Marked point processes and intensity ratios for limit ...

Japanese Journal of Statistics and Data Sciencehttps://doi.org/10.1007/s42081-021-00137-9

ORIG INAL PAPER

Marked point processes and intensity ratios for limit orderbook modeling

Ioane Muni Toke1 · Nakahiro Yoshida2

Received: 2 October 2020 / Revised: 21 June 2021 / Accepted: 5 July 2021© The Author(s) 2021

AbstractThis paper extends the analysis of Muni Toke and Yoshida (2020) to the case ofmarked point processes. We consider multiple marked point processes with intensitiesdefined by three multiplicative components, namely a common baseline intensity, astate-dependent component specific to each process, and a state-dependent componentspecific to eachmarkwithin each process.We show that for specificmark distributions,this model is a combination of the ratio models defined in Muni Toke and Yoshida(2020).Weprove convergence results for the quasi-maximumand quasi-Bayesian like-lihood estimators of this model and provide numerical illustrations of the asymptoticvariances.We use these ratio processes to model transactions occurring in a limit orderbook. Model flexibility allows us to investigate both state-dependency (emphasizingthe role of imbalance and spread as significant signals) and clustering. Calibration,model selection and prediction results are reported for high-frequency trading dataon multiple stocks traded on Euronext Paris. We show that the marked ratio modeloutperforms other intensity-based methods (such as “pure” Hawkes-based methods)in predicting the sign and aggressiveness of market orders on financial markets.

Keywords Marked point processes · Quasi-likelihood analysis · Limit order book ·High-frequency trading data · Trade signature · Trade aggressiveness

This work was in part supported by Japan Science and Technology Agency CREST JPMJCR14D7; JapanSociety for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (ScientificResearch) and by a Cooperative Research Program of the Institute of Statistical Mathematics.

B Nakahiro [email protected]

Ioane Muni [email protected]

1 Université Paris-Saclay, CentraleSupélec, Mathématiques et Informatique pour la Complexité etles Systèmes, France, Bâtiment Bouygues, 3 Rue Joliot Curie, 91190 Gif-sur-Yvette, France

2 Graduate School of Mathematical Sciences, University of Tokyo, Japan, 3-8-1 Komaba, Meguro-ku,Tokyo 153-8914, Japan

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http://crossmark.crossref.org/dialog/?doi=10.1007/s42081-021-00137-9&domain=pdf

Japanese Journal of Statistics and Data Science

1 Introduction

The limit order book is the central structure that aggregates buy and sell intentions ofall the market participants on a given exchange. This structure typically evolves at avery high frequency: on the Paris Euronext stock exchange, the limit order book of acommon stock is modified several hundreds of thousand times per day. Among thesechanges, thousands or tens of thousand events account for a transaction between twoparticipants. The rest of the events indicate either the intention to buy/sell at a limitprice lower/higher than available, or the cancellation of such intentions (Abergel et al.2016).

Empirical observation of high-frequency events on a limit order book may revealirregular interval times (durations), clustering, intraday seasonality, etc. (Chakrabortiet al. 2011). Stochastic point processes are, thus, natural candidates for the modelingof such systems and their time series (Hautsch 2011). In particular, Hawkes processeshave been successfully suggested for themodeling of limit order book events (Bowsher2007;Large2007;Bacry et al. 2012, 2013;MuniToke andPomponio 2012;Lallouacheand Challet 2016; Lu and Abergel 2018).

One drawback of such models is the difficulty to account for high intraday variabil-ity. Another drawback of such models is the lack of state-dependency: the observedstate of the limit order book does not influence the dynamics of the events. One maytry to include state-dependency by specifying a fully parametric model (Muni Tokeand Yoshida 2017), which is a cumbersome solution. Another solution is to extendthe Hawkes framework with marks (Rambaldi et al. 2017) or with state-dependentkernels (Morariu-Patrichi and Pakkanen 2018). Muni Toke and Yoshida (2020) hasshown that state-dependency can be efficiently tackled by a multiplicative model withtwo components: a shared baseline intensity and a state-dependent process-specificcomponent. An intensity ratio model can then allow for efficient estimation of state-dependency. Several microstructure examples are worked out, including a ratio modelfor the prediction of the next trade sign1.

In this work, we extend the framework of Muni Toke and Yoshida (2020) to somecases of marked point processes, by adding a third term to the multiplicative definitionof the intensity, which accounts for some mark distribution. We use this extension todeepen our investigation of limit order book data. In financial microstructure, one ofthe characteristics of an order sent to a financial exchange is its aggressiveness (Biaiset al. 1995; Harris and Hasbrouck 1996). We will say here that an order is aggressiveif it moves the price. A ratio model with marks can, thus, be used to analyze both theside (bid or ask) and aggressiveness of market orders.

The rest of the paper is organized as follows. In Sect. 2, we show that some markedmodels can be viewed as combinations of intensity ratios of non-marked processes.Section 3 defines the quasi-likelihood maximum and Bayesian estimators and pro-ceeds to the analysis of the estimation. Theorem 1 states the convergence result and anumerical illustration follows. We then turn to the main financial application in Sect.4, and show how the two-step ratio model can efficiently predict (in a theoretical set-

1 When characterizing a market order, we use indistinctly the terms side (bid/ask) or sign (−1,+1) toindicate if a transaction occurs at the best bid or best ask price of the limit order book.

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ting) the sign and aggressiveness of the next trade. Finally, the full proof of Theorem1 is given in Sect. 5, and for completeness elements on quasi-likelihood analysis arerecalled in Sect. 6.

2 Marked process models as two-step ratio models

Let I = {0, 1, ..., i}.We consider certainmarkedpoint processes Ni = (Nit )t∈R+ , i ∈ I

andR+ = [0,∞). For each i ∈ I, let ki be a positive integer, and letKi = {0, 1, ..., ki }be a space of marks for the process Ni . We denote by Ni,ki = (Ni,ki

t )t∈R+ the processcounting events of type i with mark ki ∈ Ki . We have obviously Ni = ∑

ki∈KiN i,ki .

Let I = ∪i∈I({i} ×Ki

). We assume that the intensity of the process Ni with mark ki ,

i.e., the intensity of Ni,ki , is given by

λi,ki (t, ϑ i , �i ) = λ0(t) exp

(∑

j∈Jϑ ij X j (t)

)

pkii (t, �i )

at time t for (i, ki ) ∈ I, where ϑ i = (ϑ ij ) j∈J (i ∈ I) and �i (i ∈ I) are unknown

parameters.More precisely, given a probability space (Ω,F , P) equippedwith a right-continuous filtration F = (Ft )t∈R+ , λ0 = (λ0(t))t∈R+ is a non-negative predictableprocess, X j = (X j (t))t∈R+ is a predictable process for each j ∈ J = {1, ..., j}, andpkii (t, ρi ) is a non-negative predictable process for each (i, ki ) ∈ I. Later, we will puta condition so that the mapping t �→ λi,ki (t, ϑ i , �i ) is locally integrable with respectto dt , and we assume that Ni,ki

0 = 0, and for each (i, ki ) ∈ I, the process

N i,kit = Ni,ki

t −∫ t

0λi,ki (s, (ϑ i )∗, (�i )∗)ds

is a local martingale for a value((ϑ i )∗, (�i )∗

)of the parameter

(ϑ i , �i

). We assume

that the counting processes Ni,ki (i ∈ I; ki ∈ Ki ) have no common jumps.In what follows, we consider the processes pkii (t, �i ) such that

∑

ki∈Ki

pkii (t, �i ) = 1 (2.1)

for i ∈ I. Then, the ki -dimensional process (pkii (t, �i ))ki∈Ki gives the conditionaldistribution of the event ki when the event i occurred. Under (2.1), the intensity processof Ni becomes

λi (t, ϑ i ) =∑

ki∈Ki


(∑

j∈Jϑ ij X j (t)

)

. (2.2)

The process λ0 is called a baseline intensity, whose structurewill not be specified, inother words, λ0 will be treated as a nuisance parameter, differently from the use of Cox

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regression as in Muni Toke and Yoshida (2017). The baseline intensity may representthe globalmarket activity in finance, for example, and its irregular changemay limit thereliability of estimationprocedures andpredictions for anymodel fitted to it.MuniTokeand Yoshida (2020) took an approach with an unstructured baseline intensity processand showed advantages of suchmodeling. Statistically, the processX(t) = (X j (t)) j∈Jis an observable covariate process. Since the effect of these covariate processes to theamplitude of λi (t, ϑ i ) is contaminated by the unobservable and structurally unknownbaseline intensity, a more interesting measure of dependency of λi (t, ϑ i ) to X(t) isthe ratio

λi (t, ϑ i )/∑

i ′∈Iλi

′(t, ϑ i ′)

for i ∈ I. Thus, we introduce the difference parameters θ ij = ϑ ij − ϑ0

j (i ∈ I, j ∈ J),

(θ0j = 0 in particular) and consider the ratios

r i (t, θ) =exp

(∑

j∈J ϑ ij X j (t)

)

∑i ′∈I exp

(∑

j∈J ϑ i ′j X j (t)

) =exp

(∑

j∈J θ ij X j (t)

)

1 +∑i ′∈I0 exp

(∑

j∈J θ i′j X j (t)

) (2.3)

for i ∈ I, where θ = (θ ij )i∈I0, j∈J with I0 = I \ {0} = {1, ..., i}.In this paper, we further assume that the factor pkii (t, �i ) is given by

pkii (t, �i ) =exp

(∑

ji∈Ji �i,kiji

Y iji(t)

)

∑k′i∈Ki

exp

(∑

ji∈Ji �i,k′

iji

Y iji(t)

)

for (i, ki ) ∈ I, Ji = {1, ..., ji }. Obviously, pkii (t, �i ) = qkii (t, ρi ) defined by

qkii (t, ρi ) =exp

(∑

ji∈Ji ρi,kiji

Y iji(t)

)

1 +∑k′i∈Ki,0

exp

(∑

ji∈Ji ρi,k′

iji

Y iji(t)

) (2.4)

for (i, ki ) ∈ I, where ρi,kiji

= �i,kiji

− �i,0ji

(ki ∈ Ki , j ∈ Ji , i ∈ I), ρi,0ji

= 0 in

particular, and ρi = (ρi,kiji

)ki∈Ki,0, ji∈Ji (i ∈ I) withKi,0 = Ki \ {0} = {1, ..., ki }. Thepredictable processes (Y i

ji(t))t∈R+ (i ∈ I, ji ∈ Ji ) are observable covariate processes,

Ji being a finite index set. This is a multinomial logistic regression model.Let Θ be a bounded open convex set in R

p with p = i j . For each i ∈ I, Ri

denotes a bounded open convex set in Rpi with pi = ji ki . Write ρ = (ρi )i∈I. Let

R = Πi∈IRi . We will consider Θ × R as the parameter space of (θ, ρ).

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Remark 1 The marked ratio model


(∑

j∈Jϑ ij X j (t)

) exp

(∑

ji∈Ji �i,kiji

Y iji(t)

)

∑k′i∈Ki

exp

(∑

ji∈Ji �i,k′

iji

Y iji(t)

)

is in general not equivalent to a non-marked ratio model in larger dimension, in whichwe would write the intensity of the counting process of events of type i ∈ Iwith markki ∈ Ki as

λi,ki (t, ϑ i,ki ) = λ0(t) exp

(∑

j∈Jϑi,kij Z j (t)

)

for some covariate processes Z j , j ∈ J. Equivalence of the models would requirethese expressions to coincide for some sets of covariates and parameters. However, ifZ j (t) = 0 for all j ∈ J, then necessarily X j (t) = 0 for all j ∈ J and Y i

ji(t) = 0 for

all i ∈ I and ji ∈ Ji . This in turn implies 1|Ki | = λ0(t)

λ0(t)for all i ∈ I, which is generally

not true. In Sect. 4.5, a non-marked ratio model is used as a benchmark to assess theperformances of the marked ratio model. Prediction performances are indeed shownto be different.

3 Quasi-likelihood estimation of two-step ratio model

3.1 Quasi-maximum likelihood estimator and quasi-Bayesian estimator

The two step marked ratio model consists of the two kinds of ratio models (2.3)and (2.4). Estimation of this model can be carried out with multiple successive ratiomodels.

In the first step, we consider the parameter θ = (θ ij )i∈I0, j∈J and the ratios (2.3) fori ∈ I. The quasi-log-likelihood based on observations on [0, T ] for this ratio modelis

HT (θ) =∑

i∈I

∫ T

0log r i (t, θ) dNi

t . (3.1)

This comes from the multinomial logistic regression. A quasi-maximum likelihoodestimator (QMLE) for θ is a measurable mapping θM

T : Ω → Θ satisfying

HT (θMT ) = max

θ∈Θ

HT (θ)

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for all ω ∈ Ω .2

In the second step, we consider the ratios (2.4) and the associated quasi-log-likelihood

H(i)T (ρi ) =

∑

ki∈Ki

∫ T

0log qkii (t, ρi ) dNi,ki

t (3.2)

for i ∈ I. Then, a measurable mapping ρi,MT : Ω → Ri is called a quasi-maximum

likelihood estimator (QMLE) for ρi if

H(i)T (ρ

i,MT ) = max

ρi∈Ri

H(i)T (ρi ).

It is possible to pool these estimating functions by the single estimating function

HT (θ, ρ) = HT (θ) +∑

i∈IH

(i)T (ρi ). (3.3)

In other words,

HT (θ, ρ) =∑

i∈I

∑

ki∈Ki

∫ T

0log

(r i (t, θ)qkii (t, ρi )

)dNi,ki

t . (3.4)

The collection of QMLEs(θMT , (ρ

i,MT )i∈I

)is a QMLE forHT (θ, ρ). Use ofHT (θ, ρ)

is convenient when we consider asymptotic distribution of the estimators θMT and ρ

i,MT

(i ∈ I) jointly.The quasi-Bayesian estimator (QBE)

(θ BT , (ρ

i,BT )i∈I

)is defined by

θ BT =

[ ∫

Θ×Rexp

(HT (θ, ρ)

)�(θ, ρ) dθdρ

]−1

×∫

Θ×Rθ exp

(HT (θ, ρ)

)�(θ, ρ) dθdρ (3.5)

and

ρi,BT =

[ ∫

Θ×Rexp

(HT (θ, ρ)

)�(θ, ρ) dθdρ

]−1

2 Originally, θMT is defined on a sample space ST expressing all the possible outcomes of

(λ0(t), X j (t), Yiji(t); t ∈ [0, T ], i ∈ I, j ∈ J , ji ∈ Ji ). If (Ω,F , P) is an abstract space used for defining

the true probability measure P∗T on ST by some random variable VT : Ω → ST (i.e. P∗

T = PV−1T ), then

treating θMT as a function on Ω conflicts with the definition of θMT . However, what we want to investigate is

concerning the distribution of θMT (defined on ST ) under P∗T , and then we can pull back θMT on ST to Ω by

VT if P∗T = PV−1

T . For this reason, we can identify θMT with θMT ◦ VT , and may regard θMT as defined onΩ . This remark makes sense especially when one treats a weak solution of a stochastic differential equationfor a covariate.

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×∫

Θ×Rρi exp

(HT (θ, ρ)

)�(θ, ρ) dθdρ (3.6)

for a prior probability density �(θ, ρ) on Θ ×R. We assume that � : Θ ×R → R+is continuous and

0 < inf(θ,ρ)∈Θ×R

�(θ, ρ) ≤ sup(θ,ρ)∈Θ×R

�(θ, ρ) < ∞. (3.7)

Since HT (θ) and H(i)T (ρi ) have no common parameters, the maximization of

HT (θ, ρ) with respect to the parameters θ and ρi (i ∈ I) can be carried out sepa-rately. However, these components are not always individually treated for the QBE.If �(θ, ρ) is a product of prior densities as �(θ, ρ) = � ′(θ)Πi∈I� i (ρi ), then theeach integral in (3.5) and (3.6) is simplified and we can compute θ B

T and ρi,BT (i ∈ I)

separately:

θ BT =

[ ∫

Θ

exp(HT (θ)

)� ′(θ) dθ

]−1 ∫

Θ

θ exp(HT (θ)

)� ′(θ) dθ

and

ρi,BT =

[ ∫

Ri

exp(H

(i)T (ρi )

)� i (ρi ) dρi

]−1 ∫

Ri

ρi exp(H

(i)T (ρi )

)� i (ρi ) dρi

for i ∈ I.

3.2 Quasi-likelihood analysis

Let X(t) = (X j (t)) j∈J and let Yi (t) = (Y iji(t))ji∈Ji for i ∈ I. We consider the

following conditions.

[M1] The process(λ0(t),X(t),Y(t)

)is a stationary process and the randomvariables

λ0(0), exp(|X j (0)|) and exp(|Y iji(0)|) are in L∞– = ∩p>1L p for j ∈ J, ji ∈ Ji

and i ∈ I.

Condition [M1] is not restrictive since the covariates can often be regarded as boundedin applications.

The alpha mixing coefficient α(h) is defined by

α(h) = supt∈R+

supA∈B[0,t]

B∈B[t+h,∞)

∣∣P[A ∩ B] − P[A]P[B]∣∣,

where for I ⊂ R+, BI denotes the σ -field generated by(λ0(t), (X j (t)) j∈J,

(Y i,kiji

(t))i∈I, ji∈Ji ,ki∈Ki

)t∈I .

[M2] The alpha mixing coefficient α(h) is rapidly decreasing in that α(h)hL → 0 ash → ∞ for every L > 0.

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In the two-step ratios model, the category (i, ki ) is selected with twofold multi-nomial distributions of sample size equal to 1. First the class i ∈ I is selected whenξi = 1 for some random variable

ξ = (ξ0, ..., ξi ) ∼ Multinomial(1;π0, ...., πi ).

If ξi = 1 for a class i ∈ I, then the class ki ∈ Ki is chosen as ki = k when ηik = 1 forsome independent random variable

ηi = (ηi0, ..., ηiki

) ∼ Multinomial(1;π ′0, ..., π

′ki

).

Denote by V (x, θ) the variance matrix of the (1 + i)-dimensional multinomialdistribution Multinomial(1;π0, π1, ..., πi ) with πi = r i (x, θ), i ∈ I, where

r i (x, θ) =exp

(∑

j∈J ϑ ij x j

)

∑i ′∈I exp

(∑

j∈J ϑ i ′j x j

) =exp

(∑

j∈J θ ij x j

)

1 +∑i ′∈I0 exp

(∑

j∈J θ i′j x j

) , x = (x j ) j∈J.

Denote by V i (yi , ρi ) the variance matrix of the (1 + ki )-dimensional multinomialdistribution Multinomial(1;π ′

0, π′1, ..., π

′ki

) with π ′ki

= qkii (yi , ρi ), ki ∈ Ki , where

qkii (yi , ρi ) =exp

(∑

ji∈Ji �i,kiji

yiji

)

∑k′i∈Ki

exp

(∑

ji∈Ji �i,k′

iji

yiji

)

=exp

(∑

ji∈Ji ρi,kiji

yiji

)

1 +∑k′i∈Ki,0

exp

(∑

ji∈Ji ρi,k′

iji

yiji

) , yi = (yiji ) ∈ Rji (i ∈ I).

Let us introduce some notations used in the following analysis. For a tensor T =(T i1,...,ik )i1,...,ik , we write

T [u1, ..., uk] = T [u1 ⊗ · · · ⊗ uk] =∑

i1,...,ik

T i1,...,ik ui11 · · · uikk (3.8)

for u1 = (ui11 )i1 ,..., uk = (uikk )ik . Brackets [ , ..., ] stand for a multilinear mapping.We denote by u⊗r = u ⊗ · · · ⊗ u the r times tensor product of u.

Denote by ∂(θ,ρ) the differential operator with respect to (θ, ρ). Let

ΓT (θ, ρ) = −T−1∂2(θ,ρ)HT (θ, ρ)

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and let ΓT = ΓT (θ∗, ρ∗). Then, as detailed in Section A.2,

ΓT (θ, ρ) = diag[ΓT (θ), Γ 1

T (ρ1), ..., Γ iT (ρ i )

],

where

ΓT (θ)[u⊗2] = 1

T

∫ T

0

(

V 0(X(t), θ) ⊗ X(t)⊗2)

[u⊗2]∑

i∈IdNi

t (u ∈ Rp) (3.9)

with V 0(x, θ) = (V (x, θ)i,i ′)i,i ′∈I0 , and

Γ iT (ρi )[(ui )⊗2] = 1

T

∫ T

0

(

V i0(Y

i (t), ρi ) ⊗ Yi (t)⊗2

)

[(ui )⊗2]dNit (ui ∈ R

pi )

with V i0(y

i , ρi ) = (V i (yi , ρi )ki ,k′i)ki ,k′

i∈Ki,0.

Let

Λ(w, x) = w∑

i∈Iexp

(x[ϑ∗i ]) (3.10)

for w ∈ R+ and x ∈ Rj .

We have

V (x, θ)i,i ′ = 1{i=i ′}r i (x, θ) − r i (x, θ)r i′(x, θ).

Therefore,

V (X(t), θ)i,i ′ = 1{i=i ′}r i (t, θ) − r i (t, θ)r i′(t, θ) (3.11)

and V 0(X(t), θ)i,i ′ = V (X(t), θ)i,i ′ for i, i ′ ∈ I0. Write V 0(x) = V 0(x, θ∗).We have

V i (yi , ρi )ki ,k′i= 1{ki=k′

i }qki (yi , ρi ) − qki (yi , ρi )qk

′i (yi , ρi ).

Hence,

V i (Yi (t), ρi )ki ,k′i= 1{ki=k′

i }qkii (t, ρi ) − qkii (t, ρi )q

k′i

i (t, ρi ) (3.12)

and V i0(Y

i (t), ρi )ki ,k′i

= V i (Yi (t), ρi )ki ,k′ifor ki , k′

i ∈ Ki,0. We denote V i0(y

i ) =V i

0(yi , (ρi )∗).

The symmetric matrices Γ (θ) and Γ i (ρi ) are defined by

Γ (θ)[u⊗2] = E

[(

V 0(X(0), θ) ⊗ X(0)⊗2)

[u⊗2]Λ(λ0(0),X(0))

]

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for u ∈ Rp, and

Γ i (ρi )[(ui )⊗2] = E

[(

V i0(Y

i (0), ρi ) ⊗ Yi (0)⊗2

)

[(ui )⊗2]Λ(λ0(0),X(0))r i (0, θ∗)]

for ui ∈ Rpi , i ∈ I, respectively. Let p = p + ∑

i∈I pi = i j + ∑i∈I ki ji . The full

information matrix is the p × p block diagonal matrix

Γ (θ, ρ) = diag[Γ (θ), Γ 0(ρ0), Γ 1(ρ1), ..., Γ i (ρ i )

],

and in particular set

Γ = Γ (θ∗, ρ∗). (3.13)

An identifiability condition will be imposed.

[M3] infθ∈Θ

infu∈Rp: |u|=1

Γ (θ)[u⊗2] > 0 and infρi∈Ri

infu∈Rpi : |ui |=1

Γ i (ρi )[(ui )⊗2] > 0 for

every i ∈ I.

For the QMLE ψMT = (θM

T , ρMT ) and the QBE ψ B

T = (θ BT , ρB

T ) of ψ = (θ, ρ) =(θ, ρ1, ..., ρ i ), let

u AT = T 1/2(ψ A

T − ψ∗) (A ∈ {M, B}).

Theorem 1 Suppose that Conditions [M1], [M2] and [M3] are satisfied. Then,E[ f (u A

T )] → E[ f (Γ −1/2ζ )]

as T → ∞ for A ∈ {M, B} and every f ∈ C(Rp) of at most polynomial growth,where ζ is a p-dimensional standard Gaussian random vector.

Example 1 As an illustration we consider the case with two processes (I = {0, 1}),and two marks for each process (K0 = K1 = {0, 1}). The first state-dependent termtakes into account one covariate X1 (i.e., J = {1}). The mark distributions bothdepend on another covariate Y1 (i.e. J0 = J1 = {1}). In this example, we assumethat X1 and Y1 are independent Markov chains with values in {−1, 1} and constanttransition intensities λX andλY .We assume that λ0 is the intensity of aHawkes process(Ht )t≥0 with a single exponential kernel, i.e., λ0(t) = μ + ∫ t

0 αe−β(t−s) dHs , with(α, β) ∈ (R∗+)2, α

β< 1.

The two-step ratio model estimates the parameters (θ11 , ρ0,11 , ρ

1,11 ) defined as θ11 =

ϑ11 − ϑ0

1 and ρi,11 = �

i,11 − �

i,01 , i = 0, 1. In this specific case, the matrix Γ of Eq.

(3.13) is a 3 × 3-diagonal matrix, and a direct computation shows that the diagonalcoefficients are

Γ0,0 = μ

1 − αβ

eθ11

1 + eθ11

(cosh ϑ0

1 + cosh ϑ11

),

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Table 1 Numerical results forthe estimation of the model ofExample 1

T True value θ11 ρ0,11 ρ

1,11

1.500 1.000 2.000

10 Estimator mean 1.817 1.576 5.146

Estimator sd 1.829 2.875 5.491

T− 12 Γ

− 12

i,i 0.509 0.627 0.858

30 Estimator Mean 1.541 1.044 2.402

Estimator sd 0.324 0.610 2.096

T− 12 Γ

− 12

i,i 0.294 0.362 0.495

100 Estimator Mean 1.508 0.999 2.011

Estimator sd 0.164 0.201 0.289

T− 12 Γ

− 12

i,i 0.161 0.198 0.271

300 Estimator Mean 1.502 1.005 2.013

Estimator sd 0.094 0.114 0.159

T− 12 Γ

− 12

i,i 0.093 0.114 0.157

1000 Estimator Mean 1.501 1.000 2.009

Estimator sd 0.052 0.065 0.085

T− 12 Γ

− 12

i,i 0.051 0.063 0.086

3000 Estimator Mean 1.498 1.001 1.999

Estimator sd 0.029 0.038 0.053

T− 12 Γ

− 12

i,i 0.029 0.036 0.050

Γ1,1 = μ

1 − αβ

eρ0,11

1 + eρ0,11

eθ11 /2

1 + eθ11

(

coshϑ01 + ϑ1

1

2+ cosh

3ϑ11 − ϑ0

1

2

)

,

Γ2,2 = μ

1 − αβ

eρ1,11

1 + eρ1,11

eθ11 /2

1 + eθ11

(

coshϑ01 + ϑ1

1

2+ cosh

3ϑ11 − ϑ0

1

2

)

.

We run 1000 simulations of the processes (N 0, N 1) with their marks for variousvalues of horizon T . Numerical values used in these simulations are the following:μ = 0.5, α = 1.0, β = 2.0, λX = λY = 0.5, ϑ0

1 = −0.75, ϑ11 = 0.75, �0,0

1 = −0.5,

�0,11 = 0.5, �

1,01 = −1.0, �

1,11 = 1.0. For each simulation, we compute the quasi-

maximum likelihood estimators (θ11 , ρ0,11 , ρ

1,11 ) with the two-step ratios described

above. Table 1 gives the mean estimators and the true values of the parameters, as well

as the empirical standard deviation, compared to the theoretical values T− 12 Γ

− 12

i,i ,i = 0, 1, 2 from Theorem 1, for various values of T . For completeness, Figure 1also plots the empirical standard deviations of the three estimators and the theoretical

standard deviation T− 12 Γ

− 12

i,i , i = 0, 1, 2 of Theorem 1, as a function of the horizonT .

123


●

●

●

●

●

●

10 50 200 10001e−0

31e

−01

1e+0

1 θ11

Horizon T

Stan

dard

dev

iatio

n

● Empirical sdTheoretical sd

●

●

●

●

●

●

10 50 200 10001e−0

31e

−01

1e+0

1 ρ10,1

Horizon T


●

●

●

●

●

●

10 50 200 10001e−0

31e

−01

1e+0

1 ρ11,1

Horizon T


Fig. 1 Empirical and theoretical standard deviation of the quasi-maximum likelihood estimators θ11 (left),

ρ0,11 (center) and ρ

1,11 ) (right)

Asymptotic values predicted by Theorem 1 are indeed empirically retrieved, whichends this numerical illustration.

4 Modeling and predicting sign and aggressiveness of market orders

4.1 Intensities of the processes countingmarket orders

We consider the market orders submitted to a given limit order book. Let N 0 be theprocess counting the market orders submitted on the bid side (sell market orders) andN 1 the process counting the market orders submitted on the ask side (buy marketorders). On each side, we further consider whether the order is an aggressive orderthat moves the price (labeled with mark 1), or a non-aggressive order that does notmove the price (labeled with mark 0).

We assume that the intensity of an order of type i ∈ I = {0, 1}with mark ki ∈ K =K0 = K1 = {0, 1} is


⎛

⎝∑

j∈Jϑ ij X j (t)

⎞

⎠exp

(∑ji∈Ji �

i,kiji

Y iji(t))

∑k′i∈Ki

exp(∑

ji∈Ji �i,k′

iji

Y iji(t)) . (4.1)

In the following applications, we will consider several possible models definedwith various sets of covariates X j , j ∈ J and Y i

j , j ∈ Ji , i = 0, 1. The tested sets of

covariates X j , j ∈ J and Y ij , j ∈ Ji , i = 0, 1 will all be subsets of the following list

of possible covariates (besides Z0 = 1 common to all models):

– Z1:qB (t)−q A(t)qB (t)+q A(t)

where qB(t) (resp. q A(t)) is the quantity available at the best bid(resp.ask) at time t (i.e., the imbalance);

– Z2: ε(t), where ε(t) is the sign of the last market order at time t (1 for an askmarket order, −1 for a bid market order ;

– Z3: s(t)ε(t) the signed spread, where s(t) is the observed spread in currency attime t ;

123


– Z4: H0,1(t) = log(μ0,1 + ∫ t

0 α0,1e−β0,1(t−s)dN 0,1s

)(Hawkes covariate for

aggressive bid market orders)

– Z5: H0,0(t) = log(μ0,0 + ∫ t

0 α0,0e−β0,0(t−s)dN 0,0s

)(Hawkes covariate for non-

aggressive bid market orders)

– Z6: H1,1(t) = log(μ1,1 + ∫ t

0 α1,1e−β1,1(t−s)dN 1,1s

)(Hawkes covariate for

aggressive ask market orders)

– Z7: H1,0(t) = log(μ1,0 + ∫ t

0 α1,0e−β1,0(t−s)dN 1,0s

)(Hawkes covariate for non-

aggressive ask market orders)

– Z8: H0(t) = log(μ0 + ∫ t

0 α0e−β0(t−s)dN 0s

)(Hawkes covariate for bid market

orders)

– Z9: H1(t) = log(μ1 + ∫ t

0 α1e−β1(t−s)dN 1s

)(Hawkes covariate for ask market

orders).

With these Hawkes covariates, the ratio model can actually be seen as a kind of non-linear Hawkes process. When the theory applied, the ergodicity is an assumption. Inthe present model, it depends on the nature of the process λ0, that was set generally.Brémaud andMassoulié (1996) treated a stability problem of a nonlinear Hawkes pro-cess. If the system has a Markovian representation, there is a possibility of applyinga drift condition like Abergel and Jedidi (2015) and Clinet and Yoshida (2017). Onthe other hand, the intraday stationarity (ergodicity) is not essentially important. Asdescribed in Section 3.2 of Muni Toke and Yoshida (2020), in quite parallel to thesimple stationary case, we can relax the assumption of intraday stationarity by consid-ering a repeated measurements model. Then, we only need a more realistic ergodicityof the data across the long-run repeated measurements, and after all, we can validatethe methods.

4.2 Limit order book data

We use tick-by-tick data for 36 stocks traded on Euronext Paris. The sample spansthe whole year 2015, i.e., roughly 200 trading days for each stock, although somedays are missing for some stocks. Table 3 in Sect. 7 lists the stocks investigatedand the number of trading days available. Rough data consist in a TRTH (Thomson-Reuters Tick History) database: for each trading day and each stock, one file lists thetransactions (quantities and prices) and one file lists themodifications of the limit orderbook (level, price and quantities). Timestamps are given with a millisecond precision.Synchronization of both files and reconstruction of the limit order book are carried outwith the procedure described in Muni Toke (2016). One strong advantage of the ratiomodel is that it does not require precise timestamps in itself, since timestamps do notappear explicitly in the quasi-likelihood of the ratios,while fitting other intensity-basedmodels (e.g., Hawkes processes) requires unique precise timestamps for log-likelihoodcomputation. Here, if Hawkes fits are used as covariates (covariates Z4 to Z9 in ourapplication), then we choose to consider only unique timestamps, i.e., we aggregateorders of the same type occurring at the same timestamp.

123


4.3 Estimation procedure of the two-step ratio model

Following Sects. 2 and 3, estimation of the model defined at Eq. (4.1) can be carriedout with multiple successive ratio models. In the first step, we consider the differenceparameters θ ij = ϑ i

j − ϑ0j , i ∈ I \ {0}, j ∈ J and the ratios (i ∈ I \ {0}):

r i (t, θ) =exp

(∑j∈J ϑ i

j X j (t))

∑i ′∈I exp

(∑j∈J ϑ i ′

j X j (t)) =

⎡

⎣∑

i ′∈Iexp

⎛

⎝∑

j∈J(θ i

′j − θ ij )X j (t)

⎞

⎠

⎤

⎦

−1

.

(4.2)

The quasi-log-likelihood based on the observation on [0, T ] for this ratio model isdefined at Eq. (3.1). In the second step, we consider the ratios

pkii (t, �i ) =

exp(∑

ji∈Ji �i,kiji

Y iji(t))

∑k′i∈Ki

exp

(∑

ji∈Ji �i,k′

iji

Y iji(t)

) =⎡

⎢⎣∑

k′i∈Ki

exp

⎛

⎝∑

ji∈Ji(�

i,k′i

ji− �

i,kiji

)Y iji(t)

⎞

⎠

⎤

⎥⎦

−1

,

(4.3)

and the associated quasi-log-likelihood of Eq. (3.2). Consistency and asymptotic nor-mality of the quasi-maximum likelihood estimators are guaranteed by Theorem 1.

4.4 In-samplemodel selection with QAIC and QBIC

In this first application, we perform in-sample model selection to assess the relevanceof the different possible sets of covariates. For each stock and each trading day, we fixa set of covariates. We use the indices of the tested covariates to name the models: themodel 146 is, thus, the model with covariates (Z1, Z4, Z6). If required, we estimatethe parameters of all the Hawkes covariates on the previous day and then compute theHawkes covariates using these fitted parameters. This procedure ensures that the pre-dictability of the covariates is not violated. We finally fit three ratio models followingthe above procedure : one for the processes (N 0, N 1) (signature of the marker orders),one for the processes (N 0,0, N 0,1) (aggressiveness of the bid market orders) and onefor the processes (N 1,0, N 1,1) (aggressiveness of the ask market orders).

For each trading day, we then select the model minimizing some information cri-terion. For the ratio for the side determination, the criterion is

− 2HT (θMT ) + aT |J|, (4.4)

where |J| is the cardinality of the set of J and aT = 2 for the QAIC criterion, andaT = log(T ) for the QBIC criterion. For the aggressiveness ratios, the criterion is

− 2H(i)T (�i ) + aT |Ji | (i ∈ I). (4.5)

123


Fig. 2 Side of market orders—Frequency of selection of each model by the QAIC and QBIC criteria, foreach stock

We finally compute for each stock the frequencies of selection of different sets ofcovariates (i.e., the number of trading days in which a model is selected by QAIC orQBIC over the total number of trading days in the sample for this stock). Figures 2,3 and 4 plot the results as a model × stock heatmap for each of these three ratios3.For completeness, Tables 4, 5 and 6 in Sect. 8 list for each ratio model (side, bidaggressiveness, ask aggressiveness) the frequency of selection averaged across stocksfor each model and each information criterion.

For side determination, the models 14689, 124689, 134689 and 1234689 are thefour most often chosen models: the selected model is among these four models morethan 80% of the time in average across stocks using QAIC, and close the 90% of thetime using QBIC. As expected, QBIC favors the smallest model 14689. Imbalance,Hawkes covariates for bid and askmarket orders, andHawkes covariates for aggressivebid and ask market orders, thus, appear to be the most informative covariates.

For aggressiveness determination, the model 146 is the most often selected byQBIC. This is in line with intuition: imbalance is known to be a significant proxyfor price change (see, e.g., Lipton et al. 2013) and Hawkes covariates for aggressivebid and aggressive ask are specific to the targeted events. QAIC selection is morewidespread and favors a larger model (as expected), namely 12346. Note also that forseveral stocks, models with “symmetric” sets of covariates can also be chosen: for askaggressiveness, 1679 is often selected, i.e., imbalance and all available ask Hawkes

3 In these figures, we use for QBIC the proxy aT = log(nT ), where nT is the number of events in thesample of length T , but results are actually very similar if we simply set aT = log(T ).

123


Fig. 3 Aggressiveness of bid market orders—Frequency of selection of each model by the QAIC and QBICcriteria, for each stock

Fig. 4 Aggressiveness of ask market orders—Frequency of selection of each model by the QAIC criterionand QBIC criteria, for each stock

123


Fig. 5 Frequency of spread selection among all stocks and trading days in the aggressiveness ratio modelas a function of the mean observed spread in ticks (5%-quantile bins)

covariates; symmetrically, 1458 is selected for ask aggressiveness, i.e., imbalance andall available bid Hawkes covariates.

One may in particular observe that these results confirm the primary role of thespread measured in ticks in the theory of financial microstructure. Stocks for whichthe observed spread ismostly equal to one tick are labeled ’large-tick stocks’, implyingthat market participants are constrained by the price grid when submitting orders to thelimit order book. Other stocks may be labeled ’small-tick stocks’ (Eisler et al. 2012).Using our sample, we compute the mean observed spread in ticks for each stock andeach available trading day, and group these values in bins of equal sizes. Then insideeach bin, we compute the frequency of selection of the covariate Z3 (signed spread) byQBIC for the aggressiveness ratio estimation of Equation (4.3). Bar plot is providedin Fig. 5 (left). We observe an increase of the frequency of the selection of the spreadcovariate when the mean observed spread increases from 1 tick (its minimal possiblevalue) to roughly 2.5 ticks. For larger spread values, frequency remains high then seemsto decrease at high values. This indicates that the significancy of covariates, especiallythe spread, is not the same for large-tick and small-tick stocks, and that even forsmall-tick stocks, dependency is not constant/uniform. This visual observation can, forexample, be complemented by the following statistical test. For all stocks and tradingdays, we compute the empirical cumulative distributions functions of the daily meanspread in ticks (i) when the spread covariate is selected by QBIC in the aggressivenessratios, and (ii) when the spread covariate is not selected. A one-sided Kolmogorov–Smirnov test rejects (with p-value 10−53) the fact that both distributions are identical,and chooses the alternative hypothesis that the spread covariate is more selected forlarger observed spreads. Recall that many microstructure models are developed forlarge-tick stocks, since assuming a constant spread equal to one tick often simplifies theanalysis of the limit order book dynamics. Our observation advocates for the definitionof specific microstructure models for small-tick stocks, taking into account the spreaddynamics.

123


Model selection consistency validates the use of QBIC. See Eguchi and Masuda(2018), or follow Muni Toke and Yoshida (2020) for a direct proof for consistencyincluding other criteria. However, the real performance in prediction of a selectedmodel is more important than themodel selection consistency. It is worth tryingQAIC,or the consistent QAIC.

4.5 Out-of-sample prediction performance

In this section, we use intensity and ratio models to predict the sign and aggressivenessof an incoming market order. For all tested models, the procedure is the following. Ona given trading day, themodel is fitted. Fitted parameters are then used on the followingtrading day (available in the database) to compute the intensities (or ratios for ratiomodels), at all time. The type of an incoming event is then predicted to be the type ofhighest intensity or ratio. The exercise is theoretical in the sense that we assume thatthese computations are instantaneous, so that intensities or ratios are available at alltimes.

Recall the notation N = (Ni,ki )i∈{0,1},ki∈{0,1} for the four-dimensional point pro-cess counting bid aggressive market orders, bid non-aggressive market orders, askaggressive market orders and ask non-aggressive market orders. We use two bench-mark models.

The first benchmark model is the Hawkes model. Here, N is assumed to be a four-dimensional Hawkes process with a single exponential kernel. In vector notation, theintensity is written as

⎛

⎜⎜⎜⎝

λ0,0H (t)

λ0,1H (t)

λ1,0H (t)

λ1,1H (t)

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎝

μ0,0

μ0,1

μ1,0

μ1,1

⎞

⎟⎟⎠+

∫ t

0

⎛

⎜⎝

. . .... . .

.

α(i,ki ),( j,k j )e−β(i,ki ),( j,k j )(t−s)

. .. ...

. . .

⎞

⎟⎠ ·

⎛

⎜⎜⎝

dN 0,0(s)dN 0,1(s)dN 1,0(s)dN 1,1(s)

⎞

⎟⎟⎠ .

Estimation and ratio computation can be found in, e.g., Bowsher (2007); Muni Tokeand Pomponio (2012). This model is labeled ‘Hawkes’.

The second benchmark model is the four-dimensional ratio model without marks(Muni Toke and Yoshida (2020)). In this model, the intensity of the counting process(i, ki ) is

λi,kiR (t) = λ0,R(t) exp

(∑

j∈Jϑi,kij X j (t)

)

,

with some unobserved baseline intensity λ0,R(t). Given the previous observations,we choose the set of covariates (Z1, Z4, Z6, Z8, Z9) for this benchmark. It is naturalto choose these covariates (imbalance, Hawkes for aggressive orders and Hawkesfor all orders) given the results on model selection of Sect. 4.4. Estimation and ratiocomputation are detailed in Muni Toke and Yoshida (2020). This model is labeled’Ratio-14689’.

123


Fig. 6 Out-of-sample prediction performances for the benchmark models and the marked ratio models.Label explanation is in the text

These two benchmarks are used to assess the performances of two marked ratiomodels (or two-step ratio models) described in this paper. The first marked ratio modeluses the covariates (Z4, Z5, Z6, Z7) for both steps. These covariates are based on theHawkes processes of the benchmark Hawkes model. The second marked ratio modeluses the covariates (Z1, Z4, Z6, Z8, Z9) for the first-step ratio (side determination)and (Z1, Z4, Z6) for both second-step ratios (bid and ask aggressiveness). Again,these choices are natural given the results on model selection of Sect. 4.4. Thesemodels are labeled ’MarkedRatio-4567-4567-4567’ and ’MarkedRatio-14689-146-146’, respectively.

Figure 6 plots the results for each stock for the two benchmark models and the twomarked ratio models. For completeness, the partial performances for side determina-tion and aggressiveness determination of the trades are provided on Fig. 7. Finally,Table 2 lists the partial and global prediction performances of these models aver-aged across stocks. The benchmark Hawkes model correctly predicts the sign andaggressiveness of an incoming order with an accuracy in the range [40%, 60%] forall stocks, with a 50% average. The marked ratio model with only Hawkes param-eters (’MarkedRatio-4567-4567-4567’) and no dependency on the state of the limitorder book actually reproduces closely these performances. The non-marked ratiomodel ’Ratio-14689’ improves slightly the global performances of the two previousmodels. When looking at the partial accuracies, we observe that this improvementis mainly due to a better side prediction. Finally, the ’MarkedRatio-14689-146-146’,which appeared to be in average the best model with respect to the QBIC selection,results strongly outperforms all other models. The global accuracy is in the range[60%, 80%] for all stocks, with a 67% average, i.e., we are theoretically able to cor-rectly predict both the sign and aggressiveness of an incoming market order two timesout of three. Finally, we observe by comparing side determination of ’Ratio-14689’and ’MarkedRatio-14689-146-146’ that the decoupling of the side and aggressiveness

123


Fig.

7Out-of-samplepartialpredictio

nperformancesfortheside

predictio

n(left)andaggressiveness

predictio

n(right),forthebenchm

arkmod

elsandthemarkedratio

mod

els.Labelexplanationisin

thetext

123


Table 2 Prediction performances of selected models averaged across stocks.

Accuracy Hawkes Ratio-14689 MarkedRatio 4567-4567-4567 MarkedRatio 14689-146-146

Partial - side 0.781 0.808 0.776 0.877

Partial - agg. 0.634 0.658 0.668 0.774

Global 0.503 0.533 0.516 0.667

Side accuracy gives the fraction of correctly signed trades. Aggressiveness accuracy gives the proportionof trade with a correctly predicted accuracy. Global accuracy gives the fraction of orders with correctlypredicted side and aggressiveness

ratios in the marked ratio model significantly improves the prediction performanceover the one-step four-dimensional case, while using the same covariates.

These results show that the two-step ratio model for marked point processes is asignificant improvement to existing intensity models. As in the standard ratio modelof Muni Toke and Yoshida (2020), this provides an easy way to have both clusteringand state-dependency. However, it is important to note that the two-step ratio stronglyimproves the performance of the standard ratio model in multidimensional setting. Inthis example, flexibility in the choice of covariates allows for precise model selectionfor both sign and aggressiveness.

5 Proof of Theorem 1

The convergence given in Theorem 1 can be obtained by the quasi-likelihood analysis,which we recall in Section 6. We will apply Theorems 3 and 5 in Sect. 6 to the doubleratio model. In the present situation, the scaling factor is bT = T , the joint parameter(θ, ρ) is for θ in Section 6, and the dimension of the full parameter space is p inplace of p of Section 6. Fix a set of values of parameters (α, β1, β2, ρ, ρ1, ρ2) so thatCondition [L1] (Section 6) is met with ρ = 2.

5.1 Score functions and a central limit theorem

The score function for ρi is given by

F (i)T (ρi ) = ∂ρiH

(i)T (ρi ) =

∑

ki∈Ki

∫ T

0∂ρi log qkii (t, ρi )dNi,ki

t .

Then,

F (i)T (ρi ) =

∑

ki∈Ki

∫ T

0

(1{ki }(·) − q�

i (t, ρi ))⊗ Y

i (t)dNi,kit , (5.1)

where q�i (t, ρ

i ) = (qkii (t, ρi ))ki∈Ki,0 , Yi (t) = (Y i

ji(t)) ji∈Ji and 1{ki }(κ) =

(1{κ=ki }

)κ∈Ki,0

. By some calculus with (2.1) and pkii (t, �i ) = qkii (t, ρi ), we see

123


F (i)T := F (i)

T ((ρi )∗) =∑

ki∈Ki

∫ T

0

(1{ki }(·) − q�

i (t, (ρi )∗)

)⊗ Yi (t)d N i,ki

t . (5.2)

We are assuming that the counting processes Ni,ki (i ∈ I; ji ∈ Ki ) have no commonjumps. Then, the pi × pi ′ matrix valued process

〈F (i), F (i ′)〉T = 0 (i, i ′ ∈ I, i �= i ′) (5.3)

and

〈F(i)〉T =∑

ki∈Ki

∫ T

0

{(1{ki }(·) − q�

i (t, (ρi )∗)

)⊗ Yi (t)

}⊗2r i (t, θ∗)Λ(λ0(t),X(t))q

kii (t, (ρi )∗)dt

=∫ T

0V i0(Y

i (t), (ρi )∗) ⊗ (Yi (t))⊗2 Λ(λ0(t),X(t))r i (t, θ∗)dt (i ∈ I).

Therefore, the mixing property [M2] gives the convergenceT−1〈F (i)〉T →p Γ (i)((ρi )∗)

= E

[

V i0(Y

i (0), (ρi )∗) ⊗ (Yi (0))⊗2 Λ(λ0(t),X(0))r i (0, θ∗)

]

(5.4)

as T → ∞, with the aid of [M1].The score function for θ is the p-dimensional process

FT (θ) = ∂θHT (θ) =∑

i∈I

∫ T

0∂θ log r

i (t, θ)dNit

=∑

i∈I

∫ T

0

(1{i}(·) − r �(t, θ)

)⊗ X(t)dNit , (5.5)

where r �(t, θ) = (r i (t, θ))i∈I0 . Evaluated at θ∗,

FT = FT (θ∗) =∑

i∈I

∫ T

0

(1{i}(·) − r �(t, θ∗)

)⊗ X(t)d N it

=∑

i∈I

∑

ki∈Ki

∫ T

0

(1{i}(·) − r �(t, θ∗)

)⊗ X(t)d N i,kit . (5.6)

Then, the p × p matrix valued process 〈F〉 has the expression

〈F〉T =∑

i∈I

∑

ki∈Ki

∫ T

0

(1{i}(·) − r�(t, θ∗)

)⊗2 ⊗ X(t)⊗2r i (t, θ∗)Λ(λ0(t),X(t))qkii (t, (ρi )

∗)dt

=∑

i∈I

∫ T

0

(1{i}(·) − r�(t, θ∗)

)⊗2 ⊗ X(t)⊗2r i (t, θ∗)Λ(λ0(t),X(t))dt

=∫ T

0V 0(X(t)) ⊗ X(t)⊗2Λ(λ0(t),X(t))dt .

123


Then, the mixing property [M2] provides the convergence

T−1〈F〉T →p Γ (θ∗) = E

[(

V 0(X(0)) ⊗ X(0)⊗2)

Λ(λ0(0),X(0))

]

(5.7)

as T → ∞.For i ∈ I,

〈F, F(i)〉T =∑

ki∈Ki

∫ T

0

(1{i}(·) − r�(t, θ∗)

)⊗ (1{ki }(·) − q�

i (t, (ρi )∗)

)⊗ X(t) ⊗ Yi (t)

×r i (t, θ∗)Λ(λ0(t),X(t))qkii (t, (ρi )

∗)dt

= 0 (5.8)

since

∑

ki∈Ki

(1{ki }(·) − q�

i (t, (ρi )∗)

)qkii (t, (ρi )

∗) = 0.

The full information matrix is the p × p block diagonal matrix

Γ = Γ (θ∗, ρ∗) = diag[Γ (θ∗), Γ 0((ρ0)∗), Γ 1((ρ1)∗), ..., Γ i ((ρ i )∗)

].

Let ΔT = T−1/2(FT , (F (i)

T )i∈I). Now, by the martingale central limit theorem, it is

easy to obtain the convergence

ΔT →d Γ 1/2ζ (T → ∞),

where ζ is a p-dimensional standard Gaussian random vector. The joint convergence(ΔT , Γ ) →d (Γ 1/2ζ, Γ ) is obvious since Γ is deterministic.

5.2 Condition [L4]

According to (6.2), we define the random field YT : Ω × Θ × R → R by

YT (θ, ρ) = T−1(HT (θ, ρ) − HT (θ∗, ρ∗)

)

for HT (θ, ρ) given in (3.3). From the expression (3.4) of HT (θ, ρ), we have

T−1HT (θ, ρ) = T−1

∑

i∈I

∑

ki∈Ki

∫ T

0log


)dNi,ki

t

= T−1∑

i∈I

∑

ki∈Ki

∫ T

0log


)d N i,ki

t

123


+T−1∑

i∈I

∑

ki∈Ki

∫ T

0

{log


}

×λ0(t) exp

(∑

j∈J(ϑ∗)ij X j (t)

)

pkii (t, (�∗)i,ki )dt .

By definition,

∣∣∂�

(θ,ρ) log(r i (t, θ)qkii (t, ρi )

)∣∣ ≤ C

(

1 +∑

j∈J|X j (t)| +

∑

i∈I

∑

ji∈Ji|Y ki

ji(t)|

)

(� = 0, 1),

where C is a constant depending on the diameters of Θ and R. Therefore, underCondition [M1],

E

[∣∣∣∣∂

�(θ,ρ)T

−1/2∫ T

0log


)dN i,ki

t

∣∣∣∣

2k]

<∼ E

[(

T−1∫ T

0

∣∣∂�


)∣∣2dNi,ki

t

)2(k−1)]

<∼ E

[

T−1∫ T

0

∣∣∂�


)∣∣2

k

λi,ki (t, (ϑ i )∗, (�i,ki )∗)dt]

+T−2k−2E

[(

T−1/2∫ T

0

∣∣∂�


)∣∣2d N i,ki

t

)2(k−1)]

= O(1) + T−2k−2E

[(

T−1/2∫ T

0

∣∣∂�


)∣∣2d N i,ki

t

)2(k−1)]

for k ∈ N, where the constant appearing at each <∼ depends only on p, k and theconstant of the Burkholder–Davis–Gundy inequality. By induction, we obtain

sup(θ,ρ)∈Θ×R

supT≥1

∥∥∥∥∂

�(θ,ρ)T

−1/2∫ T

0log


)d N i,ki

t

∥∥∥∥p

< ∞ (5.9)

for every p > 1 and � ∈ {0, 1}. Then, Sobolev’s inequality gives

supT≥1

∥∥∥∥ sup

(θ,ρ)∈Θ×R

∣∣∣∣T

−1/2∫ T

0log


)d N i,ki

t

∣∣∣∣

∥∥∥∥p

< ∞ (5.10)

for every p > 1.Let

123


Φ(t, θ, ρ) =∑

i∈I

∑

ki∈Ki

{

r i (t, θ∗)pkii (t, (�i )∗) logr i (t, θ)qkii (t, ρi )

r i (t, θ∗)qkii (t, (ρi )∗)

}

×λ0(t)∑

i ′∈Iexp

(∑

j∈J(ϑ∗)i ′j X j (t)

)

.

Then, Conditions [M1] and [M2] imply a Rosenthal type inequality under the mixingcondition (cf. Rio Rio (2017))

sup(θ,ρ)∈Θ×R

supT≥1

∥∥∥∥T

−1/2∫ T

0∂�(θ,ρ)

(Φ(t, θ, ρ) − E[Φ(t, θ, ρ)])dt

∥∥∥∥p

< ∞

for every p > 1 and � ∈ {0, 1}. This entails

supT≥1

∥∥∥∥T

1/2 sup(θ,ρ)∈Θ×R

∣∣∣∣T

−1∫ T

0

(Φ(t, θ, ρ) − E[Φ(t, θ, ρ)])dt

∣∣∣∣

∥∥∥∥p

< ∞ (5.11)

for every p > 1.Combining (5.11) with (5.10), we obtain

supT≥1

E

[(

T 1/2 sup(θ,ρ)∈Θ×R

∣∣YT (θ, ρ) − Y(θ, ρ)

∣∣)p]

< ∞ (5.12)

for every p > 1, if we set

Y(θ, ρ) = E[Φ(0, θ, ρ)]

= E

[∑

i∈I

∑

ki∈Ki

{

r i (t, θ∗)pkii (0, (�i )∗) logr i (0, θ)qkii (0, ρi )

r i (0, θ∗)qkii (0, (ρi )∗)

}

×λ0(0)∑

i ′∈Iexp

(∑

j∈J(ϑ∗)i ′j X j (0)

)]

.

This verifies Condition [L4](ii).As (6.1), we define ΓT (θ, ρ) by

ΓT (θ, ρ) = −T−1∂2(θ,ρ)HT (θ, ρ).

From (5.1),

∂2ρiH

(i)T (ρi ) = −

∑

ki∈Ki

∫ T

0∂ρi q�

i (t, ρi ) ⊗ Y

i (t)dNi,kit .

123


More precisely,

∂ρi,kiji

∂

ρi,k′ij ′i

H(i)T (ρi ) = −

∑

k′′i ∈Ki

∫ T

0

{

1{ki=k′i }qkii (t, ρi ) − q

kii (t, ρi )q

k′ii (t, ρi )

}

Yiji

(t)Yij ′i

(t)dNi,k′′it

= −∑

k′′i ∈Ki

∫ T

0

{


kii (t, ρi )q

k′ii (t, ρi )

}

Yiji

(t)Yij ′i

(t)dNi,k′′it

−∫ T

0

{


kii (t, ρi )q

k′ii (t, ρi )

}

Yiji

(t)Yij ′i

(t)

×ri (t, θ∗)Λ(λ0(t),X(t))dt

= −∑

k′′i ∈Ki

∫ T

0V i0(Y

i (t), ρi )ki ,k′iYiji

(t)Yij ′i

(t)d Ni,k′′it

−∫ T

0V i0(Y

i (t), ρi )ki ,k′iYiji

(t)Yij ′i

(t)Λ(λ0(t),X(t))ri (t, θ∗)dt

for ki , k′i ∈ Ki,0, ji , j ′i ∈ Ji and i ∈ I, where (3.12) was used. Similarly, from (5.5),

∂2θHT (θ) = −∑

i∈I

∫ T

0∂θr

�(t, θ) ⊗ X(t)dNit ,

equivalently,

∂θ ij∂θ i

′j ′HT (θ) = −

∑

i ′′∈I

∫ T

0V 0(X(t), θ)i,i ′ X j (t)X j ′(t)d N

i ′′t

−∫ T

0V 0(X(t), θ)i,i ′ X j (t)X j ′(t)Λ(λ0(t),X(t))dt

for i, i ′ ∈ I0 and j, j ′ ∈ J. Obviously,

∂θ∂ρiH(i)T (ρi ) = 0 and ∂

ρi ′ ∂ρiH(i)T (ρi ) = 0 (i ′, i ∈ I : i ′ �= i).

In a way similar to the derivation of (5.12), as a matter of fact it is easier, we can show

supT≥1

E[(T 1/2|ΓT (θ∗, ρ∗) − Γ |)p] < ∞

for every p > 1 under Conditions [M1] and [M2]. Therefore, Condition [L4](iv) forβ1 = 1/2 was verified. It is also possible to show [L4](iii) in a similar fashion usingthe mixing property and Sobolev’s inequality. Condition [L4](i) is already checked in(5.9). Thus, Condition [L4] has been verified.

123


5.3 Conditions [L2] and [L3]

We see

∂2(θ,ρ)Y(θ, ρ) = Γ (θ, ρ),

and by [M3], we concludeY(θ, ρ) is strictly convex function onΘ×R = Θ×Πi∈IRi .For some neighborhood U of (θ∗, ρ∗) and some positive number χ1,

Y(θ, ρ) ≤ −χ1|(θ, ρ) − (θ∗, ρ∗)|2 ((θ, ρ) ∈ U

)

by the non-degeneracy of Γ (θ∗, ρ∗). Moreover, sup(θ,ρ)∈(Θ×R)\U Y(θ, ρ) < 0. Infact, if there was a point (θ+, ρ+) /∈ U such that Y(θ+, ρ+) = 0, then at a point onthe segment connecting (θ∗, ρ∗) and (θ+, ρ+), Γ (θ, ρ) would degenerate, and thiscontradicts [M3]. As a consequence, Condition [L2] is verified for ρ = 2 and some(deterministic) positive number χ0 since the parameter space is bounded. Condition[L3] is now obvious.

5.4 Proof of Theorem 1

Wehave verifiedConditions [L1]-[L4] in the present situation. Theorem1now followsfrom Theorems 3 and 5. ��

6 Quasi-likelihood analysis

This section recalls the quasi-likelihood analysis. Let Θ be a bounded open set in Rp.Given a probability space (Ω,F , P), suppose that HT : Ω × Θ → R is of class C3,that is, the mapping Θ � θ �→ HT (ω, θ) ∈ R

p is continuously extended to Θ and ofclass C3 for every ω ∈ Ω , and the mapping Ω � ω �→ HT (ω, θ) ∈ R

p is measurablefor every θ ∈ Θ . Let Γ be a p × p random matrix.

Let θ∗ ∈ Θ . For a sequence aT ∈ GL(p) satisfying limT→∞ |aT | = 0, let

ΔT = ∂θHT (θ∗)aT and ΓT (θ) = −a�T ∂2θHT (θ)aT , (6.1)

where � denotes the matrix transpose. We consider a random field

YT (θ) = b−1T

(HT (θ) − HT (θ∗)

), (6.2)

which will be assumed to converge to a random field Y : Ω × Θ → R. Only forsimplifying presentation, we will assume that aT = b−1/2

T Ip for diverging sequence(bT )T>0 of positive numbers, where Ip is the identity matrix. In what follows, we fixa positive number L .

Wewill give a simplified exposition of Yoshida (2011) on the polynomial type largedeviation inequality. Let α, β1, β2, ρ, ρ1 and ρ2 be numbers.

123


[L1] The numbers α, β1, β2, ρ, ρ1 and ρ2 satisfy the following inequalities:

0 < α < 1, 0 < β1 < 1/2, 0 < ρ1 < min{1, α(1 − α)−1, 2β1(1 − α)−1},αρ < ρ2, β2 ≥ 0 and 1 − 2β2 − ρ2 > 0.

Let β = α(1 − α)−1.[L2] There is a positive random variable χ0 such that

Y(θ) = Y(θ) − Y(θ∗) ≤ −χ0|θ − θ∗|ρ

for all θ ∈ Θ .[L3] There exists a CL such that

P[χ0 ≤ r−(ρ2−αρ)

] ≤ CL

r L(r > 0)

and

P[λmin(Γ ) < 4r−ρ1

] ≤ CL

r L(r > 0).

[L4] (i) For M1 = L(1 − ρ1)−1, sup

T>0E[|ΔT |M1

]< ∞.

(ii) For M2 = L(1 − 2β2 − ρ2)−1,

supT>0

E

[(

suph:|h|≥b−α/2

T

b12−β2T

∣∣YT (θ∗ + h) − Y(θ∗ + h)

∣∣)M2

]

< ∞.

(iii) For M3 = L(β − ρ1)−1,

supT>0

E

[(

b−1T sup

θ∈Θ

∣∣∂3θHT (θ)

∣∣)M3

]

< ∞.

(iv) For M4 = L(2β1(1 − α)−1 − ρ1

)−1,

supT>0

E

[(

bβ1T

∣∣ΓT (θ∗) − Γ

∣∣)M4

]

< ∞.

Let UT = {u ∈ Rp; θ∗ + aT u ∈ Θ} and VT (r) = {u ∈ UT ; |u| ≥ r} for r > 0.

Theorem 2 (Yoshida (2011)) Suppose that Conditions [L1]-[L4] are satisfied. Then,there exists a constant C such that

P

[

supu∈VT (r)

ZT (u) ≥ exp(− 2−1r2−(ρ1∨ρ2)

)]

≤ C

r L

123


for all T > 0 and r > 0. Here, the supremum of the empty set should read −∞ byconvention.

We comment some points. Parameters satisfying [L1] exist. Nondegeneracy con-ditions in [L3] are obvious in ergodic cases. In this paper, we will apply Theorem 2under ergodicity of the stochastic system. Theorem 2 asserts a polynomial type largedeviation inequality can be obtained once the boundedness of moments of some ran-dom variables is verified. Condition [L4] is easy to obtain because each variable isusually a simple additive functional. The polynomial type large deviation inequalityin Theorem 2 enables us to easily apply the scheme by Ibragimov and Has’minskiı(1981) and Kutoyants (1984, 2012) to various dependence structures.

Let u ∈ Rp. Define rT (u) (u ∈ UT ) by

ZT (u) = exp

(

ΔT [u] − 1

2Γ [u⊗2] + rT (u)

)

(u ∈ UT ). (6.3)

It is said that ZT is locally asymptotically quadratic (LAQ) at θ∗ if rT (u) →p 0 asT → ∞ for every u ∈ R

p, and hence logZT (u) is asymptotically approximated by arandom quadratic function of u.

We will confine our attention to a very standard case where ZT is locally asymp-totically mixed normal, though the general theory of the quasi-likelihood analysis isframed more generally.

Any measurable mapping θMT : Ω → Θ is called a quasi-maximum likelihood

estimator (QMLE) for HT if

HT (θMT ) = max

θ∈Θ

HT (θ).

WhenHT is continuous on the compact Θ , such a measurable function always exists,which is ensured by the measurable selection theorem. Let uM

T = a−1T (θM

T − θ∗) forthe QMLE θM

T .

Theorem 3 Let L > p > 0. Suppose that Conditions [L1]-[L4] are satisfied and that(ΔT , Γ ) →d (Γ 1/2ζ, Γ ) as T → ∞, where ζ is a p-dimensional standard Gaussianrandom vector independent of Γ . Then,

E[f (uM

T )] → E

[f (u)

](T → ∞)

for u = Γ −1/2ζ and for any f ∈ C(Rp) satisfying lim|u|→∞ |u|−p| f (u)| < ∞.

Proof Wewill sketch the proof to convey the concepts of the quasi-likelihood analysisto the reader. See Yoshida (2011) for details. The space C(Rp) is the linear spaceof all continuous functions f : Rp → R satisfying lim|u|→∞ f (u) = 0. The spaceC(Rp)becomes a separableBanach space equippedwith the supremumnorm‖ f ‖∞ =supu∈Rp | f (u)|. Moreover, C(Rp) is regarded as a measurable space with the Borel

123


σ -field. Let

Z(u) = exp

(

Γ 1/2ζ [u] − 1

2Γ [u⊗2]

)

(6.4)

for u ∈ Rp.

The term rT (u) admits the expression

rT (u) =∫ 1

0(1 − s)

{Γ [u⊗2] − ΓT (θ∗ + saT u)[u⊗2]}ds (6.5)

for u such that |u| ≤ b(1−α)/2T and T such that B(θ∗, b−α/2

T ) ⊂ Θ . In this situation, wecan apply Taylor’s formula even though thewholeΘ is not convex. Condition [L4] (iii)and the convergence of ΔT ensures tightness of the random fields

{ZT |B(0,R)

}T>T0

for every R > 0, where B(0, R) = {u ∈ Rp} and T0 is a sufficiently large number

depending on R. Combining this property with the polynomial type large deviationinequality given by Theorem 2, we obtain the convergence ZT → Z in C(Rp) forthe random field ZT extended as an element of C(Rp) so that supRp\UT

ZT (u) ≤supu∈∂UT

ZT (u). Consequently, uT → u = argmaxu∈RpZ(u). It is known that ameasurable version of extension of ZT exists.

A polynomial type large deviation, evenweaker than the one in Theorem2, serves toshow Lq -boundedness of {|uT |q} for L > q > p. Then, the family {uT } is uniformlyintegrable, and hence we obtain the convergence of E[ f (uT )]. ��Remark 2 In Theorem 3, if ΔT →d Γ 1/2ζ F-stably, then (ΔT , Γ ) →d (Γ 1/2ζ, Γ )

and uMT → u F-stably.

An advantage of the quasi-likelihood analysis is that the asymptotic behavior ofthe quasi-Bayesian estimator can be obtained as well as that of the quasi-maximumlikelihood estimator and its moments convergence. The mapping

θ BT =

[ ∫

Θ

exp(HT (θ)

)�(θ)dθ

]−1 ∫

Θ

θ exp(HT (θ)

)�(θ)dθ

is called a quasi-Bayesian estimator (QBE) with respect to the prior density � . TheQBE θ B

T takes values in the convex-hull of Θ . We will assume � is continuous and0 < infθ∈Θ �(θ) ≤ supθ∈Θ �(θ) < ∞. We will give a concise exposition in thefollowing among many possible ways. The reader is referred to Yoshida (2011) forfurther information.Recall thatp is the dimensionofΘ , and B(R)denotes the openballof radius R centered at the origin. C(B(R)) is the space of all continuous functions onB(R), and it is equippedwith the supremumnorm.RecallVT (r) = {u ∈ UT ; |u| ≥ r}.As before, u = Γ −1/2ζ with a p-dimensional standard Gaussian random vector ζ

independent of Γ . Write u BT = a−1

T (θ BT − θ∗).

Theorem 4 Let p ≥ 1, L > p+1, D > p+ p. Suppose that (ΔT , Γ ) →d (Γ 1/2ζ, Γ )

as T → ∞, where ζ is ap-dimensional standardGaussian randomvector independentof Γ . Moreover, suppose the following conditions are satisfied.

123


(i) For every R > 0,

ZT |B(R) →dZ|B(R) in C(B(R)) (6.6)

as T → ∞, where Z is given in (6.4).(ii) There exist positive constants T0, C1 and C2 such that

P

[

supVT (r)

ZT ≥ C1r−D

]

≤ C2r−L (6.7)

for all T ≥ T0 and r > 0.(iii) For some T0 > 0,

supT≥T0

E

[(∫

UT

ZT (u)du

)−1]

< ∞. (6.8)

Then,

E[f (u B

T )] → E

[f (u)

](6.9)

as T → ∞ for any continuous function f : Rp → R satisfying supu∈Rp

{(1 +

|u|)−p| f (u)|} < ∞.

Proof We will give a brief summary of the proof; see Yoshida (2011) for details. Thevariable u B

T has the expression

u BT =

[ ∫

UT

ZT (u)�(θ∗ + aT u)du

]−1 ∫

UT

uZT (u)�(θ∗ + aT u)du.

By (6.7) and the properties of � , we can approximate u BT by

uT =[ ∫

B(R)

ZT (u)du

]−1 ∫

B(R)

uZT (u)du

for paying small error when R is large. By (6.6),

uT →d[ ∫

B(R)

Z(u)du

]−1 ∫

B(R)

uZ(u)du =: u(R).

The random fieldZ inherits a tail estimate from (6.7), and hence u(R) is approximatedby

[ ∫

RpZ(u)du

]−1 ∫

RpuZ(u)du = Γ −1/2ζ = u.

123


Combining these estimates, we can conclude u BT →d u as T → ∞. Convergence of

the expectation is a consequence of uniform integrability of |u BT |p ensured by (6.7). ��

Remark 3 (a) It is possible to relax the conditions of Theorem 4 to only ensure theconvergence u B

T → u. (b) In Theorem 4, if ΔT →d Γ 1/2ζ F-stably, then u BT → u

F-stably. (c) Usually, the condition (iii) of Theorem 4 is easily verified; See Lemma2 of Yoshida (2011). (d) We refer the reader to Yoshida (2021) for a simplified quasi-likelihood analysis for a locally asymptotically quadratic random field.

The following result follows from Theorem 4.

Theorem 5 Let p > p and

L > max

{

p + 1, p(β − ρ1), p(2β1(1 − α)−1 − ρ1)

}

.

Suppose that Conditions [L1]-[L4] are satisfied and that E[|Γ |p] < ∞. (ΔT , Γ ) →d

(Γ 1/2ζ, Γ ) as T → ∞, where ζ is a p-dimensional standard Gaussian random vectorindependent of Γ . Then,

E[f (u B

T )] → E

[f (u)

](T → ∞)

for u = Γ −1/2ζ and for any f ∈ C(Rp) satisfying lim|u|→∞ |u|−p| f (u)| < ∞.

Proof The convergence (6.6) holds, as shown in the proof of Theorem 3. The polyno-mial type large deviation inequality (6.7) is a consequence of Theorem 2; the numberD is arbitrary. Fix δ > 0. Then, there exists T0 > 0 such that B(δ) ⊂ Θ . In particular,rT (u) admits the representation (6.5) for all u ∈ B(δ). Since M3 = L(β −ρ1)

−1 > p,M4 = L(2β1(1−α)−1 −ρ1)

−1 > p and p > p, we have p′ := min{M3, M4, p} > pand

E[|rT (u)|p′ ] ≤ C0|u|p′(u ∈ B(δ))

for some constant C0. Then Lemma 2 of Yoshida (2011) gives the estimate

E

[(∫

B(δ)

ZT (u)du

)−1]

≤ C1

by a constant C1 depending on (p′,p, δ,C0) and the supremums appearing in[L4](i),(iii),(iv), but C1 is independent of T ≥ T0. Therefore (6.8) holds true. Thus,we can apply Theorem 4 to conclude the proof. ��

7 List of stocks

Table 3 lists all the stocks investigated in the paper. For each stock, the total numberof days available in the sample is given. Note that for lack of usage time allotment

123


Table 3 List of stocks investigated in this paper. Sample consists of the whole year 2015, representingroughly 230 trading days for all stocks except LAGA.PA and PEUP.PA which are missing roughly 70trading days

RIC Company Sector Number oftrading daysin sample

Number oftrading daysused in QAIC

AIRP.PA Air Liquide Healthcare/Energy 238 238

BNPP.PA BNP Paribas Banking 224 62

EDF.PA Electricite deFrance

Energy 236 236

LAGA.PA Lagardère Media 142 142

CARR.PA Carrefour Retail 229 229

BOUY.PA Bouygues Construction/Telecom 228 228

ALSO.PA Alstom Transport 229 229

ACCP.PA Accor Hotels 227 227

ALUA.PA Alcatel Networks /Telecom

234 234

AXAF.PA Axa Insurance 236 131

CAGR.PA Crédit Agri-cole

Banking 235 235

CAPP.PA Cap Gemini TechnologyConsulting

232 232

DANO.PA Danone Food 229 229

ESSI.PA Essilor Optics 228 228

LOIM.PA Klepierre Finance 221 221

LVMH.PA Louis Vuit-ton MoëtHennessy

Luxury 233 198

MICP.PA Michelin Tires 229 229

OREP.PA L’Oréal Cosmetics 233 233

PERP.PA PernodRicard

Spirits 224 224

PEUP.PA Peugeot Automotive 151 151

PRTP.PA Kering Luxury 227 227

PUBP.PA Publicis Communication 223 223

RENA.PA Renault Automotive 228 172

SAF.PA Safran Aerospace /Defense

232 232

TECF.PA Technip Energy 225 225

TOTF.PA Total Energy 232 75

VIE.PA Veolia Energy / Envi-ronment

234 234

VIV.PA Vivendi Media 234 234

VLLP.PA Vallourec Materials 228 228

VLOF.PA Valeo Automotive 221 212

123


Table 3 continued

RIC Company Sector Number oftrading daysin sample

Number oftrading daysused in QAIC

SASY.PA Sanofi Healthcare 229 97

SCHN.PA SchneiderElectric

Energy 224 164

SGEF.PA Vinci Construction 229 229

SGOB.PA Saint Gobain Materials 234 180

SOGN.PA SociétéGénérale

Banking 229 103

STM.PA ST Micro-electronics

Semiconductor 227 227

on the computational resources used for this paper, some trading days for few veryliquid stocks were not used for some of the marked ratio models tested in Sect. 4.4.In this case, only the trading days where all models have been computed have beenused. This is the last column of the table.

8 QAIC and QBIC selection: detailed results

See Tables 4, 5, 6.

Table 4 Sidedetermination—Frequency ofQAIC and QBIC selection (andtheir difference) of each testedmodel, averaged across stocks

QAIC QBIC diff

46 0.000 0.000 0.000

146 0.000 0.000 0.000

246 0.000 0.000 0.000

346 0.000 0.000 0.000

1246 0.000 0.000 0.000

1346 0.000 0.000 0.000

2346 0.000 0.000 0.000

12346 0.000 0.000 0.000

89 0.000 0.000 0.000

189 0.000 0.003 0.003

289 0.000 0.000 0.000

389 0.000 0.000 0.000

1289 0.000 0.002 0.002

1389 0.001 0.002 0.001

2389 0.000 0.000 0.000

123


Table 4 continued QAIC QBIC diff

12389 0.000 0.000 −0.000

4567 0.000 0.000 0.000

14567 0.001 0.002 0.001

24567 0.000 0.000 0.000

34567 0.000 0.000 0.000

124567 0.004 0.005 0.001

134567 0.003 0.004 0.001

234567 0.000 0.000 0.000

1234567 0.005 0.004 −0.001

4689 0.000 0.000 0.000

14689 0.181 0.289 0.108

24689 0.000 0.000 0.000

34689 0.000 0.000 0.000

124689 0.205 0.213 0.007

134689 0.184 0.194 0.010

234689 0.000 0.000 0.000

1234689 0.249 0.195 −0.054

456789 0.000 0.000 0.000

1456789 0.033 0.019 −0.015

2456789 0.000 0.000 0.000

3456789 0.000 0.000 0.000

12456789 0.038 0.022 −0.016

13456789 0.036 0.016 −0.019

23456789 0.000 0.000 −0.000

123456789 0.058 0.029 −0.029

1679 0.000 0.000 0.000

12679 0.000 0.000 0.000

123679 0.000 0.000 0.000

1458 0.000 0.000 0.000

12458 0.000 0.000 0.000

123458 0.000 0.000 0.000

Table 5 Bid aggressiveness determination—Frequency of QAIC and QBIC selection (and their difference)of each tested model, averaged across stocks.

QAIC QBIC diff

46 0.000 0.000 0.000

146 0.033 0.265 0.232

246 0.000 0.000 0.000

346 0.000 0.000 0.000

123


Table 5 continued

QAIC QBIC diff

1246 0.009 0.006 −0.004

1346 0.062 0.085 0.023

2346 0.000 0.000 0.000

12346 0.154 0.106 −0.048

89 0.000 0.000 0.000

189 0.020 0.141 0.121

289 0.000 0.000 0.000

389 0.000 0.000 0.000

1289 0.005 0.002 −0.003

1389 0.104 0.132 0.029

2389 0.000 0.000 0.000

12389 0.095 0.068 −0.027

4567 0.000 0.000 0.000

14567 0.027 0.009 −0.018

24567 0.000 0.000 0.000

34567 0.000 0.000 0.000

124567 0.004 0.000 −0.004

134567 0.064 0.014 −0.050

234567 0.000 0.000 0.000

1234567 0.050 0.010 −0.041

4689 0.000 0.000 0.000

14689 0.028 0.011 −0.017

24689 0.000 0.000 0.000

34689 0.000 0.000 0.000

124689 0.004 0.000 −0.004

134689 0.097 0.025 −0.071

234689 0.000 0.000 0.000

1234689 0.050 0.009 −0.041

456789 0.000 0.000 0.000

1456789 0.005 0.000 −0.005

2456789 0.000 0.000 0.000

3456789 0.000 0.000 0.000

12456789 0.001 0.000 −0.001

13456789 0.021 0.002 −0.019

23456789 0.000 0.000 0.000

123456789 0.008 0.001 −0.007

1679 0.006 0.005 −0.001

12679 0.002 0.000 −0.001

123679 0.016 0.003 −0.013

123


Table 5 continued

QAIC QBIC diff

1458 0.039 0.075 0.036

12458 0.010 0.002 −0.008

123458 0.087 0.030 −0.057

Table 6 Ask aggressivenessdetermination—Frequency ofQAIC and QBIC selection (andtheir difference) of each testedmodel, averaged across stocks

QAIC QBIC diff

46 0.000 0.000 0.000

146 0.048 0.312 0.264

246 0.000 0.000 0.000

346 0.000 0.000 0.000

1246 0.015 0.010 −0.005

1346 0.063 0.062 −0.001

2346 0.000 0.000 0.000

12346 0.129 0.086 −0.043

89 0.000 0.000 0.000

189 0.035 0.178 0.144

289 0.000 0.000 0.000

389 0.000 0.000 0.000

1289 0.010 0.005 −0.005

1389 0.105 0.119 0.013

2389 0.000 0.000 0.000

12389 0.075 0.047 −0.028

4567 0.000 0.000 0.000

14567 0.031 0.011 −0.020

24567 0.000 0.000 0.000

34567 0.000 0.000 0.000

124567 0.006 0.000 −0.006

134567 0.057 0.011 −0.047

234567 0.000 0.000 0.000

1234567 0.039 0.006 −0.033

4689 0.000 0.000 0.000

14689 0.045 0.015 −0.030

24689 0.000 0.000 0.000

34689 0.000 0.000 0.000

124689 0.008 0.001 −0.008

123


Table 6 continued QAIC QBIC diff

134689 0.079 0.014 −0.064

234689 0.000 0.000 0.000

1234689 0.040 0.006 −0.034

456789 0.000 0.000 0.000

1456789 0.014 0.000 −0.014

2456789 0.000 0.000 0.000

3456789 0.000 0.000 0.000

12456789 0.003 0.000 −0.003

13456789 0.024 0.002 −0.023

23456789 0.000 0.000 0.000

123456789 0.010 0.000 −0.010

1679 0.064 0.085 0.021

12679 0.014 0.004 −0.010

123679 0.059 0.016 −0.043

1458 0.009 0.007 −0.003

12458 0.003 0.000 −0.003

123458 0.013 0.003 −0.010

Acknowledgements The authors thank the reviewers for their careful reading and valuable comments tothe paper.

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