Can Global Stock Liquidity Be Measured?*

Transcription

1 Can Global Stock Liquidity Be Measured?* Kingsley Fong University of New South Wales Craig W. Holden Indiana University Charles A. Trzcinka** Indiana University February 200 Abstract A growing literature uses percent cost or cost per volume liquidity proxies constructed from daily stock data to conduct research in international asset pricing, international corporate finance, and international market microstructure. Relatively little is known about how well these proxies are related to transactions costs in the global setting. We compare these proxies to actual transaction costs using the microstructure data in the Reuters Datascope Tick History dataset by the Securities Industry Research Centre of Asia-Pacific. Our sample contains 8. billion trades and 2.2 billion quotes representing 6,096 firms on 4 exchanges around the world from 996 to We evaluate nine percent cost proxies (including two new ones) relative to four percent cost benchmarks: percent effective spread, percent quoted spread, percent realized spread, and percent price impact. We examine twelve cost per volume proxies (including two new ones) relative to a cost per volume benchmark: the slope of the price function lambda. We test three dimensions: () higher average cross-sectional correlation with the benchmarks, (2) higher portfolio correlation with the benchmarks, or (3) lower prediction error relative to the benchmarks. We find that a new measure, FHT, is the best proxy for percent effective spread, percent quoted spread, and percent realized spread. We find that FHT is the best proxy for percent price impact, but is only moderately-well correlated with it and does not capture the correct scale. We find that LOT Mixed Impact, Zeros Impact, and Zeros2 Impact are the best proxies for lambda, but they are only moderately-well correlated with it and do not capture the correct scale. JEL classification: C5, G2, G20. Keywords: Liquidity, transaction costs, effective spread, price impact, asset pricing. * We thank seminar participants at Hong Kong University, Hong Kong University of Science and Technology, Indiana University, Michigan State University, University of New South Wales, University of Sydney Microstructure Meeting, and University of Technology, Sydney. We are solely responsible for any errors. ** Corresponding author: Charles A. Trzcinka, Kelley School of Business, Indiana University, 309 E. Tenth St., Bloomington, IN Tel.: ; fax: : address: ctrzcink@ indiana.edu

2 . Introduction A rapidly growing literature uses percent cost or cost per volume liquidity proxies constructed from daily stock data to conduct research in international asset pricing, international corporate finance, 2 and international market microstructure. 3 For example, Bekaert, Harvey, and Lundblad (2007) use the proportion of zero returns as a percent cost proxy to test the relationship between liquidity cost and asset pricing in 9 emerging markets. Karolyi, Lee and Van Dijk, (2008) use the Amihud (2002) measure as a cost per volume proxy to analyze the commonality patterns of cost per volume liquidity, returns, and turnover across 40 countries. Lang, Lins, and Maffett (2007) use the proportion of zero returns to examine the relationship between earnings smoothing, governance, and liquidity cost in 2 countries. Levine and Schmukler (2006) use both the proportion of zero returns and the Amihud measure to examine the relationship between home market liquidity and cross-listed trading across 3 home countries. Despite the widespread use of various percent cost and cost per volume liquidity proxies in international research, relatively little is known about how well these proxies are related to actual transactions costs in the global setting. To fill this gap in the literature, we compare these proxies to actual transaction costs using the Reuters Datascope Tick History (RDTH) dataset by the Securities Industry Research Centre of Asia-Pacific (SIRCA), which contains more than a decade of global intraday Reuters data. Our sample contains 8. billion trades and 2.2 billion quotes representing 6,096 firms on 4 exchanges around the world from 996 to We evaluate nine percent cost proxies (including two new ones that we introduce) relative to four percent cost benchmarks: percent effective spread, percent quoted spread, percent realized spread, and percent price See Jain, Xia, and Wu (2004), Stahel (2005), Liang and Wei (2006), Lee (2006), Griffin, Kelly, and Nardari (2007), and Bekaert, Harvey, and Lundbland (2007). 2 See Bailey, Karolyi, and Salva, (2005), LaFond, Lang, and Skaife (2007) and Lang, Lins, and Maffett (2009). 3 See Jain (2005), Levine and Schmukler (2006), Henkel, Jain, and Lundblad (2008), Henkel (2008), Karolyi, Lee, and van Dijk (2008), and DeNicolo and Ivaschenko (2009).

3 impact. We examine twelve cost per volume proxies (including two new ones that we introduce) relative to a cost per volume benchmark: the slope of the price function lambda. In each case, we test three performance dimensions: () higher average cross-sectional correlation with the benchmarks, (2) higher portfolio correlation with the benchmarks, or (3) lower prediction error relative to the benchmarks. We find that global stock liquidity can be measured using percent cost and cost per volume liquidity proxies and we identify the best proxies in each case. Percent cost and cost per volume liquidity proxies provide an enormous advantage for international research by spanning a large cross-section of countries over a long time-series. The daily stock data from which they are constructed are available from various vendors. For example, Thompson Financial s DataStream provides daily stock data on tens of thousands of stocks from 64 countries. Their daily stock price data, from which percent cost proxies are computed, goes back to the 980 s for 2 countries, back to the 970 s for 6 additional countries, and back to the 960 s for 2 more countries. Their daily stock volume data, which is a required variable to compute cost per volume proxies, goes back to the 990 s for 9 countries, back to the 980 s for 23 additional countries, and back to the 970 s for 2 more countries. Three studies test the performance of available percent cost and cost per volume liquidity proxies using U.S. data. Lesmond, Ogden, and Trzcinka (999) test how three annual percent cost proxies are related to the annual quoted spread as computed from daily closing quoted spreads in U.S. data. Hasbrouck (2009) tests how three annual percent cost proxies and one annual cost per volume proxy are related to the benchmarks: percent effective spread and the slope of the price function lambda as computed from high-frequency U.S. trade and quote data. Goyenko, Holden, and Trzcinka (2009) test how nine annual and monthly percent cost proxies are related to the annual and monthly percent cost benchmarks: percent effective spread and percent realized spread as computed from high-frequency U.S. trade and quote data. Goyenko, Holden, and 2

4 Trzcinka also test how twelve annual and monthly cost per volume proxies are related to the annual and monthly benchmarks: the slope of the price function lambda and percent price impact as computed from high-frequency U.S. trade and quote data. Lesmond (2005) tests how three quarterly percent cost proxies and two quarterly cost per volume proxies are related to average daily spreads in 23 emerging markets. In contrast, we test nine percent cost proxies and twelve cost per volume proxies computed on a monthly basis against five benchmarks computed from intraday trade and quote data. We run these tests for 4 exchanges in both developed and emerging countries. Our benchmarks are more precise measures of transactions costs because they are computed from high frequency data rather than end of day spreads. We can safely assert that the international literature has generally not been mistaken in using Zeros and the LOT measures as proxies for percent effective spread, percent quoted spread, percent realized spread, or percent price impact around the world. However, we show that a new proxy, which is a simplification of the LOT measure, performs significantly better. Further, we show that the Amihud measure is only moderately-well correlated with lambda (the slope of the price function) around the world and does not capture the correct scale. However, we show that other proxies perform somewhat better than Amihud. We find that a new measure that we introduce, FHT, is the best and most consistent proxy for percent effective spread, percent quoted spread, and percent realized spread. We find FHT is the best proxy for percent price impact, but is only moderately-well correlated with it and does not capture the correct scale. We find that LOT Mixed Impact, Zeros Impact, and Zeros2 Impact are the best proxies for lambda, but they are only moderately-well correlated with it and do not capture the correct scale. The paper is organized as follows. Section 2 explains the high-frequency percent cost and cost per volume benchmarks. Section 3 explains the low-frequency percent cost proxies. Section 3

5 4 explains the low-frequency cost per volume proxies. Section 5 describes the dataset and methodology used. Section 6 presents our empirical results. Section 7 concludes. An online appendix with 20 additional tables is available at 2. High Frequency Percent Cost and Cost Per Volume Benchmarks We analyze four high frequency percent cost benchmarks. Our first percent cost benchmark is percent effective spread. For a given stock, the percent effective spread on the trade is defined as Percent Effective Spread 2 ln P ln M, () k k k th k where P k is the price of the k th trade and M k is the midpoint of the consolidated BBO prevailing at the time of the k th trade. Aggregating over month i, a stock s Percent Effective Spread i is the dollar-volume-weighted average of Percent over all trades in month i. Effective Spread k computed Our second percent cost benchmark is percent quoted spread. For a given time interval s, the percent quoted spread is defined as Percent Quoted Spread ( Ask Bid )/(( Ask Bid )/2), (2) s s s s s where Ask s is the best ask quote Bid s is the best bid quote in that time interval. Over month i, the stock s Percent Quoted Spread i is the time-weighted average of the time interval values during the month. Our third percent cost benchmark is realized spread, which is the temporary component of the percent effective spread. For a given stock, the percent realized spread on the defined as th k trade is Percent Realized Spreadk 2 ln( Pk) ln( M k 5), (3) 4

6 where M (k+5) is the midpoint five-minutes after the th k trade. Aggregating over month i, a stock s Percent Realized Spread i is the dollar-volume-weighted average of Percent Realized Spread k computed over all trades in month i. Our fourth percent cost benchmark is percent price impact, which is the percent change in quote midpoint caused by a trade. For a given stock, the percent price impact on the defined as th k trade is Percent Price Impact 2 ln( M ) ln( M ), (4) k k5 k where M k+5 is the midpoint of the consolidated BBO prevailing five minutes after the k th trade, and M k is the midpoint prevailing at the time of the k th trade. For a given stock aggregated over a month i, the Percent Price Impact i is the dollar-volume-weighted average of Percent Price Impact k computed over all trades in month i. We also analyze a cost per volume benchmark,, which is the slope of the price function. We follow Hasbrouck (2009) and calculate as the slope coefficient of the regression r S u, (5) n n n where, for the n th five-minute period, r n is the stock return, n S = k Sign v kn v kn is the signed square-root dollar volume, v kn is the signed dollar volume of the th n fiveminute period, and u n is the error term. th k trade in the 3. Low Frequency Percent Cost Proxies Nine low frequency percent cost proxies are explained below. For each measure, we require that the measure always produces a numerical result. If a measure cannot be computed we substitute a default value. 5

7 3.. Roll Roll (984) develops an estimator of the effective spread based on the serial covariance of the change in price as follows. Let V t be the unobservable fundamental value of the stock on day t. Assume that it evolves as V V e, (6) t t t where e t is the mean-zero, serially uncorrelated public information shock on day t. Next, let P t be the last observed trade price on day t. Assume it is determined by P V SQ, (7) t t 2 t where S is the effective spread and a buy and for a sell. Assume that Q t is a buy/sell indicator for the last trade that equals for Q t is equally likely to be or, is serially uncorrelated, and is independent of e t. Taking the first difference of Eq. (7) and combining it with Eq. (6) yields P SQ e (8) t 2 t t where is the change operator. Given this setup, Roll shows that the serial covariance is or equivalently 2, Cov P P S (9) t t 4 S 2 Cov Pt, Pt. (0) When the sample serial covariance is positive, the formula above is undefined and so we substitute a default numerical value of zero. We therefore use a modified version of the Roll estimator: 6

8 t t t t Cov P P 2 Cov P, P When Cov P, P 0 Roll. () 0 When t, t Extended Roll Holden (2009) extends the Roll model by developing a more precise version of Roll. Individual stock returns, that are adjusted for splits and dividends, can be decomposed into three parts: () systematic value innovations generated by the market, (2) idiosyncratic value innovations generated by the firm, and (3) bid -ask bounce. From a signal extraction perspective, the bid-ask bounce is the signal to be extracted and the systematic value innovations and idiosyncratic value innovations are both noise terms. It is useful to remove the systematic value innovations, because the residuals that are left have a higher signal-to-noise ratio. where The empirical procedure is to perform a market model regression t f mt f t ar r r r z, (2) ar t is the adjusted return on date t which accounts for splits and dividends, r f is the daily riskfree rate, and are the regression coefficients, r mt is the value-weighted market return on date t, and z t is the regression residual. Then use the residual to compute the idiosyncratic adjusted price change as * P t P z P. (3) * t t t where Pt is the unadjusted price on date t. The Extended Roll model uses zero when the serial covariance is positive as follows * * 2 Cov Pt, Pt * * when CovPt, Pt 0 Extended Roll P * * 0 when CovPt, Pt 0. (4) 3.3. Effective Tick 7

9 Holden (2009) and Goyenko, Holden, and Trzcinka (2009) jointly develop a proxy of the effective spread based on observable price clustering as follows. Let S t be the realization of the effective spread at the closing trade of day t. Assume that the realization of the spread corresponding to the closing trade of day is randomly drawn from a set of possible spreads s, j, 2,, J (ordered from smallest to largest) with corresponding probabilities j j, j, 2,, J. For example on a $ 8 price grid, S t is modeled as having a probability of s $ spread, 2 of s $ spread, 3 of s $ spread, and 4 of s4 $ spread. 8 Let N j be the number of trades on prices corresponding to the th j spread j, 2,, J using only positive-volume days in the month. In the $ 8 price grid example (where J 4 ), N through N 4 are the number of trades on odd prices, the number of trades on odd 8 4 prices, the number of trades on odd 2 prices, and the number of trades on whole dollar prices, respectively. Let F j be the probabilities of trades on prices corresponding to the j th spread j, 2,, J. These empirical probabilities are computed as F j J N j j N j for j, 2,, J. (5) Let U j be the unconstrained probability of the j th spread j, 2,, J. The unconstrained probability of the effective spread is 2 Fj j U j 2Fj Fj j 2,3,, J. (6) Fj Fj j J. The effective tick model directly assumes price clustering (i.e., a higher frequency on rounder increments). However, in small samples it is possible that reverse price clustering may 8

10 be realized (i.e., a lower frequency on rounder increments), which could cause an unconstrained probability to go above or below 0. Let ˆ j be the constrained probability of the jth spread j, 2,, J. It is computed in order from smallest to largest as follows ˆ j j ˆ j k k Min Max U j,0, j Min Max U,0, j 2,3,, J. (7) Finally, the effective tick measure is simply a probability-weighted average of each effective spread size divided by P i, the average price in time interval i Effective Tick J j ˆ s P i j j. (8) 4.5. Zeros Lesmond, Ogden, and Trzcinka (999) introduce the proportion of days with zero returns as a proxy for liquidity. Two key arguments support this measure. First, stocks with lower liquidity are more likely to have zero volume days and thus are more likely to have zero-return days. Second, stocks with higher transaction costs have less private information acquisition (because it is more difficult to overcome higher transaction costs), and thus, even on positive volume days, they are more likely to have no-information-revelation, zero-return days. Lesmond, Ogden, and Trzcinka define the proportion of days with zero returns as Zeros = (# of days with zero returns)/t, (9) where T is the number trading days in a month. An alternative version of this measure, Zeros2, is defined as Zeros2 = (# of positive volume days with zero return)/t. (20) 4.6. LOT 9

11 Lesmond, Ogden, and Trzcinka (999) develop an estimator of the effective spread based on the idea that stocks with larger transaction costs pose a larger hurdle for potential traders (both informed and liquidity traders). Thus, a larger segment of potential true returns are incorporated into zero returns. The LOT model assumes that the unobserved true return * R jt of a stock j on day t is given by R R, (2) * jt j mt jt where j is the sensitivity of stock j to the market return R mt on day t and jt is a public information shock on day t. They assume that jt is normally distributed with mean zero and variance. Let 0 be the percent transaction cost of selling stock j and 2 0 be the 2 j j j percent transaction cost of buying stock j. Then the observed return R jt on a stock j is given by R R when R * * jt jt j jt j * * jt jt when j jt 2 j * * jt jt 2 j when 2 j jt. R R R R R R (22) The LOT liquidity measure is simply the difference between the percent buying cost and the percent selling cost: LOT (23). j 2 j Lesmond, Ogden, and Trzcinka develop the following maximum likelihood estimator of the model s parameters: 0

12 L,,, R, R j 2 j j j jt mt jt j j mt 0 R n j R R n 2 j j ST.. 0, 0, 0, 0, jt 2 j j mt j j2 j j j 2 j jr mt j jr mt N N j j R (24) where N is the cumulative normal distribution. A key issue for LOT is the definition of the three regions over which the estimation is done. The original LOT (999) measure, now called LOT Mixed, distinguishes the three regions based on both the X-variable and the Y-variable (region 0 is R jt 0, region is R jt 0 and R 0, and region 2 is R 0 and R 0). Goyenko, Holden, and Trzcinka (2009) developed mt jt mt a new version of the measure, which they called LOT Y-split, that breaks out the three regions based on the Y-variable (region 0 is R jt 0, region is R jt 0, and region 2 is R jt 0 ) FHT This paper develops a new estimator of the effective spread, which is a simplification of the LOT Mixed and Y-split measures. First, it assumes that transaction costs are symmetric. Let S 2 be the percent transaction cost of buying a stock and S 2 be the percent transaction cost of selling the same stock. Substituting this assumption into equation (22) and suppressing the subscripts, the observed return R on an individual stock is given by 2 when 2 * * R R S R S * * R R S R S when 2 2 * * RR S 2 when S 2 R. (25)

13 Note that observed return is zero in the middle region where * S 2 R S 2. Secondly, the new estimator simplifies LOT by focusing on the return distribution of an individual stock and providing no role for the market portfolio. Specifically, the unobserved true return * R of an individual stock on a single day is assumed to be normally distributed 2 with mean zero and variance. Thus, the theoretical probability of a zero return is given by S S N N. 2 2 (26) The empirically observed frequency of a zero return is given by the measure Zeros. Let z Zeros. Equating the theoretical probability of a zero return to the empirically-observed frequency of a zero return, we obtain S S N N z 2 2 (27) By the symmetry of the cumulative normal distribution, equation (27) can be rewritten as S S N N z 2 2 (28) Solving for S, we obtain + z FHT S 2 N, 2 (29) where N is the inverse function of the cumulative normal distribution. The FHT measure is a simple, analytic measure that can be computed with a single line of SAS code FHT2 This paper develops a second new estimator of the effective spread, which takes a slightly more involved approach to simplifying the LOT Mixed and Y-split measures. Like FHT 4 Specifically: Sigma=Std(Returns); Zeros=ZeroReturnDays/TotalDays; FHT = 2*Sigma*Probit((+Zeros)/2); Run; If the daily mean return is not negligible, simply add mean(returns) to FHT. 2

14 it assumes that the true return * R jt of a stock j on day t is normally distributed with mean zero 2 and variance j. Thus, the theoretical probability that the observed return is zero and the true * return is positive R 0 is 2 N. 2 (30) Similarly, the theoretical probability that the observed return is zero and the true return is * negative R 0 is N. 2 (3) Unlike FHT, the new measure doesn t impose symmetry of buy and sell transaction costs. Instead, it empirically identifies asymmetric costs by distinguishing between zero returns that take place when the market return is positive vs. negative. Define the percentage of days with zero returns on positive market return days PZ as # of zero returns on positive market return days PZ Min,0.49, # of open-exchange days in the month (32) where the cap at 0.49 is required for numerical reasons that will be explained below. Similarly, define the percentage of days with zero returns on negative market return days NZ as # of zero returns on negative market return days NZ Min,0.49, # of open-exchange days in the month (33) where the cap at 0.49 is also required for numerical reasons that will be explained below. Equating the theoretical probability of a positive true return and zero observed return to its empirically-observed counterpart, we obtain 2 N PZ. 2 (34) 3

15 Similarly, equating the theoretical probability of a negative true return and zero observed return to its empirically-observed counterpart, we obtain N NZ. 2 (35) Solving for 2, we obtain 2 2 N PZ. (36) Notice that in the limit as PZ, 2 2. To avoid this problem, a cap is imposed of PZ 0.49, which implies a large maximum value of Solving for, we obtain 2 N NZ. (37) In the limit as NZ 2,. To avoid this problem, a cap is imposed of NZ 0.49, which implies a large minimum value of Combining these results, we obtain the new measure FHT N PZ N NZ (38) 5. Low Frequency Cost Per Volume Proxies Next, we explain twelve low-frequency cost per volume proxies. As before, we require that each measure always produce a numerical result. 5.. Amihud Amihud (2002) develops a cost per volume measure that captures the daily price response associated with one dollar of trading volume. Specifically, he uses the ratio Illiquidity rt Average, (39) Volume t 4

16 where r t is the stock return on day t and Volume t is the currency value of volume on day t expressed in units of local currency. The average is calculated over all positive-volume days, since the ratio is undefined for zero-volume days Extended Amihud Proxies Goyenko, Holden, and Trzcinka (2009) develop a new class of cost per volume proxies by extending the Amihud measure. They decompose the total return in the numerator of the Amihud model into a liquidity component and a non-liquidity component. By assumption, the non-liquidity component is independent of the liquidity component and can be eliminated as being unrelated to the variable of interest. The remaining numerator value can be approximated by any percent cost proxy over month i and the average daily currency value of volume in units of local currency over the same time interval as follows Percent Cost Proxy i Extended Amihud Proxy i =. Average Daily Currency Volumei (40) The equation above defines a class of cost per volume proxies depending on which percent cost proxy is used. For example, one member of this class uses the Roll measure for month i as follows: Roll Impact i Roll Average Daily Currency Volume i. (4) i We test nine versions of this class of cost per volume measures based on nine different percent cost proxies. The nine measures we test are: Roll Impact, Extended Roll Impact, Effective Tick Impact, LOT Mixed Impact, LOT Y-split Impact, Zeros Impact, Zeros2 Impact, FHT Impact, and FHT2 Impact Pastor and Stambaugh 5

17 Pastor and Stambaugh (2003) develop a measure of cost per volume called Gamma by running the regression e e r t r t Gamma sign rt Volume t (42) t where e r t is the stock s excess return above the value-weighted market return on day t and Volume t is the currency value of volume on day t expressed in units of local currency. Intuitively, Gamma measures the reverse of the previous day s order flow shock. Gamma should have a negative sign. The larger the absolute value of Gamma, the larger the implied price impact Amivest Liquidity The Amivest Liquidity ratio is a volume per cost measure Liquidity Volume Average. (43) t r t The average is calculated over all non zero-return days, since the ratio is undefined for zero return days. A larger value of Liquidity implies a lower cost per volume. This measure has been used by Cooper, Groth, and Avera (985), Amihud, Mendelson, and Lauterback (997), and Berkman and Eleswarapu (998) and others. 6. Data To compute our benchmarks (effective spread, realized spread, quoted spread, percent price impact, and lambda), we use data from the Reuters Datascope Tick History (RDTH) supplied by the Securities Industry Research Centre of Asia-Pacific (SIRCA). The RDTH is a commercial product launched in June 2006 and its subscribers include central banks, investment 6

18 banks, hedge funds, brokerages, and regulators. 5 RDTH provides millisecond-time-stamped tick data since January 996 of over 5 million equity instruments worldwide. RDTH data is sourced from the Reuters IDN (Integrated Data Network) which obtains the feeds directly from the exchanges. We verified the quality of RDTH by double-checking its Finland data against the Nordic Security Depository, which is the central clearing agency for all trading in Finland. It includes the complete, official trading records of all trading in securities listed on the Helsinki Stock Exchange. The random checks we performed showed the trades agree so that if a trade of 200 shares at $0 shows in the RDTH database, we will see a purchase of 200 shares at $0 and a corresponding sale of 200 shares in the Depository data. Our non-u.s. dataset covers 39 countries, including 20 developed countries and 9 emerging countries. These countries are selected on the basis of Reuters intraday and Datastream daily price, volume and market capitalization data availability. We analyze the leading exchange by volume in each country, plus two exchanges in Japan (the Tokyo Stock Exchange and the Osaka Securities Exchange) and two exchanges in China (the Shanghai Stock Exchange and Shenzhen Stock Exchange). Altogether we analyze 4 exchanges. We select all firms listed on those 4 exchanges at any time from 996 to We impose two activity filters on each stockmonth in order to have reliable and consistent proxy estimates. We require that a stock have at least five positive-volume days in the month. And we require that a stock have at least five nonzero return days in the month. Our final sample has 8,78,077,962 trades and 2,255,684,75 quotes. We compute the percent cost benchmarks and proxies and cost per volume benchmarks 5 Prior to July 0, 2009, the same underlying tick history data has been supplied by SIRCA to academics subscribers via the interface called Taqtic, which is a more restricted version of the commercial RDTH. Taqtic was decommissioned on July 7, For more information on the RDTH which is the current platform for academic and commercial users to access the global tick history, see 7

19 and proxies for 6,096 firms in,93,909 stock-months. For the proxies that require a market we use the local value-weighted market portfolio. Table provides the mean of the monthly percent cost benchmarks and proxies. Panel A is for developed markets and Panel B is for emerging markets. Each row represents a different exchange. For example, looking at the first row, the country is Australia and the exchange code is AX, which stands for the Australian Stock Exchange. Panel C maps each exchange code into the full name of the exchange. Table 2 provides the mean of the monthly cost per volume benchmarks and proxies. Again, Panel A is for developed markets and Panel B is for emerging markets. 7. Results 7.. The Spread Results Tables 3 6 report monthly percent cost proxies compared with percent cost benchmarks (percent effective spread, percent quoted spread, and percent realized spread). Tables 3, 4, and 5 each report on one of three performance dimensions (average cross-sectional correlation, timeseries correlation, and average root mean squared error) and Table 6 reports breakouts by size and turnover quintiles. Turning to Table 3, the performance criterion is the average cross-sectional correlation between the percent cost proxy and the percent cost benchmark based on individual firms. This is computed in the spirit of Fama and MacBeth (973) by: () calculating for each month the crosssectional correlation across all firms on a given exchange, and then (2) calculating the average correlation value over all months available for that exchange. Panel A reports each percent cost proxy compared to percent effective spread in developed countries and Panel B compares to percent effective spread in emerging countries. A 8

20 solid box is placed around the highest correlation in the row. A dashed box is placed around correlations that are statistically indistinguishable from the highest correlation in the row at the 5% level. 6 In the first row for the Australian Stock Exchange (AX), the new proxy FHT has the highest average cross-sectional correlation with percent effective spread at and the other new proxy FHT2 is statistically indistinguishable from the highest correlation at All of the rest of the correlations in the first row are statistically lower than the highest correlation. Boldfaced correlations are statistically different from zero at the 5% level. 7 Nearly all correlations in the table are statistically different from zero. There is considerable variation from exchange-to-exchange. In Panel A, the correlations range from.305 for Luxembourg to.822 for Ireland. The large majority exchanges have best correlations in the 0.50 s and 0.60 s. In Panel B, the large majority exchanges also have best correlations in the 0.40 s, 0.50 s, and 0.60 s. However, both Chinese exchanges seem to be outliers. The best correlation on the Shanghai Stock Exchange (SS) is and the best on the Shenzhen Stock Exchange is 0.4. None of the percent cost proxies do well on these two exchanges and a few correlations are even negative. It is not clear why these two exchanges are so different. For a subset of firms, China has a system of A shares that can only be traded by domestic investors and B shares that can 6 In Tables 3, 6-7, 0, and 3, we test whether the cross-sectional correlations are different between proxies on the same row by t-tests on the time-series of correlations in the spirit of Fama-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 of correlations of each proxy is 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. 7 We test all correlations in Tables 3, 4, 6-8, 0-, and 3 to see if they are statistically different from zero and highlight the correlations that are significant in boldface. For an estimated correlation, Swinscow (997, Ch. ) gives the appropriate test statistic as where D is the sample size. D 2 t 2, 9

21 only be traded by foreigner investors. We report the A share only results, but we verified that the inclusion of B shares leads to very similar results. Looking at both Panels A and B, we see that FHT and FHT2 either have a solid box (being the winner) or a dashed box (being insignificantly different from the winner) for nearly all exchanges. On average across all 2 exchanges in developed countries, FHT and FHT2 tie with an average cross-sectional correlation of On average across all 20 exchanges in emerging countries, FHT has an average cross-sectional correlation of and FHT2 has a correlation of For nearly all exchanges, the average cross-sectional correlations for FHT and FHT2 are much higher than the correlations for any other proxy. FHT and FHT2 dominate the effective spread comparisons. Panel C reports each percent cost proxy is compared to percent quoted spread. In Panel C, FHT2 has the best correlation for average developed at and FHT is insignificantly different at For average emerging FHT and FHT2 tie at In both cases, the FHT and FHT2 correlations are much higher than the correlations for any other proxy. An online appendix with 20 additional tables is available at In online Appendix Table, the percent quoted spread results by exchange are qualitatively similar to the percent effective spread results by exchange in Panels A and B. For nearly all exchanges, the average cross-sectional correlations for FHT and FHT2 are much higher than the correlations for any other proxy. Panel D reports each percent cost proxy is compared to percent realized spread. Again, FHT and FHT2 have the highest correlations for both average developed and average emerging. Again, they are much higher than the correlations for any other proxy. In online Appendix Table 2, the percent realized spread results by exchange show that, for nearly all exchanges, the 20

22 average cross-sectional correlations for FHT and FHT2 are much higher than the correlations for any other proxy. To summarize Table 3, FHT and FHT2 do a good job of capturing percent effective spread, percent quoted spread, and percent realized spread and they dominate all other proxies. We conducted a comparison of the distribution of the maximum correlation in each row of the developed exchanges with the distribution of the maximum correlation in each row of the emerging exchanges. The developed exchanges stochastically dominated the emerging exchanges. The x percentile of the twenty-one developed market s highest correlation is always higher than the same percentile for the twenty emerging markets. This is true for the effective spread results reported in table 3 and the timeweighted quote and realized spread results reported in Appendix Tables and 2. The later tables show stronger correlations than in Table 3 and the developed markets are more dominant. This suggests that proxies are stronger for firm level data in developed markets. Next, we form equally-weighted portfolios across all stocks on a given exchange for month i. Then, we compute a portfolio percent cost proxy in month i by taking the average of that percent cost proxy over all stocks on a given exchange in month i. Table 4 reports the timeseries correlation between each portfolio percent cost proxy and the portfolio percent cost benchmarks. A solid box identifies the highest correlation in the row and a dashed box indicates correlations that are statistically indistinguishable from the highest correlation in the row at the 5% level. 8 Boldfaced correlations are statistically different from zero at the 5% level. Panel A reports each portfolio percent cost proxy compared to portfolio percent effective spread in developed countries and Panel B compares to portfolio percent effective spread in emerging countries. In these two panels, there is huge variation in the best time-series correlation values. In Luxembourg, the best time-series correlation is only and it is not significantly 8 In Tables 4, 8, and, we test whether time-series correlations are statistically different from each other using Fisher s Z-test. 2

23 different from zero! By contrast, in Indonesia the best time-series correlation is Six proxies are frequently the best or insignificantly different from the best. Specifically, FHT is the best or insignificantly different from the best on 32 exchanges. The same is true for FHT2 on 30 exchanges, Roll on 9 exchanges, LOT Mixed on 8 exchanges, Zeros on 7 exchanges, and LOT Y on 6 exchanges. For all of the developed countries, FHT was the best with an average time-series correlation of For all of the emerging countries, Effective Tick was the best with an average time-series correlation of FHT was close behind with an average timeseries correlation of Panel C reports each percent cost proxy is compared to percent quoted spread. FHT has the best correlation for average developed at and FHT2 is close at FHT has the best correlation for average emerging at and Effective Tick is close at Panel D reports each percent cost proxy is compared to percent realized spread. FHT has the best correlation for average developed at 0.48 and FHT2 is close at FHT has the best correlation for average emerging at Comparing the distribution of the maximum correlation in each row for the developed markets versus the emerging markets does not show the clear results of Table 3. The emerging markets are often but not always, stochastically dominant. To summarize Table 4, FHT and FHT2 do a good job of capturing percent effective spread with Roll, LOT Mixed, Zeros, and LOT Y close behind. FHT does a good job of capturing percent quoted spread and percent realized spread with FHT2 close behind. Table 5 reports the average root mean squared error (RMSE) between each percent cost proxy and percent cost benchmarks based on individual firms. The root mean squared error is calculated every month for a given exchange and then averaged over all sample months. A solid box identifies the lowest RMSE in the row and a dashed box indicates RMSEs that are tatistically 22

24 indistinguishable from the lowest RMSE in the row. 9 Boldfaced RMSEs are statistically significant at the 5% level. 0 Panel A reports each percent cost proxy compared to percent effective spread in developed countries and Panel B compares to percent effective spread in emerging countries. Strikingly, very few of the RMSEs are statistically significant (boldfaced). In other words, most proxies lack explanatory power regarding the level of percent effective spread. FHT is statistically significant on 22 exchanges, followed by FHT2 on 9 exchanges, and then the next best is Roll on 5 exchanges. FHT has the lowest average RMSE on 26 exchanges. For average developed, FHT was the best with an average RMSE of , which was much lower than any other proxy. For average emerging, FHT was the best with an average RMSE of , which was much lower than any other proxy. Panel C reports each percent cost proxy is compared to percent quoted spread. FHT was the best for both average developed and average emerging. In online Appendix Table 5, FHT was the best on 26 exchanges. Panel D reports each percent cost proxy is compared to percent realized spread. Again, FHT was the best for both average developed and average emerging. In online Appendix Table 6, FHT was the best on 29 exchanges. Summarizing Table 5, FHT is the only proxy that is regularly capturing the level of percent effective spread, percent quoted spread, and percent realized spread. Compared to other proxies, it is very dominant. 9 In Tables 5-6, 9, 2, and 3, we test whether RMSEs are statistically different from each other using a paired t-test. 0 In Tables 5-6, 9, 2, and 3, we test whether RMSEs are statistically significant using the U-statistic developed by Theil (966). Here, if U 2 =, then the proxy has zero predictive power (i.e., it is no better at predicting the benchmark than the sample mean). If U 2 = 0, then the proxy perfectly predicts the benchmark. We test if U 2 is significantly less than based on an F distribution where the number of degrees of freedom for both the numerator and the denominator is the sample size. 23

25 To examine the robustness of our results, Table 6 breaks out the comparison of percent cost proxies to percent effective spread by country type (developed vs. emerging), turnover, and size. Panel A reports the average cross-sectional correlation for stocks in developing countries broken out into turnover subsamples and Panel B reports the same for stocks in emerging countries broken out by turnover. In both panels, average cross-sectional correlations are generally higher in high turnover stocks and lower in low turnover stocks. FHT is the best in 7 of the 0 subsamples with correlations in the 0.50 s and 0.60 s in most cases. Panel C reports the average RMSE for stocks in developing countries broken out into size subsamples and Panel D reports the same for stocks in emerging countries broken out by size. In both panels, very few RMSEs are statistically significant. Average RMSEs are generally lower for large stocks and higher for small stocks. FHT is the best in 7 of the 0 subsamples. Effective Tick is best in 4 subsamples. To summarize Table 6, FHT dominates compared to other proxies. Its correlation with percent effective spread is strong and robust by turnover and size subsamples, but its explanatory power in capturing the level of percent effective spread is quite modest. To summarize Tables 3 6, the new measure FHT is the best and most consistent proxy for percent effective spread, percent quoted spread, and percent realized spread. FHT ties for the best in Table 3, leads Table 4, heavily dominates Table 5, and strongly leads Table 6. FHT produces high correlations and low RMSEs on nearly all exchanges and in nearly all subsamples. Its explanatory power in capturing the level of percent effective spread is modest, but is far greater than any other proxy. The turnover of stock j in month i is defined as the share volume of stock j in month i divided by the number of shares outstanding of stock j in month i. Size is market capitalization. 24

26 Online Appendix Table 7 contains additional results based on the same U.S. data used by Goyenko, Holden, and Trzcinka (2009) and by Holden (2009). Their sample spans inclusive. It consists of 400 randomly selected stocks with annual replacement of stocks that don t survive resulting in 62,00 firm-months. The online appendix table compares the proxies in this paper plus the Holden proxy from Holden (2009) and the Gibbs proxy from Hasbrook (2004). 2 FHT has the highest correlation at for the pooled time-series, cross-sectional correlations between the proxies and percent effective spread computed from TAQ. Extended Roll has the highest correlation at for the time-series correlations based on equallyweighted portfolios. FHT is not far behind at , FHT is the lowest average RMSE at This excellent showing for FHT means that researchers can be confident of the performance of FHT on U.S. data, as well as non-u.s. data. An interesting aside is to compare the Zeros vs. the LOT proxies vs. FHT. All of these proxies are fundamentally driven by the frequency of zero returns. Prior research 3 has established and this paper confirms that LOT Mixed and LOT Y-split have greater explanatory power for effective spread than Zeros itself. But the source of the LOT proxy advantage has been hard to pin down, since the LOT proxies do multiple things as the same time. Specifically, the LOT proxies have three likelihood function regions, impose the limited dependent variable functional form, and use the market portfolio as part of the estimation. We find that FHT has greater explanatory power for effective spread than either the LOT proxies or Zeros. Since the FHT proxy focuses on the zero return region of the three LOT regions, it is clear that the zero return region is the key source of extra explanatory power. Since FHT throws away the other two 2 We do not compare the Holden or Gibbs proxies in this paper because both proxies are very numerically intensive, which would make them infeasible to compute on a scale as large as we are analyzing. 3 Specifically, Lesmond, Ogden, and Trzcinka (999), Lesmond (2005), Goyenko, Holden, and Trzcinka (2009), and Holden (2009). 25

27 regions, drops the limited dependent variable functional form, and drops the market portfolio, it is clear that none of these items were essential to the success of the LOT proxies. Indeed, these items most likely added noise, since FHT gains extra explanatory power when they are dropped. The use of market returns in FHT2 occasionally improves performance on a specific exchange over FHT. It is possible that a different market index may improve the performance of FHT2. It is unlikely that a different likelihood function will improve the performance of LOT Mixed or LOT-Y split. Finally, zeros are clearly dominated but are still a significant correlate with effective spreads. It is not surprising that the many studies using them find significant results. The findings of Tables 3 6 suggest that the results will be stronger with FHT The Percent Price Impact Results Tables 7 9 report nine percent cost proxies and three cost per volume proxies (Amihud, Pastor and Stambaugh, and Amivest) 4 compared with the percent cost benchmark: percent price impact. Tables 7, 8, and 9 each report on one of three performance dimensions. Table 7 reports the average cross-sectional correlation between each proxy and percent price impact based on individual firms. Panel A reports for developed countries and Panel B for emerging countries. In both panels, the best correlations are typically in the 0.30 s, 0.40 s, or 0.50 s. Portugal (LS) is the only exchange where the best correlation is not significantly different from zero. FHT is the best on 9 exchanges and LOT Mixed is the best on 5 exchanges. For Average Developed, the best correlation is by FHT. For Average Emerging, the best correlation is 0.39 by FHT. To summarize Table 7, FHT and LOT Mixed do moderately-well at capturing percent price impact. 4 We tested all cost per volume proxies compared with percent price impact (see online Appendix Tables 8, 9, and 20). We found that they performed much worse than the percent cost proxies. Indeed, the best correlations using percent cost proxies are typically twice as large as the best correlation using cost per volume proxies. 26

28 Table 8 reports the time-series correlation between each portfolio proxy and the portfolio percent price impact. Panel A reports for developed countries and Panel B for emerging countries. In both panels, the best correlations are typically in the 0.50 s or 0.60 s. Several exchanges have very high correlations, such as Japan (T) at 0.864, Sweden (ST) at 0.837, and the U.K. (L) at FHT has the best correlation on exchanges and Roll has the best on 7 exchanges. For Average Developed, the best correlation is by Effective Tick with FHT, FHT2, and Zeros in the leadership group. For Average Emerging, the best correlation is 0.50 by FHT with Effective Tick, FHT2, and Zeros in the leadership group. To summarize Table 8, FHT leads, but Roll, Effective Tick, FHT2, and Zeros also do well at capturing percent price impact. Table 9 reports the average RMSE between each proxy and percent price impact based on individual firms. Panel A reports for developed countries and Panel B for emerging countries. None of the RMSEs are statistically significant (i.e., none are bold-faced). Therefore, none of the price impact proxies does a good job capturing the level of percent price impact. To summarize Tables 7 9, we find FHT is the best proxy for percent price impact. However, it is only moderately-well correlated with percent price impact and does not capture the correct scale The Lambda Results Tables 0 3 report monthly cost per volume proxies compared with the benchmark lambda (the slope of the price function). Tables 0,, and 2 each report on one of three performance dimensions and Table 3 reports breakouts by size and turnover quintiles. Table 0 reports the average cross-sectional correlation between the cost per volume proxy and lambda based on individual firms. Panel A reports for developed countries and Panel B for emerging countries. In both panels, the best correlations are relatively low typically in the 27

29 0.30 s or 0.40 s. LOT Mixed Impact is the best on 4 exchanges and Zeros Impact is the best on 2 exchanges. For Average Developed, the best correlation is by LOT Mixed Impact. For Average Emerging, the best correlation is also by LOT Mixed Impact. To summarize Table 0, LOT Mixed Impact does moderately-well at capturing lambda. Similar to Table 8, we form equally-weighted portfolios across all stocks on a given exchange for month i. Then, a portfolio cost per volume proxy in month i is computed by taking the average of that cost per volume proxy over all stocks on a given exchange in month i. Table reports the time-series correlation between each portfolio cost per volume proxy and the portfolio lambda benchmark. Panel A reports for developed countries and Panel B for emerging countries. In both panels, the best correlations are typically in the 0.30 s or 0.40 s. Several exchanges have very high correlations, such as the UK (L) at 0.839, China Shanghai (SS) at 0.802, and China Shenzhen (SZ) at Zeros2 Impact has the best results on 9 exchanges and Amihud has the best results on 8 exchanges. For Average Developed, the best correlation is 0.39 by Zeros2 Impact. For Average Emerging, the best correlation is 0.30 by FHT Impact. To summarize Table, Zeros2 Impact does moderately-well at capturing lambda. Table 2 reports the average RMSE between each cost per volume proxy and the Lambda benchmark based on individual firms. As before, the RMSE is calculated every month for a given exchange and then averaged over all sample months. Panel A reports for developed countries and Panel B for emerging countries. Nearly all of the RMSEs are statistically insignificant. Only two proxies are significant on one exchange. To summarize Table 2, nothing captures the level of lambda. Table 3 checks the robustness of our results by breaking out the comparison of cost per volume proxies to lambda by country type, turnover, and size. Panel A reports the average cross- 28