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RESEARCH includes research articles that focus on the analysis and resolution of managerial and academic issues based on analytical and empirical or case research Executive Summary Execution Times of Small Limit Orders: A Simpler Modeling Approach Devlina Chatterjee and Chiranjit Mukhopadhyay In an electronic stock market, an equity trader can submit two kinds of orders: a market order or a limit order. In a market order, the trade occurs at the best available price on the opposite side of the book. In a limit order, on the other hand, the trader specifies a price (lower limit in case of sell orders and higher limit in case of buy orders) beyond which they are not willing to transact. Limit orders supply liquidity to the market and aid in price discovery since they indicate the prices that traders are willing to pay at any point of time. One of the risks that a trader placing a limit order faces is the risk of delayed execution or non-execution. If the execution is delayed, then the trader also faces a “picking-off” risk, in the event of the arrival of new information. With these issues in the background, a trader placing a limit order at a certain price, given various economic variables such as recent price movements as well as characteristics of the company in question, is interested in the probability of execution of the order as a function of subsequent elapsed time. For example, if she places a small sell order at 0.5 percent above the last traded price for a given stock, what is the probability that the order will be executed in the next t minutes? With this motivation, this paper considers execution times of small limit orders in an electronic exchange, specifically the National Stock Exchange (NSE) of India. Order execution times have been studied in several other works, where they are modeled by reconstructing the history of the order book using high-frequency data. Here, for the first time, the much simpler approach of small hypothetical orders placed at certain prices at certain points of time has been used. Given that an order has been placed at a certain price, subsequent price movements determine the lower and upper bounds of the time to execution based on when (and if) the order price is first reached and when it is first crossed. Survival analysis with interval censoring is used to model the execution probability of an order as a function of time. Several Accelerated Failure Time models are built with historical trades and order book data for 50 stocks over 63 trading days. KEY WORDS Electronic Stock Exchange Market Microstructure Hypothetical Order Survival Analysis Interval Censoring Additionally, choice of distributions, relative importance of covariates, and model reduction are discussed; and results qualitatively consistent with studies that did not use hypothetical orders are obtained. Interestingly, for the data, the differences between the above-mentioned bounds are not very large. Directly using them without interval censoring gives survival curves that bracket the correct curve obtained with interval censoring. The paper concludes that this approach, though data- and computation-wise much less intensive than traditional approaches, nevertheless yields useful insights on execution probabilities of small limit orders in electronic exchanges. VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 49 E lectronic stock exchanges worldwide have brought online share trading to retail investors. Unlike large institutional investors who can affect the market, the retail investor trading a few shares at a time is typically interested in a single, one-way interaction with the market. Nevertheless, he or she still must choose between limit and market orders, weighing the risk of non-execution against a sub-optimal price1. With this motivation, this paper makes a small order assumption and performs survival analysis using data from the National Stock Exchange of India (December 2007-February 2008), to find the probability of execution of a given order as a function of time. Execution times of limit orders in stock exchanges have traditionally been modeled by looking at actual histories of real orders. Such histories follow each individual order from submission, through modification or partial execution, ending with full execution, cancellation or expiry as the case may be (Cho & Nelling, 2000; AlSuhaibani & Kryzanowski, 2000; Omura, Tanigawa & Uno, 2000; Lo, MacKinlay & Zhang, 2002; Gava, 2005; Wen, 2008). Such approaches require reconstruction of the history of the entire order book using high frequency data, and are computationally highly demanding. A much simpler approach, admittedly with some restrictions, is developed in this paper using the idea of hypothetical orders (proposed in a different context by Handa & Schwartz, 1996). The present approach assumes that snapshots of the limit order book are available at certain discrete points of time (e.g., the National Stock Exchange of India provides such snapshots four times a day). These snapshots provide the price points and quantities of all orders in the book extant at that time. The hypothetical order considered in this paper is placed at one of those price points P, at the instant of the snapshot. Subsequently, if the actual stock price p(t) reaches P after a time duration t1, and crosses P after a time duration t2 > t1, then the execution time lies between t1 and t2. As there is no priority information about the order, the execution time can be bound only between t1 and t2 . Such data can be handled in survival analyses using interval censoring. Sometimes the price p(t) does not dwell at P for any length of time, and t2 = t1. In such cases, the event time is exactly t1 and is not censored. Finally, some price points P are 1 Order submission strategies have been studied by several authors (Verhoeven, Ching & Ng, 2004; Ranaldo, 2004). 50 never crossed within the window of observation. In such cases, the data point is right censored. These issues will be discussed in further detail below. In an early study of limit order executions, Lo, MacKinlay and Zhang (2002) observed that the mean values of t1 and t2 could differ substantially (they considered executed orders; so these were conditional mean values). They concluded that t1 and t2 were poor proxies for actual execution times. Perhaps as a result, subsequent authors have not modeled execution times using hypothetical orders. A demonstration of such use with interval censoring, along with a discussion of the results obtained, is the primary contribution of this paper. The paper notes the main restriction in using hypothetical orders that is, the orders must be small. A large order can, in principle, affect price movement. Another restriction is that the effect of modifications or cancellations or partial execution on the overall order execution time cannot be monitored. These restrictions are offset by the resulting simplification in data requirements and modeling. For small traders interested in a one-way interaction with the market, these restrictions are not serious. LITERATURE REVIEW For a broad overview of work in the area of limit order markets, see Parlour and Seppi (2008) and references therein. The focus of this paper is on directly relevant empirical studies of order execution times. The results are discussed in light of the findings of the papers cited below. Omura, Tanigawa and Uno (2000) estimated a probit model with data for December 1998 from the Tokyo Stock Exchange to analyse the probability of execution of limit orders within one day. They reconstructed the history of each order from the order book, an approach also adopted by other researchers in this area. Cho and Nelling (2000) analysed data from 10 NYSE companies, from November 1990 through January 1991 by using survival analysis and estimating an accelerated failure time model with the Weibull distribution for survival times. Al-Suhaibani and Kryzanowski (2000) analysed limit order data from the relatively illiquid Saudi Stock Market from October 1996 to January 1997, using survival analysis (again an AFT Weibull model) to study execution times. EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH In an influential work that was eventually published somewhat later, Lo, MacKinlay and Zhang (2002) used the generalized gamma distribution (which includes the Weibull as a special case) to study order survival times in the NYSE. They built a pooled model for 100 stocks and individual models for 16 stocks, using 13 months’ data from 1994-95. In particular, they reported wide mismatches in mean times for hypothetical order completions, depending on whether one used the lower bound t1 or the upper bound t2, as discussed above. More recently, Gava (2005) analysed data from the Spanish Stock Exchange for 34 companies, reconstructed the history of executed orders, and reported somewhat similar findings as Lo, MacKinlay and Zhang (2002) and Cho and Nelling (2000). Wen (2008) analysed data from the London Stock Exchange for 38 stocks, and also tracked individual actual orders. The generalized gamma model was used in both studies. To summarize, no published studies have used hypothetical orders within survival analysis models to study order execution probabilities (with or without interval censoring). In this sense, the main contribution of this paper is the demonstration of a substantially simpler approach to modeling order execution times that may be used, in future work, to probe other important questions. SURVIVAL ANALYSIS Survival analysis is the statistical approach to modeling the probability of some event occurring as a function of elapsed time. The theoretical background essential for the paper is outlined below, mostly following Klein and Moeschberger (2003). In survival analysis, the time T at which an event (say, the death of a patient) occurs is assumed to be a realization of some random process. So, T, the time for an event to occur is a random variable having a certain probability distribution. Different methods are used to model survival data depending on the kind of distribution that the survival time T follows. Survival models can handle data which may be right censored (event is not observed within some specified time), left censored (event has already occurred before some specified time), or interval censored (event has occurred within a specified interval of time). The present analysis includes events occurring at specific times as well as right censored and interval censored data. There is no left censored data. VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 The survival function represents the unconditional probability of surviving longer than t time units, and is written as: S(t) = Pr(T > t) = 1 – F(t) (1) where F(t), is the cumulative distribution function for the random variable time to failure. S(t) lies between 0 and 1, and is given by the integral of the pdf of failure times, S(t) = Pr(T > t) = ∞ ∫t f(u)du (2) with S(0) = 1. It follows that the probability of failure occurring between times t1 and t2 is S(t1) – S(t2). A related idea is that of the hazard function, which is the instantaneous probability density of failure given survival up to time t, given by h(t) = lim P(t < T < t + dt|T> t) = f(t) =— d logS(t) dt→0 dt S(t) dt (3) Four commonly used survival functions are given in Table 1. In the Table, λ, α, μ, and σ are parameters within the respective models. Of these, the log-logistic will turn out to be most suitable for our data, and so, some plots depicting S(t), f(t), and h(t) for the log-logistic case are given in Appendix A. Having understood survival functions and related quantities, the authors next consider modeling of data where individuals differ. For example, the expected remaining life of an individual may depend on the present age of that individual; the present age would be a covariate in the survival model. When the data consists of observations of survival times as well as additional continuous covariates, then such covariates must be included in the survival model. The two main approaches for doing this are: • Cox Proportional Hazards model, which is semi-parametric • Accelerated Failure Time (AFT) models, which are fully parametric. Consistent with prior studies of order execution times, this paper presents results from only parametric AFT models. The Cox proportional hazards model was investigated elsewhere, but dropped due to violation of the proportionality assumption (Chatterjee, 2010). 51 Table 1: Different Choices of S(t) and Corresponding Hazard Functions Distribution Shape of Hazard Function Survival Function 1 Exponential Constant hazard rate S(t) = exp(–λt) 2 Weibull Hazard rate is either monotone increasing, decreasing or constant with time S(t) = exp(–λtα) 3 Log-Logistic Hazard rate is either hump-shaped or always decreases S(t) = 4 Log-Normal Hazard is hump-shaped, rises to a peak before decreasing to 0 as t approaches infinity S(t) = 1 – Φ 1 1 + λtα log(t) – μ σ with Φ the cumulative distribution of a standard normal variable For parametric AFT models with covariates, it is assumed that: β’Ζ)t) S(t|Z) = S0(exp(β (4) where Z = vector of fixed-time explanatory covariates, β’= vector of regression coefficients, and S0(t) is an appropriate survival function depending on the underlying distribution of the survival times as shown in Table β’Z) is called an acceleration factor 1. The factor exp(β which stretches or accelerates the baseline time scale. Parameters β along with any parameters implicit in S0(.) can be simultaneously estimated using the method of maximum likelihood. The probability density function of failure time T is given by f(t|Z) = —S’0 (exp(β’Z)t) exp(β’Z) L(β,α α) = i D+i|ZD+i) i (6) i=1 where α represents the parameters appearing in S0(.). For example, α =(λ) for the exponential case, α =(λ)’ for the Weibull and log-logistic cases, and α = (μ, σ)’ for the log-normal case. Maximum likelihood estimation of the parameters β and α is implemented numerically in many statistical software packages. 52 a) Summarized daily information for each security including opening, closing, high, low, and last traded prices; total number of shares traded; total value of shares traded; and the total number of trades. b) Four snapshots daily of the Limit Order Book, taken at 11 am, 12 noon, 1 and 2 pm that record the kind of order (buy or sell), quantity, price, and time of order placement. c) Detailed daily information on every trade of every security, indicating time, price, and quantity. This dataset does not have information on whether the trade was initiated by a buyer or a seller. The sample used in the study included all such data for 50 stocks over 63 trading days from December 4, 2007 to February 29, 2008. Randomly selected subsets of this data (5%, 7%, and then 10%) were used to build two survival models across all 50 stocks, for buy and sell orders, with interval censoring using t1 and t2 as explained above. Response Variables in the AFT Model n-D Π f(t |Z ) Π S(t i=1 For the study, the following data from the NSE were used: (5) Let the time of death of D individuals be denoted by t1 ≤ t2 ≤ ...<tD with associated covariate vectors Z1, ..., ZD respectively. Additionally, suppose there are observations on n—D individuals whose death times are right censored at tD + 1, ..., tn with covariates denoted by ZD+1, ..., Zn. For such data, the likelihood function is D DATA The paper considers the execution of hypothetical small orders that are assumed to be executed if and when the transaction price reaches and crosses the order price. The observation window is one hour for each data point. At the beginning of the hour, all non-executed orders, then extant in the limit order book, were taken as data points. There was no information regarding the relative priorities of these orders. Each distinct limit price (or price point) reflected in the order book constituted a separate EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH observation. To illustrate, price points at an arbitrarily chosen snapshot of a typical stock are given in Box 1. These price points are real; and in the model building, a hypothetical order was assumed at each of these distinct price points. Box 1: Bank of Baroda - Buy Order Prices Recorded at 11 am snapshot of the LOB 386.35, 386.3, 386.25, 386.2, 386, 385.4, 385.05, 385, 384.15, 384, 383, 382.2, 382.1, 382, 381.5, 381.2, 381.1, 381, 380.5, 380.4, 380.25, 380.1, 380.05, 380, 379, 378, 377, 376.5, 376, 375.1, 375.05, 375, 374, 373, 372, 371.6, 371, 370.5, 370.1, 370, 369.7, 369, 368, 367, 366, 365, 364, 363, 362.1, 362, 361.15, 361, 360, 359.95, 359, 357, 355, 353, 351.65, 351, 350.35, 350, 349, 347, 346.65, 345, 335.65, 335.25, 335, 326.65, 325.15,325.1, 325, 320, 316, 312.65 shares in the order book) 5. Volatility (standard deviation of trade-to-trade price changes in the last hour) 6. Relative activity (ratio of daily traded volume to total value of outstanding shares, normalized using a large fixed number) 7. Previous day’s closing price (different for each stock; not normalized) 8. Firm size (market capitalization) 9. Time of day (the snapshot hour) 10. Industry sector the firm belonged to (included as a factor), from among • auto • energy • manufacturing • consumer goods The stock price movement was then monitored over the next 60 minutes. For each such limit price, four possibilities are listed in Table 2. The actual realized values for t1 and t2 are denoted as T1 and T2. It is clear that for each such “hypothetical” data point, the real values of t1 and t2 as determined from price movements, are the response variables in the model. Table 2: Four Possibilities for Events or Censoring using the Hypothetical Order When Price is First Reached When Price is First Crossed Classification 1 T2 = T1 mins Event Interval censored T1 mins 2 T1 mins T2 > T1 mins 3 T1 mins NA (not crossed in 60 mins) Right censored at T1 4 NA (not reached in 60 mins) NA (not crossed in 60 mins) Right censored at 60 Covariates • financial services • information technology or biotech • pharmaceutical Of these ten covariates, logarithms of the first eight were used because of large variations in magnitudes. The time of day (11 am, 12 noon, 1 pm, and 2 pm) was coded using the number of hours elapsed from 10 am (i.e., 1, 2, 3, and 4 respectively). Finally, the industry sector was directly incorporated as a factor (alternatively, it could have been coded using six binary dummy variables). The choice of covariates was guided by a detailed review of the related empirical literature mentioned earlier. It may be noted that the first covariate is a choice made by the trader; covariates 2 through 6 are stock-specific, but affected by short-term market dynamics; covariates 7, 8, and 10 are stock-specific variables which are less affected by short-term market dynamics; and covariate 9 represents both market conditions as well as a choice made by the trader. The following ten covariates were included: 1. Order price, taken as a normalized price premium above the existing mid-quote (mean of best-bid and best-ask) 2. Bid-ask spread normalized by the mid-quote 3. Depth (volume of orders on the same side of the order book at the same or more competitive prices) 4. Order imbalance (the number of shares on the same side of the order book, divided by the total number of VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 The first step was to build a composite dataset using data from all 50 companies in the study. For this purpose, 5 percent of the available data for each company (separately for buy and sell orders) was randomly selected and collated to create a combined data subset. Two more such datasets (one with a fresh 7% sample and one with a fresh 10% sample from each company) were created to check the consistency between models built for different data subsets. 53 DESCRIPTIVE STATISTICS Independent Covariates Response Variable As mentioned earlier, the response variable is the time for execution of a small hypothetical order. Based on the time t1 when the order price is first reached and the time t2 when the order price is first crossed, the response may be an event, or interval censored, or right censored. Lo, MacKinlay and Zhang (2002) had reported large differences in the mean values of t1 and t2 in their dataset. The summary statistics for t1 and t2 in the dataset are presented in Table 3. From the Table, it can be observed that a small percent of the data is interval censored and, for such data, t2 and t1 differ by about 15-18 percent. Lo, MacKinlay and Zhang (2002) reported the mean values of t1 and t2 for the data used in their study (their Table 5). The percentage difference between t1 and t2 in their data was found to be on average about four times the difference noted in the dataset of the present study. There could be a couple of reasons for these large differences in the means of t1 and t2 noted in their data. One is the fact that the source of their data was an institutional brokerage firm (ITG) that typically handled orders from other institutional investors or brokers and dealers but not many retail orders (see section 3.1 of Lo, et al, 2002). Also, ITG often handled trades from other trading platforms that had not been filled in one shot and hence there was a sampling bias inherent in their data (with a greater percentage of more illiquid orders). The data for the present study, drawn from a completely electronic exchange, did not have such a priori bias. Summary statistics for the independent covariates for buy and sell orders, from the 10 percent data subset, are given in Table 4. The number of observations in the sell dataset is larger than in the buy dataset, reflecting the often observed asymmetry in the order book. The proportion of data points that are uncensored differ for the two sides as well, possibly because many sell orders are optimistic and do not get executed within the window of observation. Interestingly, a substantial proportion of executed orders (nonright censored data points) actually require interval censoring, bearing out the necessity of the present approach. For example, for the sell side, 90.35 percent of the data points are right censored. But 3.37 percent of the total data points, or about 35 percent of the non-right censored data points, need interval censoring. The average values of the covariates (from buy and sell sides) are close in most cases; the difference is significant for order imbalance and depth, because of the asymmetry in the order book as mentioned above. BUILDING THE AFT MODEL FOR TIME TO EXECUTION This section describes the steps followed to build the AFT model for order execution times. The most appropriate distribution for survival times was first chosen using the Akaike information criterion. Then models including all covariates were built, and variables that could be dropped to develop more compact (or parsimonious) models sought using backward elimination. Having chosen the variables to retain, models with the different data subsets (5%, 7%, and 10%) were built to check for consist- Table 3: Summary Statistics for t1 and t2 and for Buy and Sell Orders Mean t1 given t1 < 60 (mins) Standard Dev (t1) (as on left) Mean t2 given t2 < 60 (mins) Standard Dev (t2) (as on left) Percent cases* where t1< t2 < 60 Mean t2–t1 , t2 given t1< t2 < 60 Buy Dataset 1 (5%) 21.52 17.89 23.91 17.95 3.92% 17.16% Buy Dataset 2 (7%) 21.53 17.78 23.87 17.85 3.89% 17.20% Buy Dataset 3 (10%) 21.65 17.95 24.06 17.98 3.83% 17.08% Sell Dataset 1 (5%) 22.15 18.11 24.65 18.16 3.32% 15.84% Sell Dataset 2 (7%) 22.42 18.16 24.87 18.14 3.37% 15.00% Sell Dataset 3 (10%) 22.05 18.12 24.50 18.13 3.37% 15.42% ( ) * relative to all points in the dataset, censored and uncensored. Also see Table 4. 54 EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH Table 4: Summary Statistics of Independent Covariates for Buy and Sell Orders Buy Orders Sell Orders Average Std. dev. Average 1 Log Price Premium -3.509 1.189 -3.445 1.147 2 Log Volatility -7.599 0.686 -7620 0.671 3 Log Relative Activity -5.555 0.841 -5.544 0.842 4 Log Order Imbalance -0.784 0.350 -0.649 0.332 5 Log Depth 10.021 1.634 10.321 1.684 6 Log Bid-Ask Spread -6.476 0.743 -6.486 0.741 7 Log Firm Size 25.648 1.454 25.666 1.452 8 Log Volume Traded 11.501 1.601 11.542 1.582 9 Log Number of Trades 7.554 1.250 7.573 1.228 6.607 0.977 6.593 0.989 10 Log Closing Price 11 Number of Observations 12 Percentage of Observations that are uncensored events 8.39% 6.28% 13 Percentage of Observations that are right censored 87.78% 90.35% 14 Percentage of Observations that are interval censored 3.84% 3.37% 181543 ency in the coefficients. For those models, an analysis of deviance (as outlined below in Section 6.3) was then conducted to gauge the relative importance of each of the covariates in the model fit. The consistency between the results obtained for different datasets suggests that the model is robust. Selection of an Appropriate Distribution for the AFT Models First, the appropriate underlying distribution of survival times was considered for the data. In previous work, most authors had a priori assumed that the data fitted a Weibull distribution (Cho & Nelling, 2000; Al-Suhaibani & Kryzanowski, 2000) or a generalized gamma distribution (Lo, et al, 2002; Gava, 2005; Wen, 2008). Both of these distributions (the former is a special case of the latter) had survival probabilities that decreased exponentially with time. This study considered the exponential, Weibull, logistic, log-logistic, log-normal, and Gaussian Std. dev. 216019 distributions (described in Table 1). For each of these six, and for each dataset, separately for buy and sell sides, survival models (36 in all) were built and the Akaike information criterion (AIC) computed (Table 5). For all the six cases, the log-logistic distribution gives the lowest AIC (given in boldface), indicating that it fits the data best. Consequently, the log-logistic distribution was adopted for the study. Preliminary Models The results of one buy side model built with the 5 percent data subset are presented in Table 6. The covariates are referred to using shortened names for ease of reference. Thus “Lpprem” is log of price premium, “Lvoltlt” is log of volatility, “Lrelactvt” is log of relative activity, “Lordimbalance” is log of order imbalance, “Ldepth” is log of depth, “Lbidask” is log of bid-ask spread, “Lfirmsize” is log of firm size, “Lcloseprice” is log of closing price, “Timeofday” is time of day in hours since Table 5: Comparison of AIC for Models assuming Different Distributions of S(t) Exponential Weibull Logistic Loglogistic Log-normal Gaussian 1 Buy Dataset 1 (5%) 127562 125983 140476 124208 124765 139497 2 Buy Dataset 2 (7%) 176094 174216 193860 171918 172723 192561 3 Buy Dataset 3 (10%) 252184 249163 276758 245401 246501 274864 Exponential Weibull Logistic Loglogistic Log-normal Gaussian 1 Sell Dataset 1 (5%) 124088 122942 135844 121200 121431 134850 2 Sell Dataset 2 (7%) 174559 172798 189946 169958 170139 188548 3 Sell Dataset 3 (10%) 248570 245856 271941 242426 242741 270089 VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 55 10 am, and finally, there are six binary dummy variables for seven industry sectors. The industry factor has been coded to have the following values: 1-auto, 2-consumer goods, 3-energy, 4-financial services, 5-information technology/biotech (IT/BT), 6-pharmaceutical, and 7-manufacturing. Analysis of Deviance Column 2 of Table 6 gives the values of each estimated coefficient for the β corresponding covariate. Column 3 gives the standard error in the estimate of each β; these error estimates are obtained in R via the Fisher information matrix (for a discussion of the theory, see Lawless, 2002). The error estimates are converted to a corresponding normally distributed z value in column 4, leading to a p value in column 5 for the null hypothesis that β = 0. Some of the p values are numerically so small that they have been rounded off to zero by R. Most other p values are extremely small as well. The conclusion is that the model is robust. It began with the explanatory variables arranged in the order given above, in turn guided by separate univariate analyses not reported here (see Chatterjee, 2010). First, a model was built with only the first covariate in the list; then another model was built with the first two covariates, followed by a third which was built with the first three; and so on. In that procedure, each additional covariate was added sequentially in the same order as originally specified. As successive models were built, the decrease in –2 LL was monitored, where LL was the log likelihood for each successive model. This decrease is the quantity reported as deviance in column 3 of Table 7. Table 6 shows that several of the covariates have very low p values. But there is a need to gauge the relative importance of each significant covariate in the model. One way of doing this is to look at an analysis of deviance table that is generated as follows. Table 6: Summary Output for Buy Side Model with All Covariates Value Std. Error Z P 11.28371 0.26971 41.8357 0 Lpprem 1.70208 0.02304 73.8856 0 Lvoltlt -1.09786 0.03413 -32.1675 5.03E-227 Lrelactvt -0.87986 0.01974 -44.5646 0 (Intercept) Lordimbalance 0.0021 0.03731 0.0564 9.55E-01 Ldepth 0.38562 0.01298 29.7179 4.51E-194 Lbidask -0.35589 0.02325 -15.3073 6.84E-53 Lfirmsize -0.78972 0.01867 -42.2934 0 Lcloseprice 0.57099 0.02102 27.1705 1.45E-162 Timeofday -0.12138 0.01158 -10.479 1.08E-25 as.factor(findustry)2 -0.17082 0.06094 -2.8031 5.06E-03 as.factor(findustry)3 -0.18774 0.0627 -2.9943 2.75E-03 as.factor(findustry)4 -0.35816 0.05759 -6.2185 5.02E-10 1.46E-02 as.factor(findustry)5 -0.1385 0.05673 -2.4413 as.factor(findustry)6 -0.15776 0.06859 -2.3 2.15E-02 as.factor(findustry)7 -0.25202 0.0561 -4.4924 7.04E-06 Log(scale) 0.02496 0.00813 3.0699 2.14E-03 Scale= 1.03 Loglikelihood of the model = -61080.5 Loglikelihood of the null model (intercept only)= -80006.9 Note: Numerical results for an AFT log-logistic model using the covariates listed, following the methods described above (see Equations 5 and 6). Thus Z consists of the elements of column 1, and the vector of estimated coefficients β appears in column 2. The Z in column 4 used for estimating p-values should not be confused with the covariates. There is a negative sign convention in the coefficients below (which R uses). Thus, β’Z of Equation 5 here corresponds to: –11.28371 – 1.70208 Lpprem + 1.09786 Lvoltlt + 0.87986 Lrelactvt – ... 56 EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH In order to quantify the relative contribution of each variable, the “% deviance explained” was computed as follows: cause many stocks with widely varying prices are included in the study). Models with Fewer Covariates % Deviance Explained Deviance Reduction by the nth Variable = Null Deviance of the Model x 100 (7) The results obtained are given in Table 7. As emphasized above, the Table actually presents results from a sequence of models, each with one added covariate: the first model has no covariates, the second has only “Lpprem”, the third has “Lpprem and Lvoltlt”, and so on. A key idea in the analysis of deviance for such models is that quality of the model fit is measured through log likelihood, which is maximized. The percentage deviance explained by each covariate is therefore a relative measure of that covariate’s sequential contribution to the quality of the model fit. In Table 7, the relatively large contribution made by price premium is not surprising because price is the single most powerful determinant of the probability of order execution. Such relatively large contributions from price premium were observed consistently in many more models built as a part of this overall research, not all reported here in detail. The authors emphasize that, since the importance of price premium is obvious a priori, it is the contributions of the other covariates that are of actual interest in this study. Other variables that appear important are volatility, relative activity, bid-ask spread, firm size, and previous day’s closing price (which here is a proxy for stock price, be- More compact models were then considered. By conducting backward elimination, it was found that on the buy side, the least important variable was order imbalance, and the next least important variable was the industry sector (see Appendix B). Both these variables could therefore be dropped. However, since the p value for the financial services sector was exceptionally small (below 10-19; see industry factor 4, Table 6), the explicit modeling choice of incorporating a binary dummy variable was made to indicate whether a firm was in that particular sector. With the resulting reduced set of variables, a more parsimonious model was built. A similar exercise was carried out for the sell side, where order imbalance was found to play some role, and so, it was retained; and again only a binary variable for the financial services sector was retained (see the Table in Appendix B). Consistency was checked for each set of fewer covariates (buy-side or sell-side) by using all the three independently sampled data subsets in three different models. Results are given for three data sets each in Tables 8 and 9 for buy and sell sides respectively. The overall picture remains the same. In both Tables 8 and 9, the last column shows percentage of deviance explained, as discussed above. The coefficients for each specific covariate differ from each other across the three data sets by amounts comparable to their estimated standard errors, indicating that the models are mutually consistent, and that a larger data subset is not needed. Table 7: Analysis of Deviance of the Model in Table 6 NULL Df Deviance Resid. Df -2*LL P(>|Chi|) NA NA 90759 160013.8 NA % Deviance Explained Lpprem 1 32645.58 90758 127368.2 0.00E+00 20.40% Lvoltlt 1 914.5062 90757 126453.7 6.89E-201 0.57% Lrelactvt 1 1409.28 90756 125044.4 0.00E-01 0.88% Lordimbalance 1 11.01061 90755 125033.4 9.06E-04 0.01% Ldepth 1 282.332 90754 124751.1 2.33E-63 0.18% Lbidask 1 529.0991 90753 124222 4.44E-117 0.33% Lfirmsize 1 1170.91 90752 123051.1 1.28E-256 0.73% Lcloseprice 1 728.122 90751 122322.9 2.29E-160 0.46% Timeofday 1 110.7422 90750 122212.2 6.74E-26 0.07% as.factor(findustry) 6 51.28657 90744 122160.9 2.59E-09 0.03% VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 57 Interpretation of Coefficients The authors finally considered the estimated values of the coefficients β for each of the covariates, using numerical values from the third (10%) model of Table 8. These coefficients are interpreted as follows. If “Lpprem” increases by unity, then the time corresponding to a given execution probability increases by a factor of e1.693= 5.436. Moreover, since “Lpprem” is the logarithm of price premium, another simple interpretation is possible. If the price premium asked in a limit order is increased by a factor of k, then the time corresponding to a Table 8: Results of Models for Buy Orders (3 different data subsets) Value Std. Error z p 42.92 0.00E+00 % Deviance Explained Buy Dataset 1 (5%) (Intercept) 11.347 0.264 Log of price premium 1.699 0.023 73.99 0.00E+00 20.40% Log of volatility -1.099 0.034 -32.77 1.78E-235 0.57% Log of relative activity -0.889 0.019 -47.99 0.00E+00 0.88% Log of firm size -0.802 0.018 -43.81 0.00E+00 0.65% Log of depth 0.388 0.013 30.13 1.96E-199 0.28% Log of closing price 0.565 0.020 27.83 1.91E-170 0.58% Log of bid ask spread -0.357 0.023 -15.38 2.22E-53 0.18% Time of day -0.121 0.012 -10.49 9.38E-26 0.07% Financial Services (factor) -0.180 0.033 -5.5 3.78E-08 0.02% Log(scale) 0.025 0.008 3.09 2.03E-03 Scale= 1.03 Buy Dataset 2 (7%) (Intercept) 11.284 0.221 51.115 0.00E+00 Log of price premium 1.688 0.019 87.707 0.00E+00 20.78% Log of volatility -1.069 0.028 -38.027 0.00E+00 0.57% Log of relative activity -0.858 0.015 -55.854 0.00E+00 0.85% Log of firm size -0.790 0.015 -51.603 0.00E+00 0.63% Log of depth 0.380 0.011 35.194 2.49E-271 0.25% Log of closing price 0.574 0.017 33.888 1.01E-251 0.62% Log of bid ask spread -0.370 0.019 -19.158 8.34E-82 0.20% Time of day -0.112 0.010 -11.636 2.71E-31 0.06% Financial Services (factor) -0.116 0.027 -4.263 2.02E-05 0.01% Log(scale) 0.002 0.007 0.281 7.78E-01 Scale= 1.00 Buy Dataset 3 (10%) (Intercept) 10.937 0.186 58.66 0.00E+00 Log of price premium 1.693 0.016 104.16 0.00E+00 20.52% Log of volatility -1.075 0.024 -44.62 0.00E+00 0.56% Log of relative activity -0.884 0.013 -67.74 0.00E+00 0.90% Log of firm size -0.772 0.013 -59.64 0.00E+00 0.59% Log of depth 0.391 0.009 43.35 0.00E+00 0.31% Log of closing price 0.547 0.014 38.29 0.00E+00 0.56% Log of bid ask spread -0.343 0.016 -21.03 3.73E-98 0.17% Time of day -0.116 0.008 -14.17 1.44E-45 0.06% Financial Services (factor) -0.169 0.023 -7.32 2.54E-13 0.02% Log(scale) 0.018 0.006 3.09 1.99E-03 Scale= 1.02 58 EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH fixed execution probability increases by a factor of k1.693; for example, if the price premium asked is doubled (k = 2), then the time corresponding to a fixed execution probability increases by a factor of 21.693 = 3.233. Similarly, because the coefficient of “Lvoltlt” is negative, the model predicts that if, all other things held fixed, the volatility increases by a factor of 2, then the time corresponding to a fixed execution probability decreases by a factor of 21.075 = 2.107. Correspondingly, by holding the time fixed, one can ask what role each coefficient has on the probability of execution within that time. It is clear that a positive coefficient indicates a decrease in the probability of execution when the corresponding covariate increases, while a negative coefficient indicates an increase in the probability of execution when the corresponding covariate increases. Thus, higher price premium, higher depth, and higher closing price of the previous day, all lower the probability of execution within a fixed time (seen in another way they increase the probability that the order will take longer to execute). Similarly, higher volatility, higher relative activity, larger firm size, higher bid ask spread, along with lateness in the day and the firm being in financial services, due to their negative coefficients, increase the probability of execution within a fixed time. It is important to note that there is no simple scaling rule for how much the probability may increase or decrease, because it depends on the details of the distribution being used (here, log-logistic). In that sense, the mathematically simpler interpretation of the coefficients is the first one above, namely the effect on the time corresponding to a given execution probability. Finally, the quantity “scale” reported in Tables 8 and 9 at the bottom of each model’s output is actually the reciprocal of the parameter α in the log-logistic survival 1 function, S(t) = 1+λ tα as mentioned in Table 1. The parameter λ now incorporates all the covariates using the coefficients in column 2. The estimated value of scale is 1.02; hence the log of scale is small in magnitude, and the corresponding p value is relatively quite large. In other models built using this data set and with a time frame of 60 minutes, the parameter “scale” was consistently found to be close to 1. Other than slightly different numerical values of various quantities of interest, results for the sell side are quite VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 similar to those for the buy side. Therefore, these results are included here (Table 9) for completeness, but not discussed in detail. In particular, interpretation of these results proceeds very similarly to those for buy orders. The only point of interest is that there is an asymmetry in buy versus sell sides, e.g., in the role played by order imbalance. Comparison of the Results with other Studies The coefficient of log of price premium in the present models is positive indicating that an order that quotes a higher price premium takes a longer time to get executed. This is similar to findings from Cho and Nelling (2000), Al-Suhaibani and Kryzanowski (2000), Lo, MacKinlay & Zhang (2002), Gava (2005), and Wen (2008). In fact it is found to be the single most important covariate in the model explaining about 20 percent of the deviance in the model (whereas the other covariates individually explain something on the order of 1% of the deviance). Log of volatility is an important covariate in these models. The coefficient is negative indicating that the time to execution decreases when the volatility is greater. This effect was also noted by Cho and Nelling (2000), Lo, MacKinlay and Zhang (2002), Gava (2005), and Wen (2008). The study by Al-Suhaibani and Kryzanowski (2000) was the only exception to find time for execution increasing with volatility in the Saudi stock exchange. Log of relative activity is an important covariate in the present models with a negative coefficient indicating that as the level of trading activity increases, the time for execution decreases. This is an intuitively logical effect and was noted by Gava (2005) and Wen (2008). Log of depth has a positive coefficient in both buy and sell side models, indicating that the execution time of a particular order increases with a greater depth on the same side. This is due to the increased competition faced by the order; the effect was also observed by Omura, Tanigawa and Uno (2000), Al-Suhaibani and Kryzanowski (2000), and Lo, MacKinlay and Zhang (2002). Log of bid ask spread is consistently present in the models, though the contribution of this variable to the model fit in terms of percentage deviance explained is small. The negative coefficient indicates that the time to execute a particular order decreases with a bigger spread. These 59 Table 9: Results of Models for Sell Orders (3 different data subsets) Value Std. Error z p % Deviance Explained Sell Dataset 1 (5%) (Intercept) 11.30437 0.26274 43.025 0.00E+00 Log of price premium 1.46737 0.02263 64.842 0.00E+00 20.18% Log of relative activity -0.96876 0.01875 -51.676 0.00E+00 0.53% Log of firm size -0.92625 0.01816 -51.014 0.00E+00 0.11% Log of volatility -1.14014 0.03457 -32.985 1.32E-238 1.82% Log of depth 0.53269 0.01307 40.745 0.00E+00 0.48% Log of closing price 0.77836 0.0202 38.538 0.00E+00 1.06% Log of bid ask spread -0.1847 0.02267 -8.147 3.74E-16 0.06% Time of day -0.14908 0.01148 -12.985 1.48E-38 0.11% Financial Services (factor) -0.1608 0.03248 -4.951 7.38E-07 0.02% Log(scale) -0.00184 0.00842 -0.219 8.27E-01 11.9736 0.22357 Log of price premium 1.50171 Log of relative activity -0.98195 Log of firm size -0.95837 Log of volatility -1.17398 0.02942 -39.9 0.00E+00 1.81% Log of depth 0.51668 0.01097 47.11 0.00E+00 0.40% Log of closing price 0.80975 0.01717 47.15 0.00E+00 1.13% Log of bid ask spread -0.18429 0.01922 -9.59 9.04E-22 0.06% Time of day -0.17526 0.00977 -17.94 5.59E-72 0.15% Financial Services (factor) -0.16222 0.02728 -5.95 2.75E-09 0.02% Log(scale) -0.00457 0.00714 -0.64 5.22E-01 Scale=0.998 Sell Dataset 2 (7%) (Intercept) 53.56 0.00E+00 0.01933 77.67 0.00E+00 19.94% 0.01588 -61.85 0.00E+00 0.57% 0.01535 -62.43 0.00E+00 0.08% Scale=0.995 Sell Dataset 3 (10%) (Intercept) 11.8953 0.18973 62.7 0.00E+00 Log of price premium 1.5409 Log of relative activity -0.9778 0.01655 93.13 0.00E+00 20.11% 0.0134 -72.99 0.00E+00 0.55% Log of firm size -0.9443 0.01295 -72.9 0.00E+00 0.08% Log of volatility -1.2064 0.02515 -47.97 0.00E+00 1.88% 0.37% Log of depth 0.496 0.00934 53.11 0.00E+00 Log of closing price 0.7759 0.01455 53.34 0.00E+00 1.03% Log of bid ask spread -0.2017 0.01648 -12.24 1.85E-34 0.07% Time of day -0.1664 0.00826 -20.14 3.04E-90 0.13% Financial Services (factor) -0.1599 0.02335 -6.85 7.44E-12 0.01% 0.016 0.00595 2.69 7.04E-03 Log(scale) Scale=1.02 findings are similar to those of Omura, Tanigawa and Uno (2000); Cho and Nelling (2000). Al-Suhaibani and Kryzanowski (2000), Gava (2005), and Wen (2008) found an opposite effect. Log of closing price has a positive coefficient in both buy and sell side models indicating that all other things held 60 constant, small orders for a stock with a higher price have a lower probability of execution. The effect of this variable is relatively large in the models developed here. This variable was considered by Lo, MacKinlay, and Zhang (2000), but they found an opposite effect (note that they studied large orders). EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH Some covariates that are found to be important in the present models, viz., log of firm size and (to a smaller extent) and the industry sector that the firm belongs to, have not been considered by any of the other studies. Log of firm size has a negative coefficient indicating that orders for larger firms (which are presumably more liquid) have shorter execution times. In terms of the industry sector, orders for stocks in the financial services sector is found to have a shorter time for execution on both the buy and sell sides. Survival Curve Plots The probability of survival of a given order (with specific values of covariates) can be plotted against time. The aim is to present a representative plot, for which the covariate values have to be chosen. Clearly, median values can be used for most of the covariates. However, since most limit orders are priced with high premiums and therefore have miniscule probabilities of execution, the 5th percentile value for price premium will be used. Also, since the binary variable marking financial services companies cannot in principle be given an intermediate value, it is set to zero. Perhaps more interestingly, exploratory models can also be built with right censored data only, with either the lower bound time t1 or the upper bound time t2. In this way, three distinct survival curves can be computed and plotted on the same axes. Two such plots, based on models built from the 10 percent data subset, and for buy and sell sides, are given in Figures 1 and 2. Figure 1: Comparison of S(t) for Buy Side Models using t1, t2 and with Interval Censoring Figure2: Comparison of S(t) for Sell Side Models using t1, t2 and with Interval Censoring VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 61 It must be noted that the survival curves shown in Figures 1 and 2 are representative curves for an order with certain covariate values. The values of the covariates used in all the three curves are the same and so they are comparable. It can be seen that the difference in survival probabilities at a certain point of time, say 20 minutes or 30 minutes is not very different in the three cases. In particular, the arithmetic mean of t1 and t2 curves is close to the interval censored curve. This paper has demonstrated the use of hypothetical orders and survival analysis with interval censoring, in modeling the time for execution of small limit orders in an electronic stock exchange, specifically the NSE. Several prior studies have addressed this problem, but always using the much more laborious and resourceintensive method of reconstructing the entire history of actual limit orders. The use of hypothetical orders in this context leads to imprecise knowledge of the execution time, which is merely bound between two instants t1 and t2. In an influential early work in this area, Lo, MacKinlay and Zhang (2002) had noted large differences in the mean values of t1 and t2 (conditional on execution). Here, for the first time, their critique has been addressed and it has been shown that models using interval censoring can in fact be built quite easily with freely available software and readily available data. Moreover, though apparent differences between t 1 and t 2 may mislead modelers into thinking that these cannot in fact be used individually, it has been noted that the survival curves obtained from such models are not too different from the survival curve for the more correct, interval censored approach. In particular, the latter curve lies about midway between the former two. Finally, the signs and relative proportions of the fitted coefficients of this study seem largely consistent with those obtained in several other studies that reconstructed the entire order history. The managerial contribution of this study is, first of all, an improved understanding of the factors that affect execution time of a given small limit order in the Indian setting. It has been found that the probability of order execution of a given order is higher when: 62 recent trading activity in the stock relative to the size of the company is high iii) there has been greater volatility in the stock price in the preceding hour iv) closing price of the stock on the previous day is low (more important on sell side) v) depth in the order book on the same side is small (indicating less competition). Some other statistically significant covariates that have lower predictive value are: CONCLUSIONS i) ii) vi) bid-ask spread (higher value increases probability of execution) vii) time of day (orders placed later in the day have higher chance of execution) viii) order imbalance (higher value lowers probability of execution) ix) industry sector (orders for financial services firms have higher probability of execution). If a trader wants to devise a simple order submission strategy, then he or she may concentrate on only the first five covariates that seem to affect the probability to a much greater extent. Firm size and industry sector have not been previously considered in similar studies. These have been found statistically significant, though they have little incremental predictive value. Another interesting finding is that order imbalance on the sell side (but not the buy side) delays order execution, reflecting the asymmetry between buying and selling. In addition to the above specific insights, this study offers a more academic contribution as well. It has demonstrated that the hypothetical order approach can be used with interval censoring instead of the time consuming process of reconstruction of the order book, and that too using high frequency data that is not easily available to all. Thus models can be built easily with readily available and current data. This, in our opinion, is probably the more practical contribution of this study and may guide future research in this area. price premium asked is small (order price is close to the best bid or best ask price) EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH Appendix A: Accelerated Failure Time Model with the Log-logistic Distribution For the log-logistic distribution, the survival function is 1 S(t) = 1 + λtα the probability density function is f(t) = αλtα–1 (1 + λtα)2 and the hazard function is h(t) = αλtα–1 1 + λtα Plots of S(t) for two values of λ and three values of α are given in figure A1 below. It is seen that larger α leads eventually to faster decay of S(t), while changing λ leads to a stretching or compression of the S(t) in the horizontal (or time) direction. The hazard curves for λ = 1, for three different values of α are given in figure A2. It is seen that the hazard function is either steadily decreasing or hump shaped, as mentioned in the main text (Table 1). When there are covariates, the parameter λ is not explicitly retained any more but instead absorbed into the acceleration factor exp(–β’Z) which multiplies the time t. Note that R actually reports the reciprocal of the parameter α, and calls it “scale”. Figure A1: Log-logistic Survival Curves for Different λ and α Figure A2: Log-logistic Hazard Curves for λ=1 and Different α VIKALPA • VOLUME 38 • NO 1 • JANUARY - MARCH 2013 63 Appendix B: Results from Backward Elimination for Dropping Some Variables Buy Side 10% Dataset Df <none> Sell Side 10% Dataset AIC Df 241679 <none> AIC 236941 - Lordimbalance 1 241679 - as.factor(findustry) 6 237054 - as.factor(findustry) 6 241753 - Lordimbalance 1 237090 - Timeofday 1 241876 - Lbidask 1 237102 - Lbidask 1 242137 - Timeofday 1 237324 - Lcloseprice 1 243045 - Lvoltlt 1 238944 - Ldepth 1 243487 - Ldepth 1 239433 - Lvoltlt 1 243516 - Lcloseprice 1 239668 - Lfirmsize 1 244995 - Lfirmsize 1 241824 - Lrelactvt 1 245949 - Lrelactvt 1 241939 - Lpprem 1 257505 - Lpprem 1 249093 REFERENCES Al-Suhaibani, M., & Kryzanowski, L. (2000). An exploratory analysis of the order book, and order flow and execution on the Saudi stock market. Journal of Banking & Finance, 24, 1323-1357. Chatterjee, D. (2010). Studies on some aspects of liquidity of stocks: Limit order executions in the Indian stock market. Ph.D thesis, Indian Institute of Science, Bangalore. Cho, J., & Nelling, E. (2000). The probability of limit order execution. Financial Analysts Journal 56(5), 28-33. Gava, L. (2005). The speed of limit order execution in the Spanish stock exchange. Working Paper, Universidad Carlos III de Madrid. available on http://hdl.handle.net/ 10016/118 Lo, A.W., MacKinlay, A. C., & Zhang, J. (2002). Econometric models of limit-order executions. Journal of Financial Economics, 65(1), 31-71. Omura, K., Tanigawa, Y., & Uno, J. (2000). Execution probability of limit orders on the Tokyo stock exchange, SSRN, available on http://papers.ssrn.com/sol3/papers. cfm?abstract_id=252588 Parlour, C. A., & Seppi, D.J. (2008). Limit order markets: A survey. in Thakor, A.V. & Boot, A.W.A. (Eds.) Handbook of Financial Intermediation and Banking. North-Holland. Verhoeven, P., Ching, S., & Ng, H.G. (2004). Determinants of the decision to submit market or limit orders on the ASX. Pacific-Basin Finance Journal, 12(1), 1-18. Handa, P., & Schwartz, R. (1996). Limit order execution. Journal of Finance, 51, 1835–1861. Ranaldo, A. (2003). Order aggressiveness in limit order book markets. Journal of Financial Markets, 7, 53-74. Klein, J. P., & Moeschberger, M.L. (2003). Survival Analysis, 2nd Edition, New York: Springer. Wen, Q. (2008). Econometric models of limit-order completion time. Vast Data Conference, Oxford-Man Institute of Quantitative Finance, September 2008. Lawless, J.F. (2003). Statistical models and methods for lifetime data. 2nd ed., Hoboken: John Wiley and Sons. Devlina Chatterjee is Assistant Professor in the IME Department at IIT Kanpur. She has a Ph.D in Management Studies from IISc, Bangalore. Her prior degrees were from IIT Kharagpur (B.Tech., Agricultural Engineering), IIM Ahmedabad (PGDM, Agribusiness Management), and Penn State University, (MS, Agricultural Engineering and Agricultural Economics). She has worked at IFCI, GE Capital Analytics, and Antrix Corporation; and taught at VGSOM, IIT Kharagpur. Her current research interests are in applied econometrics, empirical finance, rural economics, and consumer behaviour. Chiranjit Mukhopadhyay is Professor of Statistics in the Department of Management Studies at the Indian Institute of Science, Bangalore. He obtained his Ph.D. in Statistics from the University of Missouri, Columbia after receiving B-Stat and M-Stat from the Indian Statistical Institute, Calcutta. Among his many research interests, the two major ones are Reliability Theory and Empirical Finance. He has more than fifty technical publications in national and international journals, edited volumes, and refereed conference proceedings. email: [email protected] email: [email protected] 64 EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH