Frequent Batch Auctions Under Liquidity Constraints

We exploit European regulatory interventions to investigate the effects of sub-second periodic auctions on market quality under dark trading restrictions. The restrictions are linked to an observable increase in periodic auctions and an economically meaningful loss of liquidity. While periodic auctions ameliorate illiquidity, their effects are significantly less than those of the restrictions; therefore, the combined effects of periodic auctions’ increases and the restrictions are general declines in liquidity and informational efficiency. However, consistent with theory, periodic auctions are linked to reductions in adverse selection costs, thereby underscoring their potential to address latency arbitrage and the technological arms race.

…stop this nonsense by moving from continuous trading to frequent batch auctions. To human eyes trading will be essentially continuous, but the robots will effectively gather in a room every second (or 100ms, if that seems too glacial for the financial terminators) for a brief blind auction Financial Times, 21 st February, 2014

Introduction
While a large number of market microstructure studies suggest that algorithmic and high frequency trading (AT and HFT) benefit market quality (see as examples, Brogaard et al., 2014;Harris, 2013;Hasbrouck and Saar, 2013;Hendershott et al., 2011;Ibikunle, 2018), several others report their tendency to induce extreme and destabilizing events, such as "flash crashes" (see as examples, Easley et al., 2011;Kirilenko et al., 2017;Ibikunle and Rzayev, 2020). Others note their propensity to induce a greater price impact on large institutional orders (see Putnins and Barbara, 2016). Raman et al. (2014) and Anand and Venkataraman (2016) also find that endogenous HFT liquidity providers destabilise markets during stressful periods. Two additional consequences of trading at high speeds are latency arbitrage, involving the exploitation of a trading time disparity between fast and slow traders (see Rzayev and Ibikunle, 2019), and the technological arms race (see Diaz-Rainey et al., 2015), a negative externalityinducing development (see Menkveld, 2014). Budish et al. (2015) argue that the technological arms race is a symptom of a flawed market design and they propose the frequent batch auctions (FBA)/sub-second periodic auctions mechanism (hereafter referred to as periodic auctions), which divides trading into intervals of very short lengths, for example, every tenth of a second, as an antidote. In effect, this treats time as discrete instead of continuous and orders are processed in a batch auction rather than serially. While frequent batch auctions are not yet widely used globally, according to the UK's Financial Conduct Authority (FCA), periodic auctions have recently experienced two significant spurts of growth due to the implementation of the provisions of the EU's Markets in Financial Instruments Directive (MiFID) II. Although periodic auctions account for less than 5% of the average trading volume in UK markets during the MiFID II era, the mechanism's significance has recently become apparent. This results from the commencement of the double volume cap (DVC) measure, a MiFID II provision designed to restrict dark trading in European markets (see Ibikunle et al., 2019 for a detailed discussion of the DVC), in place since early 2018. Evidence suggests that a non-negligible portion of volumes, otherwise destined for dark pools in stocks under the DVC restriction, are executed through periodic auctions and other non-continuous trading mechanisms, the so-called 'quasi-dark' mechanisms (see Johann et al., 2019).
In recognition of their relevance, the European Securities and Markets Authority (ESMA) called for evidence of the effects of periodic auctions on market quality. 1 However, there have been very limited attempts to investigate the direct market quality effects of periodic auctions as a trading mechanism thus far. In addition, if periodic auctions are to serve as a means of addressing the twin issues of latency arbitrage and the technological arms race, they must be shown to, at a minimum, have a benign or positive effect on market quality. Therefore, in this study, we investigate the effects of periodic auctions on market quality characteristics in Europe's most active equity market, the UK market.
In line with Johann et al. (2019), we find that stocks experiencing DVC-imposed dark trading restrictions experience higher periodic auctions volumes and that the overall market quality effects of periodic auctions on the market are limited and mixed. Specifically, we find that periodic auctions have a generally positive effect on liquidity and a largely benign effect on adverse selection costs, except in the case of the most liquid stocks, where adverse selection declines with the use of periodic auctions. The adverse selection costs effect is explained by the predictions of Budish et al. (2015): periodic auctions offer a safe haven for slower traders who are susceptible to the latency arbitrage strategies deployed by faster traders. Thus, increasing use of periodic auctions lowers the incidence of slower traders being adversely selected, especially through the latency arbitrage strategy typically deployed by HFTs. The mixed nature of the evidence uncovered is underscored by the estimated impact of periodic auctions on informational efficiency. The overall effect of periodic auctions trading, especially during the DVC window, is a deviation from a random walk or reduction in informational efficiency. This finding is consistent with periodic auctions slowing down the price discovery process. While trading in dark pools implies a degree of delay (as in Menkveld et al., 2017;1 The ESMA report calling for evidence can be accessed here: https://www.esma.europa.eu/sites/default/files/library/esma70-156-785_call_for_evidence_periodic_auctions_for_equity_instruments.pdf Zhu, 2014), periodic auctions in an otherwise high frequency trading environment will inevitably slow down trading.
To the best of our knowledge, this is the first study to provide empirical evidence on the market quality implications of periodic auctions in European markets. Although Johann et al. (2019) include periodic auctions in their list of 'quasi-dark' trading mechanisms, the identification of periodic auctions mechanisms as quasi-dark is slightly problematic given that a key feature of auctions is transparency -specifically, the fact that all orders are reflected in the observable indicative auction price and volume as they arrive in the market and prior to uncrossing. European exchanges, such as Cboe, also claim that the orders submitted to their periodic auctions mechanisms are sent to lit order books as they arrive. Our paper also isolates the effects of periodic auctions on market quality characteristics, while controlling for volumes attributed to other trading mechanisms.
In general, deploying latency arbitrage-based trading strategies is more suited to a fragmented market environment given the likelihood of disparities in reaction times among different venues (Wah and Wellman, 2013). However, HFT makes these viable strategies within a single venue because it allows for a significant difference in the trading speeds of slow and fast traders (Menkveld, 2014;Wah and Wellman, 2016). By doing so, according to Aquilina et al. (2017b), HFT imposes adverse selection costs on slower traders (see also Rzayev and Ibikunle, 2019). The quest for faster trading speeds has resulted in the technological arms race, a competition driven by investment in hardware and software (see Biais and Woolley, 2011).
Both are sustained by fast traders' need for the retention of their speed advantage over slow traders and the pursuit of parity or the eclipsing of fast traders by the slow traders. Menkveld (2014) argues that the arms race raises the spectre of negative externality and waste in financial markets. Thus, if the importance of speed is reduced, the activities driving the arms race and latency arbitrage should decline, consequently leading to a reduction in both phenomena. The FBA, as proposed by Budish et al. (2015), could significantly reduce the influence of speed on the price discovery process. This could shift investor focus from the acquisition of speedenabling transactions to obtaining better prices, implying that the introduction of an FBA-type mechanism could offer price efficiency (see Madhavan, 1992).
Cboe's periodic auction mechanism, which was introduced on 19 th October 2015 and currently accounts for about 70% of the periodic auction volume in Europe, is largely consistent with the structure of the FBA proposed by Budish et al. (2015). Its auction book provides both pre-trade and post-trade transparency, thus meeting MiFID II's regulatory technical standards (RTS). Although periodic auctions have been the subject of some academic studies, they have only focused on long interval auction lengths and not the FBA-type periodic auctions we examine in the context of AT/HFT. For example, Madhavan (1992) argues that periodic auctions offer greater price efficiency than the more common continuous order-driven trading mechanism. This is due to the pooling effect of the periodic auctioning system, allowing for simultaneous execution. The pooling of orders for simultaneous execution addresses the problem of information asymmetry that the sequential trading system of the continuous orderdriven trading mechanism induces (see also Barclay et al., 2008). The simultaneous executions in classical auctions could also positively affect the pricing process when they are deployed in conjunction with continuous order-driven trading. Amihud et al. (1997) show that an iterated continuous trading process preceded by a call auction on the Tel Aviv Stock Exchange is linked to improvements in the price discovery process.
Evidence from other studies that broadly examine the implications of call auctions for market quality characteristics is more nuanced. Sarkar (2016) investigates midday auctions at the London Stock Exchange (LSE) and reports that the use of the mechanism is linked to a larger spread and increased price volatility. However, the most common use of call auctions in financial markets is as market opening or closing mechanisms, and this has been the focus of a stream of literature (see as examples, Bellia et al., 2020;Chang et al., 2008;Cordi et al., 2015;Ibikunle, 2015). For example, Barclay et al. (2008) and Chang et al. (2008) report on the positive effects of the use of the opening call auction for market opening. Opening call auctions can help market participants build a consensus on an opening price ahead of the continuous trading phase, and thus offer informational efficiency benefits. The work of Barclay et al. (2008) and Chang et al. (2008) nevertheless contrasts with the findings of Ibikunle (2015), who reports a high rate of failure to open, and low levels of informational efficiency for low volume stocks on the LSE when compared to the levels of informational efficiency recorded for the continuous trading period. The study also finds that although the closing call auction offers higher informational efficiency levels than the opening call auction, it is still lower than the continuous trading phase attains. This finding is linked to the fact that the advantages of transparency and liquidity the call auction offers cannot necessarily be regarded as such in an era where HFT guarantees high levels of trading activity during the continuous trading phase of the market. Cordi et al. (2015) find positive links between market quality characteristics and the use of the closing call auction, while Comerton-Forde et al. (2007) argue that the use of the closing call auction could reduce price manipulation. Chelley-Steeley (2008) and Chelley-Steeley (2009) in turn investigate the market quality impact of the introduction of the closing auction on the LSE. Both studies report market quality improvements. These findings are consistent with those of Pagano and Schwartz (2003) and Comerton-Forde et al. (2007), who examine the introduction of the closing auction on the Paris Bourse and the Singapore Stock Exchange respectively. Thus, with the exception of the evidence from the LSE (for smaller stocks), there appears to be a consensus in the literature on the links between the deployment of the call auctions and market quality characteristics. This may have implications for the use of periodic auctions at high frequency.
Finally, more recently, using data from the Taiwan Stock Exchange (TWSE) Indriawan et al. (2020) investigate a transition from batch auctions to continuous trading and find that the move is linked to an increase in adverse selection and a liquidity decline. Our study differs from theirs in at least two respects. The first relates to significant market structure differences: while they focus on a transition from batch auctions to continuous trading, we investigate an event that should lead to an increased use of periodic auctions within a hybrid trading system. Second, TWSE's batch auctions are distinct from the periodic auctions we examine in that the former operates five-second interval auctions, while the periodic auctions systems we examine operate maximum intervals of 100 milliseconds.

Periodic auctions and market quality: the literature and hypotheses
Although periodic auctions are mainly discussed in the context of addressing the technological arms race and its potential welfare externalities (see Menkveld, 2014), its deployment should first be viewed in less ambitious terms. This is because the overriding question when designing markets is that of the system of exchange, specifically, how the decision could either enhance or hinder the evolution of market quality characteristics, such as liquidity and informational efficiency. Periodic auction trading systems are structurally distinct from the other auction types that have been studied extensively in the literature. Firstly, periodic auctions have smaller intervals; Budish et al. (2015) suggest that the interval should be smaller than one second and, in line with this, the leading global system operated by Cboe provides 100ms-level auction intervals. Secondly, periodic auctions are typically conducted alongside continuous trading, and currently the volume of periodic auctions only captures a small amount of the total volume in the market. Therefore, trading in periodic auctions might be more influenced by the market's main trading system than vice versa. Thirdly, the main aim of periodic auctions has thus far been to de-emphasise the influence of speed in trading, i.e. the activities of HFT, while opening and closing call auctions aim to provide more efficient prices.
There are limited existing studies on periodic auctions. An FCA (2018) investigation of the growth of periodic auctions in UK stocks finds little difference in growth between stocks experiencing dark trading caps and those that are not. Johann et al. (2019) investigate the shift of dark pool volume to other non-continuous trading mechanisms following the imposition of dark trading restrictions on some stocks. They find that only a small proportion of the hitherto dark volume shift into such markets, including periodic auctions. They also find limited changes in overall market quality. Aquilina et al. (2017b) investigation of the impact of periodic auctions on adverse selection costs is limited by the fact that it is based on a small pre-MiFID II sample and volume. Thus, the effects of periodic auctions remain largely unexplored and unclear in the empirical literature, which underscores the ESMA call for more evidence given the concerns of various stakeholders (see McDowell, 2019). The growth also appears to be due to the migration of trading from other trading mechanisms (see Figure 2). However, the overall picture presented in Figure 1 and by the FCA (2018) fails to account for the differences in the growth of periodic auctions in stocks experiencing dark trading restrictions and those that are not. Nevertheless, it is logical to expect that there would be a difference in the periodic auctions volume growth trajectories for stocks facing dark trading restrictions and those that are not. Therefore, we propose the following hypothesis:

INSERT FIGURE 1 AND FIGURE 2 ABOUT HERE
Hypothesis 1: Following the implementation of the DVC, stocks with DVC-imposed dark trading restrictions will experience a higher periodic auctions volume when compared to those with no DVC-related restrictions.
The implementation of the DVC might lead to the shift of hitherto dark trading volume to quasi-dark markets, including the periodic auction book (see Johann et al., 2019).
Our next few hypotheses relate to the market quality effects of periodic auctions. The call auction is widely employed as an opening and closing mechanism in financial markets, with implications for market quality during the continuous trading period. Indeed, Ibikunle (2015) identifies distinctions in the effects of the call auction depending on its positioning relative to the continuous trading period and the activeness of the stocks (see also Cao et al., 2000;Jiang et al., 2012;Chang et al., 2008;Cordi et al., 2015). Therefore, it is rational to expect that periodic auctions interact with the continuous trading mechanism when deployed concurrently, and that this has implications for market quality. However, since market quality characteristics, such as liquidity, are functions of trading activity (see Chordia et al., 2001;Chordia et al., 2008), these implications are likely linked to periodic auctions volume. Although periodic auctions were deployed in European markets prior to the MiFID II era, as shown in Figure 2, they captured only a very small percentage of the overall daily market volume before the implementation of MiFID II, less than 0.1% of the total trading volume (see FCA, 2018;. The implementation of MiFID II provisions, especially the imposition of the DVC dark trading restrictions, changed this, leading to a substantial growth in period auctions volume, as seen in Figure 1. The above suggests that the potential effects of periodic auctions on market quality characteristics are more likely to be empirically evidenced following the implementation of MiFID II provisions, i.e. a period with relatively sufficient volumes. The crucial question here is whether more periodic auctions enhances or impairs market quality characteristics. Given the evolution of periodic auctions around the DVC implementation, the market quality effects of any changes in the volume of periodic auctions could be linked to the effects of dark trading restrictions. Ibikunle et al. (2019) find that MiFID II dark trading halts are linked to a general decline in market quality, while Johann et al. (2019) find that the market quality effects of any MiFID-II-induced shift in trading volume from dark to other venues is negligible. Therefore, it is useful to employ a framework that distinguishes the effects of periodic auctions on market quality characteristics while controlling for the effects of the DVC. Liquidity and informational efficiency are crucial characteristics that indicate the quality of the trading process. While an implementation of the DVC is expected to adversely impact liquidity in the affected stocks , periodic auctions should alleviate some of the liquidity constraints the DVC's implementation imposes. Information might also be released in a timelier manner when traders migrate from dark pools to more transparent trading mechanisms, such as periodic auctions. An improvement in transparency could inform an improvement in the price discovery process through the formation of more efficient prices, which in turn could encourage the submission of more orders and liquidity improvements in the aggregate market (see Amihud et al., 1997;Madhavan, 1992;Bloomfield et al., 2015). Therefore, we propose the following hypotheses: Hypothesis 2: The implementation of the DVC impairs liquidity for stocks with dark trading restrictions.

Hypothesis 3: An increase in periodic auctions alleviates the liquidity constraints induced by DVC implementation.
The argument with regard to the effects of periodic auction on liquidity related to DVC implementation is linked to transparency, i.e. the dynamics of a component of the spread and adverse selection costs. In the classical call auctions literature, congregating all available market liquidity at a single point for price determination purposes is a central theoretical argument. Schwartz (2012) asserts that doing so enhances the accuracy of the price discovery process, while Madhavan (1992) argues that since all traders are given access to the same prices at the same time, call auctions reduce information asymmetry. Schnitzlein (1996) also finds that there is a reduction in adverse selection costs incurred by uninformed traders under a call auction. Therefore, the structural similarities between the periodic auction and the call auction lead us to expect periodic auctions to be negatively related to adverse selection costs:

Hypothesis 4: Periodic auctions are negatively linked to adverse selection costs.
With respect to the DVC itself, the implementation of a dark trading halt in stocks will force a transfer of slow traders from dark pools to more transparent ones using trading mechanisms, such as continuous and periodic auctions (see Johann et al., 2019). An increase in the volume of slow (uninformed) traders in lit venues, or at least less dark venues, will lessen the concentration of informed traders in these venues, resulting in lower risk of uninformed traders being adversely selected by informed or faster ones at more transparent venues: Hypothesis 5: The implementation of the DVC leads to a reduction in adverse selection costs.
Although the implementation of the DVC implies a shift of trading activity from dark to more transparent venues, the overall impact of periodic auctions on informational efficiency is likely to be a weakness. This is because, while trading in dark pools signifies a degree of delay due to informed traders facing higher non-execution risk (see Zhu, 2014) and execution delays (see Menkveld et al., 2017) than in more transparent venues, periodic auctions are intentionally designed to slow down trading, and often to counter the effects of speed in trading. Therefore, one anticipated effect of an increased use of periodic auctions on the price discovery process is making it less efficient, i.e. the price formation process becomes slower: Hypothesis 6: Periodic auctions are negatively linked to informational efficiency.

Periodic auctions in Europe
Cboe launched its periodic auctions trading mechanism in October 2015, using both the BXE and DXE order books. The stated aim of the periodic auctions book is to provide a trading environment with reduced emphasis on speed, instead enhancing the importance of price.
Periodic auctions orders at Cboe are accepted from 08:00 to 16:30 London time during trading days. Combined orders are not allowed in the submitting processes, meaning that orders in different directions must be submitted separately. Auctions are also conducted continuously and consecutively throughout the trading day. Traders are able to submit market, limit and pegged orders in the books accepting periodic auctions orders. Orders with the so-called minimum acceptable quantity (MAQ) rule are also accepted. MAQ orders are only executable when the referenced MAQ size is fulfilled. In contrast to the FBA design envisaged by Budish et al. (2015), the duration of each auction is randomized, however, it is less than the maximum limit, which is 100ms. Each auction is split into two stages. The first is the price determination stage, when the auction price is formed; the second is the execution allocation stage. To determine the auction prices, four criteria must be met: naming maximum executable volume, minimum surplus, market pressure and reference price. The most important point here is ensuring that, for each auction, the mechanism selects the equilibrium price where the executed volume is maximised. The basis of 'price/size/time' is followed during the price determination process; this means that the importance of price is directly enhanced in the auctions.
Furthermore, in order to ensure an orderly price formation process, the EBBO (European best bid and offer) collar is introduced. By ensuring that the auction prices fall within the collar, this move protects against the auction prices, leading to best execution issues.
During the order allocation process, orders in Cboe periodic auctions, the allocation priority order is 'broker (optional)/price/size/time'. The broker preference feature is optional and refers to single broker paired transactions. The feature supports attracting broker trading activity; according to Cboe data, broker priority orders have been contributing about 20% of total periodic auction volumes since 2018 Q2. 2 In order to ensure that this feature does not interfere with price formation, it is only available at the execution stage. In line with MiFID II requirements, the Cboe periodic auctions book offers pre-trade transparency. 3 The London Stock Exchange Group (LSEG) also recently introduced its own periodic auctions book called Turquoise Plato Lit Auctions, which has been in operation since 2017 Q4.
Although the Turquoise periodic auctions book came into the market later than Cboe, it has a lot of the same features as the former, including order type, member/price priority, 4 allocation, and price formation. However, the Turquoise auction interval is slightly different from that of Cboe. In Turquoise, the interval is divided into two parts: a 50-millisecond fixed interval and a randomized interval with a maximum 50-millisecond duration. Hence, the interval durations vary from 50 to 100 milliseconds.

Sample, matching process, and data
We employ the constituents of the FTSE 250 index of stocks, which includes 250 of the largest 350 UK firms' stocks as listed on the LSE. The decision to use the FTSE 250 stocks is driven by our empirical framework, which involves deploying two estimation approaches. The first is a difference-in-differences (DiD) framework used to estimate the relative evolution in periodic auctions trading activity in stocks affected by the DVC relative to those not affected.
This approach requires the matching of the affected (treated) stocks with those that are unaffected (control stocks). The selection of the larger FTSE 100 stocks would have made pairing for a sufficient number of stocks impossible -given that a significant proportion of For the DiD estimation, we first match the sample of DVC-affected stocks with those that are unaffected. Consistent with Shkilko and Sokolov (2020), we match every stock in the treated group with a stock with dark trading privileges using total volume, a liquidity proxy (relative spread) and information efficiency proxy (5-second autocorrelation of intraday stock returns) for the first empirical framework. We compute matching error for a given number of pairs as follows: (1) where corresponds to the matching criteria, including stock price, currency volume and market value, and and represent a pair of stocks. Variables are sampled no later than a month prior to announcement of the DVC suspensions in order to ensure that they are not directly influenced by the shock. The matching process yields 57 control stocks and 57 treated stocks. The success of the matching approach is underscored by the observation that the control and treated group of stocks are not economically or significantly (in statistical terms) different from one another with respect to the variables employed in the matching prior to implementation of the DVC (see Panel B of Table 2).
Transactions in the periodic auctions of FTSE 250 stocks mainly occur at Turquoise and Cboe, with the two exchanges capturing more than 85% of the periodic auctions transactions in the market. Therefore, the intraday data we obtain for our sample of stocks includes trading activity recorded for the LSE, Turquoise and Cboe, containing data for all the trading mechanisms deployed on all three exchanges over the sample period. We also note that, based on aggregate trading data from Cboe, the three venues account for more than 95% of all trading activity in the FTSE 350 stocks.

Market quality metrics
In this section, we discuss our estimation of the market quality variables. All market quality variables are estimated using data from the continuous trading mechanism deployed by the main market for FTSE 250 stocks, the LSE's Stock Exchange Electronic Trading Service (SETS). 5 We proxy liquidity with relative spread for stock at time estimated as follows: where the , is the average of , and , for stock at time , and , and , correspond to the ask and bid prices for stock at time .
, is then computed as the daily volume-weighted value of , for stock on each day .
In addition to liquidity, we also proxy adverse selection cost as a component of the bid-ask spread. Adverse selection cost reflects the level of latency arbitrage in the market, and it is also employed by related studies, such as Aquilina et al. (2017b) and Shkilko and Sokolov (2020).
The entrance of fast traders could potentially lead to losses for the liquidity supplier, because fast traders can react more rapidly to new information, thereby inducing latency arbitrage. In this situation, irrespective of their analytical abilities, faster traders will be the informed traders and slower traders will be the uninformed traders. In response to this exposure, liquidity suppliers are likely to expand the spread by imposing higher adverse selection costs, thereby protecting themselves from being adversely selected. This is in line with Budish et al. (2015) and Rzayev and Ibikunle (2019), who argue that latency arbitrage is a form of adverse selection.
Therefore, the evolution of adverse selection in the market could be an indicator of changes in the use of latency arbitrage as a trading strategy caused by fast traders. We estimate adverse selection costs for stock in time as: where , is the midpoint price for stock at time and , +15 is the midpoint price for stock at time + 15 seconds; the 15-second window is in line with existing studies, such as Conrad and Wahal (2020) and Shkilko and Sokolov (2020). , indicates the trade direction for stock at time and corresponds to +1 for buyer-initiated trades and -1 for seller-initiated ones; we use the Lee and Ready (1991) algorithm to determine , , setting the interval at 15-seconds. In the regression models, we employ daily volume-weighted estimates of , for stock at day ; this is denoted as , .
Informational efficiency is an important market quality characteristic because it indicates the level of efficient incorporation of information into instrument prices. Therefore, we follow Boehmer et al. (2018), Ibikunle et al. (2019), and Foley and Putniņš (2016) in employing the absolute value of the autocorrelation of midpoint (average of the ask and bid prices) returns as a proxy for the test of informational efficiency. We estimate this proxy at the 5-second frequency and then aggregate across the day as a measure of short-term informational efficiency. Estimates close to zero indicate that the pricing process follows a random walk; hence, the market has a higher level of informational efficiency: , is the absolute value for the 5-second midpoint return autocorrelation for stock on day . In the formula, , , is the th of the 5-second length midpoint return of stock on day , and , , −1 is the ( − 1) t h of the 5-second length midpoint return of stock on day . Utilizing the absolute value of autocorrelation allows for easier capturing of both the under-and over-reaction of returns to information, with higher values suggesting lower efficiency.
For robustness, we also employ an additional proxy for informational efficiency: variance ratio. According to Chordia et al. (2008) and Comerton-Forde and Putniņš (2015), markets with higher levels of pricing efficiency should generate prices that follow the random walk, which suggests that variance should have a linear relation to return frequency. We estimate the measure, as outlined in Equation (5): where , is the variance ratio for stock on day , and , ,1− 2 and , ,5− 2 are the variance estimates of midpoint stock returns over 1 minute and 5 minutes respectively. In an efficient market, , ,5− 2 should be about five times the value of , ,1− 2 . As an absolute value, , is equal to or larger than zero; higher values imply worse informational efficiency.

Other variables
The other variables work as proxies for periodic auctions and variables employed as controls in our models. Periodic auctions proxies include , , , and , and they are defined as trading volume, currency value of traded volume and transactions of periodic auction books for stock on day respectively. The constructed control variables include , , which is defined as the volume of all transactions using all non-periodic auctions trading mechanisms across exchanges where stock is traded on day . , is the end-of-day close price of stock on day , and , is the end-of-day market value of stock on day .
, is the proxy for order imbalance for stock on day , computed as defined in Chordia et al. (2008), i.e. as the absolute value of the buyer-initiated volume for stock on day minus the amount of seller-initiated volume for stock on day divided by the sum of buyer and seller-initiated volume for stock on day . , is the proxy for return volatility for stock on day , and this is calculated as the variance of 1minute intervals mid-price returns. , is a proxy for momentum for stock on day , and this is estimated as the 3-day cumulative abnormal return on closing price. Table 1 defines all the variables employed in our study.

Descriptive statistics
Panel A of Table 2 presents the summary statistics for all the variables employed in the study. Mean and standard deviation estimates are presented for the full sample of stocks and stock terciles in terms of trading activity. A few estimates are of particular interest. Firstly, over the sample, the most active stocks appear to be more liquid. This is consistent with the literature on the links between trading activity and liquidity (see as an example, Chordia et al., 2001).
Interestingly, however, the more active stocks appear to perform worse in terms of informational efficiency. This is perhaps linked to the fact that these stocks are also more likely to be traded via periodic auctions, which would suggest a measure of delay in order execution since batching needs to precede uncrossing during the auctions process. Secondly, the tendency for the more active stocks to be more likely to be traded via periodic auctions than the less active stocks is explained by the former being more likely to be traded via other off-main exchange trading facilities, such as dark pools, due to the need to avoid queues (see . Table 2 reports the pre-DVC comparative estimates of the microstructure variables used in matching the stocks included in the DiD estimations (see Section 4.1). The estimates and statistical tests show minimal differences for all the variables and none of these differences are statistically significant at conventional levels.

Effects of dark trading bans on periodic auctions
Our starting hypothesis is that the imposition of the DVC will lead to an increase in the volume of transactions executed via a periodic auction mechanism. Therefore, we begin by testing whether this dynamic is observed in the data. Our first examination of this question employs univariate analysis testing for differences in trading activity on either side of the DVC coming into effect. The results presented in Table 3 include estimates for nominal stock volume, currency volume and the number of transactions. The estimates are presented separately for the control and treated groups of stocks. In all cases there are statistically significant increases in trading activity following the DVC; however, the increases are far more pronounced for the treated stocks. This is unsurprising given that following the DVC, the treated stocks lose the opportunity to trade in dark pools -an increasingly popular trading mechanism. This is also consistent with the FCA (2018) and Johann et al. (2019) INSERT where , corresponds to one of the log-formal periodic auctions proxies, i.e.
, , Finally, it is essential that the parallel trend assumption holds in the case of the dependent variables, i.e. , , , and , . In particular, the three variables need to have parallel trends in the treatment and control groups in the absence of an event.

INSERT FIGURE 3 HERE
Panels A, B and C in Figure 3 clearly show that the three variables employed in Equation (6) exhibit similar trends during the pre-treatment period and this is also confirmed by statistical tests. This implies that our treatment and control groups can be used in the DiD framework and our modeling approach satisfies the parallel trend assumption requirement. respectively and they are all statistically significant at the 0.01 level. These estimates indicate 236%, 338% and 123% increases in periodic auctions trading volume, currency volume and number of transactions respectively for the event period relative to the period preceding the imposition of the DVC. The significance of these estimates is underscored by the fact that the volume of trading occurring via trading mechanisms other than periodic auctions is controlled for and highly statistically significant in each of the regression estimations. The estimates also suggest a rise in the average execution sizes of period auctions transactions. This is because, while the number of periodic auctions transactions increases during the event period, the relative increase is much lower than that observed for trading volume and currency trading volume. These observations are consistent in the cases of the terciles as well.

INSERT TABLE 4 HERE
However, there is an area of inconsistency when considering the terciles, and this affects the 2 estimates. While for the full sample and the highest and lowest terciles, the treated group of stocks are generally traded more via periodic auctions than the control stocks, this is not the case for the middle tercile. There is no obvious or theoretically relevant explanation for this. What is interesting and theoretically relevant, however, is that once the coefficient has interacted with the coefficient, the deficit is eliminated. This is in line There is another interesting observation to be noted here. 3 estimates are generally higher for the terciles than for the full sample, except in one notable, and consistent, instance -the largest group of stocks. This suggests that the effect of the DVC is weakest in large stocks.
This phenomenon could be linked to the much higher proportions of dark trading activity typically observed among smaller stocks in the London market. In the London market, lower trading stocks are known to be frequently traded away from the downstairs continuous (lit) market, with most of their trades by value taking place in the 'dark' LSE-operated (upstairs) broker-dealer market (see Armitage and Ibikunle, 2015). In the LSE's broker-dealer market, publishing of orders is not mandatory and executed orders can go unreported for up to three minutes, with only the order submitters and attending broker-dealers aware of their existence until reported. Thus, it appears that small UK stocks are mainly traded in opaque venues. For example, Armitage and Ibikunle (2015) find that more than 62% of the orders executed by value in the smallest FTSE 250 stocks are in the LSE's broker-dealer market. This 'dark' trading facility can only remain an option for such stocks following DVC in the cases of disproportionately large orders. Therefore, when dark trading privileges are halted in small stocks, they are more likely than large stocks to pivot to using periodic auctions, a quasi-dark option. This explains the monotonic decline by stock size grouping in 3 s observed in Table   4. The estimates are larger in all three panels for the smallest stocks and lowest for the largest stocks.

Periodic auctions and market quality 4.2.1 Liquidity analysis
We now investigate how changes in the levels of periodic auctions in stock affects its market quality-related characteristics, such as liquidity and informational efficiency. Since our focus is on estimating the effects of the increase in periodic auctions trading on market quality characteristics, rather than the comparative effects between stocks experiencing dark trading restrictions and those that are not, we estimate a fixed effects panel regression model. This also allows us to expand our sample size to 158 stocks with varying levels of periodic auction trading over the full sample period.
The multivariate regression model we estimate is as follows: where all variables are as previously defined. The main variable of interest is the interaction variable, × , , which is introduced to capture the effects of the reported increase in periodic auctions following the DVC's implementation of the liquidity proxy, , . The standard errors are robust to autocorrelation and heteroscedasticity. Table 5 reports the estimation results for Equation (7) A second potential driver is that dark trading restrictions imply the loss of a potential trading mechanism, which then leads to counterparties having to queue for liquidity. However, in line with Hypothesis 3, the negative and statistically significant 2 estimates for all but one of the regressions reported in the table suggest that liquidity constraints during the sample period are alleviated by the opportunity to increase the use of periodic auctions as a trading outlet. These estimates apply to the entire sample time series and so they cannot be completely disentangled from the effects of the DVC-induced increases in periodic auctions activity despite controlling for the DVC. A theoretical explanation for the estimates is that more information is released when traders migrate from dark pools to more transparent venues due to the improvement of transparency, which in turn triggers more efficient prices and further liquidity improvements in the aggregate market (see Amihud et al., 1997;Madhavan, 1992;Bloomfield et al., 2015).

INSERT TABLE 5 HERE
Nevertheless, the improvement in liquidity appears to be less pronounced than the loss of liquidity induced by the DVC. For example, the middle and lowest terciles' 1 ( 2 ) estimates are 0.020 (-0.001) and 0.015 (-0.004) respectively and they are all statistically significant at the 0.01 level. The estimates show that exploiting periodic auctions only ameliorates the liquidity constraints to a small degree. Therefore, it is unsurprising that the 3 estimate for the full sample is positive (0.002) and statistically significant at the 0.01 level, indicating that the combined effects of the DVC and the increase in periodic auctions during the dark trading restrictions period reflects, on average, a worsening of liquidity for the full sample of stocks.
The only stocks to buck this trend are the most liquid tercile stocks. This can be explained by the fact that the most liquid stocks are usually the most active ones and therefore they are likely to be more affected by restrictions on dark trading. Periodic auctions thus provide an opportunity to shift unfulfilled hitherto dark orders.
A similar argument can be made with regard to the effect of trading using the other nonperiodic auctions trading mechanisms captured by the volume variable: the coefficient estimates, although generally negative and statistically significant at the 0.01 level, are very small in comparison to the liquidity constraining effects of the DVC. Taken together, the estimates presented in Table 5

Adverse selection analysis
We next investigate how the periodic auctions dynamics around the DVC impact adverse selection costs, a component of the spread. Periodic auctions are often touted as a countermeasure against the technological arms race for speed (see Budish et al., 2015;Cboe, 2018). The arms race in itself has given rise to adverse selection-inducing latency arbitrage (see Shkilko and Sokolov, 2020;Ibikunle and Rzayev, 2020;Indriawan et al., 2020), which suggests that, consistent with improving liquidity (as shown in Section 4.2.1), a rise in periodic auctions across the full sample period could be linked to a reduction in adverse selection costs.
In order to test this, we estimate the following panel regression model: where , is the daily volume-weighted adverse selection in stock on day . All other variables are as previously defined, and standard errors are robust to autocorrelation and heteroscedasticity. Table 6 reports the estimated coefficients for Equation (8) Schwartz (2012) asserts that this enhances the accuracy of the price discovery process, while Madhavan (1992) argues that since all traders are given access to the same prices at the same time, call auctions reduce information asymmetry. Schnitzlein (1996)  that is typically associated with a significant reduction in the proportion of transactions executed in lit markets (see Aquilina et al., 2017b). The nonlinear effect is also consistent with Eom et al. (2007) argument that market quality is an increasing concave function of transparency. It is important to note that this effect is not observed in the least liquid stocks tercile, with its 3 estimate returning a negative value of -0.03% (significant at a 0.1 level). This is also in line with Aquilina et al. (2017b) reporting a much higher inflection point where the negative relationship between adverse selection risk and dark trading in lower volume stocks turns positive.

Information efficiency analysis
We now address the question of how the evolution of periodic auctions related to dark trading halts can drive the efficiency of the price discovery process. Evidence on the direct effects of periodic auctions on informational efficiency is sparse. However, the extensive body of research on the effects of the longer duration call auctions offers some indication of what we might expect. In particular, both theoretical and empirical studies (Amihud et al., 1997;Madhavan, 1992;Chang et al., 2008;Comerton-Forde et al., 2007) suggest that call auctions improve the efficiency of the price discovery process. In order to ascertain how periodic auctions impact informational efficiency, we estimate the following panel regression model: where , corresponds to one of , or , . All other variables are as previously defined, and the standard errors are robust to autocorrelation and heteroscedasticity. Table 7 reports the regression results for Equation (9). For the sake of clarity, we examine the results presented in both panels of the table in tandem. The first observation is that all but two of the eight 1 coefficient estimates in both panels are negative and statistically significant at conventional levels; the full sample estimates in Panels A and B are -4.50% and -10% respectively, and both are statistically significant at the 0.01 level. This suggests that, contrary to its effects on liquidity, the imposition of the DVC appears to improve the efficiency of the price discovery process. This is likely to be linked to informed traders' reactions to the increased level of transparency induced by the forced migration of hitherto dark order flow to more transparent venues. In contrast to the arguments of Chowdhry and Nanda (1991) and Madhavan (1995) that informed traders trade more slowly in transparent markets, the increased transparency in the aggregate market is linked to improvements in market efficiency. One factor that makes the arguments of Chowdhry and Nanda (1991) and Madhavan (1995) rather invalid in this case is that informed traders generally face a higher risk of non-execution (see Zhu, 2014) and delays in darker/less-transparent markets (see Menkveld et al., 2017). Hence, any caution informed traders may exhibit in markets that are more transparent is less of an impediment to the price discovery process than the non-execution risk and delay associated with trading in dark pools. The economic significance of the coefficient estimates is also indicating that the effect of periodic auctions on informational efficiency is benign at best; hence, the 3 coefficients reflect the effects of the DVC on , . Another theoretical explanation is that more information is released when orders are shifted from dark pools to more transparent venues, thereby increasing transparency (see Amihud et al., 1997;Madhavan, 1992;Bloomfield et al., 2015). In contrast, in Panel A, which shows the results suggests a degree of delay (as in Menkveld et al., 2017;Zhu, 2014), periodic auctions are deliberately designed as mechanisms to make HFT less threatening and slow down trading. It is therefore unsurprising that their main effect on the price discovery process is making it less efficient, and Hypothesis 6 is upheld.

Conclusion
According to Budish et al. (2015), frequently batching orders and auctioning instruments offers an effective solution to address the latency arbitrage and the technological arms race in financial markets, as well as the externalities they induce (see Menkveld, 2014). The question of how frequently batching and uncrossing needs to take place to maintain or enhance market quality remains largely unanswered. In this paper, we exploit recent regulatory developments in Europe to investigate the effects of sub-second periodic auctions on market quality characteristics in UK-listed stocks. The UK financial markets -the most active trading environment in Europe -offer a unique opportunity to assess the direct effects of a shift of trading volume towards periodic auctions following the imposition of dark trading restrictions on the market. This is crucial because frequent auctioning remains uncommon in financial markets. The evidence of the impact of periodic auctions on informational efficiency is also mixed.

Consistent with
Periodic auctions, especially during the DVC window, are linked to a deviation from a random walk or reduction in informational efficiency. This finding is in line with periodic auctions slowing down the price discovery process. While trading in dark pools implies delays relative to trading in lit venues (as in Menkveld et al., 2017;Zhu, 2014), periodic auctions often deliberately design a mechanism to slow down trading, i.e. in response to the technological arms race or latency arbitrage.
The mixed nature of the evidence on frequent discrete trading systems is underscored by the rise in periodic auctions in Europe, while in Taiwan the TWSE has replaced its discrete system with a continuous one (see Indriawan et al., 2020). Therefore, the insights on a new breed of discrete trading systems that this study presents demand attention. Indeed, this is also a valuable early reference for regulators when considering the trade-offs between continuous and discrete trading mechanisms, especially given that the debate on the societal welfare effects of technological arms race speed in financial markets continues unabated. Our study offers some tentative evidence of the relevance of periodic auctions as a mechanism for addressing latency arbitrage and, by extension, the technological arms race.

Figure 1. Trading in periodic auctions books and the implementation of the double volume cap
The

Figure 3. Evolution of outcome variables for treated and control groups
The figures plot the evolution of three outcome variables ( , , , and , ) prior to and following the implementation of the double volume cap (DVC) mechanism.
, , , and , are as defined in Table 1

%
Adverse selection costs for stock on day , computed as the volume-weighted average of the difference between the effective spread and the realized spread divided by the average of bid and ask prices. Trade direction is estimated using the Lee and Ready (1991) classification algorithm.
, Variance ratio for stock on day , and a proxy for informational efficiency. Computed by taking the absolute value of 1 minus a long-term midpoint return variance (5 minutes) divided by a short-term midpoint return variance (1 minute) multiplied by five, which is the quotient between the long-term and the short-term. Midpoint is the average of bid and ask prices.

Table 2. Statistics
In this table, Panel A reports the summary statistics (mean and standard deviation) for all the variables employed in the study, while Panel B presents the results of a statistical comparison of the matching criteria for stocks employed in a difference-in-differences estimation. , , , and , are proxies for periodic auctions activities and are the volume, currency volume and number of transactions in periodic auctions for stock on day . , is the endof-day close price of stock on day .
, is the midpoint return volatility for stock on day . , proxies order imbalance for stock on day .
, is the volume of trading (excluding periodic auctions) in stock on day . , is the end-of-day market value of stock on day .
, is the three-day cumulative abnormal return on closing price for stock on day . , is the daily volume-weighted average of relative quoted spread for stock on day .
, is the daily volume-weighted average of adverse selection costs for stock on day . , is the variance ratio for stock on day , and , is the autocorrelation for stock on day . The sample consists of 215 FTSE 250 stocks trading in London's trading venues between 3 rd January and 29 th June 2018. The stocks are divided into terciles using currency volume in GBX.

Table 3. Trading activity in periodic auctions order books
This table presents estimates of trading activity in periodic auctions order books. , , , and , are proxies for periodic auctions' trading activities and are the volume, currency volume and number of transactions in periodic auctions for stock on day . t-statistics for two-sample tests of differences between average trading activity of the pre-and post-event periods are also presented. The sample period is from 3 rd January to 29 th June 2018 and the event date is     where , is the daily volume-weighted average of adverse selection costs for stock on day , , is the number of periodic auctions transactions in stock on day .
takes the value of one for 12 th March 2018 and subsequent days in the sample and zero otherwise, and and are stock and time fixed effects variables respectively. , contains a series of control variables for stock on day . The variables include the log of , , which is the volume of trading (excluding periodic auctions) in stock on day , log of , , the end-of-day market value of stock on day , , , which proxies order imbalance for stock on day , , , the midpoint return volatility for stock on day , and , , the three-day cumulative abnormal return on closing price for stock on day . The others include the log of , , the end-of-day closing price for stock on day , and , , a proxy for the level of liquidity in stock on day . The sample consists of 158 FTSE 250 stocks trading in London's trading venues between 3 rd January and 29 th June 2018 that are affected by the double volume cap mechanism triggered on 12 th March 2018. The stocks are divided into terciles using currency volume in GBX. ***, ** and * correspond to statistical significance at the 0.01, 0.05 and 0.1 levels respectively.   where , corresponds to one of two proxies for informational efficiency for stock on day ; the two proxies are , (Panel A) and , (Panel B). , is the number of periodic auctions transactions in stock on day . takes the value of one for 12 th March 2018 and subsequent days in the sample and zero otherwise, and and are stock and time fixed effects variables respectively. , contains a series of control variables for stock on day . The variables include the log of , , which is the volume of trading (excluding periodic auctions) in stock on day , log of , , the end-of-day market value of stock on day , , , which proxies order imbalance for stock on day , , , the midpoint return volatility for stock on day and , , the three-day cumulative abnormal return on closing price for stock on day . The others include the log of , , the end-of-day closing price for stock on day and , , a proxy for the level of liquidity in stock on day . The sample consists of 158 FTSE 250 stocks trading in London's trading venues between 3 rd January and 29 th June 2018 that are affected by the double volume cap mechanism triggered on 12 th March 2018. The stocks are divided into terciles using currency volume in GBX. ***, ** and * correspond to statistical significance at the 0.01, 0.05 and 0.1 levels respectively.