LEM

WORKING PAPER SERIES

Directed Acyclic Graph based Information
Shares for Price Discovery
Sebastiano Michele Zema aaInstitute of Economics, Scuola Superiore Sant’Anna, Pisa, Italy.

An updated version of this manuscript is published in the Journal of Economic
Dynamics and Control.

2020/28

October 2020
ISSN(ONLINE) 2284-0400

Directed Acyclic Graph based Information
Shares for Price Discovery
Sebastiano Michele Zema*
Institute of Economics, Scuola Superiore Sant’Anna
Piazza Martiri della Libertà 56127, Pisa
sebastianomichele.zema@santannapisa.it
August 7, 2021

Abstract
The possibility to measure the contribution of agents and exchanges to the priceformation process in financial markets acquired increasing importance in the literature. In this paper I propose to exploit a data-driven approach to identify structuralvector error correction models (SVECM) typically used for price discovery. Exploiting the non-Normal distributions of the variables under consideration, I propose avariant of the widespread Information Share measure, which I will refer to as the
Directed Acyclic Graph based-Information Shares(DAG-IS), which can identify theleaders and the followers in the price formation process through the exploitation ofa causal discovery algorithm well established in the area of machine learning. The
approach will be illustrated from a semi-parametric perspective, solving the identification problem with no need to increase the computational complexity which usuallyarises when working at incredibly short time scales. Finally, an empirical application
on IBM intraday data will be provided.

Keywords: Structural VECM; Information Shares; Microstructure noise; Independent Component Analysis; Directed acyclic graphs.
JEL classification: C32, C58, G14.
Declaration of interest: none

* I am grateful to Alessio Moneta, Giacomo Bormetti, Fulvio Corsi, and Mattia Guerini for their precious

comments. I gratefully acknowledge Joel Hasbrouck for sharing his data and useful clarifications. I am
also particularly grateful to Marcelo Fernandes for the valuable suggestions, as well as to the participantsof the 13th Annual SoFiE Conference for Young Scholars. All errors are my own.

11

Introduction

The past decades have been characterized by dramatic changes in financial markets, where
the proliferation of algorithmic trading strategies put aside the intervention of human agentsin the price formation process. These algorithms execute orders at incredibly short timescales and there is no doubt anymore they account for most of the trading volumes indeveloped markets. In addition, processes of market fragmentation have been carried out
jointly with the rising of high-frequency trading. This doubly increased the complexityof financial markets, since quotes and trades might be dispersed across different listingvenues and at heterogeneous time scales which mix the slower dynamic of humans withthe faster dynamic of machines. The possible benefits of fragmented versus consolidated
markets have been object of debates for both economists and regulators also in recent
times (O’Hara and Ye, 2011; Kwan et al., 2015; Hatheway et al., 2017). As a consequence,the possibility to measure the relative contribution of each exchange in which the asset
is listed, to the price formation process, acquired increasing importance in the research
environment. In this article I propose to adopt a completely data driven strategy based on
Independent Component Analysis (ICA) to identify the SVECM widely adopted in the pricediscovery context, proposing a solution for the identification problem of the Information
Share measures (Hasbrouck, 1995). The proposed methodology exploits the non-Normal
distributions of the variables to identify the transitory shocks, and the associated mixing
matrix according to which the observed model residuals correlate across markets.
Another popular measure widely adopted in price discovery analyses that worth to bementioned is the Component Share (CS) based on the permanent-transitory (PT) decomposition introduced in Gonzalo and Granger (1995) (Harris et al., 1995; Booth et al., 1999;
Hansen and Lunde, 2006). Both the IS and CS measures build their fundamentals upon
the modeling of price changes through VECMs, with the substantial difference that while
the CS is defined only in terms of speeds of adjustment toward the common trend (i.e.
markets with lower cointegration loadings rapidly adjust and are thus more informative),
the IS measure is more concerned with variations in the prices and seeks to measure theamount of information generated by each market. Both approaches have their merits andlimits which have been documented by comprehensive discussions in the literature (Baillie2et al., 2002; De Jong, 2002; Harris et al., 2002a,b; Hasbrouck, 2002b; Lehmann, 2002). The
IS approach, compared to the CS one, has a richer specification since it considers the speedof adjustment together with the relative share of variance of the efficient price process
accounted by each market.
Still, from a microstructural modeling point of view, the IS can be uniquely determined
only when the VECM residuals are uncorrelated given that the presence of substantial contemporaneous correlations hampers the correct identification of the shocks occurred in eachmarket. Hasbrouck’s suggested solution was, in absence of an established theory providing
the causal chain to correctly order the variables in the model, to identify the SVECM using
the Choleski decomposition and going through all the possible permutations of the variables so to get upper and lower bounds for the IS of each market. In empirical applications
upper and lower bounds are often very wide giving rise to interpretative ambiguities about
the real allocation of information between the analyzed variables, making impossible todistinguish between the exchanges which lead the price formation process and exchanges
that follow it.
From a recent data-driven perspective instead, Hasbrouck (2019) proposed to exploit
the high frequency at which quotes and trades occur, modeling thus in natural time to
drastically reduce the range obtained by permuting the variables. Sampling prices at very
short time scales, even from microseconds to nanoseconds precision, heavily reduces contemporaneous cross correlations between the listing venues indeed, which by construction
leads to narrower IS bounds and allow to discard any interpretive ambiguity. To deal withthe enormous amount of coefficients to be estimated in such a natural time framework,
the author handled the problem by adopting the heterogeneous autoregressive approach(HAR) proposed by Corsi (2009). Nevertheless, this modeling approach raised interesting
and useful comments and discussions in the literature, in some cases controversial, directly
related to the econometric model specification, treatment of the high level of data sparsity
in natural time, and subsequent identification of where price discovery occurs (Brugler andComerton-Forde, 2019; Buccheri et al., 2019; De Jong, 2019; Ghysels, 2020). Despite theidentification issue above mentioned and even if other measures of price discovery havebeen proposed in the literature (see Lien and Shrestha, 2009; De Jong and Schotman,3

2010; Yan and Zivot, 2010; Putniņs̆, 2013), the IS is still one of the most widely used measures for price discovery as documented by its adoption in recent works as well (Chen andTsai, 2017; Kryzanowski et al., 2017; Lin et al., 2018; Ahn et al., 2019; Baur and Dimpfl,
2019; Brogaard et al., 2019; Hagströmer and Menkveld, 2019; Entrop et al., 2020). The
idea to exploit the non-Normal distribution of financial returns to identify the IS measure
via machine-learning based causal search algorithms, directly arises from the possibilityof introducing a purely data-driven technique in a research field in which is very hard toprovide general and robust theory-driven identification strategies. This will lead to theintroduction of the Directed Acyclic Graph based-Information Shares(DAG-IS).
The idea of identifying the IS by means of the distributional properties of the variables
was firstly introduced by Grammig and Peter (2013). The authors exploited the conceptof tail dependence through the adoption in the modeling procedure of different varianceregimes, inspired by Rigobon (2003), to identify the contribution of each market to theprice discovery process. The intuition was that differences between tail and center correlations, caused by the occurrence of extreme price changes, could be exploited to reach full
identification. In particular, following Lanne and Lütkepohl (2010), they assume price innovations to emerge as a mixture of two serially uncorrelated Normal random vectors withdifferent covariance matrices, where one is the identity and the other is a diagonal matrix
associated to different variance regimes. Still providing a solution based on the exploitationof the statistical properties of the variables of interest, the methodology proposed in thisarticle differs under many aspects. First, the methodology which I am going to propose canwork in principle under any non-Normal distribution, with no need of introducing different volatility regimes to identify the model. Second, keeping Hasbrouck (2019) as a clear
benchmark, the strategy proposed in this article is found to provide coherent empiricalresults under different time specifications when identifying the leaders and the followers in
the price formation process. For all of these reasons the solution proposed in this article canbe appealing, at the cost of introducing the assumption of independent structural shocks
in place of uncorrelated ones. Together with the assumption of the presence of an acyclic
contemporaneous causal structure (Shimizu et al., 2006; Hyvärinen, 2013), I show we canconsistently identify the causal chain in the system and thus the correct permutation of4the variable in the VECM with subsequent unique identification of the IS measures.
Recent developments about the ICA approach can be found particularly in macroeconometrics where the identification issue of structural VAR (SVAR) models is pervasive(Moneta et al., 2013; Gouriéroux et al., 2017; Lanne et al., 2017) but applications can befound also in financial econometric and forecasting studies (Audrino et al., 2005; Garcı́aFerrer et al., 2012; Fabozzi et al., 2016; Hafner et al., 2020) as well as in the empiricalvalidation of simulated models (Guerini and Moneta, 2017). Here its potential effectiveness in the identification of SVECM models for price discovery purposes will be addressed.
The article is organized as follows. In section 2 the general setting is provided, showing thebaseline model with its identification issues for price discovery. In section 3, the model andassumptions are illustrated explaining the identification scheme and a simulation exercise
is provided to clarify the methodology. Section 4 provides an empirical application on IBM
3 October 2016 intraday data, in order to have the results of Hasbrouck (2019) as a clear
benchmark to compare with. Conclusion and discussions are provided in section 5.

2

General setting

In this section I briefly go back to the microstructure setting introduced in Hasbrouck(1995), which exploits the vector error correction representation of Engle and Granger(1987), and repeated in Hasbrouck (2019). The starting point is to consider a vector oftime series log-prices pt ={p1t , p2t , ... , pnt } observed in n different exchanges but pertaining
the same security, thus all arbitrage linked and whose dynamic are modeled by VECM:∆pt = αβ 0 pt−1 +kX

Φi ∆pt−k + t

(1)

i=1

where the matrix β ∈ Rn×n−1 contains the n − 1 cointegrating vectors specified as p1 − p2 ,p1 − p3 , p1 − pn since all price series naturally cointegrate each other, and α ∈ Rn×n−1 is aloading matrix. The system in equation 1 is covariance stationary, with Cov(t ) = Ω, andthus admits a VMA(∞) representation∆pt = Ψ(L)t

5

(2)

with
Ψ(L) =∞
X

Ψi Li ,

(3)i=0

and also possess, as implied by the Granger representation theorem, a common trend
representation given by
pt = p0 + Ψ(1)tX

i + Ψ∗ (L)t

(4)

i=1

where holds the decomposition Ψ(L) = Ψ(1) + (1 − L)Ψ∗ (L), which can be seen as the multivariate generalization of the decomposition introduced in Beveridge and Nelson (1981).
The second term on the right hand side of equation 4 is the random walk component driving all prices in the system, and thus can be identified as the latent efficient price process,while the last term is the transitory component admitting the VMA(∞). The matrix Ψ(1)can be computed as (Johansen, 1991):"0I−Ψ(1) = β⊥ α⊥

#−1

!

k
X

β⊥

Φi

0

α⊥

(5)

i=1

and has rank equal to one which is the dimension of the efficient price process behind theobserved series, thus all rows of Ψ(1) are identical. The information share measure formarket j is the share of variance of the common component which is induced by the jthmarket, which meansψj2 Ωjj
ISj =
ψΩψ 0

(6)

with ψ being the common row of Ψ(1) and ψj denoting the j-th element of ψ corresponding
to market j. The above definition uniquely allocate the total variance across markets only
if the covariance matrix of the innovations Ω is diagonal, while an identification issue arises
when price innovations are correlated. To deal with a non-diagonal Ω two practical solutions
have been proposed. The first is to rewrite t in terms of orthogonal innovations ut ast = Cut(7)

where C is the Choleski decomposition of Ω. The IS thus can be computed in terms of the
new orthogonal innovations ut , that is

ISj =

[ψC]j

2

ψΩψ 0
6

.

(8)

This allocation mechanism defined through the causal chain implied by the lower triangularstructure of C depends on the particular order in which the variables are inserted in the
VECM, thus the heuristic solution was to consider upper and lower bounds for the IS ofeach market by considering all the possible variable permutations.
The second practical solution consists in drastically reducing the gap between upperand lower bounds, in order to eliminate interpretative ambiguities, estimating the modelin natural time at very high resolutions. Non zero cross correlations in Ω naturally arise
as the sampling interval increases indeed (Hasbrouck, 2019; Dias et al., 2020), thus they
can be minimized by sampling at higher frequencies. This clearly comes at costs, including
both the computational aspect of dealing with such a number of observations characterizedby high level of sparsity and a suitable model specification to estimate the coefficients still
considering a sufficiently long lag-structure in the data.
As explained also in Hasbrouck (2003), the upper and lower bounds of the IS measures
cannot be interpreted as confidence interval but rather as an identification problem. In thenext section I will propose a methodology to uniquely identify, under few assumptions, thepermutation of the variables in the system to recover the exchanges which lead the pricediscovery and the following ones.

3

Model and assumptions

Consider the n-dimensional vector of price innovations t = [1t , 2t , ..., nt ] characterized
by the non-diagonal covariance matrix Ω. Assume these observed signals to be a linear
mixture of hidden components ηt , which can be modeled as
t = A0 ηt ,(9)where A0 is a n×n mixing matrix through which the latent structural shocks ηt are revealedin each market. The equation 9 can be estimated up to permutation, sign, and scaling undersome assumptions (Comon, 1994).
Assumption 3.1. The sequence of hidden sources, with finite and non-zero variance, ofmarket microstructure noise ηt possess at most one Normal marginal distribution;
7

Assumption 3.2. Independence of the latent shocks: p(η1 , η2 , ..., ηn ) =Qni p(ηi ).

Market microstructure noises embed a variety of frictions in the trading process, inherent
not only to investment scheme strategies but also to market and asset specific factors andfundamentals. As evidenced by Aı̈t-Sahalia and Yu (2009) market liquidity risk can leadto further adjustments, not explainable by asset specific fundamentals, in the asset bid-ask
spread of the assets. Then, from a price discovery perspective the independence assumption
in 3.2 would imply market microstructure noise to be market specific and independent
from the efficient price process of the asset. Still, observed price innovations are allowed tocorrelate each other by means of the mixing matrix A0 (for example as a consequence of thetime aggregation in the sampling process previously mentioned). However, since we directly
observe only the mixtures, the independence of the hidden sources cannot be tested and has
to be assumed. Concerning the non-normality assumption of financial returns it is morea stylized fact rather than assumption. The independence of the non-Normal structural
shocks ηt is a stronger concept than uncorrelatedness which is not sufficient alone to get
rid of all the dependence information in the data. This additional information is what willallow to reach full identification of the model if there exists a contemporaneous causal chain
between the variables in the system, leading to the third and last assumption.
Assumption 3.3. The observed price innovations t can be arranged in a causal chain,meaning that their data generating process possesses a directed acyclic graph structure(DAG) (Spirtes et al., 2000).
Under assumption 3.3 we can model the system in equation 9 as the following structuralmodel,
t = B0 t + ηt

(10)where A0 = (I − B0 )−1 and the assumption of acyclical contemporaneous causal structure
implies there exists an appropriate ordering of the variables according to which B0 isstrictly lower triangular. We refer to this model as the Linear Non-Gaussian Acyclic Model(LiNGAM) firstly introduced by Shimizu et al. (2006) in the research field of non-Normal
Bayesian networks.
To understand why non-normality is fundamental in the above specified model, consider for simplicity the two-dimensional case with 1 and 2 . The structural model should
8

identify one variable as the exogenous and the other as the dependent one, which triviallymeans we should be able to choose between 1 = b2 2 + η1 and 2 = b1 1 + η2 . In thegaussian case independence and uncorrelatedness coincide and there would be no way tounderstand whether 1 → 2 , or vice versa, relying on the covariance matrix of η. Thisbecause the spherical symmetry of the joint normal distribution of the random vector ηtmakes impossible to uniquely identify the matrix A0 , which could be estimated only up
to an arbitrary orthogonal transformation as usually done with the PCA approach in theGaussian case (Hyvärinen, 2013; Moneta et al., 2013). This directly relates on the necessity
to rely on a suitable economic theory which allow us to decide what the causal structureand subsequent shock propagation would be. However, under non-normal distributions of
the structural shocks in η, the structural models will not be perfectly symmetric anymore
and can be in principle distinguished and reformulated either as

  
 

1 0
η
 1 =   1
a1 1
2
η2
or
  
 

a 1
η
 1 =  2   1 ,
1 0
2
η2
which means the model selection simply reduce to the choice of one of the two models
above which will be now theoretically distinguishable.
The LiNGAM procedure exactly aims at inferring the underlying causal network structure which is more consistent with the statistical dependencies observed in the data, andthe assumption 3.3 of an acyclical causal structure is introduced exactly to identify themodel. In the Normal case the identification of the ISs would come through the impositionof particular Choleski ordering restrictions which in turn imply both independence and an
acyclical causal structure among the variables (either one variable causes the other or viceversa according to the economic theory we rely on). Thus, in this context, the value added ofassuming independence and acyclical causal relations arises indeed by taking them jointlywith the assumption of non-Normality. In what follows, the estimation steps necessary
to achieve the structural identification of the VECM with associated information shares
9are illustrated. The first step is the estimation of the hidden non-normal and independent
components ηt .

3.1

Quantifying non-normality and recovering the independent
components

The first step necessary for the identification process is to recover the non-Normal andstatistically independent sources ηt from the observed price innovations t . This requires
the adoption of suitable measures which quantify the non-normality of a random variable.
The estimation can thus proceed by estimating the mixing matrix A0 such that the nonnormality of ηt is maximized. There are many approaches to estimate the model in 9
based for instance on the maximization of the kurtosis, negentropy, or minimization of themutual information between the random variables. All methodologies are closely related
and exploit the central limit theorem. The additive mixture t of independent and nonnormal components ηt , is always closer to a Normal distribution than the latter. Thus,
maximizing the non-normality of ηt directly relates to finding a direction of the space
through the inverse of A0 such that their mutual dependence is minimized. Going tothe optimization schemes implemented so far in the literature, in this work the FastICAalgorithm of Hyvärinen and Oja (2000) is adopted being one of the most popular algorithm
whose performances have been assessed theoretically and empirically, and for which efficientvariants of the algorithm have been also provided (Reyhani et al., 2012; Koldovsky et al.,2006; Miettinen et al., 2017). The optimization problem is solved quantifying the nonnormality in terms of approximated negentropy. The entropy (amount of information) fora continuous random variables x is defined as
Z
H(x) = − f (x) log f (x) dx.

(11)

Given that a normal variable has the largest entropy among random variables of equalvariance (Cover and Thomas, 1991), one could optimally quantify, at least theoretically,
the non-normality of a random variable by looking at the difference between its entropy
and the one of a normal variable with the same variance. The so called negentropy is thus

10

defined as
J(x) = H(N ) − H(x).

(12)

However, this would require in practice the knowledge of the probability density function
from which the data are generated. For this reason the algorithm deals with an usefulapproximation of the negentropy of a random variable which takes the form
J(x) ≈ [E (g(x)) − E (g(Z))]2 ,(13)

where Z is a standardized normal and g(·) is any suitable non-quadratic function used to ap2

proximate the negentropy given the data (Hyvärinen and Oja, 1998), here g(x) = −e−x /2 .
What is important is to choose g(·) in a way that important regularity conditions, here
briefly discussed, are satisfied to guarantee the convergence of the algorithm and related
asymptotic properties. First, the mixtures are centered to be zero mean and whitened (i.e.
uncorrelated and with their variances equal to one) which means I work with the quantities
z = P D−1/2  as done also by Fernandes and Scherrer (2018), where P DP t is the spectral
decomposition of the covariance matrix of the mixtures Ω. The algorithm searches for avector w which maximizes the non-normality of wt z measured as shown by equation 13,
that isŵ = argmaxw E(J(wt z)).

(14)

Proposition 3.1. Suppose that assumptions 3.1 and 3.2 hold true and that the following
regularity conditions are satisfied:i E(z) = 0;
ii All moments of z up to the fourth exist;
iii Both g 0 (·) and g 00 (·) are Lipschitz continuous. That is, there exist δ1 , δ2 < ∞ suchthat ||g 0 (x1 ) − g 0 (x2 )|| ≤ δ1 ||x1 − x2 || and ||g 00 (x1 ) − g 00 (x2 )|| ≤ δ2 ||x1 − x2 || ;iv g 00 (·) is bounded;
Then, being E(zg(wt z) = 0 the first order optimality condition of the maximization problem0in (14), the estimator ŵ = {w : E(zg (wt z) = 0))} is consistent and asymptotically normal,√d− N (0, Ω).
that is n(ŵ − w) →11

Proposition 3.1 summarizes the regularity conditions needed to establish the asymptotical properties of the estimates. The asymptotic normality of the ICA estimates havebeen already proven for a variety of different optimization procedures. A comprehensive
theoretical discussion on the statistical properties of the FastICA estimator can be found in
Reyhani et al. (2012). It should be mentioned that also other studied proposed to use nonNormal distributions to identify structural shocks in SVAR models (Lanne and Lütkepohl,
2010; Lanne et al., 2017; Gouriéroux et al., 2017) by assuming specific density functions
for the structural shocks.

3.2

Identifying the acyclical causal structure

Until now I made use only of assumptions 3.1 and 3.2 to estimate ηt and the mixing matrix
A0 up to permutation, sign, and scaling. The permutation indeterminacy in particularprevent the possibility to determine an appropriate order for the variables. I thus introduce
at this point assumption 3.3 to identify the structural model by adapting to our context
an heuristic causal search algorithm, well established in the machine learning research area
(Shimizu et al., 2006; Hyvärinen et al., 2010), in which the acyclicity assumption makes
possible to exploit statistical dependencies to recover a unique causal chain between priceinnovations in the SVECM. As a consequence I will be able to impose a specific order of thevariables in the Choleski decomposition. In algorithm 1, the whole procedure to finally get
the IS measure for each market without permutation indeterminacy is illustrated. While
step 3 deals with the scaling indeterminacy of the ICA estimation, steps 2 and 4 deal withthe sign and permutation indeterminacy which is the crucial problem we have when wewant to identify the IS measures for each market, leading to proposition 3.1.
Proposition 3.2. Suppose that assumptions 3.1, 3.2 and 3.3 hold true. Then the Information Shares computed by following algorithm 1 are uniquely identified.
Proof. See Appendix A.
The identification scheme proposed ensures the uniqueness of the permutation according
to which the price innovations in t are mapped in a one-to-one correspondence with theshocks ηt . Assuming a causal chain among the variables, searching for the implied DAG
12

Algorithm 1 VECM-LiNGAM algorithm for IS measures
1: Estimate the VECM equation by equation given the known cointegrating relationships,
and perform the ICA estimation on the model residuals (any suitable ICA estimator)
to recover A0 and ηt .
2: Given the unmixing matrix W = A−1
0 , find the permutation of the rows of W such thatthe permuted version W ∗ minimize

Pn∗i 1/|Wii |. The objective function to minimize inthis steps can be derived from maximum likelihood approach assuming a generalized
normal distribution for the errors (see Shimizu et al., 2006) .
3: Divide each row of W ∗ by its diagonal element so to get a matrix W̃ with ones in themain diagonal.
4: Let B̃0 = I − W̃ be the estimate of B0 . Find a permutation matrix Z such that Z B̃0 Z

0as close as possible to be strictly lower triangular. Set the upper triangular elements tob0 containing the directed acyclical graphicalzero and permute back to get the matrix B
b0 indicates the variable in position
structure (DAG). A non zero element bij in matrix B
j to cause the variable in position i.
5: Thus, order the variable in the VECM according to the DAG structure obtained andperform Choleski on the estimated price innovations. Compute the IS measures.
It is useful to note that a test of statistical significance for the non zero elements of B̃0 canbe performed following if a sufficiently long time series is available, which is the case forhigh-frequency data. Code implementation of the pruning edges method publicly in the
online LiNGAM code repository.

13

0

structure through the algorithm, clearly comes at cost. In principle the matrix Z B̃0 Z might
be such that no lower triangular matrix can be obtained by permutation. In that case theassumption of a recursive structure would not be adequate, and forcing the algorithm to0find the permutation such that Z B̃0 Z is as closest as possible to lower triangular would
lead to biased results.
Rejecting the assumption of a recursive structure would have much severe consequencesthat go beyond the identification of the IS through the DAG structure. When no recursive
structure is detected in the data the Choleski decomposition itself would not be reliableconsequently, intrinsically hampering the validity of the IS approach whenever the assumption of a diagonal covariance matrix of the error is violated. A first heuristic check for
P c2
bij < 0.2,
the matrix to be close to a lower triangular one is to fulfill the conditioni≤jhowever the null hypothesis for the coefficients to be zero can be statistically tested bybootstrap (Shimizu et al., 2006).
In the next section, a simulation exercise is provided to clarify the methodology. An
empirical application will follow afterward.
3.2.1

An illustrative simulation exercise

Here I present the proposed identification mechanism on simulated data. In light of assumptions 3.1 and 3.2 I generate samples of T=5000 observations of independent sources
ηt from an Exponential Power Distribution (EPD) whose density function is defined as

p
1 η−µ
p
exp −
(15)
f (η | p, µ, σp ) =2σp p1/p Γ(1 + 1/p)
p σp
where
Z ∞
Γ(1 + 1/p) =η 1/p e−η dx

0

(16)

= (1/p)!
is the gamma function. The variances are governed through the scale parameter σp according to
σp2 Γ(3/p)σ =(17)
Γ(1/p)
Since we need ηt to be non-Normal, I choose to simulate from the EPD density (see Nardon
2

and Pianca, 2009; Kalke and Richter, 2013, for extensive discussions about simulation
14

methodologies) to have flexibility in modeling through the additional shape parameter p.
The EPD become a normal when p = 2 and allows for fat tails by setting p < 2 (DiCiccio
and Monti, 2004; Nadarajah, 2005), which is useful in the present setting to simulate data
displaying excess kurtosis as financial price changes do. When p = 1 the distribution
converges to a Laplace, I start with a shape parameter p = 1.2 which implies an excess
kurtosis of 1.8 according tok=

Γ(1/p)Γ(5/p)− 3.
Γ(3/p)2(18)

Typically, intraday financial returns display higher levels of volatility both at the beginningand at the end of the trading day, and lower levels of volatility in the middle. For this reason
I let the variance of the distributions from which I simulate ηt vary over time, modelling it
through the diurnal U-shape pattern (Hasbrouck, 2002a; Andersen et al., 2012; Bollerslev
et al., 2016).
σηt = C + Ae−at + Be−b(1−t)(19)

where parameters are set as in Andersen et al. (2012), that is C = 0.88929198, A = 0.75,
B = 0.25, a = 10, and b = 10. In the light of the empirical application provided in thenext section, in which no more than 4-variables will be contemporaneously considered, I
simulate a 4-dimensional VECM process driven by only one common stochastic trend. The
signals t are obtained by mixing the simulated non-Normal and independent shocks ηtthrough the matrix


0.9 0
0
0


0.4 0.6 00A0 = 
,
0.5 0.2 0.7 0 
0.3 0.5 0.3 0.1(20)whose lower triangular structure implies a causal chain from the first to the forth variables
passing through the second and the third ones. The shocks in ηt are set to be independentand such that Cov(ηt ) = Σt is diagonal with equal variances, the information shares of
the two markets are affected by the speed of adjustments in α as well. Details about
the simulation setting and parameters can be found in Appendix B. With the specified
parameters, the true IS measures are IS1 = 0.58, IS2 = 0.01, IS3 = 0.39, and IS4 = 0.02.

15

The identification procedure yields the following acyclic structure.


0
0
0 0


0.44 000c0 = 
B,
0.42 0.43 0 0

0.2 0.68 0.43 0(21)which means the estimated DAG structure consistently recover the causal chain from thefirst variable to the fourth, passing before through the second and third variables. Figure
1 shows the scatter plots for the residuals t , clearly correlated as imposed in the data
generating process (DGP), and the recovered independent structural sources ηt . Note thatthe estimated mixing matrix, upon which the causal search algorithm 1 is performed, closely
resemble the true A0 up to sign indeterminacy as shown below

−1 0.01 0.03 0.004

−0.43 0.69 0.04 0.01 
c0 = 
A.
−0.59 0.26 −0.75 0.005

−0.34 0.58 −0.3
0.1

(22)

The computation of the ISs going through all the possible permutations would provide
us with IS1 = [0.1, 0.58], IS2 = [0.01, 0.32], IS3 = [0.1, 0.6], and IS4 = [0.01, 0.31], whichmake impossible to correctly disentangle the contribution of each market to the varianceof the efficient price process. However, recovering the correct causal chain by means of the
proposed identification strategy we are able to correctly permute the variables to get thetrue IS measures. In the next section, an empirical application based on IBM data keeping
previous results in the literature as a benchmark will be provided.

4

Empirical application

4.1

Benchmarking the model

Bringing the procedure on high-frequency data exposes to several caveats, mostly related tothe sparsity of the data and to model specification issues. To have a benchmark to compare
with, I empirically test the proposed methodologies on the same IBM data adopted by Hasbrouck (2019), for the day 3 October 2016, which have been shared under the authorization16

of the NYSE making this analysis possible. I thus try to disentangle the relative contribution to the price discovery process of primary listing and non-primary listing exchanges,
participant-based and SIP-based quotes, trades and quotes. As previously illustrated, themain power of the approaches relies in the exploitation of the non-Normal distributions
to separate the sources of noise in each variable. In this respect it becomes interesting totest the model stability both in natural and event time, adopting a relatively low level ofresolution (i.e. second precision) in the data for the natural time specification. This toeventually check the consistency of the obtained results in both time specifications without
increasing the computational complexity and data sparsity introduced when working atvery high frequencies.

4.2

IBM, 3 October 2016

The empirical application focuses on some detailed analyses already conducted in the literature in order to have a direct comparison which makes clearer the interpretation of theobtained results. The econometric analysis is performed on IBM’s quotes and trades forthe day 3 October 2016, with each record reporting both participant-based and SIP-based
timestamps. The final whole sample for the day consists of around 30.000 observations.
VECM models are thus estimated both in natural-time and event-time with a maximum
lag k = 10, and then the data-driven identification strategies for the IS measures are implemented. The first study disentangles the impact of time reporting differentials on thequantification of price discovery measures, through the estimation of a 4-variables VECM
including national best bids (NBBs) and offers (NBOs) constructed from both participant
and Securities Information Processor (SIP) timestamps. The purpose of the SIP is to establish a consolidated and transparent way to view the market activity for all US equities.
Starting from the participant trades and quotes, the Security Information Processor compute and publicly disseminate national best bids and offers at which broker are required
to trade, by the regulation, when acting in the interest of their customers. Given that the
SIP timestamps are by construction delayed signals of the participant ones, one expects toattribute the price discovery to the participant-based data. I then proceed with the second
analysis which consists in quantifying the price discovery in both the primary listing and17other exchanges. The VECM will include bids and offers placed on the primary listing,plus best bids and offers taken from all the exchanges except the primary one. Finally, thethird study is aimed at determining the relative contributions of trades and quotes. I thus
insert in the model trades occurred on lit and dark pools separately, plus NBBs and NBOsquotes from participant timestamps. Dark pools are private trading venues, alternative to
public accessible exchanges which are defined here as lit pools (examples are the NYSE,
NASDAQ, or LSE among others), with no regulatory transparency requirements. This
allows institutional investors to trade large securities volume without making their hands
visible, thus avoiding possible adverse price effects for their trades when huge volumes are
involved since there is no order book visible to the public. To schematically summarize
the empirical application, three separate VECMs will be estimated and identified by the
proposed methodology containing respectively:
1.

=
pModel1thiParticipantsParticipants
SIP
SIP
NBBt
, NBOt, NBBt , NBOt ;2.

pModel2=thiOtherExchanges
OtherExchanges
Primary
Primary
NBBt, NBOt, Bidt, Askt;ihParticipantsParticipants
DarkPools
LitPools
.
,
Trade
=
NBB
,
NBO
,
Trade
3. pModel3
t
t
t
t
t
In figure 2, the quantile-quantile plots for the VECM residuals are displayed. It can benoticed they are visibly leptokurtic as expected (the normality hypothesis was soundly rejected at the 1% by different tests usually adopted as the Jarque-Bera and the Shapiro-Wilk
tests). The residuals of the models estimated for the participant versus SIP timestamps
are not reported in the quantile-quantile plots to avoid useless redundancies, given thatthe variables would be again NBBs and NBOs with just the time-delays differentials inreporting them. For each model related to a given price discovery analysis, the identification procedure leading to the DAG-IS measures is performed and compared with theapproach in which upper and lower bounds are computed by going through all the possible
permutations and applying the Choleski decomposition. While table 1 shows the estimatedcoefficients of the structural matrix A0 for each analysis, table 2 summarizes the information shares estimated for each variable. The autoregressive and loading coefficients, foreach estimated VECM, are not reported here for the sake of brevity and can be found in18

the supplemental online appendix. However, as also reported in Hasbrouck (2019), estimates are mostly insignificant at the 1-second resolution while they are very significant in
the event-time specification. As illustrated in the previous section the underlying acyclicalcausal structure is encoded in the instantaneous effect matrix A0 , where non-zero elements
represent the links among the variables involved. Given the estimated results, the following
acyclical structures have been recovered
1. N BBparticipants −→ N BBSIP −→ N BOparticipants −→ N BOSIP

in natural time (1second);2. N BOparticipants −→ N BBSIP −→ N BOSIP −→ N BOparticipants
3. Bidprimary −→ Askprimary −→ N BBothersin event time

N BOothers ;4. N BBparticipants −→ T radesLit −→ T radesDark −→ N BOparticipants .

For the participant versus SIP timestamps the recovered acyclical structure changes withthe time framework adopted, but most importantly participants are always placed in thefirst position and this is the reason why the DAG-IS is able to identify them as the leaders inboth cases. Surprisingly, the DAG structures recovered in the primary versus non-primarylisting exchanges analysis and quotes versus trades analysis is stable and consistent acrossthe natural and event time settings. When theis present in place of the straight arrow→ it simply means that the recovered coefficient associated to the causal relations is notstatistically significant, meaning that the causal chain is interrupted in that specific point.
This is the case for the primary versus non-primary listing exchange analysis for example,
where no statistically significant relation is detected among shocks in different exchanges
other than the primary one and the shocks propagate only from the primary listing tothe others. While the DAG-IS measure is able to identify the participant timestamps as
the dominating ones, suggesting the correct variable’s order in the system even in the low
resolution case (1-second precision), the permutation approach would not solve the identification issue given the very wide upper and lower bounds. There is no doubt in the event
time specification instead, where also the approach based on all the possible permutations
identify the participant timestamps as the variables leading the price formation process.
Also in the price discovery across exchange analysis, the DAG-IS consistently identify the19primary listing exchange as the leader both in natural and event-time. This would not be
possible using the heuristic solution with upper and lower bounds. It has to be noticed,
however, that the DAG-IS works by finding a permutation respecting the most the statistical dependencies of the data but does not solve the temporal aggregation issue we havewhen using low levels of resolution. This means that if we discard price variations in eachmarket by aggregating over seconds, the measurement will be obviously overestimated or
viceversa but we will still be able to correctly identify the leaders (primary listing) and thefollowers (other exchanges). Finally, no sound difference has been detected, surprisingly,
when measuring the informational content of quotes and trades in the natural and event
time settings. Quotes are more informative than trades and the finding is consistently
reported by the DAG-IS measure. Since the contribution of dark trades turns out to be
negligible, their shares have been put together with the ones of lit trades differentiating
only between trades and quotes. Overall, the results obtained in the empirical application just illustrated are coherent in choosing the leaders in the price formation process,and in line with the results of Hasbrouck (2019) but without increasing the modeling andcomputational complexity introduced by working at incredibly short time-scales.

5

Conclusion

Measuring the informational content of fragmented financial markets acquired increasing
importance over time for both academics and practitioners. This article proposes a datadriven methodology with the roots in the machine learning research field, exploiting thetypical non-Normal distributions of financial returns, to uniquely identify one of the mostwidely adopted measures for price discovery and for which no identification solutions had
been proposed for almost twenty years until the first approach proposed by Grammig andPeter (2013). Differently from the cited approach, with this article I put forward an identification procedure in which the Information Shares measures can be always determined,
under some statistical and structural assumptions, with no need of exploiting the possible
presence of different volatility regimes caused by extreme price changes, thus providing ageneral identification framework for price discovery analyses. To this purpose, the DAG-IS
measure is introduced. The new estimation procedure has been discussed both theoretically20and empirically, with an illustrative simulation exercise. Keeping the empirical analysis of
Hasbrouck (2019) as a direct benchmark to compare with, the proposed procedure is foundto yield coherent results even across different time specifications, being able to correctly
identify the leaders in the price formation process. Given the flexibility of the modelingstrategy which can be assessed from a semiparametric perspective, future applications in
the field might benefit from the revisited Information Share measures here introduced,
when the assumption of a causal structure among the data is plausible to exist but nosound theory is provided to decide the direction of causality a-priori.

Appendix A
Proof of Proposition 2.2. Let σ = {σ1 , ..., σn! }, with


1
2
...
n

σi = σi (1) σi (2) . . . σi (n)and σi (·) : {1, ..., n} → {1, ..., n}, be the set of all possible permutations of the n variables in
the model. Consider the set of the Cholesky factors, of the covariance matrices, associatedto each permutation of the variables C(σ) = {C(σ1 ) , ..., C(σn! ) }. The uniqueness of theInformation Share follows directly from the fact that given the estimates of the independent
components, there is only one permutation, among the possible ones, yielding a strictly
lower triangular matrix B̂0 representing the DAG structure of the variables in the model(result proven in Shimizu et al., 2006). Then, being σi∗ and C(σi∗ ) unique solutions, theidentified Information Shares given the estimated DAG structure and computed as 2ψC(σi∗ ) j
DAG − ISj =
ψΩψ 0

(23)

are unique.

Appendix B
Data for the illustrative exercise are simulated from the equivalent VAR representation of
the VECM adopted in the paper as follows
21Π(L)pt = t
where

k
X

(24)

Πi Li

(25)

k
X
αβ 0 = (Πi − In )(26)

Π(L) ≡ In −iiφs = −(Πs+1 + Πs+2 + ... + Πk )(27)

for s = 1, 2, ..., k − 1, and such that |In − Π1 z − Π2 z 2 − ... − Πk z k | = 0 has only one
unit root since the system is driven by only one common stochastic trend. Consequently,
the matrix β contains the known cointegrating vectors and has rank equal to n-1. In thetwo-dimensional case the parameters are 
1 0.450.1,α =  , Ω = 
0.45 0.32
0.5
β0 = 

1

φ1 = 

,−1


−0.6 −0.3Π2 = 
0.7
0.9



0.6

0.3




−0.7 −0.9



1.7

0.2



,
Π1 = 
−0.2 −0.4

while in the four-dimensional case are
1 0.45 0.57 0.34
0.025 0.05 0.03

 0.08 0.07 0.06
0.45 0.67 0.4 0.54




α=
, Ω = , 0.1 0.01 0.04
0.57 0.4 0.98 0.58

0.34 0.54 0.58 0.56
0.09 0.06 0.09



0.2

0.1

φ1 = 
0.6

0.4

−0.2

−0.7

0.4





1.305 −0.225

 0.310.1  , Π1 =  0.750.35 0.55 −0.1
−0.9 −0.25 0.3
0.64
0.35

0.6

22

1.270
0.25
−0.99

−0.75

0.37




0.04 
,1.54 −0.14

0 − .31 1.21
0.53






−0.4


1
−0.1 −0.35 −0.6 −0.1






..
0
Π2 = 
 , β =  . −In−1  .
−0.6 −0.35 −0.55 0.1 




1
−0.4 0.9
0.25 −0.3
−0.2

0.2

0.7

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Table 1: Estimated instantaneous effect matrices A0 .
Participant VS SIP timestamps

natural-time

1

0

0

0

0.34

1

-0.36

0

-0.99

0

1

0

0.016

-1.001-0.0161

event-time

1-0.038-0.05-0.046

0

1

0

0

0

0.063

1

0

0

0.13

-0.12

1

1

0

-0.33

-0.0120.08

1-0.015-0.034

0

0

1

0

0

0

-002

1

1

0

0

0

-0.011

1-0.0083

0.019-0.0320

10-0.0330-0.0281Non-primary VS Primarynatural-time

1

0.026

-0.45

-0.22

0

1

-0.23

-0.45

0

0

1

0

0

0

-0.35

1

event-time

Quotes VS Trades

natural-time

1

0-0.0013

0

0.012

1

0

0.039

-0.0620

1

0-0.051

0

0.071

1

event-timeNotes : Coefficients in bold are significants at the 1% level in both LiNGAM and MLE t-student
approaches. For the LiNGAM approach, statistical significance has been tested using standard
errors from 1000 bootstrap samples.

30

Table 2: Information shares: Summary results.

DAG-IS
participants

All permutations
SIP

participants

SIP

Min

Max

Min

Max

1-sec

0.999

0.001

0.002

0.999

0.001

0.998

Event time

0.962

0.038

0.943

0.999

0.001

0.057

primary

non-primary.

primary

non-primary

Min

Max

Min

Max

1-sec

0.994

0.006

0.12

0.994

0.006

0.88

Event time

0.56

0.44

0.46

0.56

0.44

0.54

QuotesTradesQuotesTrades

Min

Max

Min

Max

1-sec

0.67

0.33

0.39

0.979

0.021

0.61

Event time

0.64

0.36

0.61

0.67

0.33

0.39

Notes : Information shares measures obtained for each identification procedureand for each price discovery analysis across participants and SIP timestamps,
trades and quotes, and exchanges. In the natural-time(1-sec) setting the mostrecent price observed in a given second interval is taken. In the event time
specification, the time counter is incremented whenever there is an update toany variable in the system instead. Trades comprises both lit and dark trades,
given that the contribution of the latter to the IS measure is negligible.

31

Figure 1: Scatter plots for the simulated residuals (top half) and estimated latent structural
shocks (bottom half).

32

Figure 2: Quantile-quantile plots of the VECM residuals. In Panel (a) are displayed the model residuals
related to the price discovery analysis across trades and quotes, while in panel (b) the one across exchanges
using quotes.

33