Relative Liquidity and Future Volatility

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Relative Liquidity and Future Volatility Marcela Valenzuela, Ilknur Zer, Piotr Fryzlewicz, and Thorsten Rheinlander 2014-045 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.

Relative Liquidity and Future Volatility∗ Marcela Valenzuela†, Ilknur Zer‡, Piotr Fryzlewicz§, Thorsten Rheinl¨ander¶ January 2015 Abstract The main contribution of this paper is to identify the strong predictive power of the relative concentration of depth provision, rather than volume of orders, over volatility. Tothisend,weproposeanewmeasure,relativeliquidity,whichextracts information from a limit order book distribution and captures the level of consensus on a security’s trading price. Higher liquidity provision farther away from the best quotes, relative to the rest of the book, is associated with a disagreement on the current price and followed by high volatility. The relationship is robust to the inclusion of several alternative measures. Keywords: order-driven markets, limit order book distribution, volatility predictability, liquidity JEL Classification: G1, G20 ∗We sincerely thank Vicente Cunat, Jon Danielsson, Dobrislav Dobrev, Christian Julliard, Richard Payne, and an anonymous referee for their valuable comments. We would like to thank to Coskun Gunduz, RecepBildik, andHuseyinEskiciforprovidingusdataandthesupportforunderstandingthe market mechanisms. Valenzuela acknowledges the support of Fondecyt Project No. 11140541. †University of Chile, Department of Industrial Engineering, Republica 701, Santiago, Chile. ‡Federal Reserve Board, 20th Street and Constitution Avenue N.W. Washington, D.C. 20551, USA. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. §London School of Economics, Department of Statistics, Houghton Street, London, WC2A 2AE, UK. ¶Vienna University of Technology, Research Group Financial and Actuarial Mathematics (FAM), Wiedner Hauptstrasse 8-10/105-1, A-1040, Vienna, Austria. 1

1 Introduction Thispaperexaminesthelinkbetweentwocentralconceptsinfinancialmarkets: liquidity andvolatility. Liquidity, theeasewithwhichanassetcanbetradedwithoutaffectingthe asset’sprice,isessentialforwell-functioningfinancialmarkets. Hence,understandingthe effects of liquidity provision on market dynamics has gained an increased attention from regulators, market participants, and academics alike. On the other hand, information on volatility, variation in trade prices, is one of the main ingredients in assessing risk-return trade-off for portfolio valuation, derivatives pricing models, and it is important for the calibration of execution probability of limit orders. In this paper, we propose a new way of summarizing the distribution of liquidity in a limit order book and examine its informativeness on future volatility. Our empirical investigation is motivated by the theoretical predictions of Goettler et al. (2005, 2009), in which the relative concentration of depth provision at each quote reveals information of the disagreement on the true price. Their model predicts that higher liquidity provision around the best quotes relative to the rest of the book is associated with a consensus on the current price; whereas the accumulation of orders at a quote fartherawayfromthebestpricessignalstothemarketthatcurrentquotesaremispriced. We argue that, in the latter case price movements are more plausible, creating higher future volatility. To examine the effects of the relative accumulation of orders on future volatility, we construct a measure, relative liquidity (RLIQ), which is the first principal component of an aggregate limit order book. To calculate RLIQ, we first obtain the empirical probability density function of a limit order book for a given stock. We then calculate the cross-sectional average of individual stock distributions to reach the aggregate distribution. In other words, we measure the proportions of orders waiting at each price level in the market. Finally, we employ a principal component analysis to summarize this information in as few interpretable quantities as possible. As a summary measure, RLIQ has three ingredients: first it accounts for the relative distribution of orders waiting to be traded, which reveals information on the disagreement of the true price. Second it includes the information contained beyond the best quotes, and third it weighs this information based on price distances. The weighting scheme captures the different levels of informativeness of the quotes as depths at and farther away from the best quotes play different roles in traders’ order choices. Instead 1

of imposing an exogenous weighting scheme, we employ the loadings of the first principal component of the aggregate distribution to weigh the information provided by different price levels. Thus the principal component analysis enables us to avoid the subjective judgments regarding the relative importance of quotes. We evaluate the predictive power of relative liquidity over both market and individual stock volatilities at an intraday level, with a particular interest in the former. It is challenging to estimate intraday volatility of the price process due to the microstructure noise inherent in the high-frequency data, such as the informational effects, bid-ask bounces, or data recording errors. It has been recognized in the literature that the standard estimator—realized volatility—can be highly unreliable under the microstructure noise, especially if the sampling frequency is high. Zhang et al. (2005) and Ait-Sahalia et al. (2011) address this specific problem and provide the volatility proxy that we use in this study: the two scales realized volatility estimator (TSRV). The order and trade books of the largest 30 stocks from the Istanbul Stock Exchange (ISE) form our dataset. By matching these two books and removing the executed orders, we dynamically reconstruct the limit order book. That is, for a given time we have the best bid and ask prices, all of the orders waiting to be executed, their submitted prices, and their corresponding volumes. Since ISE is a fully centralized purely order driven market and operates with a single trading platform, our data contains the entire order flow in the public domain, which brings a major advantage compared to the main European and U.S. market exchanges. We show that relative liquidity (RLIQ) is the strongest among standard liquidity and trading activity measures, in explaining the in-sample variations in market volatility. On average, one standard deviation increase in RLIQ decreases the 15-minutes-ahead volatility by 4.4 basis points. The results are robust to the inclusion of alternative controls and jump-robust volatility measures. Out-of-sample forecasting tests provide evidence for substantial forecasting power of relative liquidity. It predicts 15-minutesahead market volatility at a 5% level with an out-of-sample R2 of 12.9%, where the forecastingpowerlastsupto75minutesahead. RLIQcanbethoughtasacomplementof the bid-ask spread as it is related to the depth dimension of liquidity and by construction does not include the spread. Further analysis reveals that capturing relative liquidity along with the tightness dimension of liquidity delivers an out-of-sample R2 of over 24%. It can be argued that the documented relationship is “mechanical”: when the limit order book is thin, i.e., when the depth at the best quotes is low, any market order causes a 2

price impact or a bid-ask bounce. However, unlike standard depth measures, relative liquidityfocusesontherelative distributionoforders, ratherthantheabsolutevolumeof orderswaitingateachquote. Inotherwords, althoughathinbookisassociatedwithlow depth, it is not necessarily associated with low RLIQ since the latter measures whether the orders are concentrated or spread over a book. The documented predictive power of RLIQ after controlling for standard depth measures suggests that the information content of RLIQ over volatility cannot be explained by absolute depth. Finally, we show that the relationship between RLIQ and market volatility is not driven byvariationsinaparticularstockorindustry, butratherthatitissharedbythemajority of the stocks. We find a significant relationship between the individual stock level RLIQ and future volatility for 87% of the stocks in our sample. This paper relates to the literature that attempts to measure the liquidity provision considering the whole book. Domowitz et al. (2005) propose an illiquidity measure based on the supply and demand step functions and conclude that the stock correlated movements in liquidity, i.e., the liquidity commonality, is priced in stock returns. Marshall (2006) defines liquidity by the weighted order value, which depends on the execution rate of orders waiting in each price band and their corresponding prices and volumes. The author documents a negative association between liquidity and monthly returns. In another related study, Naes and Skjeltorp (2006) introduce the slope of the book, which describes the average elasticity across all price levels with the corresponding volumes. This paper is part of the market microstructure literature that examines the predictive power of liquidity on intraday volatility. In an early empirical work, Ahn et al. (2001) analyze the interactions between transitory volatility and order flow composition. They conclude that the transitory volatility arises mainly from the scarcity of limit orders at the best quotes. Pascual and Veredas (2010) show that trade size and quoted depth both at the best and away from the quotes have predictive power for individual volatility. Duong and Kalev (2008) investigate the forecasting power of the Naes and Skjeltorp (2006)’s definition of order book slope. By using data from the automated futures market,Coppejansetal.(2001)studythedynamicrelationshipbetweenliquidity,return, and volatility in a vector autoregressive framework. Finally, the paper is related to the few studies that use intraday data from the Istanbul Stock Exchange (ISE). Ekinci (2008) and Koksal (2012) provide descriptive analyses of the intraday liquidity patterns of the ISE by focusing on the behavior of spreads, depths, and trading volume. Using different stocks and time periods, both studies 3

conclude that the liquidity related variables follow the usual U-shape pattern for both of the trading sessions. Valenzuela and Zer (2013) study how the market characteristics and information content of a limit order book affects the order choice of investors. Our contribution to the literature is threefold: first, we provide a new variable that summarizes the information provided by a limit order book. Contrary to the aforementioned measures, which focus on the absolute volume of the orders waiting in a given book, RLIQ is based on the relative distribution of volume at a given time. Second, we show that relative liquidity contains information on future volatility that cannot be explained by the standard predictors of volatility. Finally, in contrast to these former studies, which examine the volatility–liquidity relationship at an individual stock level, we focus on the link between aggregate liquidity and future market volatility. The rest of the paper is organized as follows: the next section describes data and the trading structure in our market. Section 3 explains the construction of our measure in detail. Section 4 introduces the econometric methodology and variables included in the analysis. The in-sample and out-of-sample predictive results, and robustness checks are given in Section 5. Finally, Section 6 concludes. 2 The Market and Data Our dataset comprises order and trade books of the individual constituents of the Istanbul Stock Exchange ISE–30 index for the period of June and July 2008. The index corresponds to almost 75% of the total trading volume of the ISE for the sample period. The ISE is a fully computerized as well as a fully centralized purely order driven stock exchange, i.e., the trading of the listed stocks has to be executed in the ISE via electronic order submissions without a market maker. Hence, our data fully captures the order flow. The trading occurs between 09:30am and 5:00pm, with a lunch break. Similar to all other major exchanges, a trading day starts with a call market matching mechanism of 15 minutes to determine the opening price. In contrast to the opening session, during the continuous double auction, all of the orders submitted are either matched instantaneously based on the usual price and time priorities or booked until the corresponding match order arrives to the system. A submitted order is valid for a given session or for a day. All brokers have access to the full book. Prior to the submission of an order, 4

they can see the quantity available at different prices, not limited to the best five or ten quotes. The order book data consists of information regarding the orders submitted for a given stock, whereas the trade data records the executed orders, both time-stamped at the accuracy of 1 second. The order and trade ID numbers generated by the exchange system allow us to identify the priority of orders submitted in the same second, to match orders in the order and trade books, and finally to track any order through submission to (possible) execution or modification. By using the order and trade books, we first reconstruct the limit order book dynamically for each stock and obtain relevant information, such as the bid and ask prices and corresponding volumes at a given time. Hence, the reconstruction methodology enables us to obtain snapshots of a limit order book at any given time. In particular, we have the same information that a trader observes: the volume of orders waiting to be executed for the entire price range. We use this information to calculate the relative frequency of orders waiting in every price level. 3 The Limit Order Book Distribution and Relative Liquidity 3.1 The limit order book distribution The limit order book distribution is obtained by employing the following steps, which are illustrated with an example in Appendix A: 1. For each security and each day, we sample the limit order books every 15 minutes, excluding the lunch break and the opening session. The first snapshot of the book contains the unexecuted orders submitted until 10:00, whereas the last one contains all of the unexecuted orders submitted until 17:00. Hereafter, the time subscript τ indexes these trading intervals, with τ = 1,2,...,21. We repeat the empirical analysis with 30–minute sampling frequencies as a check of robustness. The results are presented in Section 5.6. 5

2. We calculate the (tick-adjusted) price distance of each limit order relative to the best limit price in each snapshot. In other words, for each order i in the limit order book at τ, we define the price distance ∆ as: ∆buy = (pB −pbuy)/tick, i,τ τ i ∆sell = (psell −pA)/tick, i,τ i τ where pB (pA) is the best bid (ask) price in interval τ and pbuy (psell) is the limit τ τ i i price of the ith order. 3. For each side of the book, day, and limit order book at τ, we get the limit order book probability density function (LOB–PDF) by calculating the percentage of total volume supplied/demanded at a given ∆ for ∆ = 0,1,2,..,∆ , where ∆ is c c the maximum price distance considered. Therefore, LOB–PDF summarizes both the relative magnitude of the depth provision and its price location. 4. We calculate the equally-weighted cross-sectional average of individual LOB–PDFs to obtain the aggregate LOB–PDF (avgPDF). In other words, we take the average ofthefrequenciesineachcorrespondingbinsofthe30individualstockLOB–PDFs. That represents the proportion of orders waiting at each price level in the market. For a given trading interval τ and price distance ∆, S 1 (cid:88) avgPDFbuy = fbuy(∆), (1) ∆,τ S s,τ s=1 where fbuy(∆) is the buy side LOB–PDF of stock s and S is the total number of s,τ stocks. The measure for the sell side is calculated analogously. In order to consider the possible impact of bigger or more actively traded stocks, we also calculate the value-weighted and number-of-trades-weighted averages of the individual LOB– PDFs to obtain the avgPDF. We reach qualitatively similar results, which are presented in Section 5.6. Figure 1 reveals that for both sides of the market, the frequency of orders submitted at the second best quote is the highest and the limit order book distribution is positively skewed. The liquidity provision is concentrated closer to the best quotes for the buy side compared to the sell side, which can be observed by comparing either the mean or the skewness of the distribution presented in Table 1. The mean of the distribution, for all 6

0.25 Buy Side Sell Side 0.2 y 0.15 c n e u q e Fr 0.1 0.05 0 0 5 10 15 20 25 30 35 Price Distance Figure 1: Limit order book probability density function (LOB–PDF) for the period of June and July 2008, averaged across thirty stocks, twenty-one 15-minutes trading intervals, and thirty-nine days considering the whole book. of the time intervals, is higher for the sell side than the buy side. Wilcoxon rank sum test concludes that the difference is statistically significant at a 5% level. The cumulative frequency of orders waiting after 5 ticks away from the quotes is 28% for the sell side, whereas it is only 18% for the buy side. Moreover, the average variance of the sell side is 36% higher than the average variance of the buy side. Both observations indicate that the buy side is less dispersed compared to the sell side of the market. Indeed, the Kolmogorov-Smirnov test rejects the hypothesis that the buy-side and sellside distributions are equal, at a 1% significance level. About 90% of the submitted orders are waiting within the 10 best prices for both sides of the market. Hence, we only consider the information contained in the book up to the 10th best quotes. In other words, the main discussions presented in the rest of the paper are obtained by setting ∆ = 10. However, we examine the robustness of our findings c when ∆ is equal to 20 and 30, i.e., when we consider the whole book, in Section 5.6. c 3.2 Summarizing the limit order book distribution: RLIQ The shape of the limit order book distribution at time τ is given by the proportion of volume waiting to be traded at different price distances ∆. There are several ways to summarize this information. We want our summary measure to weigh the information provided in different quotes based on price distances to capture the different levels of informativeness of the quotes. One, for example, could assume exogenously given weights 7

or give equal weights to the frequency of orders waiting at each price distance ∆. We instead employ the principal component analysis (PCA), which extracts the most important (uncorrelated) sources of variation in the LOB distribution. The advantage of this approachisthat, itassignsanobjectiveweightingscheme, whichaimstoencodeasmuch information about the LOB distribution in as few quantities (principal components) as possible. The principal component analysis applied on the ten bins of avgPDF defined in (1) produces ten uncorrelated principal components. The first principal component is the leading eigenvector in the spectral decomposition of the covariance matrix of avgPDF, which explains the highest variation in the limit order book distribution. Relative liquidity summary measures, RLIQbuy and RLIQsell, are chosen to be the first principal components of avgPDF for both sides of the market. In Section 5.6, we discuss the sensitivity of the findings by (a) considering the first three and five principal components, (b) using the empirical frequencies of orders waiting at each price distance separately, and (c) employing LASSO (Tibshirani, 1996), a shrinkage and variable selection technique. Figure 2 plots the loadings of RLIQbuy and RLIQsell that are used to weigh the information provided by different price levels. The signs of the loadings of the first principal component are chosen so that the sign corresponding to first price distance (∆ = 0) is positive, both for the buy and sell sides of the market. The figure reveals that if the frequency of orders waiting around the best quotes increases, this increases RLIQ due to positive loadings assigned to the information provided by the top of the book. On the other hand, an increase in the proportion of orders waiting farther away from the best quotes translates into a decrease in RLIQ. This easy-to-interpret pattern in the loadings provides further justification for the use of this summary measure as a possible predictor of volatility. 8

0.4 0.2 0 -0.2 -0.4 0 2 4 6 8 10 sgnidaoL Buy Side Sell Side Price Distance Figure 2: Loadings of the first principal component of the aggregate limit order book distribution (avgPDF) defined in (1), when ∆ =10. c 4 Predictive Analysis 4.1 Methodology To evaluate the information content of a limit order book on future volatility, we rely on a standard predictive regression model of intraday volatility: 20 (cid:88) σM = a +a σM +a RLIQbuy+a RLIQsell+ b D (2) τ+1 0 1 τ 2 τ 3 τ j j,τ j=1 + controls+ε , τ where for a given interval τ, σM is the mid-quote-volatility of the value-weighted index, τ and RLIQbuy and RLIQsell are the proposed relative liquidity summary measures, calcu- τ τ lated as the first principal component of the aggregate limit order book distribution for buy and sell sides of the market, respectively. D is the intraday dummy that equals j,τ to 1 if j = τ. We include the lagged volatility, σM, and intraday dummies in the set of τ explanatory variables to control the well-known systematic intraday patterns and clustering in volatility. Furthermore, we employ both the standard predictors of volatility and other liquidity measures as control variables, which are introduced in Section 4.3. 9

4.2 Measuring volatility: the two scales realized volatility estimator (TSRV) A common practice to estimate return volatility is to use the sum of squared returns. However, it is not entirely appropriate to use the realized variance under high-frequency dataduetomicrostructurenoiseandjumpsembeddedindata. Relevantstudieshaveaddressed this issue and proposed ways to improve the estimator. Ait-Sahalia et al. (2005), Bandi and Russell (2009), Ghysels and Sinko (2011), among others, focus on optimal sampling frequency to smooth the noise. Zhou (1996) and Hansen and Lunde (2006) consider a first-order autocorrelation to bias-correct the realized variance. By focusing on assets with large tick sizes, Delattre et al. (2013) propose a statistical methodology to estimate the efficient price of an asset through the order flow. Barndorff-Nielsen and Shephard (2004), Andersen et al. (2012), among others propose volatility estimators that are robust to the presence of jumps in data. The volatility estimator that we employ in this study, the two scales realized volatility (TSRV), is proposed by Zhang et al. (2005) and Ait-Sahalia et al. (2011). In the presence of the microstructure noise, the TSRV estimator gives an unbiased and consistent estimate of volatility, under the assumptions that the log-price process follows a geometric Brownian motion and that the noise term is i.i.d. and independent of the price process. Although TSRV is a noise-robust estimator, it is not robust under the presence of jumps. In Section 5.6 we test the sensitivity of the results under alternative jump-robust volatility estimators. In financial data, we can only observe prices, either in a form of transaction or quoted price, which are often modeled as a linear combination of the true price process and some noise. Under the assumptions that the log-true price process follows a geometric Brownianmotionandthenoisetermisi.i.d. andindependentfromthetruepriceprocess, the two-scales realized market volatility is obtained as follows: (cid:118) (cid:117) K (cid:117) 1 (cid:88) 1 σM = (cid:116) [MQ,MQ]sparse,k − [MQ,MQ](all), (3) τ K τ K τ k=1 where [MQ,MQ](all) is the realized variance of the mid-quote returns of the value- τ weighted index calculated in a trading interval τ with a sample size of T. To obtain [MQ,MQ]sparse,k, we first divide the sample into K moving window subsamples (following Ait-Sahalia et al., 2011, K is set to be 5 minutes) with a fixed length of N, 10

where N = T − K. For example, the first subsample starts with the first and ends with the Nth observation, whereas the second subsample starts with the second and ends with (N+1)th observation. Then, we sample sparsely with a 30-seconds frequency. Thus [MQ,MQ]sparse,k is the realized variance estimator of the kth 30-seconds-sampled mid-quote returns. 4.3 Control variables Our first set of covariates includes the variables that have been shown as predictors of volatility in the current literature. First, consistent with Bollerslev and Domowitz (1993), Jones et al. (1994), and Foucault et al. (2007), the number of trades occurring in interval τ, NT, and the average trade size, AQ, are included to capture the trading activity. In a related study, Foucault et al. (2007) show that the bid-ask spread is informative of future individual stock volatility. Hence, we also include the relative spread, relSPR , which is calculated as the ratio of the bid-ask spread to the mid-quote τ pricesforeachinterval. Finally,weconsidertheslopeofalimitorderbook,SLOPE,asan explanatory variable following Naes and Skjeltorp (2006) and Duong and Kalev (2008). SLOPE aggregates the price-quantity information in different quotes and measures the sensitivity of the quantity supplied in the book with respect to the prices. Oursecondsetofcovariatesincludesotherliquiditymeasures. Wefirstconsiderstandard depthmeasures. Thedepth,definedasthetotalvolumeavailabletobetradedatthebest bidoraskprices, isoneofthetraditionalmeasuresofliquidity. WecalculateDEPTHibuy (DEPTHisell) for i = 1,2,...,5, which denotes the volume of orders waiting at the ith best bid (ask) to capture the volume available at and beyond the best quotes for the buy and sell sides of the market, respectively. Second, we employ the Amihud (2002)’s illiquidity measure, AMR, which is calculated as the ratio of absolute stock return to the turnover. Another related illiquidity measure is the log quote slope, logQS, which is introduced by Hasbrouck and Seppi (2001). A decrease in the logQS means that the slope of the best quotes is flatter and the market is more liquid. Finally, we consider the illiquidity measure proposed by Domowitz et al. (2005), DHW, which measures the cost of buying and selling Q shares of the stock, simultaneously. In this paper, we set Q as the median of the accumulated volume of orders waiting in the book for a given stock. All of the control variables are calculated as the equal-weighted cross-sectional average of the individual stock measures. As a robustness check, we repeat the analysis by calculating the value-weighted and trade weighted averages of the explanatory variables 11

to proxy the aggregate measures. The results presented in Section 5.6 reveal that the main findings are confirmed. Compared to standard liquidity measures like spread, depth, and ratios based on both spread and depth, RLIQ provides a more complete picture of the liquidity provision by considering the book beyond the best quotes. Moreover, instead of focusing on the volume of the orders waiting, RLIQ is based on the distribution of volume at a given time, hence, it reveals information of how quoted depths are spread or concentrated throughout a limit order book. 5 Empirical Findings 5.1 One-period-ahead predictive regressions The first focus of our analysis is to examine the predictive power of relative liquidity, RLIQ, for the 15-minutes-ahead market volatility. To account for the intraday patterns, all of the specifications include 21 trading intervals as intraday dummies. To conserve space, we do not report the estimated coefficients of the dummy variables. To improve the ease of interpretation of the estimated coefficients, all of the explanatory variables are standardized to have mean zero and unit variance, and the dependent variable is presented in percentage terms. t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and presented in parenthesis. Table 2 reveals that relative liquidity variables are significantly and negatively related to the one period ahead market volatility at a 5% level. The proportion of the variation in market volatility explained by our measures is almost 22%. When the intraday dummies are not included in the specification, RLIQbuy and RLIQsell alone explain around 16% of τ τ the variation in volatility. As Figure 2 shows, the loadings of RLIQ are positive for the proportion of volume around the best quotes and turn to negative for the orders waiting away from the best five quotes. Hence, the negative sign of the coefficients indicates that an increase in liquidity beyond the best quotes relative to the top of the book is likely to be followed by a higher level of volatility in the next period. If the volume of orders waiting to be executed is more accumulated beyond the best prices, incoming investors may interpret this as mispricing of the current quotes. Hence, large price movements are more likely to happen creating higher future volatility. 12

Asexpected, laggedvolatilityishighlyandpositivelyrelatedtoone-periodaheadvolatility. However, the predictive power of RLIQbuy is higher than the lagged volatility. One τ standard deviation increase in RLIQbuy decreases 15-minutes ahead volatility by 4.4 ba- τ sis points, whereas one standard deviation increase in volatility increases the next period volatility by 3.7 basis points. Column III shows that when relative liquidity variables and lagged volatility are included in the specification together, the adjusted R2 increases to over 25%. Columns IV and V confirm the robustness of the predictive power of RLIQ for the oneperiod-ahead market volatility when the standard predictors of volatility and alternative liquidity measures are considered. Not surprisingly, relative spread (relSPR ) is infor- τ mative of future volatility at a 5% level. Note that by construction, RLIQ is related to the depth dimension of liquidity and it can be thought as a complement of spread. Moreover, the slope of the book (SLOPE ), and the slope of the best quotes (logQS ) τ τ are positively and significantly correlated with future volatility. On the other hand, the significance of RLIQsell decreases when all of the control variables are included in the τ setting. This result is consistent with the literature documenting that buy orders are more information-driven than sell orders, i.e., the informed traders may exploit their informational advantage by submitting buy orders (see, for instance, Burdett and O’Hara (1987), Griffiths et al. (2000), and Duong and Kalev (2008), among others). Our results further extend the findings of Duong and Kalev (2008) and Foucault et al. (2007), who document that the slope of the book and relative spread, respectively, have explanatory power for future individual stock volatility. We show that the cross-sectional average of both measures (SLOPE and relSPR ) have explanatory power for the market volatility τ τ as well. Moreover, we provide new empirical evidence that the measure of Hasbrouck and Seppi (2001), log quote slope, is significantly and positively related to the subsequent market volatility. Yet, the estimated (standardized) coefficients and t-statistics of RLIQbuy are always the highest among the alternative variables and they are robust and τ stable in all of the specifications examined. The adjusted R2 increases from 25.15% to only 29.53% when all of the set of controls are included in addition to RLIQ variables. It is important to mention that the documented results are not spurious since the augmented Dickey-Fuller and Phillips-Perron tests reject the null of a unit-root in volatility and in all of the explanatory variables at a 5% level. Moreover, the estimated coefficient of the non-standardized lagged volatility, i.e., the AR(1) coefficient, equals to 0.34. We conduct an analysis of the persistency of volatility and find that the estimated AR(1) coefficient increases to 0.42 and 0.60 for half-an-hour and daily sample periods, respec- 13

tively. Persistency is decreasing with the sampling frequency, i.e., it is the lowest for the 15-minutes volatility and the highest for the daily volatility. Volatility is a latent variable, rather than an observable one. This means that any integrated volatility estimator is subject to an estimation error. Although this problem is partially alleviated by subsampling and averaging in TSRV, this result suggests that it is still subject to estimation error, especially for 15-minutes trading interval. However, even in the presence of possibly high estimation error, we document a robust and strong dependency between relative liquidity and future volatility. 5.2 Relative liquidity vs. standard depth measures Standard depth measures consider the volume of orders waiting at a given price. In a thin limit order book, i.e., when the volume at the best quotes is small, any “large” market order has a price impact and changes the quotes. Hence, it is not surprising to expect a link between the standard depth measures and future price movements. On the other hand, relative liquidity (RLIQ) extracts the relative accumulation of orders rather than the absolute volume of orders. Hence, a thin book is not necessarily associated with low RLIQ. RLIQ is lower when the orders are accumulated further away from the best quotes. However, as it is constructed from the distribution of volume at a given time, it may share common information with standard depth variables. Thus we next examine whether RLIQ is still significant in explaining subsequent volatility under the presence of depth variables. To this end, we include the volume of orders at different prices along with the RLIQ measures in our analysis. Similarly, all of the specifications include the interval dummies and lagged volatility as control variables. buy sell Table 3 shows that DEPTH1 and DEPTH1 , the total volume of orders waiting at the best bid and ask prices, respectively, significantly explain future market volatility at a 5% level. A decrease in the volume of orders at the best quotes creates higher subsequent volatility. However, when relative liquidity measures are included in the specification, DEPTH1 variables are no longer significant. The highest adjusted R2 is only 26.11% even when the total depth up to the third and up to the fifth quotes are included in the analysis. That is, by including 10 depth variables in addition to our measures, we only increase the adjusted R2 by less than 1%. In summary, we conclude that the relative concentration of depth provision, rather than the absolute volume, reveals more information about future volatility. RLIQ has a superior in-sample predictive power compared to the standard depth measures. 14

5.3 Predicting further horizons In this section, we examine the informativeness of the limit order book distribution at time τ on multiple-period-ahead volatility. Specifically, we run the baseline regression model (2), while we calculate the dependent variable as the mid-quote volatility of the index at time τ + h, with h = 1,2,...,11, i.e., up to 165 minutes ahead. Whereas the independent variables are calculated based on the limit order book information at trading interval τ. For example, τ +2 refers to volatility in a 30-minutes ahead trading interval. Table 4 shows that the significance of the estimated coefficients as well as the predictive power of relative liquidity measures are (almost) monotonically decreasing with the prediction horizon. RLIQbuy has a significant forecasting power with respect to market volatility up to 150-minutes-ahead. Moreover, the slope of the book, the relative spread, and the quote-slope significantly predict volatility at longer horizons. However, RLIQ is both economically and statistically the strongest predictor. 5.4 Out-of-sample tests In this section, we evaluate the out-of-sample forecasting ability of RLIQ compared to historical volatility. Specifically, for a subsample of observations up to a given time interval τ, we compare the h-period-ahead squared forecast errors with the squared difference between the realized value at τ +h and the sample mean value up to time τ. To do so, we split our data into two subsample periods: L is the training period and train L is the testing period with L +L = L, the total number of time intervals. We test train test then re-estimate the parameters of the model, in which we use the variable of interest as the predictor. Recursive estimators of h-period-ahead forecasts are based on the sample starting from L up to L − h. For L equals to 350 and 400 observations, we train train calculate the following error terms: ε = σM −σ(cid:100)M , 1,τ+h τ+h τ+h ε = σM −σM, 2,τ+h τ+h τ where σM is the two scales realized market volatility, σM is the mean value of the τ+h τ market volatility up to time τ, and σ(cid:100)M is the fitted market volatility obtained by τ+h regressing volatility on the variable of interest such as RLIQ or other liquidity measures. 15

We evaluate the comparison by using two different metrics: the difference in meansquared errors (∆MSE) and the out-of-sample R2. If the proposed measure has superior out-of-sample forecasting ability relative to the average of past data, then both of these measures will be positive. We employ the Diebold and Mariano (1995) predictive ability test (DM) to test the significance of ∆MSE. Finally, the out-of-sample R2 is calculated as follows: (cid:80)Ltest−hε2 R2 = 1− τ=1 1,τ+h . (4) (cid:80)Ltest−hε2 τ=1 2,τ+h Table 5 reveals that forecasts based either on relative liquidity, relative spread, number of trades, or slope of the best quotes increase the predictive power relative to forecasts based only on the sample mean of past volatility. Moreover, the predictive power of the variables is decreasing almost monotonically with the prediction horizon. When L = 400, RLIQbuy delivers out-of-sample R2’s from 12.9% when forecasting onetrain period-ahead market volatility up to 5.5% when predicting 90-minutes-ahead market volatility. On the other hand, the results are stronger for both relative spread and log quote slope when L = 350. Note that both relSPR and logQS are the variables that capture the train liquidity at the best quotes, i.e., the tightness dimension of liquidity only. Thus as a furtheranalysis, weexaminewhetherincludingrelativeliquidity, inadditiontothetightness dimension of liquidity, produces better forecasts. To do so, the first forecast errors are calculated from the model where RLIQbuy and relSPR (logQS) are the explanatory variables, whereasthesecond(benchmark)forecasterrorsarecalculatedfromthemodel, in which relative spread (log quote slope) is the only explanatory variable. Similarly, we repeat the analysis for two different estimation window sizes; 350 and 400 observations. The results show that including RLIQ into the analysis increases the out-of-sample R2 by 13.9% and 11.7% in addition to using only relative spread and the log quote slope, respectively, for L = 400. Three variables together deliver an out-of-sample R2 of train over 24% when forecasting one-period-ahead market volatility relative to forecasts based only on the sample mean of volatility. We conclude that capturing both the tightness and the depth dimension of liquidity significantly increases the out-of-sample forecasting power. 16

5.5 Predicting individual stock volatilities This section examines the in-sample predictive power of a limit order book distribution over future volatility on an individual stock level. To this end, we first run the following predictive regression in a pooled data with stock fixed effects: 20 (cid:88) σ = a +a σ +a RLIQind,buy +a RLIQind,sell + b D (5) s,τ+1 0 1 s,τ 2 s,τ 3 s,τ j j,τ j=1 30 (cid:88) + c FE +controls+ε , s s s,τ s=1 where for stock s and interval τ, σ is the mid-quote two scales realized volatility, s,τ RLIQind,buy and RLIQind,sell are the first principal components of the individual stock s,τ s,τ limit order book distributions for the buy and sell sides of the market, respectively. D j,τ is the intraday dummy that equals to 1 if j = τ, and FE are stock-specific dummy s variables allowing for stock fixed effects. Figure 3 reveals that the loadings of the first principal components differ slightly from one stock to another stock and they are similar to the loadings of the first principal component of the aggregate limit order book distribution presented in Figure 2. 0.4 0.2 0 -0.2 -0.4 0 2 4 6 8 10 sgnidaoL 0.4 0.2 0 -0.2 -0.4 0 2 4 6 8 10 Price Distance (a) Panel A: Buy Side sgnidaoL Price Distance (b) Panel B: Sell Side Figure 3: Loadings of the first principal component for the buy side of the market (Panel A) and the sell side of the market (Panel B). The loadings of a stock are presented only if the relative liquidity of the given stock is a significant predictor of 15-minutes-ahead volatility. Table 6 Columns I to IV report the estimated coefficients for (5) with the corresponding t-statistics. To take into account the possible cross-sectional variations that cannot be captured by the stock fixed effects, we as well estimate the predictive regressions for 17

each stock s separately. The summary of these results are presented in Columns V to VIII. Ourmainresultisconfirmedintheseindividualvolatilityregressions. RLIQisnegatively related to future volatility for 87% of the stocks for the buy side of the market at a 5% level. We conclude that the time-series relationship between the aggregate liquidity and market volatility is not driven by variations in a particular stock or industry, but rather it is shared by the majority of the stocks. The results reveal the asymmetry between the buy and sell sides of RLIQ at the individual stock level as well. The sell side of the market is informative only for 37% of the stocks in the individual regressions at a 5% level, in contrast to the informativeness for 87% of the stocks on the buy side. Similarly, both RLIQind,buy and RLIQind,sell are significant in the pooled regression but the estimated coefficients of the buy side are at least two times greater than the sell side. 5.6 Robustness 5.6.1 Relative liquidity: alternative measures We define relative liquidity as the first principal component of the limit order book distribution. In this section we examine the sensitivity of the presented findings when the definition of the proposed measure, RLIQ, is changed. First, instead of using the first principal component, we consider the first three and five principal components of the aggregate limit order book distribution introduced in (1), which explain 65% and 77% of the variation, respectively. Table 7 shows that the first principal components for both buy and sell sides have the leading explanatory power for future market volatility and including the first three and five principal components, in addition to the first component, increases the adjusted R2 by only 0.23% and 0.61%, respectively. Moreover, they do not have significant predictive power over volatility when other control variables are included. Onecouldeasilyargueinfavorofusingtheempiricalfrequenciesoforderswaitingateach price distance instead of summarizing this information. Hence, as a second robustness testweconsiderthevolumedistributionfor∆ = 10separatelyforeachbinaspredictors c of volatility and exclude the last bin to avoid multicollinearity. In unreported results, we find that, by including 20 variables, instead of using only RLIQbuy and RLIQsell, the adjusted R2 increases only by 1.01%. 18

Finally, we adopt another variable selection technique, least absolute shrinkage and selection operator (LASSO, Tibshirani, 1996), to reduce the dimensionality instead of employing principal component analysis. LASSO finds the coefficients of a model by minimizing the sum of squared residuals plus an I1-norm penalty function. We reach the sparse model, which corresponds to minimum mean squared errors by employing a 10-fold cross validation technique. Several conclusions arise from Figure 4. First, the estimated coefficients are negative for the bins closer to the best quotes, suggesting that higher liquidity around the best quotes are associated with lower subsequent volatility. On the other hand, the coefficients switch sign after the five best quotes, similar to the loadings of the first principal component plotted in Figure 2. Note that the signs of the loadings of RLIQ and those of the estimated LASSO coefficients are opposite: on the one hand, the loadings are summarizing the liquidity provision in the limit order book; the higher RLIQ, the higher the liquidity around the best quotes. On the other hand, the LASSO coefficients summarize the relationship between liquidity and volatility; the higher the liquidity around the best quotes, the lower the volatility. Second, we see that the estimated coefficients of the sell side distribution are smaller in absolute terms compared to the buy side, suggesting that the buy side is more informative on volatility compared to the sell side of the market, in line with the results presented in Section 5. 0.02 0.01 0 -0.01 -0.02 -0.03 0 1 2 3 4 5 6 7 8 9 stneiciffeoC Buy Side Sell Side Price Distance Figure 4: Estimated LASSO coefficients when ∆ =9. c 19

5.6.2 Alternative volatility measures In this section, we investigate whether the predictive power of relative liquidity is driven by the volatility estimator employed. To this end, instead of calculating volatility via TSRV, we start by employing the realized volatility (RV), calculated through the sum of squared mid-quote returns of the value weighted index. Moreover, for such high frequencies, the return variation caused due to discontinuities or jumps can dominate the continuous part of volatility. Hence, we further examine whether the effect of relative liquidity is on jumps or on the continuous part of volatility. To this end, we consider four jump-robust volatility estimators: minimum RV (minRV) and median RV (medRV) of Andersen et al. (2012), realized bipower variation (BPV) and tripower variation (TPV) of Barndorff-Nielsen and Shephard (2004). The former two measures use the squared returns of minimum and median of the two or three consecutive returns, which are obtained by using nearest neighbor truncation, whereas in BPV and TPV, volatility is estimated through the cumulative sum of products of adjacent absolute returns. However,asmarketmicrostructurenoiseisstillaconcern,weemploythe“pre-averaging” approach introduced by Podolskij and Vetter (2009) prior to the application of both RV and jump-robust volatility estimators. Pre-averaging enables data to produce a set of non-overlapping noise-reduced returns obtained via a kernel function. Table 8 shows that the predictive power of RLIQ is robust to alternative volatility measures and jumps do not significantly affect the liquidity-volatility relationship we are investigating. 5.6.3 Further robustness tests We perform four sets of additional robustness tests. The results presented so far consider the orders to be traded up to the 10 best quotes, i.e., setting ∆ = 10 in (1). First RLIQ c is re-calculated when the information up to the 20 and 30 best quotes (considering the whole book) is used. Second, instead of sampling the trading day using the 15-minute snapshots, wetestthepredictivepowerofthelimitorderbookdistributionovervolatility using 30-minute intervals. In the analysis, to proxy the aggregate level of liquidity, we first calculate RLIQ for each stock and get the cross-sectional average. The next robustness check includes the re-calculation of the aggregate measures by using value-weighted and number-of-tradesweightedcross-sectionalaverages. Theformerweightsarecalculatedbyusingthemarket 20

capitalization values of the individual stocks at the end of the sample period, whereas we calculate the latter by using the daily average number of trades. Finally, we perform a robustness test on the specification of the regression model. We reestimate the benchmark specification in equation (2) with the log-transformed variables to allow the left-hand side of the equation to include potentially both positive and negative numbers. Table 9 confirms the robust relationship between relative liquidity of the buy side of the market and future volatility. We observe that considering the whole book instead of the first 10 best quotes, the sampling frequency, and the regression specification do not change the results. Interestingly, we find that the sell side of the market turns to be significant when the aggregate sell side RLIQ is approximated as the value-weighted or trade-weighted average of the individual stocks. It suggests that bigger and more actively traded stocks are the ones that are informative on future volatility. 6 Conclusion As of today, most of the equity and derivatives exchanges around the world are either pure order driven or at least allowing limit orders in addition to the on-floor market making. The role of limit orders in trading processes expanded progressively over the last decade. This shift in trading is followed by a growing academic literature. This paper contributes to the literature, which studies the informativeness of a limit order book on future volatility. However, this is the first study that examines the predictive power of aggregate liquidity for intraday market volatility. It identifies a strong and robust link between volatility and the relative depth provision, rather than the absolute volume of orders submitted. To measure the relative depth provision, we propose a new way of summarizing the distribution of liquidity in a limit order book, while taking into account the relative magnitude and the location of quoted depth. Our summary measure relative liquidity, RLIQ, accounts for how liquidity is distributed in the whole book and assigns weights to the information provided by different quotes. By using high-frequency data from the Istanbul Stock Exchange, we show that RLIQ has a strong in-sample and out-of-sample predictive power with respect to market volatility, where the relationship is significant for up to 75 minutes ahead volatility. 21

Theevidencepresentedinthispapersuggeststhatthestateofalimitorderbookcontains non-negligible information about the short-term volatility. In a market microstructure context, information on future volatility is important because the execution probability of a limit order increases with volatility. Put differently, the probability that the current price hits the pre-determined limit price increases when volatility is higher. Hence, the presented relationship can be used to design trading strategies that may allow market participants to submit less aggressive orders and reduce execution costs. 22

Appendix A. Calculation of Relative Liquidity (RLIQ) Suppose that the limit order book for stock X at 11:00am is as follows: Ordertype Volume Limitprice Time BestBid BestAsk Sell 50,000 8.4 09:30:00 - 8.2 Buy 10,000 7.6 09:30:01 7.9 8.2 Sell 1,800 8.3 09:30:02 7.9 8.2 . . . Sell 3,334 8.05 10:58:17 8 8.05 Buy 25,000 8 10:58:20 8 8.05 Buy 50,000 7.9 10:58:38 8 8.05 Sell 1 8.1 10:58:50 8 8.05 The first step in the calculation of RLIQ involves the calculation of the tick-adjusted price distance ∆ of each limit order in the given book relative to the best limit price: ∆buy = (pB −pbuy)/tick, i,τ τ i ∆sell = (psell−pA)/tick, i,τ i τ where pB (pA) is the best bid (ask) price in interval τ. In this example pB=8 and pA=8.05. τ τ τ τ On the other hand, pbuy (psell) is the limit price of the ith order. i i Say the tick size is 0.05. Then we have the following price distances for each order: Ordertype Volume LimitPrice Time BestBid BestAsk ∆ Sell 50,000 8.4 09:30:00 - 8.2 7 Buy 10,000 7.6 09:30:01 7.9 8.2 8 Sell 1,800 8.3 09:30:02 7.9 8.2 5 . . . . . . . . . Sell 3,334 8.05 10:58:17 8 8.05 0 Buy 25,000 8 10:58:20 8 8.05 0 Buy 50,000 7.9 10:58:38 8 8.05 2 Sell 1 8.1 10:58:50 8 8.05 1 Next, we obtain of the percentage of total volume supplied/demanded at a given ∆ for ∆ = 0,1,2,..,30. Thisway, wereachthelimitorderbookprobabilitydensityfunction(LOB–PDF). That is: Byrepeatingtheprocedureforeachofthetimeintervalτ,weendupatimeseriesoffrequencies for each price distance ∆. Hence, the LOB–PDF of stock X for a given day and for the sell side of the market looks as follows, TheavgPDFisobtainedastheequally-weightedcross-sectionalaverageoftheindividualLOB– PDFs. In other words, we obtain the previous table for all of the stocks in our sample and 23

Buy side Sell side ∆ TotalVolume Frequency TotalVolume Frequency 0 78,500 0.270 68,400 0.186 1 52,575 0.181 71,602 0.194 2 58,440 0.201 54,588 0.148 3 45,579 0.156 62,068 0.168 . . . . . . 29 0 0.000 0 0.000 30 0 0.000 0 0.000 Frequencies Time/∆ 0 1 2 3 . . . 29 30 10:00 0.212 0.259 0.182 0.133 . . . 0.000 0.000 10:15 0.214 0.249 0.183 0.120 . . . 0.000 0.000 10:30 0.180 0.243 0.184 0.122 . . . 0.000 0.000 10:45 0.194 0.230 0.160 0.124 . . . 0.000 0.000 11:00 0.186 0.194 0.148 0.168 . . . 0.000 0.000 . . . . . . . . . . . . 16:30 0.213 0.223 0.146 0.112 . . . 0.000 0.000 16:45 0.213 0.224 0.156 0.122 . . . 0.000 0.000 17:00 0.188 0.240 0.171 0.118 . . . 0.000 0.000 calculate the cross-sectional average of frequencies to have a distribution of the market for a given delta. RLIQ is the summary measure of this aggregate limit order book distribution τ obtained as the first principal component of the frequency of orders waiting at different price levels at interval τ. 24

References Ahn, H., K. Bae and K. Chan, “Limit Orders, Depth, and Volatility: Evidence from the Stock Exchange of Hong Kong,” Journal of Finance 56 (2001), 767–788. Ait-Sahalia, Y., P. Mykland and L. Zhang, “How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise,” Review of Financial Studies 18 (2005), 351–416. ———, “Ultra high frequency volatility estimation with dependent microstructure noise,” Journal of Econometrics 160 (2011), 160–175. Amihud, Y., “Illiquidity and stock returns: cross-section and time-series effects,” Journal of Financial Markets 5 (2002), 31–56. Andersen, G., T., D. Dobrev and E. Schaumburg, “Power and bipower variation with stochastic volatility and jumps,” Journal of Econometrics 169 (2012), 75–93. Bandi, M., F. and R. Russell, J., “Microstructure Noise, Realized Variance, and Optimal Sampling,” The Review of Economic Studies 75 (2009), 339–369. Barndorff-Nielsen, E., O.andN.Shephard,“Powerandbipowervariationwithstochastic volatility and jumps,” Journal of Financial Econometrics 2 (2004), 1–37. Bollerslev, T. and I. Domowitz, “Trading Patterns and Prices in the Interbank Foreign Exchange Market.,” Journal of Finance 48 (1993), 1421–1443. Burdett, K. and M. O’Hara, “Buildingblocks: Anintroductiontoblocktrading,” Journal of Banking and Finance 11 (1987), 193–212. Coppejans, M., I. Domowitz and A. Madhavan, “Liquidity in an Automated Auction,” (2001), working paper, BlackRock; Barclays Global Investors. Delattre, S., Y. Robert, C. and M. Rosenbaum, “Estimating the efficient price from the order flow: a Brownian Cox process approach,” (April 2013), working paper. Diebold, F. X. and R. S. Mariano,“ComparingPredictiveAccuracy,”Journal of Business and Economic Statistics 13 (1995), 253–263. Domowitz, I., O. Hansch and X. Wang, “Liquidity commonality and return comovement,” Journal of Financial Markets 8 (2005), 351–376. Duong, H. and P. Kalev, “Order Book Slope and Price Volatility,” (2008), working paper, Monash University. 25

Ekinci, C., Intraday Liquidity in the Istanbul Stock Exchange, , chapter 6 (John Wiley & Sons,, 2008), 77–94. Foucault, T., S. Moinas and E. Theissen, “Does Anonymity Matter in Electronic Limit Order Markets?,” Review of Financial Studies 20 (2007), 1707–1747. Ghysels, E. and A. Sinko, “Volatility forecasting and microstructure noise,” Journal of Econometrics 160 (2011), 257–271. Goettler, R., C. Parlour and U. Rajan, “Equilibrium in a Dynamic Limit Order Market,” Journal of Finance 60 (2005), 2149–2192. ———,“Informedtradersandlimitordermarkets,”JournalofFinancialEconomics 93(2009), 67–87. Griffiths, M., B. Smith, A. Turnbull and R. White, “The costs and determinants of order aggressiveness,” Journal of Financial Economics 56 (2000), 65–88. Hansen, R., P. and A. Lunde, “Realized Variance and Market Microstructure Noise,” Journal of Business & Economic Statistics 24 (2006), 127–161. Hasbrouck, J. and J. Seppi, D., “Common Factors in Prices, Order Flows, and Liquidity,” Journal of Financial Economics 59 (2001), 383–411. Jones, C., G. Kaul and M. Lipson, “Transactions, volume, and volatility,” Review of Financial Studies 7 (1994), 631–651. Koksal, B., “An analysis of intraday patterns and liquidity on the Istanbul stock exchange,” Working paper, Central Bank of Turkey, February 2012. Marshall, R., B., “Liquidity and stock returns: Evidence from a pure order-driven market using a new liquidity proxy,” International Review of Financial Analysis 15 (2006), 21–38. Naes, R. and J. Skjeltorp, “Order book characteristics and the volume-volatility relation: Empirical evidence from a limit order market,” Journal of Financial Markets 9 (2006), 408–432. Pascual, R. and D. Veredas, “Does the Open Limit Order Book Matter in Explaining Informational Volatility?,” Journal of Financial Econometrics 8 (2010), 57–87. Podolskij, M. and M. Vetter, “Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps,” Bernoulli 15 (2009), 634–658. 26

Tibshirani, R., “Regression Shrinkage and Selection via the LASSO,” Journal of the Royal Statistical Society 58 (1996), 267–288. Valenzuela, M. and I. Zer, “Competition, signaling and non–walking through the book: Effects on order choice,” Journal of Banking and Finance 37 (2013), 5421–5435. Zhang, L., P. Mykland and Y. Ait-Sahalia, “A tale of two time scales: determining integrated volatility with noisy high-frequency data,” Journal of the American Statistical Association 100 (2005), 1394–1411. Zhou, B., “High-Frequency Data and Volatility in Foreign-Exchange Rates,” Journal of Business & Economic Statistics 14 (1996), 45–52. 27

Table 1: Summary Statistics: The Limit Order Book Distribution Descriptivestatisticsfortheempiricallimitorderbookdistributionforbothsidesthemarket. Themean, variance, skewness, and the fraction of number of shares accumulated up to a given price distance ∆ arereported. Thefirstcolumnshowsthesummarystatisticsofthelimit order bookdistribution,which areobtainedbyaveragingacrossintervals, days, andstocks. Thelastfourcolumnsreportthestatistics for four limit order book distributions (averaged across stocks) at 10:00 (beginning of the day), 12:00 (end of the morning session), 14:15 (beginning of the afternoon session) and 17:00 (end of the trading day). uncond. 10:00 12:00 14:15 17:00 Buy side mean 3.43 3.64 3.32 3.41 3.42 variance 18.42 20.06 17.67 17.83 17.52 skewness 2.41 2.34 2.60 2.33 2.35 up to 1 ∆ 0.40 0.38 0.40 0.41 0.41 up to 3 ∆ 0.68 0.66 0.69 0.68 0.68 up to 5 ∆ 0.82 0.81 0.84 0.82 0.82 up to 10 ∆ 0.93 0.92 0.94 0.93 0.93 up to 20 ∆ 0.99 0.99 0.99 0.99 0.99 up to 30 ∆ 1.00 1.00 1.00 1.00 1.00 Sell side mean 4.63 4.68 4.64 4.56 4.73 variance 25.16 27.51 25.77 23.73 24.20 skewness 1.84 1.83 1.89 1.77 1.74 up to 1 ∆ 0.30 0.31 0.29 0.30 0.28 up to 3 ∆ 0.55 0.56 0.55 0.55 0.53 up to 5 ∆ 0.72 0.73 0.72 0.72 0.70 up to 10 ∆ 0.88 0.87 0.88 0.89 0.88 up to 20 ∆ 0.98 0.98 0.98 0.98 0.98 up to 30 ∆ 1.00 1.00 1.00 1.00 1.00 28

Table 2: Predictability Regressions Estimatedcoefficientsoftheregressionmodeldefinedin(2). Thedependentvariableisthe15-minutesahead market volatility, σM , calculated as the mid-quote volatility of the value-weighted index via τ+1 the TSRV estimator (multiplied by 100). RLIQbuy (RLIQsell) is the first principal component of the respectiveaggregatelimitorderbookdistributionforthebuy(sell)sideasoutlinedinSection3.2. Allof thecontrolvariablesareconstructedasthecross-sectionalaverageofthecorrespondingindividualstock measures. SLOPE is the slope of the limit order book, relSPR is the relative spread, NT is the number of trades and AQ is the average trade size. AMR is the Amihud (2002) illiquidity measure. The logQS isthelogquoteslope,introducedbyHasbrouckandSeppi(2001). Finally,DHWistheDomowitzetal. (2005) illiquidity measure. All of the explanatory variables are standardized. t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and reported in parenthesis. For the sake of brevity, the estimated coefficients of the intraday dummies are omitted. dep. Var.: σM I II III IV V VI τ+1 RLIQbuy -0.044 -0.035 -0.031 -0.029 -0.027 (-7.38) (-7.47) (-7.16) (-6.29) (-5.70) RLIQsell -0.028 -0.022 -0.017 -0.010 -0.010 (-4.25) (-4.14) (-3.45) (-1.75) (-1.73) SLOPE 0.009 0.016 0.014 τ (1.45) (2.90) (2.38) relSPR 0.025 0.013 0.012 τ (4.89) (1.77) (1.54) NT 0.007 0.007 τ (1.27) (1.28) AQ 0.000 0.002 τ (0.10) (0.36) AMR 0.001 0.001 τ (0.33) (0.45) logQS 0.024 0.025 τ (2.13) (2.16) DHW 0.001 0.001 τ (0.17) (0.20) σM 0.037 0.023 0.015 0.015 0.012 τ (6.28) (5.04) (3.32) (3.39) (2.51) adj. R2(%) 21.95 17.25 25.15 28.48 29.45 29.53 29

Table 3: RLIQ vs standard depth measures Comparison of the in-sample predictive power of RLIQ with respect to the depth measures, i.e., the quoted volume of orders waiting at a given threshold. The dependent variable is the 15-minutes-ahead marketvolatility,σM ,calculatedasthemid-quotevolatilityofthevalue-weightedindexviatheTSRV τ+1 estimator (multiplied by 100). RLIQbuy (RLIQsell) is the first principal component of the aggregate buy limitorderbookdistributionforthebuy(sell)sideasoutlinedinSection3.2. DEPTHi isthequoted τ depth at price distance i. All of the explanatory variables are standardized. t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and reported in parenthesis. For the sake of brevity, the estimated coefficients of the intraday dummies are omitted. dep. Var.: σM I II III IV V τ+1 RLIQbuy -0.035 -0.032 -0.033 -0.036 (-7.47) (-7.02) (-6.84) (-7.04) RLIQsell -0.022 -0.018 -0.016 -0.008 (-4.14) (-3.03) (-2.54) (-1.10) buy DEPTH1 -0.012 -0.003 -0.001 0.000 τ (-2.26) (-0.66) (-0.21) (0.00) sell DEPTH1 -0.011 -0.006 -0.003 -0.003 τ (-2.50) (-1.55) (-0.51) (-0.46) buy DEPTH2 -0.007 -0.008 τ (-1.11) (-1.34) sell DEPTH2 0.001 0.000 τ (0.14) (0.06) buy DEPTH3 -0.001 0.010 τ (-0.15) (1.29) sell DEPTH3 -0.001 -0.011 τ (-0.10) (-1.42) buy DEPTH4 -0.017 τ (-2.67) sell DEPTH4 0.012 τ (1.38) buy DEPTH5 -0.005 τ (-0.84) sell DEPTH5 0.002 τ (0.23) σM 0.023 0.032 0.022 0.022 0.021 τ (5.04) (6.31) (4.96) (5.00) (4.73) adj. R2(%) 25.15 19.94 25.32 25.08 26.11 30

Table 4: Predictability Regressions–Further Horizons Estimatedcoefficientsoftheregressionmodeldefinedin(2). Thedependentvariableisthemarketvolatility,σM ,calculatedasthemid-quote τ+h volatility of the value-weighted index via the TSRV estimator (multiplied by 100) in period τ +h, for h=1,2,...,11. RLIQbuy (RLIQsell) is thefirstprincipalcomponentoftheaggregatelimitorderbookdistributionforthebuy(sell)sideasoutlinedinSection3.2. Allofthecontrol variables are constructed as the cross-sectional average of the corresponding individual stock measures. SLOPE is the slope of the limit order book, relSPR is the relative spread, NT is the number of trades and AQ is the average trade size. AMR is the Amihud (2002) illiquidity measure. ThelogQSisthelogquoteslope,introducedbyHasbrouckandSeppi(2001). Finally,DHWistheDomowitzetal.(2005)illiquidity measure. In Panel A, for every time horizon, we report the “simple” regressions, where relative liquidity measures along with the lagged volatility and interval dummies are used as regressors. On the other hand, Panel B reports the results when all of the control variables are included in the regression equation. All of the explanatory variables are standardized. t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and reported in parenthesis. For the sake of brevity, the estimated coefficients of the intraday dummies are omitted. dep. var.: σM Panel A: “simple” regressions Panel B: multiple regressions τ+h 0–15 15–30 30–45 . 120–135 135–150 150–165 0–15 15–30 30–45 . 120–135 135–150 150–165 RLIQbuy -0.035 -0.032 -0.031 . -0.030 -0.028 -0.025 -0.027 -0.023 -0.023 . -0.03 -0.029 -0.017 (-7.47) (-5.97) (-5.28) . (-3.95) (-3.21) (-2.77) (-5.70) (-5.33) (-4.48) . (-3.87) (-3.92) (-1.98) RLIQsell -0.022 -0.025 -0.023 . -0.016 -0.014 -0.015 -0.010 -0.011 -0.008 . 0.00 -0.003 0.003 (-4.14) (-3.78) (-3.12) . (-1.92) (-1.66) (-1.85) (-1.73) (-2.26) (-1.22) . (0.16) (-0.38) (0.33) SLOPE . 0.014 0.021 0.019 . 0.01 0.012 0.010 τ . (2.38) (3.04) (2.67) . (1.44) (1.39) (1.34) relSPR . 0.012 0.015 0.015 . 0.03 0.045 0.010 τ . (1.54) (2.44) (2.30) . (3.06) (3.79) (1.07) NT . 0.007 0.010 0.007 . 0.00 0.001 0.011 τ . (1.28) (1.59) (1.13) . (0.25) (0.08) (1.42) AQ . 0.002 0.004 0.002 . 0.00 -0.009 0.011 τ . (0.36) (0.89) (0.54) . (0.04) (-1.46) (1.54) AMR . 0.001 0.007 -0.001 . 0.00 -0.001 0.001 τ . (0.45) (5.98) (-0.85) . (0.20) (-0.22) (0.69) logQS . 0.025 0.030 0.031 . 0.00 -0.017 0.028 τ . (2.16) (4.13) (3.85) . (0.29) (-1.23) (2.30) DHW . 0.001 0.001 0.003 . 0.02 0.015 0.013 τ . (0.20) (0.21) (0.64) . (2.08) (1.94) (2.14) σM 0.023 0.016 0.006 . 0.005 0.011 0.011 0.012 0.001 -0.007 . 0.00 0.007 -0.006 τ (5.04) (4.17) (1.24) . (1.00) (1.66) (1.62) (2.51) (0.32) (-1.26) . (-0.76) (1.02) (-0.88) adj. R2(%) 25.15 20.93 16.73 . 12.61 13.36 14.01 29.53 28.03 23.57 . 19.20 20.00 21.81 31

Table 5: Out-of-Sample Forecasting Evaluation Out-of-sample forecasting experiment results. The h-period-ahead forecast error is obtained as the difference between the realized volatility at τ +h and the fitted value of the predictive regression estimated up to time τ with the variable of interest listed in the first column used as a regressor. On the other hand, the competing error is calculated from the sample mean volatility up to time interval τ. The dependent variable is the market volatility, σM , calculated as the mid-quote volatility of the value-weighted index via the TSRV estimator (multiplied τ+h by 100) in period τ +h, for h=1,2,...,6. RLIQbuy (RLIQsell) is the first principal component of the aggregate limit order book distribution for the buy (sell) side as outlined in Section 3.2. relSPR , NT , and logQS are the cross-sectional averages of the relative spread, number of τ τ τ trades, and log quote slope, respectively. Although we examine the out-of-sample forecasting power of all of the control variables introduced in Section 4.3, for the sake of brevity we report only the ones with significant forecasting power. The out-of-sample R2(%) and the difference in mean-squared errors (∆MSEx1000) are reported. Finally, DM denotes the Diebold and Mariano (1995) predictive ability test. Panels A and B report the results when the training period is set to 400 and 350 observations, respectively. Forecasting variable 0–15min 15–30min 30–45min 45–60min 60–75min 75–90min Panel A: Training Period: 400 obs. RLIQbuy Out-of-sample R2(%) 12.86 10.13 9.20 8.25 7.08 5.50 ∆MSE 1.91 1.50 1.36 1.23 1.06 0.83 DM t-stat 2.68 2.45 2.74 2.46 2.35 1.86 relSPR Out-of-sample R2(%) 10.85 9.95 8.94 9.13 9.48 10.57 τ ∆MSE 1.61 1.47 1.32 1.36 1.42 1.59 DM t-stat 2.24 2.18 1.94 1.94 1.99 2.30 NT Out-of-sample R2(%) 12.89 8.32 0.69 -0.54 0.38 2.16 τ ∆MSE 1.91 1.23 0.10 -0.08 0.06 0.32 DM t-stat 2.46 2.00 0.31 -0.49 0.25 0.87 logQS Out-of-sample R2(%) 14.83 12.59 10.57 10.54 10.04 11.83 τ ∆MSE 2.20 1.86 1.56 1.57 1.50 1.78 DM t-stat 2.92 2.66 2.50 2.43 2.24 2.41 Panel B: Training Period: 350 obs. RLIQbuy Out-of-sample R2(%) 11.23 8.70 7.95 7.19 6.32 4.93 ∆MSE 1.80 1.40 1.28 1.16 1.03 0.80 DM t-stat 2.76 2.44 2.69 2.46 2.37 1.90 relSPR Out-of-sample R2(%) 13.35 12.38 11.66 11.68 11.89 12.59 τ ∆MSE 2.14 1.99 1.88 1.89 1.93 2.05 DM t-stat 2.98 2.93 2.75 2.73 2.74 3.03 NT Out-of-sample R2(%) 11.11 6.91 0.73 -0.41 0.03 1.24 τ ∆MSE 1.78 1.11 0.12 -0.07 0.00 0.20 DM t-stat 2.43 1.97 0.35 -0.44 0.03 0.60 logQS Out-of-sample R2(%) 15.67 12.89 11.10 10.66 11.34 12.56 τ ∆MSE 2.51 2.07 1.79 1.73 1.84 2.05 DM t-stat 3.60 3.22 3.12 2.92 2.95 3.00 32

Table 6: Individual Stock Predictability Regressions Estimated coefficients of the regression model defined in (5). RLIQind,buy (RLIQind,sell) is the first principal component of the individual stock limit order book distribution, for the buy (sell) side. In a given trading interval τ, SLOPE is the slope of the limit order book, relSPR is the relative spread, NT is the number of trades and AQ is the average trade size. AMR is the Amihud (2002) illiquidity measure. The logQS is the log quote slope, introduced by Hasbrouck and Seppi (2001). Finally, DHW is the Domowitz et al. (2005) illiquidity measure. All of the explanatory variables are standardized. The dependent variable is σ , which is the volatility calculated using the mid-quotes τ+1 of the orders originated in interval τ +1 via the TSRV estimator (multiplied by 100). Panel A shows the results for the pooled regression. t-statistics based on cluster robust standard errors on stock level are reported. Panel B summarizes the results when the model is estimated for each stock separately. The cross-sectional median of the estimated significant coefficients at a 5% level is reported. In brackets, first, the percentage of the stocks with a significant coefficient at a 5% level and second, the percentage of the positive estimates, are reported. t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals. For the sake of brevity, the estimated coefficients of the intraday dummies and stock fixed effects are omitted. dep. var.: σ Panel A: Pooled regression Panel B: Summary of individual regressions s,τ+1 I II III IV V VI VII VIII RLIQind,buy -0.057 -0.058 -0.058 -0.055 -0.065 -0.057 -0.055 -0.052 (-13.07) (-12.95) (-12.60) (-11.94) [87/0] [87/0] [87/0] [80/0] RLIQind,sell -0.024 -0.026 -0.019 -0.019 -0.039 -0.046 -0.048 -0.042 (-4.99) (-5.36) (-3.56) (-3.66) [37/9] [33/0] [27/0] [27/0] SLOPE -0.019 0.031 0.022 -0.057 0.054 0.039 τ (-2.47) (4.47) (3.58) [43/8] [30/78] [20/67] relSPR 0.040 -0.001 -0.007 0.088 -0.104 -0.164 τ (4.09) (-0.14) (-0.67) [37/91] [30/33] [23/14] NT 0.037 0.040 0.048 0.057 τ (9.70) (11.18) [57/100] [57/100] AQ -0.005 0.001 0.023 0.036 τ (-1.02) (0.20) [17/60] [17/100] AMR 0.000 0.003 0.007 0.011 τ (-0.02) (0.93) [20/50] [20/50] logQS 0.091 0.094 0.112 0.126 τ (10.13) (10.22) [53/100] [50/100] DHW 0.013 0.014 0.055 0.055 τ (2.61) (2.79) [33/90] [33/90] σ 0.082 0.055 0.064 0.042 0.083 0.068 0.065 0.060 τ (16.19) (9.59) (14.29) (7.62) [97/100] [50/100] [77/100] [37/100] adj. R2(%) 13.24 14.75 15.50 15.95 10.27 13.13 14.08 14.95 33

Table 7: One Period Ahead Predictive Power–Principal Components In-samplepredictivepowerofotherprincipalcomponentsover15-minutes-aheadmarketvolatility,σM , τ+1 calculated as the mid-quote volatility of the value-weighted index via the TSRV estimator (multiplied by 100). PCi is the ith principal component of the aggregate limit order book distribution for the buy (sell) side as outlined in Section 3.2. Columns I–III report the estimated coefficients of the principal componentswhenthelaggedvolatilityandintradaydummiesareincludedinthespecificationascontrol variables. In columns IV to VI the other control variables defined in Table 2 are included in addition. Alloftheexplanatoryvariablesarestandardized. t-statisticsarecalculatedusingNewey-Weststandard errors to capture possible autocorrelation in the residuals and reported in parenthesis. For the sake of brevity, the estimated coefficients of the intraday dummies are omitted. dep. Var.: σM I II III IV V VI τ+1 PC1buy (RLIQbuy) -0.035 -0.034 -0.035 -0.027 -0.028 -0.029 (-7.47) (-7.27) (-7.42) (-5.70) (-5.63) (-5.85) PC1sell (RLIQsell) -0.022 -0.022 -0.022 -0.010 -0.010 -0.012 (-4.14) (-3.73) (-3.77) (-1.73) (-1.71) (-2.03) PC2buy 0.003 0.002 0.000 -0.001 (0.86) (0.73) (-0.13) (-0.42) PC2sell -0.003 -0.003 0.001 0.001 (-0.78) (-0.81) (0.21) (0.26) PC3buy -0.002 -0.002 0.001 0.001 (-0.54) (-0.57) (0.40) (0.25) PC3sell 0.007 0.008 0.003 0.004 (2.14) (2.50) (0.89) (1.17) PC4buy 0.005 0.004 (1.51) (1.13) PC4sell 0.000 -0.002 (-0.07) (-0.50) PC5buy -0.008 -0.007 (-2.09) (-1.83) PC5sell -0.002 -0.003 (-0.59) (-0.82) SLOPE 0.014 0.014 0.016 τ (2.38) (2.37) (2.69) relSPR 0.012 0.013 0.013 τ (1.54) (1.63) (1.68) NT 0.007 0.007 0.007 τ (1.28) (1.26) (1.35) AQ 0.002 0.001 0.001 τ (0.36) (0.34) (0.15) AMR 0.001 0.001 0.001 τ (0.45) (0.42) (0.34) logQS 0.025 0.024 0.022 τ (2.16) (1.94) (1.85) DHW 0.001 0.000 0.001 τ (0.20) (0.07) (0.25) σM 0.023 0.022 0.021 0.012 0.012 0.011 τ (5.04) (4.82) (4.54) (2.51) (2.48) (2.33) adj. R2(%) 25.15 25.38 25.76 29.53 29.27 29.42 34

Table 8: Robustness–Volatility This table presents the results when alternative volatility measures are employed to approximate the 15-minutes market volatility. In Columns I to II, volatility is calculated through realized variance (RV) of 30 seconds sampled squared mid-quote returns. Columns III through VI show the results whenBarndorff-NielsenandShephard(2004)’srealizedbipowervariation(BPV)andtripowervariation (TPV)areemployed,respectively. ColumnsVIIthroughXpresenttheresultswhenminRVandmedRV of Andersen et al. (2012) are used as volatility proxies, respectively. To account for the noise, the “preaveraging” approach of Podolskij and Vetter (2009) is employed in all of the volatility estimators. All of the explanatory variables are standardized. In all of the specifications t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and for the sake of brevity, the estimated coefficients of the intraday dummies are omitted. All of the explanatory variables are defined in Table 2. σM σM σM σM σM τ+1,RV τ+1,BPV τ+1,TPV τ+1,minRV τ+1,medRV I II III IV V VI VII VIII IX X RLIQbuy -0.047 -0.039 -0.041 -0.035 -0.042 -0.037 -0.037 -0.032 -0.041 -0.035 (-6.66) (-5.43) (-6.03) (-4.68) (-6.06) (-5.00) (-5.58) (-4.10) (-6.09) (-4.79) RLIQsell -0.028 -0.011 -0.028 -0.010 -0.028 -0.011 -0.027 -0.009 -0.029 -0.011 (-3.76) (-1.41) (-3.52) (-1.22) (-3.57) (-1.44) (-3.47) (-1.07) (-3.77) (-1.42) SLOPE 0.024 0.022 0.022 0.018 0.024 τ (2.37) (2.45) (2.64) (1.91) (2.85) relSPR 0.023 0.022 0.026 0.019 0.024 τ (2.07) (1.90) (2.29) (1.62) (2.12) NT 0.008 0.008 0.011 0.008 0.010 τ (1.04) (1.00) (1.33) (0.94) (1.23) AQ 0.005 0.001 -0.002 0.001 0.002 τ (0.74) (0.21) (-0.43) (0.15) (0.33) AMR 0.006 0.001 0.002 -0.001 0.001 τ (0.97) (0.26) (0.77) (-0.42) (0.29) logQS 0.029 0.026 0.021 0.023 0.026 τ (1.65) (1.41) (1.26) (1.24) (1.49) DHW 0.007 0.013 0.013 0.015 0.014 τ (1.01) (1.68) (1.82) (1.85) (1.96) σM 0.020 0.007 0.025 0.010 0.028 0.014 0.025 0.012 0.028 0.012 τ (3.56) (1.14) (4.81) (1.78) (5.02) (2.20) (5.02) (2.21) (5.26) (2.15) adj. R2(%) 22.16 26.81 23.14 28.00 26.24 30.98 21.17 25.42 23.54 28.68 35

Table 9: Other Robustness Tests Columns I through IV present the results when ∆ is equal to 20 and 30, respectively. The following two columns show the results when c the sampling period is 30 minutes instead of 15 minutes. Columns VII through X report the results when the explanatory variables are aggregatedviavalue-weightedandtrade-weightedcross-sectionalaverages. Finally,thelasttwocolumnspresenttheestimatedcoefficientsfor the log-transformed variables. All of the explanatory variables are standardized. In all of the specifications t-statistics are calculated using Newey-West standard errors to capture possible autocorrelation in the residuals and for the sake of brevity, the estimated coefficients of the intraday dummies are omitted. All of the variables are defined in Table 2. ∆ =20 ∆ =30 30–min sampling value–weighted trade–weighted log transform. c c I II III IV V VI VII VIII IX X XI XII RLIQbuy -0.033 -0.026 -0.032 -0.025 -0.053 -0.043 -0.034 -0.025 -0.038 -0.028 -0.031 -0.024 (-7.71) (-5.65) (-7.43) (-5.50) (-6.26) (-6.22) (-7.23) (-5.09) (-7.33) (-5.30) (-7.33) (-5.98) RLIQsell -0.017 -0.007 -0.016 -0.006 -0.036 -0.012 -0.023 -0.012 -0.027 -0.016 -0.022 -0.008 (-3.04) (-1.12) (-2.83) (-0.91) (-3.35) (-1.67) (-4.53) (-2.32) (-4.81) (-2.65) (-3.87) (-1.55) SLOPE 0.015 0.014 0.034 0.019 0.021 0.015 τ (2.51) (2.39) (3.20) (2.34) (2.23) (2.70) relSPR 0.012 0.013 0.025 -0.017 -0.006 0.011 τ (1.58) (1.67) (2.49) (-1.69) (-0.60) (1.43) NT 0.006 0.006 0.006 0.010 0.007 0.006 τ (1.06) (1.15) (0.62) (2.30) (1.57) (0.99) AQ 0.003 0.003 0.011 -0.003 -0.004 -0.001 τ (0.74) (0.77) (1.52) (-0.84) (-1.06) (-0.26) AMR 0.002 0.002 0.009 0.001 0.002 0.001 τ (0.65) (0.58) (1.71) (0.26) (0.77) (0.44) logQS 0.022 0.022 0.056 0.046 0.041 0.029 τ (1.89) (1.86) (4.03) (3.55) (2.96) (2.49) DHW 0.001 0.001 0.005 -0.001 -0.003 -0.002 τ (0.21) (0.16) (0.72) (-0.16) (-0.59) (-0.33) σM 0.022 0.012 0.023 0.012 0.033 0.004 0.024 0.013 0.023 0.015 0.025 0.015 τ (4.83) (2.56) (4.90) (2.58) (4.25) (0.51) (5.09) (2.68) (4.94) (2.95) (5.45) (3.15) adj. R2(%) 25.82 29.49 25.55 29.33 31.21 41.71 25.32 28.56 25.61 28.58 24.55 28.81 36