# Do Proprietary Algorithmic Traders Withdraw Liquidity during Market Stress?

The debate surrounding the usefulness of machine traders has not reached a final verdict and therefore warrants greater research (O' Hara, 2015). One of the few stylized facts established about algorithmic trading (AT) and its subset high-frequency trading (HFT) is that they generate a lot of limit orders in the market and have a high order-to-trade ratio, thereby playing an instrumental role in the supply of liquidity in limit order markets. What is in dispute, however, is the quality and robustness of the liquidity supplied. Although many argue that machine traders have improved the overall liquidity of the market (Hendershott, Jones, and Menkveld, 2011; Jovanovic and Menkveld, 2011; Boehmer, Fong, and Wu, 2014), others point out that the liquidity is "phantom" (Shorter and Miller, 2014) and nonexistent during market crashes (Kirilenko, Kyle, Samadi, and Tuzun, 2017).The literature suggests that HFT has taken over the mantle of market making (Hagstromer and Norden, 2013; Menkveld, 2013), but the focus has been on general liquidity rather than on episodic liquidity. Although a traditional market maker increases the bid-ask spread when he sees market stress, he remains in the market with his supply of liquidity. A natural question thus arises whether a similar behavior is observed for machine traders, which endogenously supply liquidity to the market. To comprehend the reliability and robustness of the liquidity provided by computer-generated algorithms, it is critical to observe its supply at times of stress. Academic research on this feature has been limited and primarily concentrated on a single event, the 2010 US flash crash (Madhavan, 2012; Kirilenko et al., 2017).

How algorithms contribute to liquidity can depend on their type. Trading algorithms can be used in either an agency or a proprietary context (Hasbrouck and Saar, 2013). When institutional investors use AT to minimize the cost of trading sizable amounts of shares through brokers or "agents," it is known as agency algorithmic trading (AAT). AAT mainly corresponds to using algorithms to break up the required order into smaller pieces to achieve an average price better than some benchmark (such as volume-weighted average price). Proprietary algorithmic trading (PAT) is known for its subclass HFT, which reacts extremely rapidly to market events. (1) Relying on speed, HFT applies algorithms to use very short-lived profit opportunities generated in the trading environment (Hagstromer and Norden, 2013; Biais and Foucault, 2014). Although our focus in this article is on PAT because recent research and arguments about algorithmic trading have centered on HFT, we also document the liquidity-supplying behavior of AAT.

Most of the empirical research on AT/HFT has been using data from developed markets. Biais and Foucault (2014) note that one of the major problems with this research is explicit identification. Most of the research is done through either some proxy for machine trading, (e.g., message traffic as in Hendershott et al., 2011; RunslnProcess using the number of linked messages per 10-minute interval as in Hasbrouck and Saar, 2013) or some specific but not exhaustive data set (NASDAQ data as in Brogaard, Hendershott, and Riordan, 2014). (2) Another concern expressed by Biais and Foucault (2014) is that most existing studies rely on a single market or a single asset and that the lack of cross-market data can affect results because high-frequency traders are likely to take positions in multiple markets at the same time. The data we employ addresses both of these concerns. We conduct our empirical study using data from the National Stock Exchange (NSE) of India; NSE itself has flagged every order and every trade based on whether it is generated from an algorithmic terminal or not and whether it is for a client account or a proprietary account. Also, the stock market in India is largely unfragmented. In the Indian cash equities segment, the NSE records nearly 75% of the traded volume (3). The only other major exchange in the same segment is the Bombay Stock Exchange, which is a long way behind the NSE regarding turnover. The situation provides an ideal setting to trace AT/PAT behavior, as this trading activity would be primarily concentrated in the market we examine. Although the exchange flags help us to exactly identify orders and trades from three mutually exclusive and exhaustive trading classes (PAT, AAT, and nonalgorithmic trading [NAT]), we also apply a criterion to identify order messages from HFT in an approximate way. Wherever possible, we perform our tests on the identified "proxy" HFT group, and in almost all cases, the results for the HFT subgroup mimics the results of its superset, PAT. Our technique for identifying HFT, which is based on the time elapsed between two occurrences (submission, modification, or cancellation) of the same order number, ensures that we detect almost all liquidity-supplying orders from HFT; however, it also ensures that we miss most of the liquidity-demanding orders from this group. Still, this limitation should have little effect on our analysis because our focus is on examining the liquidity-supplying behavior of various trading groups. Since we have both the liquidity-demanding and liquidity-supplying orders from PAT, we also ensure that our analysis remains robust to controlling for the liquidity-demanding orders of this group.

Providing liquidity is one of the two primary functions of a capital market (O'Hara, 2003). Commonly, liquidity is measured by relative spread or effective spread. For our purposes though, we do not need a general measure of liquidity: rather, we need to capture the variation in the supply of liquidity coming from each trading group. Our measure of liquidity is thus computed from the limit order book (LOB) snapshots as the sum of the size of the orders (depth) from each trading group, stationed at first few steps of the LOB. The orders near the top of the book represent genuine trading interest compared to orders away from the top, because the execution probabilities of limit orders fall significantly as we move away from the top (Aitken and Comerton-Forde, 2003). We reconstruct the LOB snapshots at one-minute interval for the 50 largest stocks on the NSE for all trading days of 2013.

Employing several definitions of market stress such as high volatility, extreme stock movement, and so on, we examine how PAT changes its supplied depth. We use two measures to determine short-term volatility: the absolute return and the residual volatility from a market model. Following the technique employed by Ahn, Bae, and Chan (2001), we explore how the depth supplied by PAT reacts to short-term volatility and find that PAT depth near the top of the book is positively related to stock-specific volatility under either definition. The change in depth supplied by PAT in reaction to marketwide volatility is positive under the first definition and is neutral under the second. This behavior, though, is limited only to PAT as AAT and NAT do not increase the supply of liquidity in times of stress. The presence of faster traders in the market possibly instills the fear of adverse selection in AAT and NAT. Furthermore, in the spirit of Bae, Jang, and Park (2003), we distinguish between informational and transitory volatility and find that PAT increases limit order supply more during high transitory volatility than during high informational volatility. These results indicate that proprietary algorithmic traders, in general, do not panic in times of high volatility but react more positively to transitory volatility because this volatility has less of an adverse selection component.

To examine whether PAT provides liquidity at all times, specifically at the most difficult times, we look at the extreme positive and extreme negative return minutes. When any stock is experiencing a high, negative, short-term return, limit buy orders become critical to the functioning of the market. When everyone is looking to sell, there must be someone who is looking to buy--a role served by market makers in traditional markets. We find that the proportion of buy-side depth supplied by PAT in the top few quotes increases during these times. Similar behavior is observed in sell-side depth when any stock is experiencing a high, positive, short-term return. These results indicate that PAT supports the market when price movement is extreme.

Order imbalance is known to affect stock returns (Chordia, Roll, and Subrahmanyam, 2002; Chordia and Subrahmanyam, 2004). Narayan, Narayan, and Westerlund (2015) document these effects for intraday data. We test whether PAT order imbalance is related to future returns, as high-frequency traders are considered informed and sophisticated and are often accused of causing abrupt price fluctuations and trading ahead of the market. Although we find a negative long-term price impact for PAT order imbalance, AAT order imbalance has significant positive predictive power for future returns. The evidence suggests that proprietary algorithmic traders do not seem to be informed traders, whereas agency algorithmic traders, who primarily execute the orders of institutional traders, seem to be informed.

Our work contributes to the literature in several ways. First, we show that in a modern stock market, a positive relation between short-term volatility and depth (Ahn et al., 2001) holds for one trading class (PAT) but not for the other two (AAT and NAT). We then segregate the effect of transitory and informational volatility on depth supplied by PAT and show that PAT depth supply is abundant for volatility driven by noise or liquidity-based trading (transitory) and slightly restricted for volatility driven by efficient prices (informational). This finding corroborates the assertion of Bae et al. (2003) about the general behavior of limit order traders, that they increase their supply of liquidity more in reaction to transitory volatility. Although the positive relation between HFT limit order supply and volatility is supported by Anand and Venkataraman (2016) and Subrahmanyam and Zheng (2016), the differential reaction of PAT to transitory and informational volatility is novel. This work should further consolidate the role of proprietary algorithmic traders as a modern market maker, building on the work by Menkveld (2013) and Hagstromer and Norden (2013).

Second, we distinguish between the limit order trading behavior of PAT and AAT groups. Although a few papers mention agency algorithms (Hagstromer and Norden, 2013; Hasbrouck and Saar, 2013), none specifically discusses their behavior. Thus, little is known about AAT trading in the literature. Nimalendran and Ray (2014) find that trades by crossing network members (similar to proprietary algorithmic traders) are informative whereas trades by crossing network agency desks (similar to agency algorithmic traders) are not. Our finding that agency algorithmic traders react in the opposite direction of proprietary algorithmic traders (i.e., they reduce limit order supply with increasing volatility and reduce the market supporting limit orders during extreme return minutes) could call into question the role of agency algorithmic traders in the market.

Third, we find that during the most extreme positive (negative) return minutes, there is an increase in the proportion of sell (buy) side depth at the top few quotes contributed by PAT. This finding should largely allay fears that the liquidity supplied by HFT is inconsistent, and it supplements the results of Brogaard, Carrion, Moyaert, Riordan, Shkilko, and Sokolov (2018), who find that HFT absorbs the imbalances caused by non-HFT during extreme price movements. Brogaard et al.'s (2018) results are based on NASDAQ HFT data, which provide information on liquidity-supplying trades. Our findings, in contrast, are based on an analysis of order book depth at the top few quotes.

The rest of the article is organized as follows. Section I looks at the related literature and hypothesis development. Section II describes the data, and Section III provides summary statistics. Section IV discusses the results. Section V presents a robustness check using an extended definition of liquidity. Section VI concludes.

I. Motivation and Hypothesis

AT/HFT accounts for a significant fraction of trading volume in global markets. Brogaard et al. (2014) find that HFT accounts for approximately 42% of volume in large-capitalization NASDAQ stocks. Kirilenko et al. (2017) report that HFT was involved in 34% of trading volume in S&P futures in May 2010. Hagstromer and Norden (2013) reveal that HFT in OMXS 30 (a leading Swedish stock index) stocks ranges from 25% to 50% of total activity. Hendershott et al. (2011) suggest AT accounted for as much as 73% of trading volume in the United States in 2009. Hendershott and Riordan (2013) find that AT accounts for 50% to 60% of trading volume on DAX 30 (a stock market index for the Frankfurt Stock Exchange) stocks. Markets in the developing world have also seen a substantial increase in HFT over the last few years. In our sample from the NSE of India, AT contributes about 43% of the trading volume.

Although AT volume is growing, regulators are doubtful about the efficacy and usefulness of fast algorithmic traders. The sheer presence of high-frequency traders and their organized and strategic interactions with the market lead some to believe that they exert a certain manipulative influence on the market. The Securities and Exchange Board of India (SEBI) is contemplating the introduction of a lock-in period for HFT orders. SEBI is presently seeking feedback from stock exchanges and select institutions on other possible steps that could slow down "blackbox trading." (4) A couple of the proposals under consideration include randomization of orders and batch auction of trades. The Reserve Bank of India (RBI) has also observed that volumes associated with algorithmic traders give rise to concerns relating to systemic issues. Another negative externality of AT/HFT is the repeated episodic spikes in quoting activity involving very frequent submissions and cancellations, giving a false sense of the supply of liquidity to the market (Egginton, Van Ness, and Van Ness, 2016). But before any sanction is imposed on AT/HFT, it is critical to understand their contribution to the functioning of modern-day markets. One way to do this is to check their contribution to market liquidity during short-term market stress.

The literature has argued that many high-frequency traders have taken over the role of modern market makers (Menkveld, 2013; Hagstromer and Norden, 2013). (5) Yet concerns remain about the reliability of HFT liquidity supply, as the act of market making without market-making obligations is questionable. Anand and Venkataraman (2016) find that endogenous liquidity providers pull back in unison when market conditions become unfavorable. Related research on AT/HFT includes the study by Hendershott and Riordan (2013). They suggest that algorithmic traders monitor market liquidity more actively than do human traders; ATs consume liquidity when it is cheap (i.e., when the bid-ask spreads are narrow) and supply liquidity when it is expensive. However Hendershott and Riordan's (2013) analysis is based on data for only 13 days on DAX30 constituents. Using order-level data on S&P 500 exchange-traded funds on NASDAQ, Scholtus Van Dijk and Frijns (2014) assess the effect of algorithmic trading on market quality around macroeconomic news. They find that in the minute following a macroeconomic news arrival, algorithmic activity increases trading volume and depth at the best quotes, but it also increases volatility and leads to a drop in overall depth.

Handa and Schwartz (1996) suggest that short-term volatility attracts limit order traders as gains from supplying liquidity surpasses possible losses from trading against information. Fou-cault (1999), in his limit order model, shows that when asset volatility increases, traders find it more optimal to place limit orders than costly market orders. Hoffmann (2014) builds on Foucault's (1999) model by endowing a fixed proportion of traders with a relative speed advantage. Although he finds that traders with the relative speed advantage face a reduced risk of being picked off (because of their ability to revise quotes after the entry of news), other traders submit limit orders with lower execution probability (because of their increased adverse selection risk). Biais, Foucault, and Moinas (2015) analyze the welfare consequences of HFT due to the unequal speed of access to information among all traders. They suggest that HFT simultaneously provides profitable trading opportunities for other investors and raises adverse selection. Furthermore, it is possible that the liquidity-supplying behavior of proprietary algorithmic traders and high-frequency traders differs with respect to single-asset volatility and marketwide volatility. This may be the case because marketwide volatility can pose potentially more challenging market conditions for proprietary algorithmic traders and high-frequency traders to continue supplying liquidity. Consistent with these theories, we offer the following hypothesis:

H1: Short-term volatility encourages proprietary algorithmic traders and high-frequency traders to increase liquidity but induces other traders to reduce liquidity as they fear being picked off.

Proprietary algorithmic traders and high-frequency traders may (per H1) increase or decrease liquidity when faced with high short-term volatility, but does their behavior differ based on the type of volatility? Bae et al. (2003) suggest that limit order traders differentiate between volatility arising out of noise trading and volatility arising out of information trading. This is because there is greater adverse selection in the latter. They find that a rise in transitory volatility encourages limit order placement, whereas a rise in informational volatility appears to have no effect on limit order placement. Along similar lines, we attempt to divide stock volatility (from HI) into transitory and informational components and examine the reaction of proprietary algorithmic traders and high-frequency traders to each. Thus, we offer the following hypothesis:

H2: Proprietary algorithmic traders and high-frequency traders increase liquidity more in reaction to transitory volatility than in reaction to informational volatility.

An important property of market makers is to provide support during extreme price movements. During these times, market makers need to offer liquidity in the opposite direction of the price movement. For example, for a sudden fall in stock prices, providing limit buy orders becomes a crucial role for market makers as there may not be many takers of the stock at that point. Because the market-making role of high-frequency traders has been emphasized in literature, we next seek to examine whether proprietary algorithmic traders serve this act. Kirilenko et al. (2017) provide evidence that during the May 6, 2010, S&P flash crash, high-frequency traders stopped providing limit orders and competed for liquidity with fundamental sellers. This action amplified the selling pressure and exacerbated the volatility. Madhavan (2012) argues that a combination of fragmented trading and HFT can cause the withdrawal of liquidity in times of market stress, as happened during the flash crash. Raman, Robe, and Yadav (2014) find in the US futures markets, electronic market makers reduce their liquidity provision in periods of significantly high customer imbalances. Only one study (Brogaard et al., 2018) finds a positive contribution by high-frequency traders in this regard. Using 2008-2009 NASDAQ data, Brogaard et al. (2018) find that high-frequency traders' participation in trades as suppliers of liquidity increases during extreme price movements. We thus offer the following hypothesis:

H3: Proprietary algorithmic traders and high-frequency traders reduce buy (sell) depth when the stock is experiencing high negative (positive) returns.

PAT/HFT is often accused of causing abrupt short-term price fluctuations in the market. Golub, Keane, and Poon (2012) suggest that stock mini-crashes have increased in recent years, and they link these crashes to HFT. Brogaard et al. (2018) find that HFT liquidity demand increases during extreme price movements. Hirschey (2017) finds that HFT anticipates non-HFT order flow and trades ahead of them. The Commodity Futures Trading Commission chairman expressed the growing concern among regulators over sudden large price movements in recent times and linked them to high-speed trading. (6) Media and industry reports have often commented along similar lines. We examine whether there is any merit in this accusation and study whether the marketable orders of PAT have any relation to short-term stock returns. Because order imbalances are known to be related to stock returns (Chordia et al., 2002; Chordia and Subrahmanyam, 2004; Narayan et al., 2015), we try to draw a causal link between order imbalances of the three trading groups and short-term returns.

H4: Proprietary algorithmic traders' order imbalances are related to short-term returns more than the order imbalances of agency algorithmic traders and nonalgorithmic traders.

II. Data

The NSE is a pure order-driven market without any designated market maker. Trading takes place by an order-matching mechanism--matching marketable sell orders with the best available limit buy orders and matching marketable buy orders with the best available limit sell orders. After a 15-minute preopening session, the exchange is ready for continuous trading at 9:15 AM. Trading is conducted on weekdays excluding public holidays in a single continuous session from 9:15 AM to 3:30 PM. For all traded securities, the exchange freely displays live, on its website, the five best bid and ask quotes and the number of shares demanded or offered at those quotes.

The NSE's pure order-driven market structure is similar to some of the world's most important non-US stock exchanges such as the Paris Bourse, Tokyo Stock Exchange, and Hong Kong Stock Exchange. Even NASDAQ, after the change in dealer market-making obligations in 2007, has become mostly reliant on endogenous liquidity providers. Another reason for the decline in importance of market makers in NASDAQ is the increase in competition from high-frequency traders (Egginton, 2014). The NSE, like NASDAQ, operates an electronic LOB and execution priority follows price and time.

Direct market access and algorithmic trading have been allowed in India since April 2008. HFT and automated trading started to take off in India with the launch of the NSE's colocation services in January 2010. (7) To complement colocation, the exchange provides (for a fee) live tick-by-tick market data generating a broadcast for every order message. For academic research purposes, the NSE provides historical tick-by-tick order and transaction data.

Such data are available for the entire universe of stocks on the NSE and for all trading days in 2013. (8) The data contain an algorithmic flag (whether the order or trade is generated from an algorithmic terminal or not) and a client flag (whether the order or trade is from a proprietary or a client account). Combining the two flags, we can segregate traders into three groups: PAT, AAT, and NAT. We identify all orders in an exact sense from PAT (the superset of HFT), AAT, and NAT, and our assertions in this article are about these three trading groups. We expect that results we get for PAT should provide us with an indication of the behavior of HFT. We also try to approximately identify orders from HFT using a reasonable criterion and additionally provide the results wherever possible for the "proxy" HFT group. The consensus in the literature is that high-frequency traders have a very high order-to-trade ratio and most of their orders get modified or canceled very quickly. Ye, Yao, and Gai (2013) find evidence of this phenomenon for their NASDAQ sample. The minimum time between two subsequent messages for each order number is used to identify orders from high-frequency traders. We define orders from the HFT group as those orders from PAT for which this minimum time is less than three milliseconds.

By construction, the HFT group would miss almost all the liquidity-demanding orders from high-frequency traders because liquidity-demanding orders usually appear once in the order messages. However, it captures the majority of liquidity-supplying orders from high-frequency traders because the bulk of HFT orders are expected to be revised or canceled very quickly. Thus, we are able to provide these extra tests of HFT for HI and H2 (involving liquidity-supplying orders) but are not able to provide them for H4 (involving liquidity-demanding orders). We rely on the results of the PAT group to estimate the behavior of high-frequency traders when we are unable to test directly for it.

We analyze the component stocks of the Nifty 50 index (the NSE's benchmark stock market index), as they are the most liquid and comprise more than 60% of the total market capitalization of the NSE. We keep the list of stocks constant for our analysis, although the index was reconstituted four times during 2013. For our study, we need to generate order book snapshots for each of the 50 stocks at the end of each one-minute interval. Each message-by-message order file comprises on average about 150 million observations per day. (9) In the generated order book, we retain the original AT and the client flag provided by the exchange.

III. Summary Statistics

Table I provides the number and type--cancellation, modification, and new order--of messages per minute. We see that the most frequent messages on the exchange are modification messages (87%). Cancellation and new order messages constitute 5% and 8%, respectively. The proportion of modification messages is higher for PAT (93%) and highest for its subclass HFT (96%). Table I also provides details about the share of messages for each trading category: PAT contributes the most (83%o), followed by AAT (12%). Thus, algorithmic trading, in all, contributes 95% of all messages in the NSE cash market segment. For some stocks, this number even increases to 99%.

Table II displays information on the contribution of trades by different groups of traders. PAT contributes about 15% of trades, in terms of both volume and number of shares traded, and AAT rises from 27% to 36% if the number of trades, rather than volume, is considered. This suggests that AAT breaks up the required order into smaller pieces during trading.

In the generated order book at one-minute intervals for 50 stocks, we examine the percentage of depth supplied by the different groups of traders at various levels, such as total depth, depth at top five quotes, and depth at top three quotes (see Table III). Considering the full order book, we observe that NAT contributes 73% of total depth followed by PAT (25%) and AAT (2%). The situation is different when looking at the top of the book, where trading is more active and quotes more likely to be traded (top five or top three positions). AT is much more active here compared to the whole book, contributing 52% (48%) of the depth of the top five (three) quotes. Placing a quote beyond the top five positions may not prompt AT because the quote is neither likely to be traded nor helpful for any market manipulative purposes. Although the share of the depth supplied by algorithmic traders increases when top three or top five quotes are considered, the pattern of the increase is not symmetric between the AAT and PAT. AAT increases its share of depth monotonically from 2% to 18% to 24%, when moving from the full book to top five quotes to top three quotes. The corresponding numbers for PAT are 25%, 34% and 24%--not quite monotonic. These figures indicate that PAT seems to store many orders at positions 4 and 5 of the order book. These two positions are significant for two reasons: 1) any quote at positions 4 and 5 is less likely to be traded instantaneously compared to quotes in the top three positions, and 2) the NSE displays the top five positions live to all traders free of charge. Therefore, substantial orders placed at positions 4 and 5 can be used for market manipulative purposes, and we leave it for future research to investigate whether PAT does indeed follow this strategy.

We look at the variation in depth supplied over the day at the top three and top five quotes, and the share of the depth supplied by AAT, PAT, and NAT. The overall depth is a little elevated at the end of the day and a little reduced at the start of the day. It remains approximately at a constant level for the rest of the day. Although the share of AAT remains almost constant throughout the day, the share of PAT decreases and the share of NAT increases. This pattern holds true for both the top three and top five quotes (see Figure 1).

We also provide summary statistics and correlation coefficients of all raw variables used in the regression analysis of Section IV (see Table IV).

IV. Results

In H1, we test the effect of short-term volatility on the depth supplied by various groups of traders. Ahn et al. (2001) and Bae et al. (2003) look at a trading day by dividing it into intervals. They argue that the interval should be neither too long nor too short, so that short-term volatility movement can be captured. (10) However, the theory is not conclusive on how short the interval should be. We analyze our data using an interval length of one minute, which from the HFT point of view may be too long, but for our analysis may be just long enough to capture short-term volatility and price movement.

For this analysis, the two primary exogenous variables of interest are short-term stock volatility (RISK) and short-term market volatility (MKT_RISK). Although there are few choices available to measure volatility, we use a measure that is devoid of microstructure noise and is popular in the literature (Ghysels, Santa-Clara, and Valkanov, 2006; Forsberg and Ghysels, 2007): the absolute return. [RISK.sub.i,t] for stock i at minute t is thus defined as the absolute return for stock i at minute t. MKT_[RISK.sub.i,t] is defined the sum of the [RISK.sub.i,t] variables for the other 49 stocks in our sample of 50 stocks. (11) Our focus here is to regress current-minute stock depth on RISK and MKT_RISK recorded in the previous minute. We consider the depth supplied at the top three quotes by the trading group (PAT/AAT/NAT) to gauge the supply of market liquidity. (12) As a robustness check, we conduct the same analysis with the depth supplied at the top five quotes, and those results are provided later in the robustness section. Depth recorded at the previous minutes for the same (other two) trading group(s) is included as a control variable to adjust for autocorrelation (possible cross-correlation) in the depth variable. Trading volume serves as a natural control because the literature has argued that depth and volume are related (Chung, Van Ness, and Van Ness, 1999). Because our variables are in different units (i.e., depth is in number of shares and volatility is a pure number), we standardize all variables. Given the serial dependence nature of our high-frequency regressand (depth) we add lags of order up to n for each independent variable, where n is determined by the Schwartz Bayesian information criterion (BIC). We also define 13 half-hour dummies, representing six and half hour trading day. (13)

The empirical model considered here is:

[mathematical expression not reproducible] (1)

where [DEPTH.sub.i,t] is the sum of the size of the orders due to PAT/HFT/AAT/NAT at the bid and ask quotes for stock i at the end of interval t considering the top three quotes; [RISK.sub.i,t] is the absolute return for stock i at interval t; MKT-[RISK.sub.i,t] is the sum of RISK variables for the other 49 stocks in the sample of 50 stocks; [NTRADE.sub.i,t] is the number of shares that were traded for stock i at time t; and Time dummies is the 13 indicator variables each representing one half an hour of trading.

We estimate our model using panel regression methods, and the t-statistics are provided using panel-corrected standard errors, which are robust to heteroskedasticity and dependence across both time and cross-section (Beck and Katz, 1995).

Table V provides the estimates for Equation (1). The number of lags used in the regression (n) is 12. However, for brevity we provide the coefficients for first 3 lags in the table, as our primary variables of interest are first lags of RISK and MKT_RISK. The results for the full model including all lags are provided in Appendix A. In support of HI, we find that for both PAT and its subclass HFT, the coefficients of the first lags of RISK and MKT_RISK are both positive and highly significant. For PAT (HFT), the coefficient [[beta].sub.1] is 1.03 (1.23) and the t-statistic is 20.07 (23.48). The coefficient [[beta].sup.M.sub.1] is also positive and significant for both PAT and HFT, although it is a little lower in value than [[beta].sub.1].The coefficient [[beta].sup.M.sub.1], is 0.63 (0.77) for PAT (HFT), and the t-statistic is 7.77 (9.39). In contrast to the results for PAT, the [[beta].sub.1] for both AAT and NAT is negative with respective t-statistics -18.99 and -26.46. [[beta].sup.M.sub.1] is insignificant for AAT and NAT.

The results in Table V support the theoretical predictions of both Hoffman (2014) and Biais et al. (2015). Hoffman's (2014) suggestion that traders with a relative speed advantage face a reduced risk of being picked off is confirmed in the results for the PAT (HFT) group, which shows that this group does not fear adverse selection and provides increased limit orders when volatility is high. This is true for both stock-specific volatility and market volatility. Biais et al.'s (2015) suggestion that HFT generates a negative externality by increasing adverse selection risk for other traders is reflected in the results for AAT and NAT. The coefficients for depth autocorrelation and the number of trades ([theta]) are both positive and significant for the first few lags. The time dummies are also significant (consistent with Figure 1), but we omit the results for brevity.

Absolute return is arguably the most easily computable and widely used measure of volatility; however, the RISK variable computed by absolute return may be influenced by overall market movement. To ensure stock-specific volatility orthogonal to marketwide volatility, we estimate a market model at a five-second frequency, compute the idiosyncratic and market volatilities from the same market model, and then reestimate Model (1) using these as definitions of RISK and MKT_RISK. The market model estimated for each stock is:

[R.sub.i,t] =[[alpha].sub.i]+[[beta].sub.i] * [R.sup.M.sub.t] + [[epsilon].sub.i,t], (2)

where ([R.sup.M.sub.t]) is the return on stock i (market) at the five-second interval t, and [[epsilon].sub.i,t] is the residual. The standard deviation of 20 residual observations in one minute is taken as the measure of idiosyncratic volatility of the stock, and the standard deviation of 20 market return observations is taken as the measure of market volatility. Using these new measures as definitions of RISK and MKT_RISK in Model (1), we find that for both PAT and HFT, [[beta].sub.1] is positive and significant (Table VI) with a t-statistic 15.68 (20.22). The coefficient [[beta].sup.M.sub.1], though, is insignificant for both PAT and HFT. Thus using the alternate definition of volatility, we find that both PAT and HFT increase their supply of liquidity in reaction to stock-specific volatility but remain neutral to market volatility. Still, it must be pointed out that PAT and HFT do not withdraw liquidity under any definition of stress. The results for AAT and NAT remain similar to Table V with negative and significant [[beta].sub.1]. Full-model results are available in Appendix B.

Bae et al. (2003) partition short-term volatility into informational and transitory components, and seek to determine the limit order placement tendency of traders in reaction to either type of short-term volatility. However, for a short interval (one minute in our case), we do not want to segregate volatility into two components using mathematical filters. Rather, we introduce an information component along with volatility, which provides a segregation. We assume the volatility in high-information intervals can be linked more to informational volatility than to transitory volatility, and similarly, the volatility in low-information intervals can be linked more to transitory volatility.

We use volume synchronized probability of information trading (VPIN) to categorize each interval as either a high-information interval or a low-information interval. The VPIN metric was introduced by Easley, Lopez de Prado, and O'Hara (2012) to compute a high-frequency estimate of the probability of informed trading and since then has been used by many researchers (Cheung, Chou, and Lei, 2015; Easley, Lopez de Prado, and O'Hara, 2016; Bjursell, Wang, and Zheng, 2017). Heuristically, VPIN measures the proportion of volume-weighted trade coming from informed traders and is computed by the ratio of average unbalanced volume to total volume. Informed traders generally trade in one direction, leading to unbalanced volume. Periods characterized by a high amount of information-based trading have high values of VPIN.

We use the algorithm provided in the working paper version of Easley et al. (2012) to compute VPIN. Computing the VPIN metric requires selecting the volume in every bucket (V), the length of time bars within each bucket, and the number of buckets (n) used to approximate the expected trade imbalance. We use parameters similar to those used by Easley et al. (2012); that is, our V equals 1 /50th of average daily volume, time bars are of one minute, and n equals 50. On an average trading day, the VPIN metric is updated 50 times. However, on an active day that sees above-average trading volume, the VPIN metric is updated more frequently.

After the VPIN values are updated for each stock for the entire sample, we introduce a VPIN dummy for each of our one-minute intervals. The variable VPIN_DUM equals one if the existing VPIN level at the interval is more than its day's average for a particular stock. We then interact RISK with VP[GAMMA]N-DUM and examine whether the coefficients of RISK and RISK*VPIN_DUM have the opposite sign in the following model:

[mathematical expression not reproducible] (3)

where VPIN_[DUM.sub.i,j] is the dummy variable that equals one when the VPIN level for stock i at interval t is greater than the average value over the stock day and the other variables are as defined previously.

The results in Table VII show that PAT (and its subclass HFT) distinguishes between informational and transitory volatility. (14) [[beta].sub.1] is now the coefficient of transitory volatility and is highly significant with a t-statistic 12.97 (16.64). Conversely, [[gamma].sub.1] is negative and significant with a t-statistic -2.37 (-2.93). Negative [[gamma].sub.1], however, in no way suggests that PAT (HFT) decreases limit order supply in reaction to informational volatility. It only suggests that that the increase in depth in reaction to informational volatility is significantly less than the increase in depth in reaction to transitory volatility. Separate unreported analysis, which divides the sample into two parts (one where VPIN_DUM equals one and the other where VPIN_DUM equals 0) and runs Model (1) on each, also suggests that PAT (HFT) increases liquidity supply in reaction to both informational volatility and transitory volatility, but the coefficient is much larger for the latter. The positive association between past volatility and increased depth (results of HI) at the top three quotes by PAT (HFT) is primarily driven by volatility with low adverse selection. The result that Bae et al. (2003) obtain for general limit order traders is true for both PAT and its subset HFT: they increase limit orders more in reaction to transitory volatility than in reaction to informational volatility. Full-model results are provided in Appendix C.

Although from H1 we find that PAT increases liquidity supply in reaction to increased absolute returns, it is not obvious what PAT behavior would be after extreme values of absolute returns are observed. We thus examine the behavior of PAT during times of extreme returns. As before, we work with one-minute intervals. The bound on absolute returns should be sufficiently high to qualify returns crossing it as extreme, but sufficiently low to produce an adequate number of observations to perform statistical tests. Based on these criteria, we choose 0.25% and 0.50% as the two bounds to define extreme returns. We employ a t-test to determine whether there is any significant difference in the proportion of buy (sell) depth supplied by PAT (in the top three quotes) in the minute following the extreme negative (positive) return minutes. The proportion of intervals for which the negative return is less than -0.25% (-0.5%) is close to 2% (0.2%) for all 50 stocks. Thus, the bounds reasonably satisfy the definition of extreme returns. The numbers are similar when we consider the high positive return intervals. The results are presented in Table VIII.

For all definitions of extreme returns, we find that the proportion of PAT depth increases significantly during stress periods. For example, the proportion of PAT buy-side depth at the top three quotes increases from 23% (when the previous-minute return is greater than -0.25%) to 27% (when the previous-minute return is less than -0.25%), with a t-statistic difference in means for the two sample periods equal to 4.7 (the t-statistics for 39 of 50 stocks are positive and significant). Similarly, the proportion of PAT sell-side depth at the top three quotes increases from 24% (when the previous-minute return is less than 0.5%) to 31 % (when the previous-minute return is greater than 0.5%), with a t-statistic difference in means for the two sample periods equal to three. Our results find a positive role for PAT in supporting the market during stress situations, which is contrary to our hypothesis (H3) and popular belief. This finding could be linked to the evolving literature on contrarian liquidity provision (Nagel, 2012; So and Wang, 2014), which suggests that efficient market makers may benefit by providing liquidity during return reversals but we fall short of explaining the rationale. However, in Table VIII, we also find that the share of AAT depth is reduced during stress situations.

Our final test (H4) examines whether the liquidity-reducing orders of proprietary algorithmic traders are related to short-term returns. Here, we employ PAT order imbalance in one minute, defined as the number of buy marketable and market orders minus the number of sell marketable and market orders from the trading group, and use it to regress future stock returns. Because the order imbalances of all trading groups can affect short-term returns, we include a similar order imbalance measure for AAT and NAT in the regression. To capture the permanent impact that each trading group has on future returns, we employ a vector autoregression (VAR) model with n lags, where n is optimally chosen with BIC. Formally, the VAR model considered here is:

[mathematical expression not reproducible] (4)

where [r.sub.i,t] is the return for stock i at interval t; [PATOI.sub.i,t] is the number of buy marketable and market orders minus number of sell marketable and market orders from PAT for stock i at interval t; [AATOI.sub.i,t] is the number of buy marketable and market orders minus the number of sell marketable and market orders from AAT for stock i at interval t; and [NATOI.sub.i,t] is the number of buy marketable and market orders minus the number of sell marketable and market orders from NAT for stock i at interval t.

The results are provided in Table IX. BIC suggests that the optimal n in this model is 10. Our main coefficients of interest are from the first equation of the VAR, where the current-period return is regressed on the previous-period order imbalances of three trading groups (PAT, AAT, and NAT). We find that the first-period lagged order imbalance from both PAT and AAT is positively associated with future returns. However, the first lag is where the similarity ends for these two groups. For PAT, this relation reverses from lag 2 onward and the price impact of PAT order imbalance on future returns measured by [[SIGMA].sup.10.sub.1] [[gamma].sup.P.sub.j] is negative and significant with a t-statistic of -2.66. But the impact of AAT order imbalance on future returns is permanent, as [[SIGMA].sup.10.sub.1] [[gamma].sup.A.sub.j] is positive and significant with a t-statistic of 21.97. The coefficient of NAT, [[SIGMA].sup.10.sub.1] [[gamma].sup.N.sub.j], is not positive. Thus, only AAT trades in the same direction as the market movement before the rest of the market. This result demonstrates that although AAT is the informed group whose order imbalance has a long-term impact on prices, the PAT group does not seem to possess similar ability.

Our four hypotheses, in general, indicate that PAT plays a positive role in modern markets. For HI and H2, we are also able to test the "proxy" HFT group and we find that behavior of HFT mimics that of PAT. Please recall that our identification technique for HFT captures almost all of the liquidity-supplying orders for this group, but it misses almost all of the liquidity-demanding orders. Because we are primarily concerned with the liquidity-supplying behavior of PAT/HFT in stress situations, this should not affect our main results.

However, the possibility remains that PAT/HFT could consume more liquidity during stress situations so that their net effect on liquidity is negative. As we have the exact classification for both liquidity-supplying and liquidity-demanding orders for the three trading groups (PAT, AAT, and NAT), we test how their net realized liquidity changes during stress periods. We consider two definitions of stress period to conduct this test: extreme stock movement in either a positive or a negative direction. Similar to our earlier criteria, we use: 1) returns in a one-minute interval that are less than -0.5% and 2) returns in a one-minute interval that are greater than 0.5%. As before, we employ a t-test to determine whether there is any significant difference in net realized liquidity supplied by PAT in the minute following the extreme return interval. For the first case where stress is defined as extreme negative returns, market-supporting liquidity is from limit buy orders, and market-demanding liquidity is from the market and marketable sell orders. Thus, here net realized liquidity is the number of shares traded as limit buy orders minus the number of shares traded as liquidity-demanding sell orders. Similarly, for the second case where stress is defined as extreme positive returns, net realized liquidity is the number of shares traded as limit sell orders minus the number of shares traded as liquidity-demanding buy orders. For both cases, we find that net realized liquidity for PAT is significantly greater during stress periods compared to normal periods. In fact, proprietary algorithmic traders go from being net liquidity demanders during nonturbulent times to net liquidity suppliers during turbulent times. Table X presents the results.

There are well-known HFT strategies other than market making, such as directional trading, arbitrage, and so on. These strategies use liquidity-demanding orders. Baron, Brogaard, and Kirilenko (2012) analyze transaction data for E-mini S&P futures contracts and find that on average, one-fourth of the HFT firms in their sample complete approximately 90% of their trades with liquidity-demanding orders, and one-third of the HFT firms complete 90% of their trades with liquidity-supplying orders. Baron et al. (2012) also find that the HFT firms with aggressive strategies contribute 15% to trading volume, whereas the HFT firms with passive strategies contribute 8% to trading volume. Hence, our finding that in normal market conditions, PAT is a net demander of liquidity is not surprising.

However, what strengthens our assertions is that in stress situations, PAT net realized liquidity becomes positive and significant. This indicates that the PAT market makers become active during stress situations and support the market when it most needs the liquidity. Our results also suggest that AAT creates the maximum imbalance in stressful times as it becomes a heavy net demander of liquidity. Net realized liquidity for NAT increases in stressful periods compared to normal periods. However, this increase is statistically weaker compared to the findings for PAT. This result is in line with Brogaard et al. (2018), who find that the net realized liquidity of HFT increases during extreme price movements and that HFT absorbs the imbalances caused by non-HFT

V. Robustness Checks

We estimate Model (1) with another definition of liquidity: depth supplied at the top five quotes. This definition is particularly significant because the NSE freely displays on its website the live best five bid and ask quotes and the number of shares demanded or offered at those quotes. Thus, for retail investors, the trading decision is often affected by the depth available at the top five quotes. The results (Table XI) are qualitatively similar to our main results using the top three quotes. PAT (HFT) depth is positively and AAT depth is negatively affected by volatility. Full-model results are provided in Appendix D.

H3 is tested using the share of depth for each trading group (PAT, AAT, and NAT). However, it may be that the actual depth numbers of all trading groups each decrease immediately after extreme return intervals although the proportion of depth for one group increases. Thus, as robustness check of the results of H3, we use absolute depth numbers rather than percentages of depth. The results in Table XII confirm the results of H3 and show that PAT increases and AAT reduces market-supporting depth immediately following extreme return intervals.

VI. Conclusion

We contribute to the ongoing debate around the usefulness of HFT in limit order markets. We use NSE-marked flags to identify orders from PAT and then apply our logic to find the majority of liquidity-supplying orders from its subclass, HFT. We find that 83% of the messages that appear on the exchange are from PAT, and most of those are modification messages.

Contrary to popular belief, we find that both PAT and its subclass HFT increase liquidity provision in times of market stress. When short-term stock volatility is high, PAT/HFT reacts by supplying more liquidity at the top few quotes of the order book. The reaction to marketwide volatility also is not negative. We find that this increase is greater when volatility is transitory rather than informational. We also find that in the minute following extreme stock returns, the proportion of market-supporting depth increases from PAT. Also, PAT order flow imbalance is not positively related to short-term stock returns, whereas AAT order flow imbalance is. Overall, our results strengthen the position of PAT/HFT as modern market makers.

These desirable properties are not found for the AAT group. AAT is an important trading group as it contributes 25% of the trading volume in the large-capitalization segment (compared to 15% for PAT). AAT reduces both depth in reaction to short-term volatility and market-supporting liquidity in times of extreme stock price movements. One possible reason could be that the presence of HFT imposes adverse selection risk on AAT, but this issue is left to future research.

Our result provides a two-sided dilemma for regulators. On one hand, we find that proprietary algorithmic traders supply liquidity in times of stress, but on the other hand, their presence may cause other traders to fear adverse selection. A tough approach restricting high-frequency traders may not be the answer, however, as this may cause disruptions to market liquidity. Rather, efforts should be focused on ensuring that other traders are not the victims of adverse selection.

Appendix A: Full Model Coefficients for Table V PAT stands for proprietary algorithmic trading, HFT stands for high-frequency trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. PAT HFT AAT Coeff. Estimate t-Stat Estimate t-Stat Estimate [[beta].sub.1] 1.03 20.07 1.23 23.48 -0.86 [[beta].sub.2] 0.39 7.51 0.59 11.20 -0.52 [[beta].sub.3] 0.10 1.99 0.25 4.66 -0.29 [[beta].sub.4] -0.05 -0.93 0.03 0.51 -0.18 [[beta].sub.5] 0.02 0.44 0.12 2.36 -0.17 [[beta].sub.6] 0.04 0.76 0.10 1.91 -0.11 [[beta].sub.7] 0.00 -0.04 0.01 0.14 -0.10 [[beta].sub.8] -0.06 -1.17 -0.05 -0.91 -0.09 [[beta].sub.9] 0.01 0.22 0.00 -0.04 -0.03 [[beta].sub.10] 0.08 1.52 0.09 1.66 0.01 [[beta].sub.11] 0.07 1.39 0.10 1.86 -0.06 [[beta].sub.12] 0.08 1.69 0.09 1.77 0.03 [[beta].sup.M.sub.1] 0.63 7.77 0.77 9.39 0.02 [[beta].sup.M.sub.2] 0.35 4.19 0.49 5.83 0.00 [[beta].sup.M.sub.3] -0.12 -1.47 0.09 1.11 0.29 [[beta].sup.M.sub.4] 0.01 0.13 0.12 1.40 0.11 [[beta].sup.M.sub.5] -0.06 -0.68 0.09 1.10 0.19 [[beta].sup.M.sub.6] -0.23 -2.67 -0.07 -0.85 0.26 [[beta].sup.M.sub.7] 0.00 -0.03 0.13 1.52 0.09 [[beta].sup.M.sub.8] -0.11 -1.29 -0.05 -0.59 0.03 [[beta].sup.M.sub.9] 0.02 0.18 0.11 1.26 -0.01 [[beta].sup.M.sub.10] 0.06 0.74 0.09 1.12 -0.08 [[beta].sup.M.sub.11] -0.01 -0.10 0.09 1.08 -0.07 [[beta].sup.M.sub.12] 0.02 0.22 0.02 0.23 -0.31 [[theta].sub.1] 0.77 14.59 0.64 11.84 0.44 [[theta].sub.2] 0.31 5.76 0.18 3.26 0.04 [[theta].sub.3] 0.16 3.01 0.10 1.78 0.12 [[theta].sub.4] -0.04 -0.76 -0.05 -0.95 0.25 [[theta].sub.5] 0.13 2.39 0.10 1.74 0.07 [[theta].sub.6] 0.05 0.85 -0.01 -0.26 0.13 [[theta].sub.7] -0.01 -0.23 -0.03 -0.50 0.13 [[theta].sub.8] 0.11 2.12 0.06 1.15 0.23 [[theta].sub.9] 0.15 2.80 0.11 2.06 0.31 [[theta].sub.10] 0.08 1.52 0.00 -0.06 0.27 [[theta].sub.11] 0.09 1.62 0.06 1.05 0.07 [[theta].sub.12] 0.07 1.34 0.00 -0.08 0.35 [[rho].sup.PAT.sub.1] 18.95 379.19 17.08 342.83 -0.88 [[rho].sup.PAT.sub.2] 10.45 205.48 9.50 188.09 -0.24 [[rho].sup.PAT.sub.3] 7.49 146.52 6.86 135.30 0.07 [[rho].sup.PAT.sub.4] 5.75 112.18 5.23 102.95 0.10 [[rho].sup.PAT.sub.5] 4.66 90.83 4.50 88.41 0.05 [[rho].sup.PAT.sub.6] 4.07 79.18 3.86 75.97 0.17 [[rho].sup.PAT.sub.7] 3.65 71.15 3.55 69.77 0.04 [[rho].sup.PAT.sub.8] 3.36 65.38 3.31 65.24 0.13 [[rho].sup.PAT.sub.9] 2.94 57.39 2.95 58.05 0.04 [[rho].sup.PAT.sub.10] 2.96 57.96 2.86 56.53 0.09 [[rho].sup.PAT.sub.11] 3.05 60.10 2.91 57.78 0.06 [[rho].sup.PAT.sub.12] 3.49 69.94 3.25 65.52 0.02 [[rho].sup.AAT.sub.1] -0.45 -8.10 -0.27 -4.72 30.91 [[rho].sup.AAT.sub.2] -0.19 -3.19 -0.22 -3.54 12.30 [[rho].sup.AAT.sub.3] -0.03 -0.48 -0.05 -0.86 7.77 [[rho].sup.AAT.sub.4] -0.05 -0.76 -0.06 -0.90 5.39 [[rho].sup.AAT.sub.5] 0.12 1.91 0.02 0.25 4.95 [[rho].sup.AAT.sub.6] 0.01 0.14 0.00 -0.01 3.91 [[rho].sup.AAT.sub.7] 0.04 0.71 -0.08 -1.31 3.36 [[rho].sup.AAT.sub.8] 0.08 1.28 0.04 0.58 2.12 [[rho].sup.AAT.sub.9] 0.05 0.75 -0.03 -0.44 2.25 [[rho].sup.AAT.sub.10] 0.13 2.01 0.05 0.76 3.21 [[rho].sup.AAT.sub.11] 0.12 1.92 0.11 1.69 2.95 [[rho].sup.AAT.sub.12] -0.02 -0.36 -0.03 -0.42 3.15 [[rho].sup.NAT.sub.1] -0.26 -4.79 -0.22 -3.95 0.52 [[rho].sup.NAT.sub.2] 0.04 0.78 0.06 1.11 -0.03 [[rho].sup.NAT.sub.3] 0.16 2.78 0.10 1.74 0.11 [[rho].sup.NAT.sub.4] 0.42 7.14 0.45 7.52 0.11 [[rho].sup.NAT.sub.5] 0.29 4.88 0.29 4.80 -0.21 [[rho].sup.NAT.sub.6] 0.14 2.35 0.14 2.31 0.00 [[rho].sup.NAT.sub.7] 0.25 4.22 0.21 3.43 -0.02 [[rho].sup.NAT.sub.8] 0.27 4.60 0.26 4.35 0.11 [[rho].sup.NAT.sub.9] 0.22 3.71 0.25 4.04 -0.03 [[rho].sup.NAT.sub.10] 0.37 6.16 0.34 5.62 -0.05 [[rho].sup.NAT.sub.11] 0.31 5.18 0.23 3.85 0.03 [[rho].sup.NAT.sub.12] 0.38 6.71 0.40 6.96 0.15 [alpha] 4.73 26.84 -0.73 -4.11 3.81 AAT NAT Coeff. t-Stat Estimate t-Stat [[beta].sub.1] -18.99 -1.29 -26.46 [[beta].sub.2] -11.48 -0.85 -17.33 [[beta].sub.3] -6.29 -0.41 -8.31 [[beta].sub.4] -3.91 -0.34 -6.94 [[beta].sub.5] -3.77 -0.37 -7.49 [[beta].sub.6] -2.46 -0.28 -5.75 [[beta].sub.7] -2.24 -0.27 -5.59 [[beta].sub.8] -1.89 -0.29 -6.00 [[beta].sub.9] -0.67 -0.28 -5.85 [[beta].sub.10] 0.27 -0.27 -5.61 [[beta].sub.11] -1.25 -0.32 -6.64 [[beta].sub.12] 0.71 -0.24 -5.08 [[beta].sup.M.sub.1] 0.26 0.02 0.32 [[beta].sup.M.sub.2] 0.05 -0.32 -4.03 [[beta].sup.M.sub.3] 3.61 -0.34 -4.25 [[beta].sup.M.sub.4] 1.31 -0.32 -3.93 [[beta].sup.M.sub.5] 2.34 -0.14 -1.78 [[beta].sup.M.sub.6] 3.24 -0.04 -0.48 [[beta].sup.M.sub.7] 1.18 -0.08 -0.97 [[beta].sup.M.sub.8] 0.31 0.04 0.49 [[beta].sup.M.sub.9] -0.17 0.22 2.71 [[beta].sup.M.sub.10] -1.08 0.17 2.18 [[beta].sup.M.sub.11] -0.91 0.15 1.91 [[beta].sup.M.sub.12] -4.09 0.35 4.67 [[theta].sub.1] 9.52 0.17 3.35 [[theta].sub.2] 0.82 0.45 8.89 [[theta].sub.3] 2.53 0.25 4.90 [[theta].sub.4] 5.27 0.36 7.05 [[theta].sub.5] 1.41 0.28 5.53 [[theta].sub.6] 2.84 0.33 6.42 [[theta].sub.7] 2.76 0.37 7.24 [[theta].sub.8] 4.77 0.38 7.49 [[theta].sub.9] 6.62 0.34 6.77 [[theta].sub.10] 5.78 0.40 7.87 [[theta].sub.11] 1.60 0.47 9.32 [[theta].sub.12] 7.69 0.71 14.23 [[rho].sup.PAT.sub.1] -20.02 0.06 1.22 [[rho].sup.PAT.sub.2] -5.30 0.34 7.10 [[rho].sup.PAT.sub.3] 1.49 0.27 5.57 [[rho].sup.PAT.sub.4] 2.31 0.27 5.52 [[rho].sup.PAT.sub.5] 1.06 0.24 5.00 [[rho].sup.PAT.sub.6] 3.80 0.46 9.38 [[rho].sup.PAT.sub.7] 0.92 0.19 3.90 [[rho].sup.PAT.sub.8] 2.80 0.27 5.66 [[rho].sup.PAT.sub.9] 0.86 0.19 3.94 [[rho].sup.PAT.sub.10] 1.90 0.32 6.59 [[rho].sup.PAT.sub.11] 1.27 0.34 7.06 [[rho].sup.PAT.sub.12] 0.53 0.25 5.18 [[rho].sup.AAT.sub.1] 601.01 0.36 6.72 [[rho].sup.AAT.sub.2] 225.39 0.05 0.88 [[rho].sup.AAT.sub.3] 140.77 0.27 4.68 [[rho].sup.AAT.sub.4] 96.57 0.01 0.12 [[rho].sup.AAT.sub.5] 87.90 0.12 2.05 [[rho].sup.AAT.sub.6] 68.79 0.10 1.62 [[rho].sup.AAT.sub.7] 58.59 0.11 1.93 [[rho].sup.AAT.sub.8] 36.79 -0.06 -1.05 [[rho].sup.AAT.sub.9] 38.90 0.10 1.73 [[rho].sup.AAT.sub.10] 55.43 0.04 0.63 [[rho].sup.AAT.sub.11] 51.15 0.21 3.56 [[rho].sup.AAT.sub.12] 57.39 0.14 2.43 [[rho].sup.NAT.sub.1] 11.08 31.82 611.78 [[rho].sup.NAT.sub.2] -0.67 11.83 213.79 [[rho].sup.NAT.sub.3] 2.07 7.12 126.69 [[rho].sup.NAT.sub.4] 2.21 4.79 84.70 [[rho].sup.NAT.sub.5] -4.05 3.14 55.21 [[rho].sup.NAT.sub.6] 0.00 2.83 49.59 [[rho].sup.NAT.sub.7] -0.30 2.64 46.14 [[rho].sup.NAT.sub.8] 2.04 1.82 31.68 [[rho].sup.NAT.sub.9] -0.65 1.63 28.44 [[rho].sup.NAT.sub.10] -0.91 1.48 25.78 [[rho].sup.NAT.sub.11] 0.62 2.05 35.96 [[rho].sup.NAT.sub.12] 3.01 2.90 52.92 [alpha] 22.32 12.08 69.60 Appendix B: Full Model Coefficients for Table VI PAT stands for proprietary algorithmic trading, HFT stands for high-frequency trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. PAT HFT AAT Coeff. Estimate t-Stat Estimate t-Stat Estimate [[beta].sub.1] 0.85 15.68 1.12 20.22 -1.02 [[beta].sub.2] 0.15 2.70 0.45 7.97 -0.39 [[beta].sub.3] 0.03 0.47 0.27 4.77 -0.20 [[beta].sub.4] 0.09 1.63 0.18 3.09 -0.12 [[beta].sub.5] 0.10 1.69 0.21 3.61 -0.01 [[beta].sub.6] 0.08 1.49 0.22 3.79 0.04 [[beta].sub.7] 0.03 0.56 0.08 1.33 0.05 [[beta].sub.8] 0.07 1.30 0.11 1.84 0.00 [[beta].sub.9] 0.06 1.14 0.18 3.18 -0.02 [[beta].sub.10] 0.20 3.52 0.21 3.65 0.09 [[beta].sub.11] 0.20 3.59 0.23 4.09 -0.14 [[beta].sub.12] 0.28 5.26 0.32 5.89 -0.12 [[beta].sup.M.sub.1] 0.00 0.02 0.03 0.58 -0.03 [[beta].sup.M.sub.2] 0.07 1.45 0.03 0.54 0.02 [[beta].sup.M.sub.3] 0.05 1.14 0.00 0.06 -0.03 [[beta].sup.M.sub.4] 0.10 2.00 0.06 1.32 0.05 [[beta].sup.M.sub.5] -0.08 -1.70 -0.08 -1.75 0.02 [[beta].sup.M.sub.6] -0.01 -0.31 -0.01 -0.28 0.09 [[beta].sup.M.sub.7] 0.03 0.66 0.05 0.97 -0.06 [[beta].sup.M.sub.8] 0.07 1.42 0.03 0.59 0.04 [[beta].sup.M.sub.9] -0.01 -0.11 -0.02 -0.34 0.03 [[beta].sup.M.sub.10] 0.14 2.85 0.12 2.46 0.01 [[beta].sup.M.sub.11] 0.06 1.26 0.09 1.85 0.06 [[beta].sup.M.sub.12] -0.10 -2.11 -0.15 -3.14 0.05 [[theta].sub.1] 0.93 17.67 0.84 15.64 0.41 [[theta].sub.2] 0.39 7.38 0.29 5.24 0.01 [[theta].sub.3] 0.18 3.38 0.13 2.39 0.11 [[theta].sub.4] -0.06 -1.11 -0.05 -0.89 0.24 [[theta].sub.5] 0.11 2.09 0.10 1.88 0.05 [[theta].sub.6] 0.03 0.49 -0.02 -0.45 0.11 [[theta].sub.7] -0.03 -0.56 -0.04 -0.64 0.10 [[theta].sub.8] 0.08 1.50 0.04 0.70 0.21 [[theta].sub.9] 0.14 2.54 0.09 1.71 0.31 [[theta].sub.10] 0.06 1.09 -0.01 -0.23 0.26 [[theta].sub.11] 0.05 0.99 0.04 0.76 0.09 [[theta].sub.12] 0.02 0.44 -0.04 -0.73 0.39 [[rho].sup.PAT.sub.1] 18.96 379.37 17.10 343.15 -0.87 [[rho].sup.PAT.sub.2] 10.45 205.25 9.50 187.91 -0.21 [[rho].sup.PAT.sub.3] 7.49 146.35 6.86 135.08 0.08 [[rho].sup.PAT.sub.4] 5.74 111.89 5.22 102.63 0.11 [[rho].sup.PAT.sub.5] 4.65 90.48 4.48 88.11 0.05 [[rho].sup.PAT.sub.6] 4.05 78.81 3.85 75.60 0.17 [[rho].sup.PAT.sub.7] 3.64 70.85 3.53 69.45 0.04 [[rho].sup.PAT.sub.8] 3.34 65.03 3.30 64.90 0.13 [[rho].sup.PAT.sub.9] 2.93 57.10 2.93 57.73 0.04 [[rho].sup.PAT.sub.10] 2.95 57.61 2.84 56.18 0.09 [[rho].sup.PAT.sub.11] 3.03 59.59 2.89 57.36 0.06 [[rho].sup.PAT.sub.12] 3.47 69.30 3.23 64.98 0.04 [[rho].sup.AAT.sub.1] -0.44 -7.89 -0.25 -4.32 30.90 [[rho].sup.AAT.sub.2] -0.19 -3.12 -0.20 -3.31 12.29 [[rho].sup.AAT.sub.3] -0.02 -0.35 -0.04 -0.59 7.77 [[rho].sup.AAT.sub.4] -0.04 -0.63 -0.04 -0.65 5.38 [[rho].sup.AAT.sub.5] 0.13 2.05 0.03 0.49 4.95 [[rho].sup.AAT.sub.6] 0.02 0.26 0.01 0.21 3.91 [[rho].sup.AAT.sub.7] 0.05 0.84 -0.07 -1.10 3.36 [[rho].sup.AAT.sub.8] 0.09 1.41 0.05 0.79 2.12 [[rho].sup.AAT.sub.9] 0.06 0.90 -0.01 -0.20 2.25 [[rho].sup.AAT.sub.10] 0.14 2.16 0.06 0.97 3.21 [[rho].sup.AAT.sub.11] 0.13 2.14 0.13 1.97 2.94 [[rho].sup.AAT.sub.12] 0.00 -0.08 -0.01 -0.09 3.14 [[rho].sup.NAT.sub.1] -0.26 -4.81 -0.23 -4.15 0.52 [[rho].sup.NAT.sub.2] 0.03 0.61 0.05 0.88 -0.04 [[rho].sup.NAT.sub.3] 0.15 2.57 0.09 1.52 0.11 [[rho].sup.NAT.sub.4] 0.42 7.09 0.44 7.42 0.12 [[rho].sup.NAT.sub.5] 0.29 4.93 0.29 4.79 -0.20 [[rho].sup.NAT.sub.6] 0.14 2.36 0.14 2.28 0.01 [[rho].sup.NAT.sub.7] 0.25 4.20 0.20 3.36 0.00 [[rho].sup.NAT.sub.8] 0.27 4.60 0.26 4.28 0.12 [[rho].sup.NAT.sub.9] 0.22 3.76 0.25 4.04 -0.03 [[rho].sup.NAT.sub.10] 0.37 6.21 0.34 5.60 -0.04 [[rho].sup.NAT.sub.11] 0.31 5.25 0.23 3.78 0.03 [[rho].sup.NAT.sub.12] 0.39 6.85 0.40 6.88 0.15 [alpha] 4.74 27.03 -0.55 -3.12 4.10 AAT NAT Coeff. t-Stat Estimate t-Stat [[beta].sub.1] 21.04 -0.68 -13.19 [[beta].sub.2] -7.82 -0.28 -5.32 [[beta].sub.3] -4.03 -0.18 -3.40 [[beta].sub.4] -2.40 -0.14 -2.70 [[beta].sub.5] -0.27 -0.13 -2.48 [[beta].sub.6] 0.80 -0.03 -0.60 [[beta].sub.7] 1.02 -0.09 -1.75 [[beta].sub.8] -0.04 0.00 0.00 [[beta].sub.9] -0.37 -0.02 -0.32 [[beta].sub.10] 1.91 0.01 0.14 [[beta].sub.11] -2.91 -0.04 -0.76 [[beta].sub.12] -2.59 -0.06 -1.18 [[beta].sup.M.sub.1] -0.65 -0.04 -0.84 [[beta].sup.M.sub.2] 0.50 0.04 0.91 [[beta].sup.M.sub.3] -0.70 0.03 0.71 [[beta].sup.M.sub.4] 1.05 0.01 0.16 [[beta].sup.M.sub.5] 0.45 0.01 0.31 [[beta].sup.M.sub.6] 2.00 0.09 1.93 [[beta].sup.M.sub.7] -1.22 -0.06 -1.35 [[beta].sup.M.sub.8] 0.80 0.05 1.09 [[beta].sup.M.sub.9] 0.67 -0.03 -0.58 [[beta].sup.M.sub.10] 0.21 0.03 0.62 [[beta].sup.M.sub.11] 1.33 0.00 0.04 [[beta].sup.M.sub.12] 1.14 0.06 1.29 [[theta].sub.1] 8.91 -0.11 -2.20 [[theta].sub.2] 0.21 0.22 4.37 [[theta].sub.3] 2.44 0.01 1.92 [[theta].sub.4] 5.15 0.23 4.47 [[theta].sub.5] 0.99 0.16 3.13 [[theta].sub.6] 2.39 0.21 4.16 [[theta].sub.7] 2.18 0.26 5.19 [[theta].sub.8] 4.48 0.27 5.35 [[theta].sub.9] 6.63 0.25 4.86 [[theta].sub.10] 5.58 0.30 5.96 [[theta].sub.11] 2.00 0.39 7.68 [[theta].sub.12] 8.41 0.67 13.44 [[rho].sup.PAT.sub.1] -19.78 0.01 0.29 [[rho].sup.PAT.sub.2] -4.79 0.31 6.37 [[rho].sup.PAT.sub.3] 1.77 0.24 5.02 [[rho].sup.PAT.sub.4] 2.50 0.25 5.09 [[rho].sup.PAT.sub.5] 1.14 0.22 4.59 [[rho].sup.PAT.sub.6] 3.83 0.44 8.99 [[rho].sup.PAT.sub.7] 0.88 0.17 3.54 [[rho].sup.PAT.sub.8] 2.79 0.26 5.27 [[rho].sup.PAT.sub.9] 0.92 0.17 3.49 [[rho].sup.PAT.sub.10] 1.93 0.30 6.15 [[rho].sup.PAT.sub.11] 1.45 0.32 6.64 [[rho].sup.PAT.sub.12] 0.92 0.23 4.88 [[rho].sup.AAT.sub.1] 600.73 0.37 7.01 [[rho].sup.AAT.sub.2] 225.22 0.07 1.23 [[rho].sup.AAT.sub.3] 140.66 0.28 4.97 [[rho].sup.AAT.sub.4] 96.50 0.02 0.36 [[rho].sup.AAT.sub.5] 87.88 0.13 2.26 [[rho].sup.AAT.sub.6] 68.81 0.11 1.88 [[rho].sup.AAT.sub.7] 58.59 0.13 2.18 [[rho].sup.AAT.sub.8] 36.79 -0.05 -0.79 [[rho].sup.AAT.sub.9] 38.83 0.12 1.98 [[rho].sup.AAT.sub.10] 55.39 0.05 0.91 [[rho].sup.AAT.sub.11] 51.01 0.23 3.82 [[rho].sup.AAT.sub.12] 57.19 0.16 2.73 [[rho].sup.NAT.sub.1] 10.92 31.89 612.77 [[rho].sup.NAT.sub.2] -0.85 11.89 214.71 [[rho].sup.NAT.sub.3] 2.10 7.17 127.56 [[rho].sup.NAT.sub.4] 2.26 4.84 85.38 [[rho].sup.NAT.sub.5] -3.95 3.18 55.82 [[rho].sup.NAT.sub.6] 0.21 2.87 50.22 [[rho].sup.NAT.sub.7] -0.07 2.68 46.74 [[rho].sup.NAT.sub.8] 2.22 1.85 32.28 [[rho].sup.NAT.sub.9] -0.53 1.67 29.07 [[rho].sup.NAT.sub.10] -0.78 1.52 26.43 [[rho].sup.NAT.sub.11] 0.67 2.09 36.63 [[rho].sup.NAT.sub.12] 2.92 2.95 53.70 [alpha] 24.20 12.22 70.61 Appendix C: Full Model Coefficients for Table VII PAT stands for proprietary algorithmic trading, and HFT stands for high-frequency trading. PAT HFT Coeff. Estimate t-Stat Estimate t-Stat [[beta].sub.1] 0.98 12.97 1.28 16.64 [[beta].sub.2] 0.24 3.06 0.51 6.40 [[beta].sub.3] 0.00 0.02 0.20 2.46 [[beta].sub.4] 0.04 0.51 0.13 1.60 [[beta].sub.5] 0.07 0.90 0.18 2.25 [[beta].sub.6] 0.08 1.00 0.24 2.95 [[beta].sub.7] 0.10 1.26 0.19 2.45 [[beta].sub.8] 0.14 1.79 0.13 1.60 [[beta].sub.9] 0.05 0.63 0.17 2.16 [[beta].sub.10] 0.31 3.98 0.28 3.50 [[beta].sub.11] 0.20 2.54 0.29 3.71 [[beta].sub.12] 0.22 2.98 0.28 3.69 [[beta].sup.M.sub.1] 0.00 0.01 0.03 0.58 [[beta].sup.M.sub.1] 0.07 1.46 0.03 0.54 [[beta].sup.M.sub.2] 0.05 1.15 0.00 0.06 [[beta].sup.M.sub.3] 0.10 2.01 0.06 1.33 [[beta].sup.M.sub.4] -0.08 -1.70 -0.08 -1.75 [[beta].sup.M.sub.5] -0.01 -0.31 -0.01 -0.28 [[beta].sup.M.sub.6] 0.03 0.66 0.05 0.97 [[beta].sup.M.sub.7] 0.07 1.42 0.03 0.60 [[beta].sup.M.sub.8] 0.00 -0.10 -0.02 -0.34 [[beta].sup.M.sub.9] 0.14 2.86 0.12 2.46 [[beta].sup.M.sub.10] 0.06 1.27 0.09 1.86 [[beta].sup.M.sub.11] -0.10 -2.10 -0.15 -3.14 [[lambda].sub.l] -0.58 -1.27 -0.41 -0.88 [[lambda].sub.2] 0.60 0.96 -0.25 -0.39 [[lambda].sub.3] -1.23 -1.95 -0.79 -1.24 [[lambda].sub.4] 0.01 0.01 0.18 0.28 [[lambda].sub.5] 1.42 2.25 1.16 1.82 [[lambda].sub.6] -1.15 -1.82 -1.44 -2.24 [[lambda].sub.7] 0.33 0.52 1.27 1.98 [[lambda].sub.8] 0.06 0.09 -0.61 -0.95 [[lambda].sub.9] -0.19 -0.30 0.15 0.23 [[lambda].sub.10] -0.26 -0.41 -0.41 -0.64 [[lambda].sub.l1] 0.59 0.93 0.73 1.14 [[lambda].sub.l2] 0.18 0.40 0.19 0.41 [[gamma].sub.1] -0.25 -2.37 -0.31 -2.93 [[gamma].sub.2] -0.17 -1.58 -0.10 -0.89 [[gamma].sub.3] 0.05 0.50 0.16 1.45 [[gamma].sub.4] 0.11 1.00 0.10 0.94 [[gamma].sub.5] 0.05 0.48 0.06 0.53 [[gamma].sub.6] 0.02 0.14 -0.03 -0.28 [[gamma].sub.7] -0.13 -1.21 -0.23 -2.10 [[gamma].sub.8] -0.13 -1.19 -0.04 -0.36 [[gamma].sub.9] 0.03 0.32 0.03 0.24 [[gamma].sub.10] -0.22 -2.00 -0.13 -1.19 [[gamma].sub.11] 0.01 0.14 -0.11 -1.05 [[gamma].sub.12] 0.13 1.22 0.09 0.85 [[theta].sub.1] 0.93 17.70 0.84 15.66 [[theta].sub.2] 0.40 7.39 0.29 5.34 [[theta].sub.3] 0.19 3.56 0.14 2.58 [[theta].sub.4] -0.05 -0.93 -0.04 -0.73 [[theta].sub.5] 0.11 2.04 0.10 1.87 [[theta].sub.6] 0.03 0.60 -0.01 -0.25 [[theta].sub.7] -0.02 -0.45 -0.03 -0.57 [[theta].sub.8] 0.09 1.60 0.05 0.83 [[theta].sub.9] 0.14 2.65 0.10 1.81 [[theta].sub.10] 0.07 1.28 0.00 -0.04 [[theta].sub.11] 0.06 1.06 0.05 0.84 [[theta].sub.12] 0.02 0.45 -0.04 -0.72 [[rho].sup.PAT.sub.1] 18.96 379.35 17.10 343.14 [[rho].sup.PAT.sub.2] 10.45 205.25 9.5 187.90 [[rho].sup.PAT.sub.3] 7.49 146.34 6.86 135.07 [[rho].sup.PAT.sub.4] 5.74 111.88 5.22 102.62 [[rho].sup.PAT.sub.5] 4.65 90.48 4.48 88.10 [[rho].sup.PAT.sub.6] 4.05 78.80 3.85 75.60 [[rho].sup.PAT.sub.7] 3.64 70.84 3.53 69.44 [[rho].sup.PAT.sub.8] 3.34 65.03 3.30 64.90 [[rho].sup.PAT.sub.9] 2.93 57.10 2.93 57.73 [[rho].sup.PAT.sub.10] 2.95 57.61 2.84 56.17 [[rho].sup.PAT.sub.11] 3.03 59.58 2.89 57.35 [[rho].sup.PAT.sub.12] 3.47 69.29 3.23 64.97 [[rho].sup.AAT.sub.1] -0.44 -7.90 -0.25 -4.32 [[rho].sup.AAT.sub.2] -0.19 -3.12 -0.20 -3.31 [[rho].sup.AAT.sub.3] -0.02 -0.35 -0.04 -0.60 [[rho].sup.AAT.sub.4] -0.04 -0.63 -0.04 -0.65 [[rho].sup.AAT.sub.5] 0.13 2.05 0.03 0.48 [[rho].sup.AAT.sub.6] 0.02 0.26 0.01 0.20 [[rho].sup.AAT.sub.7] 0.05 0.84 -0.07 -1.10 [[rho].sup.AAT.sub.8] 0.09 1.41 0.05 0.79 [[rho].sup.AAT.sub.9] 0.06 0.90 -0.01 -0.21 [[rho].sup.AAT.sub.10] 0.14 2.16 0.06 0.97 [[rho].sup.AAT.sub.11] 0.13 2.13 0.13 1.96 [[rho].sup.AAT.sub.12] -0.01 -0.09 -0.01 -0.10 [[rho].sup.NAT.sub.1] -0.26 -4.83 -0.23 -4.18 [[rho].sup.NAT.sub.2] 0.03 0.60 0.05 0.87 [[rho].sup.NAT.sub.3] 0.15 2.56 0.09 1.51 [[rho].sup.NAT.sub.4] 0.42 7.09 0.44 7.42 [[rho].sup.NAT.sub.5] 0.29 4.92 0.29 4.78 [[rho].sup.NAT.sub.6] 0.14 2.35 0.14 2.27 [[rho].sup.NAT.sub.7] 0.25 4.20 0.20 3.35 [[rho].sup.NAT.sub.8] 0.27 4.59 0.26 4.27 [[rho].sup.NAT.sub.9] 0.22 3.75 0.24 4.03 [[rho].sup.NAT.sub.10] 0.37 6.21 0.34 5.60 [[rho].sup.NAT.sub.11] 0.31 5.25 0.23 3.77 [[rho].sup.NAT.sub.12] 0.39 6.84 0.40 6.87 [alpha] 4.82 26.87 -0.47 -2.60 Appendix D: Full Model Coefficients for Table XI PAT stands for proprietary algorithmic trading, HFT stands for high-frequency trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. PAT HFT AAT NAT Coeff. Estimate t-Stat Estimate t-Stat Estimate [[beta].sub.1] 0.90 19.64 1.11 23.93 -0.73 [[beta].sub.2] 0.25 5.47 0.42 9.12 -0.42 [[beta].sub.3] 0.02 0.41 0.11 2.37 -0.25 [[beta].sub.4] -0.08 -1.67 -0.04 -0.75 -0.11 [[beta].sub.5] 0.00 -0.10 0.06 1.21 -0.08 [[beta].sub.6] -0.03 -0.62 -0.02 -0.36 -0.12 [[beta].sub.7] -0.10 -2.23 -0.11 -2.38 -0.10 [[beta].sub.8] -0.11 -2.36 -0.12 -2.56 -0.01 [[beta].sub.9] -0.08 -1.81 -0.13 -2.84 -0.03 [[beta].sub.10] 0.02 0.52 0.00 0.04 0.01 [[beta].sub.11] -0.03 -0.58 -0.01 -0.21 -0.07 [[beta].sub.12] -0.05 -1.15 -0.03 -0.61 0.05 [[beta].sup.M.sub.1] 0.55 7.55 0.74 9.97 -0.06 [[beta].sup.M.sub.1] 0.39 5.14 0.43 5.76 0.06 [[beta].sup.M.sub.2] -0.09 -1.18 0.05 0.60 0.32 [[beta].sup.M.sub.3] 0.04 0.52 0.16 2.16 0.05 [[beta].sup.M.sub.4] -0.09 -1.14 0.01 0.12 0.10 [[beta].sup.M.sub.5] -0.15 -1.98 -0.07 -0.96 0.24 [[beta].sup.M.sub.6] 0.02 0.32 0.09 1.16 0.06 [[beta].sup.M.sub.7] -0.13 -1.67 -0.09 -1.13 0.04 [[beta].sup.M.sub.8] 0.05 0.64 0.10 1.38 -0.02 [[beta].sup.M.sub.9] 0.06 0.76 0.11 1.49 -0.20 [[beta].sup.M.sub.10] 0.03 0.39 0.04 0.51 -0.09 [[beta].sup.M.sub.11] 0.04 0.60 0.03 0.40 -0.33 [[theta].sub.1] 0.54 11.38 0.47 9.75 0.02 [[theta].sub.2] 0.26 5.46 0.19 3.90 0.06 [[theta].sub.3] 0.09 1.92 0.09 1.84 0.30 [[theta].sub.4] -0.04 -0.74 -0.03 -0.71 0.24 [[theta].sub.5] 0.07 1.43 0.03 0.69 0.10 [[theta].sub.6] 0.00 -0.03 -0.02 -0.36 0.10 [[theta].sub.7] -0.04 -0.92 -0.09 -1.74 0.24 [[theta].sub.8] 0.16 3.23 0.11 2.33 0.15 [[theta].sub.9] 0.19 3.88 0.14 2.91 0.36 [[theta].sub.10] 0.07 1.48 0.02 0.41 0.19 [[theta].sub.11] 0.01 0.29 0.00 -0.02 0.00 [[theta].sub.12] 0.08 1.70 -0.01 -0.10 0.27 [[rho].sup.PAT.sub.1] 24.70 477.24 24.19 466.70 -0.88 [[rho].sup.PAT.sub.2] 12.69 238.17 12.17 228.34 -0.13 [[rho].sup.PAT.sub.3] 8.55 159.23 8.18 152.30 0.11 [[rho].sup.PAT.sub.4] 6.34 117.66 6.11 113.55 0.24 [[rho].sup.PAT.sub.5] 5.17 95.79 5.01 93.06 0.12 [[rho].sup.PAT.sub.6] 4.32 80.06 4.20 77.94 0.22 [[rho].sup.PAT.sub.7] 3.89 72.10 3.78 70.16 0.16 [[rho].sup.PAT.sub.8] 3.48 64.61 3.51 65.28 0.22 [[rho].sup.PAT.sub.9] 3.17 58.99 3.17 59.12 0.05 [[rho].sup.PAT.sub.10] 3.16 58.91 3.07 57.38 0.10 [[rho].sup.PAT.sub.11] 3.29 61.89 3.28 61.73 0.17 [[rho].sup.PAT.sub.12] 3.91 75.74 3.83 74.15 0.02 [[rho].sup.AAT.sub.1] -0.19 -3.53 -0.10 -1.86 35.42 [[rho].sup.AAT.sub.2] -0.19 -3.30 -0.22 -3.77 14.37 [[rho].sup.AAT.sub.3] -0.21 -3.54 -0.21 -3.42 7.62 [[rho].sup.AAT.sub.4] 0.09 1.45 0.01 0.13 4.72 [[rho].sup.AAT.sub.5] 0.04 0.68 -0.06 -0.99 4.09 [[rho].sup.AAT.sub.6] 0.09 1.49 0.06 0.91 3.62 [[rho].sup.AAT.sub.7] 0.15 2.49 0.13 2.00 2.95 [[rho].sup.AAT.sub.8] 0.10 1.68 0.05 0.81 1.74 [[rho].sup.AAT.sub.9] -0.01 -0.23 -0.09 -1.43 2.06 [[rho].sup.AAT.sub.10] 0.13 2.11 0.10 1.51 3.12 [[rho].sup.AAT.sub.11] 0.10 1.68 0.08 1.21 2.70 [[rho].sup.AAT.sub.12] 0.00 -0.07 -0.08 -1.28 3.18 [[rho].sup.NAT.sub.1] -0.56 -10.67 -0.48 -8.93 0.48 [[rho].sup.NAT.sub.2] 0.03 0.49 0.00 0.01 -0.12 [[rho].sup.NAT.sub.3] 0.17 2.88 0.15 2.57 0.04 [[rho].sup.NAT.sub.4] 0.33 5.75 0.29 4.97 0.20 [[rho].sup.NAT.sub.5] 0.20 3.44 0.21 3.58 -0.12 [[rho].sup.NAT.sub.6] 0.20 3.52 0.18 3.00 0.05 [[rho].sup.NAT.sub.7] 0.07 1.12 0.03 0.55 -0.05 [[rho].sup.NAT.sub.8] 0.27 4.71 0.26 4.46 0.11 [[rho].sup.NAT.sub.9] 0.19 3.30 0.22 3.74 -0.06 [[rho].sup.NAT.sub.10] 0.32 5.51 0.30 5.04 -0.06 [[rho].sup.NAT.sub.11] 0.26 4.44 0.23 3.98 -0.02 [[rho].sup.NAT.sub.12] 0.35 6.39 0.29 5.23 0.29 [alpha] 1.51 9.33 -2.83 -17.05 2.90 Coeff. t-Stat Estimate t-Stat [[beta].sub.1] -17.25 -1.20 -26.47 [[beta].sub.2] -9.81 -0.76 -16.75 [[beta].sub.3] -5.89 -0.29 -6.46 [[beta].sub.4] -2.55 -0.30 -6.60 [[beta].sub.5] -1.76 -0.36 -7.87 [[beta].sub.6] -2.74 -0.25 -5.57 [[beta].sub.7] -2.33 -0.24 -5.27 [[beta].sub.8] -0.18 -0.26 -5.77 [[beta].sub.9] -0.69 -0.24 -5.38 [[beta].sub.10] 0.16 -0.25 -5.52 [[beta].sub.11] -1.57 -0.29 -6.56 [[beta].sub.12] 1.19 -0.28 -6.30 [[beta].sup.M.sub.1] -0.91 -0.08 -1.13 [[beta].sup.M.sub.1] 0.77 -0.31 -4.20 [[beta].sup.M.sub.2] 4.29 -0.45 -6.13 [[beta].sup.M.sub.3] 0.62 -0.21 -2.76 [[beta].sup.M.sub.4] 1.30 -0.07 -0.92 [[beta].sup.M.sub.5] 3.20 -0.11 -1.50 [[beta].sup.M.sub.6] 0.75 -0.14 -1.86 [[beta].sup.M.sub.7] 0.57 0.02 0.29 [[beta].sup.M.sub.8] -0.30 0.21 2.91 [[beta].sup.M.sub.9] -2.78 0.12 1.67 [[beta].sup.M.sub.10] -1.22 0.22 3.07 [[beta].sup.M.sub.11] -4.68 0.33 4.65 [[theta].sub.1] 0.35 -0.04 -0.94 [[theta].sub.2] 1.38 0.30 6.25 [[theta].sub.3] 6.68 0.20 4.21 [[theta].sub.4] 5.38 0.27 5.68 [[theta].sub.5] 2.26 0.27 5.71 [[theta].sub.6] 2.14 0.29 6.11 [[theta].sub.7] 5.29 0.37 7.73 [[theta].sub.8] 3.33 0.33 7.02 [[theta].sub.9] 8.00 0.30 6.22 [[theta].sub.10] 4.19 0.42 8.76 [[theta].sub.11] 0.01 0.44 9.36 [[theta].sub.12] 6.14 0.69 14.81 [[rho].sup.PAT.sub.1] -18.89 -0.12 -2.41 [[rho].sup.PAT.sub.2] -2.65 0.27 5.17 [[rho].sup.PAT.sub.3] 2.22 0.27 5.25 [[rho].sup.PAT.sub.4] 4.90 0.31 5.90 [[rho].sup.PAT.sub.5] 2.43 0.20 3.82 [[rho].sup.PAT.sub.6] 4.48 0.26 4.93 [[rho].sup.PAT.sub.7] 3.39 0.10 1.97 [[rho].sup.PAT.sub.8] 4.46 0.26 4.91 [[rho].sup.PAT.sub.9] 1.05 0.12 2.30 [[rho].sup.PAT.sub.10] 2.04 0.28 5.40 [[rho].sup.PAT.sub.11] 3.67 0.27 5.24 [[rho].sup.PAT.sub.12] 0.40 0.21 4.08 [[rho].sup.AAT.sub.1] 680.34 0.14 2.55 [[rho].sup.AAT.sub.2] 256.35 0.21 3.73 [[rho].sup.AAT.sub.3] 132.86 0.36 6.07 [[rho].sup.AAT.sub.4] 81.49 -0.14 -2.40 [[rho].sup.AAT.sub.5] 70.05 0.26 4.37 [[rho].sup.AAT.sub.6] 61.63 0.06 1.01 [[rho].sup.AAT.sub.7] 49.91 -0.01 -0.20 [[rho].sup.AAT.sub.8] 29.24 0.02 0.34 [[rho].sup.AAT.sub.9] 34.59 0.08 1.38 [[rho].sup.AAT.sub.10] 52.36 0.08 1.29 [[rho].sup.AAT.sub.11] 45.58 0.14 2.29 [[rho].sup.AAT.sub.12] 56.98 0.06 0.96 [[rho].sup.NAT.sub.1] 10.06 37.48 716.38 [[rho].sup.NAT.sub.2] -2.25 12.97 229.92 [[rho].sup.NAT.sub.3] 0.84 6.84 119.64 [[rho].sup.NAT.sub.4] 3.75 4.47 77.70 [[rho].sup.NAT.sub.5] -2.21 3.73 64.70 [[rho].sup.NAT.sub.6] 0.97 2.22 38.33 [[rho].sup.NAT.sub.7] -0.99 2.26 39.00 [[rho].sup.NAT.sub.8] 2.01 1.49 25.65 [[rho].sup.NAT.sub.9] -1.15 1.5 25.78 [[rho].sup.NAT.sub.10] -1.08 1.64 28.16 [[rho].sup.NAT.sub.11] -0.34 1.81 31.23 [[rho].sup.NAT.sub.12] 5.90 2.73 49.98 [alpha] 18.44 12.06 73.78

References

Ahn, H.J., K.H. Bae, and K. Chan, 2001, "Limit Orders, Depth, and Volatility: Evidence from the Stock Exchange of Hong Kong," Journal of Finance 56, 767-788.

Aitken, M. and C. Comerton-Forde, 2003, "How Should Liquidity Be Measured?" Pacific-Basin Finance Journal 11, 45-59.

Anand, A. and K. Venkataraman, 2016, "Market Conditions, Fragility, and the Economics of Market Making," Journal of Financial Economics 121, 327-349.

Bae, K.H., H. Jang, and K.S. Park, 2003, "Traders' Choice between Limit and Market Orders: Evidence from NYSE Stocks," Journal of Financial Markets 6, 517-538.

Baron, M., J. Brogaard, and A. Kirilenko, 2012, "The Trading Profits of High Frequency Traders," Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi= 10.1.1.471.9434&rep=rep1&type=pdf.

Beck, N. and J.N. Katz, 1995, "What to Do (and Not to Do) with Time-Series Cross-Section Data," American Political Science Review 89, 634-647.

Biais, B. and T. Foucault, 2014, "HFT and Market Quality," Bankers, Markets & Investors, 128, 5-19.

Biais, B., T. Foucault, and S. Moinas, 2015, "Equilibrium Fast Trading," Journal of Financial Economics 116, 292-313.

Bjursell, J., G.H. Wang, and H. Zheng, 2017, "VPIN, Jump Dynamics and Inventory Announcements in Energy Futures Markets," Journal of Futures Markets 37, 542-577.

Boehmer, E., K.Y. Fong, and J.J. Wu, 2014, "International Evidence on Algorithmic Trading," Presented at the American Finance Association Annual Meeting, San Diego, CA.

Brogaard, J., A. Carrion, T. Moyaert, R. Riordan, A. Shkilko, and K. Sokolov, 2018, "High-Frequency Trading and Extreme Price Movements," Journal of Financial Economics 128, 253-265.

Brogaard, J., T. Hendershott, and R. Riordan, 2014, "High-Frequency Trading and Price Discovery," Review of Financial Studies 27, 2267-2306.

Cheung, W.M., R.K. Chou, and A.C. Lei, 2015, "Exchange-Traded Barrier Option and VPIN: Evidence from Hong Kong," Journal of Futures Markets 35, 561-581.

Chordia, T, R. Roll, and A. Subrahmanyam, 2002, "Order Imbalance, Liquidity, and Market Returns," Journal of Financial Economics 65, 111-130.

Chordia, T. and A. Subrahmanyam, 2004, "Order Imbalance and Individual Stock Returns: Theory and Evidence," Journal of Financial Economics 72, 485-518.

Chung, K.H., B.F. Van Ness, and R.A. Van Ness, 1999, "Limit Orders and the Bid-Ask Spread," Journal of Financial Economics 53, 255-287.

Easley, D., M. Lopez de Prado, and M. O'Hara, 2012, "Flow Toxicity and Liquidity in a High-Frequency World," Review of Financial Studies 25, 1457-1493.

Easley, D., M. Lopez de Prado, and M. O'Hara, 2016, "Discerning Information from Trade Data," Journal of Financial Economics 120, 269-285.

Egginton, J.F., 2014, "The Declining Role of NASDAQ Market Makers," Financial Review 49, 461-480.

Egginton, J.F., B.F. Van Ness, and R.A. Van Ness, 2016, "Quote Stuffing," Financial Management 45, 583-608.

Forsberg, L. and E. Ghysels, 2007, "Why Do Absolute Returns Predict Volatility So Well?" Journal of Financial Econometrics 5, 31-67.

Foucault, T., 1999, "Order Flow Composition and Trading Costs in a Dynamic Limit Order Market," Journal of Financial Markets 2, 99-134.

Ghysels, E., P. Santa-Clara, and R. Valkanov, 2006, "Predicting Volatility: Getting the Most Out of Return Data Sampled at Different Frequencies," Journal of Econometrics 131, 59-95.

Golub, A., J. Keane, and S.H. Poon, 2012, "High Frequency Trading and Mini Flash Crashes," Available at: https://arxiv.org/pdf/1211.6667.pdf.

Hagstromer, B. and L. Norden, 2013, "The Diversity of High-Frequency Traders," Journal of Financial Markets 16, 741-770.

Handa, P. and R.A. Schwartz, 1996, "Limit Order Trading," Journal of Finance 51, 1835-1861.

Hasbrouck, J. and G. Saar, 2013, "Low-Latency Trading," Journal of Financial Markets 16, 646-679.

Hendershott, T., C.M. Jones, and A.J. Menkveld, 2011, "Does Algorithmic Trading Improve Liquidity?" Journal of Finance 66, 1-33.

Hendershott, T. and R. Riordan, 2013, "Algorithmic Trading and the Market for Liquidity," Journal of Financial and Quantitative Analysis 48, 1001-1024.

Hirschey, N., 2017, "Do High-Frcquency Traders Anticipate Buying and Selling Pressure?" Available at SSRN: https://doi.org/10.2139/ssrn.2238516.

Hoffmann, P., 2014, "A Dynamic Limit Order Market with Fast and Slow Traders," Journal of Financial Economics 113, 156-169.

Jovanovic, B. and A.J. Menkveld, 2011, "Middlemen in Limit Order Markets," Presented at the Western Finance Association Annual Meeting, Santa Fe, New Mexico.

Kirilenko, A., A.S. Kyle, M. Samadi, and T. Tuzun, 2017, "The Flash Crash: High-Frequency Trading in an Electronic Market," Journal of Finance 72, 967-998.

Madhavan, A., 2012, "Exchange-Traded Funds, Market Structure, and the Flash Crash," Financial Analysts Journal 68, 20-35.

Menkveld, A.J., 2013, "High Frequency Trading and the New Market Makers," Journal of Financial Markets 16, 712-740.

Nagel, S., 2012, "Evaporating Liquidity," Review of Financial Studies 25, 2005-2039.

Narayan, P.K., S. Narayan, and J. Westerlund, 2015, "Do Order Imbalances Predict Chinese Stock Returns? New Evidence From Intraday Data," Pacific-Basin Finance Journal 34, 136-151.

Nimalendran, M. and S. Ray, 2014, "Informational Linkages between Dark and Lit Trading Venues," Journal of Financial Markets 17, 230-261.

O'Hara, M., 2003, "Presidential Address: Liquidity and Price Discovery," Journal of Finance 58, 1335-1354.

O'Hara. M., 2015, "High Frequency Market Microstructure," Journal of Financial Economics 116, 257-270.

Raman, V, M. Robe, and P. Yadav, 2014, "Electronic Market Makers, Trader Anonymity And Market Fragility," Available at SSRN: https://doi.org/10.2139/ssrn.2445223.

Scholtus. M., D. Van Dijk, and B. Frijns, 2014, "Speed, Algorithmic Trading, and Market Quality around Macroeconomic News Announcements," Journal of Banking & Finance 38, 89-105.

G. Shorter and R.S. Miller, 2014, "High-Frequency Trading: Background, Concerns, and Regulatory Developments," Congressional Research Service Report (June 14), Available at: http://www.fas.org/sgp/crs/misc/R43608.pdf.

So, E.C. and S. Wang, 2014, "News-Driven Return Reversals: Liquidity Provision Ahead of Earnings Announcements," Journal of Financial Economics 114, 20-35.

Subrahmanyam, A. and H. Zheng, 2016, "Limit Order Placement by High-Frequency Traders," Borsa Istanbul Review 16, 185-209.

Ye, M., C. Yao, and J. Gai, 2013, "The Externalities of High Frequency Trading," Available at SSRN: https://doi.org/10.2139/ssrn.2066839.

Samarpan Nawn and Ashok Banerjee (*)

We thank Bing Han (Editor) and an anonymous referee for their useful comments, which have made the paper qualitatively superior. An earlier version of the paper was presented under a different title, "Market Liquidity: A Study from Proprietary Algorithmic Traders' Perspective " at the 28th Asian Finance Association Annual Meeting and the 5th India Finance Conference. We thank the conference participants and discussants for their valuable comments.

(*) Samarpan Nawn is an Assistant Professor in the Department of Finance and Accounting at the Indian Institute of Management Udaipur in Ra!asthan, India. Ashok Banerjee is a Professor in the Department of Finance and Control at the Indian Institute of Management Calcutta in West Bengal. India.

(1) PAT can trade at lower frequencies; thus, not all PAT is HFT. However, because our data do not explicitly identify HFT, we concentrate on PAT to get a sense of the behavior of HFT. We also introduce a technique to identify orders from HFT and we include the results of the "proxy" HFT group in our analysis wherever possible.

(2) The term "message," here and in the rest of the article refers to the order messages and includes new orders, cancellations, and revisions.

(3) Handbook of Statistics on Indian Securities Market 2014 published by Securities and Exchange Board of India (SEBI).

(4) Economic Times, September 8, 2015 (https://economictimes.indiatimes.com/markets/stocks/policy/sebi-taking-feedbacks-on-slowing-down-high-frequency-trading-curbs-may-hit-volumes/articleshow/48849489.cms).

(5) There are other high-frequency traders who strategize on directional trading, arbitrage, and so on.

(6) Gregory Meyer and Joe Rennison, "US Regulator Signals Bid to Curb High-Speed Trading," Financial Times (October 21, 2015), https://www.ft.com/content/590c0dd2-7801-IIe5-933d-efcdc3cllc89.

(7) Colocation allows renting rack space with low latency connectivity to the exchange with the mandatory power supply, cooling, and security requirements of the industry.

(8) The data comprise two files for each trading day. The larger file contains each order message for each stock in the exchange displaying its symbol, price, quantity, and time stamp in jiffies ([2.sup-16]) jiffies = 1 second). It also includes supplementary information such as hidden quantity, stop loss price, and so on. The smaller file contains similar information for each trade and is time stamped to a jiffy.

(9) We omit a special type of message--stop loss message--for computational ease. These messages account for about 0.15% of the messages sent. Within this omitted message type, PAT accounts for around 6% of the messages.

(10) Ahn et al. (2001) and Bae et al. (2003), respectively, suggest 15- and 30-minute intervals. The lengths chosen were adequate as there was very limited computerized trading in the early 2000s. With the influx of AT, we must choose a shorter interval.

(11) To avoid multicolinearity, we include only 49 out of the 50 stocks in MKT-RISK.

(12) Please recall from Section III that PAT keeps a comparatively high percentage of orders in positions 4 and 5 of the order book, which might suggest a market manipulation motive. Thus, to gauge the supply of true market liquidity, it is better to restrict our study to the top three quotes.

(13) We include only 12 time-of-the-day effects in our model to avoid multicolinearity.

(14) Table VII shows the results using the definition of RISK, as the residual standard deviation of the market model. However, the result is invariant when we use the definition of RISK as the absolute return.

Table I. Descriptive Statistics This table provides daily cross-sectional average statistics on our sample of Nifty 50 Index component stocks. Cross-sectional averages arc computed over daily averages per slock. Our sample period spans 12 months, from January to December 2013. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, NAT stands for nonalgor!thmic trading, and HFT stands for high-frequency trading. Number of Percentage of Percentage of Percentage Messages per New Order Cancel of Revision Class of Minute Messages Messages Message Traders ALL groups 5,072.47 7.96% 5.03% 87.01% PAT 4,210.28 4.12% 3.30% 92.58% AAT 607.69 20.20% 11.25% 68.55% NAT 254.17 44.68% 10.13% 45.19% HFT (subset of 3,144.59 2.40% 2.06% 95.54% PAT) Table II. Share of Trades This table provides daily cross-sectional average statistics for our sample of Nifty 50 index component stocks. Cross-sectional averages are computed over daily averages per stock. Our sample period spans 12 months, from January to December 2013. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. Class of Traders Share of Volume Share of Number Share of Quantity of Trades Traded PAT 15.12% 13.03% 15.01% AAT 26.59% 36.43% 27.47% NAT 58.29% 50.54% 57.52% Table III. Share of Depth This table provides daily cross-sectional average share of depth among the trading groups on our sample of Nifty 50 index component stocks. Cross-sectional averages are computed over daily averages per stock. Our sample period spans 12 months, from January to December 2013. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, NAT stands for non algorithmic trading, and HFT stands for high-frequency trading. Level of the Book PAT AAT NAT HFT (Subclass of PAT) Top 3 positions 24.22% 24.18% 51.60% 15.27% Top 5 positions 33.78% 18.09% 48.13% 24.30% Full book 24.55% 2.21% 73.24% 7.17% Table IV. Summary Statistics and Correlations Panel A provides cross-sectional distributions of the average of the variables used in regression analysis on our sample of Nifty 50 index component stocks. For a given stock, we compute the average for 12 months, from January to December 2013. DEPTH (3)/(5) variables are the sum of the size of the orders due to proprietary algorithmic trading (PAT), high-frequency trading (HFT), agency algorithmic trading (AAT), or nonalgorithmic trading (NAT), respectively, at the bid and ask quotes at the end of each minute considering the top 3/5 quotes; RISK (1) = absolute return of each stock at each minute; RISK (2) = standard deviation of residuals for stock i in one minute of a market model at a five-second frequency regressing stock i return on market return; MKT_RISK (1) = sum of RISK (1) variables for the other 49 stocks in the sample of 50 stocks; MKT_RISK (2) = standard deviation of market return in each minute where market return is recorded at a five-second frequency; NTRADE = number of shares that were traded in one minute; VP[GAMMA]N_DUM = dummy variable that equals one when the volume synchronized probability of information trading level for stock i at time t is greater than the average value over the stock day; RET = return at each minute; and PATOI/AATOI/NATOI = number of buy marketable and market orders minus the number of sell marketable and market orders from PAT/AAT/NAT in one minute. Panel B provides the cross sectional average pairwise correlations of the variables used in the regressions. Panel A. Summary Statistics Variable Mean Std. Dev. Q1 Median [DEPTH.sup.PAT] (3) 2,401.07 9,410.93 144.12 290.18 [DEPTH.sup.AAT] (3) 795.31 1,960.09 105.74 227.41 [DEPTH.sup.NAT] (3) 4,684.95 17,710.17 367.89 766.87 [DEPTH.sup.HFT] (3) 1,582.38 6,741.89 90.43 162.40 [DEPTH.sup.PAT] (5) 6,501.38 21,009.90 424.06 868.51 [DEPTH.sup.AAT] (5) 1,144.00 2,742.73 161.32 347.53 [DEPTH.sup.NAT] (5) 8,156.15 3,0956.86 626.27 1,297.21 [DEPTH.sup.HFT] (5) 4,236.61 13,735.65 299.41 555.61 RISK (1)(x 10,000) 6.73 1.46 5.69 6.31 RISK (2) (x 10,000) 15.71 57.62 2.89 3.22 MKT_RISK (1) (x 10,000) 329.61 1.46 328.79 330.02 MKT_RISK(a) (2)(x 10,000) 311.62 -- -- -- NTRADE 8,747.46 11,740.46 3,088.63 6,048.49 RET (in basis points) -0.03 0.03 -0.06 -0.03 VPIN_DUM 0.50 0.01 0.49 0.50 PATOI -0.04 0.06 -0.06 -0.03 AATOI 0.17 1.15 -0.64 0.17 NATO I 0.00 0.94 -0.34 -0.04 Variable Q3 Min. Max. [DEPTH.sup.PAT] (3) 1,366.18 38.37 66,239.44 [DEPTH.sup.AAT] (3) 665.60 15.69 13,419.95 [DEPTH.sup.NAT] (3) 2,431.28 82.51 124,655.62 [DEPTH.sup.HFT] (3) 699.61 17.43 47,474.83 [DEPTH.sup.PAT] (5) 4,566.26 83.16 145,163.50 [DEPTH.sup.AAT] (5) 1,031.03 28.71 1,8862.51 [DEPTH.sup.NAT] (5) 4,235.34 144.56 218,194.10 [DEPTH.sup.HFT] (5) 2,864.88 44.63 94,437.59 RISK (1) (x 10,000) 7.55 3.93 11.44 RISK (2) (x 10,000) 4.07 1.82 328.03 MKT_RISK (1) (x 10,000) 330.65 324.90 332.41 MKT_RISK(a) (2) (x 10,000) -- -- -- NTRADE 10,189.05 193.25 77,986.28 RET (in basis points) 0.00 -0.11 0.03 VPIN_DUM 0.51 0.48 0.52 PATOI 0.00 -0.21 0.10 AATOI 0.88 -2.37 2.87 NATO I 0.31 -2.25 3.81 Panel B. Correlations Variable 1 2 3 4 5 6 1. [DEPTH.sup.PAT] (3) -- 2. [DEPTH.sup.AAT] (3) 0.04 -- 3. [DEPTH.sup.NAT] (3) 0.04 0.11 -- 4. [DEPTH.sup.HFT] (3) 0.84 0.03 0.03 -- 5. [DEPTH.sup.PAT] (5) 0.74 0.06 0.06 0.62 -- 6. [DEPTH.sup.AAT] (5) 0.05 0.84 0.11 0.04 0.07 -- 7. [DEPTH.sup.NAT] (5) 0.04 0.09 0.80 0.03 0.05 0.11 8. [DEPTH.sup.HFT] (5) 0.62 0.04 0.05 0.70 0.87 0.05 9. RISK(l) 0.09 -0.01 -0.03 0.09 0.09 -0.01 10. RISK (2) 0.11 -0.03 -0.03 0.10 0.12 -0.03 11. MKT_RISK(1) 0.05 0.00 -0.01 0.06 0.05 0.00 12. MKT_RISK (2) 0.00 0.00 0.00 0.00 0.00 0.00 13. NTRADE 0.08 0.09 0.10 0.06 0.08 0.10 14. RET 0.00 0.00 0.02 0.00 -0.01 0.00 15. VPIN_DUM 0.01 -0.01 -0.01 0.01 0.01 -0.01 16. PATOI 0.00 0.00 0.00 0.00 0.01 0.00 17. A ATO I 0.00 0.01 0.01 0.00 -0.01 0.01 18. NATOI 0.00 -0.01 -0.03 0.01 0.01 -0.01 Variable 7 8 9 10 11 12 1. [DEPTH.sup.PAT] (3) 2. [DEPTH.sup.AAT] (3) 3. [DEPTH.sup.NAT] (3) 4. [DEPTH.sup.HFT] (3) 5. [DEPTH.sup.PAT] (5) 6. [DEPTH.sup.AAT] (5) 7. [DEPTH.sup.NAT] (5) -- 8. [DEPTH.sup.HFT] (5) 0.04 -- 9. RISK(l) -0.03 0.09 -- 10. RISK (2) -0.03 0.12 0.47 -- 11. MKT_RISK(1) -0.02 0.06 0.34 0.33 -- 12. MKT_RISK (2) 0.00 0.00 0.00 0.00 0.01 -- 13. NTRADE 0.11 0.07 0.28 0.26 0.18 0.00 14. RET 0.02 -0.01 0.01 -0.01 -0.01 0.00 15. VPIN_DUM -0.02 0.01 0.02 0.02 0.01 0.00 16. PATOI 0.00 0.01 0.00 0.00 -0.01 0.00 17. A ATO I 0.01 -0.01 0.00 0.00 0.00 0.00 18. NATOI -0.04 0.02 0.02 0.02 0.01 0.00 Variable 13 14 15 16 17 18 1. [DEPTH.sup.PAT] (3) 2. [DEPTH.sup.AAT] (3) 3. [DEPTH.sup.NAT] (3) 4. [DEPTH.sup.HFT] (3) 5. [DEPTH.sup.PAT] (5) 6. [DEPTH.sup.AAT] (5) 7. [DEPTH.sup.NAT] (5) 8. [DEPTH.sup.HFT] (5) 9. RISK(l) 10. RISK (2) 11. MKT_RISK(1) 12. MKT_RISK (2) 13. NTRADE -- 14. RET 0.02 -- 15. VPIN_DUM 0.02 0.00 -- 16. PATOI 0.00 0.02 0.00 -- 17. A ATO I 0.01 0.00 0.00 -0.01 -- 18. NATOI -0.02 -0.04 0.01 0.13 -0.05 -- (a) The variable MKT. RISK (2) by definition has same values across stocks for a given time. Table V. Change in Liquidity Supply in Reaction to Change in Volatility (Absolute Return) This table presents panel regression estimates with t-statistics computed using panel-corrected standard errors (Beck and Katz, 1995) for the following model: [mathematical expression not reproducible] where [DEPTH.sub.i,t] = sum of the size of the orders due to proprietary algorithmic trading (PAT), high-frequency trading (HFT), agency algorithmic trading (AAT), or nonalgorithmic trading (NAT), respectively, at the bid and ask quotes for stock i at the end of interval t considering the top three quotes; [RISK.sub.i,t] = absolute return for stock i at interval t; MKT_[RISK.sub.i,t] = sum of RISK variables for the other 49 stocks in the sample of 50 stocks; [NTRADE.sub.i,t] = number of shares that were traded for stock i at interval t; and Time dummies = 13 indicator variables each representing one interval of half an hour. We include 12 indicator variables here to avoid multicollinearity. All variables in the regression are standardized. The sample includes 12 months of data covering the calendar year 2013 and comprises the Nifty 50 index component stocks. Only the estimates of each variable up to 3 lags are provided here; full-model estimates comprising all 12 lags of each variable is available in Appendix A. All estimates are multiplied by 100 for representation purposes. PAT HFT Coeff. Estimate t-Stat. Estimate t-Stat. [[beta].sub.1] 1.03 20.07 1.23 23.48 [[beta].sub.2] 0.39 7.51 0.59 11.20 [[beta].sub.3] 0.10 1.99 0.25 4.66 [[beta].sup.M.sub.1] 0.63 7.77 0.77 9.39 [[beta].sup.M.sub.2] 0.35 4.19 0.49 5.83 [[beta].sup.M.sub.3] -0.12 -1.47 0.09 1.11 [[theta].sub.1] 0.77 14.59 0.64 11.84 [[theta].sub.2] 0.31 5.76 0.18 3.26 [[theta].sub.3] 0.16 3.01 0.10 1.78 [[rho].sup.PAT.sub.1] 18.95 379.19 17.08 342.83 [[rho].sup.PAT.sub.2] 10.45 205.48 9.50 188.09 [[rho].sup.PAT.sub.3] 7.49 146.52 6.86 135.30 [[rho].sup.AAT.sub.1] -0.45 -8.10 -0.27 -4.72 [[rho].sup.AAT.sub.2] -0.19 -3.19 -0.22 -3.54 [[rho].sup.NAT.sub.3] -0.03 -0.48 -0.05 -0.86 [[rho].sup.NAT.sub.2] -0.26 -4.79 -0.22 -3.95 [[rho].sup.NAT.sub.2] 0.04 0.78 0.06 1.11 [[rho].sup.NAT.sub.3] 0.16 2.78 0.10 1.74 AAT NAT Coeff. Estimate t-Stat. Estimate t-Stat. [[beta].sub.1] -0.86 -18.99 -1.29 -26.46 [[beta].sub.2] -0.52 -11.48 -0.85 -17.33 [[beta].sub.3] -0.29 -6.29 -0.41 -8.31 [[beta].sup.M.sub.1] 0.02 0.26 0.02 0.32 [[beta].sup.M.sub.2] 0.00 0.05 -0.32 -4.03 [[beta].sup.M.sub.3] 0.29 3.61 -0.34 -4.25 [[theta].sub.1] 0.44 9.52 0.17 3.35 [[theta].sub.2] 0.04 0.82 0.45 8.89 [[theta].sub.3] 0.12 2.53 0.25 4.90 [[rho].sup.PAT.sub.1] -0.88 -20.02 0.06 1.22 [[rho].sup.PAT.sub.2] -0.24 -5.30 0.34 7.10 [[rho].sup.PAT.sub.3] 0.07 1.49 0.27 5.57 [[rho].sup.AAT.sub.1] 30.91 601.01 0.36 6.72 [[rho].sup.AAT.sub.2] 12.30 225.39 0.05 0.88 [[rho].sup.NAT.sub.3] 7.77 140.77 0.27 4.68 [[rho].sup.NAT.sub.2] 0.52 11.08 31.82 611.78 [[rho].sup.NAT.sub.2] -0.03 -0.67 11.83 213.79 [[rho].sup.NAT.sub.3] 0.11 2.07 7.12 126.69 Table VI. Change in Liquidity Supply in Reaction to Change in Volatility (Market Model Residual) This table presents the panel regression estimates with i-statistics computed using panel-corrected standard errors (Beck and Katz, 1995) for the following model: [mathematical expression not reproducible] where [DEPTH.sub.i,j] = sum of the size of the orders due to proprietary algorithmic trading (PAT), high-frequency trading (HFT), agency algorithmic trading (AAT), or nonalgorithmic trading (NAT), respectively, at the bid and ask quotes for stock i at the end of interval / considering the top three quotes; [RISK.sub.i,j] = standard deviation of residuals for stock i at interval t of a market model at a five-second frequency regressing stock i return on market return; MKT_[RISK.sub.i,j] = standard deviation of market return at interval t where market return is recorded at a five-second frequency; [NTRADE.sub.i,j] = number of shares that were traded for stock i at interval t; and Time dummies = 13 indicator variables each representing one interval of half an hour. We include 12 indicator variables here to avoid multicollinearity. All variables in the regression are standardized. The sample includes 12 months of data covering the calendar year 2013 and comprises the Nifty 50 index component stocks. Only the estimates of each variable up to 3 lags are provided here; full-model estimates comprising 12 lags of each variable are available in Appendix B. All estimates are multiplied by 100 for representation purposes. PAT HFT Coeff. Estimate t-Stat Estimate t-Stat. [[beta].sub.1] 0.85 15.68 1.12 20.22 [[beta].sub.2] 0.15 2.70 0.45 7.97 [[beta].sub.3] 0.03 0.47 0.27 4.77 [[beta].sup.M.sub.1] 0.00 0.02 0.03 0.58 [[beta].sup.M.sub.2] 0.07 1.45 0.03 0.54 [[beta].sup.M.sub.3] 0.05 1.14 0.00 0.06 [[theta].sub.1] 0.93 17.67 0.84 15.64 [[theta].sub.2] 0.39 7.38 0.29 5.24 [[theta].sub.3] 0.18 3.38 0.13 2.39 [[rho].sup.PAT.sub.1] 18.96 379.37 17.10 343.15 [[rho].sup.PAT.sub.2] 10.45 205.25 9.50 187.91 [[rho].sup.PAT.sub.3] 7.49 146.35 6.86 135.08 [[rho].sup.AAT.sub.1] -0.44 -7.89 -0.25 -4.32 [[rho].sup.AAT.sub.2] -0.19 -3.12 -0.20 -3.31 [[rho].sup.AAT.sub.3] -0.02 -0.35 -0.04 -0.59 [[rho].sup.NAT.sub.1] -0.26 -4.81 -0.23 -4.15 [[rho].sup.NAT.sub.2] 0.03 0.61 0.05 0.88 [[rho].sup.NAT.sub.3] 0.15 2.57 0.09 1.52 AAT NAT Coeff. Estimate t-Stat. Estimate t-Stat. [[beta].sub.1] -1.02 -21.04 -0.68 -13.19 [[beta].sub.2] -0.39 -7.82 -0.28 -5.32 [[beta].sub.3] -0.20 -4.03 -0.18 -3.40 [[beta].sup.M.sub.1] -0.03 -0.65 -0.04 -0.84 [[beta].sup.M.sub.2] 0.02 0.50 0.04 0.91 [[beta].sup.M.sub.3] -0.03 -0.70 0.03 0.71 [[theta].sub.1] 0.41 8.91 -0.11 -2.20 [[theta].sub.2] 0.01 0.21 0.22 4.37 [[theta].sub.3] 0.11 2.44 0.10 1.92 [[rho].sup.PAT.sub.1] -0.87 -19.78 0.01 0.29 [[rho].sup.PAT.sub.2] -0.21 -4.79 0.31 6.37 [[rho].sup.PAT.sub.3] 0.08 1.77 0.24 5.02 [[rho].sup.AAT.sub.1] 30.90 600.73 0.37 7.01 [[rho].sup.AAT.sub.2] 12.29 225.22 0.07 1.23 [[rho].sup.AAT.sub.3] 7.77 140.66 0.28 4.97 [[rho].sup.NAT.sub.1] 0.52 10.92 31.89 612.77 [[rho].sup.NAT.sub.2] -0.04 -0.85 11.89 214.71 [[rho].sup.NAT.sub.3] 0.11 2.10 7.17 127.56 Table VII. Change in Liquidity Supply in Reaction to Change in Informational and Transitory Volatility This table presents the panel regression estimates with t-statistics computed using panel corrected standard errors (Beck and Katz, 1995) for the following model: [mathematical expression not reproducible] where [DEPTH.sub.i,j] = sum of the size of the orders due to proprietary algorithmic trading (PAT), high-frequency trading (HFT), agency algorithmic trading (AAT), or nonalgorithmic trading (NAT), respectively, at the bid and ask quotes for stock i at the end of interval t considering the top three quotes; [RISK.sub.i,j] = standard deviation of residuals for stock i at interval t of a market model at a five-second frequency regressing stock i return on market return; MKT_[RISK.sub.i,j] = standard deviation of market return at interval t where market return is recorded at a five-second frequency; VPIN_[DUM.sub.i,j] = dummy variable that equals 1 when the VPIN level for stock i at interval t is greater than the average value over the stock day; [NTRADE.sub.i,j] = number of shares that were traded for stock i at interval t; and Time dummies = 13 indicator variables each representing one interval of half an hour. We include 12 indicator variables here to avoid multicollinearity. All variables in the regression are standardized. The sample includes 12 months of data covering the calendar year 2013 and comprises the Nifty 50 index component stocks. Only the estimates of each variable up to 3 lags are provided here; full-model estimates comprising all 12 lags of each variable are available in Appendix C. All estimates are multiplied by 100 for representation purposes. PAT HFT Coeff. Estimate t-Stat. Estimate t-Stat. [[beta].sub.1] 0.98 12.97 1.28 16.64 [[beta].sub.2] 0.24 3.06 0.51 6.40 [[beta].sub.3] 0.00 0.02 0.20 2.46 [[beta].sup.M.sub.1] 0.00 0.01 0.03 0.58 [[beta].sup.M.sub.2] 0.07 1.46 0.03 0.54 [[beta].sup.M.sub.3] 0.05 1.15 0.00 0.06 [[lambda].sub.1] -0.58 -1.27 -0.41 -0.88 [[lambda].sub.2] 0.60 0.96 -0.25 -0.39 [[lambda].sub.3] -1.23 -1.95 -0.79 -1.24 [[gamma].sub.1] -0.25 -2.37 -0.31 -2.93 [[gamma].sub.2] -0.17 -1.58 -0.10 -0.89 [[gamma].sub.3] 0.05 0.50 0.16 1.45 [[theta].sub.1] 0.93 17.70 0.84 15.66 [[theta].sub.2] 0.40 7.39 0.29 5.34 [[theta].sub.3] 0.19 3.56 0.14 2.58 [[rho].sup.PAT.sub.1] 18.96 379.35 17.10 343.14 [[rho].sup.PAT.sub.2] 10.45 205.25 9.50 187.90 [[rho].sup.PAT.sub.3] 7.49 146.34 6.86 135.07 [[rho].sup.AAT.sub.1] -0.44 -7.90 -0.25 -4.32 [[rho].sup.AAT.sub.2] -0.19 -3.12 -0.20 -3.31 [[rho].sup.AAT.sub.3] -0.02 -0.35 -0.04 -0.60 [[rho].sup.NAT.sub.1] -0.26 -4.83 -0.23 -4.18 [[rho].sup.NAT.sub.2] 0.03 0.60 0.05 0.87 [[rho].sup.NAT.sub.3] 0.15 2.56 0.09 1.51 Table VIII. Share of Depth after Extreme Price Movements This table shows depth share in the top three quotes comparing extreme (second column) and nonextreme (third column) returns in our sample of Nifty 50 index component stocks. The returns are computed for one-minute intervals. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. The average t-statistics of the difference in means of the two situations are provided, along with the number of stocks having positive significant and negative significant coefficients at the 10% level (in parentheses). Panels A-D characterize the four definitions of extreme returns. Panel A. Extreme Return Definition: Return [less than or equal to] -0.5% Variable Return Return > Average [less than or equal to] -0.5% f-Stat. of -0.5% Difference of Means PAT buy depth share 31.27% 23.84% 3.20 (40,3) AAT buy depth share 21.02% 25.10% -1.92 (3,29) NAT buy depth share 47.71% 51.06% -1.06 (5,21) Panel B. Extreme Return Definition: Return [less than or equal to] -0.25% Variable Return Return > Average [less than or equal to] -0.25% t-Stat. of -0.25% Difference of Means PAT buy depth share 27.37% 23.20% 4.70 (39,5) AAT buy depth share 22.12% 24.73% -3.92 (1,42) NAT buy depth share 50.51% 52.07% -0.59 (12,19) Panel C. Extreme Return Definition: Return [greater than or equal to] 0.5% Variable Return Return 0.5% Average [greater than or equal to] t-Stat. of 0.5% Difference of Means PAT sell depth 31.25% 24.00% 3.00 share (40,4) AAT sell depth 18.13% 23.96% -2.82 share (0,41) NAT sell depth 50.62% 52.04% -0.21 share (7,13) Panel D. Extreme Return Definition: Return [greater than or equal to] 0.25% Variable Return Return [greater than or equal to] [less than or equal to] 0.25% 0.25% PAT sell depth 28.44% 23.60% share AAT sell depth 20.77% 24.08% share NAT sell depth 50.79% 52.32% share Variable Average t-Stat. of Difference of Means PAT sell depth 4.28 share (37,4) AAT sell depth -5.08 share (1,44) NAT sell depth 0.48 share (17,16) Table IX. Order Imbalance and Returns This table presents estimates and t-statistics from the following vector autoregression model estimated at one-minute intervals: [mathematical expression not reproducible] where [r.sub.i,t] = return tor stock i at interval t; [PATOI.sub.i,t] = number of buy marketable and market orders minus number of sell marketable and market orders from proprietary algorithmic trading (PAT) for stock i at interval t; [AATOI.sub.i,t] = number of buy marketable and market orders minus the number of sell marketable and market orders from agency algorithmic trading (AAT) for stock i at interval t; and [NATOI.sub.i,t] = number of buy marketable and market orders minus the number of sell marketable and market orders from nonalgorithmic trading (NAT) for stock i at interval t. All variables in the regression are standardized. In addition to the individual coefficients, the sum of the coefficients over 10 lags of one variable is provided for ease of interpretation. All estimates are multiplied by 100 for representation purposes. Dependent Variable: [r.sub.it] Coeff. Est. t-Stat. Est. [[alpha].sub.1] 0.00 -2.85 [[beta].sub.11] -6.68 -80.17 [[gamma].sub.1.sup.A] 0.76 [[beta].sub.12] -1.30 -15.23 [[gamma].sub.2.sup.A] 0.42 [[beta].sub.13] -0.34 -3.97 [[gamma].sub.3.sup.A] 0.21 [[beta].sub.14] -0.42 -4.93 [[gamma].sub.4.sup.A] 0.05 [[beta].sub.15] -0.26 -3.11 [[gamma].sub.5.sup.A] 0.04 [[beta].sub.16] 0.02 0.24 [[gamma].sub.6.sup.A] -0.03 [[beta].sub.17] -0.30 -3.50 [[gamma].sub.7.sup.A] -0.15 [[beta].sub.18] -0.15 -1.81 [[gamma].sub.8.sup.A] -0.01 [[beta].sub.19] -0.19 -2.26 [[gamma].sub.9.sup.A] -0.08 [[beta].sub.110] 0.10 1.18 [[gamma].sub.10.sup.A] -0.19 [10.summation over 1] -9.52 -29.45 [10.summation over 1] 1.03 [[beta].sub.ij] [[gamma].sup.A.sub.j] [[gamma].sub.1.sup.P] 0.99 8.51 [[gamma].sub.1.sup.N] -0.21 [[gamma].sub.2.sup.P] -0.26 -2.15 [[gamma].sub.2.sup.N] 0.02 [[GAMMA].sub.3.sup.P] -0.35 -2.90 [[gamma].sub.3.sup.N] -0.15 [[gamma].sub.4.sup.P] -0.58 -4.82 [[gamma].sub.4.sup.N] 0.01 [[gamma].sub.5.sup.P] 0.13 1.04 [[gamma].sub.5.sup.N] -0.05 [[gamma].sub.6.sup.P] -0.69 -5.64 [[gamma].sub.6.sup.N] -0.11 [[gamma].sub.7.sup.P] 0.08 0.69 [[gamma].sub.7.sup.N] -0.13 [[gamma].sub.8.sup.P] -0.14 -1.14 [[gamma].sub.8.sup.N] 0.02 [[gamma].sub.9.sup.P] 0.06 0.47 [[gamma].sub.9.sup.N] 0.03 [[gamma].sub.10.sup.P] 0.16 1.32 [[gamma].sub.10.sup.N] 0.04 [10.summation over 1] -0.60 -2.66 [10.summation over 1] -0.55 [[gamma].sup.P.sub.j] [[gamma].sup.N.sub.j] Dependent Variable: [PATOI.sub.it] Coeff. t-Stat. Coeff. Est. [[alpha].sub.1] [[alpha].sub.2] -0.91 [[beta].sub.11] 16.35 [[beta].sub.21] -1.26 [[beta].sub.12] 8.37 [[beta].sub.22] -0.20 [[beta].sub.13] 4.22 [[beta].sub.23] -0.13 [[beta].sub.14] 0.97 [[beta].sub.24] -0.16 [[beta].sub.15] 0.84 [[beta].sub.25] -0.19 [[beta].sub.16] -0.58 [[beta].sub.26] -0.11 [[beta].sub.17] -2.94 [[beta].sub.27] -0.15 [[beta].sub.18] -0.11 [[beta].sub.28] -0.13 [[beta].sub.19] -1.51 [[beta].sub.29] -0.13 [[beta].sub.110] -4.05 [[beta].sub.210] -0.14 [10.summation over 1] 21.97 [10.summation over 1] -2.60 [[beta].sub.ij] [[beta].sub.2j] [[gamma].sub.1.sup.P] -4.16 [[lambda].sub.1.sup.P] 25.96 [[gamma].sub.2.sup.P] 0.29 [[lambda].sub.2.sup.P] 10.12 [[GAMMA].sub.3.sup.P] -2.93 [[lambda].sub.3.sup.P] 6.12 [[gamma].sub.4.sup.P] 0.19 [[lambda].sub.4.sup.P] 4.88 [[gamma].sub.5.sup.P] -1.04 [[lambda].sub.5.sup.P] 3.44 [[gamma].sub.6.sup.P] -2.11 [[lambda].sub.6.sup.P] 2.92 [[gamma].sub.7.sup.P] -2.56 [[lambda].sub.7.sup.P] 2.47 [[gamma].sub.8.sup.P] 0.33 [[lambda].sub.8.sup.P] 1.90 [[gamma].sub.9.sup.P] 0.51 [[lambda].sub.9.sup.P] 2.67 [[gamma].sub.10.sup.P] 0.91 [[lambda].sub.10.sup.P] 2.57 [10.summation over 1] -8.92 [10.summation over 1] 63.05 [[gamma].sup.P.sub.j] [[lambda].sup.P.sub.j] Dependent Variable: [PATOI.sub.it] Coeff. t-Stat. Est. t-Stat. [[alpha].sub.1] -3.29 [[beta].sub.11] -54.93 [[lambda].sub.1.sup.A] -0.14 -5.57 [[beta].sub.12] -8.85 [[lambda].sub.2.sup.A] -0.13 -4.93 [[beta].sub.13] -5.59 [[lambda].sub.3.sup.A] -0.06 -2.39 [[beta].sub.14] -6.81 [[lambda].sub.4.sup.A] -0.04 -1.52 [[beta].sub.15] -8.04 [[lambda].sub.5.sup.A] 0.03 1.24 [[beta].sub.16] -4.62 [[lambda].sub.6.sup.A] -0.03 -0.98 [[beta].sub.17] -6.61 [[lambda].sub.7.sup.A] 0.01 0.50 [[beta].sub.18] -5.83 [[lambda].sub.8.sup.A] 0.01 0.50 [[beta].sub.19] -5.85 [[lambda].sub.9.sup.A] 0.06 2.18 [[beta].sub.110] -6.15 [[lambda].sub.10.sup.A] 0.01 0.23 [10.summation over 1] -33.48 [10.summation over 1] -0.28 12.51 [[beta].sub.ij] [[lambda].sup.P.sub.j] [[gamma].sub.1.sup.P] 384.92 [[lambda].sub.1.sup.N] 0.94 40.36 [[gamma].sub.2.sup.P] 144.75 [[lambda].sub.2.sup.N] 0.10 3.99 [[GAMMA].sub.3.sup.P] 86.96 [[lambda].sub.3.sup.N] 0.03 1.03 [[gamma].sub.4.sup.P] 68.95 [[lambda].sub.4.sup.N] -0.05 -2.05 [[gamma].sub.5.sup.P] 48.43 [[lambda].sub.5.sup.N] -0.01 -0.57 [[gamma].sub.6.sup.P] 40.99 [[lambda].sub.6.sup.N] -0.09 -3.58 [[gamma].sub.7.sup.P] 34.62 [[lambda].sub.7.sup.N] 0.00 0.11 [[gamma].sub.8.sup.P] 26.73 [[lambda].sub.8.sup.N] -0.14 -5.35 [[gamma].sub.9.sup.P] 37.60 [[lambda].sub.9.sup.N] -0.14 -5.64 [[gamma].sub.10.sup.P] 37.16 [[lambda].sub.10.sup.N] -0.05 -1.95 [10.summation over 1] 511.04 [10.summation over 1] 0.59 22.36 [[gamma].sup.P.sub.j] [[lambda].sup.N.sub.j] Dependent Variable: [AATOI.sub.it] Coeff. Est. t-Stat. Est. [[alpha].sub.3] 1.07 1.56 [[beta].sub.31] 2.01 40.21 [v.sub.1.sup.A] 40.73 [[beta].sub.32] 0.14 2.84 [v.sub.2.sup.A] 11.82 [[beta].sub.33] -0.07 -1.44 [v.sub.3.sup.A] 8.66 [[beta].sub.34] -0.18 -3.61 [v.sub.4.sup.A] 6.11 [[beta].sub.35] -0.16 -3.25 [v.sub.5.sup.A] 3.12 [[beta].sub.36] -0.14 -2.76 [v.sub.6.sup.A] 2.99 [[beta].sub.37] -0.19 -3.77 [v.sub.7.sup.A] 3.27 [[beta].sub.38] -0.20 -3.99 [v.sub.8.sup.A] 2.97 [[beta].sub.39] -0.24 -4.83 [v.sub.9.sup.A] 3.09 [[beta].sub.310] -0.18 -3.72 [v.sub.10.sup.A] 5.07 [10.summation over 1] 0.79 4.46 [10.summation over 1] 87.83 [[beta].sub.3j] [v.sup.A.sub.j] [v.sub.1.sup.P] -1.48 10.97 [v.sub.1.sup.N] -0.47 [v.sub.2.sup.P] -0.20 -1.41 [v.sub.2.sup.N] -0.19 [v.sub.3.sup.P] -0.10 -0.74 [v.sub.3.sup.N] -0.04 [v.sub.4.sup.P] -0.19 -1.37 [v.sub.4.sup.N] 0.01 [v.sub.5.sup.P] 0.23 1.64 [v.sub.5.sup.N] 0.00 [v.sub.6.sup.P] 0.24 1.68 [v.sub.6.sup.N] 0.04 [v.sub.7.sup.P] 0.32 2.27 [v.sub.7.sup.N] 0.03 [v.sub.8.sup.P] 0.11 0.78 [v.sub.8.sup.N] -0.14 [v.sub.9.sup.P] 0.09 0.65 [v.sub.9.sup.N] -0.12 [v.sub.10.sup.P] 0.03 0.24 [v.sub.10.sup.N] -0.01 [10.summation over 1] -0.94 -3.89 [10.summation over 1] -0.88 [v.sup.P.sub.j] [v.sup.N.sub.j] Dependent Variable: [NATO.sub.it] Coeff. t-Stat. Coeff. Est. t-Stat. [[alpha].sub.3] [[alpha].sub.4] -8.11 -10.60 [[beta].sub.31] 664.80 [[beta].sub.41] -10.28 -154.07 [[beta].sub.32] 178.45 [[beta].sub.42] -6.52 -96.83 [[beta].sub.33] 129.96 [[beta].sub.43] -4.63 -68.68 [[beta].sub.34] 91.46 [[beta].sub.44] -3.29 -48.93 [[beta].sub.35] 46.55 [[beta].sub.45] -2.73 -40.68 [[beta].sub.36] 44.68 [[beta].sub.46] -2.05 -30.53 [[beta].sub.37] 48.87 [[beta].sub.47] -1.68 -25.13 [[beta].sub.38] 44.34 [[beta].sub.48] -1.04 -15.60 [[beta].sub.39] 46.41 [[beta].sub.49] -0.75 -11.34 [[beta].sub.310] 81.95 [[beta].sub.410] -0.08 -1.20 [10.summation over 1] 1,648.04 [10.summation over 1] -33.05 -141.92 [[beta].sub.3j] [[beta].sub.4j] [v.sub.1.sup.P] -10.58 [[eta].sub.1.sup.P] 3.60 22.05 [v.sub.2.sup.P] -3.91 [[eta].sub.2.sup.P] 0.74 4.36 [v.sub.3.sup.P] -0.73 [[eta].sub.3.sup.P] -0.05 -0.31 [v.sub.4.sup.P] 0.27 [[eta].sub.4.sup.P] -0.60 -3.47 [v.sub.5.sup.P] 0.01 [[eta].sub.5.sup.P] -0.72 -4.15 [v.sub.6.sup.P] 0.76 [[eta].sub.6.sup.P] 0.32 1.83 [v.sub.7.sup.P] 0.55 [[eta].sub.7.sup.P]" -0.58 -3.35 [v.sub.8.sup.P] -2.81 [[eta].sub.8.sup.P] -1.43 -8.24 [v.sub.9.sup.P] -2.58 [[eta].sub.9.sup.P] -0.10 -0.56 [v.sub.10.sup.P] -0.13 [[eta].sub.10.sup.P] -1.03 -6.16 [10.summation over 1] -16.57 [10.summation over 1] 0.14 0.52 [v.sup.P.sub.j] [[eta].sup.P.sub.j] Dependent Variable: [NATO.sub.it] Coeff. Est. t-Stat. [[alpha].sub.3] [[beta].sub.31] [[eta].sub.1.sup.A] -0.45 -7.88 [[beta].sub.32] [[eta].sub.2.sup.A] -0.58 -9.43 [[beta].sub.33] [[eta].sub.3.sup.A] -0.31 -5.06 [[beta].sub.34] [[eta].sub.4.sup.A] -0.13 -2.02 [[beta].sub.35] [[eta].sub.5.sup.A] 0.01 0.23 [[beta].sub.36] [[eta].sub.6.sup.A] -0.13 -2.06 [[beta].sub.37] [[eta].sub.7.sup.A] 0.15 2.34 [[beta].sub.38] [[eta].sub.8.sup.A]" 0.11 1.71 [[beta].sub.39] [[eta].sub.9.sup.A] -0.02 -0.35 [[beta].sub.310] [[eta].sub.10.sup.A] -0.09 -1.64 [10.summation over 1] [10.summation over 1] -1.45 -27.46 [[beta].sub.3j] [[eta].sup.A.sub.j] [v.sub.1.sup.P] [[eta].sub.1.sup.N] 41.50 597.29 [v.sub.2.sup.P] [[eta].sub.2.sup.N] 9.44 125.79 [v.sub.3.sup.P] [[eta].sub.3.sup.N] 6.73 89.28 [v.sub.4.sup.P] [[eta].sub.4.sup.N]" 4.25 56.30 [v.sub.5.sup.P] [[eta].sub.5.sup.N] 3.37 44.64 [v.sub.6.sup.P] [[eta].sub.6.sup.N] 3.07 40.73 [v.sub.7.sup.P] [[eta].sub.7.sup.N] 2.62 34.76 [v.sub.8.sup.P] [[eta].sub.8.sup.N] 2.81 37.46 [v.sub.9.sup.P] [[eta].sub.9.sup.N] 2.65 35.49 [v.sub.10.sup.P] [[eta].sub.10.sup.N] 4.02 58.70 [10.summation over 1] [10.summation over 1] 80.45 1,050.42 [v.sup.P.sub.j] [[eta].sup.N.sub.j] Table X. Change in net Realized Liquidity after Extreme Price Movements This table shows the cross-sectional average (median in parentheses) net realized liquidity comparing extreme (second column) and nonextreme (third column) returns in our sample of Nifty 50 index component stocks. The returns are computed for one-minute intervals. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. The average t-statistics of the difference in means of the two situations are provided along with number of stocks having positive significant and negative significant coefficients at the 10% level (in parentheses in last column). Panels A and B characterize two definitions of extreme returns. For Panel A, the net realized liquidity for a trading group is the number of shares traded as limit buy orders minus the number of shares traded as liquidity-demanding sell orders. For Panel B, the net realized liquidity for a trading group is the number of shares traded as limit sell orders minus the number of shares traded as liquidity-demanding buy orders. Panel A. Extreme Return Definition: Return [less than or equal to] -0.5% Class of Return Return > -0.5% Average t-Stat. Traders [less than or equal to] of Difference -0.5% of Means PAT 351.82 (222.91) -280.00 (-157.76) 1.72 (25,3) AAT -1,327.55 (-779.34) 70.34 (32.44) -1.83 (1,26) NAT 975.81 (183.19) 209.80 (50.37) 0.51 (12,5) Panel B. Extreme Return Definition: Return [greater than or equal to] 0.5% Class of Return Traders [greater than or equal to] 0.5% PAT 545.21 (318.58) AAT -2,323.42 (-1,188.00) NAT 1,778.22 (495.71) Class of Return Traders [less than or equal to] 0.5% PAT -288.89 (-162.31) AAT -36.10 (-7.56) NAT 325.04(171.63) Panel B. Extreme Return Definition: Return [greater than or equal to] 0.5% Class of Average t-Stat. of Traders Difference of Means PAT 1.89 (28,5) AAT -2.51 (1,36) NAT 1.02 (21,8) Table XI. Change in Liquidity Supply (Top 5 Quotes) in Reaction to Change in Volatility (Absolute Return) This table presents the panel regression estimates with t-statistics computed using panel-corrected standard errors (Beck and Katz, 1995) for the following model: [mathematical expression not reproducible] where [DEPTH.sub.i,t] = sum of the size of the orders due to proprietary algorithmic trading (PAT), high-frequency trading (HFT), agency algorithmic trading (AAT), or nonalgorithmic trading (NAT), respectively, at the bid and ask quotes for stock i at the end of time t considering the top five quotes; [RISK.sub.i,t] = absolute return for stock i at time t; [MKT_RISK.sub.i,t] = sum of RISK variables for the other 49 stocks in the sample of 50 stocks; [NTRADE.sub.i,t] = number of shares that were traded for stock i at time t; Time dummies = 13 indicator variables each representing one interval of half an hour. We include 12 indicator variables here to avoid multicollinearity. All variables in the regression arc standardized. The sample includes 12 months of data covering the calendar year 2013 and comprises the Nifty 50 index component stocks. Only the estimates of each variable up to 3 lags are provided; full model estimates comprising all 12 lags of each variable are available in Appendix D. All estimates are multiplied by 100 for representation purposes. PAT HFT AAT Coeff. Estimate t-Stat. Estimate t-Stat. Estimate [[beta].sub.1] 0.90 19.64 1.11 23.93 -0.73 [[beta].sub.2] 0.25 5.47 0.42 9.12 -0.42 [[beta].sub.3] 0.02 0.41 0.11 2.37 -0.25 [[beta].sup.M.sub.1] 0.55 7.55 0.74 9.97 -0.06 [[beta].sup.M.sub.2] 0.39 5.14 0.43 5.76 0.06 [[beta].sup.M.sub.3] -0.09 -1.18 0.05 0.60 0.32 [[theta].sub.1] 0.54 11.38 0.47 9.75 0.02 [[theta].sub.2] 0.26 5.46 0.19 3.90 0.06 [[theta].sub.3] 0.09 1.92 0.09 1.84 0.30 [[rho].sup.PAT.sub.1] 24.70 477.24 24.19 466.70 -0.88 [[rho].sup.PAT.sub.2] 12.69 238.17 12.17 228.34 -0.13 [[rho].sup.PAT.sub.3] 8.55 159.23 8.18 152.30 0.11 [[rho].sup.AAT.sub.1] -0.19 -3.53 -0.10 -1.86 35.42 [[rho].sup.AAT.sub.2] -0.19 -3.30 -0.22 -3.77 14.37 [[rho].sup.AAT.sub.3] -0.21 -3.54 -0.21 -3.42 7.62 [[rho].sup.NAT.sub.1] -0.56 -10.67 -0.48 -8.93 0.48 [[rho].sup.NAT.sub.2] 0.03 0.49 0.00 0.01 -0.12 [[rho].sup.NAT.sub.3] 0.17 2.88 0.15 2.57 0.04 AAT NAT Coeff. t-Stat. Estimate t-Stat. [[beta].sub.1] -17.25 -1.20 -26.47 [[beta].sub.2] -9.81 -0.76 -16.75 [[beta].sub.3] -5.89 -0.29 -6.46 [[beta].sup.M.sub.1] -0.91 -0.08 -1.13 [[beta].sup.M.sub.2] 0.77 -0.31 -4.20 [[beta].sup.M.sub.3] 4.29 -0.45 -6.13 [[theta].sub.1] 0.35 -0.04 -0.94 [[theta].sub.2] 1.38 0.30 6.25 [[theta].sub.3] 6.68 0.20 4.21 [[rho].sup.PAT.sub.1] -18.89 -0.12 -2.41 [[rho].sup.PAT.sub.2] -2.65 0.27 5.17 [[rho].sup.PAT.sub.3] 2.22 0.27 5.25 [[rho].sup.AAT.sub.1] 680.34 0.14 2.55 [[rho].sup.AAT.sub.2] 256.35 0.21 3.73 [[rho].sup.AAT.sub.3] 132.86 0.36 6.07 [[rho].sup.NAT.sub.1] 10.06 37.48 716.38 [[rho].sup.NAT.sub.2] -2.25 12.97 229.92 [[rho].sup.NAT.sub.3] 0.84 6.84 119.64 Table XII. Change in Actual Depth after Extreme Price Movements This table shows depth numbers in top three quotes comparing extreme (second column) and nonextreme (third column) returns in our sample of Nifty 50 index component stocks. The returns are computed for one-minute intervals. PAT stands for proprietary algorithmic trading, AAT stands for agency algorithmic trading, and NAT stands for nonalgorithmic trading. The average t-statistics of the difference in means of the two situations are provided, along with number of stocks having positive significant and negative significant coefficients at the 10% level (in parentheses in the last column). Panels A and B characterize two definitions of extreme returns. Panel A. Extreme Return Definition: Return [less than or equal to] -0.25% Variable Return Return > -0.25% [less than or equal to] -0.25% PAT buy depth number 178.42 142.79 AAT buy depth number 92.89 115.01 NAT buy depth number 356.55 359.52 Variable Average t-Stat. of Difference of Means PAT buy depth number 2.61 (34,6) AAT buy depth number -1.52 (6,34) NAT buy depth number 0.11 (8,12) Panel B. Extreme Return Definition: Return [greater than or equal to] 0.25% Variable Return Return [greater than or equal to] [less than or equal to] 0.25% 0.25% PAT sell 224.70 150.51 depth number AAT sell 91.58 103.77 depth number NAT sell 377.76 384.84 depth number Variable Average t-Stat. of Difference of Means PAT sell 4.23 depth number (38,1) AAT sell -1.36 depth number (7,26) NAT sell 1.90 depth number (21,8)

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Author: | Nawn, Samarpan; Banerjee, Ashok |
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Publication: | Financial Management |

Geographic Code: | 9INDI |

Date: | Jun 22, 2019 |

Words: | 24215 |

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