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Scaling of inefficiencies in the U.S. equity markets: Evidence from three market indices and more than 2900 securities

Using the most comprehensive, commercially-available dataset of trading activity in U.S. equity markets, we catalog and analyze latency arbitrage opportunities and realized opportunity costs (ROC) incurred by market participants. We find that latency arbitrage opportunities are common, observing a total of over 3.1 billion latency arbitrage opportunities in the Russell 3000 during trading in 2016, or roughly 525 per second of trading. Up to 23% of observed trades may have contributed the the measured inefficiencies, leading to a ROC greater than $2 billion USD. A subset of the constituents of the S&P 500 index experience the greatest amount of ROC and may drive inefficiencies in other stocks. In addition, we identify fine structure and self-similarity in the intra-day distribution of latency arbitrage opportunity start times. These results point to universal underlying market mechanisms arising from the physical structure of the U.S. National Market System.


INTRODUCTION
Securities markets utilize auction mechanisms to facilitate the valuation and trade of assets [1][2][3][4]. The operational structure of these markets including the auction mechanism, physical infrastructure through which the market is implemented, and endogenous information asymmetries are thus intrinsic factors in their efficiency [5,6]. While some authors have largely neglected or minimized these factors in analyses of the efficiency of financial markets [7,8], others have recognized that these so-called microstructure variables are central to the performance of modern-day markets [9][10][11].
A. Modern U.S. market We focus our investigation on equities traded in the U.S. National Market System (NMS), a network of privately-owned and operated stock exchanges locate in the U.S., since it is the proverbial center of the world equity markets. In particular, we turn our attention to the roughly 3000 largest equities traded on the NMS, compiled by FTSE International Ltd. as the Russell 3000 index. These securities represent the vast majority of the securities traded in the U.S. and can serve as a nearly comprehensive cross-section of publicly-traded securities from which the observation and assessment of microstructure quantities can be made.
Using the most comprehensive commercially-available dataset of NMS messages available, we enumerate and describe latency arbitrage opportunities and realized opportunity costs, defined in Section III, in Russell 3000 securities and a selected subset of exchange-traded funds (ETFs) during calendar year 2016. Extending earlier work by the present authors and others [12], we find that latency arbitrage opportunities and differing trades resulting in realized opportunity cost occur frequently in the National Market System, with over 3 × 10 9 latency arbitrage opportunities and a total realized opportunity cost of $2,051,916,739.66 in 2016 among securities studied. As shown in Table I, a significant portion of all trades studied here (23.71%) were trades for which there existed a different price elsewhere in the NMS at which the trade did not execute. These findings correspond with, and extend, the findings presented in [12].

B. Scaling in finance
There is widespread agreement that [13,14] was one of the first to characterize the scaling properties of price returns in modern markets. The scaling of returns was later revisited [15] and formalized and extended [16,17]. Beyond just price fluctuations (i.e., returns in price time series), additional financial variables have been found to display scaling properties. For some representative examples, market indices and foreign exchange rates [18] as well as share volume and number of trades [19] adhere to scaling properties.
With the dramatic increase in the number of securities under study and concomitant increase in the range of market capitalization, we examine scaling relationships between microstructure variables, such as latency arbitrage opportunities and realized opportunity cost, and market capitalization and its derivative statistics. La-TABLE I. Summary statistics of the realized opportunity cost for all studied securities that traded in 2016. Realized opportunity cost across all equities in the year is over $2B USD. We discuss statistical characteristics of realized opportunities extensively in Section IV. We note that ratio of the percent of differing trades to the percent of differing traded values is discussed as a statistic of the relative value "on the table" per differing trade-a proxy for the potential profitability of latency arbitrage strategies.  tency arbitrage opportunities occur in equities of all sizes. While they are more frequent in equities with large market capitalizations, the distributions of their qualities, such as their size (magnitude) and duration (how long they lasted), are more extreme among equities with smaller market capitalizations. The majority of realized opportunity cost is generated by equities in the S&P 500 that are not also in the Dow (termed the SPexDow). The SPexDow also Granger-causes realized opportunity costs in other mutually-exclusive market categories (Dow 30 and Russell 3000 less the S&P 500, or RexSP), pointing to its centrality in the U.S. equities market. When realized opportunity cost is analyzed at marketplace (all exchanges and the aggregation of dark pools or alternative trading facilities-ATFs-which is collectively known as FINRA) granularity, the marketplaces with highest volume also have the highest realized opportunity cost.
In the following sections, we first provide a brief overview of the U.S. National Market System for the unfamiliar reader. We then detail our data, the available and used fields, and summarize the equities studied. After describing statistics of latency arbitrage opportunities, including distributions of start times and durations, we move to analysis of realized opportunity cost, providing summary statistics, comparisons across exchanges, and correlation and Granger-causality analyses. We close with a brief exploration of exchange-traded funds (ETFs), a discussion of results, and possibilities for future work.

II. MARKET OVERVIEW
We provide a brief overview of the U.S. equities market for the unfamiliar reader; a more comprehensive summary is given in [12], along with formal set of definitions that is used in this work. The U.S. equities market, known as the National Market System (NMS), is composed of 13 unique, privately-owned exchanges, platforms on which price of equity securities are discovered by market participants using a continuous or discrete auction mechanism. Another core component of the market infrastructure making up the NMS are on the order of 40 privately-owned alternative trading systems (ATSs) [20], also known as dark pools, on which market participants can trade but usually cannot participate in price discovery [21]. Each exchange and ATS keeps its own book of orders submitted by agents; trading facilities (exchanges and ATSs) attempt to execute orders at the "top" of the book (the highest-priced bid orders, or orders to buy securities, and the lowest-priced ask orders, or orders to sell securities) along with "market" buy and sell ordersorders that should be executed immediately, without regard for execution price. The top of the book at each exchange is termed the best bid and offer (BBO). All BBOs are aggregated over the entire NMS and the best bid and offer among all BBOs is calculated and termed the national best bid and offer (NBBO). By law, trades must execute at the NBBO, albeit with certain exceptions. [22] FIG. 1. Relationships between market capitalization and different dependent variables: ROC, differing trades, and total trades. It is immediately apparent that there is a strong positive relationship between market cap and realized opportunity cost. The data exhibits interesting nonlinearity and heteroskedasticity. Equities with smaller market capitalization have higher variability in each dependent variable, while equities with larger Market Capitalization (MC) have generally lower variability in each dependent variable. Also note the equities in the financial sector have a consistantly lower ROC relative to MC while equities in the energy sector have a consistantly higher ROC relative to MC. The data are well-fit by linear and quadratic functions in doubly-logarithmic space. Error ellipses are 95% confidence intervals for the true regression curve calculated using bootstrapping techniques.
There are different methods by which these trading facilities communicate with one another. The Securities Information Processor (SIP) provides a regulationrequired communication system by which trades are reported and price information is disseminated. There also exist privately-owned direct information feeds that provide faster information updates than the SIP data feeds. This differential information quality can contribute to price discrepancies and latency arbitrage opportunities that sometimes exist between trading facilities. The NMS is regulated by the U.S. Securities and Exchange Commission (SEC), a federal government agency, and self-regulated by the Financial Industry Regulatory Authority (FINRA), a professional organization. FINRA self-polices its members and attempts to ensure they adhere to SEC rules and other professional guidelines, while SEC designs, implements, and enforces rules that are intended to promote market stability and economic efficiency.
The physical structure of the NMS, in conjunction with the existence and usage of information feeds of differing speeds, leads to market inefficiencies [23] in the form of latency arbitrage opportunities and realized opportunity costs; on Dow 30 equities, over 120 million latency arbitrage opportunities and over $160M USD in realized opportunity cost were cataloged during calendar year 2016 [12]. These are not the first violations of the so-called "law of one price" that have been detected in financial ©2018 The MITRE Corporation. All Rights Reserved. Approved for Public Release; Distribution Unlimited Case 18-3296 markets; multiple violations in international and domestic financial markets have been identified for more than a decade [24,25]. However, our results here present one of the more egregious violations of this "law".

A. Data
Our data is the most comprehensive set of market data commercially available and is effectively identical to the SEC's MIDAS dataset [26]. It is available for purchase from Thesys Technologies [27] and is comprised of every message, quote, and trade that occurred on the SIP and of direct feeds in the NMS during the time period of study. Information on individual companies, such as the indexes into which companies fall, their Global Industry Classification Standard (GICS) sector, and market capitalization were gathered using the standard commercial Bloomberg Terminal.
The indices we consider are subject to change daily. In order to have consistent sets of companies to study we consider the Dow 30 and S&P 500 as they stood on Jan 1 2016. Additionally we consider the Russell 3000 2016 (i.e., the Russell 3000 constructed in June 2016) excluding components that were not publicly traded on a major exchange on Jan 1 2016.
Each index under study was curated to only include companies that survived as an individual, publicly traded entity on a major exchange for the entire calendar year of 2016. In other words, companies that were delisted from NYSE or NASDAQ for any reason (e.g. Chapter 11 bankruptcy or buyout) were removed. Companies that merged with another member of our dataset remained under study while those who merged with a firm outside of our dataset were removed from consideration. This curation process allows us to avoid several edge cases of market behavior including IPOs and de listings.
Many companies in our dataset changed their ticker symbol over the course of the calendar year and thus appear as a different entity in the data. To study a company over a long time period it is necessary to know all tickers it traded under and when the ticker changes occurred. As there is an absence of a consolidated public record of these symbol changes, we tracked them through an extensive manual review of press releases. These symbol changes were then validated by comparison with the Thesys data feed. Specifically, we confirmed that on the date of the change, trading activity ceased on the old ticker and began on the new symbol.
This curation reduced the Russell 3000 from 3005 companies to 2903 and the S&P 500 from 500 companies to 472. No companies were removed from the Dow 30 set. Note that the mutually exclusive indices are constructed using the curated sets and the curated sets are designated by appending a prime to the respective base index (e.g. Dow 30 → Dow 30 ). Finally all companies in our dataset were classified by their market capitalization as it stood in the beginning of Q4 2016. There is no industry standard on either the range of market caps that define a class or the number of classes. Furthermore these definitions change over time as companies reach larger market caps with the overall growth of the economy. Our definitions for these classes are displayed in Table IV [28].
Our work here focuses on dislocations (and their constituent latency arbitrage opportunities) and the related realized opportunity costs arising from price discrepancies between the SIP and direct feeds. Both concepts have been discussed throughout the empirical market microstructure literature [29][30][31][32], though definitions of these quantities are not entirely settled, we will abide by those set in [12]. We briefly review these definitions and detail the calculation of dislocations, latency arbitrage opportunities, and realized opportunity cost from our data.
Suppose that there exist two sources of information I 1 and I 2 on market data displaying respective prices for an asset p 1 (t) and p 2 (t) observed from a single, welldefined location. We will term a dislocation between these sources of data as the amount of time during which the price information given by the information feeds differs. A related but distinct concept is that of a latency arbitrage opportunity, defined as an amount of time during which the price information given by the information feeds differs in a single direction (i.e., we must have ∆p > 0 or ∆p < 0 during the entire time of the latency arbitrage opportunity). The realized opportunity cost of using I 1 instead of I 2 over an amount of time [0, T ] is calculated as follows: for every trade that occurs during [0, T ], the difference in execution price |p 1 − p 2 | is calculated and multiplied by the number of shares in the trade n. The total realized opportunity cost over the interval [0, T ] is then given by the sum over the realized opportunity cost associated with each trade in [0, T ].
Equities contained in our dataset represent approximately 98% of all publicly-traded equities in the U.S. by market capitalization [33]. Tables III -V provide summary statistics and distribution of these equities across business sector, market cap, and market category, for several indices.

B. Methods
We first compute basic summary statistics and qualitative descriptions of the distributions of latency arbitrage opportunities and realized opportunity costs. In addition to computing summary statistics, the larger sample of equities (all equities traded on the National Market System during 2016) allows us to conduct a cross-sectional study of the effect of company "size" on these microstructure quantities. We quantify the notion of size of a company by both its market capitalization and its rank in relation to other companies. In addition, we create disjoint sets of equities and compute aggregate statistics across these sets. Since the S&P 500 is a strict superset of the Dow 30 and the Russell 3000 is a strict superset of the S&P 500, the natural division of the superset of all equities under study is into three distinct classes: the Dow 30, the S&P 500 excluding the Dow 30 (SPexDOW), and the Russell 3000 excluding the S&P 500 (RexSP). We compute measures of correlation between these disjoint subsets, and characterize the statistical properties of the time series of latency arbitrage opportunities and realized opportunity cost across these disjoint categories. We further explore the relationship between these categories by conducting a Granger causality analysis of aggregated realized opportunity cost time series [34].
We also analyze the distribution of microstructure quantities across exchanges. Since different exchanges may have substantially different hardware (such as fiber-optic cables and computers) and software (such as matching engines), it is possible that latency arbitrage opportunity and realized opportunity cost distributions vary significantly across exchanges; examples of the hardware and software used in exchanges vary in type and complex-ity [35][36][37][38]. We compute summary statistics and provide qualitative comparisons of the results.
The next section gives results on latency arbitrage opportunities, including summary statistics overall and across mutually-exclusive market category, and regressions of latency arbitrage opportunities against market cap. We then discuss structure in the intra-day distribution of latency arbitrage opportunity start times and latency arbitrage opportunity duration. Following this, we provide statistics of the realized opportunity cost across the market as a whole and again within mutually-exclusive market categories. We then explore statistical properties of the ROC time series and conduct a Granger-causality analysis of ROC by mutually-exclusive market category. We close with an overview of the statistics of ETF latency arbitrage opportunities and realized opportunity costs, contrasting these with those of the market as a whole.
©2018 The MITRE Corporation. All Rights Reserved. Approved for Public Release; Distribution Unlimited Case 18-3296 IV.

A. Latency arbitrage opportunities
Latency arbitrage opportunities can occur when prices propagated by distinct information feeds differ. We cataloged all latency arbitrage opportunities occurring in the equities studied here and present summary statistics and qualitative comparisons of their distributions and higherorder moment statistics. Tables VIII -XVI display means of summary statistics of latency arbitrage opportunities for each mutually-exclusive subset of the equities under study. We will use the notation f A to denote an average of the quantity f conditioned on the condition A. These averages are interpreted as the quantity f conditioned on condition A averaged over all securities and all times of observation; defining the number of instances of the quantity f having condition A as N A , we have On average, there were more latency arbitrage opportunities in Dow 30 securities ( N uncond However, the average maximum magnitude of latency arbitrage opportunities in the Dow30 in each conditioning class (unconditioned, conditioned only on duration, and conditioned on both duration and magnitude) is lower than those of the SPexDow, which in turn are lower than those of the RexSP. In particular, actionable latency arbitrage opportunities (those with duration > 545µs) with magnitude > $0.01 exhibit more extreme behavior in the SPexDow and RexSP than in the Dow. On average, the median maximum magnitude in the Dow 30 among actionable latency arbitrage opportunities was median max mag duration,magnitude $0.023, while in the SPexDow we observed median max mag duration,magnitude $0.034 and in the RexSP median max mag duration,magnitude $0.045, a roughly one-cent increase in the median maximum magnitude of a latency arbitrage opportunity in each mutually-exclusive category. Examples of distributions of these quantities are given in Figure 2, where the distributions of the means of minimum magnitude, maximum magnitude, and duration are plotted for the RexSP.
Taken as a whole, these results provide evidence for the existence of a market capitalization scaling effect in latency arbitrage opportunities: larger securities are on average traded more and hence have a higher average number of latency arbitrage opportunities, but exhibit on average less extreme behavior in moment and quan-tile statistics of these opportunities; more frequent trading implies a lower probability that prices across differing information feeds will diverge by large magnitudes.
Since latency arbitrage opportunities are not distributed evenly throughout the day in the Dow 30 [12], we examine their distribution in the SPexDow and the RexSP as well. Appendix B contains figures displaying the distribution of latency arbitrage opportunity start times plotted modulo day and aggregated over the year and figures displaying the distribution of latency arbitrage opportunity durations for each mutually exclusive market category. Distributions are plotted both without conditioning and when conditioned on duration or magnitude (or both).
Distributions of start times display predictable structure. In all market categories, there are large peaks at the very beginning and end of the trading day (circa 9:30 AM and 4:00 PM), along with a noticeable and sudden increase in density around 2:00 PM. The peak in density that occurs at the end of the day is most noticeable when the distribution of start times is not conditioned on latency arbitrage opportunity size; when the distribution is conditioned, this peak is present but the density is much lower. These observations correspond with the results found for the Dow 30 in [12]. However, along with these granular observations, there also exists structure on shorter timescales. The distribution exhibits selfsimilarity on the half-hour timescale, with large peaks every half-hour and decreasing density toward a sudden peak at the next half-hour. There is also structure at the five-minute timescale that is noticeable before the 2:00 PM spike in density but does not appear to be present after the spike. (Future work could statistically test for the presence of this structure and for its persistence across multiple timescales.) The structure on shorter timescales is present in all distributions but, again, is more pronounced in distributions not conditioned on magnitude.
Distributions of latency arbitrage opportunity duration also exhibit definite structure, though it is realized in a different way. These distributions are plotted logtransformed, as they are leptokurtic. All distributions exhibit one or more peaks in the range 10 −4 s ≤ log 10 duration ≤ 10 −3 s, but there is also a distinct and much lower peak in the distribution near approximately one second in length.
As a visual aid to these results, we have included circle plots, as introduced in [12], to demonstrate the nonuniform distribution of latency arbitrage opportunities in both time-and event-space that can occur. We construct these circle graphs for the equities with the smallest market cap in our sample (GALE), highest market cap in our sample (AAPL), and the highest-volume S&P 500 ETF (SPY) for comparison. for the same tickers but aggregated over a year (modulo day). There is substantial variation between the circle graphs of these tickers. AAPL displays intricate structure in event space, while GALE display similar structure but with much lower density, as it has far fewer latency arbitrage opportunities. In time-space, AAPL and GALE display similar nonuniform density of events, with both exhibiting notable peaks near the beginning of the trading day. AAPL also exhibits large spikes in density near 12:30 PM and 2:30 PM. In contrast, SPY displays time-space event density that is far more uniform than either AAPL or GALE.

B. Market capitalization
Further evidence for scaling behavior arises from analysis of market capitalization. Tables III and V display market capitalization statistics broken down by industry sector and categorical size, e.g., micro-cap, mega-cap, etc. Market capitalization is significantly positively correlated with realized opportunity cost. Tables XXXII -XXXV display results from ordinary least squares regressions predicting realized opportunity cost using predictors including market capitalization. A linear fit predicting log 10 ROC from log 10 market capitalization, log 10 total trades, and log 10 differing trades gives R 2 0.908, with a positive coefficient relating log 10 ROC to log 10 market capitalization; higher market capitalization is associated with higher ROC. A similar regression is computed including quadratic terms in log 10 market capitalization, which has a significant, but weak, negative association with ROC. Similar relationships hold for both the linear and quadratic models when the dependent variable is instead chosen to be total or differing trades.
Though behavior of ROC as a function of market capitalization is generally similar when equities are stratified by sector, some sectors display lower average levels of ROC, differing trades, or total trades when market capitalization is held constant. Equities classified as being in the financial sector generally have a smaller amount of ROC, while equities classified as being in the energy sector exhibit a higher amount of ROC on average. However, there is no clear general trend linking sectors to market capitalization or to ROC.

C. Realized opportunity cost
As expected with an increase in the number of analyzed equities from 30 to more than 2900, the amount of realized opportunity cost rose substantially from the quantity reported in an earlier study by the present authors and others, from $160M to $2.05B USD. Realized opportunity cost (ROC) clearly displays sublinear scaling with number of studied equities; we do not observe a thousandfold increase in the amount of ROC with a thousandfold increase in the number of equities. The information advantage afforded traders with access to direct feed information is not uniform; though a vast majority of the ROC ($1. ©2018 The MITRE Corporation. All Rights Reserved. Approved for Public Release; Distribution Unlimited Case 18-3296 fraction of "differing traded value"-the nominal market value of all differing trades-was slightly higher (25.25%) than the fraction of all trades that were differing trades. The fractional part of the ratio of these fractions (1.0651) can be interpreted as a rough estimate of a rate of profitability of latency arbitrage trading strategies.
SPexDow securities account for a plurality of all trades under study that resulted in realized opportunity cost, with the median number of differing trades per day that occurred in the SPexDow calculated to be 2,006,091, in contrast to the 309,158 in the Dow 30 or 1,921,121 in the RexSP. The median differing traded value per day in the SPexDow was also the highest among the three categories, totaling approximately $14.07T versus the ReXSP's total of $6.7T and the Dow's total of $3.27T. Realized opportunity cost per share differed across the three categories, with median ROC per share per day on the Dow calculated to be $0.011, on the SPexDow to be $0.015, and on the RexSP to be $0.021. This result is not surprising, as this particular measure of "market inefficiency" (ROC per share) increases with the decrease in average size of equity, with lowest ROC per share occurring in the Dow and highest ROC per share occurring in the set of equities ranked 501 and above. Median total ROC per day on the Dow amounted to $514.8K, while median total ROC per day on the SPex-Dow totaled $3.384M and on the RexSP amounted to $3.564M. Summary statistics for distributions of ROC for each mutually-exclusive category are given in Tables  XXV, XXIV and XXVI. It is interesting to consider the distribution of both total ROC and ROC per share by both equity and mutuallyexclusive category. Figure 6 displays ROC of the top 30 and bottom 30 of all securities under study when ranked by ROC. Included in this figure for comparison is the exchange-traded fund SPY, an ETF that tracks the S&P 500. (Selected ETFs are also treated separately in Section IV D.) It is notable that the equity with the largest ROC, Bank of America (BAC), has more than twice the ROC of the equity with the second-largest amount of ROC, Verizon (VZ). Though not an equity and not included in the rest of this study, it is also notable that SPY, one of the most heavily traded securities on the NMS along with BAC, is close to BAC in amount of ROC. Of the top 30 securities with most ROC, eight of the 30 are Dow 30 equities; only four out of 30 are RexSP equities, while the other 17 non-ETF securities are SPexDow equities. Since the S&P 500 appears to be the primary driver of ROC across all equities (c.f. below), we find the top 30 and bottom 30 S&P 500 securities ranked by ROC, including Dow 30 securities, and plot their ROC in Figure 7. Even in this subset, only 10 of the top 30 equities are Dow 30 securities. However, when the unit of analysis changes to ROC per share, as in Figure 8, we find that RexSP equities fill 27 out of 30 top ranks, which corresponds with the aggregated statistics reported in Table   XXVI when compared with Tables XXIV and XXV. Since there appear to be differences between the (stationary) summary statistics of the mutually-exclusive market categories, it is reasonable that there may be significant differences between the ROC statistics considered as time-dependent stochastic processes and simply considered as random variables decoupled from time. Within each category, the ROC was computed for all equities in that category for each day. Each ROC series is then normalized as r i → ri− ri √ Var(ri) , which allows direct comparison of the series. Figure 16 displays a quantile-quantile plot of the Dow, SPexDow, and RexSP ROC distributions. The Dow distribution is plotted as linear and the other two distributions are compared with it. It is immediately obvious that the left tails of the SPexDow and RexSP distributions are heavier than that of the Dow; this also appears to be the case for the right tails of the distributions, but there is little sampling in this region and so no conclusion can be drawn. This similarity of the SPexDow and RexSP distributions is also striking; when normalized they appear almost identical. We also note that, in accordance with the QQ plot of the time-decoupled distributions above, the DFA exponents of the SPexDow and RexSP-and thus their corresponding dynamical behavior-are closer than they are to the Dow DFA exponent.
A review of the above results points to the SPexDow as being the "dominant" mutually-exclusive category in some sense: it accounts for a plurality of differing trades, differing traded value, and total ROC, while also having a DFA exponent lower than that of the Dow and close in value to that of the RexSP, meaning that its timeseries of ROC is strongly mean-reverting. The amalgamation of these facts can be interpreted as evidence that the SPexDow ROC time series is possibly least likely to be influenced by the other series of ROC. To test this hypothesis, we conduct a number of Granger causality tests on the time series of ROC. Granger causality is the notion that, beyond mere correlation in time series, past values of one time series may be useful in predicting current and future values of another time series [34]. A maximum lag of 40 days was set and four tests were calculated pairwise between each of the three mutuallyexclusive categories: sum of squared residuals χ 2 -test, a likelihood ratio test, sum of squared residuals F -test, and a Wald test. We consider there to be a significant Granger causality between series when all four tests indi-©2018 The MITRE Corporation. All Rights Reserved. Approved for Public Release; Distribution Unlimited Case 18-3296 cate significant Granger causality at the p = 0.05/N lags confidence level. The correction for multiple comparisons is done using the most conservative estimate, the Bonferroni correction, to minimize the probability of Type I error [40]. Figure 5 displays the results of these tests graphically as a directed network. The direction of edges denotes the direction of the Granger-causal relationship between the categories, while the weights on the edges denote the total number of lags for which the relationship was significant. The SPexDow is shown to significantly influence both the Dow and RexSP while not being significantly influenced by either category; this provides strong evidence to support our above hypothesis. We note that the SPY tracks the S&P 500, is one of the most heavilytraded securities, and has the second-highest amount of ROC of the securities under study here. The SPY's price dynamics and ROC may thus have a material effect on the relationships between the S&P 500's ROC and those of the other market categories, providing a partial confounding effect to the Granger-causal relationship determined here; there may be a mutually-causal relationship between the real S&P 500 and the ETF that tracks it. The RexSP and Dow have a mutually Granger-causal relationship, with the Dow exerting more influence on the RexSP than the other way around. This finding corresponds with the ranking of categories on a total shares traded per number of equities basis; this is not a surprising result. We also find that the SPexDow exerts far less influence on the RexSP than does the Dow (four total lags for the SPexDow versus 23 total lags for the Dow), a fact for which we do not have a ready explanation. Providing further evidence for the above hypothesis, we compute Pearson correlations between pairs of mutually exclusive categories for both ROC and ROC per share; these results are displayed in Table VI. ROC correlations are strongest between SPexDow and RexSP (ρ = 0.72) and SPexDow and Dow (ρ = 0.45), while the correlation between the RexSP and Dow is lower (ρ = 0.31). ROC per share correlations are universally lower than those for ROC, but the correlations between SPexDow and RexSP (ρ = 0.41) and SPexDow and Dow (ρ = 0.10) are still higher than that between RexSP and Dow (ρ = −0.01), which is actually negative.
Finally, we break ROC out by both exchange and (not mutually-exclusive) market category. Figures 9, 10, and 11 display the distributions of total ROC per day in 2016 by exchange; the vertical axis is transformed as x → log 10 (|x| + 1) for ease of viewing. (We choose this transformation for its usefulness in the case where data has both a large range and many values near zero. Since log(|x| + 1) = ∞ n=1 (−1) n+1 |x| n n for |x| < 1, near zero we have log(|x| + 1) |x|, while at large values of |x| it is essentially indistinguishable from log |x|.) In all three categories, the exchanges with the most total ROC are NYSE and NASDAQ, followed by ARCA, BATS, and EDGX. Consistent with all summary statistics reported in previous work, CHX and NSX have by far the least ROC across all three categories [12]. Turning our analysis to ROC per share by exchange, Figures 12, 13, and 14 display daily ROC per share by exchange. The left panel of each figure shows the entire distribution, while the right panel displays the distribution truncated to within the 99th and 1st percentiles. The non-truncated distributions are almost-comically heavy-tailed-the excess kurtosis (kurtosis minus 3) of the EDGX -Russell distribution is κ 55, 652-hence the truncation for the purposes of analysis; it seems likely that the kurtosis of the theoretical distributions do not exist, implying tail exponent γ < 4 in the distribution Pr(X > x) ∼ x −(γ−1) .
(Table XXVII displays the kurtosis and skew for each non-truncated distribution.) The means of the truncated distributions are significantly above one cent per share favoring the direct feeds (i.e., significantly below −$0.01) in all distributions. Among Dow 30 equities, CHX and NYSE have exceptionally long tails favoring the direct feeds. In the case of S&P 500 tickers, FINRA (alternative trading facilities / dark pools) have a distribution significantly more toward the direct feed than any exchange; its mean ROC per share is above $0.20, while when considering the entire superset of all equities in the Russell 3000 NYSE, AMEX and FINRA are all leptokurtic favoring the direct feeds.

D. ETFs
Exchange traded funds (ETFs) are securities that trade on the NMS and are designed to mimic as closely as possible a particular portfolio of other securities. They are thus governed by the same price discovery mechanism as other securities that trade on the NMS, as opposed to the end-of-day price discovery mechanism to which mutual funds are subjected, but also allow investors to own a portion of potentially many underlying assets (or at least a simulacrum of such), similar to a mutual fund. Here, we briefly remark on the similarities and differences between ETFs designed to track subsets of the market and those subsets of the market themselves. We calculated statistics on the latency arbitrage opportunities and realized opportunity cost attributed to ten ETFs, the descriptions of which are given in

V. CONCLUSION
In sum, we have demonstrated that the existence of latency arbitrage opportunities and realized opportunity cost is not restricted to Dow 30 securities. Furthermore, we have established that these microstructure quantities occur with non-negligible frequency and size; we show that total realized opportunity cost in Russell 3000 securities was well in excess of $2 billion USD during 2016. Compounding these results, we provide strong statistical evidence that the S&P 500 excluding Dow 30 securities, to which we refer as the SPexDow, is the primary driver of realized opportunity cost among the three mutually exclusive categories of equities (Dow 30, SPexDow, and Russell 3000 excluding S&P 500 securities, or the RexSP).
Compounding the above results, we find that structure in the distributions of latency arbitrage opportunity start times and duration persist across the entire Russell 3000, indicating some broader microstructure-based proximate cause of this structure. Distributions of latency arbitrage duration exhibit a large peak between 10 −4 and 10 −3 seconds (100 microseconds to one millisecond), but also exhibit a second smaller, yet distinct, peak near one second. This separation of timescales in the distribution provide evidence for the existence of at least two distinct proximate causes of latency arbitrage opportunities. Distributions of latency arbitrage opportunity start times display even more intricate structure, with large peaks at the beginning and end of the trading day, self-similarity on the half-hour and ten-minute timescales, and a large spike at 2:00 PM.
Realized opportunity cost was highest among SPexDow securities, but realized opportunity cost per share was highest among RexSP securities, which were also the most lightly-traded securities. All time series of realized opportunity cost exhibit behavior of anti-autocorrelation, meaning that they are mean-reverting. Realized opportunity costs in the SPexDow Granger-cause realized opportunity costs in the other market categories, but the converse is not true; while the Dow Granger-causes the RexSP, the RexSP only weakly Granger-causes the Dow and does not have any effect on the SPexDow. When considering realized opportunity cost by exchange (including the aggregation of all alternative trading facili-©2018 The MITRE Corporation. All Rights Reserved. Approved for Public Release; Distribution Unlimited Case 18-3296 ties, denoted as FINRA), the largest exchanges by market volume (NYSE, INET, ARCA, BATS, and EDGX) capture a large majority of the realized opportunity cost. However, passing to analysis of realized opportunity cost per share, NYSE, CHX, and FINRA are clear outliers in the Dow, S&P. and Russell; AMEX is a clear outlier when considering the Russell superset.
Taken together, these results paint the picture of a NMS the physical structure of which generates effects that are persistent across size of equity and exchange. Amplifying these persistent effects is the apparent central role of the SPexDow; in number of latency arbitrage opportunities, amount of realized opportunity cost, spectral properties of realized opportunity cost time series, and Grangercausal relationships, the story emerges of the SPexDow's characteristics being generated by largely-endogenous factors and subsequently influencing the characteristics of the Dow and RexSP. Future work could explore in more depth the extent to which microstructure effects arising first in the SPexDow then spread to other mutually exclusive market categories and propagate through time. This work could also explore the evolutionary dynamics of the modern NMS from its birth following the financial crisis of 2007/8 to the present day. The NMS may not have remained static, with a constant number of market centers and a stationary distribution of market agents and trading strategies, but rather may have experienced fluctuations in the number of exchanges, in the regulatory environment, and in strategy profiles of trading agents. Such an analysis could pave the way for more well-informed modelling efforts and development of financial economic theory.