What Does the Prevalence of Zero Returns Tell Us about Jump Identification? Evidence from U.S. Treasury Securities. Seung-Oh Han.

Transcription

1 What Does the Prevalence of Zero Returns Tell Us about Jump Identification? Evidence from U.S. Treasury Securities Seung-Oh Han Sahn-Wook Huh [Current Version: July 29, 2018] School of Management, University (SUNY) at Buffalo, Buffalo, NY School of Management, University (SUNY) at Buffalo, Buffalo, NY Corresponding Author We thank Chunchi Wu for providing useful advice on the U.S. Treasury databases, George Jiang for valuable assistance on programming issues related to jump-identification methods, and Craig Holden for providing us with the SAS code for the Holden-Jacobsen (2014) algorithm. All errors are solely the authors responsibility.

2 What Does the Prevalence of Zero Returns Tell Us about Jump Identification? Evidence from U.S. Treasury Securities Abstract We document that each trading day U.S. Treasury notes have a large proportion of zero returns, mainly because virtually all trades are executed at the best ask or bid quote and most of quoted spreads are set close to the minimum tick. The proportion of zero returns is highly negatively correlated with commonly used illiquidity measures, suggesting that it may not capture illiquidity in the Treasury market. Given the prevalence of zero returns, we provide evidence that conventional jumpdetection methods are vulnerable to biases, leading to falsely identifying jumps in Treasury notes. We propose a method to significantly improve the performance of jump detection in the Treasury market. Our results are robust to using models with stochastic volatility, lower sampling frequencies, datasets from electronic platforms, quote midpoint returns, and discrete price grids for simulations. JEL Classification: G12; G14; G17 Keywords: U.S. Treasury Notes; Zero Returns; Trade Execution; Monte Carlo Simulations; Discrete Price Grids; Jump Identification; Combined Methods

3 1. Introduction TheU.S.Treasurymarketisknownasoneofthelargestandmostliquidfinancial markets in the world. According to the CME Group (2016), daily trading volume of U.S. Treasury notes and bonds in the secondary market was well above $400 billion all throughout the 2000s, which compares to the daily trading volume of $110 billion in 2006 in the U.S. stock market. U.S. Treasury securities are traded overseas as well as in the U.S. around the clock. Competition among dealers and interdealer brokers leads to narrow bid-ask spreads for most Treasury securities. This is especially true for on-the-run Treasury securities. On-the-run Treasuries are the most recently issued Treasury bonds or notes of a particular maturity. On the other hand, off-the-run Treasuries refer to Treasury securities that have been issued before the most recent issue and are still outstanding. The on-the-run note or bond is the most frequently traded Treasury securities of its maturity. Since on-the-run issues are the most liquid, they generally trade at a small premium and therefore yield slightly less than their off-the-run counterparts. Since the Treasury market is very liquid, one would expect that when the on-the-run trading data of Treasury securities are sampled at 3- to 15-minute frequencies, intraday intervals that have a return of zero are rare. Contrary to this assumption, however, we observe zero-return intervals quite often in the Treasury market. For example, when the trading data for the 2- year Treasury note are sampled at a 5-minute frequency, which is commonly used in intraday analyses, about 55.2% of intervals within a given trading day have a zero return. This means that the number of intervals with a zero return is as large as 63 out of the total minute intervals on each trading day. This is surprising, given our perception that the U.S. Treasury market (especially for on-the-run notes) is one of the most liquid markets in the world. Why does a high proportion of zero returns matter in finance studies? First, the proportion of zero returns is often used as a measure of illiquidity for financial assets. Lesmond, Ogden, and Trzcinka (1999) develop a measure of illiquidity based on the frequency of zero returns using a limited dependent variable procedure. The intuition in their study is that arbitrageurs trade only if the value of accumulated information exceeds the marginal cost of trading. If trading costs are relatively high, new information should accumulate longer before investors 1

4 engage in trading. Therefore, the proportion of the zero-return days can be a proxy for the length of information accumulation. They provide evidence that the frequency of zero returns is positively correlated with conventional measures of illiquidity such as the quoted spread and the Roll (1984) measure in the U.S. equity market. Since the study of Lesmond, Ogden, and Trzcinka (1999), the proportion of zero-return days has often been used as a measure of illiquidity for financial assets (e.g., Bekaert, Harvey, and Lundblad, 2007; Goyenko, Holden, and Trzcinka, 2009; Goyenko and Sarkissian, 2014; Hou and Loh, 2016). Second, when zero returns are observed often, this may induce errors in studies that examine the behavior of asset-price movements. Recent empirical and theoretical studies have shown that asset prices are characterized by jumps, which have implications for portfolio management, asset pricing, and risk management or hedging (e.g., Merton, 1976; Bates, 1996; Bakshi et al., 1997; Duffie and Pan, 2001; Johannes, 2004; Lee and Mykland, 2007). Jumps can be caused by arrivals of important news that affects the value of assets, extreme events such as political upheavals, shocks to macroeconomic variables such as interest rates and energy prices, or global perturbation of recessions. Despite the developments of more elaborate asset-pricing models and their inference techniques, studies show that jumps are by nature difficult to identify. Especially, when zero returns occur frequently in the price process of any asset, commonly used methods for jump identification may spuriously detect a small price move as a jump. In this study, we first document using long and comprehensive on-the-run Treasury data (about 15 years: GovPX for 1993: :09 and BrokerTech for 2001: :11) that in the U.S. Treasury market, the proportions of zero-return intervals are unexpectedly high when measured by 1 and 2. The former metric ( 1) is the ratio of the number of intraday intervals with a zero return to the total number of intraday intervals within a day. The latter ( 2) is the ratio of the number of positive-volume intervals with a zero return to the total number of intraday intervals within a day. Even when the more conservative measure ( 2) is used, the proportion of zero returns is significantly high in U.S. Treasury notes. For example, when the data are sampled at a 5-minute frequency, 2 is as high as 30.7% for the 3-year note and even higher 39.1% for the 2-year note. The prevalence of zero returns is not sensitive to using different sampling frequencies, notes with different maturities, different 2

5 trading platforms, and returns based on quote midpoints. Then, a natural question is whether the two zero-return metrics, 1 and 2, can capture illiquidity and hence be used as measures of illiquidity for Treasury securities, as they are for equity securities. To examine this issue, we compute correlations of the two metrics with typical illiquidity measures such as the quoted spread, the effective spread, and the two components of trading costs (adverse-selection and non-information components). The results show that 1 and 2 are highly negatively correlated with all of the traditional measures of illiquidity. 1 We provide plausible explanations for the prevalence of zero returns and its implications. The high proportions of zero returns are caused mainly by the fact that unlike equity securities, Treasury notes are executed almost exclusively (99.9%) at a prevailing ask or bid quote (not inside/outside the quoted spread), and most of spreads are set at values close to the minimum tick. Thus, a higher proportion of zero returns does not necessarily mean the market is illiquid. No study, so far as we are aware, has documented the above empirical aspects in the U.S. Treasury market. While our findings suggest that researchers should exercise caution when selecting appropriate measures of illiquidity for studies on U.S. Treasury securities, our focus is more on the effects that the prevalence of zero returns might have on other finance studies. Since the influential study of Merton (1976) on the application of jump processes in valuing options, financial modeling based on discontinuous price changes or jumps has attracted much attention not only from academicians but also from practitioners. The fact that zero returns are very common in Treasury notes has important implications for jump-identification studies. In particular, the prevalence of zero returns in the Treasury market may potentially induce significant biases in existing jump-detection procedures. In the later part of the paper, therefore, we evaluate how existing jump-identification methods perform for Treasury securities and propose a method to improve the performance of jump-detection procedures. Barndorff-Nielsen and Shephard (BNS, 2004) developed a jump-detection method by comparing the realized volatility and the bi-power variation. Since their study, a variety of non- 1 Unreported results show that both 1 and 2 are also strongly negatively correlated with the Amihud (2002) measure constructed with Treasury note data. 3

6 parametric jump-identification tests have been proposed to address the shortcomings of BNS. Aït-Sahalia and Jacod (2009) construct a jump-testing procedure under both null and alternative hypothesis because the BNS test does not allow for an asymptotic theory under the alternative hypothesis of the presence of jumps. Jiang and Oomen (JO, 2008) build upon the intuition that a swap-replication strategy is imperfect in the presence of jumps. The JO-type methods allow for handling market microstructure effects. 2 In the same vein, Andersen, Dobrev, and Schaumburg (2012, ADS) propose two integrated variance estimators, which can be used in the BNS approach. These alternatives can deal with the presence of jumps and microstructure noise stemming from stale quotes, rounding to a discrete price grid, and bid-ask bounce. Nevertheless, jump-detection methods with efficiency properties often falsely identify jumps. Apparently, spuriously detected jumps can induce biases in studies on the effect of jumps and market dynamics around jump occurrences. A simple way to reduce the probability of incorrectly identifying jumps (i.e., a Type I error) is to choose relatively conservative significance levels [e.g., Bollerslev, Law, and Tauchen (2008) employs a 0.1% critical value]. Another way of handling spurious jumps is to apply thresholds. Bajgrowicz, Scaillet, and Treccani, (2016) propose a formal thresholding approach. Jiang, Lo, and Verdelhan (2011) use a jump-size restriction, in which price movements are identified as a jump only if the size of the price change is larger than four times the standard deviation of daily returns. Dumitru and Urga (2012) suggest that combining single-procedure methods can reduce the number of spuriously identified jumps that are caused by microstructure noise, providing empirical evidence using five-year data ( ) on five NYSE-listed stocks. Unlike their study, our paper is motivated by the prevalence of zero returns in U.S. Treasury notes, which is potentially a very important aspect for studies on the U.S. Treasury market. We find few studies that explicitly examine the impact of frequent zero returns in the Treasury market on jump-identification methods. To evaluate the performance of existing nonparametric jump-identification methods, we conduct Monte Carlo simulations. We consider a standard logarithmic Brownian motion and a stochastic volatility model, each of which generates 10,000 replications. Model parameters are 2 Many studies document the market microstructure effects arising from stale quotes and non-synchronous trading. See, Epps (1979) and Lo and MacKinlay (1990) for the non-synchronous trading that causes potentially serious biases. 4

7 calibrated based on the 5-year Treasury note at a 5-minute sampling frequency. Our results show that without the presence of zero returns, the 12 BNS/ADS-type and two JO-type methods perform properly, with the probability of falsely identifying jumps (the size of the tests) being lower than 5%. However, a large proportion of zero returns has a serious consequence. That is, with the presence of frequent zero returns, the size of tests surges in an unrestrainable manner. In the cases where the fraction of zero returns is set at the same level as the actual data, the probability is substantially high in most of the single-procedure methods, with the size (distortion) of the tests being often above 30% and, for some methods, even above 90%. Given the prevalence of zero returns in the U.S. Treasury market, none of the 14 single-procedure methods is suitable for jump identification purposes. Now a more important question is how we can improve the performance in jump identification for Treasury notes. We propose that by combining any two of the single-procedures, we can significantly improve the performance. The simulation results show that the size of the tests in any of the 17 combined-procedure methods is less than 10% even when the zero-return fraction is set at the same level as the actual data. Especially, the Med/Med & Ratio method exhibits the size of only 5.5%, which compares to 21.2% or 15.9% when one of the corresponding single-procedure methods is used. Whereas most of the combined-procedure methods exhibit significantly improved performance, the Min/Med & Diff and Min/Med & Ratio methods perform best in terms of reducing the size distortion, without losing the power of tests. Our results are robust to using models with stochastic volatility, lower sampling frequencies, datasets from electronic platforms, quote midpoint returns, and a discrete grid approach. 2. Data and Variable Construction 2.1. Trade and Quote Data for U.S. Treasury Securities For this study we use trade and quote data on on-the-run 2-, 3-, 5-, and 10-year notes for about 15 years in the secondary U.S. Treasury market. Tick-level transaction data are obtained from two data sources: from January 4, 1993 to September 30, 2001 we use GovPX, which consol- 5

8 idates data from voice-assisted brokers, and from October 1, 2001 to November 30, 2007 we use BrokerTec, which consolidates data from electronic interdealers. Electronic communications networks (ECNs), which first appeared on the Treasury markets in early 2000s, quickly dominated most of the transactions since its introduction. But before 2000 all voice-assisted brokers reported their transactions to GovPX. 3 Barclay, Hendershott, and Kotz (2006) document that on-the-run securities with all maturities transacted in were mostly traded on two electronic trading platforms, BrokerTec and espeed. According to Mizrach and Neely (2006), BrokerTec is the only data provider that covers more than 60% of trading activities in the secondary over-the-counter (OTC) market for on-the-run Treasury securities in the third quarter of Furthermore, some recent studies (e.g., Dunne, Li, and Sun, 2015; Fleming and Nguyen, 2013) that use the two electronic platforms show that BrokerTec has a greater market share for the four types of securities analyzed in this paper. Because GovPX and BrokerTec contain all market activity information such as trade directions, using trade-classification algorithms is not necessary. Since the secondary U.S. Treasury market trades 24 hours a day, the choice of sample hours can be flexible, depending on the purpose of research design. We choose normal trading hours from 7:30 A.M. to 5:00 P.M. EST. to examine the effects of liquidity on the proportion of intraday zero returns and potential biases in jump-identification methods due to the prevalence of zero returns. The tick size for the 2-, 3-, and 5-year Treasury notes is 2/256ths of 1% (or basis points), while that of the 10-year note is 4/256ths of 1% (or basis points). The sample period in this paper is about 15 years from January 1993 to November 2007 and this period is comparable to the sample periods of previous studies on the market microstructure of the U.S. Treasury market Measures for the Proportion of Zero Returns Lesmond, Ogden, and Trzcinka (1999) provide a theoretical framework for evaluating the proportion of days with a zero return as a degree of illiquidity, and thus the proportion of zero-return 3 See Mizrach and Neely (2006) for the introduction of electronic trading, and Fleming, Mizrach, and Nguyen (2017) for details on the microstructure of the U.S. Treasury market following its migration to electronic trading. 4 For example, see Brandt and Kavajecz (2004), Green (2004), Dungey, Mckenzie, and Smith (2009), Jiang, Lo, and Verdelhan (2011), and Dumitru and Urga (2016). 6

9 days in a month has been used as an illiquidity proxy in the U.S. and other global equity markets (e.g., Goyenko, Holden, and Trzcinka, 2009; Bekaert, Harvey, and Lundblad, 2007; Hou and Loh, 2016). Following the intuition of Lesmond, Ogden, and Trzcinka (1999), we construct two daily zero-return-related measures and investigate whether these measures can proxy for illiquidity in the Treasury market as well. We define the first zero-return measure, 1, as follows: 1= Number of Intervals with a Zero Return (1) where is the total number of intervals within a day ( = 189, 113, 56, and 37 at a 3-, 5-, 10-, and 15-minute sampling frequency, respectively). Thus, 1 measures the proportion of intraday intervals that have a zero return within a given trading day. An alternative measure, 2, is similarly defined as: 2= Number of Positive Volume Intervals with a Zero Return (2) 2 is more conservative in the sense that it is the proportion of intraday intervals in which at least one trade occurs and at the same time have a zero return on a given trading day. In Table 1 we report the time-series average of the two measures ( 1 and 2) for the 2-, 3-, 5-, and 10-year U.S. Treasury securities. Panel A1 shows the average of the daily proportion of intervals with a zero return ( 1) based on transaction prices, while Panel A2 does the same for the average daily proportion of positive-volume intervals with a zero return ( 2). The statistics contained in Panels A1 and A2 are obtained from GovPX for the 1993: :09 period and from BrokerTec for the 2001: :11 period. Panel A1 shows that the proportions of zero-return intervals within a day are substantial for all four maturities, regardless of sampling frequencies including the 5-minute frequency, which is most commonly used for intraday analyses: 1 in Panel A1 ranges from 17.8% to 65.4%. To be more specific, for the 2-year note in Panel A1, 55.2% of intervals sampled at a 5- minute frequency have a zero return, indicating that the number of intervals with a zero return is approximately 62 out of the total minute intervals within a given day. One way to decrease the zero-return ratios is to take a sample at a lower frequency, thereby reducing the 7

10 number of intervals (hence observations) in the intraday analyses. We find that zero-return intervals are still prevalent even at a 15-minute sampling frequency, in which case the total number of intraday intervals is reduced to 37. At a 15-minute frequency, the 2-year note has the highest zero-return ratio ( 1 = 34.1%), while the 5-year note has the lowest ratio ( 1 = 17.8%). These high ratios of zero returns are counter-intuitive, given that U.S. Treasury securities are known as one of the most liquid asset classes in financial markets. Under what circumstances do we observe a zero return in a trading interval? First, a zero return can occur when there is no transaction in the interval. In this case, it is a convention (e.g., Chan, Chung, and Johnson, 1993; De Jong and Nijman, 1997; Andersen et al., 2007; Evans, 2011) that the prevailing price at the end of the previous interval is carried over to the current interval as its closing price, which makes the return in the current period zero. Second, even though there are transactions in the current interval, a zero return can occur when the last trade of the current interval is executed at the same price as the last transaction price of the previous interval. 1 reflects both cases described above. On the other hand, 2 captures the second case only. When we use the more conservative measure ( 2) in Panel A2, the proportion of zeroreturn intervals are still significantly high. We find in Panel A2 that 2 ranges from 15.3% to 39.1%. At a 5-minute frequency, 2 is as high as 23.3% for the 5-year note and even higher at 39.1% for the 2-year note. This implies that zero returns are frequently observed in U.S. Treasury securities, more likely because transactions are executed at the same price across intervals than because there is no transaction in the trading intervals. Now we check whether the proportion of zero-return intervals decreases when we use BrokerTec, which is more recent (2001: :11) and records data from an electronic trading platform. Thus, we compute 1 and 2 in Panel B using the BrokerTec database only. As Panel B1 shows, 1 decreases slightly, compared to the corresponding values reported in Panel A1. In Panel B2, however, 2 does not decrease much but rather increases in many cases, relative to the proportions in Panel A2. For instance, at the 5-minute frequency for the 3-year note, 1 decreases from 50.5% in Panel A1 (where the older GovPX data- 8

11 base is jointly used) to 46.6% in Panel B1. However, 2 increases from 30.7% in Panel A2 to 32.9% in Panel B2. This demonstrates that by choosing one database over another, we do not see the proportion of zero-return intervals change substantially. In other words, even with the dataset consolidated from an automated electronic trading platform, zero returns are a phenomenon that is common in U.S. Treasury notes. Using the returns based on quote midpoints (as opposed to transaction prices) may decrease the proportion of zero returns, since quotes are more frequently reported than trades themselves (and quote midpoint returns are free from bid-ask bounce). Therefore, we obtain the two measures in Panel C using the midpoint returns that are computed with the best bid and ask quotes submitted to BrokerTec for the same period (2001: :11). Panels C1 and C2 presents that both 1 and 2 tend to decrease in general compared to the corresponding values in Panels B1 and B2. However, the proportion of zero-return intervals at a 5-minute frequency are still substantially high, often hovering above 20%-40%. For comparison purposes, we report zero-return statistics for the U.S. stock market in Table X1 in the Appendix. To compute the time-series average of cross-sectional statistics in the U.S. stock market, we use the Trades and Automated Quotations (TAQ) database for the 30 component stocks included in the of the Dow Jones Industrial Average Index (DJIA30) from October 2001 to November 2007, which is the same sample period (BrokerTec Era) as that in Panels B and C of Table 1. It is undoubtedly the case that the proportion of zero returns in the U.S. equity securities is much smaller than that in the Treasury securities. For example, when sampled at the 5-minute frequency, the U.S. equity securities have zero returns of 9.8% (measured by Zeros1 in Panel A1) and 9.4% (measured by Zeros2 inpanela2)onaveragein Table X1, when transaction prices are used. These compare to the proportions of 26.2%-51.3% (measured by Zeros1 in Panel B1) and 23.4%-42.3% (measured by Zeros2 inpanelb2) intable 1 for the Treasury securities. When calculating the proportions based on quote midpoints in Panel B of Table X1, we find a similar pattern: 8.2% vs. 18.5%-42.2% (measured by Zeros1) and 7.8% vs. 18.2%-42.0% (measured by Zeros2). To get a sense of how the proportion of zero-return intervals change over time in the Treasury 9

12 market, we obtain the monthly average of daily values for 2 sampled at a 5-minute frequency. We then plot the time series of 2 in Figure 1 for each of the four different types of Treasury notes over the sample period (1993: :11). 5 We find several interesting aspects in Figure 1. First, the proportion of zero returns ( 2) substantially decreases around the Asian financial crisis and the Russian default (late August 1998). It also shows a significantly lower levels during the recession in the early 2000s. Inthesamevein, 2 sharply decreases again toward the recent financial crisis period (2007). The above aspects strongly suggest that the common denominator that drives the decreasing (not increasing) trends in the zero-return ratio appears to be greater uncertainty embedded in the adverse economic events, which leads to the flight to quality" in the U.S. Treasury market. Whatever the reason, the above results leave no doubt that the proportion of zero returns in the U.S. Treasury market is generally far higher than one would expect, even after the shift from the voice-assisted system to the electronic trading platform Microstructural Aspects in the U.S. Treasury Market Why are the zero-return ratios so high in the U.S. Treasury market, which is known as one of the largest and most liquid markets in the world? The reason may be that Treasury notes are almost always executed at the prevailing ask or bid quote (rather than inside or outside the quoted spread) with a spread close to the minimum tick, which would generate more zero returns, and thus a higher proportion of zero returns does not necessarily mean the market is illiquid. To examine this possibility, we classify the proportions of trades into five categories based on the execution location: that is, where (around the ask/bid quotes) trades are executed in Treasury securities during the BrokerTec era (2001: :11). Panels A and B in Table 2 show that indeed virtually all (99.9%) trades are executed at the best ask (49.6%) or best bid (50.3%) quote in the Treasury market. Regardless of maturity, it is very rare to find trades executed above the best ask, below the best bid, or inside the bid-ask spread. To facilitate comparison, we also compute in Panel B the proportions and numbers of trades 5 Unless otherwise stated, 1 and 2 used in this paper are calculated at a 5-minute sampling frequency. 10

13 (averaged across all individual stocks or maturities) for the 30 DJIA component stocks using the TAQ database, in addition to the average values for the Treasury notes during the same sample period. Considering that the DJIA component firms represent the broader U.S. economy, Panel B shows that the average number of daily trades (# ) for the stocks exceeds 11,700 per day, whereas that for the Treasury notes is about 1,900. Although U.S. Treasury notes are considered liquid, they are not comparable to the large stocks in terms of the number of trades. We are however more interested in the issue of where equity securities are executed. We find in Panel B that for the 30 DJIA stocks, as large as 24.6% of trades are executed at other levels than the ask/bid quotes: 17.9% of trades are executed inside the bid-ask spread and 6.7% are above the best ask or below the best bid. Given that all trades in the Treasury notes are executed at the best ask or bid quote, our next question is how often the prevailing bid/ask quotes matched with each trade are placed with a spread equivalent to a minimum tick size allowed in the Treasury notes. If quoted spreads are set most of the time at an amount equivalent to a minimum tick, then we would observe zero returns more frequently in the intradaily intervals. Recall that the minimum tick size for the 2-, 3-, and 5-year Treasury notes is 2/256ths of 1% (or basis points) and that of the 10-year note is 4/256ths (or basis points). As Panels A and B in Table 3 show, 86.2% of the prevailing quoted spreads are set at a one-tick equivalent, with the remaining 13.8% being at two ticks or larger for the Treasury notes during the BrokerTec era. During the same period in the U.S. stock market, the minimum tick size was decimalized (i.e., $0.01 for stocks trading above $1.00). Thus, using $0.01 as a minimum tick, we compute similar proportions for the 30 DJIA component stocks. We find in Panel B that the fraction of quoted spreads set at a minimum tick is much smaller (63.6%) on average in the stock market. About 36% of the quoted spreads are set at two ticks or larger. Especially, the fraction of spreads equal to or larger than five ticks is 7.2% in the stock market (vs. 1.9% in the Treasury market). In sum, zero returns are prevalent in the U.S. Treasury market, and this is mainly due to the facts that virtually all trades are executed at the best ask or bid quote, and most of quoted spreads are set at an amount close to a minimum tick. These aspects all point to the notion that the zero-return ratios do not necessarily capture illiquidity in the Treasury market. 11

14 3. Illiquidity and the Proportion of Zero Returns in the U.S. Treasury Market 3.1. Other Illiquidity Measures To investigate how the two zero-return measures, 1 and 2, are related to illiquidity (or liquidity) in the Treasury market, we construct with the two Treasury databases the following daily measures of illiquidity commonly used in the literature. First, we compute the proportional effective spread ( )eachdayforeachofthe four different Treasury notes (with 2- to 10-year maturities). is the daily average of intradaily proportional effective spreads ( ), which are calculated as =[2 {ln ( ) ln ( )}] 10 4 (3) where is a buy/sell indicator (+1 if buyer-initiated and -1 if seller-initiated), is the price of a Treasury note at trade, and is the quote midpoint prior to trade. Note that in the Treasury databases, prices ( ) and quotes are recorded differently from stock trading databases. 6 Since the value of the spread is too small, we multiply the raw number by 10 4 as in Eq. (3), in which case the unit of the spread can be interpreted as a basis point (bps). Similarly, the proportional quoted spread,, is the daily average of intradaily proportional quoted spreads ( ), which are given by = {ln ( ) ln ( )} 10 4 (4) where and are the prevailing best ask and bid quotes at the time of trade. In addition to the above two measures ( and )thataremostcommonly used as proxies for illiquidity in the literature, we also decompose trading costs via the Glosten and Harris (1988) and Foster and Viswanathan (1993) models. Glosten and Harris (1988) 6 For a U.S. Treasury note, GovPX and BrokerTec record its price and quote as a multiple of 1/256. Therefore, if the price of a Treasury note is recorded as, for example, 26,444, then in Eq. (3) we use =( )( 1 )= , which is now the percentage of its face value. 12

15 decompose the effective spread into two components: the adverse-selection and noninformation components. For each of the four different Treasury notes (with 2- to 10-year maturities), we estimate and each day using intraday order flows (or signed volume, )by running the following regression: = + ( 1 )+ (5) where is a price change at trade, is volume (face value) in $millions, is the adverse-selection component of trading costs, and is the noninformation component. Foster and Viswanathan (1993) use unexpected order flows to estimate the price impact of trades. Their model considers that, if order flows are serially correlated, only the unpredictable portion of order flows should affect quotes and prices. This approach is compelling, because the order-splitting practice in recent years may cause order flows to be autocorrelated. Therefore, we filter intraday order flowsbyanar(5)-process, = + P 5 =1 +,where is the residual from the time-series regression. We measure the unexpected order flows by andthenreplace in Eq. (5) with for estimating the two components each day by running the following regression: = + ( 1 )+ 0 (6) where is the adverse-selection component of trading costs, and is the noninformation component. We multiply the four daily estimated components (,,,and )by 10 4 for interpretational convenience in our analyses, since their estimates are small numbers Descriptive Statistics All of the daily measures are winsorized at the 1 st and 99 th percentiles to eliminate the effects of extreme outliers. To consider the relations between the commonly used illiquidity measures and the two zero-return measures ( 1 and 2) in terms of an innovation (or a shock), we also compute metrics for the eight daily measures obtained above (two zero- 13

16 return measures, two spread measures, and four components of trading costs) by standardizing the measures. That is, each day the difference between a daily measure and its mean over the previous five trading days (from 5 to 1) is scaled by the standard deviation (STD) as in = Mean( 5 1 ) STD( 5 1 ),where is the value in one of the eight (raw) measures on day. 7 Table 4 reports descriptive statistics for the eight variables (both raw and standardized): e.g., number of observations (days) ( ), mean ( ), standard deviation ( ), first quartile ( 1), median ( ), and third quartile ( 3). In general, the number of observations ( ) is more than 3,700 (days) for each of the different type of Treasury notes. But Panel B shows that for the 3-year note is much smaller (fewer than 2,600 days over the sample period), reflecting the fact that the U.S. Treasury Department suspended auctions for the 3-year note in 1998 and resumed in 2003 (hence no trade/quote data from September 9, 1998 to May 6, 2003). The table shows that the proportions of zero returns measured by 1 and 2 (computed by sampling at a 5-minute frequency) are generally higher for the notes with a shorter (2- or 3-year) maturity than for those with a longer (5- or 10-year) maturity, regardless of using the mean ( )ormedian( ) values. The mean value of 1 is 55.1% for the 2-year note in Panel A, while it is 33.4% for the 10-year note in Panel D. By construction, 2 is on average smaller than 1 by 23%-39%, depending on the maturity. patterns are virtually the same for the two unexpected (or standardized) measures ( 1 and 2 ). 8 We find in each panel that the effective spread ( ) and the quoted spread ( ) are very close in their magnitude, reflecting the fact that, unlike stocks, Treasury notes are traded most of the time at the best bid and ask quotes. 9 More noteworthy is that the two traditional measures of illiquidity ( and ) monotonically increases as the maturity of the notes gets longer: for instance, the mean value of PESPR for the 2-, 3-, 5-, and 10-year 7 Another advantage of using the unexpected metrics is that the variables will not have any unit root under various specifications, which precludes spurious regressions. 8 The values of zero-return based measures in Table 4 are slightly different from the ones reported in Table 1 because of winsorization. 9 Chordia, Roll, and Subrahmanyam (2008) document that the effective spread is on average 8.96 cents, which is much smaller than the quoted spread of 13.5 cents for NYSE stocks over the period, suggesting that trades in the U.S. stock market are often executed within the bid-ask spread, consistent with our analysis in Table 2. The 14

17 note is 0.88, 1.00, 1.28, and 2.22 bps, respectively. Thus, the proportional effective spread for the 10-year note is wider than that for the 2-year note by a factor of 2.5. This in turn suggests that the longer the maturity is, the more illiquid the Treasury notes are. We observe a similar pattern in the decomposed elements of trading costs. The mean value of the adverse-selection component based on Glosten and Harris (1988),, again increases monotonically from for the 2-year note to for the 10-year note, with the degree of changes being much more dramatic (more than 10 times!). So does the mean of the noninformation component,, which becomes larger from to across the notes with different maturities. We see a similar regularity for the two components estimated via the Foster-Viswanathan (1993) model, although their magnitude is relatively smaller, since the Foster-Viswanathan model uses unexpected order flows. To sum up, the two traditional measures of illiquidity ( and )increaseas the maturity of the notes becomes longer. When trading costs are decomposed into two components, the adverse-selection and noninformation components both exhibit similar patterns. This implies that Treasury notes with longer maturities are more illiquid than those with shorter maturities. By contrast, what we see in Table 1 is that the frequency of zero returns in intraday intervals measured by 1 and 2 in the Treasury market is much lower for the notes with longer maturities than for those with shorter maturities. This aspect is quite intriguing, considering the fact that the ratios have been used as measures of illiquidity in stock markets. Above findings all together suggest that the proportions of zero-return intervals computed on a daily basis (e.g., 1 and 2) may not capture illiquidity in the Treasury market Correlation Analyses To formally look into the issue of whether the proportion of zero returns behaves in a manner consistent with other illiquidity measures commonly used in the literature, we conduct correlation analyses. In doing so, we will use the more conservative (hence more relevant) measure, 2, and its standardized version, 2. To ensure the reliability of our analyses, we compute correlations in four different ways. Table 5 reports the average correlations between 15

18 the zero-return ratio ( 2) and one of the other 6 illiquidity measures for Treasury notes over the sample period. The raw and standardized measures are winsorized at the 1 st and 99 th percentiles before calculating correlation coefficients. unexpected (or standardized) measures in Panel B. We use raw measures in Panel A and First, we calculate the cross-sectional average of time-series correlations. That is, in Panels A1 and B1 we compute a time-series correlation coefficient for each of the four different types of Treasury notes over the whole sample period (about 15 years), and then obtain the crosssectional average across the four maturity types. The total number of observations (days) in Panel A1 for the 2-, 3-, 5-, and 10-year Treasury notes is 3,708, 2,558, 3,682, and 3,624, respectively, while that in Panel B1 is 3,703, 2,552, 3,677, and 3,611, respectively. A striking feature in Panel A1 is that the daily zero-return ratio ( 2) ishighlynegativelycorrelatedwith the six typical illiquidity measures. In particular, the correlation between 2 and either of the two traditional measures, and, is -56.0% and -62.5%, respectively! The correlation between 2 and the adverse-selection components ( and )orthe noninformation components ( and ) ranges from -23.6% to -49.4%. When we use the standardized measure in Panel B1, 2 is again negatively correlated with all the standardized illiquidity measures, albeit to a lesser degree (because the standardized counterparts account for the unexpected changes in the measures). Next, we compute the correlations with a pooled sample. In Panels A2 and B2, therefore, the daily values of the four different types of Treasury notes are all together pooled in the sample over the whole period, and then we compute a correlation (and its -value). In this case, the number of observations used to compute a correlations in each grid in Panels A2 and B2 is 13,572 and 13,543, respectively. We find again in Panel A2 that the daily zeroreturn ratio is highly negatively correlated with the six illiquidity measures, and the coefficients are significant in any case. The correlation between 2 and either of the two traditional measures, and, is close to -60%. The correlation between 2 and the adverse-selection components ( and ) or the noninformation components ( and ) is also above 40% in absolute terms. Panel B2 shows that 2 is negatively and significantly correlated with all the standardized illiquidity measures. Its correlations with the 16

19 two standardized spread measures ( and ) are above 25% in absolute terms. Third, we obtain the correlations by week. Thus, in Panels A3 and B3 the daily values of all the four different types of Treasury notes are pooled in each week, and we calculate a correlation coefficient for that week. Then the (weekly) correlation coefficients are averaged over the 15 years, and a -value for the average correlation is obtained. The average number of observations (maturity-days) used in each week in Panel A3 and B3 is 18 and the total number of weekly correlations used to calculate the time-series average is 751 and 749, respectively. Panel A3 presents that the correlations of 2 with the four decomposed measures tend to be strengthened. The level of correlations in Panel B3 is similar to that in Panel B1. Last, we compute the correlation coefficients by maturity and month. That is, in Panels A4 and B4 a correlation coefficient is computed for each of the four maturity types in each month. Then the correlation coefficients (obtained by maturity and month) are averaged across all maturity-months. The average number of observations used in each maturity-month in Panel A4 and B4 is 21, and the total number of maturity-month correlations used to calculate the average over the 15 years is 652. Panel A4 shows that the coefficients are likely to be lower in general than those in Panels A1 and A2, but all of them are above 13% in absolute terms. The correlations in Panel B4 are little different from the corresponding values in Panels B1 and B Zero-Return Ratios and Uncertainty in the Treasury Market As Figure 1 suggests, zero-return ratios in the Treasury market tend to fall when uncertainty in the economy increases. To further look into the relations between the zero-return ratio in the Treasury market and the U.S. market volatility over time, we plot the monthly averages of 2 and VIX in Figure 2. VIX, often termed as the fear index, is a measure of expected price fluctuations or uncertainty implied by the S&P 500 Index options, calculated by the Chicago Board Options Exchange. VIX is divided by 10 for convenience in the plot. 2 is the average of the zero-return ratios for the four different maturities plotted in Figure 1. AscanbeseeninFigure2, 2 (solid line with squares) and VIX (solid line) move in the opposite (not same) direction most of the time. This is particularly the case in the periods of 17

20 high uncertainty: e.g., the Asian financial crisis (late ), the Russian default (August 1998), the dot-com bubble coupled with the 9/11 event (March ), and the recent financial crisis ( ). These are exactly when the U.S. financial markets suffered most from high volatility and illiquidity, but 2 falls sharply in these periods (the correlation between 2 and VIX is -37.7%). To gain more insights, we add the proportional effective spread, PESPR (averaged across the notes with the four maturities), in Figure 2. We find that PESPR (dashed line with circles) moves together with VIX (with a correlation of 31.7%), indicating that liquidity generally decreases or dries up when volatility is high during the crisis periods (Brunnermeier and Pedersen, 2009; Nagel, 2012). By contrast, however, PESPR and 2 move in the opposite direction, as already shown in Table 5. Overall, the conservative measure of zero returns ( 2) is strongly negatively (not positively) correlated with commonly used illiquidity measures (as well as VIX). The above analyses confirm our conjecture that the proportions of zero returns do not properly capture illiquidity in the Treasury market. Lesmond, Ogden, and Trzcinka (1999) provide evidence that the frequency of zero returns is positively correlated with conventional measures of illiquidity such as the quoted spread and the Roll (1984) measure in the U.S. equity market, and thus the proportion has often been used as a measure of illiquidity in equity securities (e.g., Bekaert, Harvey, and Lundblad, 2007; Goyenko, Holden, and Trzcinka, 2009; Goyenko and Sarkissian, 2014; Hou and Loh, 2016). Our results suggest that we should exercise caution in choosing illiquidity measures for analyses of U.S. Treasury securities, but our focus is more on the consequences that the prevalence of zero returns might have for studies on the Treasury market. 4. Jump-Identification Methods and Model Setup We have shown that zero returns are very common in intraday intervals when Treasury trading data are sampled at 3- to 15-minute frequencies. We also find that the high proportions of zero returns in Treasury notes do not necessarily mean that the Treasury market is illiquid. One issue that arises when zero returns occur frequently is its effect on jump-identification studies. We believe that the prevalence of zero returns can induce serious biases in existing jump-detection 18

21 methods.toinvestigatethisissue,weevaluateusingmontecarlosimulationshowcommonly used jump-identification methods perform under such environments. We then propose a way to alleviate the biases and improve the performance of jump-identification methods Commonly Used Jump-Identification Methods In this subsection we consider two categories of jump-identification methods commonly used in the literature: the first based on Barndorff-Nielsen and Shephard (2004, BNS) and Andersen, Dobrev, and Schaumburg (2012, ADS); and the second based on Jiang and Oomen (2008, JO). We briefly describe these procedures, which will be evaluated later with Monte Carlo simulations. A logarithmic asset price is =ln 0, on a probability space (Ω ), fromwhich we derive two kinds of returns. The first is the simple return defined as: =( ) (exp 1) (7) where is the drift term, is the continuous variance without jumps, and is a random jump term. The second type is the log return, which follows a stochastic process: = (8) The first category of tests based on Barndorff-Nielsen and Shephard (2004, BNS) detects jumps by comparing the realized variance (RV) with the estimator of latent instantaneous variance as shown in Eq. (8), or the integrated variance (IV), where IV is defined as R 1 0 One of general IV estimators is the bipower variation (BV) proposed by BNS. In the equally-spaced discretization of time, BV with N intraday returns over the interval [0, 1] is given by = P 1 =1 +1, where log return of =ln( 1 ) for =1. The bipower 2 1 variation (BV) is one class of multipower variations (MPV), defined as: ( ; ) = +1 ( ) X =1 + 1 Z (9) 19

22 where = =[2 Γ( ( +1)) Γ( 1 2 ) ], is the number of adjacent absolute returns, and is the power of the adjacent products. For instance, the second-order power variation, ( ;2), is related to the integrated second power of volatility, or IV. We can express the realized variance as = (1; 2), and the bipower variation as = (2; 2). Similarly, the fourth-order power variation, ( ;4), is an estimator of the integrated fourth power of the volatility, or integrated quarticity. Tripower quarticity (TQ) is (3; 4), and quadpower quarticity (QQ) is denoted as (4; 4). Although the bipower variation (BV) may be an efficient and consistent IV estimator in a frictionless setting, Andersen, Dobrev, and Schaumburg (2012, ADS) propose two simple alternatives to BV by using neighbor truncation: MinRV and MedRV. In particular, their two new IV estimators have better finite-sample robustness in the presence of frequent zero returns in the stock market. These two are given by, = = X =1 min( +1 ) 2 1 X med( 1 +1 ) 2 2 =2 (10a) (10b) Andersen, Dobrev, and Schaumburg (2012, ADS) provide the joint asymptotic stable mixed normal distribution between RV, BV, MinRV, and MedRV as: (11) where MN denotes a mixed Gaussian limiting distribution conditional on the realization of the integrated quarticity = R ; and the realized variance (RV) with N intraday returns over the interval [0, 1] is given by = P = Similarly to jump-robust IV estimators, 20

23 four different IQ estimators are used: TQ, QQ, MinRQ, and MedRQ, where = = X = min( +1 ) 4 1 X med( 1 +1 ) 4 2 =2 (12a) (12b) By using Eq. (11), we can construct several jump-identification methods based on IVs and IQs. For example, the ratio test based on and with the additional maximum adjustment is given by 1 05 max(1 2 (1 ) ) (0 1) Intuitively, the prevalence of zero returns in intraday intervals can cause downward biases in the higherorder multipower variation [MPV in Eq. (9)] and hence in our three estimators (BV, TQ, and QQ) which are obtained from MPV. For assets with frequent zero returns, the product of adjacent absolute returns is more likely to be zero as the number of adjacent returns increases. In other words, the extent of having a zero value in the product is proportional to the number of neighboring returns ( ) in Eq. (9). However, the MedRV, MedRQ, MinRV, and MinRQ estimators are more robust to the existence of jumps and market microstructure noise. particular, for the MedRQ estimator, if there is a zero return in the three neighboring intervals, the median operator simply ignores the zero return. The second category of jump-identification methods is proposed by Jiang and Oomen (2008, JO). From Eqs. (7) and (8), the difference between the simple return and the log return is: 2 Z 1 0 ( )= +2 = { + Z 1 0 Z 1 0 (exp 1) 2 } + {2 Z 1 0 (exp 1) Z 1 0 In 2 } (13) Eq. (13) can be expressed in the discretization over the interval [0, 1] with N intraday returns as ( )=2 R 1 0 (exp 1) R where =2 P =1 ( ) for the simple return = 1 1 and the log return of and = + R Furthermore, the swap variance jump tests under the null hypothesis of no jumps are given by 21