When do low-frequency measures really measure transaction costs?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. When do low-frequency measures really measure transaction costs? Mohammad R. Jahan-Parvar and Filip Zikes 2019-051 Please cite this paper as: Jahan-Parvar, Mohammad R., and Filip Zikes (2019). “When do low-frequency measures really measure transaction costs?,” Finance and Economics Discussion Series 2019-051. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.051. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

When do low-frequency measures really measure transaction costs? Mohammad R. Jahan-Parvar Filip Zikes∗ June 28, 2019 Abstract Wecomparepopularmeasuresoftransactioncostsbasedondailydatawiththeirhighfrequency data-based counterparts. We find that for U.S. equities and major foreign exchange rates, (i) the measures based on daily data are highly upward biased and imprecise; (ii) the bias is a function of volatility; and (iii) it is primarily volatility that drives the dynamics of these liquidity proxies both in the cross section as well as over time. We corroborate our results in carefully designed simulations and show that such distortions arise when the true transaction costs are small relative to volatility. Many financial assets exhibit this property, not only in the last two decades, but also in the previous century. We document that using low-frequency measures as liquidity proxies instandardassetpricingtestsmayproducesizablebiasesandspuriousinferencesabout the pricing of aggregate volatility or liquidity risk. ∗Jahan-Parvar, mohammad.jahan-parvar@frb.gov, Division of International Finance, Federal Reserve Board; Zikes, filip.zikes@frb.gov, Division of Financial Stability, Federal Reserve Board. We thank Karim Abadir, Alain Chaboud, Anusha Chari, Sergio Correia, Dobrislav Dobrev, Michael Gordy, Luca Guerrieri,PeterReinhardHansen,ErikHjalmarsson,AytekMalkhozov,ValeryPolkovnichenko,DavidRappoport, Marius Rodrigues, Pawel Szerszen, Clara Vega, Lingling Zheng, Molin Zhong, and seminar participants at the University of North Carolina at Chapel Hill, Federal Reserve Board, Market Microstructure: Confronting Many Viewpoints #5, and Econometric Conference on Big Data and AI in Honor of Ronald Gallantforusefulcommentsanddiscussions. WearegratefultoMarkMorleyandCharlesHorganforexcellent research assistance and to William McClennan for helping us run our programs on a computer cluster. The views expressed in this paper are those of the authors and should not be interpreted as representing the views of the Federal Reserve Board or any other person associated with the Federal Reserve System. 1

1 Introduction Transaction costs are a key input in many financial decisions. The literature on measuring transaction costs and market liquidity more generally, has evolved into two distinct strands. With the increasing availability of high-frequency data since the 1990s, one strand emphasizes the importance of using transactions data to calculate trading costs. Chordia, Sarkar, and Subrahmanyam (2005) who study stock and bond market liquidity and Mancini, Ranaldo, and Wrampelmeyer (2013) who examine foreign exchange market liquidity are examples of papers that use high-frequency measures. The other strand argues that the availability of long histories of daily data—dating back to early 1900s—and the significant cost associated with acquiring and processing high-frequency data makes it desirabletoestimatetransactioncostsfromlow-frequency, dailydata. Roll(1984), Hasbrouck (2009), Corwin and Schultz (2012), and Abdi and Ranaldo (2017) propose low-frequency measuresthathavebecomepopularintheliterature. Asthelatterareessentiallystatistical proxies for the former, we would expect them to deliver similar results on average. In this paper, we find that, in general, the low-frequency measures of transaction costs are not good estimators for the high-frequency-based trading costs. We show that for U.S. equities and major foreign exchange rates, (i) several widely used measures suffer from a large bias that cannot be diminished by using more data; (ii) more importantly, the bias is a function of volatility, and hence it induces a positive correlation between the lowfrequency measures and volatility above and beyond what is implied by the correlation between volatility and the true transaction costs; and (iii) the volatility-induced bias has important implications for empirical asset pricing. Put simply, volatility explains the wedge between the low-frequency measures and their high-frequency counterparts. Specifically, for the S&P 1500 stocks between 2003 and 2017, we show that the average effective spread computed from the Trade and Quote (TAQ) data is around 12 basis points, while the monthly low-frequency measures–measures calculated from one month of daily data–yield spreads between 26 and 168 basis points on average. The five major foreign exchange rates trade in the Electronic Broking Services (EBS) market with an effective spread of less than one basis point on average between 2008 and 2015, but the monthly low-frequency measures deliver estimates between 13 and 48 basis points on average. When 2

we regress the monthly low-frequency measures on the true spread and realized volatility, we find that the coefficient on realized volatility is highly statistically significant. Moreover, relative to a simple regression of the low-frequency measures on the true spread alone, adding realized volatility significantly increases the R2, often more than doubling it. For the foreign exchange rates, adding realized volatility to the regression even renders the coefficient on the true spread statistically insignificant for some low-frequency measures. How does the volatility-induced bias arise? When recovering the transaction costs from daily data, one is attempting to estimate a small, positive-valued object from noisy data. This requires methods that guarantee non-negativity of the transaction costs estimates, such as censoring, truncation, or an appropriate prior distribution if one chooses a Bayesian approach. It is well-known from the study of limited dependent variable models that truncatingorcensoringarandomvariableatzerowillincreasesthemeanandthemeanbecomes a function of volatility; see Greene (2017). We show by carefully designed simulations that the same problem arises in the construction of the low-frequency measures. The magnitude of the problem depends on the relative size of the true transaction costs and the volatility of the daily asset price. The lower (higher) the transaction costs relative to volatility, the higher (lower) the bias and the stronger (weaker) the dependence on volatility are. Since many financial markets experienced a significant decline in transaction costs, both in absolute terms and relative to the volatility of asset returns, over the past two decades, theissuesweuncoverhavebecomemoreacute. Buttheyhavealsoimplicationsforthemost liquidfinancialassetsinthepreviouscentury. Jones(2002)documentsthattheproportional effective spread was only around 60 basis points on average for the Dow Jones Industrial Average 30 stocks between 1928 and 2000. Thus, there has always been a nontrivial subset of U.S. stocks for which the issues we document mattered. Bessembinder (1994) reports that the average bid-ask spread in the interdealer foreign exchange market was between 5 and 8 basis points on average for major currencies between 1979 and 1992. Although such values are much larger than in the current century, our results indicate that they are still small enough for the volatility to have a sizable effect. We use two well-known asset pricing models to illustrate how using transaction costs measuresthatsufferfromthepresenceofanon-negligible,volatility-drivenbiasmayproduce 3

misleadinginferenceandspuriousresults. First,wedocumenttheconsequencesofusinglowfrequencyliquiditymeasuresasliquidityproxiesintheliquidity-adjustedCAPMofAcharya and Pedersen (2005). Our simulation results show that both the liquidity betas and the market price of risk suffer from sizable biases; in empirically relevant calibrations, popular low-frequency measures yield negative market price of risk estimates even though the true value in the simulation is positive. Second, we study the pricing of aggregate volatility and liquidity risks in an unconditional asset pricing model motivated by Ang, Hodrick, Xing, and Zhang (2006). We demonstrate through simulations that using low-frequency measures as a proxy for aggregate liquidity may yield spurious pricing of liquidity risk due to the correlation of these low-frequency measures with volatility. The estimated price of liquidity risk can be as low as −10% per annum or as high as 2.5% per annum on average, depending onwhichlow-frequencymeasureisused, eventhoughthetruepriceofriskinthesimulation is zero. Accurate measures of transaction are essential in many other areas of empirical finance. Our findings about the upward bias of the low-frequency measures raise questions about their suitability for optimal portfolio choice problems with transaction costs, where the so-called no-trade region is a function of the level of transaction costs (Constantinides, 1986), and for evaluation of trading strategies and asset pricing anomalies, where small changes in transaction costs have a large impact on performance (Novy-Marx and Velikov, 2015, Chen and Velikov, 2018, and Patton and Weller, 2018). The dependence of the lowfrequency measures on volatility limits their use in studies of the commonality in liquidity, where liquidity proxies are used to measure co-movements in liquidity across assets and markets (Chordia, Roll, and Subrahmanyam, 2000, Hasbrouck and Seppi, 2001, Korajczyk and Sadka, 2008, Karolyi, Lee, and Van Dijk, 2012, Mancini, Ranaldo, and Wrampelmeyer, 2013). In all of these applications, using imperfect proxies for transaction costs can lead to suboptimal financial decisions. The literature on low-frequency measures of liquidity is fairly large. A number of influential studies introduce measures that build upon the seminal work of Roll (1984) and his proposed measure of transaction costs. Among these studies, we examine the measures introducedbyHasbrouck(2009), CorwinandSchultz(2012), andAbdiandRanaldo(2017). 4

Alternative measures include those based on the frequency of zero returns (Lesmond, Ogden, and Trzcinka, 1999), effective tick, which is based on the concept of price clustering (Holden, 2009 and Goyenko, Holden, and Trzcinka, 2009), or the simplified version of the Lesmond, Ogden, and Trzcinka (1999) measure introduced by Fong, Holden, and Trzcinka (2017). We do not study these measures here for two reasons. First, Corwin and Schultz (2012) and Abdi and Ranaldo (2017) show that their proposed measures outperform these alternatives, and second, these measures are less suitable for our empirical exercise. For example, we virtually never observe a zero daily return in our data. Additionally, previous literature has examined the performance of a variety of lowfrequencymeasuresagainsthigh-frequencybenchmarks. Hasbrouck(2009),Goyenko,Holden, and Trzcinka (2009), Corwin and Schultz (2012), Abdi and Ranaldo (2017) provide evidence from the U.S. stock market; Fong, Holden, and Trzcinka (2017) extend the analysis to global equity markets; Karnaukh, Ranaldo, and S¨oderlind (2015) study the foreign exchange market; Marshall, Nguyen, and Visaltanachoti (2011) study the commodity market; and Schestag, Schuster, and Uhrig-Homburg (2016) examine the U.S. corporate bond market. All of these studies conclude that the low-frequency measures are good proxies of true transactioncost, astheycloselycorrelatewiththeirhigh-frequencybenchmarks. Sowhydo wereachadifferentconclusion? Wearguethatwhilethesemeasuresarecorrelatedwiththe true spreads as the literature finds, this correlation is to a large extent driven by volatility. Since the realized volatility and the true spread are positively correlated as predicted by market microstructure theory, omitting the realized volatility from the regression inflates themagnitudeandstatisticalsignificanceofthecorrelationbetweenthelow-frequencymeasures and the true spread. To the best of our knowledge, this issue has not been recognized by the existing literature. 1 We do not claim that low-frequency measures of transaction costs are not generally useful. They are certainly useful for assets with relatively higher transaction costs and lower volatility, such as corporate bonds. However, we argue that caution is needed when applyingthesemeasures–whetherintoday’selectronicmarketsortodatafromtheprevious 1OurforeignexchangeresultsdifferfromthoseofKarnaukh,Ranaldo,andSo¨derlind(2015). Wediscuss the sources of these differences in Section 4.2.2. 5

century–unless one is reasonably confident that the transaction costs are large relative to the daily volatility. It is worth mentioning that Hasbrouck (2009) already noted that his measurerequiresasignificantamountofdataandshouldnotbeexpectedtoperformwellfor highly liquid assets. Contrary to his recommendation, however, the common practice in the literature is to construct the measures from only one month worth of data and apply them to the entire cross section of U.S. stocks and foreign exchange rates. In this study, we show that for less liquid assets, the consistent version of Corwin and Schultz (2012) performs better than other low-frequency liquidity measures, provided that one uses a fairly long window of data (at least one year) for constructing the measure. Otherwise, high-frequency measures are preferable. Our analysis bears considerable similarity to several important debates in finance and economics literature. First, recovering a small positive object from noisy data is notoriously difficult. Examples include estimating the expected stock return from low-frequency data (Merton, 1980), recovering a small predictable component of consumption growth from consumption data (Bansal and Yaron, 2004), or expected dividend growth (Lettau and Ludvigson, 2005). Second, the necessity of using high-frequency data for measuring transactioncostsresemblethediscussionintheearly2000sabouttheuseofparametric,lowfrequencydatabasedmeasuresofvolatilityagainsthigh-frequencynonparametricvolatility, the so-called realized measures (Andersen, Bollerslev, Diebold, and Labys, 2003, Andersen, Bollerslev, and Diebold, 2010, Bollerslev, Hood, Huss, and Pedersen, 2018). Spiegel and Zhang (2013) critically assess the convex relation between flow and past performance of mutual funds. They show that several studies that report a convex flow response function suffer from misspecification in their empirical models. Finally, our exercise is also similar in spirit to the paper by Farre-Mensa and Ljungqvist (2016), who study the performance of various measures of financial constraints. They find that instead of measuring financial constraints, these measures in reality capture firms’ growth and financing policies; our results show that popular low-frequency measures of transaction costs are largely driven by volatility. The rest of the paper is organized as follows. In Section 2, we describe the four lowfrequency measures of transaction costs, which have become popular in the literature. In 6

Section 3, we study the properties of these measures in simulations and document their bias and dependence on volatility. Section 4 presents our empirical results for U.S. stocks and major foreign exchange rates, and in Section 5 we discuss the implications of our results for empirical finance. Section 6 concludes. 2 Low-frequency effective spread measures The four low-frequency measures of transaction cost that we discuss in this paper are all derived within the well-known model of Roll (1984). In this section, we first present the model and then briefly describe the measures. For more details on the model, and its many generalizations, see Foucault, Pagano, and R¨oell (2013). 2.1 Notation and framework We assume there are T days with n transactions each. For every i = 1,...,n and t = 0,...,T −1, the logarithmic (log) transaction price p and the log efficient price m follow s p = m + q , (1) t+i/n t+i/n 2 t+i/n m = m +(cid:15) , (2) t+i/n t+(i−1)/n t+i/n where (cid:15), the efficient price innovation, is independently and identically distributed (iid) with variance σ2/n; q, the buy/sell indicator, is iid and q = ±1 with equal probability; t+i/n (cid:15) and q are independent at all leads and lags; and m = 0 without loss of generality. The 0 sequence of daily closing log prices is thus given by {p ,p ,...,p }, the sequence of daily 1 2 T high log prices by {h ,h ,...,h }, where h = max p , and the sequence of daily 1 2 T t 1≤i≤n t−1+i/n low log prices by {l ,...,l }, where l = min p . The sequence of daily returns is 1 T t 1≤i≤n t−1+i/n given by {r ,r ,...,r }, where r = p −p . It follows from the assumptions above that 2 3 T t t t−1 the daily efficient price innovations, m −m , are iid with zero mean and variance σ2. t t−1 The parameter of interest is s, the proportional effective spread. 7

2.2 Roll (1984) Roll(1984)exploitsthenegativeserialcorrelationinreturns,inducedbythebid-askbounce. In our framework, the first-order serial covariance is Cov(r ,r ) = −s2 . For a sample of t t−1 4 T days, the Roll measure is obtained by replacing the true covariance with its sample counterpart: (cid:118) (cid:117) (cid:40) T (cid:41) (cid:117) 1 (cid:88) R = 2(cid:116)max − r r ,0 , (3) t t−1 T −2 t=3 where the censoring at zero ensures nonnegative estimates, as the sample first-order serial covariance is not generally guaranteed to be negative. 2.3 Hasbrouck (2009) Hasbrouck (2009) proposes to estimate the Roll model by Bayesian methods. He develops a Gibbs sampler to approximate the posterior distribution of s, given a sample of T daily closing transaction prices. The prior on s is half-normal, s ∼ N (0,σ2), and the prior on σ + s is inverse Gamma, σ ∼ IG(α,β). Following Hasbrouck (2009), we set α = β = 1e−12 and σ2 = 0.052, but we also consider a tighter prior for s, namely σ2 = 0.012. We denote the s s resulting estimators by H and H , respectively. L T 2.4 Corwin and Schultz (2012) Corwin and Schultz, henceforth CS, exploit the information contained in daily high and low prices. They observe that “the sum of the price ranges over 2 consecutive single days reflects 2 days volatility and twice the spread, while the price range over one 2-day period reflects 2 days volatility and one spread”. This yields two equations and two unknowns (s and σ), which is solved analytically. The two-day CS estimator of the spread is given by 2(eαt −1) S = , (4) t 1+eαt √ √ 2β − β (cid:114) γ t t t α = √ − √ , (5) t 3−2 2 3−2 2 β = (h −l )2+(h −l )2, (6) t t+1 t+1 t t γ = (max{h ,h }−min{l ,l })2. (7) t t+1 t t+1 t 8

For a sample of T days, the Corwin and Schultz (2012) measures are obtained by averaging the 2-day estimates: (cid:40) T−1 (cid:41) 1 (cid:88) CS = max S ,0 , (8) M t T −1 t=1 T−1 1 (cid:88) CS = max{S ,0}, (9) D t T −1 t=1 T−1 1 (cid:88) CS = S I , (10) P (cid:80)T−1I t {St≥0} t=1 {St≥0} t=1 where I is an indicator function that takes the value of one if S ≥ 0. The first measure, St≥0 t CS , is the censored version of CS and thus guaranteed to be positive definite. The second M measure, CS , is constructed by censoring the 2-day spreads at zero before averaging. This D version is recommended by Corwin and Schultz for U.S. equities and is one of the most widely used measures in the literature. Finally, the last measure, CS , is obtained by only P averaging over the non-negative 2-day spreads. Karnaukh, Ranaldo, and S¨oderlind (2015) use this measure for spot foreign exchange rates, because it is more closely correlated with the true spread than CS . D The CS measures are derived under the assumption of continuous trading, that is, the overnight return between two consecutive trading days is assumed to be zero. When this is notthecase,suchasinmostequitymarkets,CSproposeasimpleadjustmentthatmitigates the effect of non-zero overnight returns (Corwin and Schultz, 2012, p. 726). We employ this correction in our empirical applications. 2.5 Abdi and Ranaldo (2017) Abdi and Ranaldo (2017), henceforth AR, combine the ideas of Roll and CS. They define the daily mid-range by η = (h +l )/2, the average of the daily high and low log prices, t t t and observe that the statistic δ = 4(p −η )(p −η ), (11) t t t t t+1 9

satisfies E(δ ) = s2, Thus, using a sample of T days, they propose to estimate s by t (cid:118) (cid:117) (cid:40) T−1 (cid:41) (cid:117) 1 (cid:88) AR = (cid:116)max δ ,0 , (12) M t T −1 t=1 T−1 1 (cid:88)(cid:112) AR = max{δ ,0}, (13) D t T −1 t=1 T−1 1 (cid:88)(cid:112) AR = δ I , (14) P (cid:80)T−1I t {δt≥0} t=1 {δt≥0} t=1 where, similar to CS, AR employ censoring or truncation to achieve nonnegative estimates. Unlike the CS measures, the AR measures are robust to the presence of overnight returns and so no adjustment is required when applying these measures to non-24-hour markets. 3 Statistical properties and the role of volatility Havingdescribedthelow-frequencymeasures, wenowdocumenttheirstatisticalproperties. First, we show by simulation that the low-frequency measures are significantly upward biasedwhenthetruespreadissmall. Second, westudythesourceofthebiasanddocument its dependence on the volatility of the efficient price. Finally, we study the implications of this dependence for the dynamics of the low-frequency measures when both volatility and spreads vary over time. 3.1 Bias and precision ForcomparisonwithCSandAR,weadopttheir“near-ideal”simulationdesign, exceptthat we consider smaller values of the effective spread, ranging from 5 to 300 basis points, as these values are more consistent with the transaction costs observed in our data and sample period. Specifically, we assume there are 390 transactions per day, the daily volatility of the efficient price innovation equals 3%, and the sample size is either 21 trading days (approximately one month) or 251 trading days (approximately one year). All simulations are based on 10,000 replications. Table1reportsthesimulationresults. Wefindthatallmeasuresaresignificantlyupward biased when the spread is small. For example, when the true spread equals 10 bps, the bias 10

ranges from 35 bps for the CS measures to 236 bps for the AR measure, or 250% and M P 2,260%ofthetruespread,respectively. Butforsomemeasures,thebiasremainssubstantial, even for moderate values of s. When the spread is large, the bias is reasonable, and this is known from previous studies as well. The measures based on censoring and truncation before averaging (the “D” and “P” measures) exhibit a much higher bias than measures obtained by censoring after averaging (the “M” measures). Hasbrouck’s measures are also highly biased when the spread is small or moderate, and the bias is highly dependent on the prior. In terms of precision measured by the root mean square error (RMSE), CS generally M performs the best, but this comes at a cost: a large fraction of the estimates is equal to zero. For example, for s = 10 bps, the measure equals zero 38% of the time. The AR M producesahigherRMSEandmanymorezeros, evenformoderatevaluesofs. The“D”and “P” measures perform better in terms of standard deviation, but the bias dominates and producesinferiorRMSEtothoseofthe“M”measures. Whenthesamplesizeincreasesfrom T = 21 to T = 251, the bias and RMSE of the consistent estimators—the “M” estimators and the Hasbrouck measures—improve significantly, as expected, while those of the “D” and “P” measures do not. 3.2 Source of bias What drives the bias when the true effective spread is small? Recall that to ensure nonnegativity, the Roll, CS, and AR measures are all based on some form of censoring or truncation either before or after averaging. These operations induce a positive bias and dependence on the volatility of the efficient price σ as we now demonstrate. To develop some intuition, consider a simple case where x is iid normal with mean µ, t µ > 0, and variance ω2. Of course, the summands (r r , S , δ ) in the various measures t t−1 t t are neither normal nor iid, but the normal iid case provides informative predictions for the actual behavior of the effective spread measures. It is well-known that in the case of iid normal x , t (cid:32) (cid:40) T (cid:41)(cid:33) (cid:32)√ (cid:33) (cid:32)√ (cid:33) 1 (cid:88) Tµ Tµ ω E max x ,0 = Φ µ+φ √ , (15) t T ω ω T t=1 11

(cid:32) T (cid:33) 1 (cid:88) (cid:16)µ(cid:17) (cid:16)µ(cid:17) E max{x ,0} = Φ µ+φ ω, (16) t T ω ω t=1 (cid:32) T (cid:33) 1 (cid:88) (cid:104) (cid:16)µ(cid:17)(cid:105)−1(cid:104) (cid:16)µ(cid:17) (cid:16)µ(cid:17) (cid:105) E x I = Φ Φ µ+φ ω, . (17) (cid:80)T I t {xt≥0} ω ω ω t=1 {xt≥0} t=1 First, these results show that all three forms of imposing non-negativity introduce a bias thatdependsonbothsandω. Second, onlycensoringafter averagingcandeliverconsistent estimatesoftheeffectivespread, equation(15). Recallthatanestimatorθˆ isconsistentfor T the true parameter θ when for T → ∞, θˆ →− p θ . Censoring or truncation before averaging 0 T 0 produces a bias that does not vanish as the sample size increases, and hence measures introduced in equations (16) and (17) are not consistent. Finally, truncation delivers a higher bias and greater dependence on volatility than censoring. Henceforth, we refer to “M” measures (CS , AR , and Roll measures) as “consistent” measures, and “D” and M M “P” measures (CS ,CS ,AR , and AR ) as “inconsistent”. D P D P Analogous analytical results for the four low-frequency measures we consider are not available in closed form, so we approximate the expectations by simulation over a grid for s and σ, where both parameters vary between 10 and 500 basis points. The simulation design is otherwise identical to that in the previous subsection. The results are summarized in Figures 1, 2, and 3. In each of these figures, the left panel plots the expectations as a function of volatility for a given spread, while the right panel plots the expectations as a function of the spread for a given level of volatility. Starting with the Roll measure shown in the top panel of Figure 1, we find that when the spread is large and/or the volatility is small, the Roll measure is fairly insensitive to volatility. However, for small values of s, the dependence on volatility becomes apparent. Forexample, whenthespreadequals50bps, themeasurevariesalmostlinearlywithvolatility even when volatility is as low as 1%. At the same time, at this level of volatility, the sensitivity to changes in the spread is significantly less than one (right panel); increasing volatility further makes this sensitivity even smaller. Turning to the CS measures shown in Figure 2, censoring after averaging (CS ) pro- M duces a positive bias that increases with σ, especially for small s, but this bias disappears as the sample size increases (not shown). Censoring before averaging (CS ) delivers a D 12

more pronounced dependence on volatility, regardless of the level of the spread. Truncation before censoring (CS ) makes the dependence on volatility even stronger. Moreover, when P the spread is small, both measures are essentially a linear function of volatility, even for low σ. Also, as shown in the right panel, the sensitivity to s declines as volatility increases. Increasing the sample size does not resolve these issues (not shown). Finally, the AR measures are shown in Figure 3. We notice two differences relative to CS. First, the dependence on volatility is less pronounced for large values of s. This comes at a cost, however, and that is a lower sensitivity to s when s is small and volatility is high. Similar to CS, when the spread is small or when the volatility is high, the dependence of the AR measures on volatility is essentially linear. In the case of the Hasbrouck measures, non-negativity is achieved by choosing an appropriate prior. It is difficult to make predictions about how exactly the choice of the prior affects the bias and dependence on volatility, if any. But a simulation can shed some light on this issue. In Figure 1, we plot the expected values of the Hasbrouck measures approximated by simulation as we did for the other measures. We consider two different priors for s, one where we set σ2 = 0.052 as in Hasbrouck’s original paper (middle panel), and s one where we consider a tighter prior by setting σ2 = 0.012 (bottom panel). In the former s case, we find a behavior that is qualitatively similar to that of the Roll measure, although the dependence on volatility is somewhat stronger. In the latter case, the dependence on volatility varies with the size of the spread; for small values of s, the dependence is positive, while for large values of s it is negative. As volatility increases, the posterior increasingly resembles the prior and, regardless of the true spread, the measure produces very similar estimates. As a result, the sensitivity of the measure to changes in the spread becomes very small (right panel). 3.3 Correlation with true spread and volatility The key finding from the simulations so far is that when the effective spread is small, the variationinthelow-frequencymeasurescanbedrivenbyvolatility. Inpractice,bothspreads andvolatilityvaryovertimeandinthecrosssection, aspredictedbymicrostructuretheory, and they tend to be contemporaneously correlated (see Foucault, Pagano, and R¨oell, 2013, 13

for a textbook treatment). In the final set of simulations, we therefore explore the implications of time-varying and correlated spread and volatility for the dynamics of the effective spread measures. Specifically, we study the correlation of the spread estimates with the true spreads and volatility by calculating simple pairwise correlations. More importantly, though, we run OLS regressions of the spread estimates on the true spread and volatility: sˆ = α+β s +β σ +u . (18) t s t σ t t Ifthelow-frequencymeasures(sˆ)trulycapturetheeffectivespread,theseregressionswould t exhibit a spread coefficient β close to unity, a volatility coefficient β close to zero, and a s σ high R2. If, on the other hand, volatility plays an important role in the dynamics of these measures, β will be different from zero, and the R2 in the bivariate regression in equation σ (18) will be substantially higher than that in a univariate regression of sˆ on s alone. t t Since we cannot calculate the population coefficients in equation (18) in closed form, we approximatethembysimulation. Asbefore,wesimulatesamplesofsizeT = 21(onemonth) or T = 252 (one year) days with n = 390 intraday returns each. All simulations are based on 10,000 replications. We keep the spread and volatility fixed within each sample (month or year). However, we allow the spread and volatility to vary randomly across samples (months or years). Both the spread and volatility are assumed to be independently lognormally distributed, either uncorrelated, or contemporaneously correlated with ρ = 0.5.2 We set the mean daily volatility of the efficient price equal to 2% and the volatility of volatility equal to 2%. A 2% average daily volatility is approximately what we observe for the S&P 1500 stocks during our sampling period. We consider three scenarios for the effective spread. First, we consider an effective spread with a mean of 10bps and spread volatility of 10 bps, i.e., the implied ratio of expected spread to expected daily volatility is 5%, which is close to what we observe for the large- and mid-cap stocks in our sample. Second,weconsideraneffectivespreadwithameanspreadof50bpsandspreadvolatilityof 50bps, sothatthesignal-to-noiseratioequals25%. Thisvalueislargeformodernelectronic 2We also run simulations with ρ=0.75. These results are available in Appendix A. 14

markets, but it is close to what the DJIA 30 stocks experienced in the previous century (see Jones, 2002). Finally, we consider an effective spread of 3% with volatility of 3%, implying a signal-to-noise ratio of 150%. We run this last simulation mainly to show that when the signal-to-noise ratio is high, the consistent low-frequency measures perform as dictated by statistical theory. Table 2 reports the simulation results for the case of no correlation between the spread and volatility. In the left part of Panel A, we show the results for the small, tranquil spread and T = 21 observations (days) used to calculate the low-frequency measures. We find that the correlation with the true spread is very small for all measures, while the correlation with volatility is high, especially for the D and P measures, where the correlation exceeds 0.9. The slope coefficient in a univariate regression on the true spread is well below one and the intercept is well above zero, consistent with the large bias documented in Table 1. The bivariate regressions on the true spread and volatility show that it is almost exclusively volatility that drives the dynamics of the low-frequency measures: the slope coefficient for volatility is well above zero and the R2 increases from essentially zero to values that range from 25% for the consistent, M, measures to over 90% for the P measures. In the right part of Panel A, we repeat the simulation with T = 251 observations. Except for CS , we find M that the performance does not really improve, and in some cases it actually deteriorates: for the inconsistent D and P measures, the correlation with volatility is now near perfect. Increasing the number of observations reduces the variance of these measures and the bias, which is a function of volatility and does not diminish asymptotically, dominates. Panel B of Table 2 reports the simulation results for the medium, moderately volatile spread. There are clear improvements in terms of the correlations with the true spread relative to Panel A, but all measures still exhibit significant correlation with volatility. This is also evident in the bivariate regressions, where, with the exception of CS , the volatility M coefficientisfarfromzeroandtheR2 ismuchhigherthanintheunivariateregressiononthe true spread alone. For example, the most widely used CS measure exhibits an R2 = 0.17 D in the univariate regression and an R2 = 0.89 in the bivariate regression when the number of observations equals 21 days, suggesting that even with a fairly large signal-to-noise ratio of 25%, the dynamics of the measures is mainly determined by volatility and not the true 15

spread. Finally, in Panel C we report the results for the high, volatile spread. We find that with such a high noise ratio of 150%, the measures generally work as expected, exhibiting a high correlation with the true spread and a slope coefficient close to unity in a univariate regression on the true spread alone (except H ). The consistent measures CS and AR T M M work particularly well, even when T = 21. The inconsistent measures, D and P, while much less plagued by volatility than in Panels A and B, still exhibit a nontrivial dependence on volatility as evidenced by a correlation between 0.10 and 0.36, respectively, when T = 21, and this correlation barely changes as T increases to 251. Whilethesimulationresultswithuncorrelatedspreadandvolatilityclearlyillustratethe impact of volatility on the low-frequency measures, in practice, the spread and volatility tend to be positively correlated. To be able to better interpret our empirical findings, we therefore repeat the simulations setting the correlation between the spread and volatility equalto0.5butleavingthesimulationdesignotherwiseunchanged. Theresultsarereported in Table 3. We first note how the contemporaneous correlation between the spread and volatility masks the problem with dependence on volatility when one judges performance by correlation with the true spread. Panel A shows that even if the true spread is very small, the correlation between the low-frequency measures and the true spread tends to be quite high. However, when one runs a univariate regression on the true spread, the issue becomes clearly apparent and manifests itself in a slope coefficient that is much larger than one. Moreover, whenvolatilityisaddedtotheregression, theR2 increasessignificantly. For example, with T = 21, the CS measure exhibits an increase in R2 from 0.26 to 0.88, while D the CS ’s R2 increases from 0.27 to 0.94; the AR and AR measures exhibit a similar P D P behavior. IncreasingT to251doesnotyieldanyimprovementforeithersetofmetrics. This problem does not go away even when we increase signal-to-noise ratio to 25%, as shown in Panel B. Including the volatility in the regression yields a small increase in R2 for the inconsistent D and P measures only with a large signal-to-noise ratio of 150%, reported in Panel C. In summary, the simulation results confirm the fact that the inconsistent low-frequency measures are contaminated by volatility, and it is primarily volatility that drives the dy- 16

namics of these measures for parametrizations that are empirically relevant in electronic markets. To end on a positive note, we find that the CS measure, while being quite M noisy when the true spread is small or when the number of observations is small, does deliver almost unbiased estimates for moderate (and larger) spreads and when the number of observations is large. Unfortunately, the measure is almost never applied in practice, as the authors themselves advise against it on the grounds that it exhibits low correlation with the true spread. Our simulations show that while the D and P measures, which are recommended by both CS and AR, are correlated with the true spread, they are correlated with the true spread for the wrong reason—they load on volatility. 4 Estimating transaction costs in equity and foreign exchange markets 4.1 U.S. equities 4.1.1 Data Our U.S. equity sample consists of the S&P 1500 constituent stocks during the period between October 2003 and December 2017. We focus on the post-decimalization period and the S&P 1500 stocks for several reasons. First, the S&P 1500 index comprises over 90% of U.S. stock market capitalization, and thus our results apply to the vast majority of economicallysignificantU.S.stocks. Second,verysmallstocksdonotnecessarilytradeoften enough (sometimes not even once a day), which necessitates imposing onerous conditions when implementing the low-frequency methods. We do not claim that these stocks do not matter. However, if we insist on incorporating them in our analysis, more robustness checks and alternative treatments for issues related to infrequent trading need to be carefully explored, which we believe is best left for a separate paper. Third, estimating realized volatility, which is a key variable in our empirical analysis, is only possible for sufficiently liquid assets. ThedataonthehistoricalS&P1500andotherindexmembershipcomefromCOMPUS- TAT.Foreachmonthinoursample, weselectallstocksthatwereincludedintheS&P1500 index between the 5th and 25th of the month and obtain daily high, low, close, bid, and 17

ask prices for these stocks from CRSP, matching by stock CUSIP. We use the daily CRSP price data to calculate the low-frequency measures at the stock-month and stock-year level. Ourhigh-frequencyeffectivespreadbenchmarksandvariousrealizedvolatilitymeasures are calculated from the Daily TAQ data accessed via Wharton Research Data Services (http://wrds-web.wharton.upenn.edu/wrds/). WeusetheHoldenandJacobsen(2014)SAS code, kindly shared by the authors on their web site, to first construct the national best bid and offer (NBBO) data and then calculate daily dollar-volume-weighted percent effective spreads for all stock in our sample. In addition to the filters employed by Holden and Jacobsen (2014), we also remove transactions where the transaction price differs from the prevailing NBBO quote by more than 10%—i.e. the implied effective spread is larger than 20%—before calculating the daily effective spreads. This helps remove obvious outliers, but it does not materially affect our results, as the effective spreads of our stocks are two orders of magnitude smaller on average. Effective spread estimates at the stock-month or stock-year level are then obtained by simply averaging the daily estimates for a given stock and month or year, respectively. Novelinourpaperistheuseofvolatilityinexplainingthebehaviorofthelow-frequency measures. We construct high-frequency-based proxies for the daily volatility of the efficient price by calculating 5-minute realized volatilities from the NBBO mid-quotes. Liu, Patton, and Sheppard (2015) and Bollerslev, Hood, Huss, and Pedersen (2018) show that the 5minuterealizedvolatilitytendstoperformverywellacrossdifferentassetclasses. Toensure that our results are not plagued by market microstructure noise (see Bandi and Russell, 2006, for an in-depth treatment), we also consider the 30-minute realized volatility. Before calculating the daily realized volatilities, we employ some additional data cleaning methods in the spirit of Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009) and Bollerslev, Hood, Huss, and Pedersen (2018). Specifically, we remove (1) quotes where the bid or ask price is missing or equal to zero, or the bid-ask spread is less than or equal to zero; (2) quotes with bid-ask spreads larger than 50 times the median bid-ask spread for a given stock and day; and (3) quotes where the absolute difference between the mid-quote and the median daily mid-quote exceeds 10 times the average value of this difference on a given day. These filters are designed to remove obvious outliers, such as spurious jumps of the 18

bid and ask prices to $10,000, for example. To calculate monthly realized volatility, we sum the daily realized volatilities within a given month and add the sum of squared overnight returns, that is, returns between 4:00pm and 9:30am the following trading day. Finally, we merge the stock-month and stock-year variables created from CRSP and TAQ by the stock CUSIP. In order for a stock-month to be included in our final sample, the CRSP and TAQ have to have data for the same number of trading days for a given stock and month. Following Abdi and Ranaldo (2017), we also eliminate stock-months with stock splits or unusally large distributions (> 20%). Finally, we drop May 2010 from our sample, as a well-known flash crash occurred on the 6th of this month and was associated with unusual price and liquidity dynamics in the U.S. equity markets (Kirilenko, Kyle, Samadi, andTuzun,2017). Thisleavesuswith241,236stock-months. Incaseofstock-years,astockyear is included in our sample, as long as the CRSP and TAQ contain data for a given stock for at least 231 (11 months) trading days and the number of days with data differs by no more than 21 (1 month) between CRSP and TAQ. We also eliminate stock-years with stock splits and unusually large distributions (> 20%), and in case of 2010, eliminate May from all stock-years. Thus, the results for 2010 are only based on 11 months of data. This leaves us with 18,239 stock-years. Table 4 reports daily summary statistics for our final sample. The daily average number of trades in our sample period is 8,606 and the average daily trading volume equals $71 million. The average quoted spread equals 16.3 basis points, while the average effective spread is 12.4 basis points. The average daily realized volatility stands at 2.17% and the signal-to-noise ratio–the ratio of effective spread to realized volatility–equals a mere 6% on average. The table also reports analogous statistics for the four sub-indices that comprise the S&P 1500 index. The largest stocks (DJIA 30) trade with an effective spread of 4.3 basis points on average and daily realized volatility of 1.42%, and a signal-to-noise ratio of 3.4%. The S&P 500 index constituents have an effective spread of 6.1 basis points, daily realized volatility of 1.83% and a signal-to-noise ratio of 3.7% on average. For the mid caps (S&P 400), these figures stand at 9.1 basis points, 2.06%, and 5%, while for the small caps (S&P 600), they are 20 basis points, 2.55%, and 8.5%, respectively. Thus, even the mid and small caps trade with a fairly small effective spread and a signal-to-noise ratio well below 19

10%. 4.1.2 Results We show the empirical results for U.S. stocks in three stages. First, we present the results for the whole sample, i.e., the S&P 1500 stocks. Since there is significant variation in the effective spread and the signal-to-noise ratio in the cross section (Table 4), we then repeat the exercise for the largest stocks, i.e., the DJIA 30 stocks, and the small caps, i.e., S&P 600 stocks. Full-sample results: Monthly Starting with the full sample, Panel A of Table 6 summarizes descriptive statistics for the TAQ effective spread and the low-frequency measures pooled over all stock-months in the sample. Clearly, all low-frequency measures exhibit a sizable upward bias relative to the TAQ spread. While the true spread equal 12.4 bps on average, the low-frequency measures average between 26.4 bps for CS and 168 bps for H . As expected, for the CS and M L AR measures, the average bias increases when one censors or truncates before averaging the daily estimates within a month; the mean CS and AR are 76.2 bps and 82.2 bps, D D respectively, while the mean CS and AR are 123 bps and 157 bps, respectively. The P P median and the RMSE exhibit a similar pattern. In Panel B, we report the cross-sectional average of time-series correlations between the low-frequency measures and the TAQ spread and realized volatility. We find that the consistent measures exhibit low time-series correlation with the TAQ spread, ranging between 0.19and0.28,whiletheinconsistentmeasuresexhibitroughlytwiceashighcorrelationwith the TAQ spread. At the same time, the inconsistent measures tend to be correlated with realizedvolatility, andthecorrelationcoefficientrangefrom0.62to0.74, roughlydoublethe correlation of the consistent measures with the realized volatility. In Panel C, we present analogous results for the cross-sectional correlations averaged over time. These correlations exhibit a similar pattern. In Table 7 we report results of regressions of the low-frequency measures on the TAQ spread and realized volatility. In Panel A, we run pooled OLS regressions and use the Driscoll and Kraay (1998) standard errors, which are robust to heteroscedasticity, serial 20

correlation,andcross-sectionaldependenceforinference. Wefindthatourestimationresults are qualitatively consistent with our simulation results reported in Table 3. First, the realized volatility is highly statistically significant for all measures; second, including RV significantlyreducestheestimatedcoefficientfortheTAQspreadandincreasestheR2. The increase in the R2 is particularly pronounced for the inconsistent measures, where it at a minimum doubles the value of the R2. For example, for the CS measure the R2 goes from D 0.26 to 0.66, while for the AR measure, it increases from 0.22 to 0.64. D In the next two panels, we examine the variation separately in the time-series and cross-section. Starting with the time-series variation, in Panel B, we report the “within” estimation, that is, pooled OLS on data that were de-meaned at the stock level, which is equivalent to running OLS with stock fixed effects. We again use the Driscoll and Kraay (1998) standard errors for inference. In Panel C, we run cross-sectional regressions for each month in the sample and report the mean cross-sectional parameter estimates together with the associated Newey and West (1987) Student’s t-statistics. The two sets of results are qualitatively consistent with the pooled results in Panel A. One notable difference between the time-series and cross-sectional estimations is that the increase in the R2 is more pronounced for the former. The results reported so far have been calculate using the 5-minute realized volatility. To check for robustness to this sampling frequency, in Table 24 in Appendix B we rerun our regression using the 30-minute realized volatility. Sampling at lower frequency should alleviate any concerns about the effect of market microstructure noise on the volatility estimates, see Bandi and Russell (2006). As is seen from the table, our results are virtually unchanged by replacing the 5-minute RV by the 30-minute RV and the point estimates, as well as the associated t-statistics, are remarkably close. Full-sample results: Annual We now turn to annual results, where all low-frequency measures are calculated using one year’s worth of daily data. Table 8 report the annual descriptive statistics. Several differences relative to the monthly results are immediately apparent. First, the bias and RMSE have significantly declined for the consistent measures: the mean H declined from 168 to L 64.2 bps, the mean CS from 26.4 to 18.3 bps, and the AR from 58.3 to 46.0 bps. In con- M M 21

trast, the annual inconsistent measures exhibit almost the same bias as the monthly ones. The correlation with the true spread also increased for the consistent measures, especially the time-series correlations (Panel C). For example, the annual CS measure exhibits an M average time-series correlation with the TAQ spread of 0.64 relative to the monthly 0.37. Similar increases are observed for the other consistent measures as well, but not for the inconsistent ones. These findings are perfectly in line with our simulation results, the asymptotic theory, and show that only the consistent measures can deliver increasingly precise estimates as the sample size increases. In Table 9 we run the regressions of low-frequency measures on the TAQ spread and realized volatility at the annual frequency. We again find a very different change relative to the monthly results for the consistent and inconsistent measures. The former now exhibit substantially smaller coefficients on realized volatility, and in some cases the realized volatility is only marginally statistically significant, especially in the cross-sectional regressions (Panel C); take, for example, the case of CS and AR , where the coefficient on RV M M is essentially zero. In contrast, the inconsistent (“D” and “P”) measures exhibit no such changes and, if anything, the R2 in the annual regressions with RV increase relative to the monthly regressions. The dependence on volatility of the inconsistent measures cannot be simply undone by increasing the sample size. Results by size Tables 10–17 report analogous monthly and annual results—descriptive statistics and regression results—for very large stocks (DJIA 30) and small caps (S&P 600). Starting with the monthly results, we find that the realized volatility plays a more important role for the very large stocks: the low-frequency measures tend to be more correlated with RV for these stocks, and adding RV to the regression on the TAQ spread increases the R2 more (see Tables 10 and 11). But even for the small stocks, the realized volatility is a significant factor. All low-frequency measures exhibit a non-negligible correlation with realized volatility, and the realized volatility is highly significant in both time-series and cross-sectional regressions (see Tables 14 and 15). In Tables 25 and 26 reported in Appendix B we again check for robustness of our results to the choice of the sampling frequency at which we calculate realized volatility. The table shows that using the 30-minute RV in place of the 5-minute 22

RV leaves our results intact. Turning to the annual results reported in Tables 12 and 13 for the DJIA 30 stocks and 16 and 17 for the S&P 600 stocks, we find that the bias of the consistent measures improves moreforthesmallcapsthanfortheverylargecaps. Forexample,theaverageCS measure M drops from 36 bps to 27 bps, which is much closer to the TAQ spread of 20 bps, while the same measure declines from 16 bps to 11 bps for the very large caps, which have a TAQ spread of mere 4 bps on average. So for the very large caps, the bias is almost 156% of the TAQ spread even at the annual frequency. The bias associated with the inconsistent measures does not, of course, change as the sample size increases. Second, the correlation of the consistent measures with the TAQ spreads increases more forthesmallcapsthanforthelargecapswhenmovingtotheannualfrequency. Bothofthese results are consistent with our simulation results and the intuition developed in equations (15)-(17) that increasing the sample size reduces the bias and increases the precision of the consistent measures more when the signal-to-noise ratio is high. However, the regression resultsshowthatevenforthesmallcapsandtheannualfrequency(Table17), allconsistent measures still exhibit statistically significant dependence on volatility both in the cross section and time series. For the inconsistent measures, this dependence actually becomes stronger at the annual frequency, as can be seen from the magnitude of the coefficients on RV, their statistical significance, and the incremental R2 in the bivariate regressions. This is not surprising; the variance of the measures is reduced in larger samples, but the bias, which is a function of volatility, is not, and hence the measures now have a less noisy relationship with volatility. In summary, our empirical results for the very large and small caps are perfectly in line with our simulation evidence. The consistent low-frequency measures perform better for less liquid assets, i.e., assets with a higher ratio of transaction costs to volatility, and this performance improves more when increasing the sample size. The inconsistent lowfrequency measures, on the other hand, exhibit no such improvement. Even for the small caps and at the annual frequency, however, all measures suffer from the dependence on volatility both in the cross section as well as in the time series. 23

4.2 Foreign exchange rates 4.2.1 Data Our foreign exchange sample consists of five major currency pairs for which we have intraday data: EUR/USD, EUR/CHF, EUR/JPY, USD/CHF, USD/JPY. The source of our intraday data is EBS, one of the largest electronic interdealer platforms for trading spot FX. Although other currency pairs are traded on this platform, it is well known that EBS is the primary source of price discovery for the euro, Japanese yen, and Swiss franc, while Reuters is the primary venue for other currencies (Hasbrouck and Levich, 2018). Thus, we exclude these currencies from our analysis because the transaction costs estimates obtained from the EBS data might not be representative of the true costs of trading these currencies. The EBS data contain transactions time stamped to the millisecond and classified as either buyer or seller initiated, and top-of-the-book quotes sampled at regular 100ms intervals. Unlike for the U.S. stocks, we cannot therefore align quotes with trades exactly, and use instead the most recent—rather than the prevailing—mid-quote when calculating dollar-volume-weighted effective spreads, as in Karnaukh, Ranaldo, and S¨oderlind (2015).3 We calculate daily realized volatility from 5-minute mid-quotes; Chaboud, Chiquoine, Hjalmarsson, and Loretan (2010) show that the 5-minute sampling frequency is sufficiently low to avoid contamination by microstructure noise. Our sample period is from January 2008 to December 2015. We drop holidays, and in case of the exchange rates involving the Swiss Franc,wealsodropthemonthswhentheexchangeratefloorwasintroducedandabandoned by the Swiss National Bank, as these were months of unusual market developments. Table 5 reports summary statistics for the foreign exchange data. The EUR/USD and USD/JPY are the most frequently traded currency pairs, with around 39,000 and 21,000 trades per day on average, and $52 billion and $28 billion in average daily volume, respec- 3An additional source of measurement error stems from the fact that trading on EBS takes place in three different locations—London, New York, and Tokyo—and the associated latencies may give rise to inaccuracies in trade time stamps. Addressing this issue directly requires information on the geographical locationoftrades,butsuchdataarecurrentlynotavailable. Asaresult,theeffectivespreadmaybenegative forsometrades. InAppendixC,weconsidertwoalternativewaysofconstructingthedollar-volume-weighted effective spreads for robustness and show that our results remain unchanged. 24

tively. Theothercurrencypairsareconsiderablylessfrequentlytraded. Theaveragequoted spread is very small for all currencies, as is the effective spread. The most liquid currency pair is the EUR/USD with an average quoted spreads of 1.07 bps and an effective spread of 0.50 bps; USD/JPY exhibits slightly higher transaction costs, 1.45 bps and 0.68 bps, respectively; the remaining currency pairs trade with an average quoted spread between 2.6 and 2.8 bps and an average effective spread between 0.69 and 0.83 bps. The average daily realized volatility ranges from 45 bps for EUR/CHF and 85 bps for EUR/JPY. These values are almost two orders of magnitude larger that the average effective spreads, producing signal-to-noise ratios between 1 and 2%. In this sense, the spot foreign exchange rates are even more liquid than the DJIA 30 stocks discussed in the previous section. Following Karnaukh, Ranaldo, and S¨oderlind (2015), we obtain daily high, low, and closing prices for the five exchange rates from Thompson Reuters. These prices are based on quotes rather than transactions, and to the best of our knowledge, daily transaction prices are not publicly available. This means that we can only calculate the Corwin and Schultz (2012) effective spread measures from these data, since other measures explicitly require daily closing transaction prices; the effective spread is otherwise not identified. We calculate the Roll, Hasbrouck, and Abdi and Ranaldo measures from the daily high, low, and closing prices extracted from our intraday EBS data. We acknowledge that this is not feasible for researchers who do not have access to the proprietary EBS data, but it is nonetheless instructive to study how these measures would perform if the daily transactionbased data were publicly available. 4.2.2 Results Table 18 summarizes our foreign exchange results. Similar to the analysis of the U.S. equities, we calculate statistics and run regressions for FX-month data pooled across the five currency pairs in our sample. We do not perform the analysis at the annual frequency because we only have five exchange rates and 8 years in our sample, i.e., only 40 exchange rate-years, whichistoosmallasampleforanymeaningfulstatisticalanalysis. Startingwith the summary statistics reported in Panel A, we find that all low-frequency measures are severely upward biased. The true effective spread is 0.68 bps on average, while the average 25

Roll measure equals 25.9 bps, the Hasbrouck measures vary between 44.8 and 46.3 bps depending on the prior, the Corwin and Schultz average measures range from 13.3 to 44.7 bps, and the Abdi and Ranaldo from 13.7 to 47.6 bps. The standard deviations of the lowfrequency measures are also two orders of magnitude higher than that of the true spread. These results are in stark contrast to those of Karnaukh, Ranaldo, and S¨oderlind (2015), who report a much smaller bias for their currencies and sample period (2007-2012). When replicating their results, we find that they report the summary statistics in percent rather than basis points, as claimed. For example, Karnaukh, Ranaldo, and S¨oderlind (2015) report a mean CS for EUR/USD of 0.476 basis points,4 while our replication shows that it P is actually 0.476%, or 47.6 basis points. Given that the effective spreads from EBS data is reported to be 0.584 basis points on average in Karnaukh, Ranaldo, and S¨oderlind (2015), the bias is actually very large, and very similar to what we obtain for EUR/USD during our sample period (2008-2015). PanelBofTable18reportsregressionresultsfromregressingthelow-frequencymeasures onthetrueeffectivespread. Wefindthattheinterceptisstatisticallyindistinguishablefrom zero, but the slope coefficient is far from unity, as would be required of a well-performing measure. The large slope coefficients reflect, of course, the large bias the low-frequency measures suffer from. The coefficient of determination is generally not very large, reaching a maximum of 0.32 for the CS measure. In Panel C of Table 18, we add realized volatility D totheregression. EchoingtheresultsfortheDJIA30stocks,threemainresultsemergefrom these regressions: (i) realized volatility is highly statistically significant in all regressions except for CS ; (ii) compared to the univariate regressions in Panel B, the R2 more than M doubles for all measures except for consistent M measures (those based on censoring after averaging); and (iii) for the Hasbrouck measures and the “P” measures, adding realized volatility renders the spread coefficient (β ) statistically insignificant. ES In summary, volatility is an important driver of the Hasbrouck and the “D” and “P” measures in the foreign exchange market, explaining why these measures exhibit a higher 4Page 3080, Table 2, “Corwin-Schultz high-low estimate/2 (LF), bps”. Since we focuson the full spread inourpaper,whileKarnaukh,Ranaldo,andSo¨derlind(2015)reporthalf-spread,multiplyingtheir0.238by two produces 0.476. 26

correlation with the true spread documented in Panel A and in previous studies. Volatility affects the “M” measures by a much smaller degree. As a result, these measures exhibit a much smaller correlation with the true spread. 5 Implications for empirical asset pricing Thus far, we have documented the statistical properties of the low-frequency measures. In thissection, wehighlighttheimplicationsofthesepropertiesforempiricalassetpricing. We provide two case studies. In the first case, we investigate how the imprecision of the lowfrequency measures affects the estimates of betas and market price of risk in the liquidityadjusted CAPM of Acharya and Pedersen (2005). In the second case, we follow Ang, Hodrick, Xing, and Zhang (2006) and add a liquidity factor to their unconditional factor model with market and aggregate volatility risks. Our goal is to examine if and to what extent the dependence of the low-frequency measures on volatility affects the estimation of liquidity and volatility betas and the associated prices of risk if one employs low-frequency measures to construct an aggregate liquidity factor. 5.1 Liquidity-adjusted CAPM We assume that stock and market returns follow: ri−si = µ +β (rm−sm)+(cid:15)i, (19) t t i i t t t rm−sm = µ +σm(cid:15)m, (20) t t m t t where ri and rm are the gross stock and market returns, and si and sm are the stock and t t t t market transaction costs. Without loss of generality, we assume that the risk-free rate is zero. We assume that stock idiosyncratic volatility is constant, but this assumption can be easily relaxed and a factor structure assumed following Herskovic, Kelly, Lustig, and Van Nieuwerburgh (2018). Acharya and Pedersen (2005) decompose the beta in equation 27

(1) (2) (3) (4) (19) into four separate betas, β = (β +β −β −β ), where i i i i i Cov(ri,rm) Cov(si,sm) Cov(ri,sm) Cov(si,ri) β (1) = t t ,β (2) = t t ,β (3) = t t ,β (4) = t t . i Var(rm−sm) i Var(rm−sm) i Var(rm−sm) i Var(rm−sm) t t t t t t t t (21) Thus, the expected return is: E(ri) = E(si)+λ (β (1) +β (2) −β (3) −β (4) ). (22) t t m i i i i The asset-pricing model in equations (19) and (20) is written in terms of monthly returns. However, to calculate the various low-frequency effective spread measures, we need to simulate intraday returns. To do so, we continue with our simulation design of Section 3 andassumethatthetransactioncostsandmarketvolatilityareconstantwithineachmonth and divide the month into n = T ×n periods each, where T is the number of days in the T month and n is the number of intraday periods as before. Consistent with equations (19) and (20), we assume that gross returns at the intraday level follow: √ ri −si/n = β (rm−sm/n )+(σi/ n )(cid:15)i , (23) tj t T i tj t T T tj √ rm−sm/n = µ /n +(σm/ n )(cid:15)m, (24) tj t T m T t T tj where j = 1,...,n and (cid:15)i and (cid:15)m are independent standard normal random sequences. T tj tj Aggregating to the monthly frequency by summing up the intraday returns within the month reproduces equations (19) and (20). The intraday efficient prices and transaction prices are then generated as in the Roll (1984) model: j (cid:88) mi = m + ri , (25) t−j/nT 0 tj l=1 si pi = mi + tqi , (26) t−j/nT t−j/nT 2 t−j/nT where qi is an independent sequences of binary trade indicators. Once we generate the t−j/nT intraday transaction prices for a given month, we then compute the high, low, and closing pricesforeachdayinthemonth,followedbythelow-frequencyspreadmeasuresandrealized 28

volatilities for each stock. Armed with the monthly time-series of effective spread estimates and stock and market returns, we first estimate the time-series regression in equation (19) separatelyforeachstock. Inthesecondstep, werunacross-sectionalregressionofthestock net returns on the estimated betas to recover the market price of risk. We do this exercise separately for each low-frequency measure. That is, we substitute si by a low-frequency t measures and substitute sm by the cross-sectional average of the measure. t We calibrate the simulations as follows. In each replication, we generate 360 months (or 30 years) worth of intraday data for 26 stocks. As in the previous simulations, we assume that each month has T = 21 trading days with n = 390 transactions each. We set the gross expected return of the market (µ ) equal to 5% per annum and let the stocks’ betas to be m equidistantly spaced between 0.5 and 1.5. The market transaction costs and volatility are iid and jointly log-normally distributed with correlation of 0.5.5 The mean and volatility of the market spread are set to 10 basis points each, and the mean and volatility of daily marketvolatilityaresetto1%each; theratioofexpectedspreadtoexpecteddailyvolatility is thus 10%. We posit that individual stocks’ transaction costs follow a factor model: si = β smui, i = 1,...,26, (27) t i t t where ui is an independent log-normal innovation with unit mean and standard deviation t equalto0.5. Sincetheaveragebetaequalsone,thecrosssectionalmeanofthestocks’spread equals the mean of the market spread, i.e., 10 bps, but the high beta stocks have a higher effective spread and the low beta stock have a lower effective spread. Finally, to calibrate the idiosyncratic volatility, σi, we observe that conditional on σm, sm, and si, the variance t t t of stock returns in equation (19) is (σi)2 = β2(σm)2+(σi)2. We set (σi)2 = 0.25β2E((σm)2) t i t i t so that, on average, the idiosyncratic variance accounts for 20% of the total stock variation. With this choice, the stock beta also governs the total stock volatility, with higher beta stocks having higher total volatility. Moreover, the ratio of the expected stock spread to the expected stock volatility is constant across stocks. We run 10,000 replications. We report the simulation results in Table 19. To save space, we only report three sets of 5We also run simulations with ρ=0.75 and these simulations are reported in Appendix A. 29

estimated betas and the associated decompositions—small (0.5), medium (1.0), and large (1.5). We report the true values of the four Acharya and Pedersen (2005) liquidity betas in the first column. The second column reports the (infeasible) estimation results when we use the true spread (s) to estimate the liquidity-adjusted CAPM. Clearly, using the true spread yields estimates which are, on average, very close to the true values of the betas and the market price of risk, λ . The remaining columns report the results for the low-frequency m measures. While, on average, the estimated β is very close to the true value, this is not the 1 case for the remaining betas. For β , the bias is uniformly positive and very large, ranging 2 from 5 times to 54 times the actual value for CS and AR , respectively. For β and M P 3 β , the H measure yields a large positive bias, while the bias associated with the other 4 L measures is either positive or negative depending on the form of censoring or truncation. The market price of risk estimates, reported in Panel D, exhibit sizable negative biases. For the consistent measures CS and AR , the bias is around -1.1% and -2.3%. The bias gets M M worsewhenthecensoringisdonebeforeaveraging: -5.6%forCS and-5.1%forAR . Still, D D moving from censoring to truncation before averaging yields even more biased estimates of market price of risk. On average, the price of risk is estimated at -5% by CS and -6.2% by P AR . The H measures produces a similarly biased price of risk estimates, about -4.6%. P L In summary, these simple simulations show that using low-frequency liquidity measures to estimatetheliquidity-adjustedCAPMcouldeasilyleadtomisleadinginferencesaboutboth the size and sign of the market price of risk and the liquidity betas. 5.2 Asset pricing with aggregate liquidity and volatility risks As mentioned earlier, we use a variation of Ang, Hodrick, Xing, and Zhang (2006) model to demonstrate the potentially spurious inference resulting from using low-frequency measures of liquidity in settings similar to theirs. We assume that stock and market returns evolve according to: ri−µ = βm(rm−µ )+βl(l −¯l)+βσ(σm−σ¯ )+(cid:15)i, (28) t i i t m i t i t m t rm−µ = σm(cid:15)m, (29) t m t t 30

where µ is stock i’s expected return, ¯l = E(l ) is the average liquidity, and σ¯ = E(σm) is i t m t the average market volatility. In the absence of arbitrage, the expected return is given by E(ri) = βmλ +βlλ +βσλ , (30) t i m i l i σ where λ , λ , and λ are the market, liquidity, and volatility prices of risk, respectively. m l σ Consistent with equations (28) and (32), we assume that at the intraday level, stock and market returns follow: √ ri = µ /n +βm(rm−µ /n )+βl(l −¯l)/n +βσ(σm−σ¯m)/n +(σi/ n )(cid:15)i , (31) tj i T i tj m T i t T i t T T tj √ rm = µ /n +(σm/ n )(cid:15)m, (32) tj m T t T tj j = 1,...,n , (cid:15)i and (cid:15)m are independent standard normal random sequences. Aggregating T tj tj tothe monthlyfrequencybysummingup theintraday returns withinthemonthreproduces equations (28) and (32). √ We set the liquidity factor in equation (32) according to l = 10 Tsm, that is, the t t liquidity factor equals the common factor driving the stock effective spreads in equation √ (27), adjusted by a factor of 10× T in order to make the liquidity and volatility factors of the same order of magnitude. Recall that σm is the monthly stock volatility and the t expected daily volatility equals 10 times the average effective spread. sm and σm are jointly t t log-normal with the same parameters as in the previous subsection, si follows the factor t model in equation (27), and σi is also set as in the previous subsection. The intraday efficient prices and transaction prices evolve according to equations (25) and (26), with ri tj and rm given in by equations (31) and (32), respectively. tj We consider five different market betas, ranging from 0.5 to 1.5, five different liquidity betas, ranging from −2 to 2, and five different volatility betas, ranging from −2 to 2. Thus, we generate data for 5×5×5 = 125 stocks, one for each combination of the three betas. We consider three scenarios for the risk prices. In the first scenario all three risks are priced with λ = 5%, λ = λ = −1% per annum. In the second scenario, only the market and m s σ volatility risk are priced, i.e. λ = 5%, λ = −1%, and λ = 0. In the third scenario, only m σ s the market and liquidity risks are prices, i.e. λ = 5%, λ = −1%, and λ = 0. m s σ 31

In each replication, we generate 360 months of data with 21 days in each month and 390 returnsforeachdayforeachofthe125stocks. Oncethedataaregenerated,wecalculatethe low-frequencymeasuresforeachstockandmonth. Foreachlow-frequencymeasure,wethen run the usual two-stage asset pricing exercise, where we use the measure’s cross-sectional √ average, multiplied by 10× T, as the aggregate liquidity factor in equation (28). To proxy for the aggregate volatility factor, we use either the monthly realized volatility based on daily closing values of the market factor or the monthly range-based volatility estimates based on the daily high and low values of the market factor (Parkinson, 1980). Thus, we only use daily data to estimating the market volatility. In the first stage, we estimate the time-series regression in equation (28) for each stock, while in the second stage, we estimate the cross-sectional regression in equation (30) using the first-stage beta estimates in order to estimate the prices of risk. We repeat this 10,000 times and report the average prices of risk obtained in the simulation. Table20summarizestheresults. InPanelA,weusetherealizedvolatilityasaproxyfor aggregate volatility, while in Panel B, we use the range-based volatility. The first column reports the true values of the price of risk parameter values. In the second column, we show that using the true effective spread in the estimation yields prices of risk that are on average very close to the true values, regardless of the volatility proxy used. Turningtotheresultsforthelow-frequencymeasures, wefindthatusingthesemeasures produceslargebiasesinestimatedpricesofrisk, whichcanleadtospuriousinferencesabout the sign and the magnitude of the risk prices. First, when only the volatility risk is priced, i.e., the price of liquidity risk is zero, all measures produce a non-zero price of liquidity risk on average, and the price of risk is typically large and negative. For example, the CS and D AR measuresyieldapriceofliquidityriskof−2.22%and−6.86%,respectively,whendaily D realized volatility is used, and 2.56% and −7.20% when the daily range is used. Second, when only liquidity risk is priced, that is when the price of volatility risk is zero, most measures produce significantly biased liquidity risk prices and often non-trivial volatility risk prices on average. Also, the sign of the estimated liquidity risk premium varies across the different measures. For example, the AR measures yield on average positive prices of liquidity risk ranging from 0.81% to 5.93% even though the true value is −1%, while the 32

CS measures produce uniformly negative prices of liquidity risk ranging from −4.18% to −11.1%. Finally, when both volatility and liquidity risks are priced in the cross section, all measures produce sizable negatively biased estimates of liquidity risk premia, while the volatility prices of risks are on average quite close to the true value. 6 Conclusion Many financial decisions crucially depend on accurate estimates of transaction costs. The availabilityoflonghistoriesandtherelativeeaseofhandlingdailydatamotivateresearchers and practitioners to employ low-frequency transaction cost measures. However, as we show in this study, a number of well-known low-frequency measures are biased and inconsistent. Thisbiasissignificant,positive,stemsfromtheconstructionofthesemeasures,isafunction of volatility, and hence it induces a positive correlation between the low-frequency measures and volatility above and beyond what is implied by the correlation between volatility and the true transaction costs. The relative size of this bias is particularly large for liquid assets such as S&P 1500 stocks or heavily traded foreign currencies. Throughcarefulsimulation,wedocumentthepropertiesandproblemsofseveralpopular low-frequency measures when they are used in analyzing highly liquid assets, where the size of transaction costs relative to volatility is small. We then document the problems that arise, including biased or outright spurious results, when one uses these measures in asset pricing applications. Often, proponents of these measures point to their applicability for historical data. However, at least two studies (Jones, 2002 and Bessembinder, 1994) report results that point to trading costs that are much lower than those implied by low-frequency measures for the largest, most liquid U.S. equities and exchange rates as early as 1920s and early 1980s, respectively. We do not claim that low-frequency measures are not applicable in general. They are useful for illiquid assets, where trading costs are relatively large in comparison to volatility. Examples may include high-yield corporate debt or infrequently traded equities. Awareness of potential pitfalls of using the low-frequency measures of transaction costs mattersinpractice. Similartotheestimationofmeanofdailyequityreturns,recoveringliquidity measures—which are small, positive-valued quantities—from noisy transaction data 33

is challenging. For highly liquid assets, using high-frequency transactions data is probably the only reasonable way for estimation of accurate liquidity measures; see Chordia, Sarkar, and Subrahmanyam (2005) and Mancini, Ranaldo, and Wrampelmeyer (2013). If one must use a low-frequency measure, then as we show in this study, for less liquid assets the consistent version of Corwin and Schultz (2012) performs better than other competing measures, provided that one uses a fairly long window of data (at least one year) for constructing the measure. 34

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fo snoitidnoc ”laedI-raeN“ eht swollof ngised noitalumis ehT .srotamitse daerps evitceffe eht fo noitalumis olraC etnoM :1 elbaT gnidart eno yletamixorppa( syad 12 = T fo sezis elpmas redisnoc eW .%3 = σ dna 093 = n tes ohw ,)2102( ztluhcS dna niwroC eht ni etamitse egareva eht si ”naeM“ ,erusaem ycneuqerf-wol hcae roF .)raey gnidart eno yletamixorppa( 152 = T dna )htnom eht ”0 ≤“ dna ,rorre erauqs naem toor eht ”ESMR“ ,setamitse eht fo noitaived dradnats gnidnopserroc eht si ”dtS“ ,noitalumis .snoitacilper 000,01 no desab era stluser ehT .noitalumis eht ni setamitse evitisop-non fo noitroporp 152= T 12= T RA RA RA SC SC SC H H R RA RA RA SC SC SC H H R s P D M P D M T L M P D M P D M T L M 63.2 81.1 53.0 60.2 12.1 81.0 19.0 99.0 36.0 63.2 81.1 46.0 60.2 12.1 43.0 74.1 30.2 51.1 naeM 50.0 31.0 01.0 04.0 01.0 90.0 51.0 13.0 53.0 27.0 74.0 53.0 57.0 73.0 13.0 04.0 14.0 46.0 63.1 dtS 13.2 41.1 05.0 10.2 71.1 91.0 19.0 00.1 39.0 63.2 81.1 59.0 50.2 02.1 05.0 74.1 80.2 57.1 ESMR 00.0 00.0 05.0 00.0 00.0 81.0 00.0 00.0 94.0 00.0 00.0 05.0 00.0 00.0 83.0 00.0 00.0 05.0 0 ≤ 63.2 81.1 53.0 70.2 22.1 91.0 19.0 99.0 26.0 63.2 81.1 46.0 70.2 22.1 53.0 74.1 30.2 41.1 naeM 1.0 31.0 01.0 04.0 01.0 90.0 51.0 13.0 53.0 37.0 84.0 53.0 57.0 73.0 13.0 14.0 14.0 46.0 63.1 dtS 72.2 90.1 74.0 89.1 31.1 81.0 78.0 69.0 98.0 13.2 41.1 39.0 00.2 61.1 84.0 34.1 40.2 17.1 ESMR 00.0 00.0 94.0 00.0 00.0 51.0 00.0 00.0 05.0 00.0 00.0 05.0 00.0 00.0 83.0 00.0 00.0 15.0 0 ≤ 73.2 91.1 63.0 21.2 62.1 52.0 19.0 00.1 26.0 73.2 91.1 66.0 21.2 72.1 93.0 74.1 40.2 51.1 naeM 2.0 31.0 01.0 14.0 01.0 90.0 61.0 13.0 53.0 37.0 84.0 53.0 57.0 83.0 13.0 34.0 14.0 46.0 63.1 dtS 71.2 99.0 44.0 29.1 70.1 71.0 87.0 78.0 48.0 22.2 50.1 88.0 59.1 11.1 74.0 33.1 59.1 66.1 ESMR 00.0 00.0 84.0 00.0 00.0 80.0 00.0 00.0 05.0 00.0 00.0 94.0 00.0 00.0 33.0 00.0 00.0 94.0 0 ≤ 14.2 22.1 84.0 82.2 34.1 25.0 49.0 30.1 96.0 04.2 22.1 27.0 92.2 44.1 95.0 84.1 60.2 02.1 naeM 5.0 31.0 01.0 44.0 11.0 90.0 71.0 23.0 63.0 57.0 94.0 63.0 87.0 83.0 43.0 05.0 14.0 56.0 93.1 dtS 19.1 27.0 44.0 97.1 49.0 71.0 55.0 56.0 77.0 79.1 08.0 18.0 38.1 99.0 15.0 60.1 96.1 55.1 ESMR 00.0 00.0 73.0 00.0 00.0 00.0 00.0 00.0 64.0 00.0 00.0 64.0 00.0 00.0 91.0 00.0 00.0 94.0 0 ≤ 35.2 23.1 19.0 85.2 47.1 99.0 60.1 71.1 39.0 35.2 13.1 39.0 95.2 47.1 00.1 25.1 41.2 33.1 naeM 0.1 41.0 11.0 34.0 11.0 11.0 81.0 73.0 24.0 18.0 05.0 83.0 68.0 04.0 83.0 95.0 24.0 76.0 64.1 dtS 35.1 43.0 44.0 85.1 57.0 81.0 83.0 54.0 28.0 16.1 94.0 68.0 46.1 38.0 95.0 76.0 33.1 05.1 ESMR 00.0 00.0 01.0 00.0 00.0 00.0 00.0 00.0 43.0 00.0 00.0 73.0 00.0 00.0 60.0 00.0 00.0 64.0 0 ≤ 45.3 14.2 00.3 48.3 22.3 29.2 25.2 67.2 39.2 45.3 04.2 09.2 48.3 22.3 29.2 98.1 49.2 16.2 naeM 0.3 61.0 51.0 91.0 31.0 41.0 81.0 36.0 16.0 36.0 55.0 35.0 57.0 74.0 05.0 36.0 45.0 98.0 09.1 dtS 65.0 16.0 91.0 58.0 62.0 91.0 08.0 56.0 46.0 77.0 97.0 57.0 69.0 55.0 46.0 32.1 09.0 49.1 ESMR 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 10.0 00.0 00.0 00.0 00.0 00.0 32.0 0 ≤ 38

ycneuqerf-wol eht fo noisserger dna ) σ( ytilitalov dna ) s( daerps eurt eht htiw serusaem ycneuqerf-wol fo snoitalerroC :2 elbaT t t htoB .%0.2 slauqe ytilitalov fo ytilitalov dna %0.2 slauqe ytilitalov egareva ehT .ytilitalov ro/dna daerps eurt eht no serusaem .snoitacilper 000,01 no desab era snoitalumis llA .tnednepedni yllautum dna detubirtsid yllamrongol dii era daerps dna ytilitalov 152= T 12= T RA RA RA SC SC SC H H R RA RA RA SC SC SC H H R P D M P D M T L M P D M P D M T L M 1.0=)s(dtS ,1.0=)s(E :daerps liuqnart ,llamS .A 20.0 40.0 41.0 50.0 80.0 24.0 70.0 50.0 50.0 10.0 20.0 50.0 30.0 60.0 71.0 30.0 10.0 20.0 s htiw .rroC t 00.1 99.0 05.0 00.1 99.0 35.0 97.0 98.0 84.0 69.0 29.0 05.0 79.0 49.0 74.0 76.0 09.0 64.0 σ htiw .rroC t 45.1 67.0 91.0 33.1 77.0 80.0 55.0 46.0 93.0 65.1 77.0 93.0 63.1 97.0 91.0 29.0 13.1 37.0 α s no .geR t 43.0 03.0 36.0 56.0 36.0 38.0 92.0 53.0 24.0 90.0 41.0 04.0 73.0 84.0 17.0 31.0 70.0 12.0 β s 00.0 00.0 20.0 00.0 10.0 81.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 30.0 00.0 00.0 00.0 2R 20.0- 20.0- 30.0- 10.0- 10.0- 20.0- 12.0 10.0- 10.0 40.0- 10.0- 30.0- 20.0- 20.0- 20.0- 65.0 51.0 70.0 α σ , s no .geR t t 32.0 42.0 16.0 55.0 75.0 38.0 52.0 34.0 34.0 62.0 22.0 54.0 25.0 75.0 37.0 61.0 71.0 23.0 β s 97.0 04.0 11.0 86.0 04.0 50.0 71.0 23.0 91.0 08.0 93.0 12.0 86.0 04.0 11.0 81.0 95.0 33.0 β σ 99.0 99.0 72.0 99.0 99.0 54.0 26.0 08.0 32.0 29.0 58.0 52.0 49.0 88.0 52.0 44.0 08.0 12.0 2R 5.0=)s(dtS ,5.0=)s(E :daerps elitalov yletaredom ,muideM .B 91.0 04.0 67.0 62.0 64.0 59.0 56.0 84.0 74.0 61.0 63.0 74.0 32.0 14.0 17.0 83.0 12.0 32.0 s htiw .rroC t 89.0 98.0 12.0 69.0 88.0 11.0 05.0 37.0 83.0 49.0 38.0 73.0 49.0 48.0 42.0 45.0 68.0 24.0 σ htiw .rroC t 24.1 16.0 80.0 72.1 07.0 40.0 04.0 94.0 52.0 84.1 46.0 62.0 43.1 37.0 31.0 38.0 81.1 95.0 α s no .geR t 85.0 46.0 49.0 27.0 87.0 69.0 07.0 37.0 08.0 45.0 46.0 48.0 76.0 67.0 19.0 04.0 15.0 56.0 β s 40.0 61.0 85.0 70.0 22.0 09.0 34.0 32.0 22.0 30.0 31.0 22.0 50.0 71.0 15.0 41.0 40.0 50.0 2R 80.0- 90.0- 50.0- 30.0- 40.0- 10.0- 31.0 80.0- 70.0- 01.0- 11.0- 80.0- 50.0- 50.0- 30.0- 25.0 60.0 20.0- α σ , s no .geR t t 75.0 36.0 49.0 17.0 87.0 69.0 96.0 27.0 18.0 95.0 66.0 58.0 17.0 87.0 29.0 04.0 35.0 66.0 β s 67.0 63.0 60.0 66.0 73.0 30.0 41.0 82.0 61.0 77.0 63.0 61.0 76.0 83.0 80.0 61.0 65.0 03.0 β σ 99.0 49.0 26.0 99.0 89.0 19.0 76.0 67.0 63.0 19.0 38.0 63.0 49.0 98.0 75.0 44.0 87.0 32.0 2R 0.3=)s(dtS ,0.3=)s(E :daerps elitalov ,egraL .C 09.0 89.0 99.0 29.0 89.0 00.1 29.0 89.0 79.0 09.0 79.0 79.0 29.0 79.0 89.0 85.0 09.0 28.0 s htiw .rroC t 83.0 01.0 00.0 73.0 91.0 00.0 80.0- 70.0 20.0 63.0 01.0 30.0 63.0 81.0 20.0 61.0- 02.0 90.0 σ htiw .rroC t 97.0 40.0 10.0- 88.0 93.0 20.0- 92.0 80.0 10.0 08.0 40.0 10.0 88.0 83.0 00.0 72.1 26.0 21.0 α s no .geR t 19.0 79.0 00.1 19.0 59.0 99.0 58.0 89.0 99.0 19.0 79.0 00.1 29.0 59.0 99.0 43.0 68.0 49.0 β s 28.0 69.0 99.0 48.0 59.0 00.1 48.0 79.0 49.0 28.0 59.0 49.0 48.0 49.0 79.0 33.0 28.0 86.0 2R 53.0- 92.0- 30.0- 42.0- 71.0- 30.0- 35.0 01.0- 90.0- 33.0- 62.0- 90.0- 42.0- 71.0- 60.0- 65.1 30.0- 81.0- α σ , s no .geR t t 19.0 79.0 00.1 29.0 59.0 99.0 58.0 89.0 99.0 19.0 79.0 00.1 29.0 59.0 99.0 43.0 68.0 49.0 β s 85.0 61.0 10.0 65.0 82.0 10.0 21.0- 90.0 50.0 75.0 51.0 50.0 65.0 82.0 30.0 51.0- 33.0 51.0 β σ 79.0 79.0 99.0 89.0 99.0 00.1 58.0 79.0 49.0 59.0 69.0 49.0 79.0 89.0 79.0 53.0 68.0 96.0 2R 39

ycneuqerf-wol eht fo noisserger dna ) σ( ytilitalov dna ) s( daerps eurt eht htiw serusaem ycneuqerf-wol fo snoitalerroC :3 elbaT t t htoB .%0.2 slauqe ytilitalov fo ytilitalov dna %0.2 slauqe ytilitalov egareva ehT .ytilitalov ro/dna daerps eurt eht no serusaem desab era snoitalumis llA .5.0 = ρ htiw detalerroc ylsuoenaropmetnoc dna detubirtsid yllamrongol dii era daerps dna ytilitalov .snoitacilper 000,01 no 152= T 12= T RA RA RA SC SC SC H H R RA RA RA SC SC SC H H R P D M P D M T L M P D M P D M T L M 1.0=)s(dtS ,1.0=)s(E :daerps liuqnart ,llamS .A 15.0 25.0 33.0 35.0 55.0 85.0 64.0 64.0 13.0 05.0 84.0 23.0 25.0 15.0 23.0 14.0 94.0 72.0 s htiw .rroC t 00.1 99.0 55.0 00.1 99.0 56.0 77.0 98.0 85.0 69.0 19.0 45.0 79.0 49.0 15.0 56.0 88.0 94.0 σ htiw .rroC t 67.0 83.0 90.0 66.0 83.0 40.0 73.0 43.0 61.0 77.0 83.0 81.0 76.0 04.0 11.0 07.0 96.0 83.0 α s no .geR t 29.7 20.4 05.1 42.7 14.4 81.1 00.2 53.3 06.2 50.8 11.4 26.2 23.7 73.4 94.1 62.2 23.6 89.3 β s 62.0 72.0 11.0 92.0 03.0 43.0 12.0 12.0 01.0 52.0 32.0 01.0 72.0 62.0 01.0 71.0 42.0 80.0 2R 00.0 10.0- 30.0- 10.0- 10.0- 10.0- 22.0 30.0 50.0- 00.0 10.0- 30.0- 20.0- 00.0 00.0 65.0 81.0 10.0 α σ , s no .geR t t 11.0 11.0 23.0 74.0 54.0 96.0 93.0 70.0 41.0 05.0 33.0 25.0 66.0 94.0 04.0 65.0 54.0 35.0 β s 97.0 93.0 21.0 86.0 04.0 50.0 61.0 23.0 32.0 77.0 83.0 12.0 86.0 93.0 11.0 61.0 55.0 63.0 β σ 99.0 99.0 13.0 00.1 99.0 15.0 06.0 97.0 43.0 29.0 38.0 92.0 49.0 88.0 72.0 34.0 87.0 42.0 2R 5.0=)s(dtS ,5.0=)s(E :daerps elitalov yletaredom ,muideM .B 95.0 66.0 87.0 56.0 57.0 59.0 76.0 56.0 15.0 65.0 06.0 15.0 36.0 07.0 27.0 94.0 55.0 83.0 s htiw .rroC t 99.0 69.0 74.0 89.0 49.0 35.0 76.0 48.0 05.0 69.0 09.0 94.0 59.0 09.0 05.0 26.0 78.0 15.0 σ htiw .rroC t 77.0 53.0 20.0 96.0 83.0 20.0 43.0 82.0 41.0 47.0 43.0 41.0 66.0 73.0 60.0 07.0 66.0 82.0 α s no .geR t 87.1 40.1 79.0 58.1 73.1 99.0 76.0 20.1 39.0 38.1 70.1 59.0 19.1 04.1 00.1 45.0 44.1 32.1 β s 43.0 44.0 16.0 24.0 65.0 19.0 54.0 24.0 62.0 13.0 73.0 62.0 93.0 94.0 15.0 42.0 03.0 41.0 2R 10.0- 20.0- 20.0- 00.0 10.0- 10.0- 42.0 20.0 10.0- 10.0- 10.0- 10.0- 10.0 00.0 10.0- 85.0 91.0 00.0 α σ , s no .geR t t 73.0 83.0 09.0 16.0 76.0 59.0 54.0 74.0 46.0 73.0 83.0 66.0 46.0 76.0 78.0 52.0 33.0 25.0 β s 57.0 53.0 30.0 66.0 73.0 20.0 11.0 62.0 41.0 57.0 53.0 51.0 56.0 73.0 70.0 31.0 25.0 23.0 β σ 99.0 79.0 26.0 00.1 99.0 19.0 95.0 77.0 53.0 39.0 58.0 43.0 49.0 09.0 45.0 24.0 77.0 82.0 2R 0.3=)s(dtS ,0.3=)s(E :daerps elitalov ,egraL .C 69.0 99.0 00.1 69.0 99.0 00.1 38.0 89.0 79.0 49.0 89.0 79.0 49.0 89.0 89.0 74.0 19.0 48.0 s htiw .rroC t 86.0 74.0 84.0 07.0 85.0 84.0 22.0 74.0 94.0 96.0 84.0 84.0 17.0 95.0 94.0 50.0 35.0 64.0 σ htiw .rroC t 53.0 80.0- 10.0- 44.0 71.0 10.0- 85.0 00.0 00.0 63.0 70.0- 40.0- 54.0 81.0 00.0 42.1 34.0 50.0- α s no .geR t 89.0 49.0 00.1 00.1 99.0 89.0 86.0 79.0 99.0 89.0 49.0 00.1 00.1 89.0 89.0 12.0 28.0 59.0 β s 29.0 89.0 99.0 29.0 89.0 00.1 96.0 79.0 59.0 88.0 59.0 49.0 98.0 59.0 69.0 22.0 28.0 07.0 2R 70.0- 50.0- 10.0- 50.0- 30.0- 10.0- 09.0 60.0 10.0- 80.0- 70.0- 50.0- 30.0- 30.0- 20.0- 04.1 53.0 01.0- α σ , s no .geR t t 48.0 59.0 00.1 58.0 29.0 89.0 08.0 99.0 99.0 38.0 49.0 00.1 48.0 19.0 89.0 72.0 97.0 49.0 β s 24.0 30.0- 10.0- 84.0 02.0 10.0- 33.0- 60.0- 00.0 54.0 00.0 00.0 94.0 12.0 20.0 71.0- 90.0 50.0 β σ 89.0 89.0 99.0 99.0 99.0 00.1 57.0 79.0 59.0 59.0 59.0 49.0 79.0 79.0 69.0 72.0 38.0 07.0 2R 40

Table 4: Daily summary statistics for the S&P 1500 data. The table reports averages over stock-days in the sample. The quoted spread, effective spread, and realized volatility were winsorized at the 99.9% level prior to calculating the means. The daily realized volatility is based on 5-minute intraday mid-quote returns and close-open mid-quote returns. The sample period is from October 2003 to December 2017. DJIA30 S&P500 S&P400 S&P600 S&P1500 (very large) (large) (mid) (small) (all) # Trades 51,070 16,691 5,555 2,132 8,606 Volume ($mn) 609.8 150.3 30.12 8.632 71.03 Quoted spread (bps) 2.920 5.617 11.01 29.17 16.35 Effective spread (bps) 4.292 6.069 9.065 19.98 12.36 Realized vol (bps) 142.3 183.5 206.2 254.6 217.1 Signal-to-noise 0.034 0.037 0.050 0.085 0.059 Table5: Dailysummarystatisticsfortheforeignexchangerates. Thetablereportsaverages overstock-daysinthesample. Thedailyrealizedvolatilityisbasedon5-minutemid-quotes. The sample period is January 2008 to December 2015. EUR/USD EUR/CHF EUR/JPY USD/JPY USD/CHF # Trades 39,390 3,635 5,529 21,033 6,241 Volume ($mn) 51,837 4,668 6,584 28,118 7,547 Quoted spread (bps) 1.071 2.579 2.819 1.452 2.722 Effective spread (bps) 0.505 0.721 0.687 0.680 0.826 Realized vol (bps) 66.31 45.24 84.58 66.68 73.55 Signal-to-noise 0.008 0.023 0.009 0.011 0.012 41

Table 6: Monthly descriptive statistics for S&P 1500 stocks. Panel A reports descriptive statistics calculated across all stock-months (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute midquotes. All variables were winsorized at the 99.9% level prior to calculating the descriptive statistics. The sample period is October 2003 to December 2017 and the sample size is 241,236 stock-months. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 12.4 217 90.0 168 107 26.4 76.2 123 58.3 82.2 157 10pct 3.68 105 0.00 56.7 54.1 0.00 33.6 57.6 0.00 33.1 70.2 25pct 5.39 134 0.00 79.0 73.0 0.00 45.0 74.6 0.00 46.1 93.1 Med 8.69 182 35.5 117 101 15.4 63.1 102 34.4 66.6 129 75pct 14.9 257 138 181 135 38.8 90.9 144 89.4 98.4 186 90pct 24.6 364 245 289 166 67.9 131 207 152 146 269 RMSE 153 261 104 35.6 78.2 132 89.8 88.7 176 B. Cross-sectional averages of time-series moments Std 5.69 105 121 198 40.1 30.7 38.1 58.2 71.5 46.2 80.3 (6.18) (55.1) (51.1) (139) (9.49) (14.3) (21.0) (33.2) (31.5) (24.8) (44.5) ρ(ES,·) 0.43 0.19 0.19 0.28 0.20 0.45 0.47 0.19 0.41 0.44 (0.28) (0.22) (0.27) (0.23) (0.23) (0.27) (0.27) (0.23) (0.26) (0.28) ρ(RV,·) 0.43 0.32 0.60 0.51 0.16 0.62 0.70 0.29 0.65 0.74 (0.28) (0.26) (0.20) (0.19) (0.26) (0.26) (0.25) (0.26) (0.23) (0.21) C. Time-series averages of cross-sectional moments Std 11.7 99.3 111 199 39.8 30.8 36.3 53.7 67.5 41.8 72.2 (3.28) (31.4) (46.3) (46.8) (4.60) (11.8) (14.9) (22.9) (28.1) (19.1) (32.4) ρ(ES,·) 0.48 0.20 0.16 0.30 0.37 0.54 0.50 0.29 0.47 0.46 (0.10) (0.08) (0.08) (0.10) (0.09) (0.07) (0.07) (0.10) (0.10) (0.09) ρ(RV,·) 0.48 0.30 0.55 0.52 0.24 0.67 0.74 0.33 0.68 0.76 (0.10) (0.08) (0.05) (0.13) (0.08) (0.07) (0.07) (0.08) (0.06) (0.06) 42

skcots 0051 P&S rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :7 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf ylhtnom eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .shtnom-kcots 632,142 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 83.7 901 57.3 4.45 06.0- 6.13 0.21 1.58 65.8 3.05 77.4 9.21 3.36 9.29 6.41- 921 87.5- 5.85 tsnoc )40.1( )7.52( )79.0( )3.52( )41.0-( )9.71( )53.2( )2.52( )69.2( )6.52( )96.3( )2.61( )3.31( )7.82( )79.0-( )8.33( )88.0-( )1.61( 64.0 09.3 25.0 42.2 50.1 51.2 25.0 00.3 66.0 80.2 18.0 90.1 41.0 41.1 97.1- 01.3 63.0 45.2 β SE )22.7( )05.6( )4.11( )60.7( )85.7( )28.7( )96.8( )92.7( )6.91( )45.8( )9.71( )7.41( )50.2( )0.03( )26.5-( )87.8( )56.3( )67.6( 66.0 33.0 12.0 84.0 72.0 50.0 91.0 49.0 24.0 β VR )7.51( )2.41( )08.7( )0.41( )6.31( )83.5( )38.9( )14.9( )37.9( 57.0 12.0 46.0 22.0 02.0 11.0 37.0 32.0 66.0 62.0 81.0 51.0 43.0 01.0 03.0 30.0 91.0 60.0 2R noitamitse nihtiW .B 20.1 22.5 77.0 09.2 50.1 05.2 49.0 39.3 38.0 55.2 37.0 90.1 21.0- 70.1 03.3- 03.3 44.0 43.3 β SE )80.7( )85.4( )93.8( )19.4( )25.5( )42.5( )1.21( )68.4( )9.51( )63.5( )5.41( )19.7( )22.1-( )8.31( )75.8-( )60.4( )35.3( )16.4( 46.0 33.0 22.0 64.0 62.0 60.0 81.0 10.1 44.0 β VR )5.41( )4.31( )82.7( )9.11( )5.11( )87.4( )4.11( )77.8( )34.9( 07.0 91.0 85.0 81.0 61.0 70.0 86.0 02.0 85.0 02.0 01.0 70.0 42.0 40.0 82.0 20.0 71.0 40.0 2R noitamitse emit-naem delooP .C 37.0 68.2 56.0 86.1 50.1 07.1 28.0 03.2 48.0 76.1 98.0 99.0 52.0 01.1 58.2- 25.2 26.0 78.1 β SE )88.9( )0.91( )2.21( )4.81( )8.01( )7.51( )6.61( )3.22( )6.02( )7.22( )3.81( )4.02( )43.7( )1.91( )1.21-( )9.42( )36.6( )2.71( 05.0 52.0 61.0 53.0 02.0 30.0 22.0 33.1 03.0 β VR )7.53( )5.33( )8.22( )3.03( )4.03( )57.8( )8.02( )6.32( )9.72( 06.0 22.0 05.0 32.0 41.0 90.0 85.0 62.0 25.0 92.0 51.0 41.0 92.0 01.0 33.0 30.0 11.0 40.0 2R 43

Table 8: Annual descriptive statistics for S&P 1500 stocks. Panel A reports descriptive statistics calculated across all stock-years (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute midquotes. All variables were winsorized at the 99.9% level prior to calculating the descriptive statistics. The sample period is October 2003 to December 2017 and the sample size is 18,239 stock-years. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 12.1 231 71.8 64.2 60.0 18.3 75.9 122 46.0 82.2 157 10pct 3.83 122 0.00 29.2 28.8 0.00 40.6 66.2 0.00 43.9 84.7 25pct 5.51 154 0.00 37.6 37.0 4.45 49.9 80.9 0.00 54.2 104 Med 8.72 203 47.3 51.8 50.3 13.3 66.1 106 34.2 70.8 136 75pct 14.9 274 110 75.7 71.8 25.3 89.5 143 68.8 97.1 185 90pct 23.9 372 187 114 104 41.5 125 198 112 135 256 RMSE 107 65.2 57.8 16.7 71.8 123 62.1 79.5 163 B. Cross-sectional averages of time-series moments Std 3.33 64.6 63.4 25.9 21.8 11.1 21.4 34.1 36.4 24.3 45.0 (4.41) (52.2) (43.6) (21.0) (16.5) (8.89) (17.2) (27.2) (27.1) (19.6) (36.2) ρ(ES,·) 0.39 0.16 0.34 0.33 0.27 0.46 0.44 0.20 0.42 0.42 (0.49) (0.48) (0.48) (0.48) (0.50) (0.47) (0.48) (0.48) (0.48) (0.48) ρ(RV,·) 0.39 0.30 0.56 0.54 0.21 0.74 0.77 0.23 0.75 0.77 (0.49) (0.52) (0.49) (0.49) (0.54) (0.43) (0.41) (0.53) (0.42) (0.41) C. Time-series averages of cross-sectional moments Std 10.5 84.6 73.1 29.5 25.5 17.4 27.7 42.7 47.5 29.9 55.4 (1.96) (22.8) (31.4) (14.4) (10.4) (6.24) (8.72) (13.9) (20.9) (11.2) (20.0) ρ(ES,·) 0.54 0.28 0.48 0.47 0.64 0.66 0.58 0.47 0.62 0.56 (0.08) (0.08) (0.08) (0.09) (0.06) (0.07) (0.07) (0.07) (0.10) (0.10) ρ(RV,·) 0.54 0.24 0.61 0.57 0.35 0.86 0.89 0.31 0.87 0.90 (0.08) (0.08) (0.09) (0.11) (0.08) (0.03) (0.04) (0.09) (0.03) (0.03) 44

skcots 0051 P&S rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :9 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf launna eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc raey-yb-raey fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .sraey-kcots 932,81 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 37.5 311 92.2 4.65 42.7- 1.61 52.6 4.78 36.3 3.15 11.0- 47.4 2.01 1.44 26.0 5.44 7.21- 8.24 tsnoc )61.1( )3.21( )28.0( )4.21( )66.1-( )49.7( )58.1( )8.11( )67.1( )0.21( )41.0-( )53.5( )10.6( )70.8( )22.0( )39.7( )73.2-( )66.4( 43.0 36.3 74.0 31.2 67.1 74.2 53.0 48.2 85.0 40.2 79.0 21.1 82.0 13.1 82.0 36.1 96.0 93.2 β SE )89.2( )56.5( )15.4( )47.5( )88.6( )81.6( )17.8( )05.6( )6.51( )20.7( )0.41( )9.01( )55.2( )7.01( )79.2( )08.6( )06.3( )78.6( 46.0 23.0 41.0 84.0 82.0 30.0 02.0 62.0 33.0 β VR )1.22( )7.02( )77.5( )5.02( )2.02( )92.4( )8.21( )00.01( )33.6( 88.0 52.0 58.0 03.0 82.0 32.0 88.0 62.0 78.0 43.0 04.0 83.0 84.0 61.0 35.0 71.0 02.0 80.0 2R noitamitse nihtiW .B 07.0 52.5 36.0 79.2 17.1 50.3 16.0 30.4 76.0 17.2 38.0 51.1 71.0 08.1 33.0 24.2 82.0 84.3 β SE )73.3( )87.3( )64.4( )10.4( )74.5( )39.4( )90.7( )99.3( )07.9( )23.4( )6.01( )95.7( )65.1( )77.4( )99.4( )80.4( )19.1( )46.3( 36.0 33.0 91.0 84.0 82.0 40.0 32.0 92.0 54.0 β VR )3.42( )3.32( )28.6( )8.91( )5.81( )60.5( )1.21( )9.11( )1.01( 68.0 12.0 28.0 32.0 22.0 21.0 68.0 22.0 38.0 52.0 42.0 81.0 54.0 11.0 25.0 31.0 32.0 50.0 2R noitamitse emit-naem delooP .C 65.0 59.2 16.0 57.1 79.1 31.2 55.0 63.2 07.0 37.1 50.1 40.1 35.0 01.1 85.0 92.1 14.1 98.1 β SE )48.3( )5.01( )12.5( )86.9( )94.7( )82.8( )13.7( )9.31( )1.01( )8.31( )6.11( )5.31( )42.5( )8.21( )31.5( )0.21( )50.6( )7.11( 45.0 62.0 40.0 14.0 32.0 00.0 31.0 61.0 11.0 β VR )7.52( )3.32( )60.2( )4.92( )9.13( )60.0( )1.61( )6.11( )48.3( 28.0 33.0 97.0 93.0 42.0 22.0 18.0 53.0 08.0 44.0 14.0 14.0 83.0 32.0 24.0 42.0 01.0 80.0 2R 45

Table 10: Monthly descriptive statistics for DJIA 30 stocks. Panel A reports descriptive statistics calculated across all stock-months (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute intraday mid-quotes. The sample period is October 2013 to December 2017 and the sample size is 4,791 stock-months. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 4.30 142 56.9 104 81.0 15.8 49.3 80.8 37.4 54.8 105 10pct 1.91 80.6 0.00 41.3 40.1 0.00 25.4 43.7 0.00 25.1 52.7 25pct 2.40 94.5 0.00 55.0 52.6 0.00 32.0 53.1 0.00 33.2 67.2 Med 3.67 118 11.4 76.1 71.0 10.5 41.5 67.5 25.3 45.1 87.3 75pct 5.48 155 88.5 110 96.6 24.2 55.2 89.1 59.1 62.9 116 90pct 7.37 221 153 174 136 39.2 76.8 124 91.3 89.5 166 RMSE 104 148 87.7 22.4 54.6 92.7 59.1 63.7 124 B. Cross-sectional averages of time-series moments Std 1.88 73.4 81.2 91.2 38.2 18.9 26.7 42.3 46.3 32.7 57.8 (0.97) (52.8) (32.1) (49.5) (9.89) (4.93) (14.3) (27.8) (13.8) (18.7) (37.4) ρ(ES,·) 0.40 0.14 0.26 0.26 0.14 0.36 0.38 0.14 0.35 0.37 (0.20) (0.21) (0.22) (0.19) (0.24) (0.26) (0.25) (0.24) (0.20) (0.24) ρ(RV,·) 0.40 0.38 0.65 0.59 0.16 0.70 0.80 0.26 0.73 0.83 (0.20) (0.34) (0.20) (0.18) (0.27) (0.23) (0.17) (0.33) (0.21) (0.15) C. Time-series averages of cross-sectional moments Std 1.57 45.1 60.6 72.7 29.7 15.2 16.2 25.1 36.8 20.9 36.8 (0.78) (39.3) (37.5) (63.0) (14.7) (7.78) (11.9) (21.4) (20.3) (15.8) (29.9) ρ(ES,·) 0.40 0.07 0.23 0.20 0.07 0.32 0.36 0.08 0.28 0.33 (0.20) (0.23) (0.23) (0.20) (0.22) (0.23) (0.21) (0.23) (0.22) (0.21) ρ(RV,·) 0.40 0.16 0.64 0.53 -0.01 0.51 0.65 0.16 0.59 0.71 (0.20) (0.28) (0.20) (0.22) (0.26) (0.22) (0.19) (0.28) (0.20) (0.16) 46

.skcots 03 AIJD rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :11 elbaT atad no snoisserger SLO deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP stroper C lenaP .sesehtnerap ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed -t ehT .setamitse noisserger lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht .setamitse lanoitces-ssorc eht fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats .shtnom-kcots 197,4 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 86.0 1.15 33.1 7.62 55.8 7.02 23.3 5.93 48.4 5.42 17.5 12.8 5.24 0.26 96.0- 7.45 01.9- 4.52 tsnoc )23.0( )88.3( )50.1( )22.4( )38.2( )41.6( )35.1( )30.4( )55.3( )27.4( )09.4( )22.6( )55.7( )3.61( )01.0-( )29.4( )25.1-( )71.3( 09.0 5.21 07.0 45.6 70.1 78.3 72.1 06.9 42.1 77.5 91.1 67.1 30.0- 44.4 53.1- 4.11 16.0- 33.7 β SE )31.2( )37.2( )74.2( )88.2( )11.2( )93.3( )14.3( )87.2( )83.4( )60.3( )22.5( )28.4( )70.0-( )24.3( )36.1-( )09.2( )65.0-( )63.2( 17.0 53.0 71.0 15.0 82.0 40.0 72.0 87.0 84.0 β VR )1.53( )6.82( )42.9( )9.22( )4.02( )89.5( )10.8( )2.31( )2.01( 48.0 91.0 27.0 71.0 21.0 40.0 38.0 02.0 27.0 02.0 70.0 50.0 43.0 70.0 14.0 70.0 42.0 40.0 2R noitamitse nihtiW .B 30.1 7.21 19.0 48.6 04.1 04.4 55.1 99.9 64.1 80.6 72.1 49.1 42.0 17.4 75.1- 6.11 60.0 23.8 β SE )53.2( )94.2( )31.3( )46.2( )36.2( )40.3( )84.3( )45.2( )63.4( )77.2( )85.4( )49.3( )64.0( )39.2( )87.1-( )45.2( )50.0( )71.2( 07.0 53.0 81.0 05.0 82.0 40.0 72.0 97.0 94.0 β VR )6.13( )8.62( )03.9( )2.12( )9.81( )24.6( )61.9( )8.31( )48.9( 18.0 61.0 96.0 51.0 21.0 40.0 18.0 81.0 96.0 81.0 70.0 40.0 13.0 60.0 93.0 50.0 32.0 40.0 2R noitamitse emit-naem delooP .C 59.0 83.8 24.0 89.3 67.0 64.2 85.1 01.6 41.1 54.3 88.0 37.0 37.0- 95.3 23.3- 6.01 49.0- 39.2 β SE )72.2( )33.8( )26.1( )72.8( )45.1( )72.5( )23.6( )84.8( )31.6( )79.9( )68.4( )22.4( )95.1-( )55.7( )64.3-( )41.7( )69.0-( )26.3( 16.0 92.0 51.0 73.0 91.0 10.0- 54.0 32.1 92.0 β VR )1.92( )8.52( )91.8( )7.02( )0.81( )96.1-( )8.51( )3.31( )94.8( 65.0 51.0 24.0 31.0 41.0 60.0 94.0 71.0 63.0 51.0 11.0 50.0 53.0 80.0 84.0 11.0 51.0 60.0 2R 47

Table 12: Annual descriptive statistics for DJIA 30 stocks. Panel A reports descriptive statistics calculated across all stock-years (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute midquotes. All variables were winsorized at the 99.9% level prior to calculating the descriptive statistics. The sample period is October 2003 to December 2017 and the sample size is 388 stock-years. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 4.32 157 45.0 43.2 41.5 11.0 49.8 81.5 32.0 55.7 107 10pct 1.97 94.3 0.00 20.6 20.5 0.00 31.9 51.2 0.00 34.5 65.7 25pct 2.43 110 0.00 26.3 25.6 3.91 36.1 59.0 0.00 40.0 76.5 Med 3.90 130 23.2 34.4 33.9 9.65 43.0 69.5 30.2 48.0 91.5 75pct 5.46 170 65.4 48.0 46.6 16.2 53.8 88.7 45.4 59.6 115 90pct 7.00 239 122 80.8 78.0 22.8 73.0 125 66.8 84.8 159 RMSE 74.5 48.0 44.5 10.9 50.4 87.1 42.7 58.1 117 B. Cross-sectional averages of time-series moments Std 1.60 70.6 47.2 21.9 19.3 6.68 17.1 30.1 25.7 20.5 40.7 (1.59) (80.4) (27.9) (14.1) (11.0) (3.19) (13.1) (26.8) (16.7) (17.9) (40.4) ρ(ES,·) 0.40 0.11 0.31 0.31 0.33 0.47 0.43 0.22 0.42 0.40 (0.33) (0.31) (0.37) (0.38) (0.47) (0.33) (0.35) (0.41) (0.35) (0.36) ρ(RV,·) 0.40 0.47 0.74 0.73 0.18 0.84 0.85 0.22 0.85 0.85 (0.33) (0.43) (0.28) (0.28) (0.51) (0.31) (0.32) (0.51) (0.30) (0.32) C. Time-series averages of cross-sectional moments Std 1.66 49.6 42.0 16.2 14.6 6.34 10.9 20.1 25.6 14.5 29.7 (1.07) (50.7) (21.9) (9.82) (7.28) (2.86) (8.11) (17.3) (14.4) (11.6) (26.6) ρ(ES,·) 0.49 0.06 0.25 0.23 0.09 0.55 0.53 0.19 0.49 0.50 (0.18) (0.22) (0.22) (0.20) (0.19) (0.12) (0.15) (0.22) (0.16) (0.17) ρ(RV,·) 0.49 0.11 0.45 0.43 -0.15 0.79 0.84 0.07 0.80 0.85 (0.18) (0.31) (0.18) (0.18) (0.22) (0.09) (0.09) (0.22) (0.11) (0.09) 48

skcots 03 AIJD rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :31 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf launna eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc raey-yb-raey fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .sraey-kcots 883 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 2.81 7.05 2.31 6.82 12.9 3.41 7.61 3.93 4.41 3.62 29.6 96.6 6.31 1.52 19.8 9.22 5.01 5.92 tsnoc )53.5( )57.4( )52.4( )65.6( )14.1( )12.2( )05.4( )95.5( )00.4( )71.8( )21.2( )48.1( )65.2( )15.6( )08.1( )35.5( )26.0( )33.2( 84.1 0.31 77.0 72.6 82.2 90.4 47.1 67.9 71.1 24.5 90.1 00.1 92.0- 97.3 03.0- 86.4 81.3- 85.3 β SE )41.2( )86.2( )96.1( )59.2( )47.1( )65.3( )88.2( )37.2( )77.2( )62.3( )18.1( )50.2( )54.0-( )15.2( )44.0-( )34.2( )13.1-( )07.1( 25.0 52.0 80.0 63.0 91.0 00.0- 91.0 32.0 13.0 β VR )9.51( )9.81( )70.7( )8.61( )3.32( )74.0-( )8.41( )7.51( )8.22( 78.0 43.0 38.0 23.0 41.0 01.0 68.0 63.0 18.0 63.0 80.0 80.0 05.0 51.0 65.0 71.0 81.0 20.0 2R noitamitse nihtiW .B 07.1 3.31 49.0 95.6 24.2 08.4 31.2 3.01 33.1 47.5 11.1 22.1 20.0 73.4 70.0 33.5 38.2- 50.5 β SE )88.2( )58.2( )80.2( )98.2( )32.2( )98.3( )96.2( )28.2( )75.2( )90.3( )09.1( )87.2( )30.0( )32.2( )90.0( )72.2( )62.1-( )92.1( 35.0 62.0 11.0 73.0 02.0 10.0 02.0 42.0 63.0 β VR )6.42( )8.72( )18.6( )9.82( )9.62( )64.0( )0.52( )4.32( )9.21( 88.0 92.0 38.0 82.0 71.0 01.0 68.0 23.0 18.0 13.0 90.0 90.0 25.0 41.0 85.0 61.0 32.0 30.0 2R noitamitse emit-naem delooP .C 69.1 07.8 10.1 91.4 07.3 62.3 19.1 52.6 53.1 46.3 79.0 15.0 52.0 21.2 52.0 54.2 34.0- 02.1 β SE )09.1( )75.5( )39.1( )45.6( )12.3( )24.3( )76.4( )18.6( )63.7( )3.11( )87.3( )19.1( )03.0( )29.2( )92.0( )79.2( )52.0-( )56.0( 25.0 52.0 30.0- 43.0 81.0 40.0- 61.0 71.0 61.0 β VR )1.31( )4.21( )39.0-( )4.31( )8.41( )69.3-( )78.8( )57.9( )88.1( 77.0 82.0 96.0 62.0 31.0 90.0 57.0 03.0 96.0 23.0 31.0 40.0 62.0 90.0 82.0 11.0 41.0 50.0 2R 49

Table 14: Monthly descriptive statistics for S&P 600 stocks. Panel A reports descriptive statistics calculated across all stock-months (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute intraday mid-quotes. The sample period is October 2003 to December 2017 and the sample size is 95,923 stock-months. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 20.1 255 108 197 119 35.5 91.3 143 73.7 96.9 182 10pct 8.22 135 0.00 69.0 64.4 0.00 42.9 72.0 0.00 40.6 86.3 25pct 10.9 167 0.00 95.1 85.1 0.00 56.8 92.2 0.00 56.3 113 Med 15.6 218 53.8 139 114 24.0 77.7 122 50.5 80.3 153 75pct 23.4 298 169 212 146 52.7 109 168 114 116 215 90pct 35.7 414 289 330 175 86.8 153 235 185 169 305 RMSE 173 298 108 42.8 86.8 146 105 97.5 195 B. Cross-sectional averages of time-series moments Std 8.05 110 133 219 40.2 35.7 40.5 59.9 80.6 49.2 83.5 (8.12) (59.5) (57.8) (161) (10.6) (16.9) (23.3) (35.8) (36.8) (27.7) (48.7) ρ(ES,·) 0.39 0.17 0.16 0.25 0.20 0.42 0.44 0.19 0.38 0.40 (0.30) (0.24) (0.28) (0.25) (0.26) (0.30) (0.30) (0.26) (0.29) (0.30) ρ(RV,·) 0.39 0.28 0.57 0.45 0.15 0.56 0.65 0.28 0.60 0.69 (0.30) (0.28) (0.22) (0.21) (0.28) (0.28) (0.26) (0.28) (0.25) (0.23) C. Time-series averages of cross-sectional moments Std 14.9 108 128 226 40.4 37.2 40.3 58.1 79.0 46.9 79.0 (5.12) (33.0) (48.7) (61.9) (4.52) (13.9) (16.3) (24.2) (31.8) (21.0) (34.8) ρ(ES,·) 0.42 0.16 0.11 0.23 0.33 0.46 0.43 0.27 0.42 0.40 (0.14) (0.09) (0.10) (0.10) (0.10) (0.11) (0.11) (0.11) (0.12) (0.12) ρ(RV,·) 0.42 0.27 0.52 0.44 0.20 0.61 0.69 0.30 0.64 0.72 (0.14) (0.10) (0.07) (0.15) (0.10) (0.09) (0.09) (0.10) (0.08) (0.07) 50

skcots 006 P&S rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :51 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf ylhtnom eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .shtnom-kcots 329,59 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 13.7 611 57.2 8.75 67.5- 2.33 4.61 4.49 2.11 8.65 27.5 0.61 6.67 401 8.11- 451 88.3- 0.46 tsnoc )27.0( )4.12( )05.0( )9.91( )19.0-( )4.01( )84.2( )9.52( )89.2( )2.62( )96.3( )8.61( )0.61( )3.43( )96.0-( )7.14( )34.0-( )5.61( 26.0 82.3 06.0 59.1 60.1 20.2 25.0 24.2 06.0 27.1 27.0 79.0 60.0 37.0 29.1- 31.2 35.0 91.2 β SE )06.6( )26.5( )79.8( )32.6( )37.7( )58.6( )3.11( )33.6( )7.02( )84.7( )9.02( )9.21( )30.1( )7.32( )88.6-( )26.5( )29.4( )57.5( 46.0 23.0 32.0 64.0 72.0 60.0 61.0 79.0 04.0 β VR )1.41( )3.31( )29.7( )1.31( )2.31( )05.6( )8.01( )08.9( )48.8( 07.0 22.0 06.0 32.0 12.0 21.0 86.0 22.0 16.0 62.0 71.0 41.0 52.0 70.0 72.0 20.0 61.0 60.0 2R noitamitse nihtiW .B 11.1 21.4 28.0 63.2 01.1 72.2 29.0 40.3 08.0 40.2 07.0 00.1 30.0- 17.0 67.2- 14.2 86.0 37.2 β SE )17.7( )90.5( )84.8( )55.5( )63.6( )18.5( )2.21( )46.5( )1.51( )73.6( )6.41( )07.9( )43.0-( )5.51( )42.8-( )92.4( )63.4( )01.5( 26.0 23.0 42.0 34.0 52.0 60.0 51.0 60.1 24.0 β VR )8.31( )3.31( )55.7( )9.11( )0.21( )80.6( )3.11( )88.8( )22.8( 66.0 22.0 55.0 12.0 71.0 90.0 46.0 32.0 55.0 32.0 11.0 80.0 91.0 40.0 62.0 10.0 41.0 50.0 2R noitamitse emit-naem delooP .C 46.0 81.2 06.0 53.1 89.0 74.1 36.0 07.1 86.0 82.1 77.0 58.0 12.0 07.0 84.2- 14.1 55.0 44.1 β SE )93.7( )0.51( )2.01( )2.51( )6.01( )9.31( )1.11( )0.71( )9.41( )8.71( )5.61( )2.71( )96.7( )4.61( )0.11-( )1.31( )54.6( )7.31( 94.0 42.0 61.0 43.0 91.0 30.0 71.0 23.1 82.0 β VR )1.13( )6.92( )9.81( )3.62( )3.52( )02.8( )2.71( )8.02( )3.32( 45.0 81.0 54.0 91.0 31.0 90.0 15.0 91.0 44.0 32.0 31.0 21.0 22.0 60.0 03.0 20.0 90.0 40.0 2R 51

Table 16: Annual descriptive statistics for S&P 600 stocks. Panel A reports descriptive statistics calculated across all stock-years (pooled). Panel B reports cross-sectional averages, and the associated standard deviations in parentheses, of stock-specific time-series standard deviation and correlation with the TAQ effective spread (ES) and 5-minute realized volatility RV. Panel C reports the time-series averages, and the associated standard devationsinparentheses,ofcross-sectionalstandarddeviationandcorrelationwiththeTAQ effective spread and realized volatility. The realized volatility is based on 5-minute midquotes. All variables were winsorized at the 99.9% level prior to calculating the descriptive statistics. The sample period is October 2003 to December 2017 and the sample size is 7,122 stock-years. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Pooled Mean 19.7 267 88.8 75.3 69.7 26.9 90.9 142 63.5 96.7 182 10pct 8.81 159 0.00 36.1 35.5 0.81 53.5 85.1 0.00 55.5 106 25pct 11.4 192 0.00 46.2 45.0 10.6 64.1 102 0.00 66.8 128 Med 16.0 242 68.5 61.8 59.4 22.2 80.0 126 52.9 84.4 161 75pct 23.2 312 139 90.1 83.6 36.7 105 164 93.8 113 211 90pct 34.4 407 218 130 117 56.1 142 221 142 152 283 RMSE 121 69.5 60.5 20.7 79.5 135 75.5 86.8 179 B. Cross-sectional averages of time-series moments Std 4.60 59.1 64.7 25.2 20.9 12.7 20.9 32.1 39.6 23.5 42.3 (5.91) (53.3) (48.6) (22.3) (17.4) (10.8) (19.4) (29.4) (31.8) (21.6) (38.4) ρ(ES,·) 0.31 0.15 0.30 0.29 0.30 0.39 0.36 0.23 0.35 0.33 (0.56) (0.53) (0.54) (0.54) (0.54) (0.54) (0.55) (0.53) (0.55) (0.56) ρ(RV,·) 0.31 0.24 0.46 0.44 0.22 0.67 0.70 0.23 0.67 0.70 (0.56) (0.56) (0.55) (0.55) (0.58) (0.50) (0.47) (0.57) (0.48) (0.46) C. Time-series averages of cross-sectional moments Std 12.8 84.1 83.7 32.2 27.6 20.7 28.9 43.5 55.3 31.6 56.5 (3.10) (23.3) (32.4) (14.5) (10.2) (7.59) (9.98) (15.1) (23.5) (12.5) (21.3) ρ(ES,·) 0.47 0.24 0.41 0.40 0.59 0.58 0.50 0.46 0.56 0.49 (0.14) (0.07) (0.08) (0.07) (0.04) (0.12) (0.12) (0.10) (0.15) (0.14) ρ(RV,·) 0.47 0.19 0.51 0.46 0.31 0.84 0.87 0.30 0.84 0.88 (0.14) (0.09) (0.11) (0.12) (0.08) (0.04) (0.04) (0.11) (0.04) (0.04) 52

skcots 006 P&S rof ytilitalov dezilaer etunim-5 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :71 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf launna eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc raey-yb-raey fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .sraey-kcots 221,7 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 93.0 321 68.2- 3.06 0.12- 6.41 80.4 8.79 97.0 7.75 30.3- 03.6 4.41 1.84 32.1 3.74 0.31- 8.44 tsnoc )50.0( )7.12( )56.0-( )6.22( )63.2-( )24.4( )18.0( )4.02( )32.0( )2.12( )94.1-( )85.6( )49.4( )5.01( )03.0( )3.11( )76.1-( )07.7( 83.0 99.2 15.0 58.1 37.1 94.2 72.0 62.2 84.0 96.1 58.0 50.1 83.0 01.1 44.0 24.1 10.1 42.2 β SE )35.2( )05.4( )91.4( )57.4( )05.6( )33.5( )97.4( )00.5( )6.11( )75.5( )8.51( )95.9( )79.4( )37.8( )21.6( )07.5( )47.6( )89.4( 56.0 43.0 91.0 05.0 03.0 50.0 81.0 42.0 13.0 β VR )1.22( )7.02( )46.6( )3.91( )0.81( )95.4( )6.31( )4.01( )01.8( 78.0 42.0 38.0 03.0 33.0 52.0 78.0 42.0 58.0 13.0 93.0 53.0 93.0 61.0 64.0 81.0 81.0 90.0 2R noitamitse nihtiW .B 76.0 40.4 36.0 04.2 07.1 19.2 25.0 90.3 85.0 71.2 37.0 01.1 13.0 93.1 84.0 39.1 18.0 99.2 β SE )82.4( )31.4( )91.5( )14.4( )27.5( )31.5( )08.6( )74.4( )6.01( )98.4( )1.01( )23.8( )23.3( )37.5( )83.5( )86.4( )39.3( )92.4( 56.0 43.0 32.0 05.0 13.0 70.0 12.0 82.0 24.0 β VR )0.03( )8.03( )50.9( )8.22( )1.02( )96.5( )8.31( )1.41( )0.41( 68.0 52.0 38.0 82.0 82.0 71.0 78.0 52.0 58.0 92.0 13.0 22.0 93.0 21.0 84.0 61.0 12.0 70.0 2R noitamitse emit-naem delooP .C 64.0 71.2 65.0 83.1 38.1 10.2 63.0 86.1 45.0 13.1 29.0 59.0 55.0 68.0 95.0 30.1 33.1 26.1 β SE )89.2( )17.7( )08.4( )75.7( )49.6( )70.7( )25.4( )45.9( )72.8( )0.01( )1.11( )6.11( )94.6( )4.21( )42.6( )3.01( )76.5( )90.8( 55.0 72.0 60.0 24.0 52.0 10.0 01.0 41.0 90.0 β VR )3.52( )7.32( )19.2( )3.52( )2.82( )40.2( )2.01( )5.01( )27.2( 97.0 62.0 57.0 43.0 32.0 22.0 87.0 62.0 67.0 53.0 53.0 53.0 82.0 61.0 23.0 81.0 80.0 60.0 2R 53

Table 18: Empirical results for foreign exchange rates. The table reports pooled descriptive statisticsforthemonthlylow-frequencymeasuresandpooledleastsquaresregressionsofthe low-frequencymeasuresonthetruespread(ES)and5-minuterealizedvolatility(RV)atthe monthly frequency. Driscoll-Kraay standard errors, which are robust to heteroskedasticity, serial correlation, and cross-sectional dependence are reported in parentheses. The sample period is January 2008 to December 2015 and the sample size is 480 exchange rate-months. ES RV R H H CS CS CS AR AR AR M L T M D P M D P A. Summary statistics Mean 0.68 70.6 25.9 46.3 44.8 13.3 29.0 44.7 13.7 24.2 47.6 10pct 0.43 34.2 0.00 15.5 15.0 0.00 14.1 20.0 0.00 7.74 17.7 25pct 0.53 52.7 0.00 27.8 27.1 4.12 19.1 30.1 0.00 14.4 29.6 Med 0.64 68.1 6.61 40.6 39.6 12.5 26.8 41.8 1.19 22.4 44.4 75pct 0.81 83.6 44.6 56.4 54.6 18.2 36.3 55.5 23.3 31.0 60.3 90pct 1.10 123 95.2 97.7 88.5 34.2 53.6 80.3 49.6 48.3 93.0 Std 0.22 32.7 34.0 32.0 31.0 11.5 13.3 21.5 19.5 14.7 26.7 RMSE 42.3 55.7 53.8 17.0 31.2 48.9 23.4 27.7 53.9 ρ(ES,·) 0.56 0.29 0.39 0.38 0.34 0.56 0.52 0.31 0.51 0.51 ρ(RV,·) 0.56 0.37 0.71 0.65 0.16 0.80 0.89 0.31 0.82 0.91 B. Regression on ES const -4.80 7.69 8.92 1.21 5.88 9.96 -4.61 1.05 5.85 (-0.74) (0.87) (1.11) (0.86) (1.59) (1.36) (-1.35) (0.22) (0.57) β 45.2 56.8 52.7 17.8 34.0 51.1 27.0 34.0 61.4 ES (4.50) (4.06) (4.16) (9.57) (6.02) (4.45) (4.91) (4.36) (3.64) R2 0.09 0.15 0.14 0.12 0.32 0.27 0.09 0.26 0.26 C. Regression on ES and RV const -9.43 -2.44 0.09 1.43 1.70 1.56 -6.37 -4.00 -4.96 (-1.89) (-0.92) (0.03) (1.02) (1.11) (0.84) (-2.09) (-2.90) (-2.61) β 19.1 -0.32 2.99 19.1 10.5 3.76 17.1 5.53 0.48 ES (2.06) (-0.06) (0.53) (7.71) (5.15) (1.31) (3.91) (2.37) (0.20) β 0.32 0.69 0.60 -0.02 0.29 0.57 0.12 0.35 0.74 RV (5.23) (16.4) (12.0) (-0.66) (8.14) (13.7) (2.56) (13.0) (26.2) R2 0.15 0.50 0.42 0.12 0.66 0.80 0.12 0.67 0.82 54

Table 19: Simulation results for the liquidity-adjusted CAPM of Acharya and Pedersen (2005). Panels A–C report the mean beta estimates obtained in the simulation using different measures of stock and market transaction costs. The column labeled “True” reports thetruebetas, thecolumnlabeled“s”reportsbetaestimateswhenthetrueeffectivespread is used in estimation, and the remaining columns show mean beta estimates when the lowfrequency measures are used for estimation. Panel D reports the mean annualized price of market risk (λ ). The simulations are based on 10,000 replications. m True s R H CS CS CS AR AR AR M L M D P M D P A. Small β β 0.50 0.50 0.49 0.51 0.50 0.50 0.50 0.50 0.50 0.50 1 β ×104 1.19 1.29 57.0 53.4 6.37 19.8 50.6 17.7 18.2 64.8 2 β ×104 1.19 1.39 -3.38 52.8 -1.53 2.65 8.17 -1.72 2.81 10.3 3 β ×104 1.19 1.31 -2.95 32.0 -1.45 2.62 8.10 -1.58 2.90 10.2 4 B. Medium β β 1.00 1.02 1.01 1.03 1.02 1.02 1.01 1.02 1.02 1.01 1 β ×104 2.38 2.66 117.3 105.3 13.1 40.4 103.5 36.4 37.3 132.2 2 β ×104 2.38 2.91 -7.59 108.0 -3.26 5.25 16.58 -3.84 5.60 20.8 3 β ×104 2.38 2.90 -7.89 106.3 -3.01 5.24 16.67 -3.68 5.65 21.2 4 C. Large β β 1.50 1.50 1.48 1.52 1.50 1.50 1.49 1.49 1.50 1.49 1 β ×104 3.57 3.96 173.8 146.1 19.5 59.4 152.2 54.0 54.9 194.7 2 β ×104 3.57 4.23 -10.4 160.2 -4.65 8.09 25.0 -5.24 8.74 31.5 3 β ×104 3.57 4.56 -9.98 188.3 -4.14 8.01 24.3 -4.64 8.62 31.5 4 D. Price of risk λ (% p.a.) 5.00 4.97 0.11 -4.58 3.91 -0.58 -5.01 2.69 -0.05 -6.21 m 55

Table 20: Simulation results for the three-factor model with aggregate liquidity and volatility risks. The table reports the mean price of market risk (λ ), aggregate liquidity risk m (λ ), and aggregate volatility risk (λ ) obtained in the simulation. The prices of risk are s σ expressed in % per annum. The column labeled “True” reports the true prices of risk, the column labeled “s” reports the estimated prices of risk when the true effective spread is used in estimation, and the remaining columns report the mean estimated prices of risk when the loq-frequency measures are used in estimation. Panel A report results based on daily realized volatility as a proxy for the true volatility, while panel B reports results based on daily range-based volatility. All results are based on 10,000 replications. True s R H CS CS CS AR AR AR M T M D P M D P A. Daily realized volatility Liquidity and volatility risks priced λ 5.00 5.02 5.01 5.00 5.01 5.01 5.01 5.01 5.01 5.01 m λ -1.00 -0.99 3.59 1.23 -7.12 -7.85 -8.50 -3.93 -1.13 -4.24 s λ -1.00 -0.93 -1.00 -0.97 -0.62 -0.34 -0.34 -1.01 -1.06 -1.16 σ Volatility risk priced λ 5.00 5.02 5.01 5.01 5.01 5.01 5.01 5.01 5.01 5.01 m λ 0.00 0.02 -23.1 -3.54 2.54 -2.22 -4.60 -9.74 -6.86 -1.53 s λ -1.00 -0.96 -0.92 -0.85 -1.14 -1.03 -1.23 -0.85 -0.82 -1.09 σ Liquidity risk priced λ 5.00 5.02 5.01 5.00 5.01 5.01 5.01 5.01 5.01 5.01 m λ -1.00 -0.99 20.4 9.42 -11.1 -7.12 -4.18 0.81 5.29 1.36 s λ 0.00 0.10 -0.03 -0.02 0.65 1.01 1.11 -0.09 -0.18 -0.17 σ B. Daily realized range Liquidity and volatility risks priced λ 5.00 5.02 5.01 5.00 5.01 5.01 5.01 5.01 5.01 5.01 m λ -1.00 -0.99 3.65 0.79 -8.50 -7.11 -8.29 -3.76 -0.73 -3.46 s λ -1.00 -0.91 -0.97 -0.92 -0.34 -0.85 -0.83 -0.97 -0.95 -0.95 σ Volatility risk priced λ 5.00 5.02 5.01 5.01 5.01 5.01 5.01 5.01 5.01 5.01 m λ 0.00 0.02 -23.0 -4.71 -4.60 2.56 -1.61 -9.74 -7.20 -0.13 s λ -1.00 -0.91 -0.90 -0.84 -1.23 -0.94 -0.89 -0.84 -0.91 -0.78 σ Liquidity risk priced λ 5.00 5.02 5.01 5.00 5.01 5.01 5.01 5.01 5.01 5.01 m λ -1.00 -0.99 20.4 9.18 -4.18 -11.1 -8.11 0.99 5.93 1.46 s λ 0.00 0.07 -0.04 0.01 1.11 0.16 0.17 -0.07 0.00 -0.07 σ 56

E(R ) as function of s E(R ) as function of s 5 M 5 M ss==00..11 s=s=00..11 ss==00..55 s=s=00..55 4 ss==11..00 4 s=s=11..00 ss==33..00 s=s=33..00 ss==55..00 s=s=55..00 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 E(H ) as function of s E(H ) as function of s 5 L 5 L 4 4 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 E(H ) as function of s E(H ) as function of s 5 T 5 T 4 4 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 Figure 1: Expectations of the low-frequency measures. The figure shows E(R ), E(H ), M T andE(H )asafunctionofvolatility(leftpanel)andtruespread(rightpanel)whenT = 21. L The expectations are approximated by simulation with 10,000 replications. 57

E(CS ) as function of s E(CS ) as function of s 5 M 5 M ss==00..11 s=s=00..11 ss==00..55 s=s=00..55 4 ss==11..00 4 s=s=11..00 ss==33..00 s=s=33..00 ss==55..00 s=s=55..00 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 E(CS ) as function of s E(CS ) as function of s D D 4 4 2 2 0 1 2 3 4 5 0 1 2 3 4 5 E(CS ) as function of s E(CS ) as function of s P P 6 6 4 4 2 2 0 1 2 3 4 5 0 1 2 3 4 5 Figure 2: Expectations of the low-frequency measures. The figure shows E(CS ), E(CS ), M D and E(CS ) as a function of volatility (left panel) and spread (right panel) when T = 21. P The expectations are approximated by simulation with 10,000 replications. 58

E(AR ) as function of s E(AR ) as function of s 5 M 5 M ss==00..11 s=s=00..11 ss==00..55 s=s=00..55 4 ss==11..00 4 s=s=11..00 ss==33..00 s=s=33..00 ss==55..00 s=s=55..00 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 E(AR ) as function of s E(AR ) as function of s 5 D 5 D 4 4 3 3 2 2 1 1 0 1 2 3 4 5 0 1 2 3 4 5 E(AR ) as function of s E(AR ) as function of s 6 P 6 P 4 4 2 2 0 1 2 3 4 5 0 1 2 3 4 5 Figure3: Expectationsofthelow-frequencymeasures. ThefigureshowsE(AR ),E(AR ), M D andE(AR )asafunctionofvolatility(leftpanel)andspread(rightpanel)whenT = 21.The P expectations are approximated by simulation with 10,000 replications. 59

A Simulation results with ρ = 0.75 In this section, we report simulations results with the correlation between the effective spread and volatility set equal to 0.75 rather than 0.5 as in our baseline results reported in Tables 3, 19, and 20. The results are reported in Tables 21, 22, and 23. 60

ycneuqerf-wol eht fo noisserger dna ) σ( ytilitalov dna ) s( daerps eurt eht htiw serusaem ycneuqerf-wol fo snoitalerroC :12 elbaT t t htoB .%0.2 slauqe ytilitalov fo ytilitalov dna %0.2 slauqe ytilitalov egareva ehT .ytilitalov ro/dna daerps eurt eht no serusaem desab era snoitalumis llA .57.0 = ρ htiw detalerroc ylsuoenaropmetnoc dna detubirtsid yllamrongol dii era daerps dna ytilitalov .snoitacilper 000,01 no 152= T 12= T RA RA RA SC SC SC H H R RA RA RA SC SC SC H H R P D M P D M T L M P D M P D M T L M 1.0=)s(dtS ,1.0=)s(E :daerps liuqnart ,llamS .A 67.0 57.0 44.0 77.0 77.0 56.0 26.0 76.0 34.0 47.0 27.0 64.0 57.0 47.0 24.0 55.0 86.0 04.0 s htiw .rroC t 00.1 99.0 55.0 00.1 99.0 96.0 77.0 98.0 06.0 69.0 29.0 75.0 79.0 49.0 25.0 56.0 98.0 15.0 σ htiw .rroC t 83.0 91.0 40.0 33.0 91.0 20.0 03.0 91.0 50.0 63.0 71.0 60.0 23.0 91.0 60.0 36.0 54.0 91.0 α s no .geR t 8.11 79.5 10.2 6.01 43.6 43.1 37.2 48.4 66.3 2.21 62.6 48.3 8.01 73.6 39.1 89.2 76.8 99.5 β s 75.0 75.0 91.0 95.0 06.0 24.0 93.0 54.0 81.0 55.0 25.0 12.0 75.0 45.0 81.0 03.0 64.0 61.0 2R 00.0 10.0- 20.0- 10.0- 00.0 10.0- 22.0 40.0 50.0- 30.0- 30.0- 50.0- 20.0- 00.0 00.0 65.0 81.0 20.0- α σ , s no .geR t t 01.0 80.0 12.0 24.0 14.0 16.0 14.0 40.0 83.0- 47.0 55.0 95.0 96.0 15.0 03.0 97.0 47.0 03.0 β s 97.0 93.0 21.0 86.0 04.0 50.0 51.0 13.0 62.0 77.0 83.0 22.0 86.0 93.0 11.0 51.0 35.0 93.0 β σ 99.0 99.0 13.0 00.1 99.0 15.0 06.0 97.0 63.0 29.0 48.0 23.0 49.0 98.0 72.0 44.0 97.0 62.0 2R 5.0=)s(dtS ,5.0=)s(E :daerps elitalov yletaredom ,muideM .B 97.0 28.0 87.0 38.0 88.0 59.0 96.0 57.0 45.0 67.0 77.0 65.0 28.0 48.0 37.0 75.0 17.0 54.0 s htiw .rroC t 99.0 99.0 16.0 99.0 79.0 57.0 37.0 78.0 45.0 69.0 29.0 45.0 69.0 39.0 26.0 46.0 88.0 25.0 σ htiw .rroC t 04.0 91.0 10.0- 63.0 02.0 00.0 13.0 71.0 90.0 73.0 71.0 50.0 23.0 81.0 20.0 56.0 54.0 21.0 α s no .geR t 94.2 33.1 79.0 15.2 27.1 00.1 66.0 71.1 79.0 55.2 63.1 60.1 85.2 57.1 50.1 36.0 58.1 15.1 β s 36.0 76.0 06.0 07.0 77.0 09.0 84.0 65.0 03.0 85.0 95.0 23.0 66.0 07.0 35.0 33.0 15.0 12.0 2R 00.0 10.0- 20.0- 10.0 00.0 10.0- 62.0 40.0 20.0 10.0- 00.0 10.0- 00.0 00.0 10.0- 85.0 91.0 10.0- α σ , s no .geR t t 13.0 82.0 29.0 06.0 56.0 59.0 23.0 43.0 85.0 23.0 13.0 76.0 76.0 76.0 88.0 42.0 33.0 15.0 β s 57.0 63.0 20.0 66.0 73.0 20.0 11.0 72.0 31.0 57.0 63.0 31.0 46.0 73.0 60.0 31.0 15.0 23.0 β σ 99.0 89.0 16.0 00.1 99.0 19.0 85.0 87.0 43.0 39.0 68.0 53.0 59.0 19.0 45.0 34.0 97.0 82.0 2R 0.3=)s(dtS ,0.3=)s(E :daerps elitalov ,egraL .C 89.0 99.0 00.1 89.0 00.1 00.1 77.0 89.0 89.0 79.0 89.0 79.0 79.0 89.0 89.0 24.0 09.0 28.0 s htiw .rroC t 48.0 96.0 37.0 58.0 87.0 37.0 93.0 07.0 47.0 38.0 86.0 17.0 58.0 87.0 37.0 81.0 86.0 36.0 σ htiw .rroC t 71.0 80.0- 10.0- 32.0 80.0 10.0- 68.0 20.0 10.0 61.0 90.0- 40.0- 42.0 90.0 00.0 72.1 44.0 60.0- α s no .geR t 00.1 19.0 00.1 40.1 00.1 89.0 55.0 69.0 89.0 00.1 19.0 00.1 40.1 99.0 89.0 41.0 67.0 49.0 β s 69.0 99.0 00.1 69.0 99.0 00.1 95.0 69.0 59.0 39.0 59.0 59.0 39.0 69.0 69.0 71.0 18.0 86.0 2R 30.0- 10.0- 10.0- 10.0- 10.0- 00.0 80.1 90.0 20.0 30.0- 30.0- 30.0- 10.0 00.0 00.0 33.1 24.0 60.0- α σ , s no .geR t t 18.0 89.0 00.1 28.0 19.0 89.0 97.0 30.1 99.0 18.0 79.0 10.1 18.0 09.0 79.0 12.0 47.0 39.0 β s 83.0 31.0- 00.0 74.0 71.0 10.0- 64.0- 41.0- 00.0 93.0 21.0- 30.0- 74.0 81.0 10.0 31.0- 30.0 10.0 β σ 99.0 99.0 00.1 99.0 00.1 00.1 76.0 79.0 59.0 69.0 69.0 59.0 79.0 79.0 69.0 12.0 18.0 86.0 2R 61

Table 22: Simulation results for the liquidity-adjusted CAPM of Acharya and Pedersen (2005). Panels A–C report the mean beta estimates obtained in the simulation using different measures of stock and market transaction costs. The column labeled “True” reports thetruebetas, thecolumnlabeled“s”reportsbetaestimateswhenthetrueeffectivespread is used in estimation, and the remaining columns show mean beta estimates when the lowfrequency measures are used for estimation. Panel D reports the mean annualized price of market risk (λ ). The simulations are based on 10,000 replications. m True s R H CS CS CS AR AR AR M L M D P M D P A. Small β β 0.50 0.50 0.49 0.51 0.50 0.50 0.50 0.50 0.50 0.50 1 β ×104 1.19 1.26 57.4 53.6 6.61 20.6 52.2 17.8 18.4 65.3 2 β ×104 1.19 1.31 -2.06 53.5 -1.38 3.10 8.90 -1.24 3.15 11.1 3 β ×104 1.19 1.34 -2.53 32.9 -1.38 3.13 8.85 -1.47 3.04 11.2 4 B. Medium β β 1.00 1.02 1.01 1.03 1.02 1.02 1.01 1.02 1.02 1.02 1 β ×104 2.38 2.60 118.1 105.5 13.6 42.16 106.5 36.6 37.6 133.4 2 β ×104 2.38 2.75 -4.73 109.1 -2.89 6.19 17.98 -2.64 6.33 22.4 3 β ×104 2.38 2.91 -4.48 108.4 -2.58 6.70 18.9 -2.44 6.60 23.1 4 C. Large β β 1.50 1.50 1.48 1.52 1.50 1.50 1.49 1.49 1.50 1.49 1 β ×104 3.57 3.87 175.0 146.2 20.2 62.1 156.7 54.3 55.4 196.2 2 β ×104 3.57 4.17 -8.14 161.4 -4.50 9.25 27.0 -4.56 9.27 33.3 3 β ×104 3.57 3.68 -5.78 189.5 -4.28 9.34 26.9 -3.25 9.71 33.1 4 D. Price of risk λ (% p.a.) 5.00 5.02 0.17 -4.53 4.02 -0.50 -4.93 2.77 0.00 -6.17 m 62

Table 23: Simulation results for the three-factor model with aggregate liquidity and volatility risks. The table reports the mean price of market risk (λ ), aggregate liquidity risk m (λ ), and aggregate volatility risk (λ ) obtained in the simulation. The prices of risk are s σ expressed in % per annum. The column labeled “True” reports the true prices of risk, the column labeled “s” reports the estimated prices of risk when the true effective spread is used in estimation, and the remaining columns report the mean estimated prices of risk when the loq-frequency measures are used in estimation. Panel A report results based on daily realized volatility as a proxy for the true volatility, while panel B reports results based on daily range-based volatility. All results are based on 10,000 replications. True s R H CS CS CS AR AR AR M T M D P M D P A. Daily realized volatility Liquidity and volatility risks priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ -1.00 -1.04 -6.57 -5.62 -4.18 -5.51 -7.28 -5.11 -3.13 -4.45 s λ -1.00 -1.00 -1.09 -1.06 -0.88 -0.80 -0.87 -1.07 -1.09 -1.18 σ Volatility risk priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ 0.00 -0.03 -25.7 -11.3 4.59 -0.75 -3.38 -9.94 -7.41 -4.16 s λ -1.00 -1.08 -0.93 -0.89 -1.28 -1.29 -1.62 -0.85 -0.80 -0.92 σ Liquidity risk priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ -1.00 -1.04 11.4 4.65 -9.80 -5.46 -3.89 0.40 3.60 2.33 s λ 0.00 0.07 -0.20 -0.18 0.43 0.58 0.72 -0.22 -0.29 -0.36 σ B. Daily realized range Liquidity and volatility risks priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ -1.00 -1.04 -5.34 -1.75 -7.28 -3.83 -5.47 -3.57 -2.11 -3.20 s λ -1.00 -1.01 -1.07 -1.04 -0.87 -1.00 -1.00 -1.06 -1.03 -0.97 σ Volatility risk priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ 0.00 -0.03 -28.8 -15.5 -3.38 4.28 -4.46 -11.5 -9.99 -13.1 s λ -1.00 -0.98 -0.90 -1.00 -1.62 -0.93 -0.77 -0.85 -0.97 -0.96 σ Liquidity risk priced λ 5.00 4.97 4.96 4.96 4.96 4.96 4.96 4.96 4.96 4.96 m λ -1.00 -1.04 16.2 14.7 -3.89 -9.12 -2.69 4.39 7.50 11.5 s λ 0.00 -0.04 -0.21 -0.06 0.72 -0.05 -0.18 -0.22 -0.08 0.01 σ 63

B Equity results with 30-minute realized volatility In this section, we re-run our monthly regressions of low-frequency measures on the TAQ effective spreads and realized volatility, calculating RV from 30-minute intraday returns rather than 5-minute returns used for our baseline results reported in Tables 7, 11, and 15. As can be clearly seen from the following tables, the results with 30-minute RV are remarkably similar to those with 5-minute RV for all stocks in our sample (Table 24) as well as for the very large caps (Table 25) and small caps (Table 26). 64

skcots 0051 P&S rof ytilitalov dezilaer etunim-03 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :42 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf ylhtnom eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .shtnom-kcots 632,142 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 9.21 901 54.6 4.45 26.0 6.13 8.61 1.58 5.11 3.05 56.5 9.21 0.56 9.29 0.11- 921 20.3- 5.85 tsnoc )69.1( )7.52( )97.1( )3.52( )51.0( )9.71( )84.3( )2.52( )22.4( )6.52( )37.4( )2.61( )7.31( )7.82( )27.0-( )8.33( )05.0-( )1.61( 66.0 09.3 26.0 42.2 01.1 51.2 96.0 00.3 67.0 80.2 48.0 90.1 02.0 41.1 56.1- 01.3 64.0 45.2 β SE )0.11( )05.6( )1.51( )60.7( )25.8( )28.7( )89.9( )92.7( )9.91( )45.8( )1.91( )7.41( )67.2( )0.03( )41.6-( )87.8( )37.4( )67.6( 96.0 53.0 22.0 05.0 82.0 50.0 02.0 20.1 54.0 β VR )9.41( )7.31( )20.8( )0.31( )6.21( )50.5( )08.9( )43.9( )28.9( 57.0 12.0 56.0 22.0 12.0 11.0 27.0 32.0 56.0 62.0 71.0 51.0 33.0 01.0 23.0 30.0 91.0 60.0 2R noitamitse nihtiW .B 12.1 22.5 68.0 09.2 90.1 05.2 21.1 39.3 49.0 55.2 77.0 90.1 70.0- 70.1 22.3- 03.3 45.0 43.3 β SE )69.9( )85.4( )9.01( )19.4( )52.6( )42.5( )8.41( )68.4( )7.81( )63.5( )6.51( )19.7( )26.0-( )8.31( )2.01-( )60.4( )05.4( )16.4( 76.0 43.0 42.0 74.0 72.0 50.0 91.0 90.1 74.0 β VR )4.31( )7.21( )45.7( )8.01( )4.01( )54.4( )6.11( )19.8( )25.9( 96.0 91.0 85.0 81.0 61.0 70.0 66.0 02.0 75.0 02.0 90.0 70.0 42.0 40.0 03.0 20.0 71.0 40.0 2R noitamitse emit-naem delooP .C 29.0 68.2 47.0 86.1 90.1 07.1 89.0 03.2 39.0 76.1 19.0 99.0 33.0 01.1 25.2- 25.2 96.0 78.1 β SE )3.41( )0.91( )1.51( )4.81( )7.11( )7.51( )8.02( )3.22( )5.32( )7.22( )6.81( )4.02( )2.01( )1.91( )8.11-( )9.42( )38.7( )2.71( 25.0 52.0 71.0 53.0 02.0 20.0 22.0 24.1 23.0 β VR )4.33( )7.13( )4.32( )9.72( )3.82( )46.7( )9.02( )0.42( )4.82( 95.0 22.0 05.0 32.0 41.0 90.0 65.0 62.0 05.0 92.0 51.0 41.0 92.0 01.0 53.0 30.0 11.0 40.0 2R 65

skcots 03 AIJD rof ytilitalov dezilaer etunim-03 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :52 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf ylhtnom eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .shtnom-kcots 197,4 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 83.3 1.15 86.2 7.62 60.9 7.02 15.5 5.93 80.6 5.42 69.5 12.8 4.34 0.26 10.1 7.45 45.7- 4.52 tsnoc )94.1( )88.3( )10.2( )22.4( )40.3( )41.6( )13.2( )30.4( )30.4( )27.4( )80.5( )22.6( )38.7( )3.61( )31.0( )29.4( )62.1-( )71.3( 20.1 5.21 57.0 45.6 60.1 78.3 14.1 06.9 33.1 77.5 22.1 67.1 20.0- 44.4 35.1- 4.11 06.0- 33.7 β SE )54.2( )37.2( )76.2( )88.2( )31.2( )93.3( )75.3( )87.2( )35.4( )60.3( )53.5( )28.4( )50.0-( )24.3( )78.1-( )09.2( )65.0-( )63.2( 57.0 83.0 81.0 35.0 92.0 40.0 92.0 48.0 25.0 β VR )2.23( )2.62( )87.9( )4.02( )7.71( )25.5( )01.8( )9.21( )1.01( 48.0 91.0 27.0 71.0 21.0 40.0 28.0 02.0 17.0 02.0 70.0 50.0 53.0 70.0 34.0 70.0 42.0 40.0 2R noitamitse nihtiW .B 81.1 7.21 99.0 48.6 14.1 04.4 47.1 99.9 75.1 80.6 13.1 49.1 72.0 17.4 37.1- 6.11 90.0 23.8 β SE )27.2( )94.2( )53.3( )46.2( )07.2( )40.3( )26.3( )45.2( )84.4( )77.2( )66.4( )49.3( )05.0( )39.2( )69.1-( )45.2( )80.0( )71.2( 47.0 83.0 91.0 35.0 92.0 40.0 82.0 58.0 35.0 β VR )4.82( )2.42( )48.9( )8.81( )4.61( )09.5( )72.9( )6.31( )16.9( 18.0 61.0 96.0 51.0 21.0 40.0 97.0 81.0 86.0 81.0 70.0 40.0 23.0 60.0 14.0 50.0 32.0 40.0 2R noitamitse emit-naem delooP .C 72.1 83.8 75.0 89.3 08.0 64.2 28.1 01.6 72.1 54.3 19.0 37.0 25.0- 95.3 59.2- 6.01 38.0- 39.2 β SE )12.3( )33.8( )52.2( )72.8( )46.1( )72.5( )55.7( )84.8( )50.7( )79.9( )30.5( )22.4( )71.1-( )55.7( )12.3-( )41.7( )78.0-( )26.3( 26.0 03.0 61.0 73.0 91.0 20.0- 74.0 72.1 03.0 β VR )7.72( )7.42( )74.8( )1.91( )0.71( )59.1-( )1.51( )8.31( )58.8( 55.0 51.0 14.0 31.0 41.0 60.0 84.0 71.0 53.0 51.0 11.0 50.0 53.0 80.0 94.0 11.0 51.0 60.0 2R 66

skcots 006 P&S rof ytilitalov dezilaer etunim-03 dna daerps evitceffe QAT no serusaem ycneuqerf-wol fo snoissergeR :62 elbaT deloop ,si taht ,setamitse nihtiw stroper B lenaP .setamitse noisserger SLO deloop stroper A lenaP .ycneuqerf ylhtnom eht ta ni detroper era scitsitats-t tsubor yaarK-llocsirD eht ,slenap htob nI .level kcots eht ta denaem-ed atad no snoisserger SLO lanoitces-ssorc htnom-yb-htnom fo segareva seires-emit eht ,si taht ,setamitse emit-naem deloop eht stroper C lenaP .sesehtnerap fo ecnairav eht fo rotamitse tseW-yeweN eht gnisu detaluclac era ,sesehtnerap ni detroper ,scitsitats-t ehT .setamitse noisserger .shtnom-kcots 329,59 si ezis elpmas eht dna 7102 rebmeceD ot 3002 rebotcO si doirep elpmas ehT .setamitse lanoitces-ssorc eht RA RA RA SC SC SC H H R P D M P D M T L M ————— ————— ————— ————— ————— ————— ————— ————— ————— noitamitse delooP .A lenaP 6.31 611 47.5 8.75 53.4- 2.33 9.12 4.49 6.41 8.65 58.6 0.61 1.87 401 68.6- 451 94.1- 0.46 tsnoc )74.1( )4.12( )41.1( )9.91( )47.0-( )4.01( )85.3( )9.52( )12.4( )2.62( )67.4( )8.61( )3.61( )3.43( )04.0-( )7.14( )81.0-( )5.61( 77.0 82.3 76.0 59.1 90.1 20.2 56.0 24.2 86.0 27.1 47.0 79.0 01.0 37.0 18.1- 31.2 85.0 91.2 β SE )35.9( )26.5( )3.11( )32.6( )94.8( )58.6( )4.41( )33.6( )1.32( )84.7( )4.12( )9.21( )45.1( )7.32( )86.7-( )26.5( )92.6( )57.5( 76.0 43.0 52.0 74.0 82.0 60.0 71.0 50.1 34.0 β VR )9.31( )3.31( )62.8( )5.21( )6.21( )61.6( )6.01( )67.9( )60.9( 17.0 22.0 16.0 32.0 22.0 21.0 86.0 22.0 06.0 62.0 61.0 41.0 62.0 70.0 92.0 20.0 71.0 60.0 2R noitamitse nihtiW .B 62.1 21.4 98.0 63.2 41.1 72.2 60.1 40.3 98.0 40.2 37.0 00.1 10.0 17.0 76.2- 14.2 37.0 37.2 β SE )67.9( )90.5( )2.01( )55.5( )01.7( )18.5( )4.41( )46.5( )9.61( )73.6( )5.51( )07.9( )80.0( )5.51( )82.9-( )92.4( )13.5( )01.5( 46.0 33.0 52.0 44.0 62.0 60.0 61.0 31.1 54.0 β VR )1.31( )7.21( )09.7( )9.01( )9.01( )45.5( )6.11( )71.9( )84.8( 76.0 22.0 65.0 12.0 81.0 90.0 36.0 32.0 45.0 32.0 11.0 80.0 91.0 40.0 82.0 10.0 51.0 50.0 2R noitamitse emit-naem delooP .C 77.0 81.2 66.0 53.1 10.1 74.1 47.0 07.1 57.0 82.1 97.0 58.0 52.0 07.0 32.2- 14.1 95.0 44.1 β SE )1.01( )0.51( )1.21( )2.51( )3.11( )9.31( )1.41( )0.71( )0.71( )8.71( )5.61( )2.71( )98.9( )4.61( )5.01-( )1.31( )86.7( )7.31( 15.0 52.0 71.0 43.0 91.0 20.0 81.0 14.1 13.0 β VR )7.92( )6.82( )8.91( )7.42( )0.42( )94.7( )6.71( )5.12( )1.42( 45.0 81.0 54.0 91.0 41.0 90.0 94.0 91.0 34.0 32.0 31.0 21.0 22.0 60.0 23.0 20.0 01.0 40.0 2R 67

C FX results with alternative high-frequency effective spread proxies AsmentionedinSection4.2.1, wedonotobservecontinuousquotesforourforeignexchange rates but only 100–millisecond snapshots. Thus, transaction prices cannot be directly compared to prevailing mid-quotes but only to the most recently observable mid-quotes. Additionally, the latencies associated with trading in three distant geographical locations— London, New York, and Tokyo—can give rise to measurement error in trade time stamps. As a result, the volume-weighted effective spread calculated for day t according to 1 (cid:88) ES = v q (p −m )/m (33) t (cid:80) i,t i,t i,t i,t i,t v i i,t i can potentially over– or underestimate the true spread depending on how the mid-quote moves between the last snaphot and the time of the transaction; in principle, q (p −m ) i,t i,t i,t can even turn negative. For robustness, we therefore consider two alternative ways of calculating the effective spread from the EBS data, where we impose the restriction that the effective spread cannot be negative for any transaction: 1 (cid:88) E(cid:100)S t = (cid:80) v i,t max{q i,t (p i,t −m i,t ),0}/m i,t , (34) v i i,t i 1 (cid:88) E(cid:103)S t = (cid:80) v i,t |p i,t −m i,t |/m i,t . (35) v i i,t i Note that the latter definition is also used to calculate the TAQ effectives spreads for U.S. equities. We then repeat the analysis with these spreads in place of ES . In Table t 27 we report some summary statistics for the alternative EBS spreads, and in Table 28 we report the regression results analogous to those reported in Table 18. Table 27: Descriptive statistics for alternative measures of effective spread for FX rates. The mean, median, standard deviation, and percentiles are expressed in basis points. The sample period is 2008–2015. Mean 10pct 25pct Med 75pct 90pct Std ρ(·,RV) ρ(·,E(cid:100)S) ρ(·,E(cid:103)S) ES 0.68 0.43 0.53 0.64 0.81 1.10 0.22 0.56 0.97 0.91 E(cid:100)S 0.86 0.52 0.66 0.82 1.03 1.37 0.28 0.64 0.99 E(cid:103)S 1.04 0.63 0.79 0.98 1.24 1.63 0.34 0.67 68

Table 28: Empirical results for foreign exchange rates with alternative EBS effective spread proxies. Thetablereportspooledleastsquaresregressionsofthelow-frequencymeasureson the EBS effective spread (E(cid:100)S or E(cid:103)S) and 5-minute realized volatility (RV) at the monthly frequency. Driscoll-Kraay standard errors, which are robust to heteroskedasticity, serial correlation, and cross-sectional dependence are reported in parentheses. The sample period is January 2008 to December 2015 and the sample size is 479 exchange rate-months. R H H CS CS CS AR AR AR M L T M D P M D P A.1. Regression on E(cid:100)S const -4.95 1.30 4.18 1.31 3.13 4.33 -4.33 -1.62 -0.19 (-0.79) (0.16) (0.56) (0.92) (1.01) (0.67) (-1.35) (-0.39) (-0.02) β 36.0 53.1 47.4 14.0 30.2 47.0 21.0 30.1 55.7 ES (4.66 ) (4.94) (5.08) (8.51) (8.11) (5.82) (5.00) (5.50) (4.46) R2 0.09 0.19 0.18 0.11 0.39 0.36 0.09 0.32 0.33 A.2. Regression on E(cid:100)S and RV const -6.81 -2.52 0.70 1.50 1.58 1.12 -5.01 -3.57 -4.40 (-1.36) (-0.88) (0.21) (1.11) (1.07) (0.62) (-1.77) (-2.63) (-2.33) β 11.1 1.98 0.81 16.4 9.35 4.19 12.0 3.90 -0.75 ES (1.38) (0.34) (0.15) (8.15) (5.65) (1.69) (3.27) (2.08) (-0.41) β 0.33 0.68 0.62 -0.03 0.27 0.57 0.12 0.35 0.75 RV (4.98) (12.9) (10.9) (-1.39) (7.73) (12.9) (2.37) (12.5) (26.7) R2 0.14 0.44 0.43 0.12 0.66 0.80 0.11 0.67 0.82 B.1. Regression on E(cid:103)S const -2.94 1.09 3.95 2.20 3.17 3.53 -2.92 -1.54 -0.74 (-0.49) (0.15) (0.59) (1.59) (1.22) (0.63) (-0.98) (-0.43) (-0.09) β 27.8 44.2 39.5 10.7 24.9 39.7 16.1 24.8 46.6 ES (4.54) (5.45) (5.63) (7.67) (9.68 (6.84) (4.77) (6.25) (4.98) R2 0.08 0.20 0.19 0.10 0.41 0.40 0.08 0.34 0.36 B.2. Regression on E(cid:103)S and RV const -4.49 -1.94 1.19 2.35 1.96 1.04 -3.50 -3.09 -4.07 (-0.94) (-0.68) (0.37) (1.82) (1.31) (0.58) (-1.31) (-2.25) (-2.15) β 5.53 0.67 -0.21 12.9 7.57 3.87 7.81 2.55 -1.29 ES (0.83) (0.13) (-0.05) (7.57) (5.66) (1.91) (2.54) (1.66) (-0.87) β 0.35 0.68 0.62 -0.03 0.27 0.56 0.13 0.35 0.75 RV (5.07) (12.4) (10.6) (-1.46) (7.65) (12.7) (2.47) (12.4) (27.3) R2 0.14 0.44 0.43 0.11 0.66 0.80 0.11 0.67 0.82 69