The Impact of Aggressive Orders in an Order
Driven Market: A Simulation Approach1
Gunther Wuyts2
University of Leuven
January 2011
1 We
would like to thank the editor Ingmar Nolte and the anonymous referee for the very
helpful and interesting comments and suggestions on an earlier version of the paper.
2 Correspondence can be sent to: gunther.wuyts@econ.kuleuven.be.
Postal address:
Gunther Wuyts, University of Leuven, Faculty of Business and Economics, Department of
Accountancy, Finance and Insurance, Naamsestraat 69, 3000 Leuven, Belgium.
Abstract
This paper investigates resiliency in an order-driven market. We simulate the e¤ect
of a large liquidity shock, measured by a very aggressive market order that uses at
least the depth at the three best prices in the book. The simulation is done on basis
of a VAR model that captures various dimensions of liquidity and their interactions,
it is estimated for data of the Spanish Stock Exchange. We show that, despite the
absence of market makers, the market is resilient. All dimensions of liquidity (depth at
the best prices, spread and order book imbalances) revert to their steady-state values
within 15 orders after the shock. For prices, a long-run e¤ect is found. Furthermore,
di¤erent dimensions of liquidity interact. Immediately after a liquidity shock, the
spread becomes wider than in the steady state, implying that one dimension of liquidity
deteriorates, while at the same time depth at the best prices increases which means
an improvement of another dimension of liquidity. In subsequent periods, the spread
reverts (declines) back to the steady-state level but also depth decreases. Also, we …nd
evidence for asymmetries in the impact of shocks on the ask and bid side. Shocks on
the ask side have a stronger impact, compared to shocks on the bid side. Finally, we
…nd that resiliency is higher for stocks which are less frequently traded and stocks with
a larger relative tick size, in line with theoretical predictions.
JEL Classi…cation: G10
Keywords: Liquidity, Liquidity Shock, Resiliency, Limit Order Market, Order
Aggressiveness
1
Introduction
This paper analyzes resiliency in an order-driven market. Resiliency refers to how fast
dimensions of liquidity, such as best prices, depths or duration, recover to their normal
(i.e. steady-state) level after the market has been hit by a liquidity shock. Reversion
in liquidity following liquidity shocks is especially crucial in order-driven markets since
no market makers are present with an obligation to maintain an orderly market by
providing liquidity. Can order-driven markets be resilient without their presence?
To answer this question, we develop a VAR model that incorporates the di¤erent
dimensions of liquidity (prices, depths and duration). The VAR model allows for
analyzing jointly these various dimensions of liquidity and capturing the interactions
between them. Models such as Rosu (2009) and Parlour (1998) show that bid and
ask side of the market do not evolve independently. Also, the ask price cannot
move arbitrary far away from the bid price, Engle and Patton (2004) …nd they are
cointegrated. Moreover, prices and depths at these prices could interact. If an arriving
trader is faced with a long queue of limit orders at the ask, she might prefer to undercut
the ask to obtain price priority and probably faster execution. Based on our VARmodel, we analyze resiliency by simulating a liquidity shock. In the literature, such
shock is typically implemented via the innovation process of the VAR-model (see e.g.
Beltran, Durré and Giot (2009)). The disadvantage of this is that little is said about the
speci…c origins of such “general” shock. Therefore, we use an alternative approach by
analyzing a well-de…ned liquidity shock having a very speci…c source. More speci…cally,
we measure a liquidity shock by a very aggressive order. In particular, a liquidity shock
on the ask (bid) side is a very aggressive buy (sell) order that consumes at least all
the depth at the three …rst ask (bid) prices in the book and part of the depth at the
next price. It is natural to specify a liquidity shock in term of a speci…c type of orders
(see also Large (2007) or Degryse et al. (2005)), since in a limit order market, liquidity
(and resiliency) is determined by the interaction between limit orders, which supply
liquidity, and market orders, which consume it. Ultimately, it is the mix between
both types of orders that shapes the liquidity of a limit order market. While VARmodels are regularly used in market microstructure research to model resiliency, other
methods have been used as well. They include the intensity model of Large (2007),
copula techniques applied to multivariate transaction processes (see e.g. Nolte (2008)
or Bien, Nolte and Pohlmeier (2009)) and event study methods (see Degryse et al.
(2005)). These methods all are alternatives to the VAR approach, that is used in
our paper. Our choice for a VAR model is economically motivated by the theoretical
framework of Foucault, Kadan and Kandel (2005) (henceforth FKK05), one of the very
few theorical contribution explicitly focussing on resiliency. Their theoretical model
de…nes resiliency as the “probability that the spread reverts to its competitive level
1
before the next transaction occurs”. We measure the competitive level of variables,
including the spread, by their steady-state value, i.e. their average value over time.
Furthermore, FKK05 de…ne a liquidity shock as “a succession of market orders that
enlarges the spread”. Our measure of a liquidity shock - a very aggressive market order
that uses several levels in the limit order book - is very close to this concept of a series
of market orders that enlarge the spread. Moreover, given that FKK05 “have checked
numerically that all our implications regarding our measure of market resiliency (for
instance Corollary 1) hold for the order-based measure of resiliency as well”(see page
1185 of FKK05), we estimate our VAR model and compute the impulse responses to
a liquidity shock in order time. In sum, we favor a VAR approach over the other
approaches because it o¤ers a convenient framework for analyzing resiliency in a way
that is closely linked to and motivated by theory, and therefore allows an intuitive
economic interpretation.
We apply our model and simulation framework to a sample of 35 Spanish index
stocks over the period July-December 2000 (124 trading days). The Spanish stock
market operates as an order-driven market. Our key results can be summarized as
follows. First, we show that in general the Spanish stock market is resilient, even
though no market makers are present to provide liquidity at all times. The bid-ask
spread, depth at the best prices and the duration between order submissions all recover
to their steady-state levels within 15 orders after the liquidity shock. For best bid and
ask prices, as well as for the midprice, a long-run e¤ect is documented. Secondly,
we …nd an interaction and complimenarity between di¤erent dimensions of liquidity.
After a liquidity shock, there is a long-run e¤ect on prices as a result of which the
spread becomes wider than in the steady state, implying that one dimension of liquidity
(spread) deteriorates. However, at the same time depth at the best prices increases
which implies an improvement of another dimension of liquidity. In subsequent periods,
we observe the opposite evolution: the spread reverts (declines) back to the steadystate level but also depth declines. Thirdly, after a liquidity shock, the limit order
book becomes imbalanced. Following a shock on the ask side, depth at the ask side
becomes considerably larger than depth at the bid side. It takes around 15 orders for
the order book to become balanced again. A similar picture is found after a shock on
the bid side. Fourthly, we …nd evidence for asymmetries in the impact of shocks on
the ask and bid side. Shocks on the ask side have a larger impact and - depending on
the liquidity measure though - it may take longer for the liquidity measure to recover,
compared to shocks on the bid side. Finally, we …nd that resiliency is higher for stocks
which are less frequently traded and stocks with a larger relative tick size (i.e. tick
size/midprice). These …ndings thus con…rm the theoretical predictions in FKK05.
The remainder of this paper is organized as follows. In Section 2 we relate our work
2
to the existing literature and elaborate on the di¤erences with and contributions to
previous studies. In Section 3, the econometric model is developed. Also, we explain the
simulation methodology for studying resiliency. Section 4 presents the dataset used. In
Section 5, …rst the estimation results of the VAR model are presented. Subesequently,
we discuss in detail our results for resiliency. Section 6 concludes.
2
Related Literature and Contributions
In empirical market microstructure research, the use of VAR-models has been pioneered
by Hasbrouck (1991). He …nds that the full impact of a trade innovation is not realized
instantaneously but with a lag. Moreover, the impact is higher if the stock is more
infrequently traded, if the trade size is large and if there exists a wide spread at the
moment of the trade. Since this seminal paper, numerous variants and extensions
have been developed. Dufour and Engle (2000) investigate the informational role of
market activity and show how the dynamics in prices and trades are a¤ected by the
information revealed by the time between transactions. In the model of Easley and
O’Hara (1992), the time between trades is an indication whether or not a news event
has occurred. Uninformed traders use this signal to revise their beliefs about the
arrival of news about which some traders might be informed. Dufour and Engle …nd
that both larger quote revisions and stronger positive autocorrelation of trades are
linked to a higher trading activity. In this case, prices converge faster to the full
information value after an unexpected trade. When combining their results with those
of Hasbrouck, they conclude that high trading activity goes together with large spreads,
high volume and a high price impact of trades. Active markets are then illiquid in the
sense that the price impact of trades is higher. Both Hasbrouck (1991) and Dufour
and Engle (2000) include the change in the quote midpoint (and the trade direction)
as endogenous variables in their VAR-models. Two recent papers, Engle and Patton
(2004) and Escribano and Pascual (2006), include both bid and ask quotes in their
model and in this way allow trades to have an asymmetric impact on bid and ask
prices. They also correct for cointegration between bid and ask. The motivation of
Engle and Patton (2004) for including both bid and ask quotes lies in a study by Jang
and Venkatesh (1991). The latter show that the most common response by liquidity
suppliers to news is no adjustment, but the second preferred response is that only one
of both quotes is changed. After a buy, it is more likely for the ask to be changed
than for the bid. This means that the dynamics of ask and bid quotes will be di¤erent
after a buy order than after a sell order. Engle and Patton (2004) …nd evidence for
a strong asymmetric impact on bid and ask prices of buyer and seller initiated trades
in the short run. Short-duration and medium-volume trades have the largest impact.
3
The order of magnitude of the e¤ects is in general larger in the lower trade frequency
deciles. Also Escribano and Pascual (2006) …nd that the responses of bid and ask
quotes to a trade are not necessarily symmetric. They show that the mean impact of
an unexpected buy on the ask quote di¤ers from the mean impact of an unexpected
sell on the bid quote. Moreover, the sensitivity of the response of the ask to a trade
innovation when market conditions are changed, di¤ers from that of the bid quote. The
impact of a buy on the ask quote depends e.g. more on the time since the previous
trade than the impact of the corresponding sell on the bid quote. However, symmetry
is more likely the more informative trades are. Coppejans, Domowitz and Madhavan
(2004) also address the issue of liquidity in a limit order market, although they use
5-minute intervals, instead of order time as we do. They use market depth at the buy
and sell side as a measure of liquidity and the change in the midquote as a return.
With data from the limit order book of the market for Swedish stock index futures,
they show that a liquidity shock on the ask (bid) side of the market lowers (increases)
returns. The e¤ects are short-lived since almost all the impact occurs during the 10
minutes following the shock. Moreover, they …nd evidence of liquidity clustering since
if there is an increase in liquidity1 on one side of the market, this leads to an increase
in liquidity at the other side of the market as well. The study most related to ours
is Degryse et al. (2005). They also analyze resiliency, but employ an event study
instead of a simulation based on a VAR-model. They …nd that depths stay around
their mean before and after aggressive orders, whereas spreads return to their mean
after about twenty best limit updates. Aggressive orders induce a a long-term price
e¤ect. To conclude this paragraph, we would like to remark that our paper does not
aim to explain why a trader chooses a certain order type. Ranaldo (2004) discusses the
complementary issue of the determinants of the choice of an order type by a trader.
We do not model this choice, but take the submission of the order as given and analyze
its aftermath. In this sense, Ranaldo focusses on the period before the submission of
an order while we investigate the period thereafter. In this way, we shed some light
on the consequences of the order choices and how the market responds to them. Note
however that we do correct in the regression models for past order ‡ow.
How does our paper di¤er from and aim to contribute to the above mentioned
literature? First, most papers in the literature employing VAR models only focus
on spreads or prices as measures for liquidity, in this way ignoring depth in the
market, another important dimension of liquidity. We therefore make a contribution
to Dufour and Engle (2000), Engle and Patton (2004) and Escribano and Pascual
(2006) by analyzing prices, depth and duration in a unifying VAR framework (in
1
They measure a liquidity shock as a shock to market depth, de…ned as total depth 6 ticks away
from the quote midpoint
4
alternative speci…cations we also look at spreads, depth imbalances and midprices), in
this way disentangling spread/price e¤ects and depth e¤ects of shocks. Recall from the
introduction that other approaches exist, next to VAR models (see e.g. Large (2007)
or Bien, Nolte and Pohlmeier (2009)). As argued, we opted for the VAR approach
because of its close relation with theory, given the research question of our paper.
Further, Engle and Patton (2004) focus on the estimation results of a VAR (VEC)
model, and do not consider the impact of liquidity shock over time as we do. Also, these
authors consider the NYSE, which is a hybrid market, while we study a pure orderdriven market without market makers that have an obligation to provide liquidity,
providing thus a more clear picture of a pure order-driven market. Secondly, our study
di¤ers from Coppejans, Domowitz and Madhavan (2004) since we investigate e¤ects
in order time, and thus o¤er a more high-frequency, less aggregate analysis. Moreover,
we use a clearly identi…able liquidity shock (a very aggressive order) instead of an
innovation in the error terms of the VAR model, which is harder to interpret from
an economic point of view. Furthermore, our paper also di¤ers from Degryse et al.
(2005) in a number of ways. Compared to Degryse et al. (2005), our shock is a much
more extreme event, since very aggressive orders are rare (less than 1% of buy or sell
orders, see Table 3) but have large price impact. Moreover, given that order types are
clustered over time2 , an event study, while having the advantage of looking directly
at what happens in the limit order book around aggressive orders, has the potential
drawback that multiple events (i.e. liquidity shocks) can occur within the event window.
We can isolate one individual shock in our simulation framework. Furthermore, their
descriptive approach does not allow for disentangling the e¤ects of order type and order
size. Therefore, our paper complements the event study in Degryse et al. (2005) by
developing a formal econometric analysis of liquidity, allowing for the accommodation
of some drawbacks of the descriptive approach. We are able to correct for the problem
of confounding events and can account for the state of the limit order book around
aggressive orders. Moreover, we can separate the impact of an aggressive order in
itself from other elements such as order size or market environment. Also interactions
between dimensions of liquidity are captured in our econometric model, in contrast
with an event study. Finally, we motivate the de…nition of our model explicitly by
the theoretical literature and in particular Foucault, Kadan and Kandel (2005). We
also contribute to the above literature by empirically testing a number of theoretical
predications about the determinants of resiliency that emerge from their model.
2
This is the so-called diagonal e¤ect (see Biais, Hillion and Spatt (1995) and Degryse et al. (2005)).
An order of a given type is more likely to be followed by an order of the same type, relative to the
unconditional probability of the order type.
5
3
Econometric Methodology
3.1
Modeling Prices, Depths and Duration
As argued in the introduction, prices and depth at the bid and ask side of the market
may not move independently, but rather interact with each other. The same holds
for prices and duration between updates of the limit order book. Therefore, we
develop in this section an empirical model which is able to capture these elements. In
empirical research, vector autoregressive (VAR) models are a popular and convenient
methodology for analyzing the interaction between variables of interest and their
behavior around shocks. In our study, which makes use of a VAR-model, the variables
of interest are the best bid and ask prices in the limit order book, the depth at these
best prices and duration. By explicitly modeling both sides of the market, we allow for
potential asymmetries between them. This approach is also in line with existing work
by e.g. Engle and Patton (2004) and Escribano and Pascual (2006) who consider both
bid and ask prices. When analyzing a limit order market, also depth at the best prices
needs to be incorporated in the model. Parlour (1998) shows that in an order-driven
market the choice between limit and market orders depends on the depth at both sides
of the book. Recall that it is precisely the mix between market and limit orders which
determines liquidity in a limit order market, since they determine respectively liquidity
demand and supply. We include also duration in our model. Theory shows that the
time between trades is informative. In the model of Easley and O’Hara (1992), the
lack of trades (or in other words, the duration since the last trade) provides a signal
about the existence of new information in the market3 . As a result, spreads are shown
to decrease as the time between trades increases. Foucault, Kadan and Kandel (2005)
show that the resiliency of a limit order market is decreasing in the order arrival rate,
i.e. a market having a lower order arrival rate, is more resilient than a fast market. The
reasoning is that, when the order arrival rate increases, limit order traders become less
aggressive in their price improvements. As a consequence, more orders are needed to
bring down the spread to its competitive level, implying resiliency declines. Empirically,
Engle and Patton (2004) con…rm that the impact of a trade is di¤erent depending on
the time elapsed since the previous trade. They …nd that short duration and medium
volume trades have the largest impacts on quote prices. Hence, we de…ne the vector of
endogenous variables yt as follows:
yt = f ln (At ) ;
3
ln (Bt ) ; ln (ADt ) ; ln (BDt ) ; ln (Durt )g
Trades themselves then reveal information about the direction of the information event.
6
with At (Bt ) the best ask (bid) price at time t, ADt (BDt ) the depth at the best
ask (bid) at time t, Durt the duration since the previous order and ln the natural
logarithm (in Section 4.2, we provide the details how these variables are measured
in our empirical implementation). In this speci…cation, we follow Harris (1990) who
considers prices and depth as two separate aspects of liquidity. We take …rst di¤erences
of the price series because the null hypothesis of a unit root in both best bid and ask
series cannot be rejected. These …rst di¤erence of log prices can be interpreted as the
return on the best bid (or ask). The time index t refers to order time, i.e. we record a
new observation each time when a new order occurs in the market. This means that
the VAR model is speci…ed in event time (as opposed to calendar time). If there are
multiple orders at the same 1/100 second, we take the values after the last order.
When modeling prices, depths and duration, we account for the characteristics of
the order ‡ow. Order ‡ow is modelled by including three sets of exogenous variables.
The …rst set describes the degree of aggressiveness of the submitted orders and consists
of dummy variables for each order type i = 1; :::; 18. We classify each order according
to a scheme which is inspired by the one proposed in Biais, Hillion and Spatt (1995)
and also used in other papers, see e.g. Gri¢ ths et al. (2000) or Ranaldo (2004). A
de…nition of each order type is provided in Table 1. From this table, it can be seen that
we use 9 di¤erent types of buy order, and 9 types of sell order. The orders are presented
in descending aggressiveness: M O_B1 can be interpreted as the most aggressive buy
order type, Canc_B3 is the least aggressive buy order. The main distinction in our
classi…cation is based on the fact whether an order is a market order or marketable
limit order (M O), a (non-marketable) limit order (LO) or a cancellation or modi…cation
(Canc). From the table, it can be seen that we use a slightly more elaborate version
of the Biais, Hillion and Spatt (1995) scheme. First, we added one category of market
order - the most aggressive - because we want to measure a liquidity shock by a very
aggressive market order M O_B1t and M O_S1t (and our dataset allows doing so, see
Section 3.2). Secondly, we also added cancellations and modi…cations. The reason
is that, in principle a cancellation can have the same impact as the submission of a
market order. For instance, the cancellation of an order at the best ask may cause the
best ask to increase, just as an aggressive market order. To make a distinction between
market orders and cancellations, we added both in the classi…cation scheme.
Please insert Table 1 around here.
The second set of order ‡ow variables is related to order size. For market
orders, orders with a higher degree of aggressiveness have a larger order size (as
can be expected from de…nition), see Table 3. For other order types (limit orders
7
or modi…cation/cancellations), the relation between aggressiveness and order size is
not monotonic. Therefore, by including both the degree of aggressiveness and size, we
are able to disentangle aggressiveness in itself and order size. Moreover, Easley and
O’Hara (1987) show that order size is important in determining price impacts. Hence,
OSize_B1t represents the order size of an order of type M O_B1; more speci…cally,
the variable is equal to the order size if order t is of type M O_B1, and zero otherwise.
For all other order types, the order size variables are de…ned in a similar way.
Thirdly, to account for possible intraday patterns in returns and depth4 , we include
time of day dummies. We take a separate dummy for the …rst and last 15 minutes of
trading because of the more pronounced trading activity during these intervals. Hence,
T _01t (T 18t ) equals 1 if the order takes place between 9:00 and 9:15 a.m.. (5:15 and
5:30 p.m.). The remainder of the trading day is divided in intervals of 30 minutes:
T 02t is 1 if the order is between 9:15 a.m and 9:30 a.m, ... , T 17t is 1 if the order is
between 4:45 p.m and 5:15 p.m and zero otherwise. It needs to be remarked that other
frequently used techniques for diurnal adjustments are piecewise linear splines (from
the duration literature) or trigonometric functions (from the volatility literature). We
have performed a robustness check and diurnally adjusted the endogenous variables and
order sizes using piecewise linear splines. Reestimating the VAR and plotting impulse
responses with the adjusted variables does not change our results. We therefore use
time of day dummies, because it they allow for a more intuitive interpretation. This
approach is also in line with other market microstructure papers using VAR models,
such as Dufour and Engle (2000), Engle and Patton (2004) or Escribano and Pascual
(2006).
Hence, the vector of exogenous variables xt is:
xt = fM O_B1t ; :::; Canc_S2t ; Size_M O_B1t ; :::; Size_Canc_S2t ; T 01t ; :::T 17t g
where we left out Canc_S3t (which is then the reference category for order types) and
the dummy for the last interval of the trading day T 18t to avoid perfect multicollinearity
in estimation.
A …nal point concerns the econometric properties of the series used. Best ask and
bid prices not only have a unit root, but can also be expected to be cointegrated. Engle
and Patton (2004) …nd empirical evidence of the presence of cointegration. In our
model, the cointegrating term has a simple interpretation, being the log spread, hence:
LSpreadt = ln (At ) ln (Bt ). This means that the VAR-model is speci…ed in error
correction form. We do not impose other cointegrating relations since only bid and ask
prices are found to be integrated of order 1 or I (1). Depths and duration are I (0).
4
This e¤ect might e.g. be due to the U-shaped pattern of trading activity during the day (see
Biais, Hillion and Spatt (1995) for an illustration of this phenomenon on the Paris Bourse).
8
Bringing all of the above together, our VAR-model is speci…ed as follows:
yt =
L
X
l=1
Al y t
l
+ A0
M
X
Bm xt
m
+ LSpreadt
1
+ ut
(1)
m=0
with ut the error term which is assumed to be white noise and A0 , Al , Bm and the
coe¢ cient matrices to be estimated. Note that for the endogenous variables, only lags
are included, while for the vector of exogenous variables, also the contemporaneous
values are considered.
3.2
Resiliency
Resiliency, a crucial aspect of liquidity in order-driven markets, refers to the ability of
a trading system to cope with liquidity shocks hitting the market. Earlier, we de…ned
resiliency as “the speed of recovery to their steady-state levels of di¤erent measures of
liquidity (i.e. prices, depths and duration) after a liquidity shock hits the market”. It
then remains to be speci…ed what comprises such liquidity shock. Most of the market
microstructure literature uses a “general” shock, implemented via the error terms in
a VAR-model (in our case this would be ut in equation (1)). In contrast, our paper
proposes an alternative approach by using a clearly identi…able and concrete event.
More speci…cally, we measure a liquidity shock on the ask (bid) side as a type M O_B1
(type M O_S1) order. From Table 1, it is clear that such orders are the most aggressive
market buy (sell) orders. Such shock widens the bid-ask spread and consumes all the
depth at the …rst three prices and some of the depth behind the third price. Note
that a distinction is made between a shock on the buy and sell side of the market to
investigate potential asymmetries. This measure is also in line with the one in the model
of Foucault, Kadan and Kandel (2005), who de…ne a liquidity shock as “a succession
of market orders that enlarges the spread”. Our measure of a liquidity shock - a very
aggressive market order that uses several levels in the limit order book - is very close
to this concept of a series of market orders that enlarge the spread.
To investigate resiliency, we start from the estimation results of the VAR model in
Equation (1). We then implement a simulation of the impact of a liquidity shock (a
type M O_B1 or M O_S1 order) on each of the endogenous variables. We then are able
to study resiliency by comparing these impulse responses of the endogenous variables
with their steady-state values, and more in particular how fast - expressed in number of
orders - variables return to their steady-state level after the shock. This methodology is
also in line with theory. Given that Foucault, Kadan and Kandel (2005) “have checked
numerically that all our implications regarding our measure of market resiliency (for
instance Corollary 1) hold for the order-based measure of resiliency as well”(see page
9
1185 of their paper), we follow their theoretical model and estimate our VAR model
and compute the impulse responses to a liquidity shock in order time. For the full
technical discussion of our simulation methodology, we refer to Appendix A. Finally,
it should be noted that we can only compute steady-state levels in a meaningful way
for variables that are stationary (hence not for prices).
4
4.1
Data
Sample
This paper uses data from the Spanish Stock Exchange SIBE, an exchange which
operates essentially as a pure order-driven market. For the institutional details of
SIBE and a description of its main features, we refer to Pardo and Pascual (2007). The
sample contains 35 stocks that were part of the IBEX35 stock index during the sample
period. The IBEX35 is composed by the 35 most liquid and active stocks, traded on
the exchange. Our sample period ranges from July 2000 - December 2000 and thus
spans 124 trading days. The data on the limit order book contain the …ve best bid and
ask prices and the displayed depth at each of these ten prices. Moreover, we have data
on all trades that were executed during the continuous trading session. Preopening or
postclosing orders are not included since the trading mechanism during this period is
di¤erent from the one during the trading day. All changes in the book (i.e. submitted
orders) are timestamped to 100th of a second. The trading data show price and size of
each trade. The index numbers and time stamps allow for a perfect matching of trade
and limit order book data. Since the sample period is before 2001, we do not have to
take into account the presence of volatility auctions (see Pardo and Pascual (2007)).
These auctions were only introduced at May 14, 2001. Finally, we …ltered the database
to exclude items that contain errors (e.g. registers out of sequence, increases in the
accumulated volume over the day that are negative, ...).
Table 2 presents the summary statistics of each of the stocks in our sample. The …rst
column shows the code of the stock. Columns 2 and 3 present the market capitalization
(expressed in 1000 euro) at the end of 2000, and the total trading volume (in 1000 euro)
during 2000, respectively. The next four columns show the average daily number of
trades, the average trade size (in shares), the average duration (in seconds) between
orders, the average bid-ask spread and the average midprice. All these average are
computed over the whole sample period. From this table, it can be seen that there is
substantial variation between the stocks with respect to market cap, trading volume
and spreads. As a …nal remark, it should be noted that the tick size for all stocks is
1 cent (0.01 euro). Hence, bid-ask spreads expressed in cents give the same results as
10
when expressed in number of ticks.
Please insert Table 2 around here.
4.2
De…nition and Summary Statistics of Endogenous and
Exogenous Variables
Table 3 presents the summary statistics for the endogenous and exogenous variables
that are used in our VAR model, speci…ed in Section 3.1. We now provide their
empirical de…nition and the unit of measurement. All endogenous variables, as well
as log order sizes and log durations are winsorized at the 99.5% level to account for
extreme outliers. In Panel A, we …rst show the mean, median and standard deviation
for the …ve endogenous variables. For all stocks, the returns on the best ask and bid,
expressed in % and denoted ln (At ) ; ln (Bt ), are small on average. This is due to
the fact that for a majority of the orders, there is no change in the best price (see the
de…nitions in Table 1), hence the return is zero for these observations. Subsequently,
the log of depths at the best bid and ask ln (ADt ) ; ln (BDt ) are shown. Depths ADt
(BDt ) are measured in euros and obtained by multiplying depth in shares at the best
ask (bid) with the prevailing best ask (bid price). Note that the average depth at
the best ask is larger than average depth at the best bid. The average log of duration
ln (Durt ) is the next variable, Durt is computed as the time in seconds between order
t 1 and order t.
The second part of Panel A contains summary statistics of variables that will be
used in transformations of our main model (see Sections 5.2.2 and 5.2.3). Spread is the
bid-ask spread in cents, it is on average 6.36 cents. The second variable, ln (BADR) is
the log of the ratio of depth at the best bid and ask (in euro) and can be interpreted as
a measure for the imbalance of the limit order book. Its value is close to zero, implying
on average a balanced book. Next, ln (M ) is the return on the midprice in % and
ln (BAD_Avg) is the log of the average of depth (in euro) at the best bid and ask.
Panel B presents the frequency (in %) of occurring of the di¤erent order types,
the average order size in shares and the log of order size per type (which are the size
variables in our VAR model), the left part of the table for buy order, the right part
for sell orders. The results reveal that the most aggressive buy and sell orders (type
M O_B1 and M O_S1) have the smallest frequency of occurring across order types.
So when these two types are used as measures for a liquidity shock, we are looking at
relatively rare events but with a large impact as they consume depth at at least the …rst
three price levels in the order book. The most frequent order type are the market orders
11
that are …lled completely and have no price impact (type M O_B3 and M O_S3). All
frequencies are in line with the …ndings of Biais, Hillion and Spatt (1995) or Degryse
et al. (2005) for the Paris Bourse and of Gri¢ ths et al. (2000) for the Toronto stock
exchange5 . Furthermore, it can be seen that in general more aggressive market orders
are also larger, with the largest orders being types M O_B1 and M O_S1. At …rst
sight, this …nding may seen to be expected from their de…nition since these are the most
aggressive orders, consuming all depth at one or more prices. However, a priori it could
also be true that orders are classi…ed as aggressive because they are submitted at times
when depths in the book are small. The fact that the average order size of aggressive
market orders is large and moreover considerably larger than the average depth shows
that this is not the case. For limit order types and cancellations/modi…cations, the
relation between aggressiveness and order size is not monotonic, but there are no
theoretical reasons to a priori expect such a monotonic relation (e.g. there is no reason
why a cancellation should be smaller than a limit order). This is our motivation for
including both the type and size of an order in the econometric model, in this way we
distinguish both e¤ects.
Please insert Table 3 around here.
5
Empirical Results
5.1
Model Estimation
Before estimating the model, we verify the econometric properties of the series.
Augmented Dickey-Fuller tests (not-reported) reveal that log best prices ln (At ) and
ln (Bt ) contain a unit root, while log depth at the best ask and bid and log duration
between orders are stationary. As stated in Section 3, we take this into account in the
model speci…cation. During the estimation of the model in equation (1), we included
…ve lags of the endogenous variables (L = 5) and zero lags of the exogenous variables
(M = 0). We aimed to make the speci…cation of the VAR as concise as possible,
while on the other hand also optimizing the AIC criterion. Our choice of L = 5 and
M = 0 balances both points and it is also in line with the literature, see e.g. Engle
and Patton (2004). The VAR model is estimated for each of the 35 stocks separately
with least squares and Newey-West standard errors are computed. Recall that the
de…nition of the variables is explained in Section 3 and the units of measurement are
shown in Section 4.2, we do not repeat them here. All endogenous variables, as well
5
Recall though that these papers use a slightly less detailed classi…cation scheme, see Section 3.1.
12
as log order sizes and log durations are winsorized at the 99.5% level to account for
extreme outliers. Finally, it should be noted that all results for resiliency (see the next
section), are robust to alternative speci…cations of the VAR-model (1). Neither adding
more lags of endogenous and exogenous variables, nor leaving out certain exogenous
variables (e.g. estimating the VAR only with order type dummies, and leave out order
sizes) do not change the main conclusions obtained in the resiliency section.
Table 4 present the estimation results of the VAR model in Equation (1) with the
vector of endogenous variables yt = f ln (At ) ; ln (Bt ) ; ln (ADt ) ; ln (BDt ) ; ln (Durt )g
and the vector of exogenous variables xt = fM O_B1t ; :::, Canc_S2t , ln (Size_M O_B1t ),
:::, ln (Size_Canc_S2t ), T 01t ; :::T 17t g. The …rst column in the table gives the righthand-side variables, the next columns are the …ve equations in the VAR, the header
shows the endogenous variable. For each coe¢ cient, we give the unweighted average
value of the coe¢ cient across the 35 stocks, followed - between brackets - by the number
of stocks out of 35 for which the coe¢ cient is signi…cantly positive (…rst element) and
negative (second element) at the 5% level. For instance, the …rst element in the table,
0:0980 (0; 35), should be interpreted as follows: the average coe¢ cient across the
35 stocks of the variable ln (At 1 ) in the equation for ln (At ) is 0:0980. For 0
stocks, the coe¢ cient is signi…cantly positive, for 35 stocks it is signi…cantly negative.
For 0 stocks (the di¤erence between the 35 stocks in our sample and the sum of the
two elements 0+35), it is insigni…cant. To save space, we do not report results in the
table for the log order size variables and time-of-day dummies, but they are included
in the estimation (and later in the simulation). Below the table, the average adjusted
R2 across stocks are reported for the di¤erent equations.
As the main interest of our analysis is in resiliency (see next subsection), we discuss
the estimation results in a relatively concise way and just highlight the main points.
We consider coe¢ cients to be signi…cant if they are so for a majority of the stocks (see
the numbers between brackets). Table 4 clearly provides evidence for the existence
of a number of relationships between the endogenous variables. First, the estimates
show a relation between both sides of the market. Ask returns are signi…cant in the
bid return equation and the other way around. Depth at the ask (bid) side of the
market is positively related to lagged depth at the bid (ask) side, in line with the
theory in Parlour (1998) who shows that traders look at both sides of the market
when determining their order choice between market and limit orders (and this choice
determines liquidity in a limit order market). Secondly, our results also demonstrate an
interaction between the dimensions of liquidity, as argued by Harris (1990). This can
be seen by noting that ln (AD) has a negative sign in the ask- and bidreturn-equations,
while ln (BD) has a positive sign. The intuition is as follows. Suppose depth at the
best ask is high and a seller arrives. Since the execution probability of an additional
13
sell limit order joining the queue at the best ask is small, she will not submit such
order. She will either improve the best ask (implying a negative return on the ask) or
submit a market order. If in the latter case all depth at the best bid is consumed, this
implies a negative return on the bid. Similarly, if depth at the best bid is higher, ceteris
paribus, a buyer will either submit a market order (increasing the best ask if all depth
at the best ask in consumed), or improve the best bid, implying a positive return on
the best bid. Furthermore, we …nd signi…cant coe¢ cients of returns in the equations
for depth, positive for the ask return, negative for the bid return. The intuition is that
an undercutting of the best ask by a limit order, implying a negative return on the
best ask, will lower depth at the best ask (since the size of the order becomes the new
depth and this is most likely smaller than the depth that was present before the order).
This in turn leads to a positive correlation between ln (A) and ln (AD). Turning
to duration, we …nd a no relation between lagged duration and returns or depth. This
is in line with Engle and Patton (2004) and the predictions in Easley and O’Hara
(1992) that long durations do not reveal information. In the duration equation, we
…nd signi…cant coe¢ cients for returns, and for the …rst lag of depth. Finally, in line
with the literature, all endogenous variables are autocorrelated, the coe¢ cients of their
own …ve lags are signi…cant. However, their magnitude quickly decreases. For returns
the autocorrelations are negative, while for depths and duration they are positive.
Secondly, as in Engle and Patton (2004), we …nd evidence of cointegrating behavior,
since the lagged log spread LSpreadt 1 is signi…cant in the return equations. Its sign is
as expected. If the spread after the previous period was large, the ask can be expected
to decrease and/or the bid to increase at the next order. This causes the spread to
become narrower and return to its equilibrium value. Also, if the spread is larger, it
is easier to undercut the best prices in the limit order book. Our results con…rm this
intuition since the lagged spread has a negative sign in the ask return equation and
a positive sign in the bid return. In the duration equation, the spread has a negative
sign, meaning that when the lagged spread was larger, the duration between orders
will become smaller.
The signs of the order type variables in the return equations are as can be expected
from their de…nition. Focusing on the types that will be used as measures for shocks, we
…nd that type M O_B1 orders have a positive sign in the ask return equation, implying
a large positive ask return of 0.4339% (recall this is on an order by order basis), while
type M O_S1 orders have a negative sign in the bid return equation, i.e. a decrease in
the best bid by 0.4078%. Now turning the e¤ect of di¤erent order types on the depth
at the best prices, we …nd that type M O_B1 orders have a positive sign in the AD
equation, while type M O_S1 orders have a positive sign in the BD equation. This
can be interpreted as evidence that the limit order book behind the three …rst prices
14
is deep. Finally, the type of the order has only a limited impact on the duration since
only few coe¢ cients of order types are signi…cant in the duration equation.
Please insert Table 4 around here.
5.2
Resiliency
In this section, we discuss the results for resiliency. As explained in Section 3.2,
resiliency is analyzed by simulating the paths of di¤erent dimensions of liquidity after
a liquidity shock. This simulation is performed on basis of the estimated VAR-model
in equation (1). We do the simulation separately for each stock and then compute
the median, and 5% and 95% percentile across the 35 stocks in our sample. In the
discussion, we make a distinction between a shock occurring on the ask side of the
market and one occurring on the bid side. Such shock is measured by a type M O_B1
order on the ask side and a type M O_S1 order on the bid side. We …rst discuss
results for our main model. Next, we transform the model to analyze the behavior
of the bid-ask spread and depth imbalance after a liquidity shock. A third subsection
investigates midprice returns and average depth. In a …nal subsection, we make a
distinction between more and less frequently traded stocks, and stocks with a large
and small relative tick size.
5.2.1
Responses a Liquidity Shock
Figure 1 presents the impulse responses of di¤erent variables after a liquidity shock,
during a period of 15 orders (x-axis) after the shock. We de…ne a liquidity shock as an
order of type M O_B1 in Panel A (which measures a shock on the ask side), and as an
order of type M O_S1 in Panel B (measuring a shock on the bid side). The …rst graph
in each panel draws the impact of such shock on the return (in %) on the best ask
price ln (A), the second graph the impact on the return (in %) on the best bid price
ln (B). The third and fourth graph display the impact of the shock on the log depth
in euro at the best ask and best bid price, ln (AD) and ln (BD) respectively. The
…fth graph shows the impact on log duration (in seconds) ln (Dur). The simulation
for the impulse responses is done for each stock separately according to the procedure
outlined in Appendix A, full lines in the graphs represent the median value of the
responses across the 35 stocks in our sample, the upper (lower) dashed lines are the
95% (5%) percentile across stocks. The dotted straight lines are the steady-state levels,
measured as the average value of the variable over the sample period. In our discussion
below, we focus mainly on the median value (full line), if not stated explicitly otherwise,
all statements should be interpreted in this way.
15
Please insert Figure 1 around here.
The return graphs demonstrate that a liquidity shock has a long-term e¤ect on the
best prices in the book. This e¤ect is however realized very quickly. More speci…cally,
after a type M O_B1 order, there is a positive return on the best ask, but in the
periods after this initial impact of the shock, there is a small reversal: in subsequent
periods, returns on the best ask are slightly negative. A similar response is found for
a shock at the bid side M O_S1: a negative return on the best bid immediately after
the shock, followed by a small reversal. One interpretation for this long-run e¤ect on
prices is the presence in the market of informed traders that are trading on perishable
information. We further …nd some evidence that the impact of a shock also a¤ects
the other side of the market than where the shock occurred. There is a small positive
return on the best bid following a type M O_B1 order, while after an aggressive sell
order (type M O_S1) a small negative return on the best ask is found. These small
e¤ects disappear after some periods. They o¤set part of the e¤ect on the own side of
the market on the spread6 . The e¤ect on the other side of the market is con…rming
the empirical predictions of the model of Rosu (2009).
The impact of a liquidity shock on log depths is found mainly at the side of the
market at which the shock occurs. More speci…cally, depth increases at that side of
the market: after an M O_B1 order, ln (AD) increases. After this initial increase,
however, ln (AD) moves gradually back to its steady state, which it achieves around
15 orders after the shock. A symmetric result applies for ln (BD) after a M O_S1
order. An interpretation can be given to this result. At …rst, just after the shock, the
limit order book behind the best prices is deep. In subsequent periods new liquidity (in
the form of new limit orders) is provided to the market after it has been consumed by
the aggressive order. These orders slighty improve the best prices, while the depth of
these orders becomes the new depth at the best prices. At the other side of the market,
depth remains more ‡at and the impact of the shock is small since median log depths
do not move far away from their steady state.
The last graphs in Figure 1 show that the duration between orders declines in the
periods immediately after the shock. However, very quickly duration increases and
stabilizes again around its steady state value.
Furthermore, our results demonstrate the interaction and complementarity between
di¤erent dimensions of liquidity. Right after the shock, there is a long-run e¤ect on
prices as a result of which the spread becomes wider than in the steady state, such that
one aspect of liquidity (spread) deteriorates. However, at the same time depth increases
which implies an improvement of another dimension of liquidity. In later periods, we
6
A negative return on the ask means an improvement of the best ask. On the other hand, a positive
return on the bid points to an improvement of the best bid. Both imply a narrower spread.
16
have the opposite evolution: depth declines to its steady-state, but also the spread
reverts and becomes smaller. Also, we …nd evidence for asymmetries in the impact
of shocks. Shocks on the ask side of the market have a larger impact than shocks on
the bid side. This can be seen by comparing the responses of ln (A), ln (AD) and
ln (Dur) in Panel A with ln (B), ln (BD) and ln (Dur) in Panel B. The median values
in Panel A are always larger, in other words, a shock resulting from an aggressive buy
order has a larger impact than a shock resulting from an aggressive sell order. The
speed of recovery of the variables, or resiliency, is very similar, however. Escribano
and Pascual (2006) also report asymmetric adjustments of ask and bid prices and
asymmetric impacts of buyer and seller initiated trades. We show that this conclusions
extends to other dimensions of liquidity such as depth and duration.
5.2.2
Spread and Depth Imbalances
An interesting transformation of our model allows for analyzing the behavior after a
liquidity shock of the bid-ask spread Spread (in cents) and the log depth imbalance,
ln (BADR), de…ned as the log of the ratio of depth at the best bid and ask (in euro)
ln (AD=BD). ln (BADR) = 0 can then be interpreted as a balanced limit order book
(i.e. depth at best bid and ask are equal), ln (BADR) > 0 means that there is an
imbalance at the bid side (i.e. larger depth at the bid side, compared to the ask side)
and ln (BADR) < 0 implies an imbalance at the ask side. We estimate a similar model
as Equation (1), but rede…ne the vector of endogenous variables as
yt = fSpreadt ; ln (BADRt ) ; ln (Durt )g
The vector of exogenous variables xt , number of lags and estimation procedure remain
the same as before. Impulse responses after a liquidity shock are computed using the
procedure outlined in Appendix A. Figure 2 draws these impulse response functions
during a period of 15 orders (x-axis) after the shock. Liquidity shocks are again
measured by an order of type M O_B1 in Panel A (shock on the ask side), and by an
order of type M O_S1 in Panel B (shock on the bid side). The …rst graph in each panel
draws the impact of such shock on bid-ask spread Spread (in cents), the second graph
the impact on ln (BADR), the log of the ratio of depth at the best bid and ask (in euro)
ln (AD=BD). The third graph shows the impact on log duration (in seconds) ln (Dur).
The simulation for the impulse responses is done for each stock separately, full lines in
the graphs represent the median value of the responses across stocks, the upper (lower)
dashed lines are the 95% (5%) percentile across stocks. The dotted straight lines are
the steady-state values, measured as the average value of the variable over the sample
period.
17
Please insert Figure 2 around here.
We …nd that the spread increases directly after the shock, to 11 cents (or ticks, since
tick size in 1 cent for all stocks), which is almost double the value of the steady-state
spread. This shows again that our measure for a liquidity shock indeed models indeed
a signi…cant event. Also, the order book becomes imbalanced: after a shock on the
ask (bid) side, ln (BADR) becomes negative (positive), implying an larger depth at
the ask (bid) side. The market is resilient, however. Both spread and depth imbalance
revert to their steady-state values within around 15 orders after the liquidity shock.
For log duration the results are in line with the main model above.
5.2.3
Midprice Returns and Average Depth
In a next subsection, we implement another transformation of our model. More
speci…cally, denote the return (in %) on the midprice M (the average of ask and bid)
as ln (Mt ), and let ln (BAD_Avgt ) be the log of the average depth at the best ask
and bid price. We then rede…ne the vector of endogenous variables as follows:
yt = f ln (Mt ) ; ln (BAD_Avgt ) ; ln (Durt )g
The vector of exogenous variables xt , number of lags, estimation procedure and
computation of impulse responses remains the same as before. Figure 3 presents the
impulse response functions, the interpration of this …gure is identical to the one in the
previous subsection.
Please insert Figure 3 around here.
From this …gure, it can be seen that a liquidity shock has a log-run impact on
midprices. After a liquidity shock on the ask, the midprice increases. In subsequent
periods, returns are slightly negative meaning a small reversal of midprices. After a
shock in the bid side, a symmetric result is found, which is slightly smaller in magnitude.
Further, a liquidity shock induces an increase in the log average depth ln (BAD_Avg).
It returns relatively quickly (within 15 orders) to its steady-state after a shock at the
bid side, but it takes longer after a shock on the ask side. This con…rms earlier …ndings
on asymmetries, i.e. that shocks on the ask side have a stronger impact.
5.2.4
Trading Frequency and Relative Tick Size
While the percentiles in the …gures above already provided an idea of the di¤erences
in impact of liquidity shocks across stocks, we discuss this issue more in detail in this
18
subsection. More in particular, we …rst divide the 35 stocks in our sample …rst on basis
of their trading activity, i.e. the average daily number of trades (see Table 2). The
most frequently traded subsample M ost_F req contains the 10 stocks that are most
frequently traded, the least frequently traded subsample Least_F req then comprises
the 10 stocks who are least frequently traded (the 15 middle stocks are not considered).
The distinction between frequently and less frequently traded stocks motivated by the
model of Foucault, Kadan and Kandel (2005), who show that resiliency is decreasing in
the order arrival rate. The reasoning is that, when the order arrival rate increases, limit
order traders become less aggressive in their price improvements. As a consequence,
more orders are needed to bring down the spread to its competitive level, implying
resiliency declines.We use trading frequency, i.e. the average number of daily trades, as
a proxy for the arrival rate. Spierdijk (2004) shows that considerable di¤erences exist
between the price impact of trades for frequently and infrequently traded stocks. For
both subsamples, we repeat the same estimation and simulation as above. Important
to stress is that the steady-state level is computed seperately for the two subsamples.
The reason is that we are interested in reversion of liquidity dimensions to their normal
level (steady state) and this level can be di¤erent across subsamples. To save space, we
do not report the estimation results or plots of the impulse responses, but just discuss
the main …ndings. The results are available upon request. The key result of this
analysis con…rms the result in the model of Foucault, Kadan and Kandel (2005): we
…nd a negative relation between resiliency and trading frequency. Although almost all
liquidity dimensions7 (i.e. spread, depth8 , depth imbalance, average depth) do revert to
their steady state within at maximum 30 periods, we …nd that reversion to the steadystate occurs much faster, i.e. it takes less orders after the shock, for less frequently
traded stocks. Given our and Foucault et al.’s de…nition of resiliency, this implies
that resiliency is thus higher for less frequently traded stocks. Only for duration, we
…nd little di¤erence. Secondly, we …nd some di¤erences in magnitude. The impact of a
shock on returns is larger for stocks in the Least_F req group, but still realized quickly.
The impact on depth however is smaller for these stocks, compared to frequently traded
stocks. The explanation for these …ndings is as follows. For frequently traded stocks,
the second best price in the book and those beyond are closer to the best one, while
for the Least_F req stocks, the di¤erence between the subsequent prices in the book
is larger. Furthermore, depth is more evenly distributed in the book for Least_F req
stocks than for M ost_F req stocks for which depth beyond the best prices is larger. For
duration, the impact of a shock is larger for Least_F req stocks. Important to note
is that while the recovery of variables, is expressed in number of orders for the two
subsamples, this takes much more time expressed in seconds for Least_F req stocks
7
8
Recall that we do not investigate reversal for prices, as they are non-stationary.
The only exception to this result is ln (BD) :
19
than in the case of M ost_F req stocks, since the time between orders is larger for the
latter. In other words the period of 15 orders is much longer (in seconds) for these
Least_F req stocks, since the duration between orders is larger.
Next, we create two subsamples based on relative tick size. The relative tick size
Rel_tick for each stock is de…ned as the ratio of the tick size (in cent) to the average
price midpoint (in euro):
Rel_ts =
tick size
average midprice
Creating separate groups based on the minimum price variation is in line with the
model of Foucault, Kadan and Kandel (2005). They show that there exists a link
between the resiliency of an order-driven market and tick size. More speci…cally, a
smaller tick size reduces resiliency9 . The reasoning is that when the tick size is larger,
traders need to improve prices by more than they would with a smaller tick size in order
to obtain price priority. This spread improvement e¤ect causes the bid-ask spread to
narrow faster between transactions, making the market more resilient. The subsample
Small_T ick contains the 10 stocks with the smallest relative tick size, the subsample
Large_T ick the 10 stocks with the smallest relative tick size (the 15 middle stocks
are again not considered). Again, we compute the steady-state level separately for
the two subsamples. To save space, we do not report the estimation results or plots
of the impulse responses, but discuss the main …ndings. We …nd that - as predicted
theoretically by Foucault - that resiliency is higher for stocks with a larger relative
tick size. For the Large_T ick subsample, all liquidity dimensions (i.e. spread, depth,
depth imbalance, average depth) return faster to their respective steady-state levels,
compared to the Small_T ick subsample. Secondly, we also …nd some di¤erences in
magnitude. The results show that the impact of an aggressive order on returns is found
to be much larger for large tick stocks. Secondly, the impact of aggressive orders on
depth is also larger for large tick stocks. This is the case both at the side of the market
at which the aggressive order was submitted as on the other side. This comparison
between both groups for returns and depth suggests that the larger relative tick size
is a binding constraint for this group of stocks. These results are in line with the
literature that investigates tick size. When faced with a smaller tick size, traders use
more intermediate prices, but the depth at these prices is lower. Finally, the impact of
a shock on duration is comparable for stocks with large and small tick size.
9
To be precise, they prove that a market having a minimum price variation is more resilient that
a market without.
20
6
Conclusion
In this paper we analyze resiliency in an order-driven market. We develop a VARmodel that captures di¤erent dimensions of liquidity, being prices, depth, duration.
Our econometric model allows for taking into account relations between bid and ask
sides of the market, as well as between liquidity dimensions. We used the estimated
VAR to simulate liquidity shocks, measured by very aggressive buy and sell orders that
at least consume all depth at the …rst three ask and bid prices in the book. A liquidity
shock is found to have a permanent e¤ect on prices at the side of the market at which
the shock occurred, but the impact is realized almost instantaneously. However, in line
with Rosu (2009), we also document an e¤ect - though smaller in magnitude - on the
best price at the other side of the market, which partly o¤sets the e¤ect of the own
side. Depth at the best prices increases after a liquidity shock, but mainly at the side
of the market of the shock. Duration declines just after a negative shock, but increases
quickly again to converges within a few orders to its average value value.
Our results clearly demonstrate the importance of incorporating di¤erent dimensions of liquidity in the analysis. After a liquidity shock, we …nd a permanent e¤ect on
prices, with returns (in absolute value) of initially around 0.4% (realized on an order
by order basis) and a widening of the spread (almost doubling compared to its steadystate value). On the other hand, depth at the best prices increases, initially with up
to 20%. This implies on the one hand an derioration of liquidity since the spread
widens, but at the same time, another dimension of liquidity improves since depth at
the best prices increases. In subsequent periods, we observe the opposite evolution: the
spread reverts (declines) back to the steady-state level but also depth declines. These
results demonstrate the need to account for interaction and complementarity between
di¤erent dimensions of liquidity. The Spanish market is found to be resilient, since all
dimensions of liquidity revert to their steady-state level within around 15 orders after
the shock. Furthermore, an analysis of liquidity should also allow for asymmetries in
dynamics at bid and ask side of the market, since we …nd that shocks at the ask side
have a larger impact compared to stocks on the bid side. These …ndings thus comprise
relevant recommendations for theoretical modeling and empirical research.
Furthermore, some di¤erences between groups of stocks are found. The e¤ect of
shocks on both returns and depth is larger for stocks with a larger relative tick size.
This may indicate that tick size is a binding constraint for these stocks. The impact
of a shock on returns is also larger for less frequently traded stocks, but still realized
relatively quickly. The impact on depth however is smaller for these stocks.
Concluding, the analysis in this paper indicates that despite the absence of market
makers with an obligation to maintain an orderly market and provide liquidity, pure
order-driven markets seem able to cope with large liquidity shocks. They recover
21
quickly from the impacts of such shocks. In other words, the limit order book is
able to provide su¢ cient liquidity to the market. As a quali…cation, it needs to be
remarked, however, that although the stocks in our sample account for both a large
part of trading and market capitalization at the Spanish Stock Exchange, we did not
consider very infrequently traded stocks.
22
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23
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24
25
M O_B1
M O_B2
M O_B3
LO_B1
LO_B2
LO_B3
Canc_B1
Canc_B2
Canc_B3
Sell orders are classi…ed symmetrically in the same way as buy orders from M O_S1 to Canc_S3:
Panel B: Sell Orders
market buy order that consumes at least all depth until the 3th ask price (or a further price) and part of subsequent level
market buy order that consumes minimum all depth at the best ask price and maximum part of the depth at the 3th ask price
market buy order that consumes part of the depth at the best ask price
limit buy order that improves the best bid price
limit buy order that increases depth at the best bid price
limit buy order that queues in the book beyond the best bid price
cancellation or modi…cation of buy order that consumes all depth at the best bid price
cancellation or modi…cation of buy order that consumes part of the depth at the best bid price1
cancellation or modi…cation of buy order beyond the best bid
Notation De…nition
Panel A: Buy Orders
Note: This table gives an overview of the various order types. Panel A (Panel B) presents the various buy (sell) order types. The notation is given in the …rst column,
the de…nition of the type in the second column.
Table 1: De…nition of the Variables
Table 2: Sample: Descriptive Statistics
Note: This table presents the descriptive statistics of the sample. The …rst column gives the code
of the stock, while columns 2 and 3 in the table present the market capitalization (in 1000 euro) at
the end of the year 2000 and the total trading volume (in 1000 euro) during 2000. The next four
columns show the average daily number of trades, the average trade size (in shares), the average
duration (in seconds) between orders, the average bid-ask spread and the average midprice, where
averages are computed over the sample period. For all stocks, the tick size is equal to 0.01 euro.
Market Cap
Annual Trading
# Trades
Trade Size
Duration
Spread
Midprice
ACE
2,590,264
1,362,467
342
1751
48.93
3.02
8.99
ACR
1,153,006
866,086
370
802
41.62
3.17
9.3
ACS
1,607,952
1,801,030
249
664
51.61
13.7
27.11
ACX
1,925,285
2,518,999
329
650
40.52
13.07
32.09
AGS
1,818,759
887,916
233
875
59.01
6.63
14.19
ALB
1,969,448
1,288,164
189
910
58.35
19.03
27.55
ALT
5,040,279
7,958,527
773
1832
20.93
4.05
16.31
AMS
4,663,754
6,940,124
925
1552
16.58
2.84
10.23
ANA
2,583,331
2,135,196
288
531
45.54
14.68
38.48
BBVA
50,654,255
61,753,870
2024
3944
8.58
1.71
16.07
BKT
2,709,682
5,406,253
559
490
26.33
11.73
44.19
CAN
2,252,616
2,861,239
148
1191
79.40
12.06
20.97
CTG
600,639
782,829
369
1316
34.48
7.29
18.94
DRC
1,999,276
1,979,024
440
1853
33.82
3.6
9.94
ELE
19,216,351
28,783,198
1496
2549
11.52
2.45
20.8
FCC
2,426,060
1,637,996
323
695
41.43
8.63
19.56
FER
1,907,601
1,144,835
397
538
40.02
4.96
13.92
IBE
12,035,682
12,600,314
758
3451
21.08
2.57
13.8
IDR
1,486,405
2,358,084
531
654
28.35
5.87
17.81
MAP
1,228,587
921,525
117
1493
101.95
12.12
18.27
NHH
1,565,881
1,780,277
232
2022
57.40
6.26
13.09
POP
8,056,418
6,826,078
464
1092
29.63
8.57
34.72
PRS
3,851,100
3,369,875
555
883
27.40
7.95
23.68
REE
1,359,464
1,075,761
280
662
51.85
4.49
10.66
REP
20,779,096
30,377,832
1655
2655
10.45
2.41
20.32
SCH
51,476,555
52,786,425
2962
3804
6.24
1.34
11.45
SGC
2,037,499
4,146,904
546
451
23.93
12.24
33.37
SOL
2,034,392
1,309,640
293
1199
49.64
4.7
10.92
TEF
76,396,509
152,323,837
6609
3505
2.70
1.56
21.9
TPI
2,098,936
5,192,664
1009
1384
15.35
2.41
8.83
TPZ
563,465
3,708,955
848
1915
19.48
1.6
4.92
TRR
7,206,684
32,774,256
4597
814
3.52
4.75
34.53
UNF
5,956,481
6,864,353
414
2159
36.33
4.4
20.47
VAL
836,328
914,042
254
1580
59.31
3.01
6.71
ZEL
1,989,150
5,817,857
1776
594
8.35
4.25
34.95
26
Table 3: Model Variables: Descriptive Statistics
Note: Panel A presents the descriptive statistics of the endogenous variables, i.e. their mean, median
and standard deviation across the stocks in our sample.
ln (A) and ln (B) are the returns of
the best ask and bid in %, ln (AD) and ln (BD) the log of depth in euro at the best ask (bid),
ln (Dur) the log of duration in seconds since the previous order. Spread is the bid-ask spread in
cents, ln (BADR) is the log of the ratio of depth at the best bid and ask, ln (M ) is the return on
the midprice in % and ln (BAD_Avg) is the log of the average of depth in euro at the best bid and
ask. Panel B shows the frequency of occurring, in % of the total number of orders, of the di¤erent
order types, and the average order size (in shares and in logs) of each order type. In both panels,
all statistics are computed after winsorizing at the 99.5% level.
Panel A: Endogenous Variables
ln(At )
ln(Bt )
Mean
Median
S.d.
-0.0004
0.0000
0.0945
0.0002
0.0000
0.0943
Mean
Median
S.d.
Spreadt
6.3600
4.9100
4.4200
ln(BADRt )
0.0150
0.0103
1.7770
ln(ADt )
ln(BDt )
ln(Durt )
9.4985
9.6060
1.2895
9.5134
9.6235
1.2684
1.7706
2.3428
2.5158
ln(Mt ) ln(BAD_Avgt )
-0.0001
9.5060
0.0000
9.6147
0.0772
1.2790
Panel B: Exogenous Variables
Buy Orders
Notation
M O_B1t
M O_B2t
M O_B3t
LO_B1t
LO_B2t
LO_B3t
Canc_B1t
Canc_B2t
Canc_B3t
Freq
Size
0.06% 11069
5.76% 2687
15.17% 1050
8.24% 2104
5.55% 1789
6.32% 1487
2.12% 1494
1.07% 1557
6.10% 1541
Sell Orders
ln(Size)
Notation
8.423
6.514
5.773
6.582
6.618
6.468
6.400
6.485
6.582
M O_S1t
M O_S2t
M O_S3t
LO_S1t
LO_S2t
LO_S3t
Canc_S1t
Canc_S2t
Canc_S3t
27
Freq
Size
0.08% 10845
6.00% 2836
17.10% 833
7.14% 1956
4.98% 1795
6.08% 1381
1.87% 1451
0.90% 1530
5.47% 1572
ln(Size)
8.455
6.534
5.543
6.516
6.555
6.371
6.316
6.403
6.543
Table 4: Estimation Results VAR-model
Note:
P5
This table present the estimation results of equation (1):
yt = A0 +
A
y
+
B
x
+
LSpread
+
u
.
The
vector
of
endogenous
variables
is yt =
0 t
t 1
t
l=1 l t l
f ln (At ) ; ln (Bt ) ; ln (ADt ) ; ln (BDt ) ; ln (Durt )gwith
ln (At ) ( ln (Bt )) the return on the
best ask (bid) price in % at time t, ln (ADt ) (ln (BDt )) the log of depth in euro
at the best ask (bid) at time t, ln (Durt ) the log of duration in seconds since the
previous order and ln the natural logarithm.
The vector of exogenous variables is
xt = fM O_B1t ; :::; Canc_S2t ; ln (Size_M O_B1t ) ; :::; ln (Size_Canc_S2t ) ; T 01t ; :::T 17t g. The
de…nition of the order types M O_B1t ; :::; Canc_S2t can be found in Table 1; the next set of
variables in xt are the corresponding log of order sizes (in number of shares); T 01t ; :::T 17t are
time-of-dummies, equal to one if the order is in period 1,...,17 of the day, respectively, and zero
otherwise. LSpreadt 1 = ln (At ) ln (Bt ). Ai , Bi and are the coe¢ cient matrices to be estimated.
Estimations are done per stock using least squares, Newey-West standard errors are computed. The
…rst column shows the right-hand-side variables, the next columns the VAR equations (endogenous
variables are shown in the header). For each variable, we show the unweighted average value of the
coe¢ cient across stocks, followed - between brackets - by the number of stocks out of 35 for which
the coe¢ cient is signi…cantly positive (…rst element) and negative (second element) at the 5% level.
All endogenous variables, log order sizes and log durations are winsorized at the 99.5% level.
ln(At )
ln(Bt )
ln(ADt )
ln(BDt )
ln(Durt )
ln(At
1)
-0.0980
(0,35)
0.0220
(35,0)
0.4030
(35,0)
0.0510
(16,0)
-1.0560
(0,35)
ln(At
2)
-0.0540
(0,35)
0.0110
(33,0)
0.2730
(35,0)
0.0630
(16,0)
0.6410
(35,0)
ln(At
3)
-0.0370
(0,35)
0.0060
(23,0)
0.1960
(35,0)
0.0530
(12,0)
0.4310
(30,0)
ln(At
4)
-0.0250
(0,35)
0.0050
(18,0)
0.1470
(33,0)
0.0500
(18,0)
0.3520
(28,0)
ln(At
5)
-0.0170
(0,35)
0.0040
(20,1)
0.0650
(24,0)
0.0290
(10,0)
0.2130
(21,0)
ln(Bt
1)
0.0220
(35,0)
-0.0990
(0,35)
-0.0600
(0,17)
-0.3970
(0,35)
0.9840
(35,0)
ln(Bt
2)
0.0110
(32,0)
-0.0530
(0,35)
-0.0840
(0,18)
-0.2660
(0,34)
-0.6750
(0,34)
ln(Bt
3)
0.0070
(28,0)
-0.0360
(0,35)
-0.0690
(0,15)
-0.1830
(0,33)
-0.4680
(0,31)
ln(Bt
5)
0.0030
(15,1)
-0.0230
(0,35)
-0.0530
(0,13)
-0.1350
(0,33)
-0.4130
(0,35)
ln(Bt
5)
0.0020
(14,0)
-0.0160
(0,35)
-0.0380
(0,9)
-0.0500
(0,13)
-0.2710
(0,29)
ln(ADt
1)
0.0020
(35,0)
0.0010
(29,1)
0.7680
(35,0)
-0.0080
(0,27)
-0.0400
(0,31)
ln(ADt
2)
-0.0010
(0,28)
0.0000
(0,4)
0.0430
(35,0)
0.0040
(11,0)
0.0080
(5,6)
ln(ADt
3)
0.0000
(0,16)
0.0000
(1,2)
0.0170
(35,0)
0.0020
(7,0)
0.0100
(8,2)
ln(ADt
4)
0.0000
(0,10)
0.0000
(1,3)
0.0140
(35,0)
0.0020
(6,0)
0.0070
(2,1)
ln(ADt
5)
-0.0010
(0,24)
0.0000
(0,2)
0.0230
(35,0)
0.0040
(16,0)
0.0020
(3,2)
ln(BDt
1)
-0.0010
(0,28)
-0.0020
(0,35)
-0.0080
(0,27)
0.7530
(35,0)
-0.0530
(0,32)
ln(BDt
2)
0.0000
(3,4)
0.0010
(26,0)
0.0040
(12,0)
0.0450
(35,0)
0.0170
(11,4)
ln(BDt
3)
0.0000
(3,3)
0.0000
(11,0)
0.0020
(7,1)
0.0170
(34,0)
0.0100
(5,0)
ln(BDt
4)
0.0000
(2,4)
0.0000
(9,0)
0.0020
(8,0)
0.0150
(35,0)
0.0040
(4,1)
ln(BDt
5)
0.0000
(5,0)
0.0000
(17,0)
0.0040
(18,0)
0.0230
(35,0)
0.0080
(7,1)
ln(Durt
1)
0.0000
(2,7)
0.0000
(11,1)
-0.0010
(1,11)
0.0000
(0,6)
0.0170
(23,6)
ln(Durt
2)
0.0000
(1,12)
0.0000
(16,0)
0.0000
(1,5)
0.0000
(3,2)
0.1080
(35,0)
ln(Durt
3)
0.0000
(0,11)
0.0000
(10,0)
0.0000
(0,7)
0.0000
(2,1)
0.0700
(35,0)
ln(Durt
4)
0.0000
(0,10)
0.0000
(15,1)
0.0000
(2,4)
0.0000
(2,3)
0.0700
(35,0)
ln(Durt
5)
0.0000
(0,4)
0.0000
(7,1)
0.0000
(2,5)
0.0000
(2,4)
0.0560
(35,0)
0.0140
(34,0)
-0.0160
(0,33)
1.2550
(35,0)
1.3770
(35,0)
1.9090
(35,0)
C
28
Table 4 (continued)
ln(At )
ln(Bt )
ln(ADt )
ln(BDt )
ln(Durt )
M O_B1t
0.4339
(35,0)
-0.0002
(0,0)
1.3310
(34,0)
-0.0230
(0,0)
-0.3433
(0,6)
M O_B2t
0.1734
(35,0)
0.1071
(35,0)
0.5631
(29,6)
-0.2867
(0,34)
-0.8168
(0,35)
M O_B3t
-0.0005
(0,0)
0.0020
(7,0)
-0.5087
(0,35)
0.0188
(11,0)
-0.5793
(0,35)
LO_B1t
0.0068
(35,0)
0.1004
(35,0)
-0.0012
(0,2)
-0.0288
(20,14)
-1.0720
(0,35)
LO_B2t
0.0032
(28,0)
0.0005
(0,0)
0.0009
(2,0)
1.0883
(35,0)
-1.3744
(0,35)
LO_B3t
0.0019
(6,0)
0.0029
(22,0)
0.0032
(2,0)
-0.0235
(0,18)
-0.7196
(0,35)
Canc_B1t
0.0017
(0,0)
-0.1502
(0,35)
0.0142
(0,0)
0.0145
(14,20)
-0.0318
(3,4)
Canc_B2t
0.0018
(0,0)
0.0006
(0,0)
0.0126
(0,0)
-0.7161
(0,35)
0.0234
(2,2)
Canc_B3t
0.0024
(15,0)
0.0022
(8,0)
0.0026
(0,0)
0.0049
(3,0)
-0.0366
(0,6)
M O_S1t
-0.0005
(0,0)
-0.4078
(0,35)
0.0018
(0,0)
1.2596
(34,0)
-0.4278
(0,6)
M O_S2t
-0.1061
(0,35)
-0.1664
(0,35)
-0.2856
(0,35)
0.5306
(29,5)
-0.8299
(0,35)
M O_S3t
0.0007
(3,0)
0.0024
(12,0)
0.0104
(6,0)
-0.5284
(0,35)
-0.4818
(0,35)
LO_S1t
-0.1027
(0,35)
-0.0046
(0,31)
-0.0204
(20,14)
-0.0036
(0,2)
-1.2030
(0,35)
LO_S2t
0.0019
(5,0)
-0.0007
(0,0)
1.1220
(35,0)
0.0006
(2,1)
-1.3410
(0,35)
LO_S3t
-0.0005
(0,1)
0.0006
(0,0)
-0.0233
(0,16)
0.0053
(2,0)
-0.5815
(0,35)
Canc_S1t
0.1604
(35,0)
0.0008
(1,0)
-0.0029
(15,20)
0.0188
(1,0)
-0.0143
(8,2)
Canc_S2t
0.0013
(2,0)
0.0009
(0,0)
-0.7542
(0,35)
0.0161
(0,0)
0.0461
(3,0)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
-1.7336
(0,35)
1.7459
(35,0)
-2.1463
(0,18)
-0.6536
(5,7)
-51.6209
(1,34)
ln(Size_M O_B1t )
...
ln(Size_Canc_S2t )
T 01t ... T 17t
LSpreadt
Adj R2
1
0.50543
0.50961
0.71097
29
0.69507
0.10115
30
Panel A: Responses to Liquidity Shock on the Ask Side M O_B1
Note: This …gure presents the impulse responses of a number of variables after a liquidity shock, during a period of 15 orders (x-axis) after the shock. This shock is
de…ned as an order of type M O_B1 in Panel A (measuring a shock on the ask side), and as an order of type M O_S1 in Panel B (measuring a shock on the bid side).
The …rst graph in each panel draws the impact of such shock on the return on the best ask price ln (A), the second graph the impact on the return on the best
bid price ln (B). The third and fourth graph display the impact of the shock on the log of depth in euro at the best ask and best bid price, ln (AD) and ln (BD)
respectively. The …fth graph shows the impact on log of duration (in seconds) ln (Dur). The simulation for the impulse responses is done for each stock separately,
full lines in the graphs represent the median value of the responses across stocks, the upper (lower) dashed lines are the 95% (5%) percentile across stocks. The dotted
straight lines are the steady-state values, measured as the average value of the variable over the sample period. The computational details for the impulse responses
can be found in Appendix A.
Figure 1: Impulse Responses after a Liquidity Shock
31
Panel B: Responses to Liquidity Shock on the Bid Side M O_S1
Figure 1 (continued)
Figure 2: Spread Model: Impulse Responses after a Liquidity Shock
Note: This …gure presents the impulse responses of a number of variables after a liquidity shock,
during a period of 15 orders (x-axis) after the shock. This shock is de…ned as an order of type
M O_B1 in Panel A (measuring a shock on the ask side), and as an order of type M O_S1 in Panel
B (measuring a shock on the bid side). The …rst graph in each panel draws the impact of such shock
on bid-ask spread Spread (in cents), the second graph the impact on ln (BADR), the log of the
ratio of depth at the best bid and ask (in euro) ln (AD=BD). The third graph shows the impact
on log of duration (in seconds) ln (Dur). The simulation for the impulse responses is done for each
stock separately, full lines in the graphs represent the median value of the responses across stocks,
the upper (lower) dashed lines are the 95% (5%) percentile across stocks. The dotted straight lines
are the steady-state values, measured as the average value of the variable over the sample period.
The computational details for the impulse responses can be found in Appendix A.
Panel A: Responses to Liquidity Shock on the Ask Side M O_B1
Panel B: Responses to Liquidity Shock on the Bid Side M O_S1
32
Figure 3: Midprice Model: Impulse Responses after a Liquidity Shock
Note: This …gure presents the impulse responses of a number of variables after a liquidity shock,
during a period of 15 orders (x-axis) after the shock. This shock is de…ned as an order of type
M O_B1 in Panel A (measuring a shock on the ask side), and as an order of type M O_S1 in Panel
B (measuring a shock on the bid side). The …rst graph in each panel draws the impact of such
shock on the return (in %) on the midprice ln (M ), with M de…ned as the average of the best ask
and bid. The second graph plots the impact on the log of the average of depth at the best bid and
ask (in euro) ln (BAD_Avg). The third graph shows the impact on log of duration (in seconds)
ln (Dur). The simulation for the impulse responses is done for each stock separately, full lines in
the graphs represent the median value of the responses across stocks, the upper (lower) dashed lines
are the 95% (5%) percentile across stocks. The dotted straight lines are the steady-state values,
measured as the average value of the variable over the sample period. The computational details for
the impulse responses can be found in Appendix A.
Panel A: Responses to Liquidity Shock on the Ask Side M O_B1
Panel B: Responses to Liquidity Shock on the Bid Side M O_S1
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Appendix A: Simulation Methodology: Computation of the Impulse Response Functions
Using the VAR model in Equation (1), the resiliency of the market can be investigated
by simulating a liquidity shock on basis of this model. More speci…cally, we start from
an order of type i, i 2 fM O_B1, M O_S1g, and compute the evolution of returns,
depths and duration during a period of h = 1::15 orders after the submission of the
initial order. In the simulation procedure, which is inspired by Escribano and Pascual
(2006), it is assumed that an aggressive order of type i is submitted in interval 9 (this
is between 12h45 and 13h15). After having estimated the VAR model in (1), the
simulation procedure proceeds through the following steps:
1. As stated, a liquidity shock will be measured by an order of type i in interval
9 of the day. This is simulated by setting the relevant order type and time of
day dummies equal to one. So, T 09t = 1, and either M O_B1 or M O_S1 = 1,
depending on the value of i. The corresponding order size is set equal to the
average order size of type i.
2. In a second step, we need to compute the initial values for the lags of the
endogenous variables (yt l ; l = 1::L), as well as the lags of the exogenous variables
(xt m , m = 1::M )10 . They are equated to their mean, conditional upon the
submission of an order of type i in period t. In this way, we consider in our
simulation the average market conditions around a liquidity shock.11
3. For each of the exogenous variables, a predicted value needs to be computed
for h = 1::15 periods after the shock (recall that one period refers to a trade).
For predicting the order type and order size variables, we calculate the average
value of each of these variables in period t + h, h = 1::15, conditional upon the
submission of a type i order in interval 9, which is indexed by t. Finally, we need
predicted values for the last group of exogenous variables, being the time of day
dummies. For them, we assume that the 15 orders following an aggressive order,
fall entirely in the ninth interval of the day (i.e. between 12h45 and 13h15), such
that T 09t+h = 1; h = 1::15. The summary statistics on duration indicate that
this is a plausible assumption. Moreover, it is not a restrictive assumption since,
as will turn out later, the coe¢ cients of the time of day dummies often are not
signi…cant, neither econometrically, nor from an economic point of view.
10
The values for x0 are those of the initial shock and are determined in step 1.
As a robustness check, we also take the unconditional mean over the sample period. The
interpretation of this alternative is that in this way the system starts from a “stationary”or “steadystate”situation, after which an unexpected shock arrives. The result are unchanged and are available
upon request.
11
34
4. On the basis of the VAR in Equation (1) and the necessary data computed in
the previous three steps, we are able to compute the impulse response functions
for bid and ask returns, depth at best bid and ask and duration for each period
h = 1; :::; 15 after the liquidity shock, i.e. the submission of the order of type i.
Note that for each period h, the value of Spread is updated in the following way:
Spreadt+h = Spreadt+h 1 + ln At+h
ln Bt+h .
35