0.162 0.267 9.01 8.96 8.86 8.66 9.50

US NASDAQ 0.428 0.435 0.430 0.300 0.380 9.69 9.65 9.61 9.41 14

US All US 0.379 0.384 0.378 0.278 0.333 18 18 18 18 23

Developed 0.303 0.314 0.324 0.277 0.368 9.03 9.02 6.92 8.88 9.68

Developing 0.392 0.382 0.402 0.325 0.443 136 135 135 127 101

Global 0.347 0.347 0.363 0.301 0.405 82 82 88 77 62

3

Table IA.3The Performance of Daily Cost-Per-Dollar-Volume Proxies Compared to Daily λ by Exchange

The high-frequency cost-per-dollar-volume benchmark, daily lambda, is calculated from every trade and corresponding BBOquote in the SIRCA Thomson Reuters Tick History database for a sample stock-day. The daily cost-per-dollar-volume proxiesare calculated from daily stock price data for a sample stock-day. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is significant at the 5%level.


HLImpact

CHLImpact

Closing %QS


HLImpact

CHLImpact

Closing %QS



Australia Australian 0.187 0.179 0.229 0.260 0.350 -0.021 0.010 -0.014 0.019 0.099

Austria Vienna 0.504 0.419 0.583 0.609 0.779 0.154 0.218 0.130 0.295 0.480

Belgium Brussels 0.487 0.465 0.580 0.590 0.765 0.170 0.193 0.317 0.315 0.483

Brazil Sao Paulo 0.313 0.252 0.348 0.345 0.426 0.240 0.166 0.243 0.196 0.312

Canada Toronto 0.610 0.496 0.651 0.658 0.818 0.383 0.375 0.426 0.508 0.695

Chile Santiago 0.220 0.158 0.302 0.294 0.310 0.196 0.217 0.442 0.404 0.363

China Hong Kong 0.336 0.301 0.422 0.467 0.608 0.194 0.199 0.322 0.226 0.387

China Shanghai 0.429 0.309 0.467 0.498 0.658 0.034 0.026 0.060 0.003 0.255

China Shenzhen 0.472 0.337 0.518 0.561 0.764 0.052 0.035 0.085 -0.013 0.372

Denmark Copenhag. 0.311 0.240 0.375 0.376 0.448 0.062 0.099 0.101 0.090 0.268

France Paris 0.559 0.480 0.660 0.659 0.825 0.328 0.330 0.424 0.446 0.643

Finland Helsinki 0.217 0.183 0.259 0.267 0.367 0.190 0.176 0.247 0.228 0.350


Greece Athens 0.570 0.415 0.600 0.575 0.750 0.372 0.355 0.223 0.492 0.597

India Bombay 0.451 0.376 0.525 0.515 0.650 0.273 0.204 0.189 0.268 0.380


Ireland Irish 0.280 0.215 0.343 0.338 0.418 0.062 0.114 0.103 0.157 0.269

Israel Tel Aviv 0.546 0.386 0.680 0.700 0.823 0.322 0.305 0.522 0.545 0.680

Italy Milan 0.403 0.362 0.459 0.451 0.624 0.235 0.353 0.399 0.361 0.601

Japan Tokyo 0.528 0.454 0.628 0.631 0.816 0.150 0.217 0.375 0.366 0.576


Mexico Mexican 0.532 0.430 0.628 0.626 0.718 0.351 0.335 0.444 0.397 0.505

Netherlands AEX 0.545 0.473 0.613 0.650 0.788 0.224 0.208 0.259 0.273 0.479

New Zeland New Zea. 0.154 0.139 0.265 0.259 0.295 0.161 0.206 0.245 0.269 0.334

Norway Oslo 0.290 0.256 0.365 0.365 0.446 0.356 0.345 0.408 0.397 0.509

Philippines Phillipine 0.299 0.227 0.370 0.358 0.459 0.140 0.093 0.155 0.106 0.130

Poland Warsaw 0.506 0.417 0.593 0.594 0.766 0.116 0.128 0.078 0.145 0.458

Portugal Lisbon 0.517 0.413 0.593 0.589 0.762 0.342 0.343 0.430 0.443 0.664

Singapore Singapore 0.452 0.354 0.527 0.481 0.628 0.366 0.290 0.462 0.334 0.483

South Africa Johannes. 0.444 0.365 0.550 0.537 0.636 0.280 0.325 0.344 0.337 0.472

South Korea Korea 0.596 0.490 0.627 0.621 0.808 0.304 0.241 0.403 0.267 0.532

Spain Barcelona 0.566 0.451 0.611 0.622 0.800 0.367 0.348 0.492 0.475 0.637



Taiwan Taiwan 0.590 0.432 0.643 0.651 0.802 0.193 0.237 0.331 0.195 0.532

Thailand Thailand 0.184 0.154 0.204 0.204 0.347 0.093 -0.021 0.124 0.144 0.195

Turkey Istanbul 0.584 0.412 0.660 0.578 0.698 0.212 0.257 0.335 0.442 0.592

UK London 0.440 0.365 0.556 0.604 0.711 0.344 0.315 0.366 0.483 0.714

US New York 0.524 0.421 0.569 0.606 0.737 0.092 0.079 0.221 0.138 0.349

US American 0.393 0.305 0.474 0.459 0.624 0.219 0.201 0.281 0.327 0.533

US NASDAQ 0.474 0.417 0.528 0.549 0.690 0.344 0.347 0.419 0.388 0.592

US All US 0.476 0.424 0.536 0.556 0.694 0.251 0.273 0.347 0.367 0.528

Developed 0.263 0.248 0.305 0.329 0.450 -0.198 -0.103 -0.156 -0.083 -0.110

Developing 0.435 0.381 0.472 0.476 0.572 0.566 0.543 0.521 0.561 0.595

Global 0.432 0.378 0.469 0.473 0.562 0.826 0.809 0.798 0.821 0.779

4

Table IA.4The Performance of Daily Cost-Per-Dollar-Volume Proxies Compared to Daily λ by Exchange

The high-frequency cost-per-dollar-volume benchmark, daily lambda, is calculated from every trade and corresponding BBOquote in the SIRCA Thomson Reuters Tick History database for a sample stock-day. The daily cost-per-dollar-volume proxiesare calculated from daily stock price data for a sample stock-day. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is significant at the 5%level.


HLImpact

CHLImpact

Closing %QS


HLImpact

CHLImpact

Closing %QS


Argentina Buenos Ar. 0.142 0.145 0.257 0.228 0.328 5.67 5.67 10 5.66 5.94

Australia Australian 0.014 0.042 0.058 0.077 0.170 23 23 29 23 221

Austria Vienna -0.003 0.082 0.087 0.091 0.340 5.00 4.95 5.57 4.94 19

Belgium Brussels 0.082 0.112 0.159 0.155 0.360 9.26 9.25 9.23 9.23 34

Brazil Sao Paulo 0.143 0.129 0.182 0.201 0.330 12 12 12 12 15

Canada Toronto 0.139 0.146 0.223 0.258 0.465 13 13 23 13 46

Chile Santiago 0.014 0.014 0.017 0.022 0.083 2.62 2.67 3.79 2.68 2.84

China Hong Kong -0.034 0.037 0.027 0.089 0.231 43 43 43 43 73

China Shanghai 0.051 0.034 0.070 0.070 0.274 2.06 2.06 2.06 2.05 5.78

China Shenzhen 0.062 0.040 0.080 0.068 0.292 2.53 2.53 2.54 2.53 4.17

Denmark Copenhag. 0.036 0.075 0.091 0.075 0.146 428 436 436 437 1320

France Paris 0.092 0.103 0.157 0.182 0.397 15 15 15 15 44

Finland Helsinki 0.001 0.015 0.029 0.044 0.118 3.29 3.31 3.31 3.31 23

Germany Frankfurt 0.034 0.071 0.118 0.141 0.251 8.03 8.05 7.98 7.98 33

Greece Athens 0.167 0.138 0.223 0.154 0.342 12 12 13 12 30

India Bombay 0.097 0.110 0.146 0.132 0.254 6.12 6.14 5.74 6.12 5.95


Ireland Irish 0.079 0.086 0.124 0.150 0.218 15 15 16 15 100

Israel Tel Aviv 0.151 0.130 0.268 0.194 0.428 7.46 7.45 14 7.44 7.73

Italy Milan 0.040 0.067 0.085 0.058 0.234 28 28 28 28 103

Japan Tokyo 0.026 0.058 0.087 0.095 0.297 9.06 9.05 12 9.05 7.58

Malaysia Kuala Lum. 0.085 0.108 0.167 0.167 0.269 6.66 6.65 7.02 6.63 12

Mexico Mexican 0.087 0.099 0.194 0.177 0.285 9.57 8.83 8.79 8.82 8.16

Netherlands AEX 0.041 0.105 0.097 0.158 0.325 13 13 13 13 53

New Zeland New Zea. 0.059 0.073 0.118 0.130 0.135 46 46 46 46 265

Norway Oslo 0.047 0.070 0.083 0.116 0.186 166 174 174 174 299

Philippines Phillipine 0.032 0.030 0.093 0.117 0.161 176 177 182 177 185

Poland Warsaw 0.138 0.134 0.204 0.198 0.430 4.58 4.58 4.29 4.57 4.92

Portugal Lisbon 0.042 0.085 0.133 0.133 0.356 6.33 6.33 6.31 6.33 10

Singapore Singapore 0.037 0.049 0.073 0.082 0.170 13 13 16 13 40

South Africa Johannes. 0.093 0.082 0.145 0.155 0.256 26 26 25 25 40

South Korea Korea 0.079 0.064 0.113 0.076 0.268 7.39 7.36 7.53 7.27 159

Spain Barcelona 0.098 0.112 0.137 0.114 0.340 17 17 17 17 47

Sweden Stockholm 0.048 0.065 0.095 0.115 0.213 654 660 886 660 925

Switzerland SWX Swiss 0.070 0.099 0.125 0.154 0.328 5.41 5.41 5.40 5.39 39

Taiwan Taiwan 0.048 0.015 0.112 0.092 0.067 11 11 20 11 26

Thailand Thailand -0.022 0.004 -0.014 0.018 0.087 85 85 101 85 81

Turkey Istanbul 0.263 0.191 0.399 0.251 0.280 3.72 3.73 3.73 3.72 3.56

UK London 0.078 0.110 0.116 0.186 0.298 46 46 45 46 36

US New York 0.055 0.041 0.075 0.090 0.196 6.36 6.36 6.27 6.35 11

US American 0.094 0.062 0.102 0.123 0.317 4.58 4.58 4.63 4.57 7.75

US NASDAQ 0.061 0.094 0.107 0.151 0.346 9.00 8.99 9.03 8.98 20

US All US 0.064 0.076 0.098 0.132 0.304 13 13 13 13 27

Developed 0.047 0.074 0.104 0.122 0.281 14 14 14 14 20

Developing 0.068 0.069 0.119 0.112 0.239 165 165 156 165 167

Global 0.058 0.072 0.112 0.117 0.260 110 111 105 111 113

5

Table IA.5The Performance of Monthly Percent-Cost Proxies Compared to Monthly Percent Effective Spread by

Exchange

The high-frequency percent-cost benchmark, monthly percent effective spread, is calculated from every trade and correspondingBBO quote in the SIRCA Thomson Reuters Tick History database for a sample stock-month. The monthly percent-cost proxiesare calculated from daily stock price data for a sample stock-month. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is signiffcant at the 5%level.


Country Exchange HL CHLClosing %

QSVoV

(% Spread)HL CHL

Closing %QS

VoV(% Spread)

Argentina Buenos Ar. 0.487 0.547 0.891 0.846 0.538 0.626 0.861 0.804

Australia Australian 0.814 0.836 0.929 0.900 0.732 0.796 0.899 0.853

Austria Vienna 0.576 0.645 0.854 0.863 0.471 0.498 0.805 0.728

Belgium Brussels 0.548 0.663 0.837 0.826 0.524 0.618 0.831 0.775

Brazil Sao Paulo 0.578 0.644 0.827 0.759 0.475 0.436 0.348 0.314

Canada Toronto 0.787 0.837 0.855 0.914 0.768 0.833 0.790 0.787

Chile Santiago 0.357 0.330 0.808 0.681 0.439 0.178 0.552 0.578

China Hong Kong 0.653 0.580 0.855 0.792 0.761 0.659 0.919 0.877

China Shanghai 0.214 0.141 0.716 0.536 0.384 0.409 0.736 0.674

China Shenzhen 0.209 0.128 0.657 0.490 0.471 0.482 0.533 0.697

Denmark Copenhag. 0.682 0.671 0.849 0.821 0.661 0.721 0.554 0.678

France Paris 0.618 0.677 0.836 0.811 0.682 0.668 0.833 0.711

Finland Helsinki 0.752 0.690 0.920 0.867 0.844 0.485 0.887 0.811

Germany Frankfurt 0.703 0.709 0.884 0.857 0.605 0.687 0.828 0.768

Greece Athens 0.612 0.681 0.801 0.732 0.576 0.588 0.759 0.715

India Bombay 0.591 0.637 0.764 0.743 0.716 0.746 0.781 0.764

Indonesia Jakarta 0.750 0.743 0.856 0.747 0.885 0.889 0.884 0.834

Ireland Irish 0.789 0.775 0.895 0.866 0.594 0.661 0.785 0.670

Israel Tel Aviv 0.651 0.558 0.811 0.833 0.346 0.079 0.583 0.721

Italy Milan 0.499 0.595 0.843 0.845 0.684 0.131 0.523 0.310

Japan Tokyo 0.404 0.481 0.915 0.860 0.771 0.771 0.929 0.893

Malaysia Kuala Lum. 0.683 0.709 0.887 0.810 0.723 0.768 0.835 0.828

Mexico Mexican 0.556 0.634 0.794 0.791 0.559 0.520 0.690 0.683

Netherlands AEX 0.692 0.772 0.905 0.904 0.656 0.766 0.840 0.753

New Zeland New Zea. 0.649 0.679 0.771 0.711 0.612 0.671 0.697 0.628

Norway Oslo 0.587 0.631 0.752 0.697 0.559 0.586 0.796 0.724

Philippines Phillipine 0.693 0.697 0.812 0.708 0.713 0.733 0.829 0.783

Poland Warsaw 0.502 0.570 0.724 0.698 0.366 0.428 0.712 0.591

Portugal Lisbon 0.805 0.838 0.922 0.856 0.816 0.781 0.886 0.792

Singapore Singapore 0.880 0.881 0.944 0.868 0.903 0.924 0.961 0.893

South Africa Johannes. 0.762 0.732 0.868 0.815 0.734 0.741 0.766 0.751

South Korea Korea 0.415 0.453 0.852 0.856 0.678 0.718 0.888 0.789

Spain Barcelona 0.624 0.514 0.870 0.831 0.560 0.467 0.414 0.728

Sweden Stockholm 0.780 0.816 0.905 0.885 0.787 0.815 0.846 0.877

Switzerland SWX Swiss 0.699 0.733 0.844 0.834 0.769 0.810 0.620 0.834

Taiwan Taiwan 0.318 0.226 0.869 0.876 0.350 0.111 0.839 0.857

Thailand Thailand 0.560 0.539 0.858 0.744 0.785 0.787 0.908 0.861

Turkey Istanbul 0.573 0.331 0.909 0.505 0.624 0.651 0.926 0.803

UK London 0.594 0.525 0.882 0.843 0.416 0.501 0.819 0.813

US New York 0.429 0.469 0.749 0.771 0.255 0.286 0.219 0.461

US American 0.636 0.700 0.830 0.830 0.533 0.624 0.453 0.579

US NASDAQ 0.609 0.666 0.885 0.871 0.659 0.663 0.849 0.721

US All US 0.660 0.711 0.861 0.871 0.727 0.740 0.822 0.749

Developed 0.733 0.737 0.848 0.835 0.901 0.837 0.879 0.915

Developing 0.723 0.705 0.866 0.813 0.851 0.802 0.904 0.933

Global 0.735 0.726 0.862 0.823 0.905 0.861 0.914 0.933

6

Table IA.6The Performance of Monthly Percent-Cost Proxies Compared to Monthly Percent Effective Spread by

Exchange

The high-frequency percent-cost benchmark, monthly percent effective spread, is calculated from every trade and correspondingBBO quote in the SIRCA Thomson Reuters Tick History database for a sample stock-month. The monthly percent-cost proxiesare calculated from daily stock price data for a sample stock-month. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at leastfive positive-volume days and eleven non-zero return days. A solid box means the highest correlation or the lowest averageroot mean squared error (RMSE) in the row. Dashed boxes mean correlations that are statistically indistinguishable from thehighest correlation or average RMSEs that are statistically indistinguishable from the lowest average RMSE in the row at the5% level. Bold-faced numbers are statistically different from zero or proxies have predictive power that is significant at the 5%level.



QSVoV

(% Spread)HL CHL

Closing %QS

VoV(% Spread)



Austria Vienna 0.221 0.264 0.341 0.415 0.012 0.011 0.009 0.011


Brazil Sao Paulo 0.131 0.140 0.331 0.308 0.020 0.018 0.015 0.016

Canada Toronto 0.273 0.284 0.419 0.424 0.017 0.015 0.015 0.014

Chile Santiago 0.154 0.116 0.245 0.231 0.015 0.028 0.013 0.011

China Hong Kong 0.239 0.217 0.487 0.346 0.014 0.015 0.012 0.011

China Shanghai 0.162 0.166 0.425 0.404 0.007 0.007 0.002 0.003

China Shenzhen 0.202 0.212 0.376 0.386 0.007 0.007 0.003 0.003


France Paris 0.258 0.296 0.383 0.323 0.014 0.013 0.010 0.012



Greece Athens 0.201 0.247 0.433 0.191 0.019 0.017 0.015 0.025

India Bombay 0.276 0.313 0.413 0.398 0.038 0.034 0.032 0.045


Ireland Irish 0.264 0.206 0.423 0.292 0.012 0.013 0.013 0.011

Israel Tel Aviv 0.237 0.185 0.483 0.322 0.027 0.028 0.027 0.018

Italy Milan 0.250 0.217 0.340 0.252 0.007 0.008 0.011 0.007

Japan Tokyo 0.237 0.287 0.549 0.486 0.006 0.007 0.004 0.006


Mexico Mexican 0.145 0.169 0.364 0.315 0.013 0.012 0.018 0.011

Netherlands AEX 0.302 0.331 0.453 0.412 0.010 0.009 0.007 0.007


Norway Oslo 0.154 0.174 0.326 0.191 0.016 0.014 0.013 0.013


Poland Warsaw 0.174 0.187 0.358 0.233 0.034 0.033 0.036 0.045

Portugal Lisbon 0.244 0.252 0.361 0.312 0.009 0.008 0.005 0.009



South Korea Korea 0.169 0.195 0.414 0.368 0.010 0.010 0.006 0.007

Spain Barcelona 0.322 0.278 0.457 0.306 0.006 0.007 0.005 0.004



Taiwan Taiwan 0.172 -0.082 0.407 0.434 0.005 0.007 0.002 0.006


Turkey Istanbul 0.204 0.150 0.601 0.367 0.005 0.005 0.002 0.005

UK London 0.118 0.123 0.301 0.252 0.018 0.018 0.016 0.013

US New York 0.123 0.144 0.143 0.224 0.008 0.008 0.008 0.005

US American 0.200 0.239 0.280 0.264 0.022 0.020 0.037 0.018

US NASDAQ 0.244 0.272 0.421 0.317 0.016 0.015 0.012 0.012

US All US 0.212 0.240 0.344 0.290 0.015 0.014 0.015 0.012

Developed 0.237 0.260 0.455 0.387 0.013 0.013 0.014 0.012

Developing 0.242 0.232 0.479 0.398 0.017 0.017 0.015 0.020

Global 0.239 0.246 0.467 0.392 0.015 0.015 0.015 0.016

7

Table IA.7The Performance of Daily Percent-Cost Proxies Compared to Daily Percent Effective Spread by Exchange

The high-frequency percent-cost benchmark, daily percent effective spread, is calculated from every trade and correspondingBBO quote in the SIRCA Thomson Reuters Tick History database for a sample stock-day. The daily percent-cost proxiesare calculated from daily stock price data for a sample stock-day. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at least five positive volume days and eleven non-zeroreturn days. A solid box means the highest correlation or the lowest average root mean squared error (RMSE) in the row.Dashed boxes mean correlations that are statistically indistinguishable from the highest correlation or average RMSEs that arestatistically indistinguishable from the lowest average RMSE in the row at the 5% level. Bold-faced numbers are statisticallydifferent from zero or proxies have predictive power that is significant at the 5% level.



QSVoV

(% Spread)HL CHL

Closing %QS

VoV(% Spread)



Austria Vienna 0.290 0.324 0.751 0.772 0.044 0.171 0.299 0.487


Brazil Sao Paulo 0.296 0.298 0.683 0.674 0.111 0.162 0.371 0.338

Canada Toronto 0.476 0.463 0.741 0.790 0.051 0.270 0.299 0.377

Chile Santiago 0.078 0.130 0.636 0.549 0.064 0.129 0.292 0.287

China Hong Kong 0.329 0.231 0.739 0.730 0.092 0.062 0.571 0.556

China Shanghai 0.082 0.042 0.473 0.629 -0.004 0.036 0.346 0.471

China Shenzhen 0.076 0.043 0.504 0.578 0.015 0.117 0.361 0.535


France Paris 0.373 0.403 0.781 0.783 0.105 0.275 0.507 0.517



Greece Athens 0.305 0.331 0.665 0.692 0.151 0.199 0.421 0.418

India Bombay 0.262 0.365 0.626 0.680 0.007 0.306 0.346 0.457


Ireland Irish 0.407 0.389 0.791 0.779 0.190 0.291 0.628 0.588

Israel Tel Aviv 0.252 0.234 0.736 0.688 0.150 0.158 0.506 0.468

Italy Milan 0.222 0.260 0.660 0.803 -0.005 0.188 0.237 0.519

Japan Tokyo 0.158 0.193 0.694 0.731 -0.089 0.153 0.387 0.496


Mexico Mexican 0.191 0.242 0.689 0.660 0.080 0.144 0.432 0.329

Netherlands AEX 0.400 0.423 0.798 0.826 0.071 0.188 0.516 0.474


Norway Oslo 0.316 0.343 0.714 0.674 0.123 0.227 0.494 0.440


Poland Warsaw 0.316 0.343 0.680 0.672 0.085 0.087 0.537 0.109

Portugal Lisbon 0.488 0.449 0.838 0.788 0.226 0.205 0.567 0.480



South Korea Korea 0.174 0.187 0.666 0.822 -0.075 0.192 0.505 0.653

Spain Barcelona 0.335 0.213 0.695 0.819 0.024 0.114 0.118 0.489



Taiwan Taiwan 0.118 0.081 0.606 0.789 -0.103 0.100 0.345 0.442


Turkey Istanbul 0.281 0.105 0.828 0.393 -0.184 0.230 0.286 0.631

UK London 0.335 0.204 0.852 0.749 0.018 0.082 0.420 0.444

US New York 0.174 0.145 0.619 0.769 0.021 -0.006 0.053 0.064

US American 0.281 0.259 0.604 0.638 0.041 0.104 0.134 0.196

US NASDAQ 0.302 0.334 0.786 0.773 0.064 0.151 0.374 0.346

US All US 0.358 0.361 0.742 0.767 0.069 0.142 0.393 0.421

Developed 0.451 0.411 0.778 0.704 0.465 0.370 0.682 0.627

Developing 0.410 0.403 0.773 0.726 0.500 0.444 0.904 0.892

Global 0.445 0.429 0.774 0.718 0.598 0.516 0.905 0.901

8

Table IA.8The Performance of Daily Percent-Cost Proxies Compared to Daily Percent Effective Spread by Exchange

The high-frequency percent-cost benchmark, daily percent effective spread, is calculated from every trade and correspondingBBO quote in the SIRCA Thomson Reuters Tick History database for a sample stock-day. The daily percent-cost proxiesare calculated from daily stock price data for a sample stock-day. The earlier (later) sample spans 42 exchanges around theworld from 1996-2007 (2008-2014). All stock-months are required to have at least five positive volume days and eleven non-zeroreturn days. A solid box means the highest correlation or the lowest average root mean squared error (RMSE) in the row.Dashed boxes mean correlations that are statistically indistinguishable from the highest correlation or average RMSEs that arestatistically indistinguishable from the lowest average RMSE in the row at the 5% level. Bold-faced numbers are statisticallydifferent from zero or proxies have predictive power that is significant at the 5% level.



QSVoV

(% Spread)HL CHL

Closing %QS

VoV(% Spread)


Australia Australian -0.026 0.130 0.243 0.224 0.024 0.029 0.022 0.019

Austria Vienna 0.000 0.118 0.273 0.311 0.010 0.014 0.008 0.007


Brazil Sao Paulo -0.015 0.043 0.157 0.170 0.025 0.028 0.020 0.020

Canada Toronto -0.014 0.101 0.205 0.320 0.022 0.026 0.019 0.017

Chile Santiago 0.029 0.026 0.287 0.135 0.016 0.017 0.016 0.013

China Hong Kong 0.001 0.075 0.281 0.281 0.023 0.028 0.016 0.016

China Shanghai 0.008 0.006 0.143 0.368 0.010 0.014 0.002 0.004

China Shenzhen 0.002 0.016 0.140 0.359 0.010 0.014 0.002 0.004

Denmark Copenhag. -0.030 0.118 0.268 0.249 0.018 0.020 0.014 0.015

France Paris -0.008 0.163 0.296 0.352 0.014 0.017 0.010 0.010

Finland Helsinki -0.017 0.127 0.244 0.274 0.013 0.017 0.010 0.011

Germany Frankfurt -0.005 0.068 0.205 0.161 0.032 0.037 0.037 0.060

Greece Athens -0.007 0.112 0.285 0.324 0.021 0.025 0.018 0.025

India Bombay -0.030 0.172 0.247 0.297 0.046 0.047 0.044 0.045


Ireland Irish -0.025 0.149 0.302 0.258 0.016 0.020 0.013 0.012

Israel Tel Aviv -0.025 0.077 0.308 0.304 0.032 0.033 0.020 0.025

Italy Milan -0.012 0.094 0.219 0.350 0.010 0.014 0.013 0.006

Japan Tokyo -0.016 0.087 0.221 0.330 0.009 0.015 0.006 0.006

Malaysia Kuala Lum. -0.014 0.125 0.338 0.309 0.018 0.023 0.015 0.016

Mexico Mexican 0.011 0.064 0.250 0.154 0.016 0.018 0.018 0.014

Netherlands AEX -0.019 0.114 0.250 0.318 0.012 0.015 0.009 0.008

New Zeland New Zea. -0.042 0.111 0.213 0.175 0.012 0.014 0.011 0.010

Norway Oslo -0.022 0.114 0.261 0.229 0.019 0.022 0.015 0.014

Philippines Phillipine -0.011 0.109 0.406 0.203 0.019 0.024 0.014 0.018

Poland Warsaw -0.014 0.104 0.267 0.308 0.025 0.029 0.024 0.031

Portugal Lisbon -0.013 0.105 0.217 0.263 0.012 0.016 0.008 0.012

Singapore Singapore -0.012 0.090 0.224 0.257 0.015 0.020 0.012 0.013

South Africa Johannes. -0.023 0.121 0.303 0.226 0.024 0.026 0.020 0.019

South Korea Korea -0.024 0.081 0.183 0.428 0.014 0.020 0.007 0.009

Spain Barcelona 0.011 0.086 0.178 0.380 0.009 0.013 0.008 0.004

Sweden Stockholm -0.021 0.105 0.220 0.249 0.020 0.024 0.016 0.016

Switzerland SWX Swiss -0.027 0.141 0.248 0.319 0.012 0.015 0.010 0.008

Taiwan Taiwan -0.032 -0.010 0.152 0.041 0.008 0.015 0.003 0.007

Thailand Thailand -0.009 0.082 0.260 0.245 0.015 0.021 0.007 0.017

Turkey Istanbul -0.028 0.058 0.170 0.378 0.009 0.014 0.002 0.007

UK London 0.007 0.025 0.215 0.124 0.021 0.024 0.018 0.016

US New York -0.006 0.023 0.036 0.123 0.009 0.014 0.008 0.003

US American 0.001 0.048 0.104 0.191 0.024 0.029 0.024 0.019

US NASDAQ -0.000 0.083 0.192 0.315 0.019 0.025 0.013 0.012

US All US -0.001 0.063 0.141 0.249 0.018 0.024 0.014 0.012

Developed -0.013 0.092 0.224 0.258 0.018 0.022 0.016 0.019

Developing -0.013 0.084 0.244 0.285 0.021 0.025 0.017 0.019

Global -0.013 0.088 0.233 0.271 0.020 0.024 0.018 0.020

9

Documents

VoV(%Spread)” for percent spread cost and a ne · 2019-08-29 · We test the monthly and daily versions of these new and existing liquidity proxies against liquidity benchmarks