Estimating permanent price impact via machine learning

Journal of Econometrics xxx (xxxx) xxx

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Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom

Estimating permanent price impact via machine learning✩ R. Philip Discipline of Finance, The University of Sydney, Camperdown, NSW, 2006, Australia

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Article history: Received 25 August 2017 Received in revised form 6 May 2019 Accepted 2 October 2019 Available online xxxx JEL classification: C45 C58 G14

a b s t r a c t In this paper, we show that vector auto-regression (VAR) models, which are commonly used to estimate permanent price impact, are misspecified and can produce conflicting and incorrect inferences when the price impact function is nonlinear. We propose an alternative method to estimate permanent price impact by modifying a reinforcement learning (RL) framework. Our approach assumes the data is stationary and Markov, but is otherwise unrestrictive. We obtain empirical estimates for our model using an iterative learning rule and demonstrate that our model captures nonlinearities and makes correct inferences. Crown Copyright © 2019 Published by Elsevier B.V. All rights reserved.

Keywords: Price impact Information content of a trade Machine learning Reinforcement learning

1. Introduction In asymmetric information models, traders interact with a specialist to set market prices. One critical component of asymmetric information models is that a trade conveys private information. To empirically estimate the private information content of a trade, researchers typically use a trade’s permanent price impact, estimated via the impulse response function of a vector auto-regression (VAR) framework. The VAR framework represents a multivariate linear time series model of trade and quote revisions. While researchers use a change in the midquote price to represent a quote revision, for the trade variable, researchers typically choose between (1) trade sign, (2) signed trade size or (3) a polynomial trade variable. Interestingly, the literature provides conflicting guidance on the choice of trade variable.1 We demonstrate that when permanent price impact has a nonlinear relation with trade size the choice of trade variable alters conclusions due to model misspecification. While the existing VAR framework may suffer from misspecification due to a nonlinear relation between price impact and trade size, the VAR framework may also not be applicable in today’s modern trading environment. Hasbrouck (1991a) ✩ This paper has benefited from the comments of Yacine Ait-Sahalia, Sean Anthonitz, Michael Goldstein, David Johnstone, Amy Kwan, Albert Menkveld, Maurice Peat, Christine Parlour, Andrew Patton, Artem Prokhorov, Talis Putnins, Ryan Riordan, Ioanid Rosu, Steve Satchell, Andriy Shkilko, Wing Wah Tham, Chen Yao, Bart Yueshen, Haoxiang Zhu and three anonymous referees. All errors remain my own. E-mail address: [email protected]. 1 The existing theoretical and empirical literature provides justification for all of the three trade variables. Theoretical models suggest that permanent price impact is linear with trade size (see Huberman and Stanzl (2004)), which supports using signed trade size as the trade variable. In contrast, the empirical literature shows that trade size contributes little incremental explanatory power above the trade direction when estimating permanent price impact (see Hasbrouck (2007), Jones et al. (1994)), which supports using trade sign as the trade variable. Hasbrouck (1991a) argues that the assumed linearity between the trade variable and quote updates may be a tenuous approximation and could lead to a misspecified model. To overcome this misspecification, Hasbrouck (1991a) advocates the use of polynomial terms as the trade variable to capture any nonlinear relations between quote updates and trade size. https://doi.org/10.1016/j.jeconom.2019.10.002 0304-4076/Crown Copyright © 2019 Published by Elsevier B.V. All rights reserved.

Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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asserts that the VAR framework is misspecified if the dealer possesses an informational advantage in their knowledge of the limit order book. In today’s trading environment, high frequency traders (HFTs) have largely replaced traditional dealers. Because of their speed advantage, HFTs have an information advantage over other traders and exploit this order book information in their trading strategies (see Brogaard et al. (2014) and Goldstein et al. (2018)). For this reason, the VAR framework can lead to incorrect conclusions in the modern trading environment. We develop a recursive model to estimate permanent price impact by modifying a reinforcement learning (RL) framework. RL offers several advantages over traditional VAR methods. RL only assumes that the data is Markov. Thus, RL models are highly flexible and can be used to model nonlinear processes, such as the relation between trades and quote updates, which are difficult to realistically model through a set of linear equations. RL is widely used in the computer science literature and its flexibility is highly suitable for modelling the limit order book (see Nevmyvaka et al. (2006)). Bertsimas and Lo (1998) and Nevmyvaka et al. (2006) demonstrate the efficacy of RL in solving a trader’s objective function. However, we modify the existing RL framework so we no longer optimize an individual’s objective function. Rather, we model the permanent price impact of a single trade. Specifically, we model a trade’s permanent price impact as the immediate price impact plus the sum of all permanent price impacts for all subsequent trades reacting to the initial trade. We empirically estimate our proposed model using an iterative learning rule and demonstrate that our model draws correct conclusions unlike the traditional VAR framework. If we only include trade sign as the trade variable, we show our recursive model yields similar empirical results to the VAR model.2 However, when the model specifications become more complex, such as including both trade size and order book variables, our recursive model yields correct conclusions whereas the VAR framework does not. Additionally, our recursive model can decompose the permanent price impact into a private information component and a component reflecting the trader’s ability to exploit public information contained in the limit order book. In contrast, a VAR framework cannot decompose these two features. Further, the computational time required to estimate our recursive model is significantly less than the computational time required to estimate a comparable VAR model. We document several findings, which have important implications for empirical and theoretical research. First, we document a strong nonlinear relation between permanent price impact and trade size. Specifically, as a function of trade size, the permanent price impact is positive, increasing and concave for all 20 sample stocks.3 We demonstrate the nonlinear relation between permanent price impact and trade size using two models. Initially, for a benchmark model, we propose the nonlinear VAR to capture nonlinear relations between permanent price impact and trade size. We then estimate permanent price impact using our proposed recursive model and show both models yield similar estimates. We report that 94% of our recursive model’s permanent price impact estimates are insignificantly different to the nonlinear VAR model’s estimates. Importantly, both models document a strong nonlinearity between trade size and permanent price impact. Second, we demonstrate that the traditional VAR framework is misspecified due to the nonlinear relation between price impact and trade size, which can result in incorrect research inferences. Specifically, when comparing price discovery across two market participants who differ in average trade size, using trade sign or signed trade size as the trade variable produces incorrect and conflicting conclusions.4 This finding is particularly important for research comparing price discovery across trading venues (e.g., Barclay et al. (2003)), between lit and dark markets (e.g., Comerton-Forde and Putnins (2015)), or between different trader types (e.g., Brogaard et al. (2019)) as erroneous research inferences can occur if any nonlinearities are not captured. Using an empirical sample of two market participants that are equally informed but differ in average trade size, we demonstrate the potential issues of a misspecified VAR model. Since both market participants are equally informed, they should have equivalent permanent price impacts. However, depending on the choice of trade variable, the traditional VAR model provides conflicting results. When we use trade sign as the trade variable in the VAR model, we find that the market participant with a larger average trade size has a larger permanent price impact. In contrast, we obtain the opposite result when we estimate a VAR model using signed trade size as the trade variable (i.e., the market participant with a smaller average trade size has a larger price impact). This discrepancy in conclusions is attributed to the nonlinear relations between permanent price impact and trade size. When we estimate the permanent price impact of a trade for both market participants using our flexible nonlinear VAR model or our recursive model, we correctly conclude that both participants cause equal permanent price impact. Third, using an empirical sample of two market participants that are equally informed but differ in the timing of trades, we demonstrate how the VAR model makes incorrect inferences.5 Since both traders have the same level of 2 For our investigation, we analyse 4 models: the traditional VAR model using trade sign as the trade variable (trade sign VAR), the traditional VAR model using signed trade size as the trade variable (trade size VAR), a flexible nonlinear VAR model (nonlinear VAR), and the recursive model. For the trade sign VAR, we use trade sign, or a +1 for buyer initiated trades and −1 for seller initiated trades, as the trade variable in the traditional VAR model proposed in Hasbrouck (1991a). Similarly, for the trade size VAR, we use signed trade size, or trade size × trade sign as the trade variable in the traditional VAR model of Hasbrouck (1991a). We extend the Hasbrouck (1991a) traditional VAR model to capture nonlinear relations and refer to this model as the nonlinear VAR. We provide a full description of the models we estimate in Appendix A. 3 These findings are consistent with Hasbrouck (1991a) and Engle and Patton (2004). 4 Using trade sign as the trade variable assumes that a step function exists between trade size and permanent price impact. On the other hand, using signed trade size as the trade variable assumes a linear relation between trade size and permanent price impact. 5 One participant times their trades based on information contained in the shape of the limit order book, while the second participant does not. We use depth imbalance, which is shown to predict short term price movements, as a proxy for limit order book information (see Engle and Patton (2004), Cao et al. (2009) and Goldstein et al. (2018)).

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private information, we expect the traders to have the same permanent price impact function. However, we show that the traditional linear VAR models (i.e. trade sign VAR or trade size VAR) again provide mixed results depending on the chosen specification. Further, the nonlinear VAR also suffers estimation issues due to overspecification.6 Additionally, the VAR framework fails to decompose the permanent price impact into a component attributed to private information, and a component reflecting the trader’s ability to exploit public information contained in the limit order book. On the other hand, we make correct conclusions using the recursive model. Specifically, both traders have equivalent permanent price impact functions, but operate at different points on the function. Further, we demonstrate the information contained in the shape of the limit order book is a significant component of permanent price impact. Notably, the cross sectional variation of permanent price impact due to the shape of the order book is approximately 3 times larger than the cross sectional variation of permanent price impact attributed to trade size. These findings highlight the importance of controlling for order book information when estimating the private information contained in a trade. Fourth, using our recursive model, we provide results that support the predictions from Mendelson and Tunca (2004), who provide one theoretical explanation for why a nonlinear relation between permanent price impact and trade size exists. They suggest agents are endogenous in the volumes they trade; Agents trade large volumes when the market is liquid, resulting in small price impacts per unit traded, and small volumes when the market is illiquid, resulting in large price impacts per unit traded. We provide empirical evidence supporting this theory and show a trade executing against the thin side of the order book has a larger permanent price impact than a trade executing against the thick side of the order book. Further, we demonstrate that traders minimize their price impact by conditioning their trades on the available liquidity. Traders are 13.7 times more likely to submit a large order when the market is liquid than when it is illiquid, and 9.9 times more likely to submit a small order than a large order when the market is illiquid. These findings extend CollinDufrense and Fos (2015), who suggest that informed traders selectively trade during times of higher liquidity. Our results show large traders also select times of higher liquidity when they trade. Finally, the concave monotonic relation we document between permanent price impact and trade size provides one possible explanation for why it is generally accepted that trade size contributes little incremental explanatory power above trade sign when estimating the VAR model (see Hasbrouck (2007)). A VAR model using signed trade size as the trade variable assumes a linear relation between order size and permanent price impact. Conversely, a VAR model estimated using trade sign as the trade variable assumes a step function, which better approximates the concavity in the price impact function we document. This paper is organized as follows. Section 2 presents the VAR framework and explains the generalization we propose. Section 3 develops the model by modifying an RL algorithm. Section 4 describes the data used in the analysis. Section 5 compares different VAR and recursive model specifications under multiple scenarios and discusses the results. Section 6 explains why permanent price impact has a nonlinear relation with size. Finally, Section 7 concludes. 2. The VAR models 2.1. VAR model with one market participant To study the information content of a trade, Hasbrouck (1991a) proposes the following VAR system: rt =

∞ ∑

αi rt −i +

∞ ∑

i=1

xt =

βi xt −i + ϵ1,t

i=0

∞ ∑

δi rt −i +

i=1

∞ ∑

φi xt −i + ϵ2,t .

(1)

i=1

Common implementations of this model define rt as the change in the natural logarithm of the midpoint that follows a trade at time t and xt is the trade indicator variable. The immediate price impact of a trade is given by β0 while the permanent price impact of a trade is obtained from the vector moving average (VMA) representation of the VAR model:

[ ] rt xt

[

a(L) = c(L)

][

b(L) d(L)

] ϵ1,t , ϵ2,t

(2)

where L is a lag operator. The permanent price impact of a trade, rtx+∞ , is measured by the cumulative impulse response function: rtx+∞ =

∞ ∑

bi ϵ2,t .

(3)

i=1

The model assumes a linear relation exists between the trade indicator variable, which is commonly the trade sign or signed trade size, and quote revisions. For clarity, when we use trade sign (signed trade size) as the trade variable, we refer to (1) as the trade sign VAR (trade size VAR). 6 A nonlinear VAR model that captures trade size, market conditions and trade lags quickly becomes overparametrized. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Hasbrouck (1991a) highlights that the model is misspecified when a nonlinear relation between quotes and the trade variable exists. As such, Hasbrouck (1991a) suggests using a quadratic VAR specification to capture any nonlinear relations between trade size and permanent price impact. To implement a quadratic VAR specification we replace the xt terms in (1) by a set {x0t , xt , x2t }, where x0t equals 1 (−1) for buyer (seller) initiated trades, xt is the signed trade size, and x2t = x0t |xt |2 . However, such a transformation raises three possible concerns. First, if the relation between trade size and permanent price impact is not quadratic, this model remains misspecified and the concerns raised in (1) persist. Second, estimating the impulse response function is not straightforward: A trade simultaneously shocks the error variables for x0t , xt and x2t . The implementation of these simultaneous shocks is constrained by the model specification and knowledge of how future residuals unfold.7 Third, there is a large number of parameters to estimate.8 Because of these potential estimation issues, for our benchmark model, we propose a similar approach to Engle and Patton (2004) to capture nonlinear relations. Specifically, we discretize trade size into n quantiles to obtain the following VAR, which we refer to as the nonlinear VAR: rt =

∞ ∑

αi rt −i +

n ∞ ∑ ∑

i=1

xkt =

∞ ∑

j

bji xt −i + ϵ1,t

j=1 i=0

δik rt −i +

i=1

n ∞ ∑ ∑

j

cjik xt −i + ϵtk

for k = 1, ..n,

(4)

j=1 i=1

j

where xt is +1 (−1) if a trade is buyer (seller) initiated and its trade size falls in quantile j, and 0 otherwise. In contrast to the previously outlined VARs, which assume a functional relation between trade size and permanent price impact, (4) makes the less restrictive assumption that trades in the same trade size quantile have equal price impact. 2.2. Nonlinear VAR model with multiple market participants The concern for misspecification due to nonlinear relations is most apparent when comparing the difference in price impact between two market participants. As an illustration, consider the scenario in which two participants differ in average trade size but have the same nonlinear permanent price impact function. Fig. 1, Panel A depicts this scenario and shows the relation between permanent price impact and trade size for both the large trader and the small trader.9 Both the large and the small trader operate at different points on the same nonlinear permanent price impact function (black line). Thus, for a correctly specified model, we should conclude that the large and the small trader have the same permanent price impact, conditional on trade size. The solid black line for the large trader in Panel B, subpanel (i), shows the permanent price impact of the large trader for a linear VAR model using trade sign as the trade variable. Similarly, Panel B, subpanel (ii) depicts the permanent price impact for the small trader. Comparing between the graphs for the large and small trader, we arrive at the incorrect inference that the large trader has a larger price impact than the small trader. In Fig. 1, Panel C, we illustrate the implications of using a linear VAR model using signed trade size by replacing the trade sign variable with signed trade size. In this scenario, we observe that the gradient of the estimated price impact function of the small trader (subpanel (ii)) is steeper than the gradient of the large trader (subpanel (i)). Again, we draw the incorrect conclusion that the small trader has a larger price impact than the large trader, for equivalent sized trades. Importantly, linear specifications of the VAR model using either trade sign or signed trade size as the trade variable lead to incorrect conclusions. Further, the result flips depending on the choice of the trade variable. If we use trade sign as the trade variable in the VAR, we find that the large trader has a larger price impact than the small trader. In contrast, when we use signed trade size in the model, we find the smaller trader has the larger price impact. To simultaneously estimate the price impact of two market participants, most researchers rely on the VAR model of Barclay et al. (2003). Barclay et al. (2003) develop a three equation VAR system, which includes a trade indicator variable for each of the market participants. A 10 lag version of their model is written as: rt =

xet =

xm t =

10 ∑

αi rt −i +

10 ∑

10 ∑

βi xet−i +

i=1

i=1

i=0

10 ∑

10 ∑

10 ∑

δi rt −i +

i=1

i=1

10 ∑

10 ∑

i=1

ϑi rt −i +

i=1

φi xet−i +

ζi x m t −i + ϵ1,t νi xm t −i + ϵ2,t

i=0

ψi xet−i +

10 ∑

η i xm t −i + ϵ3,t ,

(5)

i=1

7 A full discussion of this concern is presented in Appendix B. 8 For example, given there are 7 variables, a model with 5 lags requires 245 coefficient estimates. 9 Fig. 1 is a simplified visual representation of the potential concerns relating to model misspecification. We test whether this misspecification exists using actual empirical data in Section 5. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 1. Panel A depicts the true price impact function of two market participants who differ in their average trade sizes. Panel B shows the estimated price impact function for each participant if we use the trade sign VAR. Panel C plots the estimated price impact function for each participant if we use the trade size VAR.

e where rt is the log return during interval t, xm t (xt ) is the trade variable during interval t for market participant m (e). This specification is a structural VAR because the contemporaneous trade variable for market participant m, xm t , appears in the log return and market participant e equations. Thus, market participant m’s trades are assumed to have a contemporaneous effect on quote changes and market participant e’s trades. The contemporaneous causality assumption for quote changes

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follows from the reasoning that trades cannot instantly react to midquote changes, but trades can cause instantaneous changes to the midquote. However, the assumed contemporaneous affect of market participant m’s trades on market participants e’s trades requires theoretical justification by the econometrician In this study, we refer to (5) as the two participant trade sign VAR when using trade sign as the trade variable. In contrast, when we use signed trade size as the trade variable, we refer to the model as the two participant trade size VAR. When trade sign is the chosen trade variable, this model specification assumes either, (1) trades of different sizes all cause the same permanent price impact, or (2) both market participants have the same trade size. If assumption (1) is violated, to allow a like for like comparison between the two market participants, then assumption (2) must hold and both market participants must have the same trade size. Alternatively, if the trade variable is signed trade size then, for similar reasons, the model assumes, (1) price impact is linear with trade size, or (2) both market participants have the same trade size. Hasbrouck (1991a) and Engle and Patton (2004) demonstrate that the first assumption does not hold for either trade variable specification. Similarly, the second assumption is unlikely to hold for many empirical studies. For example, Brogaard et al. (2019) report an average trade size of $4299 and $8526 for HFT and non-HFT, respectively. Comerton-Forde and Putnins (2015) report an average size of dark trades between $10,000 to $150,000 in their sample, while lit trades have an average size of $5000 to $13,000. Last, Barclay et al. (2003) report ECN trades are smaller, on average, than market maker trades. Specifically, 51% of small trades occurs on ECNs, but this percentage declines to only 2% for large trades. To overcome these tenuous assumptions, we propose a flexible VAR system. Specifically, we extend the nonlinear VAR proposed in (4) to a multiple market participant case, which we refer to as the two participant nonlinear VAR. rt =

∞ ∑

αi rt −i +

n ∞ ∑ ∑

e,j

n ∞ ∑ ∑

beji xt −i +

i=1

j=1 i=0

j=1 i=0

e,j

∞ ∑

n ∞ ∑ ∑

e,j

n ∞ ∑ ∑

i=1

j=1 i=1

j=1 i=1

m,j

∞ ∑

n ∞ ∑ ∑

e,j

n ∞ ∑ ∑

xt =

xt

=

δij rt −i + δij rt −i +

i=1

j=1 i=1

cjie xt −i + cjie xt −i +

m,j

bm ji xt −i + ϵ1,t m,j

j

m,j

j

cjim xt −i + ϵ2,t cjim xt −i + ϵ3,t ,

(6)

j=1 i=0

y ,j

where xt is +1 (−1) if a trade has volume which lies in quantile j, and is buy (sell) initiated by market participant y, y,j or 0 otherwise, where y ∈ (e, m). The impulse response function arising from a shock to xt provides an estimate for the permanent price impact of a trade of specific size, j, initiated by market participant, y. By comparing the impulse response e,j m,j function from shocking xt with the impulse response function from shocking xt , we are able to make a fair comparison of the permanent price impacts between participants e and m for trades of equal size, j. Eq. (6) allows us to disentangle if one market participant has a larger price impact due to a bigger trade size or more private information. 3. Permanent price impact: A recursive approach In this section, we present the RL framework and discuss how we adapt the RL framework to model, and then estimate, permanent price impact. RL is a type of machine learning founded in the computer science literature, which is well suited to computing the permanent price impact of a trade. RL assumes the data is Markov, but is otherwise unrestrictive. As such, RL allows a large degree of flexibility as to what explanatory variables should be included, and is less likely to suffer from overparameterization or misspecification than a VAR model. Bertsimas and Lo (1998) first propose using RL for market microstructure problems and solve the problem of optimal order execution. Nevmyvaka et al. (2006) demonstrate RL’s efficacy for modelling the limit order book and advocate its use when solving the problem of buying a specified volume of shares in a specified amount of time. In Section 3.1 we outline the RL framework and how to empirically estimate an RL model using an iterative algorithm known as Q-learning. In Section 3.2, we demonstrate how we augment the RL framework to model permanent price impact, while Section 3.3 documents how we modify the Q-learning rule in order to obtain empirical estimates of our proposed model. In Section 3.4, we provide an illustrative example of our model and the estimation process and we conclude with a multiple market participant extension to our model in Section 3.5. 3.1. RL framework Typically, for an RL framework, an agent interacts with its environment in discrete time steps. At each time step, the agent receives an observation about the current environment, s. The agent then chooses an action, a, from a set of available actions, which is subsequently sent to the environment. The environment moves to a new state, s′ , and the agent receives the instant reward, r, associated with the transition. The goal of the agent is to collect as much reward as possible, by making choices that maximize the long run discounted sum of the instant rewards received for each action. In this section, we outline the RL framework and a corresponding algorithm that enables the agent to learn the optimal action to take at each state, which is known as Q learning. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Formally, an RL framework consists of:

• A discrete set of environment states S, with n(S) distinct states. • A set of possible actions A, with n(A) distinct actions. • A reward function R : S × A → ℜ, which maps elements in the set S × A to the real numbers, ℜ, with n(S)n(A) possible values.

• A state transition function T : S × A → Π (S), where Π is a mapping function where a member of Π (S) is a probability distribution over the set S. The environment states include any variables reflecting the current environment. The set of actions reflect the available options an agent can make. We define R(s, a) as the expected immediate reward as a function of the current state, s, and chosen action, a. Last, we define T (s, a, s′ ) as the probability of making a transition from state s to state s′ using action a. In general, for RL, the agent must choose the optimal sequence of actions, or policy, to maximize the infinite discounted sum of immediate rewards the agent receives. Defining a policy as π , the optimal value of a state is typically computed as V ∗ (s) = max E π

∞ (∑

) γ t R(st , at ) ,

(7)

t =0

where R(st , at ) is the immediate reward at time t and 0 < γ < 1 is a discount factor. V ∗ (s) is the expected infinite discounted sum of reward the agent receives if they start in state s and execute the optimal policy defined by π moving forward. We can define V ∗ (s) as the solution to the S simultaneous equations,

(

V ∗ (s) = max R(s, a) + γ

∑

a

)

T (s, a, s′ )V ∗ (s′ ) , ∀s ∈ S .

(8)

s′ ∈S

For every state, there are A possible actions, thereby resulting in S × A possible experience or state–action tuples, ⟨s, a⟩. For every experience tuple, there is an associated value Q ∗ (s, a), which is the expected infinite discounted sum of reward the agent gains if the agent takes action a in state s, then follows the optimal policy path. In other words, Q ∗ (s, a) is the immediate reward for taking action a (R(s, a)) plus all subsequent rewards the agent receives by taking the optimal action in future states. Thus, we can consider Q ∗ (s, a) as the long term reward for taking action a, whereas R(s, a) is the short term, or immediate reward for taking action a. Using (8), we note that Q ∗ (s, a) can be expressed recursively as Q ∗ (s, a) = R(s, a) + γ

∑

T (s, a, s′ ) max(Q ∗ (s′ , a′ )).

(9)

a′

s′ ∈S

Eq. (9) is the basis for Q learning and the associated Q learning rule is

(

)

Qt +1 (st , at ) = Qt (st , a) + α R(st , at ) + γ max(Qt (st +1 , a)) − Q (st , a) ,

(10)

a

where α is the learning rate. The Q learning rule is a value iteration update. Watkins and Dayan (1992) show that the Q values will converge to Q ∗ with probability 1 if all actions are repeatedly sampled in all states and the action-values are represented discretely. 3.2. Modelling price impact We demonstrate how the RL framework can be adapted to form a recursive model of permanent price impact. Using the RL framework previously defined, we consider the agent to be a trader while the environment states reflect the current market conditions, s. The actions available, a, to the agent or trader are interactions that alter the state of the limit order book, while the instant reward, R(s, a), is the demeaned change in log midpoint due to action a, while in state s.10 This model specification leaves the specific actions and market states relatively general, which provides the econometrician with the flexibility to select the actions and market states to investigate. For example, if we want to estimate the permanent price impact of a market buy or market sell order, we could consider a model with only one state, which is analogous to the market being open, and two actions available to the agent: to buy or to sell. This model specification is similar to the trade sign VAR. However, the model can be extended to a more realistic scenario by allowing for a variety of market conditions and available actions.11 Moreover, Section 5 demonstrates the importance of modelling different market conditions and available actions to reduce miss-specification and avoid erroneous conclusions. To model the permanent price impact of a single trade, we must modify the standard RL framework. In a standard RL framework, the agent makes action a, which transitions them to futures state s′ with probability T (s, a, s′ ). Once in 10 In our application, the instant reward, R(s, a), is a term consistent with the RL literature and should not be interpreted as an increase in utility for the agent or trader. Our goal is to estimate the permanent price impact of a trade. Thus, R(s, a) is the immediate price impact caused by action a, while in state s, and our model makes no assumption about the effect of the immediate price impact on the agent’s utility. 11 Brogaard et al. (2014) and Goldstein et al. (2018) show traders make trades dependent on market conditions, which are reflected in the shape of the order book. Later, we provide an example in which we extend the recursive model to account for these differing market conditions by allowing for several states and different trade sizes.

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Fig. 2. The permanent price impact of a buy (sell) is depicted in Panel A (Panel B) as the immediate impact of a buy (sell) plus the expected permanent price impact of the next action.

future state s′ the agent makes another action and picks the optimal future action a′ . The agent continues this decision making process recursively. We modify the RL framework so other market participants can make the future action a′ . In our framework, the agent makes action a, which transitions the market from state s to s′ with probability T (s, a, s′ ). Once in state s′ , the agent does not solely make the subsequent action a′ . Rather, we observe the market’s reaction a¯ . Thus, we distinguish between a′ and a¯ by the initiator of the action.12 As a result of this modification, in our model, the agent makes action a while in state s, which transitions the market to s′ where we observe action a¯ with probability T (s, a, ⟨s′ , a¯ ⟩). This modification alters T : Rather than being a transition probability function, which measures the probability of transitioning from state s to s′ via action a, T can now be represented as a new transition probability function. This new transition probability function measures the probability of transitioning from state s to experience tuple ⟨s′ , a¯ ⟩, via action a. As a result of our modification, T maps transition probabilities from n(S)n(A) experience tuples to n(S)n(A) future experience tuples. Therefore, if we order these probabilities appropriately, we can represent T (s, a, ⟨s′ , a¯ ⟩) by a transition probability of a Markov chain. By making this modification, we capture the phenomenon that a trade, or action, triggers a series of subsequent actions, which is reflected in the autocorrelation of order flow (see Hasbrouck and Ho (1987), Hasbrouck (1988) and Hasbrouck (1991a)). Further, the standard RL framework contains a maxa′ term, which is required if the trader is attempting to solve their own objective function containing a series of trades.13 However, our aim is to empirically estimate the permanent price impact of a single trade, regardless of the trader’s objective function. By modifying the RL framework and allowing other market participants to participate in future trades, we remove the maxa′ term from (9), and we are no longer required to optimize an objective function. As a result, we model the long term permanent shift in the midpoint triggered by one initial trade, which is expressed as Q ∗ (s, a)

  

permanent price impact

=

R(s, a)

  

immediate impact

+γ

∑∑

T (s, a, ⟨s′ , a⟩) Q ∗ (s′ , a).

s′ ∈S a′ ∈A





probability of observing future action a

(11)

   

permanent price impact caused by action a

′ ∗ ′ ′ ′ We simplify the right hand term of the expression in (9) from γ s′ ∈S T (s, a, s ) maxa (Q (s , a )) to γ s′ ∈S a′ ∈A T (s, a, ∗ ′ ′ ′ ⟨s , a⟩)Q (s , a). T (s, a, ⟨s , a⟩) is the probability that the subsequent experience tuple is ⟨s′ , a⟩ or more specifically, the probability that an agent’s action a transitions the market from state s to state s′ , where we observe action a. The corresponding permanent price impact of a, while in state s′ , is Q ∗ (s′ , a). We emphasize that a need not be initiated by the same market participant who initiated the original action a. Accordingly, in our model defined by (11), the permanent price impact, Q ∗ (s, a), of taking action a, while in state s, is the action’s immediate price impact plus the sum of the expected permanent price impact of all subsequent actions triggered by the initial action. Our model is a recursive Markovian model for permanent price impact that assumes the parameters are constant for all agents and across time, which is consistent with an equilibrium structure. To provide an intuitive understanding of our model defined by (11), consider a simple model setup in which there is only one market state (open) and two possible actions (buy or sell). Fig. 2, Panel A (Panel B) depicts the permanent price impact of a buy (sell). In this setup, we estimate the permanent price impact of a buy (Panel A) as the sum of three components: (1) the immediate impact of a buy, (2) the probability that the subsequent trade is a buy multiplied by the permanent price impact of a buy, (3) the probability that the subsequent trade is a sell multiplied by the permanent price impact of a sell. The permanent price impact of a sell has a similar logic as shown in Fig. 2, Panel B. The trees in Fig. 2 have a recursive nature. For Panel A, the root of the tree is the permanent price impact of a buy while the permanent price impact of a buy is also the top right node. Fig. 2, Panel B follows a similar setup for the permanent

∑

∑

∑

12 The subsequent action a¯ could be the same agent initiating another trade, or a market participant reacting to the agents initial trade. 13 For example, Nevmyvaka et al. (2006) use a similar objective function to minimize execution costs for a large order. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 3. The permanent price impact of a buy is the immediate impact of a buy plus the immediate impact of all subsequent trades, multiplied by their likelihood of occurring.

price impact of a sell. Thus, the permanent price impact of a buy (sell) can be modelled recursively. Fig. 3 depicts the recursive nature of this setup to model the permanent price impact of a buy. In doing so, we begin with Fig. 2, Panel A and replace any node that contains the permanent price impact of a buy with the whole tree in Fig. 2, Panel A, and any node that contains the permanent price impact of a sell and replace it with the whole tree in Fig. 2, Panel B. Thus, the permanent price impact of a buy is the immediate impact of a buy plus the immediate impact of all subsequent trades multiplied by their likelihood of occurring. Fig. 2 represents a simple model setup in which there is only one market state (open) and two possible actions (buy or sell). However, the model can be extended to accommodate more market states and available actions. For example, if we consider the available depth on the bid and ask as an important market state, we could define two market states, positive depth imbalance, +DI, and negative depth imbalance, -DI, where a positive (negative) depth imbalance represents market conditions when the depth on the bid is greater (less) than the depth on the ask. Moreover, we could include more available actions. For example, we could model four possible actions; small buy, small sell, large buy and large sell. Fig. 4 represents such a scenario. Specifically, the permanent price impact of an action, given the current depth imbalance or market state, is the immediate impact caused by the action plus the permanent price impact of the subsequent action given the subsequent market state multiplied by the likelihood of occurring. 3.3. Estimation procedure Next, we demonstrate how we modify the Q-learning rule to obtain empirical estimates of the permanent price impact model outlined in Section 3.2. Specifically, we augment the Q learning rule from the RL framework to obtain an iterative learning rule that converges to point estimates for permanent price impact for each experience tuple satisfying our proposed model. We obtain the permanent price impact model defined by (11) by simplifying the right hand term of the expression in (9). This simplification changes the Q-learning rule in (10) to Qt +1 (st , at ) = R(st , at ) + γ

∑

T (st , at , ⟨s′ , a⟩)Qt (s′ , a).

(12)

s′ ∈S

Given all actions are repeatedly sampled in all states, and we represent the action values discretely, the Q values will converge to Q ∗ with probability 1 as we iterate (12) (see Watkins and Dayan (1992)). To empirically estimate (12), we first define A available actions and S market states. For example, actions could be different trade size quantiles, while market states could be defined by various shapes of the limit order book. With S states and A actions defined, the parameters of the model are the elements of the immediate reward R, and the transition probability function, T . R maps the immediate reward for each element of the n(S)n(A) experience tuples, or action and market state combinations, to a real number. We empirically estimate the real number for each Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx

Fig. 4. The permanent price impact of an action is the immediate price impact caused by the action plus the permanent price impact of all subsequent actions, multiplied by their likelihood of occurring.

element as the average demeaned change in log midpoint for the corresponding observations. T contains the second set of parameters for our model. If we define the current state s, where we observe action a as experience tuple i, and the future state s′ , where we observe future action a as experience tuple j, we can represent T as a transition probability function, which measures the probability of transitioning to future experience tuples from current experience tuples. As such, T has a similar interpretation to the transition probability matrix of a Markov chain, where T (i, j) is the probability of observing experience tuple j after experience tuple i and is given by the ith row and∑ jth column. Since the transition n(S)n(A) T (i, j) = 1, which leaves probability from experience tuple i to all other experience tuples must sum to 1, for all i, j=1

(

)

n(S)n(A) n(S)n(A) − 1 parameters to estimate. Similar to the immediate reward, we empirically estimate the transition probabilities. We assume our empirical data is a realization of our model and if we further assume that such a stochastic process is a stationary Markov chain, which we do, estimation is straightforward. If we define Ni,j as the number of times i is followed by j, it is straightforward to show that the MLE estimate of T (i, j) is Ni,j T (i, j) = ∑n(S)n(A) j=1

Ni,j

.

(13)

This derivation can be done by maximizing the likelihood function subject to n(S)n(A) Lagrangians corresponding to each constraint. To estimate the permanent price impact, we initiate the permanent price impact estimates for each action and market state combination Q (s, a) to 0. Next, we use our empirically estimated immediate reward and transition probability parameters as inputs and iteratively update our permanent price impact estimates using (12) until the estimates converge. We present the pseudo code for this procedure below and provide an illustrative example of this process in Section 3.4. In our model, γ , is a discount factor where 0 <= γ < 1. This discount factor is akin to the time value of money, as the immediate price impact of trades further in the future receive more discounting. Because we are measuring the price impact of trades over extremely short time horizons, very little discounting is required and thus, it is appropriate to choose γ close to 1. If γ = 1, all environmental histories are infinitely long, and permanent price impact estimates made from additive, undiscounted immediate rewards are infinite. Accordingly, to ensure convergence, γ < 1. For these reasons, we set γ = 0.999 for our empirical investigations. In Appendix C, we show that our estimates are robust to the choice of γ .14 14 Given trades arrive asynchronously, it is common to operate in trade time rather than clock time. However, unlike the VAR model, which operates in trade time (when using trade data), our model provides the choice to operate in trade time or clock time. If γ is constant, we are Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Pseudo code procedure Estimate immediate rewards and transition probabilities for all state_action tuples do 1. Find the subset of observations, o, in the data which have this state_action. 2. Compute the immediate reward for the current state_action, R(s, a), as the average demeaned change in log midpoint of all o. 3. Compute the transition probability to a future state–action tuple, T (s, a, ⟨s′ a⟩), as the number of observations in subset o, with subsequent observations in the future state–action tuple, ⟨s′ a⟩, divided be the length of o. 4. Loop to repeat 3 for all possible future state–action tuples. end for procedure Estimate permanent price impact using the iterative learning rule Q _lag(s, a) ← 0, ⟨s, a⟩ ∈ S × A Q (s, a) ← 0, ⟨s, a⟩ ∈ S × A

▷ Initialize lagged ppi estimate to 0 for all tuples ▷ Initialize ppi estimate to 0 for all tuples

while the estimates have not converged do for all state_action tuples∑ do ∑ ′ ′ Q (s, a) ← R(s, a) + γ s′ ∈S a′ ∈A T (s, a, ⟨s , a⟩)Q _lag(s , a) end for

▷ update ppi estimate for each experience tuple

a′ ∈A |Q (s, a) − Q _lag(s, a)|< ϵ then exit while loop else Q _lag(s, a) ← Q (s, a), ⟨s, a⟩ ∈ S × A end if

if

∑

s′ ∈S

▷ where ϵ ⪆ 0 ▷ estimates have converged

∑

▷ Estimate have not converged ▷ update the ppi estimates for all tuples

end while

The transition probabilities and immediate rewards are unobserved input parameters. We empirically estimate these input parameters from finite samples of data, which can introduce estimation error and corresponding variance in the input parameters. Nilim and Ghaoui (2005) demonstrate that estimation errors in the transition probabilities can have a large impact on the optimal solutions to Markov decision problems. Therefore, variance in the input parameters causes variance in the permanent price impact estimates. To account for these estimation errors, we block bootstrap the data to obtain confidence intervals for our permanent price impact estimates. Similar to White and White (2010), we construct a bootstrap sample by concatenating n/l blocks, where each block contains l consecutive data points. Each block is chosen randomly with replacement, resulting in a time series sample of length n. We select n as the number of trades in our sample, and following Hall et al. (1995), we use a block length l of n1/3 . We obtain permanent price impacts for 1000 bootstrap samples and compute the corresponding confidence interval. 3.4. Illustrative example We provide a simple example to illustrate the iterative update process of the learning rule defined by (12). We consider a model with two possible actions (A = 2), to buy or sell shares via a market order, which we define as buy and sell, respectively. Moreover, we define two market states (S = 2) based on the shape of the order book. We refer to the first state as a positive depth imbalance, +DI, which occurs when there is more depth available on the bid compared to the ask. Similarly, we refer to the second state as a negative depth imbalance, −DI, which occurs when there is more depth available on the ask relative to the bid. This specification results in a model with four experience tuples (⟨−DI , sell⟩, ⟨+DI , sell⟩, ⟨−DI , buy⟩ and ⟨+DI , buy⟩). Next, we estimate the immediate reward for each experience tuple, which we empirically compute as the average demeaned change in log midpoint that corresponds to each experience tuple. For example, to estimate the immediate reward for the ⟨+DI , buy⟩ experience tuple, we take the average demeaned change in log midpoint for all market buy orders when the depth available at the bid is more than the depth available at the ask. In Section 5, we empirically implicitly operating in trade time. However, to operate in clock time, we could vary γ depending on the time until the next trade and apply more (less) discounting when there is a longer (shorter) wait until the next trade. Because we want to directly compare our recursive model with the traditional VAR models, we estimate the recursive model in trade time.

Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx Table 1 Input parameters. Transition probability matrix

−DI , sell +DI , sell −DI , buy +DI , buy

Immediate reward

−DI ′ sell

+DI ′ sell

−DI ′ buy

+DI ′ buy

0.65 0.07 0.35 0.04

0.06 0.55 0.03 0.25

0.25 0.03 0.55 0.06

0.04 0.35 0.07 0.65

R(−DI , sell) = −1 R(+DI , sell) = −0.5 R(−DI , buy) = 0.5 R(+DI , buy) = 1

estimate the immediate reward values from our sample data, however, to maintain simplicity in this illustrative example, we assign immediate rewards of −1, −0.5, 0.5 and 1 basis point for experience tuples ⟨−DI , sell⟩, ⟨+DI , sell⟩, ⟨−DI , buy⟩ and ⟨+DI , buy⟩, respectively. These immediate rewards are qualitatively similar to those we empirically obtain in Section 5 and reflect that a market order executing against the thin side of the order book will have larger immediate impact than an equivalent order executing against the thick side of the order book. T (s, a, ⟨s′ , a⟩) represents the probability of observing action a in state s′ , after action a occurs in state s. For example, T (+DI , buy, ⟨+DI ′ , buy⟩) is the probability a buy with more depth on the bid than the ask (i.e. a positive depth imbalance) follows a buy with more depth on the bid than the ask (i.e. a positive depth imbalance). In contrast, T (+DI , buy, ⟨−DI ′ , sell⟩) is the probability a sell with more depth on the ask than the bid (i.e. a negative depth imbalance) follows a buy with more depth on the bid than the ask (i.e. a positive depth imbalance). Similar to the estimations for the immediate reward, we compute these probabilities empirically, which we obtain using the MLE estimate defined by (13). For example, to estimate T (+DI , buy, ⟨−DI ′ , sell⟩), we observe the subsample of observations that have a market buy order initiated when a positive depth imbalance exists. Next, we compute the proportion of the observations in this subsample which are subsequently followed by a market sell order initiated when a negative depth imbalance exists. To maintain simplicity, for this illustrative example, we assign the transition probabilities presented in Table 1, which are qualitatively similar to the empirical estimates in Section 5. The transition probabilities capture the autocorrelation in trade direction (see Hasbrouck and Ho (1987), Hasbrouck (1988) and Hasbrouck (1991a)) and the observation that a market buy (sell) order is more likely when a positive (negative) depth imbalance exists, consistent with Parlour (1998). We initialize our permanent prices impact estimates for each experience tuple to zero. Specifically, Q0 (+DI , buy), Q0 (+DI , sell), Q0 (−DI , buy) and Q0 (−DI , sell) all equal zero. Using the learning rule defined by (12), we update our permanent price impact estimates for Q (+DI , buy) for the first iteration as follows: Q1 (+DI , buy) = R(+DI , buy) + γ (T (+DI , buy, ⟨+DI ′ , buy⟩)Q0 (+DI ′ buy)

+ T (+DI , buy, ⟨+DI ′ , sell⟩)Q0 (+DI ′ sell) + T (+DI , buy, ⟨−DI ′ , buy⟩)Q0 (−DI ′ buy) + T (+DI , buy, ⟨−DI ′ , sell⟩)Q0 (−DI ′ sell)) = 1 + 0.99(0.65 × 0 + 0.25 × 0 + 0.06 × 0 + 0.04 × 0) =1 Similarly, we obtain −1 for Q1 (−DIt , sellt ) and 0.5 and −0.5 for Q1 (−DIt , buyt ) and Q1 (+DIt , sellt ) respectively. On iteration two, the input values we use in the learning rule remain the same except for the permanent price impact estimates, which are updated to the values estimated in iteration 1. Thus, for iteration two, we update our permanent price impact estimate for Q (+DI , buy) as follows: Q1 (+DI , buy) = R(+DI , buy) + γ (T (+DI , buy, ⟨+DI ′ , buy⟩)Q1 (+DI ′ buy)

+ T (+DI , buy, ⟨+DI ′ , sell⟩)Q1 (+DI ′ sell) + T (+DI , buy, ⟨−DI ′ , buy⟩)Q1 (−DI ′ buy) + T (+DI , buy, ⟨−DI ′ , sell⟩)Q1 (−DI ′ sell)) = 1 + 0.99(0.65 × 1 + 0.25 × −0.5 + 0.06 × 0.5 + 0.04 × −1) = 1.50985 Table 2 reports the progression of our permanent price impact estimates for each iteration of the learning rule. After iteration 41, the permanent price impacts estimates do not change to a precision of 4 decimal places for any subsequent iterations. This stability indicates the learning rule has converged to 4 decimal places by iteration 42 and we conclude, for this example, the permanent price impact of a buy when a positive depth imbalance exists, Q ∗ (+DI , buy), is 2.7929 basis points. Consistent with expectations, the permanent price impact of a buy when a negative depth imbalance exists, Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Table 2 Estimates of the permanent price impact (in basis points) for all experience tuples at the end of each iteration of the learning rule defined by (12). The bottom row, labelled ‘‘Change’’, reports the sum of the total change in permanent price impact estimates after each iteration. Q ∗ (−DI , sell) Q ∗ (+DI , sell) Q ∗ (−DI , buy) Q ∗ (+DIbuy)

Iteration 0

Iteration 1

Iteration 2

Iteration 3

...

Iteration 40

Iteration 41

Iteration 42

Iteration 43

0.0000 0.0000 0.0000 0.0000

−1.0000 −0.5000

−1.5099 −0.4802

−1.8215 −0.3287

−2.7928

−2.7929

−2.7929

−2.7929

0.5000 1.0000

0.4802 1.5099

0.3287 1.8215

... ... ... ...

3.0000

1.0593

0.9263

...

Change

0.5650

0.5650

0.5651

0.5651

−0.5650

−0.5650

−0.5651

−0.5651

2.7928

2.7929

2.7929

2.7929

0.0002

0.0001

0.0001

0.0000

Q ∗ (−DI , buy), is lower at 0.5651 basis points. To further illustrate convergence of the learning rule, the bottom row of Table 2 reports the total change across all permanent price impact estimates after each iteration. The change in permanent price impact estimates decreases after each additional iteration until we observe a value of 0 to a precision of 4 decimal places. 3.5. Multiple participant scenario In this section, we extend the model to contain multiple market participants, c ′ ∈ C . Specifically, we modify (11) to Q ∗ (s, a, c) = R(s, a, c) + γ

∑∑∑

T (⟨s, c ⟩, a, ⟨s′ , a, c ⟩)Q ∗ (s′ , a, c).

(14)

c ′ ∈C a′ ∈A s′ ∈S

In (14), the experience tuple now includes market participant c, who makes the trade/action. As such, T (⟨s, c ⟩, a, ⟨s′ , a, c ⟩) is the probability that market participant c makes action a that transitions the market from current state s to future state s′ , where we observe market participant c make future action a. Similar to (11), we emphasize that future actions a are initiated by participant c, who does not need to be the participant who initiated the original action. Further, Q ∗ (s′ , a, c) is the permanent price impact caused by market participant c, making future action a, while in state s′ . If there are two market participants, c1 and c2 , who are equally informed about the future stock price, they should cause the same permanent price impact for trades with the same size and market conditions, i.e., Q (si , aj , c1 ) = Q (si , aj , c2 ) for all i and j. Conversely, if market participant c1 is more informed than participant c2 , we expect buys (sells) from participant c1 to have a larger positive (negative) permanent price impact than equivalent buys (sells) from participant c2 , or more formally, Q (si , aj , c1 ) > Q (si , aj , c2 ) for buys and Q (si , aj , c1 ) < Q (si , aj , c2 ) for sells. Thus, any discrepancy between respective Q values for differing market participants can be attributed to private information. To formally test whether one market participant contributes more to price discovery due to private information, we estimate the following regression: Q (si , aj , c1 ) = β Q (si , aj , c2 ) + ϵ.

(15)

If β > 1, market participant c1 has more private information and has a larger permanent price impact for trades with the same size and market conditions than market participant c2 . The converse is true if β < 1. Last, if β = 1, neither market participant has more private information.15 Consider a market in which one participant trades very large volumes while the other participant trades small volumes. Economic considerations suggest the large trader will cause more permanent price impact for their trades. However, is this larger impact due only to the trade size differential or knowledge of private information? Our proposed technique enables us to disentangle these two effects. Eq. (15) determines if two market participants differ in private information. In contrast, differences in the transition probabilities T (st ct , at , s′ ac) from (14) reflects any difference in price impact attributed to a differential in trade size or market conditions. In a similar spirit to Hasbrouck (1991b), we propose a simple and intuitive metric, which measures the proportion of total price movement attributable to a certain action or trade. The price contribution is computed as

∑ PCa =

s∈S

Q (s, a)n(s, a)

κ

,

(16)

where n(s, a) is the number of times action a is made while in state s and κ is s∈S ,a∈A |R(s, a)|, the total amount of price movement observed in the sample. For a model in which trades are the only actions available, all of the trades in the sample must account for all the price movement. This implies that the sum of the absolute value of all immediate

∑

15 Equation (15) could suffer from two problems: (1) error in variables due to estimation error of the Q values, and (2) small sample size. We argue that estimation errors are relatively innocuous given our estimates have tight bootstrapped confidence intervals. However, we can use higher moments (Dagenais and Dagenais, 1997) or characteristic function techniques (Carrasco and Kotchoni, 2017) if error in variables is of concern. Similarly, sample size is unlikely to be an issue as we can create more state–action tuples to increase the number of observations. For example, in Section 5.3.2, we estimate (14) using 80 observations. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx Table 3 Summary statistics for the 20 sample stocks. The sample contains 40 trading days from January 1, 2015 to February 28, 2015. N is the number of trades. Price is the average executed price in dollars. Spread is the average distance between the bid and ask reported in dollars. Size is the trade size reported in number of shares. We report the average, median and maximum trade size. Stock

N

Price

Spread

Average size

Median size

Max size

BHP CBA WBC ANZ NAB TLS WOW RIO CSL WPL NCM QBE AMP SUN ORG BXB MQG STO AMC IAG

235 518 287 796 184 768 179 097 186 243 86 543 201 455 241 023 242 493 222 413 129 765 114 310 61 588 107 562 91 894 85 246 200 880 118 752 115 396 77 176

30.11 88.46 35.17 33.50 35.43 6.39 31.65 59.15 86.90 35.04 13.31 11.32 5.96 14.21 11.52 10.53 63.42 7.80 13.21 6.25

0.011 0.015 0.012 0.011 0.012 0.010 0.012 0.015 0.018 0.013 0.011 0.011 0.010 0.011 0.011 0.011 0.015 0.011 0.011 0.010

852.62 244.05 663.29 732.71 688.33 6366.16 540.13 229.03 135.99 388.59 796.54 1230.89 2822.38 1017.55 1013.79 1110.69 160.13 1661.36 1043.32 2671.26

264 94 207 242 208 718 165 97 53 128 200 261 388 225 248 218 65 300 260 336

150 000 28 894 128 266 150 000 262 552 693 154 88 741 33 646 399 865 50 000 704 467 249 954 312 179 124 613 57 147 184 984 25 255 198 500 72 259 304 016

rewards or midpoint shifts in the sample must equal the∑ sum of the permanent action multiplied by ∑ price impact of each 16 the frequency of that action occurring, or formally, κ = s∈S ,a∈A |R(s, a)| = s∈S ,a∈A Q (s, a)n(s, a). We extend (16) to measure the proportional price contribution of each market participant. For example, in the case of two market participants, the permanent contribution to price made by market participant ci is

∑ PCci =

s∈S ,a∈A

Q (s, a, ci )n(s, a, ci )

κ

.

(17)

Eqs.(16) and (17) provide examples of how the recursive model attributes the proportion of price discovery coming from various sized trades, or different market participants. The ratio of total price movement attributed to a specific action or market participant to the total price movement observed in the sample determines the proportion of price discovery due to that specific action or market participant. This concept can be extended to additional actions available to the market, such as limit order submission or cancellations. We provide an example of this extension in Appendix K. 4. The data We use full order book data for the Australian Securities Exchange (ASX) extracted from the SIRCA database for the period January 1, 2015 to February 28, 2015.17 The SIRCA database contains every trade, order submission, cancellation and amendment allowing us to fully reconstruct the order book. In our analysis, we consider the 20 largest stocks by market capitalization. Table 3 presents summary statistics for the 20 stocks. For the purpose of our study, data from the ASX offers several advantages over other markets. First, the Australian market remains largely consolidated during our sample period, allowing for the correct chronological ordering of trades. Because the ASX executes almost 90% of lit volume, unobserved trades on competing exchanges are less likely to contaminate our price impact measures. Second, we can accurately infer trade direction from the reconstructed limit order book and do not need to rely on trade classification algorithms, such as Lee and Ready (1991).18 Third, most datasets report large orders that execute against several resting limit orders as multiple trades.19 Johnson et al. (2018) demonstrate that using multiple trades, rather than the single large order, can bias empirical studies. Using the granularity of our data set, we accurately group multiple trade reports into a single transaction, even when a single large order executes against multiple resting orders in the limit order book with different prices. Finally, we can accurately determine the best bid and ask prices and liquidity available immediately before a trade.20 16 Empirically, slight deviations could occur due to discretization of the data. 17 This sample is long enough for precise estimations (see Dufour and Engle (2000)). 18 Theissen (2001) outlines the concerns of trade classification algorithms. 19 Johnson et al. (2018) demonstrate that a single large order cannot be recovered from a sequence of reported trades when using data sources like NYSE daily TAQ and the consolidated tape. 20 In contrast, Lee and Ready (1991) suggest using a 5 s delay between the reported quote and trade price to reflect the fact that the quotereporting mechanism is not time synchronized with the trade-reporting mechanism. While the recommended delay has changed over time, to reflect the markets progression in technology, the problem still persists. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Table 4 The permanent price impacts estimated using the recursive model (Rec) and the trade sign VAR. Panel A (Panel B) presents the results for the recursive model (trade sign VAR). We estimate the recursive model using two actions: buy (+1) and sell (−1). This model specification is similar to the trade sign VAR. Panel A

AMC AMP ANZ BHP BXB CBA CSL IAG MQG NAB NCM ORG QBE RIO STO SUN TLS WBC WOW WPL

Panel B

I

II

III

IV

V

Rec Sell

Rec Buy

VAR

VAR Lower 5%

VAR Upper 95%

−1.74 −1.72 −1.04 −1.06 −1.92 −0.65 −1.17 −1.52 −0.85 −0.87 −2.18 −2.11 −2.17 −0.98 −3.06 −1.37 −1.10 −1.06 −1.06 −1.33

2.06 2.00 0.97 1.04 1.83 0.64 1.37 1.62 0.98 0.86 2.14 2.18 2.08 1.00 3.18 1.56 1.03 0.99 1.08 1.39

1.14 1.54 0.94 0.78 1.64 0.67 1.20 1.41 0.94 0.68 1.73 2.10 1.52 0.96 2.07 1.00 1.00 0.98 0.98 1.26

0.60 −0.55 0.72 0.46 0.63 0.57 1.01 −0.31 0.76 −0.26 1.15 1.17 0.71 0.80 1.01 0.38 −0.38 0.76 0.72 1.02

1.73 3.76 1.17 1.08 2.65 0.77 1.38 3.19 1.12 1.58 2.32 2.99 2.35 1.11 3.19 1.61 2.44 1.20 1.23 1.55

We prepare the data for our analysis as follows. First, we reconstruct the limit order book. We treat a series of consecutive trades reported with the same timestamp, trading in the same direction and initiated by the same broker as one trade. Second, for every trade, we determine the prevailing bid and ask immediately prior to the trade. Third, we only include trades during continuous trading hours and omit all trades occurring in the opening and closing auctions. Omitting the opening price avoids contamination from large price moves caused by the arrival of overnight news. This preparation process leaves a sample of more than 280,000 trade observations for the most actively trade stock (CBA) and over 61,000 trade observations for the least actively traded stock in the sample (AMP). 5. Estimation and results In this section, we compare and contrast the performance of different specifications of the VAR and recursive models. For all comparisons, we empirically estimate the different specifications of the two models using the historical trade and quote data described in Section 4. In Section 5.1, we establish that a baseline version of our recursive model and the trade sign VAR yield similar inferences. In Section 5.2, we provide empirical evidence that there is a nonlinear relation between trade size and permanent price impact. Last, in Section 5.3, we investigate the implications when we do not capture nonlinearities. 5.1. Base recursive model vs VAR model In this subsection, we compare a trade’s permanent price impact estimated via the trade sign VAR with a trade’s permanent price impact estimated via the simplest specification of the proposed recursive model. We specify a recursive model with only two possible actions, A ∈ (+1, −1), where at = +1 for buyer initiated trades and at = −1 for seller initiated trades. This specification assumes there is only one market state (i.e. open) and parallels Hasbrouck’s (1991a) VAR model using trade sign as the trade indicator variable. Table 4, Columns I and II report the recursive model’s estimated permanent price impact, for a sell and buy, respectively. For each sample stock, we also report the permanent price impact estimated using the trade sign VAR in Column III. Table 4 also provides the corresponding 95% confidence interval for the estimated permanent price impact obtained using the trade sign VAR. For all sample stocks, except AMC, the recursive model’s estimated permanent price impact for buys and sells lie within the 95% confidence interval of the trade sign VAR permanent price impact estimates. Our results demonstrate the baseline recursive model and the traditional VAR model yield similar inferences. 5.2. Controlling for trade size Next, we study the relation between trade size and permanent price impact using both the nonlinear VAR model and the recursive model. We investigate the VAR model in Section 5.2.1 and the recursive model in Section 5.2.2. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx Table 5 The first lag coefficient estimates and t-statistics (in bold) for the nonlinear VAR. xkt is +1 (−1) if a trade is buyer (seller) initiated and its trade size falls in quantile k, and 0 otherwise. rt is the change in the natural logarithm of the midpoint that follows a trade at time t. Column 1 (Column 8) presents the first lag coefficients for the smallest (largest) trade quantile equation, k = 1 (k = 8). Column 9 reports the first lag coefficients for the quote revision equation.

c1k,1 c2k,1 c3k,1 c4k,1 c5k,1 c6k,1 c7k,1 c8k,1

δ1k

I

II

III

IV

V

VI

VII

VIII

IX

xkt =1

xkt =2

xkt =3

xkt =4

xkt =5

xtk=6

xtk=7

xkt =8

rt

0.17 60.19 0.06 21.72 0.03 11.71 0.03 9.09 0.02 7.49 0.01 4.30 0.01 4.53 0.01 2.98 −65.10 −13.36

0.05 18.16 0.10 33.69 0.05 17.75 0.04 14.61 0.04 12.43 0.03 11.08 0.02 6.19 0.01 2.99 −52.67 −11.06

0.04 12.86 0.05 17.99 0.10 34.70 0.05 17.48 0.05 15.69 0.04 13.64 0.03 9.48 0.01 2.36 −61.09 −12.61

0.03 9.21 0.04 14.53 0.06 19.11 0.09 29.96 0.06 20.39 0.05 18.84 0.05 15.67 0.02 8.37 −56.72 −11.75

0.02 8.22 0.04 12.13 0.04 13.55 0.05 18.79 0.15 53.40 0.08 28.03 0.06 22.03 0.05 16.38 −62.12 −12.99

0.03 8.80 0.03 11.26 0.04 12.49 0.05 15.65 0.08 28.73 0.10 35.41 0.06 21.19 0.05 15.93 −44.22 −9.18

0.02 7.53 0.02 8.25 0.03 9.47 0.04 14.23 0.07 23.06 0.08 25.88 0.07 25.21 0.05 17.13 −30.84 −6.39

0.01 4.43 0.01 4.64 0.02 7.57 0.02 7.44 0.06 19.64 0.06 19.68 0.05 16.35 0.08 26.90 −10.68 −2.21

b1 b2 b3 b4 b5 b6 b7 b8

α

−0.00 −37.69 −0.00 −28.14 −0.00 −24.87 −0.00 −19.96 −0.00 −11.62 −0.00 −11.60 −0.00 −8.66 0.00 16.04 −0.18 −58.21

5.2.1. VAR model Using BHP as a representative stock, we estimate the nonlinear VAR using 5 lags and 8 trade size quantiles.21 Table 5, Columns I to VIII present the first lag coefficient estimates for the trade quantile equations and Column IX shows the first lag coefficients for the quote revision equation. For example, Column I reports the first lag coefficients corresponding to the smallest trade quantile (k = 1) equation defined as x1t =

5 ∑ i=1

δik rt −i +

5 ∑ i=1

1 1 c1i xt −i +

5 ∑

1 2 c2i xt −i +

i=1

5 ∑

1 3 c3i xt −i + · · · +

i=1

5 ∑

1 8 c8i xt −i + ϵt1 .

(18)

i=1

Specifically, the first lag coefficient for the smallest trade quantile variable (c11,1 ) is 0.17, the first lag coefficient for the second smallest trade quantile variable (c21,1 ) is 0.06, while the first lag coefficient for the largest trade quantile variable (c81,1 ) is 0.01. Last, the first lag coefficient for the quote revision variable (δ11 ) is −65.10. Consistent with earlier findings, our lag coefficient estimates for all trade size quantile equations are positive and significant, indicating that signed trades exhibit a strong positive autocorrelation (see Hasbrouck (1991a), Engle and Patton (2004) and Dufour and Engle (2000)). Table 5 also provides evidence of autocorrelation in trade size. Table 5, Column I, reports the first lag coefficients for the small sized trade equation (k = 1) and we find that the coefficients decrease monotonically between c1k = 0.17 and c8k = 0.01. In contrast, Column VIII reports the first lag coefficients for the large sized trade equation and find that the coefficients increase monotonically between c1k = 0.01 and c8k = 0.08. This result demonstrates that a small trade is more likely to be followed by another small trade, while a large trade is more likely to precede a large trade. We now compare our nonlinear VAR results with those obtained using the trade sign VAR or trade size VAR. Fig. 5 plots the relation between permanent price impact and trade size for our representative stock, BHP, using the three different VAR specifications. Estimates of the permanent price impact as a function of trade size using the nonlinear VAR are concave. In contrast, when we use the trade sign VAR, estimates of the permanent price impact produce a step function. This step function reflects the assumption that trades of differing size have equal permanent price impact. Alternatively, when we use the trade size VAR, we obtain a linear relation between permanent price impact and trade size. Fig. 5 demonstrates that the VAR model produces a step function when we use trade sign as the trade variable in the VAR model defined by (1). This step function partially captures the nonlinear relation between price impact and trade size and could explain why, despite economic considerations, much of the literature uses trade sign as the trade variable when estimating a VAR model. 21 Our results are robust to the choice of parameters. The choice of the number of quantiles is a tradeoff between precision and bias. In Appendix D, we show that 8 trade size quantiles provide a balance between the number of parameters to estimate and model flexibility to capture nonlinearity. For optimal lag selection, Ivanov and Kilian (2005) recommend using the Akaike information criterion. However, the literature typically uses 3 lags (see Hasbrouck (1991b)), 5 lags (see Hasbrouck (1991a) and Dufour and Engle (2000)) or 10 lags (see Barclay et al. (2003)). In Appendix E, we show that our results are similar for 3, 5 or 10 lags. For consistency with the literature we present results using 5 lags. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 5. The permanent price impact for various trade sizes using the trade sign VAR, trade size VAR and nonlinear VAR.

5.2.2. Recursive model Next, we demonstrate how the recursive model captures the nonlinear dynamics between trade size and permanent price impact and compare the recursive model’s estimates with the nonlinear VAR estimates. To capture nonlinearities using the recursive model, we expand the number of possible actions for the recursive model from buy and sell to a larger set. Specifically, we estimate the recursive model using sixteen possible actions, A ∈ (1, . . . , 16), where at = i if the directional trade size falls in quantile i. Thus, we assign a large sell order as at = 1, while we assign a large buy order as at = 16. Similar to the model estimated in Section 5.1, we select only one market state (open). Fig. 6 plots the recursive model’s estimated permanent price impact for each trade quantile for our representative stock, BHP. For comparison, Fig. 6 also depicts the nonlinear VAR’s permanent price impact estimates. Observing Fig. 6, we see that both models yield a similar relation between trade size and permanent price impact. One discrepancy between the VAR model and the recursive model is the smoothness of the relation between trade size and permanent price impact; the relation estimated by the recursive model is smoother relative to the relation estimated using the VAR model. Economic considerations suggest that permanent price impact should increase monotonically with trade size. The recursive model captures this monotonic relation for all trade quantiles. In contrast, the VAR model estimates appear noisy and do not recover a smooth monotonic relation. Finally, we compare inferences for the recursive model and the nonlinear VAR for all sample stocks. Table 6 reports the recursive model and nonlinear VAR estimates of permanent price impact for different sized trades for each sample stock. Numbers reported in bold reflect a statistical difference at the 5% level between the recursive estimate and the nonlinear VAR estimate. In general, the recursive model and the nonlinear VAR yield similar estimates. We note that only 6% of the price impact estimates are statistically different from each other, which could be attributed to type 1 errors. Further, our results confirm a nonlinear relation between permanent price impact and trade size for all sample stocks. Specifically, we show that permanent price impact increases with trade size monotonically at a decreasing rate. The results demonstrate that in the modern market environment, permanent price impact, as a function of trade size, is positive, increasing and concave, consistent with Hasbrouck (1991a) and Engle and Patton (2004). While the nonlinear VAR and recursive model yield similar estimates, one advantage of our recursive model is computational efficiency as our method does not require the inversion of large matrices. In Appendix F, we demonstrate that the computational time required to estimate the recursive model is significantly less than the computational time required to estimate an equivalent VAR model. For example, our recursive model with 16 actions takes approximately half a second to estimate, while an equivalent VAR model takes over 2 min to estimate. Moreover, while computational time increase for both models with the number of trade size quantiles, the increase is exponential for the nonlinear VAR, whereas the increase is more linear with the recursive model. 5.3. Comparing multiple market participants or trade venues In the previous section, we establish a nonlinear relation between permanent price impact and trade size. In this section, we investigate the economic implications if these nonlinear relations are not captured. In Section 5.3.1, using Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx

Fig. 6. The permanent price impact for various trade sizes using the nonlinear VAR (stars) and recursive model (circles). The dotted grey line is the upper and lower 95% confidence interval of the recursive model’s estimates.

an example of two equally informed investors, who differ only in their average trade size, we estimate their respective price impacts using (1) the traditional two participant trade sign VAR, (2) the two participant trade size VAR, (3) the two participant nonlinear VAR we propose and, (4) our suggested recursive model. In Section 5.3.2, we investigate the concern that the VAR framework is misspecified if a trader uses knowledge of the limit order book in their trading decisions (see Hasbrouck (1991a)), which is a phenomenon seen in the modern trading environment (see Goldstein et al. (2018)). 5.3.1. Two equally informed market participants with differing trade size Asymmetric information models suggest that a trade conveys private information (see Kyle (1985)). Additionally, trades with more private information cause a larger permanent shift in price. Thus, a trade’s permanent price impact reflects the amount of private information a participant possesses, and two participants with the same private information should have the same permanent price impact functions. Fig. 7, Panel A shows the price impact function for two traders with the same amount of private information, who trade with different average trade sizes. Because the traders have the same amount of private information, both traders operate on same price impact function. The large trader (stars) operates at the extremes of the price impact function and the small trader (circles) operates at the centre of the price impact function. Fig. 7, Panel B shows the price impact functions if the large trader (stars) has more private information than the small trader (circles). Because the large trader is more informed than the small trader, the large trader has a steeper price impact function than the small trader. We exploit these concepts to demonstrate the implications of model misspecification. Specifically, we create two market participants with equal amounts of private information, who differ in their average trade size.22 We define one participant as the large size trader, ls, and the other as the small size trader, ss. Because ls and ss have the same amount of private information (i.e., they are equally informed), we expect both participants to operate on the same price impact function. However, because they have different average trade sizes, we expect the participants to operate in different regions of the price impact function. This scenario is depicted in Fig. 7, Panel A.23 To create the two market participant samples, for each stock in the ASX sample, we first allocate all trades with a trade size above the median to ls, and all trades below the median trade size to ss. We then randomly select 25% of all trades from ls and re-categorize these trades to ss. Similarly, we randomly allocate 25% of ss trades to ls.24 This reallocation 22 We construct this size differential to emphasize how erroneous inferences about private information can occur in empirical studies that compare between trading venues, lit and dark markets, or trader types in which size differentials typically occur. 23 We depict the price impact function in Fig. 7 as an ‘S’ shape. However, the price impact function need not have this shape. If all trade sizes contain equivalent information, the two market participants have equivalent permanent price impacts regardless of trade size and the price impact function would be a step function as shown in the trade sign VAR of Fig. 5. However, our results in Section 5.2 show larger trades contain more information than smaller trades. As such, we expect our two equally informed market participants to have the same permanent price impacts only for equivalent sized trades. 24 In Appendix F, we also reallocate 10% and 40% of trades from ls (ss) to ss (ls). We do not reallocate 50% to ensure the two market participant samples have a trade size differential.

Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size

(VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive)

q1

q2

q3

q4

q5

q6

q7

q8

q9

q10

q11

q12

q13

q14

q15

q16

−3.39 −3.61 −6097 −5.6 −5.02 −13 618 −1.94 −1.95 −3618 −2.12 −1.95 −4958 −4.52 −4.32 −5264 −1.34 −1 −1229 −2.01 −1.54 −511 −4.36 −4.13 −12 583 −1.78 −1.35 −860 −1.73 −1.67 −3436 −4.59 −3.97 −4771 −5.17 −4.77 −5780 −4.55 −4.37 −6423 −2.04 −1.68 −1096 −6.91 −6.07 −8743

−1.82 −2.64 −1345 −3.94 −2.5 −2136 −1.15 −1.29 −850 −1.2 −1.39 −1181 −2.32 −2.7 −1132 −0.89 −0.79 −288 −1.31 −1.15 −125 −2.54 −2.12 −1784 −1.17 −1.03 −233 −1.27 −1.14 −840 −2.4 −2.61 −1143 −2.88 −3.11 −1472 −2.33 −2.89 −1360 −1.34 −1.32 −293 −3.34 −4.02 −1863

−1.72 −2.45 −628 −3.35 −1.99 −990 −1.07 −1.08 −454 −1.16 −1.22 −573 −2.29 −2.23 −527 −0.77 −0.73 −149 −1.24 −1.25 −88 −2.69 −1.63 −792 −0.77 −0.9 −114 −0.96 −1.07 −442 −1.68 −2.22 −558 −2.35 −2.86 −643 −1.99 −2.68 −599 −1.17 −1.18 −162 −2.16 −3.25 −805

−1.16 −2.32 −370 −2.71 −1.65 −442 −1.03 −1.1 −288 −0.97 −1.16 −322 −1.39 −1.7 −263 −0.76 −0.65 −102 −0.87 −1.01 −62 −3.34 −1.61 −437 −0.98 −0.89 −85 −0.86 −1.04 −263 −1.4 −2.11 −308 −1.52 −2.49 −320 −1.43 −2.26 −313 −1.07 −1.09 −109 −1.76 −3.24 −403

−1.26 −1.73 −200 −3.76 −1.13 −172 −0.6 −0.96 −171 −0.51 −1.02 −171 −1.6 −1.05 −107 −0.48 −0.67 −69 −0.73 −1.02 −40 −3.37 −1.26 −212 −0.66 −0.82 −54 −0.59 −0.82 −141 −0.54 −1.82 −171 −1.55 −1.88 −156 −1.46 −1.46 −147 −0.73 −1.09 −73 −2.28 −2.01 −161

−1.37 −1.07 −86 −4.96 −0.67 −93 −0.44 −0.86 −77 −0.49 −0.82 −78 −1.09 −0.73 −45 −0.32 −0.58 −35 −0.57 −0.95 −23 −4.02 −1.03 −116 −0.42 −0.67 −29 −0.6 −0.71 −68 −0.93 −1.29 −86 −2.21 −1.5 −66 −1.33 −1.33 −80 −0.51 −0.95 −39 −2.26 −1.27 −59

−1.37 −1.35 −34 −4.76 −0.7 −38 −0.41 −0.61 −26 −0.55 −0.72 −23 −2.19 −0.8 −13 −0.26 −0.52 −14 −0.4 −0.78 −11 −3.22 −1.08 −47 −0.33 −0.61 −11 −0.44 −0.55 −35 −1.1 −1.06 −37 −1.7 −1.26 −20 −1.46 −1.13 −28 −0.45 −0.93 −17 −1.99 −1.3 −25

−1.42 −0.05 −4 −3.26 −0.84 −10 −0.37 −0.11 −2 −0.38 −0.11 −1 −1.74 −0.05 −1 −0.15 −0.31 −2 −0.38 −0.78 −3 −2.49 −0.84 −2 −0.31 −0.41 −3 −0.29 −0.25 −4 −0.83 −1.17 −11 −1.41

1.42 1.01 8 3.26 0.4 5 0.37 0.46 11 0.38 0.61 16 1.74 0.77 9 0.15 0.36 5 0.38 0.67 2 2.49 0.41 15 0.31 0.45 2 0.29 0.47 13 0.83 0.86 3 1.41 1.27 14 1.28 0.84 4 0.28 0.81 3 1.65 1.38 9

1.37 1.28 32 4.76 0.72 20 0.41 0.61 39 0.55 0.66 49 2.19 0.96 26 0.26 0.48 19 0.4 0.73 8 3.22 0.72 59 0.33 0.54 9 0.44 0.53 41 1.1 1.24 17 1.7 1.11 40 1.46 0.93 21 0.45 0.88 15 1.99 1.72 45

1.37 1.38 89 4.96 0.53 55 0.44 0.78 91 0.49 0.83 117 1.09 0.7 67 0.32 0.53 40 0.57 0.9 20 4.02 0.64 117 0.42 0.69 24 0.6 0.78 94 0.93 1.72 57 2.21 1.14 79 1.33 1.49 56 0.51 0.93 40 2.26 2.09 109

1.26 1.81 190 3.76 0.71 135 0.6 0.98 183 0.51 0.97 231 1.6 1.24 138 0.48 0.61 70 0.73 1.13 38 3.37 1.24 203 0.66 0.88 49 0.59 0.83 187 0.54 1.85 154 1.55 1.77 163 1.46 1.81 148 0.73 1.04 75 2.28 2.38 251

1.16 1.96 348 2.71 1.62 361 1.03 1.05 293 0.97 1.05 400 1.39 1.61 284 0.76 0.64 104 0.87 1.07 62 3.34 1.36 471 0.98 0.93 88 0.86 0.95 303 1.4 2.01 278 1.52 2.36 329 1.43 2.33 310 1.07 1.1 109 1.76 2.69 493

1.72 2.17 595 3.35 1.93 955 1.07 1.07 471 1.16 1.16 670 2.29 1.98 512 0.77 0.71 156 1.24 1.25 94 2.69 1.58 854 0.77 0.92 117 0.96 0.96 488 1.68 2.34 473 2.35 2.44 644 1.99 2.49 593 1.17 1.22 163 2.16 2.74 971

1.82 2.48 1298 3.94 2.82 2427 1.15 1.27 893 1.2 1.23 1235 2.32 2.75 1120 0.89 0.75 291 1.31 1.34 150 2.54 2.21 2021 1.17 1.07 198 1.27 1.15 855 2.4 2.56 971 2.88 2.95 1393 2.33 3.11 1407 1.34 1.26 291 3.34 3.61 2129

3.39 3.55 5722 5.6 5.42 16 587 1.94 1.84 3589 2.12 1.87 4294 4.52 3.88 5766 1.34 1.06 1206 2.01 1.88 536 4.36 4.14 13 168 1.78 1.44 785 1.73 1.59 3662 4.59 3.92 4112 5.17 4.45 5312 4.55 4.39 6074 2.04 1.75 1097 6.91 6.19 8950

0.26 1 −1.28 −1.02 −7 −0.28 −0.45 −3 −1.65 −1.25 −7

R. Philip / Journal of Econometrics xxx (xxxx) xxx

Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

Table 6 The permanent price impact for trades of different size quantiles (q1, . . . , q16) estimated using the nonlinear VAR and the recursive model. We report in bold the permanent price impact estimates that differ between the nonlinear VAR and the recursive model at the 5% level. The average trade size for each quantile is also reported.

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Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size Price impact Price impact Trade size

(VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive) (VAR) (Recursive)

q1

q2

q3

q4

q5

q6

q7

q8

q9

q10

q11

q12

q13

q14

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q16

−3.06 −2.98 −3883 −3.32 −3.16 −36 987 −1.98 −1.88 −3436 −2.42 −2.08 −2389 −2.88 −2.21 −1810

−1.71 −2.17 −809 −3.79 −1.61 −4656 −1.22 −1.34 −868 −1.4 −1.34 −632 −1.76 −1.65 −485

−1.18 −1.75 −405 −3.75 −1.24 −1923 −1.06 −1.18 −442 −1.18 −1.2 −311 −1.34 −1.58 −241

−1.01 −1.73 −245 −4.24 −1.02 −867 −1.18 −1.05 −258 −1.17 −1.08 −197 −1.05 −1.46 −150

−1.16 −1.39 −130 −5.09 −0.83 −363 −0.55 −0.99 −143 −0.88 −1.02 −123 −0.65 −1.28 −87

−0.97 −0.63 −73 −6.55 −0.58 −140 −0.46 −0.81 −72 −0.53 −0.83 −68 −0.78 −0.91 −40

−0.96 −1.05 −36 −4.45 −0.47 −21 −0.48 −0.64 −22 −0.44 −0.77 −25 −0.63 −0.81 −16

−0.98 −0.9 −6 −5.34

0.98 0.72 3 5.34 0.51 37 0.26 0.55 19 0.32 0.67 7 0.47 0.92 5

0.96 1.05 18 4.45 0.36 100 0.48 0.5 49 0.44 0.81 28 0.63 0.79 19

0.97 1.01 49 6.55 0.45 207 0.46 0.87 86 0.53 0.93 73 0.78 1.22 46

1.16 1.44 142 5.09 0.69 415 0.55 0.91 164 0.88 1.05 138 0.65 1.16 93

1.01 1.86 280 4.24 0.94 1013 1.18 1.09 268 1.17 1.02 211 1.05 1.18 153

1.18 2.06 475 3.75 1.32 2293 1.06 1.14 431 1.18 1.17 339 1.34 1.32 244

1.71 2.27 988 3.79 1.53 5405 1.22 1.24 809 1.4 1.39 689 1.76 1.64 477

3.06 3.28 4281 3.32 3.14 38 612 1.98 1.79 3340 2.42 2.02 2735 2.88 2.23 1714

0.28 10 −0.26 −0.04 1 −0.32 −0.39 −4 −0.47 −0.46 −3

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Table 6 (continued).

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Fig. 7. The permanent price impact function for two market participants who differ in their average trade sizes. In Panel A, the large trader and the small trader are equally informed. In Panel B, the large trader is more informed than the small trader.

ensures ls contains some small size trades and ss contains some large size trades, which allows for a comparison between similar sized trades.25 Because of the random allocation of trades, the small trades randomly allocated to ls from ss should have the same information content as the small trades remaining in ss. Likewise, the large trades randomly allocated to ss from ls should have the same information content as the remaining trades in ls. Because of this random allocation, we argue both market participants have equivalent private information sets across the full range of trade sizes and thus, the same price impact functions. However, the two market participants operate at different regions on the price impact function, similar to Fig. 7, Panel A. These two market participants differ only by their average trade size. Therefore, if we correctly control for trade size, we should conclude they have the same permanent price impact. Next, we use the two market participant samples to investigate inferences when we (1) ignore trade size (i.e., the two participant trade sign VAR), (2) assume trade size is linear with private information (i.e., the two participant trade size VAR), or (3) assume trade size is nonlinear with private information (i.e., the two participant nonlinear VAR and recursive model). Fig. 8 plots the impulse response functions for the two market participants estimated via the two participant trade sign VAR (Panel A) and the two participant trade size VAR (Panel B) for the representative stock, BHP. Using the two participant trade sign VAR, we find that the permanent price impact for market participant ls is larger than the impact caused by a trade initiated by market participant ss (Panel A). In contrast, using the two participant trade size VAR, we show that the permanent price impact for market participant ls is now smaller than the impact caused by a trade initiated by market participant ss (Panel B).26 Fig. 8 demonstrates that our conclusions change depending on whether we use the two participant trade size VAR or the two participant trade sign VAR. Using the two participant trade size VAR, we find that the trader with the smaller order size has the larger price impact. Conversely, using the two participant trade sign VAR, we show that the larger order size trader has the larger price impact. These conflicting results arise because of the nonlinear relation between trade size and permanent price impact. These results have significant implications for research inferences when comparing trades originating from different market participants or trades across different market venues. For example, when comparing high frequency traders, who have a small average trade size, against institutional traders, who have a large average trade size, incorrect inferences could occur if the VAR model is misspecified (see Brogaard et al. (2014)). A similar concern occurs if we compare the permanent price impact of trades across different exchanges, or between lit and dark trading venues, due to a documented difference in average trade size among these venues (see Comerton-Forde and Putnins (2015) and Barclay et al. (2003)) To overcome the assumption of linearity, we estimate our proposed two participant nonlinear VAR using 5 lags and 8 trade size quantiles for BHP for the large (ls) and small (ss) trader. Fig. 9 depicts the permanent price functions for the two market participants who differ in average trade sizes. For each of the trade size quantiles, we plot the long term impulse responses for ls and ss. We observe that both market participants have similar price impact functions. Specifically, ss’s price impact function lies within the 95% confidence interval of ls’s price impact function. This result is consistent with the expectation that both ls and ss should have the same permanent price impact for comparable size trades as they are 25 To estimate the full price impact function, we need observations at all trade size quantiles. 26 We use the same data to estimate the impulse response functions in Panels A and B. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 8. The cumulative impulse response function for two uninformed market participants, who differ in their average trade size. Panel A plots the cumulative impulse response function for the large trade (solid line) and small trade (dotted line) using the trade sign VAR. Panel B depicts the cumulative impulse response function for the large and small trader using the trade size VAR.

equally informed.27 Thus, when comparing the permanent price impact between two market participants, we advocate the use of the two participant nonlinear VAR to draw correct inferences. To compare the two participant nonlinear VAR with the recursive model for our sample stock, BHP, we estimate the recursive model defined by (14) using sixteen possible actions, A ∈ (1, . . . , 16), where at = i if the directional trade size falls in quantile i.28 Thus, we assign a large sell order as at = 1 and a large buy order as at = 16. Again, we select only one market state (open). Fig. 10 plots the recursive estimated permanent price impacts for each trade size quantile for the two market participants. We observe that ss’s price impact function is visually identical to ls’s price impact function. Further, the small trader’s price impact function lies well within the 95% confidence interval of the large trader’s price impact function. Figs. 9 and 10 depict the permanent price impact caused by trades of different size for the two market participants using the two participant nonlinear VAR and our proposed recursive model, respectively. While both models draw the same inferences (i.e. that both participants have the same permanent price impact for equivalent size trades), the recursive model produces less noisy estimates. Unlike the VAR model, we observe the recursive model has a smooth monotonically increasing price impact function for both traders. Consistent with economic considerations, the two participants have nearly identical price impact functions. We conduct two experiments to further show that our two market participant samples are equally informed. First, because our choice of reallocating 25% of trades is somewhat arbitrary, we repeat the experiment by reallocating 10% and 40% of trades between ls and ss.29 Importantly, as shown in Appendix G, we recover the same permanent price impact function for ls and ss, regardless of the percentage of trades reallocated. This finding provides further evidence that ls 27 We provide more rigorous testing of these findings in Appendix I. 28 The choice of 16 actions is equivalent to the eight trade quantiles used for the two participant nonlinear VAR. 29 To ensure a trade size differential in the market participant samples, we are unable to reallocate 50% of trades. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 9. The permanent price impact for various trade sizes estimated using the nonlinear VAR. We estimate the permanent price impact for two equally informed market participants, who differ in their average trade sizes. The circles (stars) represent the large (small) trader. The dotted line represents the upper and lower 95% confidence interval for the large trader’s permanent price impact.

Fig. 10. The permanent price impact caused for various trade sizes estimated using the recursive model. There is one market state, open, with sixteen possible actions A ∈ (1, . . . , 16), where at = i if the directional trade size falls in decile i. We assign a large sell order as at = 1 and a large buy order as at = 16. The permanent price impact is estimated for two equally informed market participants, who differ in their average trade sizes. The circles (stars) represent the large (small) trader. The dotted line represents the upper and lower 95% confidence interval for the large trader’s permanent price impact.

and ss are equally informed: If ls is more informed than ss, we should observe a steeper price impact function for ss, and shallower price impact function for ls as we reallocate more trades between ls and ss.30 Second, in Appendix H, we generate two market participants who differ in their levels of private information. One market participant is informed and has knowledge of future price movements, while the other market participant is 30 As we reallocate more trades between the two samples, the price impact of the more informed sample would decrease as its information is being diluted by reallocating its informed trades and receiving less informed trades in return. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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uninformed. Specifically, we assign all trades which buy (sell) at the bottom (top) quartile of the price series to the informed market participant and allocate the remaining trades to the uninformed participant.31 We demonstrate that the two participant nonlinear VAR makes the correct inference; the more informed participant has a steeper permanent price impact function than the uninformed participant. In contrast, when we apply the two participant nonlinear VAR to our sample of two equally informed participants, we find that the participants have the same permanent price impact function. This finding provides additional reassurance that we are generating two equally informed market participant categories. 5.3.2. Two participants, one with knowledge of the order book The existing literature suggests that permanent price impact is a function of multiple factors in addition to trade size. For example, Dufour and Engle (2000) demonstrate that the time duration between trades affects the permanent price impact of trades, while Engle and Patton (2004) show the volume available at the best bid and ask impacts quote revisions. Thus, the permanent price impact of a trade is a function of both the trader’s private information and their knowledge of the limit order book. Hasbrouck (1991a) asserts that the VAR framework is misspecified if the dealer possesses an informational advantage in their knowledge of the limit order book, which is a phenomenon we know exists in today’s modern market. For example, Goldstein et al. (2018) show HFT use their speed advantage to gain an informational advantage of the limit order book and execute trades in the direction of the order book imbalance. Specifically, HFT take liquidity from the thin side of the order book and supply liquidity to the thick of the order book. As such, when comparing the private information content of a trade between market participants, the VAR model should control for the public information contained within the limit order book. To investigate this potential misspecification issue, we estimate different VAR specifications for two equally informed market participants who trade under different market conditions. Based on the findings of Goldstein et al. (2018), we investigate the permanent price impact of one participant, who executes their orders against the direction of the imbalance in the order book (ai), and another market participant, who executes their orders with the direction of the imbalance in the order book (w i). We define imbalance of the order book using the same metric as Goldstein et al. (2018):

∑5

DI = ∑5i=1 i=1

VolBid,i −

∑5

VolAsk,i

VolBid,i +

∑5

VolAsk,i

i=1

i=1

,

(19)

where VolBid,i is the total volume on the bid at price level i and VolAsk,i is the total volume on the ask at price level i. We allocate all trades that are buyer initiated when DI > 0 and seller initiated when DI < 0 to the market participant who trades with the imbalance, w i. All remaining trades in the sample trade against the imbalance and we assign these trades to market participant ai. For similar reasons presented in Section 5.3.1, we randomly select 25% of all trades in the ai (w i) category and reallocate to w i (ai). Due to the random process in our participant classification, we argue that neither market participant is more privately informed about the fundamental value of the stock. Accordingly, after controlling for the public information contained in the order book, we expect both market participants to have the same permanent price impact attributed to private information. To estimate the permanent price impact of the two market participants, we use two VAR specifications common in the literature. The first is the two participant trade sign VAR and the second includes the order book imbalance prior to the trade as an additional explanatory variable.32 Fig. 11, Panel A depicts the estimated impulse response functions for our representative stock, BHP, for both market participants using the two participant trade sign VAR. We find that w i has a larger permanent price impact than ai. This difference is because the impulse response function captures both the private information and the market participant’s ability to time trades around publicly available information contained in the limit order book. To distinguish between the private and public information content of a trade, we include the order book imbalance as an additional variable in the two participant trade sign VAR. Engle and Patton (2004) propose a similar VAR, which we define as rt =

xet =

xm t =

10 ∑

αi rt −i +

10 ∑

10 ∑

βi xet−i +

i=1

i=1

i=0

10 ∑

10 ∑

10 ∑

δi rt −i +

i=1

i=1

10 ∑

10 ∑

i=1

ϑi rt −i +

i=1

φi xet−i +

ζi x m t −i + ωt DI + ϵ1,t νi xm t −i + ψt DI + ϵ2,t

i=1

ψi xet−i +

10 ∑

η i xm t −i + Γt DI + ϵ3,t .

(20)

i=0

31 By definition, an informed participant must buy at the low prices and sell at the high prices. 32 The results for the two participant trade size VAR are similar to those reported for the two participant trade sign VAR. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 11. The cumulative impulse response function for two market participants, who differ in their trade timing. One participant trades with the order book imbalance (solid line) and the other trades against the imbalance (dotted line). Panel A plots the cumulative impulse response function for the trade sign VAR. Panel B depicts the cumulative impulse response functions for the trade sign VAR, which includes the order book imbalance as an exogenous variable.

Fig. 11, Panel B plots the estimated cumulative impulse response functions for our representative stock for both market participants using the VAR model defined by (20). In contrast to our earlier results using the two participant trade sign VAR, we find that the market participant who trades against the imbalance has the bigger permanent price impact.33 These results demonstrate how conflicting conclusions occur depending on the VAR specification and the corresponding assumptions. When using the two participant trade sign VAR, we assume the shape of the order book has no effect on quote revisions. In contrast, (20) assumes that the order book imbalance affects quote revisions linearly. However, it is possible that permanent price impact has a non nonlinear relation with depth imbalance. Previously, we demonstrate that quote revisions are nonlinear in trade size and that we can specify a VAR model to capture the nonlinear relation by discretizing trades into size buckets. Extending this concept, a VAR model that captures nonlinear terms in trade size and order book imbalance could contain discretized trade size and order book imbalance dummies along with their interactions. However, the number of coefficient estimates for such a model grows exponentially.34 Accordingly, the number of parameters to estimate becomes infeasible for large amounts of discretization. On the other hand, the recursive model does not suffer the same concerns as the VAR model. For the recursive model, we assume sixteen possible actions or trade sizes, A ∈ (1, . . . , 16) where at = i if the directional trade size falls in decile i. We also specify 5 environment, or market states s ∈ (1, . . . , 5), where st = j if the depth imbalance just before trade t lies in quantile j. In addition, we allow for two market participant categories such that C ∈ (w i, ai). These extensions result in 160 (s × a × c) experience tuples ⟨s, a, c ⟩. We estimate the recursive model using our representative stock, BHP, for ai and wi. Fig. 12, Panel A plots the recursive model’s estimated permanent price impact for every experience tuple for ai. For comparison, Fig. 12, Panel B, contains the 33 We provide more rigorous testing of these findings in Appendix J. 34 For example, a two market participant, 5 lag VAR model, that discretizes both trade size and depth imbalance into eight quantiles requires 19,125 coefficient estimates. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 12. The permanent price impact for various trade sizes and order book imbalances estimated using the recursive model. Trades are ranked into quintiles based on the depth imbalance at the time of the trade. Q1 DI (Q5 DI) represents the trades with the most negative (positive) imbalance. We estimate the permanent price impact for two market participants, who differ in their trade timing. Panel A (Panel B) plots the permanent price impact for a market participant who trades with (against) the imbalance.

permanent price impact estimates for every experience tuple for wi. Visually, our results show that the two participants have equal permanent price impacts for trades of the same size under similar market conditions. To test these differences more formally, we regress all permanent price impact tuples for wi against the equivalent tuple for ai. We find the regression coefficient is insignificantly different from 1, which demonstrates that both market participants have equivalent permanent price impacts for trades of similar conditions. Extending the analysis to all 20 sample stocks, we obtain the same result. The results from the recursive model are in stark contrast to the VAR models tested above, which show that market participants have different permanent price impact depending on the VAR specification. From Fig. 12, we note there is approximately 3 times more cross sectional variation in permanent price impact across the depth imbalance metric than the cross sectional variation in permanent price impact attributed to trade size. These results suggest that the shape of the order book has significantly more influence on permanent price impact than trade size. This finding reinforces the notion that price discovery does not just occur via market orders but also via limit orders, which supports the findings from Brogaard et al. (2019).35 Furthermore, given that HFT trade conditionally on the shape of the order book (see Goldstein et al. (2018)) and can account for almost 70% of market share (see Carrion (2013)), empirical research conducted in the modern trading environment should control for the current state of the order book. 6. Why is price impact nonlinear with size? Mendelson and Tunca (2004) suggest agents are endogenous in the volumes they trade, which provides one theoretical motivation for why a nonlinear relation between permanent price impact and trade size exists. Specifically, they conject that agents time their trades by trading large volumes when the market is liquid, resulting in small price impacts per unit 35 To provide additional support, we illustrate the recursive model with additional actions, such as limit order submissions and cancellations, in Appendix K. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. 13. The permanent price impact for various trade sizes under different states of market liquidity. Trades are ranked into quintiles based on the liquidity available at the time of trade. S1 (S5) represents trades in the most illiquid (liquid) quintile. Table 7 The probability a trader submits a trade of certain size under five different states of market liquidity. Liquid/Illiquid reports the ratio of the probability a trader submits an order under liquid market conditions to the probability a trader submits an order under illiquid conditions (i.e., Column V divided by Column I). Similarly, Small/Large reports the ratio of the probability a trader submits a small order versus a large order (i.e., Row 8 divided by Row 1). I (Illiquid)

II

III

1 (Large order) 2 3 4 5 6 7 8 (Small order)

0.003 0.003 0.013 0.032 0.044 0.042 0.035 0.029

0.003 0.048 0.045 0.022 0.023 0.020 0.019 0.020

0.037 0.030 0.026 0.021 0.022 0.020 0.022 0.021

Small/Large

9.895

6.955

IV

0.552

V (Liquid)

Liquid/Illiquid

0.041 0.022 0.024 0.020 0.021 0.021 0.025 0.026

0.041 0.021 0.022 0.019 0.021 0.023 0.027 0.027

13.727 6.984 1.697 0.606 0.475 0.542 0.769 0.920

0.627

0.663

traded, and small volumes when the market is illiquid, resulting in large price impacts per unit traded. In this section, we use the recursive model to test if traders behave consistently with Mendelson and Tunca (2004). We estimate the recursive model with 16 possible actions, A ∈ (1, . . . , 16), where at = i if the directional trade size falls in decile i. We also include 5 environment or market states, s ∈ (1, . . . , 5) to reflect the amount of liquidity available at the time of the market order submission, with st = 1(5) representing the lowest (highest) liquidity available quintile.36 Fig. 13 presents the permanent price impact for our representative stock, BHP, for each market liquidity state and demonstrates that small orders in an illiquid market cause a larger permanent price impact than large orders in a liquid market. Specifically, when the market is illiquid, a small market buy order (i.e. less than 10 shares) has a permanent price impact of approximately 2 basis points. In contrast, when the market is highly liquid, a large market buy order (i.e. approximately 4000 shares) has a permanent price impact of approximately 1 basis point. This finding supports Mendelson and Tunca’s (2004) hypothesis that a trade executing against the thin side of the order book has a higher permanent price impact than a trade executing against the thick side of the order book. Furthermore, we test whether traders change their behaviour based on the amount of liquidity available in the market as predicted by Mendelson and Tunca (2004). Consistent with their predictions, Table 7 shows that traders are more likely to trade in large sizes during periods of high liquidity and trade in small sizes during periods of low liquidity. Our results show a trader is 13.7 times more likely to submit a large market order when the market is extremely liquid compared to when the market is extremely illiquid. Similarly, when the market is extremely illiquid, traders are 9.8 times more likely to submit a small order than a large order. 36 We compute the amount of liquidity available as the total volume available at the best bid (ask) for market sell (buy) orders. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Collectively, these results demonstrate that a trade’s permanent price impact is conditional on the current liquidity available and that agents trade endogenously depending on prevailing market conditions. Specifically, traders choose to time their order submission to minimize permanent price impact by submitting large orders in times of high liquidity and small orders in times of low liquidity. These results support Mendelson and Tunca’s (2004) hypothesis that traders alter their behaviour due to the time varying nature of market liquidity and provide one possible explanation for why permanent price impact is non linear with trade size. Collin-Dufrense and Fos (2015) suggest that informed traders selectively trade during times of higher liquidity. Our results extend their findings and show large traders also selectively trade during times of higher liquidity. 7. Conclusion We demonstrate the importance of trade size when investigating the dynamics of trade and quote revisions. Specifically, we show that the (Hasbrouck, 1991a) VAR model is misspecified when we use trade sign or signed trade size as the trade variable, leading to incorrect conclusions. We address this misspecification by generalizing the (Hasbrouck, 1991a) VAR model to capture any nonlinear relations that may exist between quote updates and trade size. We demonstrate that our generalized VAR model performs well under basic specifications, arriving at correct inferences. However, the VAR framework remains misspecified if a trader uses knowledge of the limit order book in their trading decisions, which is a phenomenon that exists in today’s modern trading environment (see Brogaard et al. (2014) and Goldstein et al. (2018)). Estimation issues arise when we extend the VAR framework to control for information in the limit order book. As an alternative to the VAR framework, we propose a recursive model to estimate a trade’s permanent price impact. For low specifications, the VAR model and recursive model yield similar permanent price impact estimates. In contrast, with more complex model specifications, the recursive model outperforms the VAR model. Using the recursive model, we demonstrate the shape of the order book has strong effects on the dynamics of trade and quote revisions. Notably, a trade executing against the thin side of the order book has a higher permanent price impact than a trade executing against the thick side of the order book. We demonstrate that traders condition their trades on the liquidity available: agents trade large volumes during periods of high liquidity and small volumes when the market is illiquid. This finding is consistent with Mendelson and Tunca (2004) and provides one explanation for the nonlinear dynamic between quote revisions and trade size. Appendix A. Model summary See Table A.1. Appendix B. Polynomial VAR We use a simple example to demonstrate the difficulty in estimating a VAR model with quadratic terms. Suppose we have a VAR defined as

[ ] xt x2t

[

a = 11 a21

a12 a22

][

xt −1 ϵ + 1,t . ϵ2,t x2t −1

]

[

]

(B.1)

It follows that a21 xt −1 + a22 x2t −1 + ϵ2,t = (a11 xt −1 + a12 x2t −1 + ϵ1,t )2 . As such, while we can typically estimate the VAR model, two issues can arise. First, once we obtain coefficient estimates, it is possible to have an out of sample observation, which fails to satisfy the condition a21 xt −1 + a22 x2t −1 + ϵ2,t > 0. Second, when estimating the VAR coefficients, all observations must satisfy the condition ϵ2,t > a21 xt −1 + a22 x2t −1 . Given this VAR structure is heavily constricted, the resulting coefficients may result in a relatively poor fit to the data and may not represent the true data generating process. While (Hasbrouck, 1991a) addresses these issues by replacing x2t with sign(xt )|xt |2 , difficulties still arise when estimating the impulse response function. Once again, let us assume a simple VAR defined by x˜ (t) = Ax˜ (t − 1) + ϵ˜ (t),

(B.2)

where x˜ (t) is a 2 × 1 vector at time t, A is a 2 × 2 matrix of coefficients and ϵ˜ (t) is a 2 × 1 vector of residuals. Eq. (B.2) takes the VMA representation of x˜ (t) =

∞ ∑

Aj ϵ˜ (t − j).

(B.3)

j=0

To obtain the cumulative impulse response function, we now consider moving ϵ˜ (t − K ) to ϵ˜ (t − K ) + ∆ and consider x˜ (t + ∆) − x˜ (t), where x˜ (t + ∆) refers to the realization of the model when ϵ˜ (t − K ) is replaced by ϵ˜ (t − K ) + ∆. It is clear that x˜ (t + ∆) − x˜ (t) = AK ∆. Consider the case when

[ ] x˜ (t) =

xt x2t

∆=

[ ] ∆1 . ∆2

Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Table A.1 Summary of the various VAR models we estimate. Trade Sign VAR

rt =

∑5 ∑ α r + 5 β x + ϵ1,t ∑i5=1 i t −i ∑5i=0 i t −i xt = i=1 δi rt −i + i=1 φi xt −i + ϵ2,t

t: time of trade r: log change in midpoint x: +1(−1) for buys (sells)

Trade Size VAR

rt =

∑5 ∑ α r + 5 β x + ϵ1,t ∑i5=1 i t −i ∑5i=0 i t −i xt = i=1 δi rt −i + i=1 φi xt −i + ϵ2,t

t: time of trade r: log change in midpoint x signed trade size

Nonlinear VAR

rt =

∑∞

xkt =

=1 ∑i∞

αi rt −i +

k i=1 δi rt −i +

∑n ∑∞

j

=0 ∑jn=1 ∑i∞

bji xt −i + ϵ1,t

k j i=1 cji xt −i

j=1

+ ϵtk for k = 1, ..n

∑10 ∑10 ∑10 e m =1 αi rt −i + i=1 βi xt −i + i=0 ζi xt −i + ϵ1,t ∑i10 ∑10 ∑10 e = i=1 δi rt −i + i=1 φi xt −i + i=1 νi xm + ϵ2,t ∑ ∑10 ∑10 t −mi e = 10 i=1 ϑi rt −i + i=1 ψi xt −i + i=0 ηi xt −i + ϵ3,t

t: time of trade r: log change in midpoint n: number of trade size quantiles xkt : +1(−1) for buys (sells) whose trade size falls in quantile k, 0 otherwise

Two participant trade sign VAR

rt =

Two participant trade size VAR

rt =

∑10 ∑10 ∑10 e m i=1 αi rt −i + i=1 βi xt −i + i=0 ζi xt −i + ϵ1,t ∑ ∑ ∑ 10 10 10 e m xet = δ r + φ x + ν x i t −i i i=1 i t −i + ϵ2,t ∑i=101 ∑i=101 t −ei ∑ 10 xm = ϑ r + ψ x + η xm i t − i i i t t −i t −i + ϵ3,t i=1 i=1 i=0

t: time of trade r: log change in midpoint e: participant one m: participant 2 j xt : signed trade size for trades initiated by participant j, 0 otherwise

Two participant nonlinear VAR

r∑ t = ∞

t: time of trade r: log change in midpoint n: number of trade size quantiles m,j xt : +1(−1) for buys (sells) initiated by participant m whose trade size falls in quantile j, 0 otherwise

xet xm t

i=1

e,j

t: time of trade r: log change in midpoint e: participant one m: participant 2 j xt : +1(−1) for buys (sells) initiated by participant j, 0 otherwise

m,j

αi rt −i +

∑n ∑∞

beji xt −i +

∑n ∑∞

bm ji xt −i +ϵ1,t

δij rt −i +

∑n ∑∞

cjie xt −i +

e,j

∑n ∑∞

cjim xt −i + ϵ2,t

j=1

i=0

j=1

i=0

e,j

xt =

∑∞

i=1

j=1

i=1

j=1

i=1

m,j

j

m,j = ∑∞ j ∑n ∑∞ e e,j ∑n ∑∞ m m,j j i=1 δi rt −i + i=1 cji xt −i + j=1 j=1 i=0 cji xt −i + ϵ3,t

xt

To implement an impulse response perhaps, one may set ∆2 = 0 to yield the following:

[ ] xt x2t

[ =

] ] [ ] [ x ϵ + ∆1 ϵ1,t + A2 t2−2 . + A 1,t −1 ϵ2,t −1 ϵ2,t xt −2

(B.5)

However, this unconstrained answer does not reflect the fact that (xt +∆ )2 = x2t +∆ . A potential solution for this problem is to choose appropriate values for ∆1 and ∆2 such that (xt +∆ )2 = x2t +∆ ,

[ ] xt x2t

=

[ ] [ ] [ ] x ϵ1,t ϵ + ∆1 + A 1,t −1 + A2 t2−2 . ϵ2,t −1 + ∆2 ϵ2,t xt −2

(B.6)

Setting (xt +∆ )2 = x2t +∆ yields (ϵ1,t + a11 (ϵ1,t −1 + ∆1 ) + a12 (ϵ2,t −1 + ∆2 ) + φ1 )2 = ϵ2,t + a21 (ϵ1,t −1 + ∆1 ) + a22 (ϵ2,t −1 + ∆2 ) + φ2 , where φ1 is the first term of the vector A2

[

xt −2 x2t −2

]

(B.7)

and φ2 is the second term. Eq. (B.7) can be solved for ∆2 as a function

of ∆1 . However, inspecting (B.7), we note that the choice of the residual shock, ∆2 , chosen for period t − 1, is a function of the residuals at period t. Typically, since the concept of the impulse response function calculated at time t − k requires knowledge of the information available at time t − k, this framework fails to satisfy this requirement. Appendix C. Sensitivity to γ To investigate the sensitivity to our choice of γ , we estimate the permanent price impact for our representative sample stock, BHP, using the recursive model with different values for γ . Fig. C.1 depicts the permanent price impact estimates for various trade sizes based on three values of γ : 0.8, 0.95 and 0.9999. The grey dotted line represents the upper and lower 95% confidence interval when we use γ = 0.999, which is the value used in our empirical investigations. Our results show that all estimates lie within the 95% confidence interval for our benchmark value of 0.999, which demonstrates that our model is robust to a range of reasonable choices for γ . Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. C.1. The permanent price impact for various trade sizes using the recursive model for different γ values. The dotted grey line is the upper and lower 95% confidence interval of the recursive model’s estimates when γ = 0.999.

Fig. D.1. The permanent price impact for various trade sizes estimated using the nonlinear VAR. The model is estimated using a variety of different trade size quantiles, spanning from 2 to 32.

Appendix D. Determining the appropriate quantile size When estimating the nonlinear VAR and recursive model, we must determine the optimal number of quantiles. The choice of the number of quantiles is a trade-off between precision and bias. A smaller number of quantiles yields more precise estimates as more observations are used. However, estimates using a smaller number of quantiles will be biased if the data is nonlinear. Meanwhile, a larger number of quantiles is more likely to capture any nonlinearities that exist in the data. Unfortunately, this reduction in bias comes at a cost of less precise estimates as there are fewer observations in each quantile to estimate the coefficients. We estimate the nonlinear VAR defined by (4) using different trade size quantiles. Fig. D.1 plots the permanent price impact for trades based on the nonlinear VAR using 5 trade size quantiles: 2, 4, 8, 16, and 32. For all choices in the number of trade size quantiles, we observe a similar nonlinear relation between price impact and trade size. However, for models estimated with a low number of quantiles, we fail to capture the extent of the nonlinearities in the data: there is less curvature in the permanent price impact for large trades. In contrast, for models Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. E.1. The permanent price impact for various trade sizes estimated using the nonlinear VAR model with 3, 5 or 10 lags.

estimated with a large number of quantiles, we capture the nonlinearities in the data but our estimates are noisy and the curve is no longer monotonically increasing. Overall, we observe a similar relation between price impact and trade size for all choices of trade size quantiles. We present our results in the main text using eight quantiles, which balances the trade-off between precision and bias.37 By classifying the signed trade size into different quantiles, we assume that trade sizes in the same quantile generate the same permanent price impact. If a precise estimate of the permanent price impact for a given trade size is required, we advocate interpolation or curve fitting of the tabular representation of the permanent price impact estimates.38 Appendix E. VAR lag structure We investigate if the choice for the number of lags chosen for the VAR model alters the permanent price impacts estimates. Fig. E.1 plots the estimated permanent price impact estimates for our representative stock, BHP, when we use the nonlinear VAR model with 3, 5 and 10 lags, which are common choices in the literature. Fig. E.1 demonstrates that the 10 lag and 3 lag VAR estimates lie within the 95% confidence interval of the 5 lag VAR estimates. This finding suggests the higher order lag structure of the VAR model provides little additional economic significance relative to lower order lag structure VARs Appendix F. Computation time comparisons We investigate the computational time to estimate the recursive model with the computational time to estimate an equivalent VAR model. Because we require an iterative learning rule to estimate the recursive model, each iteration of the rule takes computational time, which can slow the estimation process, particularly if convergence requires many iterations. However, the iteration rule does not require the inversion of large matrices like the VAR model, which is computationally intensive. Accordingly, it is not immediately apparent which estimation technique requires more computational time. We compare the computational time for the recursive model and the VAR model based on different sample sizes and dimensionality of the state space. To ensure that we are comparing equivalent models, we require each model to have the same number of available actions. For example, a recursive model with two actions specified (i.e., to buy or sell) is equivalent to a VAR model with one trade size quantile (i.e., +1 and −1 for a buy and sell respectively). Next, to obtain the computational estimation time for each model, we estimate each model 50 times and compute the average time taken. 37 Herrera et al. (2013) provide a survey of 30 algorithms which are designed to optimally discretize a continuous variable and can be applied to help select the number of trade size quantiles. 38 For precise estimates of extremely large or small trades, we advocate the use of many trade size quantiles, which reduces bias. While increasing the number of quantiles increases the amount of noise in the estimates, we can remove much of the noise by fitting a curve to the noisy estimates in a two stage process. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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R. Philip / Journal of Econometrics xxx (xxxx) xxx Table F.1 Reports the average time (in seconds) to estimate the recursive model and the equivalent VAR model. Average times are reported for sample sizes containing 50,000, 100,000 and 200,000 observations and increasing dimensionality of the state space. Number of actions (trade size quantiles) 2 (1)

4 (2)

6 (3)

8 (4)

10 (5)

12 (6)

14 (7)

16 (8)

0.07 4.65

0.08 7.87

0.08 14.10

0.09 17.78

0.10 25.11

0.15 10.11

0.15 16.07

0.18 24.79

0.19 36.60

0.21 51.24

0.38 24.74

0.38 41.61

0.39 63.08

0.43 92.05

0.52 127.37

50,000 observations Recursive model VAR model

0.05 0.53

0.06 1.25

0.06 2.59

100,000 observations Recursive model VAR model

0.11 1.05

0.12 2.76

0.13 5.21

200,000 observations Recursive model VAR model

0.30 2.53

0.32 6.78

0.35 14.31

Fig. F.1. The average time (in seconds) to estimate the recursive model and equivalent VAR model as the dimensionality of the state space increases. Estimates are based on a sample of 200,000 observations.

We vary the sample sizes from 50,000 to 200,000 observations and the number of available actions (trade size quantiles) from 2 to 16 (1 to 8).39 Table F.1 reports the average estimation time for each model depending on the number of observations and the chosen number of actions or trade size quantiles. As expected, the computational time increases with the number of observations and the number of actions. Importantly, we observe that the computational time for the recursive model is significantly less than that of the VAR model. A recursive model with 16 available actions and 200,000 observations takes approximately half a second to estimate. In contrast, an equivalent VAR model with similar specification (i.e., 8 trade size quantiles and 200,000 observations) takes over 2 min to estimate. Fig. F.1 plots the computational time (in seconds) for both models for increasing dimensionality of the state space using 200,000 observations. As the dimensionality of the problem increases, the computational time increases for both models. However, the increase is exponential for the VAR model, whereas the increase is more linear with our proposed model. For example, using 200,000 observations, a recursive model with 16 actions takes approximately 1.7 times longer to estimate than a recursive model with 2 actions. In contrast, a VAR model with 8 trade size quantiles takes approximately 50 times longer to estimate than a VAR model with 1 trade size quantile.

39 We estimate the models using R 3.4.2 software on a PC with 64 GB RAM and a Xeon 3.2 GHz processor. For the VAR model estimation we use the R package ‘‘vars’’. We do not include data importing in the estimation time calculations. Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Fig. G.1. The permanent price impact caused for various trade sizes estimated using the VAR model defined by (6). The permanent price impact is estimated for two equally informed market participants. Panel A (Panel B) is a participant with a large (small) average trade size. To form the market participants, we reclassify 10%, 20% or 40% of trades from one participant to the other market participant.

Appendix G. Creating two equally informed participants To provide further confidence our two samples are equally informed, we alter the proportion of trades we reclassify to the other participant. If the participants differ in their levels of private information, we expect a change in the proportion of trades reclassified to change the permanent price impact for each participant. For example, if participant ss is more informed than participant ls then the more trades we reclassify from ss to ls, the more ls price impact should increase. The left panel of Fig. G.1 demonstrates the permanent price impact estimates for participant ls do not change regardless of the proportion of trades reclassified from ls to ss and vice-versa. This result provides more support that our two participants are equally informed. Appendix H. One informed and one uninformed market participant We investigate the two participant nonlinear VAR’s performance when one market participant is more informed about the future price than the other. To form the two market participants, we allocate all seller (buyer) initiated trades which occur at the top (bottom) quantile of midpoint prices to the informed market participant (i), and allocate all remaining trades to the uninformed market participant (u). This trade allocation method ensures the informed participant has knowledge of future price movements and buys (sells) prior to a price increase (decrease). For our representative stock, we estimate the permanent price impact for the two market participants using the two participant nonlinear VAR. Fig. H.1 displays the price impact of different trade quantiles for the two market participants and shows that buy orders of equivalent size for the informed market participant cause a larger price impact than buy orders for the uninformed market participant. Similarly for sells, the informed market participant has a larger negative permanent price impact than the uninformed market participant. This figure demonstrates the two participant nonlinear VAR correctly captures the difference in private information between the two market participants. Next, we formally test if the two participant nonlinear VAR correctly concludes the informed market participant has a larger permanent price impact than the uninformed for all sample stocks. We estimate the two participant nonlinear VAR using 5 lags and 8 trade size quantiles for all sample stocks and obtain the corresponding impulse response function for each trade size quantile for both market participants. Using these values, we estimate (I.1) and obtain β estimates that are statistically greater than one for all sample stocks. The lowest (highest) value is 1.163 (1.79) for TLS (MQG), with a sample stock average of 1.49. These results demonstrate that the two participant nonlinear VAR correctly concludes that for equivalent sized trade, the informed market participant’s trades cause a larger permanent price impact than the uninformed market participant’s trades. Appendix I. Two uninformed market participants with differing trade size To formally test whether both participants have equivalent permanent price impacts, we conduct the following regression: ilsk = β iss k + ϵ, Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

(I.1) permanent

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Fig. H.1. The permanent price impact for various trade sizes estimated using the VAR model defined by (6). We estimate the permanent price impact for an informed participant, who has private information about the future stock price and an uninformed participant, who has no information about the future stock price.

where ilsk is the permanent price impact for a trade with a size that falls into quantile k initiated by market participant ls, and iss k is the permanent price impact for a trade with a size that falls into quantile k initiated by market participant ss. In (I.1), if β > 1, then the large trade size market participant, ls, has a larger permanent price impact than the small trade size market participant, ss, for an equivalent sized trade and we conclude that ls has more private information. Conversely, if β < 1, we conclude that ss has more private information than ls. If β = 1, then neither participant has more private information. For our sample stocks the estimated coefficients range from 0.998 to 1.011. We find that β is insignificantly different from one for all sample stocks, with a maximum t-statistic of 0.34. This result confirms the two participant nonlinear VAR draws correct conclusions when comparing the information content of a trade for multiple market participants. Appendix J. Multi factor VAR framework We estimate the two participant trade sign VAR system and the VAR model defined by (20) for all sample stocks. One market participant trades with the imbalance (w i) and other market participant trades against the imbalance (ai). We expect both participants to have the same permanent price impact functions as they have the same private information. Columns 1 and 2 of Table J.1 report the long term impulse responses for each market participant for the two participant trade sign VAR, which does not control for the order book depth imbalance. For all sample stocks, we observe that w i has a larger estimated long term impulse response function than ai. In contrast, Columns 4 and 5 of Table J.1 report the long term impulse responses estimated via (20). For 18 of the 20 sample stocks we now conclude that w i has a smaller estimated long term impulse response function than ai. Column 6 reports the coefficient estimates for DI on log returns, ωt . For all sample stocks the coefficient is positive and significant, which demonstrates the importance of order book imbalance on quote revisions. Appendix K. Price contribution of various market actions We provide an illustration of how the recursive model can be extended to estimate the permanent price impact of additional actions, such as limit order submissions and cancelations. Specifically, we extend the recursive model to include six possible actions: market order submission (buy or sell), limit order submission (submitted to best bid or best ask) and limit order cancellation (from best bid or best ask). Table K.1 reports the permanent price impact attributed to each of the possible actions. Similar to Brogaard et al. (2019), we find both limit order submissions and cancellations convey information resulting in permanent price impacts. Consistent with economic considerations, buy limit order submissions and sell limit order cancellations result in a positive permanent price impact. Conversely, we observe a negative permanent price impact when a limit sell order is submitted or limit buy order cancelled. The submission of a market order has the largest permanent price impact of around 0.69 basis Please cite this article as: R. Philip, Estimating https://doi.org/10.1016/j.jeconom.2019.10.002.

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Table J.1 Permanent price impact estimates for market participants who trade with the imbalance (w i) and against the imbalance (ai). Panel A and B present the results for the VAR defined by (6) and (20), respectively. Model (20) includes the depth imbalance as an explanatory variable. Columns 1 and 4 present the estimates for w i and Columns 2 and 5 present the estimates for ai. In columns 3 and 6, ‘T’ denotes that w i < ai and F denotes that w i ≮ ai. Column 7 reports ωt , the coefficient for the depth imbalance, defined by (20). Panel A

AMC AMP ANZ BHP BXB CBA CSL IAG MQG NAB NCM ORG QBE RIO STO SUN TLS WBC WOW WPL

Panel B

1

2

3

4

5

6

7

wi

ai

w i < ai

wi

ai

wi < ai

ωt

6.94 2.41 4.90 5.19 6.53 10.51 6.09 1.98 23.67 5.10 10.84 10.65 6.00 15.55 6.01 4.93 0.73 5.42 7.46 14.71

5.75 2.24 3.77 3.48 4.78 6.76 0.84 1.79 18.67 3.19 3.20 9.36 5.58 10.97 4.68 4.36 0.60 4.37 5.39 10.68

F F F F F F F F F F F F F F F F F F F F

3.12 1.77 2.75 2.76 3.44 6.14 3.52 1.44 13.09 2.67 4.64 5.80 3.81 6.94 4.60 2.85 0.64 2.24 3.50 8.72

6.79 2.40 5.13 4.48 5.42 7.81 0.91 1.97 20.95 4.44 4.24 12.11 6.63 13.13 6.05 5.44 0.67 6.12 6.93 12.66

T T T T T T F T T T F T T T T T T T T T

2.63 2.26 0.87 1.13 2.50 0.58 1.25 2.11 0.93 0.77 2.44 3.16 2.23 1.00 1.88 1.93 1.32 1.04 1.15 1.49

Table K.1 Permanent price impact estimates and contribution to total price discovery for six different actions. The actions available are: market order submission (buy or sell), limit order submission (submitted to best bid or best ask) and limit order cancellation (from best bid or best ask).

Number of obs. Price impact (bp) Contribution (%)

Market order

Submit limit order

Cancel limit order

Buy

Sell

Buy

Sell

Buy

Sell

64 093 0.67 29.5

57 050 −0.71 27.7

171 782 0.13 15.1

154 590 −0.14 14.9

103 757 −0.10 7.3

97 769 0.07 5.2

points. The submission of a limit order causes permanent price impact of roughly 0.13 basis points, which is approximately 19% the magnitude of impact caused by a market order. Cancellations have the smallest permanent price impact of 0.085 basis point, which is only 12% of the permanent price impact of a market order. These findings are comparatively similar to those reported by Brogaard et al. (2019). Extending Brogaard et al. (2019), we use (16) to estimate the proportion of total price discovery attributable to each action. Table K.1 shows that market orders contribute 57.2% to price discovery, while limit order submissions and cancellations contribute 30.1% and 12.7%, respectively. While the average permanent price impact of a limit order (0.13 bps for submissions and 0.085 bps for cancellations) is significantly smaller than that of a market order (0.69 bps), limit order submissions and cancellations contribute close to 45% to overall price discovery. The proportionately higher marketwide contribution is due to the large number of limit order submissions and cancellations, relative to market orders submissions. References Barclay, M.J., Hendershott, T., McCormick, D.T., 2003. Competition among trading venues: Information and trading on electronic communications networks. J. Finance 58 (6), 2637–2666. Bertsimas, D., Lo, A., 1998. Optimal control of execution costs. J. Financial Mark. 1 (1), 1–50. Brogaard, J., Hendershott, T., Riordan, R., 2014. High-frequency trading and price discovery. Rev. Financ. Stud. 27 (8), 2267–2306. Brogaard, J., Hendershott, T., Riordan, R., 2019. Price discovery without trading: Evidence from limit orders. J. Finance. Cao, C., Hansch, O., Wang, X., 2009. The information content of an open limit-order book. J. Futures Mark. 29 (1), 16–41. Carrasco, M., Kotchoni, R., 2017. Efficient estimation using the characteristic function. Econom. Theory 33 (2), 479–526. Carrion, A., 2013. Very fast money: High-frequency trading on the nasdaq. J. Financial Mark. (ISSN: 1386-4181) 16 (4), 680–711. Collin-Dufrense, P., Fos, V., 2015. Do prices reveal the presence of informed trading? J. Finance 70 (4), 1555–1582. Comerton-Forde, C., Putnins, T.J., 2015. Dark trading and price discovery. J. Financ. Econ. 118 (1), 70–92.

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