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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
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Spanish stock exchange. Working Paper, Universidad
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10016/118
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Parlour, C. A., & Seppi, D.J. (2008). Limit order markets: A
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Verhoeven, P., Ching, S., & Ng, H.G. (2004). Determinants of
the decision to submit market or limit orders on the
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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,
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Wen, Q. (2008). Econometric models of limit-order completion time. Vast Data Conference, Oxford-Man Institute of
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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]
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EXECUTION TIMES OF SMALL LIMIT ORDERS: A SIMPLER MODELING APPROACH