SDS-5
The Simultaneous Multivariate Hawkes-type Point
Processes and their application to Financial Markets
Naoto Kunitomo, Daisuke Kurisu
and Naoki Awaya

November 2017

Statistics & Data Science Series back numbers:
http://www.mims.meiji.ac.jp/publications/datascience.html

The Simultaneous Multivariate
Hawkes-type Point Processes and theirapplication to Financial Markets ∗Naoto Kunitomo†
Daisuke Kurisu ‡andNaoki Awaya §October 16, 2017

Abstract
In economic and financial time series we sometimes observe sudden and large jumps.
Although these events are relatively rare, they would have significant influence not
only on a financial market but also on several diﬀerent markets and macro economies.
By using the simultaneous Hawkes-type multivariate point process (SHPP) models, it is possible to analyze the causal eﬀects of large events in the sense of theGranger-non-causality (GNC) from one market to other markets and the instantaneous Granger-non-causality (IGNC). We investigate the financial market of Tokyoand other major markets, and apply the Granger-non-causality tests to investigate
the relationships of large events among several markets. We have found several
important empirical findings among financial markets and macro economies.
Key Words
Simultaneous Hawkes-type marked point process (SHPP), Granger-non-causality,
Instantaneous non-causality, Non-causality tests, International Financial markets∗

We thank Yusuke Amano for the computational assistance and also Takaki Hayashi for usefulcomments. The research was supported by Grant-in-Aid for Scientific Research (JP25245033) fromthe JSPS and D. Kurisu has been supported by Grant-in-Aid as JSPS Research Fellow (16J06454).
†
Meiji University, School of Political Sicence and Economics‡
Graduate student, Graduate School of Economics, University of Tokyo§
Graduate student, Graduate School of Economics, University of Tokyo1

1

Introduction

In economic and financial time series, we sometimes observe sudden
and large jumps. Although they are relatively rare events, they often
have significant influence not only on a single financial market but
also on several diﬀerent markets and macro-economies. There havebeen several recent events occurred in European and Asian countries
including the major financial crisis of 2007-2008 (often called Lehman
Shock in Japan).
The standard econometric method for investigating economic andfinancial time series has been the statistical analysis of discrete timeseries analysis in econometrics. In the statistical time series analysis,
we often assume that the observed time series data are equally spaced
realizations of stochastic process and the state space is Rp in themultivariate cases. Many statistical procedures of discrete time series
analysis have been developed and applied to economic and financial
time series in the last several decades. When we do not observe events
frequently, however, the traditional use of discrete time series modelingwith continuous state space may have some limitations. For instance,
it may be diﬃcult to distinguish the major large contagious eﬀects
from small contagious events among diﬀerent financial markets acrossinternational borders.
In this paper we will propose to use an alternative way of investigating economic and financial events with time series data in macroeconomies, that is, the statistical analysis of the marked point process2approach to investigate multivariate time series events. Although it
has not been a standard approach in time series econometrics, there
have been statistical applications in statistical seismology (see Ogata(1978, 2015) and its related literature, for instance). We will showthat this approach would be an alternative useful way to investigate
multivariate economic and financial markets and shed some new lighton some aspects sometimes ignored. In particular, we shall propose to
use the simultaneous Hawkes-type multivariate point process models
and their applications in this study. It seems that they are not standard statistical models in the past econometric analyses, but there aresome reasons that they are useful in economic and financial time series
analysis. By using the standard multivariate time series analysis. it
may not be easy to distinguish the eﬀects of large jumps and small
jumps in data from one series to another one, for instance. Also it isdiﬃcult to distinguish the eﬀects of past events or contemporaneous
events in time and space. We will show in this paper that by using thesimultaneous multivariate Hawkes-type point process (SHPP) model,
which is a new multivariate point process, it is possible to investigate the causal eﬀects of sudden and large events of their magnitudein the sense of the Granger-non-causality (GNC) and the instantaneous Granger-non-causality (IGNC) through the stochastic intensity
modeling. In the econometric time series analysis, the concept ofGranger-Causality has been one of important tools to investigate therelationships among multivariate time series variables since Granger3

(1969). In econometric literature, Florens and Fougere (1996) have investigated several Granger-causality concepts in the framework of continuous time stochastic processes, but their formulation of the problem
was incomplete because they had excluded the possibility of co-jumps
in their formulation, which means that the simultaneous jumps inmultivariate times series excluded from the beginning. The problem
of co-jumps is important because we often use economic time series
data in discrete time (with the periods of every month, week, day,hour and/or minute) while the continuous stochastic process formulation has been not unusual recently in financial econometrics. We
need to unify the discrete time series analysis and the continuous
stochastic processes coherently. In this paper, we shall investigate
the possible use of co-jumps in a systematic way and will develop the
new tests of the Granger-non-causality and the instantaneous Grangernon-causality, which may give some new light on the econometric timeseries modeling.
There have been a number of recent studies in financial econometrics which have utilized the point processes and the conditional
intensity modeling. See Ait-Sahalia and Jacod (2014), Ait-Sahalia etal. (2015), Embrechts et al. (2011), Grothe et al. (2014) and others.
As Bacry et al. (2015) have discussed as a survey on these and otherworks, they are mostly on the studies of micro-market structures offinancial markets. Our approach developed in this paper is related
to these works, but the main purpose is quite diﬀerent from them4

because we are developing a new point process approach assess the relationships among diﬀerent (international financial) markets. In thisrespect, there have been also some studies on the international linkage
of financial markets in the context of economic and financial studies
such as Hamao et al. (1990), but our statistical method is quite different from theirs because they have used the standard discrete timeseries modeling.
As empirical examples, we will investigate the interactions among
Tokyo-NY (New York), Tokyo-London, and Tokyo-HK (Hong Kong)
financial markets, and then apply the Granger non-causality tests
we will develop. We have found several important empirical findings
among major financial markets.

In Section 2 we present a general formulation of the simultaneous
multivariate Hawkes-type point process (SHPP) model in this study.
Then in Section 3, we will discuss the estimation method and develop
the non-causality tests in the sense of Granger (1969). In Section 4,
we will discuss some simulation results and the empirical applications
will be given in Section 5. Finally, concluding remarks will be presented in Section 6. Some mathematical details we use will be given
in Appendix.

5

2

Simultaneous Hawkes-type Point Process

We divide the observation period [0, T ] into the discrete periods Iin =(tni−1 , tni ] (i = 1, · · · , n) and set the initial time is tn0 = 0. We interpret
Iin as the i−th day, but it is possible to use the observation periods
with finer frequency periods than the daily data in principle. Let theobservable d−dimension price process be Pj (t) (j = 1, · · · , d ; tni−1 <t ≤ tni , i = 1, · · · , n) and in s ∈ Iin we denote the (negative) log-returnof prices Xjn (s) (tni−1 < s ≤ tni ) as(2.1) Xjn (s) = − log[Pj (s)/Pj (tni−1 )] (j = 1, · · · , d; i = 1, · · · , n) .
Let the first stopping time when Xjn (s) exceeds the threshold uj ins ∈ Ii be τ n (i, j, 1). Also let the second stopping time when Xjn (s)exceeds the threshold uj in s ∈ Ii ∩ (τ n (i, j, 1), tni ] be τ n (i, j, 2) anddefine the sequence of τ n (i, j, k) (k ≥ 1). Then we have a sequence ofsets Jj (i) = #{j : τ n (i, j, k) ∈ [tni−1 , tni )} and(2.2)

Njn∗ (t) =∑1
Nj (tni−1 , tni ) (tni−1 ≤ t ≤ tni ) ,1≤l≤i−1 Jj (l)where Nj (tni−1 , tni ) is the number of counts that Xjn (s) (s ∈ (tni−1 , tni ])exceeds uj in the threshold u.
For the resulting expository purpose, we will treat as if the jumpsof the counting process Njn∗ (s, uk ) can occur at tni , the end of eachintervals (tni−1 , tni ], because the number of jumps over a threshold ina finite interval should be finite with probability one and we set thethreshold uj = u (j = 1, · · · , d). We notice that the interval length
goes to zero, that is, maxi=1,···,n |tni − tni−1 | −→ 0 as n −→ ∞ for a fixed6

T and the counting process, which is a simple point process, Njn∗ (s, u)
converges to Nj∗ (s, u) weakly. The resulting counting process can beinterpreted as the limiting process in the high frequency asymptotics,
which is not a diﬀusion type but a pure jump process. (Ikeda and
Watanabe (1989), Ait-Sahalia and Jacod (2014) for instance.)
We consider the point processes, Njn∗ (t) (j = 1, · · · , d), which aresimple and satisfy the standard condition for point processes that as∆t → 0 we have
P (Njn∗ (t + ∆t, u) − Njn∗ (t, u) = 1|Ftn ) = λn∗j (t, u)∆t + op (∆t) ,P (Njn∗ (t + ∆t, u) − Njn∗ (t, u) > 1|Ftn ) = op (∆t) ,where Ftn is the σ−field generated by the information at t, and the(conditional) intensity functions are given by(2.3)

Njn∗ (t + ∆t, u) − Njn∗ (t, u) nn∗
λj (t, u) = lim E[|Ft− ] .
∆t→0∆tn
We denote Ft for Ft−
in the following analysis whenever there is noconfusion on the notation.
n∗Next, we define the point processes, which are simple, Njk(s, u)

by the number of stopping times that Xjn (s) exceed u (j = 1, · · · , d)for a particular j and also Xkn (s) exceed uk (k = 1, · · · , d; k ̸= j) for
another k, and other Xln (s) (l ̸= j, k) do not exceed u by the time sin the interval Iin . By this construction, we can introduce the pointn∗processes Njk(t, u) with co-jumps of Nj and Nk by

P (Njn∗ (t + ∆t, u) − Njn∗ (t, u) = Nkn∗ (t + ∆t, u) − Nkn∗ (t, u) = 1|Ft )= λn∗jk (t, u)∆t + op (∆t) ,7

P (Njn∗ (t + ∆t, u) − Njn∗ (t, u) > 1|Ft ) = op (∆t) ,where λn∗jk (t, u) are the conditional intensity functions of co-jumps.
Then when we have co-jumps of two point processes, we can define
the point processes(2.4)

Njn (s, u) = Njn∗ (s, u) +

∑

n∗
Nj,k
(s, u) (j, k = 1, · · · , d)k̸=jand the corresponding conditional intensity functions are given by(2.5)

λnj (t, u) = λn∗j (t, u) +∑λn∗j,k (t, u) .

k̸=j

The resulting point processes can be interpreted as the marginal point
process for the j-th component of the vector point process Nn (s, u)with d dimension.
By extending this formulation to have more complicated co-jumps andin general we define(2.6)

Njn (s, u) =∑Jj ∈(1,···,d)Njn∗(s, u) (j = 1, · · · , p),
1 ,···,jl

where the index set Jj = {j1 , · · · , jl } ∈ {1, · · · , d} is a subset of(1, · · · , d). The index sets are defined as Ji = {i} for (i = 1, · · · , d),Ji = {1, 1+(i−d)} for (i = d+1, · · · , 2d−1), · · · , and Jp = {1, · · · , d}.
Then we sequentially define Nin (s, u) = Nin∗ (s, u) (i = 1, · · · , d),nn∗n∗and Nd+1(s, u) = N1,2
(s, u), · · · , Npn (s, u) = N1,···,d(s, u). We use theself-exciting form of conditional intensity functions for co-jumps asnλn∗j,k (t, x|Ft− ) in the same way and the marginal conditional intensity

function for the j−th components as(2.7)

λnj (t, u) =∑Jj ∈(1,···,d)8λn∗j,k (t, u) .

There is a one-to-one transformation between Njn (s, u) and Njn∗(s, u),
1 ,···,jddand λnj (t, u) and λn∗j1 ,···,jd (t, u) for j = 1, · · · , p and p = 2 − 1.

The self-exciting Hawkes-type conditional intensity functions for themarked point processes are given byn(2.8) λn∗j (t, x|Ft− ) = λj,0 +



p ∫ t
∑
i=1 −∞∗c∗ji (x)gji(t − s)NJ∗ni (ds × dx)for j = 1, · · · , p, where NJ∗ni (ds × dx) are the marked point precesses,gji (t − s) = e−γji (t−s) are the damping functions, and C(X) = (cji (x))
are the impact functions.
Since we are interested in sudden and large jumps of the underlying
price processes, it is important to use the probability functions of the
return process in the tail areas. Hence it may be appropriate to use theGeneralized Pareto distributions (GDP) as tail probability functions
for x > u (j = 1, · · · , d) as]−1/ξjξj
1 + σj y[]−1/ξjξ[(2.9) P (Xjn (s) > x|Xjn (s) > u, Fs ) =1 + σjj u−1/ξjξj= 1 + ∗ (y − u)
σj,

and we set σj∗ = ξj uj + σj (σj > 0).
(See Resnick (2007) for the details of GDP in the statistical extreme
value theory (SEVT).)In this paper we assume that given the return at s Xjn (s) the conditional density functions are given by(2.10)

−1/ξj −1

ξj1
fj (x, s) = ∗ 1 + ∗ (x − u)
σjσj
9(x > u , ξj > 0)and the conditional intensity functions are given by
p ∫ t
∑
∗n
n
(2.11) λJi (t, u) = λj0 +[Aji (Xin )c (s−)]gi (t − s)dNJ∗ni (s, u)
i=1 0
∗n
n
n
respectively, where Nd+1
(s, u) = N1,2
(s, u), · · · , Np∗n (s, u) = N1,···,d(s, u)and the parameters λj0 and γi are constants.
As the impact functions, we will mainly consider the form
Cij (X) = (Aij max xcj ) (0 ≤ c ≤ 1; i, j = 1, · · · , p).
j∈Ji

In particular when p = d and Cij = δ(i, j) (indicator functions), theycorrespond to the multivariate marked Hawkes-type processes, which
are the simple point processes without co-jumps.

Let p × 1 vector point process Nn (t, u) be partitioned as (d + (p −d)) × 1 processes as(2.12)

n N1 (t, u) Nn (t, u) = nN2 (t, u)










= 











N1n (t, u)..
.
Ndn (t, u)n
N1,2(t, u)..
.
nN1,2,···,d(t, u)











 ,










(Nn1 (t, u) is the d×1 vector of marginal point processes with p = 2d −1
and Nn2 (t, u) is the (p − d) vector of co-jump point processes) and the10corresponding conditional intensity functions as

(2.13)











n

λ
(t,
u)
 1

n
 =

λ (t, u) = 
λn2 (t, u)










λn1 (t, u)..
.
λnd (t, u)
λn1,2 (t, u)..
.
λn1,2,···,d (t, u)











 ,










and p × p matrices
C(X(s−)) = [cij (Xs− )] , G(t − s) = [diag(gj (t − s))] .
(We use the notation that λn1 (t, u) is the vector process of conditional
intensities of marginal jumps, diag(·) for diagonal matrices and we
often omit n for and λnJi (s) (i = 1, · · · , p) and NJni whenever their
meanings are clear.)
Then we rewrite (2.6) and (2.7) as
Nn1 (t, u) = D1 Nn (t, u) ,(2.14)andNn2 (t, u) = D2 Nn (t, u) ,(2.15)

where D1 is a d × p matrix as0··· 0 1

1···

0··· 1 1··· 0 1

0

···

0
..
.

···

0 ··· ··· 1 0 ··· ··· ···

1

 1



 0

D1 =  .
 .
 .



1

0 0

11

···



1 


1 


1

and D2 is a (p − d) × p matrix as D2 = [O, Ip−d ] (p ≥ d).
In this paper we call the above Hawkes-type conditional intensity
models as the simultaneous multivariate Hawkes-type point process(SHPP) models because the resulting market point processes are notnecessarily simple 1 . The classical Hawkes-type point processes havebeen useful in applications because they are simple point processes.
However, they exclude the possibility of simultaneous jumps or cojumps in consideration, and we need the possibilities of co-jumps forapplications. The above constructions of our marked point processes
can be regarded as an extension of Solo (2007).

3

Stationarity and Decomposition of Bartrett Spectrum

3.1

On Stationarity of Haykes-type Processes

In our applications, we will use the stationary self-exciting Hawkestype (marked) point processes. We take the expectation of the intensity function of (2.11) and (2.13) in (−∞, t] asn∫ t(3.1) E[λ (t, u)] = λ0 + E[−∞

C(X(s−)G(t − s)dNn (s, u)] .

We take the non-negative intensity functions and then a set of sufficient conditions for the existence of stationary point processes are

1

The definition of ”simple-point process” and other basic terminologies of point processes andtheir mathematical details are given in Dalay and Vere-Jones (2003), for instance.

12

that E[C(X(s−))] are bounded for any s and the spectral radius
sup max |µi (Ft )| < 1 ,

(3.2)

t

1≤i≤p

where µi (Ft ) is the characteristic roots of
(3.3)

Ft =

∫ t
−∞

E[C(X(s−)]G(t − s)Fs ds .

For instance, if we have a constant matrix C = E[C(X(s−)] andΓ = (diag(γi )), gi (t) = e−γi t (γi > 0; i = 1, · · · , p), then we have
Ft = F = CΓ−1 and Γ = diag(γj ). When d = p = 1 (one-dimensional
Hawkes process) in particular, C = α and and Γ = γ (> 0), thenF = α/γ.

3.2

On the Use of Bartrett Spectrum

Hawkes (1971) introduced the spectral density for the stationary vector point process N(t) = (Ni (t)), which was originally developed by
Bartlett (1963), and it is defined for the conditional intensity vector
in the form of(3.4)

λ(t) = λ0 +∫ t−∞γ(t − u)dN(u) ,where γ(u) = (γij (u)) is a d × d matrix and γ(u) = (0) (zero-matrix)for u < 0. Let the Fourier transform of γ(τ ) be
(3.5)

∗

Γ (ω) =∫ ∞−∞e−iωτ γ(τ )dτ ,where i2 = −1.
Then when p = d (there are no co-jumps), the Bartrett spectral matrix
13for frequency ω (∈ R) is given by(3.6)g(ω) =

1′[Id − Γ∗ (ω)]−1 Σ[Id − Γ∗ (−ω)]−1 ,2πwhere Γ∗ in (3.16) is a d×d matrix for the d−dimensional vector point
process. When there can be co-jumps, the Bartrett spectral matrix
for the d−dimensional marginal point process vector can be definedby(3.7) g(ω) =

Id
1
′
′
′
],
[Id , O][D − DΓ∗ (ω)]−1 Σ[D − D Γ∗ (ω)]−1 [2πOwhere g(ω) = (gij (ω)) is the d × d spectral density matrix and Σ =(σii ) is the diagonal matrix with diagonal elements of the variances
σii (i = 1, · · · , p).
Then we define the relative power contribution (RPC) of the marginal
spectral density function gii (ω) (i = 1, · · · , d) with the frequency ω canbe defined by using the joint spectral density matrix g(ω). The (i,i)component of g(ω) can be represented as(3.8)

gii (ω) =p∑|aik (ω)|2 σkk

k=1

and
|aik (ω)|2 σkk(3.9) RPCk→i (ω) =(i = 1, · · · , p; k = 1, · · · , d) ,gkk (ω)where aij (ω) (i = 1, · · · , d; j = 1, · · · , p) are the functions of complex
variables. Also the instantaneous RPC (IRPCj→i ) can be defined by(3.10)|aij (ω)|2 σjj
IRPCj→i (ω) =(j = d + 1, · · · , p) .
gii (ω)14

In this way, we can measure the relative power contributions for any
frequency ω, which corresponds to the Granger-causality measures in
the frequency domain. One important aspect of the above formulationis the fact that we have a natural definition of Instantaneous Grangercausality in the frequency domain, which is diﬀerent from the discrete
time series modeling.
3.3

On Conditional Probability Prediction

One of important applications of the conditional intensity modeling is
to asses the conditional probability of rare events in the future frompast observations. Let τ (j) (j = 1, · · · , d) be the first arrival time ofan event to be occur in the j− the market. Then we can write theprobability of the random variable τ (j) as(3.11)∫ T′N
P r(τ (j) ≥ T |FT ) = exp(−
λnj (t, u|FTN )dt)
T
′

,
′

where FTN is the σ−field of information available at time T < T and
λnj (t, u|FTN ) is the conditional intensity of the j−the variable. .
Kunitomo, Ehara, and Kurisu (2017) have conducted some experiments and suggested that some useful information on the conditional
probability of future events can be extracted from past observations.
For instance, they have given an important example on the conditional
probability prediction of Lehman Shock given past information available before that events. This would illustrate the possible usefulnessof our approach.

15

4

Estimation and Non-causality Tests

4.1

Likelihood Function

When the point process is simple and there is no co-jump, the loglikelihood function of (d-dimensional) multivariate point process has
been known (see Daley and Jones (2003)) and it is given by
(4.1)

d
∑

{−

∫ T
0

i=1

λni (s)ds +∫ T
0log(λni (s))dNin (s)} .

The log-likelihood function of the marked multivariate point processwith the density function fi (x) is given by(4.2)

log LT = L1T + LT 2 ,

where
L1T =
L2T =

d
∑

{−

∫ T

i=1
d ∫ T
∑

{

i=1

0

0

λni (s)ds +∫ T
0log(λni (s))dNin (s)} ,log fi (xni (s−))dNin (s)}and the density function for the tail probability is given by(4.3)

fi (x) =1
xi − ui − ξ1 −1(1+ξ
) i (i = 1, · · · , d) .
i
σi∗
σi∗

Then we can apply the maximum likelihood method to L1T and L2T
separately. In this formulation we use the GPD (generalized Pareto
distribution) for the marginal distributions.
When there can be co-jumps, the log-likelihood function of (d-dimensional)
marginal point process is not the above form and it should be given
by
(4.4)

log L∗T = L∗1T + L∗2T ,16

where
L∗1T

=

d
∑

{−

i=1

∫ T
n
λi (s)ds +log(λni (s))dNin (s)}
0
0

∫ T

∫ T
n
+
{−
λij (s)ds +log(λnij (s))dNijn (s)}00i̸=j=1∫ T∫ Tnn+ · · · + {−λi···d (s)ds +log(λni···d (s))dNi...d(s)} .
0
0
d
∑

∫ T

and L∗2T = L2T .
In our applications we mainly deal with the case when d = 2 andthen there is only one extra term in the likelihood function because
p = 2d − 1.
We assume the stationarity condition (3.2) and the existence of
second order moments of C(X) = cij (X(s)) in the statistical inferenceof Hawkes-type point processes without and with co-jumps. Also wetake λ(u) as the stationary conditional intensity and some q × p predictable processes ξ(t) having the second order moments.
Then, because of the resulting martingale property given the information available at each time, it is straight-forward to show the asymptotic properties as we have
(4.5)
and
(4.6)

1∫T
ξ(t)[N(t, u) − λ(t, u)]dt −→ 0 (a.s.)
T 0
1∫T
p
ξ(t)[λ(t, u) − λ(u)]dt −→ 0
T 0

as T → ∞.
For the one-dimensional point processes with the stationary intensity function, Ogata (1978) has given a set of suﬃcient conditions for17the consistency and asymptotic normality of the maximum likelihood(ML) estimation. His derivations are based on a martingale central
limit theorem (MCLT) and it is straightforward to extend his arguments to the multi-dimensional case. For the sake of completeness, wehave given some detail of our arguments based on a new MCLT in our
Appendix, which may be more general than the standard literature as
the ones given by Ogata (1978). We will also give the outline of ourproofs of Theorems used in the next subsection, which are developed
for our empirical applications as the new non-causality tests.
4.2

Non-Causality Tests

We will develop and use the Granger non-causality tests based on thelikelihood ratio principle for the Hawkes-type point processes, which
may be new. In particular, our results in this subsection, whose proofs
are given in Appendix, include not only the multivariate extension of
the existing results, but also the cases when the resulting limiting
Fisher information matrix can be random variables. We first state ourresult for the case of no co-jumps under a set of regularity conditions,which will be extended in the more general case. We summarize thebasic result.
Theorem 4.1 : Let the log-likelihood function of the Hawkes-type
point processes with true parameters be LT (θ0 ), the log-likelihood
function with the maximum likelihood estimator θ̂M L be LT (θ̂M L ) under Θ ∈ θ and the log-likelihood function with the restricted maximum
18likelihood estimator θ̂RM L be LT (θ̂RM L ) under Θ1 ∈ θ (Θ1 ⊂ Θ). We
assume that the suﬃcient conditions for the stationarity, the existenceof the second order moment condition of C(X), and the parameter
space Θ ∈ θ in Rr the parameter space and Θ1 ∈ θ in Rr1 (0 ≤ r1 < r)are compact sets. Under a set of regularity conditions (see Theorem
A-3 in Appendix), as T → ∞,
(4.7)

d

2{LT (θ̂M L ) − LT (θ̂RM L )} → χ(r − r1 ) ,where r − r1 is the number of restrictions of θ = (θk ) and χ2 (r − r1 ) isthe χ2 −random variable with r − r1 degrees of freedom.
The details of a set of regularity conditions will be discussed in Appendix. When there can be co-jumps in the Hawkes-type processes,
we cannot apply Theorem 4.1, but it is important to obtain the corresponding results in such cases for econometric applications. Especiallywhen we use the discrete versions of point processes, which would beoften the case in econometric applications, we need to consider the
existencve co-jumps. Then we will develop the non-causality tests
based on the likelihood ratio principle. In this respect we notice thatin our setting discussed in Section 2, although we allow the possible
co-jumps, it is possible to apply the martingale central limit (MCLT)
theorem for point processes. We state our result, which is an extensionof Theorem 4.1 to the case of co-jumps.
Theorem 4.2 : Let the log-likelihood function of the Hawkes-type
point processes with true parameters be LT (θ0 ), the log-likelihood
19

function with the maximum likelihood estimator θ̂M L be LT (θ̂M L ) under Θ ∈ θ and the log-likelihood function with the restricted maximum
likelihood estimator θ̂RM L be LT (θ̂RM L ) under Θ1 ∈ θ (Θ1 ⊂ Θ). We
assume that the suﬃcient conditions for the stationarity, the existenceof the second order moment condition of C(X), and the parameter
space Θ ∈ θ in Rr the parameter space and Θ1 ∈ θ in Rr1 (0 ≤ r1 < r)are compact sets. Under a set of regularity conditions (see Theorem
A-3 in Appendix), as T → ∞,Under a set of regularity conditions even when co-jumps exist, as
T → ∞,
d2{LT (θ̂M L ) − LT (θ̂RM L )} → χ(r − r1 ) ,(4.8)

where r − r1 is the number of restrictions of θ = (θk ) and χ2 (r − r1 ) isthe χ2 −random variable with r − r1 degrees of freedom.
Some details of the regularity conditions required in Theorem 4.1 andTheorem 4.2 will be discussed in Appendix.

5

Simulations

To examine the relevance of our estimation procedure proposed inthis paper we have done a set of simulations. The model we have usedin our simulations are the simultaneous Hawkes-type model with two
dimension and the intensity functions are given by
λn1 (t)

=

∫ t
∫ t
n
−γ(t−s)n
λ10 + α11 e
X1 dN1 (s) + α12 e−γ(t−s) X2 dN2n (s)00∫ t−γ(t−s)n+ α13 e[max Xi ]dN12 (s) ,0i20

λn2 (t)=λn12 (t) =

∫ t
∫ t
n
−γ(t−s)n
λ20 + α21 e
X1 dN1 (s) + α22 e−γ(t−s) X2 dN2n (s)00∫ t−γ(t−s)n+ α23 e[max Xi ]dN1,2 (s) ,
0
i
∫ t
∫ t
λn12,0 + α31 e−γ(t−s) X1 dN1n (s) + α32 e−γ(t−s) X2 dN2n (s)
0
0
∫ t
n
(s) .
+ α33 e−γ(t−s) [max Xi ]dN1,20i

We first generate the stock price returns by using the generalized
Pareto distribution as marginal and the two-dimensional Gaussian
copura. Then we use the maximum likelihood (ML) method to obtain
the estimates of the underlying parameters. We give several figures(Figures 5.1-5.6) among many results and all figures of the finite sample distributions of the ML estimator are standardized as
(5.1)

I1/2
n (θ̂ − θ) ,where θ = (θi ) is the vector of parameters and and θ̂ is the ML
estimator. This makes possible to compare them to the standard
normal distributions.

In our numerical evaluations we sometimes hit the boundaries of the
non-negativity of intensity functions with finite samples, which makethe simulation some instabilities. Thus we have set non-negativity restrictions on parameters in our simulations. Then we have reasonable
results, but then sometimes we observe that the maximum likelihood
estimators of coeﬃcients have the resulting biases, which are basicallynot very large. (Figure 5-2 is a typical example of this kind.) We
summarize the setting of our numerical experiments : the simulation
21

∗
Figure 5-1 : α12size N = 100, and for GPD(σj , ξj ) we set (σ1 , ξ1 ) = (0.007, 0.22),
and (σ2 , ξ2 ) = (0.008, 0.15). These numerical values are based on the
preliminary estimates obtained by our empirical studies, which givereasonable estimates.
∗
α11

∗
α12

∗
α13

∗
α21

∗
α22

∗
α23

True

0.57000 0.00000 0.19000 0.00010 0.71000 0.09500

Mean

0.63641 0.00259 0.12387 0.03994 0.76318 0.07905

RMSE 0.01045 0.00426 0.00913 0.00568 0.01004 0.00557
∗
α31

∗
α32

∗
α33

γ∗

λ∗1,0

λ∗2,0

λ∗3,0

True

0.05900 0.12000 0.20000 0.02700 0.00930 0.00530 0.00084

Mean

0.06748 0.13922 0.11315 0.02859 0.00853 0.00427 0.00107

RMSE 0.00272 0.00380 0.00963 0.00033 0.00019 0.00017 0.00007
Table 5-1 : Simulation results

Among many simulations we illustrate our results in Table 5-1 and∗Figures. Because we have taken α12= 0, we have a sampling dis-22

∗Figure 5-2 :α21∗
Figure 5-3 :α23

23

Figure 5-4 :γ ∗tribution around zero and the resulting estimate is not significant as
in Figure 5-1. Other estimates of αij take reasonable values on average and some of the sampling distributions are illustrated in Figures
5.2-5.4. We have found that there are some positive biases on αij
and negative biases on the initial intensities, which may be due to theresults of the non-negative constraints of the parameter restrictions.
In the ML estimation there can be some eﬀects of initial conditions and we have investigated this problem in the SHPP models. We
have confirmed that there are such eﬀects, but they are minor in our
simulations.
We will also use the χ2 −distributions as the limiting distributions of
the likelihood ratio statistics for hypotheses testing in our empirical
study. We have confirmed that the χ2 −approximations with finite
samples are often appropriate.

24

6Empirical Applications

In this section we will report two empirical examples by using the
SHPP models. The first one is the three major stock markets, namely,
Tokyo, New-York, and London. Since there are some time diﬀerences
when each markets are open and close, it is reasonable to assume
that there is no co-jumps. As the second example, we will report theempirical analysis of Tokyo and HK (Hong Kong) markets. Since thetime zones are similar when two markets are open and time diﬀerenceis negligible, it may be reasonable to use the extended Hawkes models
with co-jumps. Our data used in the first example are daily dataof Nikkei225, S&P500 and FTSE100 during 1990/1/2-2015/8/25. We
have chosen u = 2% because Kunitomo, Ehara and Kurisu (2017) haveanalyzed this case and obtained reasonable results and we will report
some of their results as Example 1 because they are only available inJapanese. The example 2 on Tokyo-Hong-Kong markets is completely
new and its empirical observation is the main reason why we havedeveloped the SHPP models in this study.
In addition to daily data, we also have used the data of the beginning of the day and the lowest of the day in two examples because
these data have been often used is financial industries and the analysis
may give us the robustness check of our results on the conditional intensity modeling including the non-causality tests. However, we haveomitted reporting the details of the results because they are basicallythe same as we will report in this section.
25

Table 6-1 : Tail Distributions

J

Log Likelihood

σi∗

ξi

-1190.72

0.00806

0.168740.000650.06431

σi∗

ξi

SD
Log Likelihood
NY-797.385

0.00765 0.21538

SDL

0.00076 0.08082
Log Likelihood

σi∗

ξi

-775.779

0.00850

0.107990.00084

0.07717SD

6.1

Example 1 (Tokyo-NY-London)

We first maximize the likelihood L2T to estimate the marginal distributions of financial market returns. As we have shown in Table 6.1, we
have confirmed that the marginal distributions of market returns havethicker tails than the normal distribution. Hence, it may be appropriate to use the generalized Pareto distribution in our estimation. The
standard deviations (SD) are estimated by the numerical evaluationof Fisher Information matrix.
As the estimated models with two dimension (d = p = 2), we takethe impact functions c(x) as Case (1) c(x) = 1, Case (2) c(x) = x,
and Case (3) c(x) = xc (0 < c < 1). The estimated values of the loglikelihood and AIC are those with the marginal distribution L1T . The
full likelihood can be calculated by using L1T and L2T . The standard
deviations of the estimated coeﬃcients are also evaluated numerically26

by using the inverse of the estimated Fisher information matrix.
Case 1
We estimated the intensity function as∫ t∫ t−γ11 (t−s)
nnndN1 (s) + α12 e−γ12 (t−s) dN2n (s) ,λ1 (t) = λ10 + α11 e
0
0
∫ t
∫ t
n
n
−γ21 (t−s)
nλ2 (t) = λ20 + α21 e
dN1 (s) + α22 e−γ22 (t−s) dN2n (s) .
00Since the maximum likelihood estimates can be sometimes unstablenumerically, we set the restriction that the discounted parameters
γij (i, j = 1, 2) have the same value γ in the following estimation.
We show our estimation results on Case 1 in Table 6.2.

Table 6-2 (1): Tokyo-NY
Log Likelihood

AIC

α11

α12

-2444.14

4902.270.014900.00452

0.0021020.00162

Tokyo-NY
SD
α21

α22

γ

λ10

λ20

0.00000

0.01796 0.0234 0.00583

0.003900.000700.00247 0.0030 0.00126

0.00093

In the above table N1 corresponds to Tokyo and N2 corresponds
to NY in Tokyo-NY markets. In Tokyo-London, N1 corresponds toTokyo while N2 corresponds to London.
The most important finding in Table 6.2 (and also in Table 6.3
below), is the fact that the coeﬃcient α12 is statistically significant
27

Table 6-2 (2) : Tokyo-London

Tokyo-London

Log Likelihood

AIC

α11

-2421.02

4856.04SD

α12

0.01692 0.00437
0.00235 0.00173

α21

α22

γ

λ10

λ20

0.000620.02028 0.02729

0.00683 0.00361

0.00073

0.00284 0.00341

0.00126 0.00087

while the coeﬃcient α21 is not statistically significant. This is a kindof the non-causality test, but as we will discuss in more formal ways.
We have found reasonable values for other parameters and they are
significant both in Tokyo-NY and Tokyo-London.
Case 2
We estimated the intensity function as
λn1 (t) = λn10 +
λn2 (t) = λn20 +

∫ t
0
∫ t
0

α11 e−γ(t−s) X1 dN1n (s) +α21 e−γ(t−s) X1 dN1n (s) +

∫ t
0
∫ t
0

α12 e−γ(t−s) X2 dN2n (s) ,α22 e−γ(t−s) X2 dN2n (s) ,and we have shown our estimation results in Table 6.3.
In the present case we have similar values for the estimated coeﬃcientsas Case 1 except α21 . It seems that we have more significance than inCase 1, which correspond to the likelihood values and their AIC.

Case 328

Table 6-3 (1) : Tokyo-NY

Tokyo-NY

Log Likelihood

AIC

α11

-2441.59

4897.19SD

α120.48308 0.13139
0.07110 0.05232

α21

α22

γ

λ10

λ20

0.00000 0.58128

0.02328 0.0067209 0.0043152

0.02584 0.08305

0.003100.001310.00095

Table 6-3 (2) : Tokyo-London

Tokyo-London

Log Likelihood

AIC-2418.77

4851.55

SD

α11

α12

0.57164 0.13416
0.08127 0.05768

α21

α22

γ

λ10

λ20

0.02901 0.68684

0.02833 0.007645

0.003880.02905 0.09947

0.00370.000890.00130

We estimated the intensity function as
∫ t
∫ t
c11
n
n
−γ(t−s)n
λ1 (t) = λ10 + α11 e
X1 dN1 (s) + α12 e−γ(t−s) X2 c12 dN2n (s) ,
0
0
∫ t
∫ t
c21
n
n
n
−γ(t−s)λ2 (t) = λ20 + α21 e
X2 dN1 (s) + α22 e−γ(t−s) X2 c22 dN2n (s) .
00The maximum likelihood estimates are sometimes unstable numerically, we have set the restriction that the discounted parameters
γij (i, j = 1, 2) have the same value γ and also we set the restriction c11 = c12 , c21 = c22 after several diﬀerent trials. We show theestimation results in Table 6.4.

In this case the values of the estimated coeﬃcients are reasonable,
29

Table 6-4 (1) : Tokyo-NY

Tokyo-NY

Log Likelihood

AIC-2440.769

4899.538

SD
γ

λ10

λ20

α11

α12

α21

0.0789 0.02271

0.0000 0.0948

0.35530.0053 0.1169

0.1005c11 = c12

c21 = c22

0.023200.006132 0.00403

0.47390.47605

0.00305

0.00159

1.28043

0.359400.00096

α22

Table 6-4 (2) : Tokyo-London
Log Likelihood
Tokyo-London-2417.74

AIC

λ10

α12

α21

α22

4853.48 0.21233

0.05087 0.00656 0.17068

0.27649

0.06779 0.01023 0.17829

SD
γ

α11

λ20

c1,1 =c1,2

c2,1 =c2,20.02800 0.00730 0.00373

0.71330

0.599620.00363 0.00133 0.00089

0.37337

0.29613but often they are not statistically significant. For instance, the estimated value of c is between zero and one. From our estimated results,
it seems that we have found that Model 2 and Model 3 are better thanModel 1. Also by using AIC Model 2 is better than Model 3 mainly
because Model 3 has too many parameters and Model 2 is better thanModel 3 as Tokyo-NY markets. Hence we have adopted Model 2 orCase 2 in the following non- causality tests.
Non-Causality Tests
When we apply the Granger Non-Causality test procedure, we have set
the impact function as c(x) = x. We report our empirical results for30the hypothesis H0 : αij = 0 by using the likelihood ratio test statistics
based on the Tokyo-NY data. For the null-hypothesis H0 : α21 = 0, thelikelihood ratio was 2 × (−2441.594 + 2441.594) ∼ 0 and we could not
reject the null-hypothesis. (The 95% upper-percentage point of χ2 (1)is 3.481 in Table 6-5(1).) This means that the change of the Japanesefinancial market has little impact on the U.S. financial market.
For testing the null-hypothesis H0 : α12 = 0, the likelihood ratio
test statistics based on the Tokyo-NY data was 2 × (−2441.594 +
2446.297) = 9.406, and then the null-hypothesis was rejected. Thismeans that there is a significant eﬀect from the U.S. market to theJapanese financial markets (see Table 6-5(2)).

Similarly, we have done the empirical analysis on Tokyo-London
markets. For the null-hypothesis H0 : α21 = 0, the likelihood ratio statistic was 2 × (−2418.773 + 2419.359) = 1.172 and the nullhypothesis was not rejected. This means that the eﬀect of Japanesefinancial market on London is rather limited.
For the null-hypothesis H0 : α12 = 0, the likelihood ratio statistic
based on the Tokyo-London data was 2 × (−2418.773 + 2422.848) =8.15 and the null-hypothesis was rejected. This means that there isan eﬀect of London market on Tokyo (see Tables 6-2(2), 6-3(2) and6-4(2)).

To summarize our findings among three major financial markets,31the eﬀects of Japanese market on the U.S. and London are rather limited while we have found significant eﬀects of U.S. financial market
and London financial market on Tokyo market, were rather statistically significant.
6.2

Example 2 : Tokyo-HK markets

For the second example, we have used daily data of Nikkei-225 andHansen Index of Hong-Kong(HK) during 1990/1/2-2015/8/25. Since
the trading periods in two financial markets are quite similar, we had
expected that there can be simultaneous movements in two markets.
Because there can be many additional parameters in Case 3, which
has the general form of impact functions, the estimated results areoften not statistically significant and we have omitted to report our
results of Case 3.
We first maximize the likelihood L∗2T to estimate the marginal distributions of financial market returns. As we have shown before, wehave confirmed that the marginal distributions of market returns havethicker tails than the normal distribution in Table 6-7. Hence, it may
be appropriate to use the generalized Pareto distribution in our estimation.
The estimated models with two dimensions (d = 2 and p = 3), we
take the impact functions c(x) as Case (1) c(x) = 1 and Case (2) c(x) =x. The estimated values of the log-likelihood and AIC are those withthe marginal distributions L∗1T . The full likelihood can be calculated
32

Table 6-5 : Tail Distributions
Log Likelihood

σi∗

ξi

-1919.307

0.00757

0.227780.00051

0.05552

σi∗

ξi

J
SD

Log Likelihood
HK-1888.716

0.00861 0.15773

SD

0.00055 0.05076

by using L∗1T and L∗2T . The standard deviations of the estimated coefficients are evaluated numerically by using the inverse of the estimatedFisher information matrix.
Case 1
We estimated the intensity function as
λn1 (t) = λn10 +
+

∫ t
0

∫ t
0

0
∫ t

∫ t
0

0

α21 e−γ(t−s) dN1n (s) +

∫ t
0
∫ t

n
α23 e−γ(t−s) dN1,2(s) ,λn12 (t) = λn12,0 ++α11 e−γ(t−s) dN1n (s) +nα13 e−γ(t−s) dN1,2(s) ,λn2 (t) = λn20 +
+

∫ t

∫ t
0

α31 e−γ(t−s) dN1n (s) +0α12 e−γ(t−s) dN2n (s)α22 e−γ(t−s) dN2n (s)∫ t0α32 e−γ(t−s) dN2n (s)nα33 e−γ(t−s) dN12(s) .

Again the maximum likelihood estimates can be sometimes unstablenumerically, we have set the restriction that the discounted parameters
γij (i, j = 1, 2, 3) have the same value γ. We show the estimation
results in Table 6-6.
33

Table 6-6 : Tokyo-HK
Log Likelihood
Tokyo-HK

AIC-3954.73

α11

α12

α13

7935.47 0.015

0.000

0.012

0.002

0.001

0.0036

SD
α31

α32

Tokyo-HK 0.0015 0.0035
SD

0.0007 0.0008
γ

α33
0.00860.0022

α21

α22

α23

λ1

λ2

λ3

Tokyo-HK

0.000

0.020

0.007

0.0262 0.0090 0.0048

0.0008SD

0.00074

0.0025 0.0033 0.0028 0.0016 0.0012

0.0007

We note that in the above table N1 corresponds to Tokyo and N2
corresponds to NY in Tokyo-NY markets.
Case 2
We estimated the intensity function as
λn1 (t)=λn2 (t) =λn12 (t) =

∫ t
∫ t
n
−γ(t−s)n
λ10 + α11 e
X1 dN1 (s) + α12 e−γ(t−s) X2 dN2n (s)00∫ t−γ(t−s)n+ α13 e[max Xi ]dN1,2 (s) ,
0
i
∫ t
∫ t
λn20 + α21 e−γ(t−s) X1 dN1n (s) + α22 e−γ(t−s) X2 dN2n (s)
0
0
∫ t
n
+ α23 e−γ(t−s) [max Xi ]dN12
(s) ,
0
i
∫ t
∫ t
n
−γ(t−s)nλ12,0 + α31 e
X1 dN1 (s) + α32 e−γ(t−s) X2 dN2n (s)00∫ t−γ(t−s)n+ α33 e[max Xi ]dN1,2 (s) .
0i

We have shown our estimation results in Figure 6.7. From our estimated results, we find that Model 2 is better than Model 1 as in34

Example 1. By comparing Table 6-7 and Table 6-8, there are several
interesting findings. The value of AIC in Case 2 is better than Case 1
as we have observed in the Tokyo-NY and Tokyo-London Data sets.
The estimates of coeﬃcients of the past eﬀects are often statisticallyin-significant while the contemporaneous eﬀects of co-jump term arestatistically significant. This aspect basically agrees with our motivations for developing the SHPP models.

Table 6-7 : Tokyo-HK
Log Likelihood

AIC

α11

α12

α13

-3944.79

7915.58

0.56750.000

0.19300.764

0.0373 0.0738

Tokyo-HK
SD

α31

α32

Tokyo-HK 0.0586 0.1242
SD
α21

α22

0.0007 0.0008
α23γTokyo-HK 0.0001 0.7147 0.0950 0.0267
SD

0.0241 0.0871 0.0701 0.0029

α33
0.05470.0022λ1

λ2

λ30.0094 0.0053 0.0008
0.0016 0.0012 0.0007

Non-Causality Tests
When we apply the Granger-causality test procedure, we have set the
impact function as c(x) = x. We report our empirical results for the
hypothesis H0 : αij = 0 by using the likelihood ratio test statistics.

For the null-hypothesis H0 : α13 = 0, the likelihood ratio based on
35

Tokyo-HK data set was 11.14 and we reject the null-hypothesis. (The95% upper-percentage point of χ2 (1) is 3.481.) This means that wehave a significant instantaneous causal relation between the Japanesefinancial market and Hong-Kong financial market.
For testing the null-hypothesis H0 : α12 = 0, the likelihood ratio test
statistics was 0.0, and then the null-hpothesis was accepted.
Also for testing the null-hypothesis H0 : α12 = 0, α13 = 0 the likelihood ratio test statistics was 11.14, and then the null-hypothesis was
rejected.
Next, for the null-hypothesis H0 : α21 = 0, the likelihood ratio was
0.006 and we reject the null-hypothesis. (The 95% upper-percentage
point of χ2 (1) is 3.481.)
For testing the null-hypothesis H0 : α23 = 0, the likelihood ratio test
statistics was 2.42, and then the null-hypothesis was accepted(see Table 6-10(5)).
Similarly, for testing the null-hypothesis H0 : α21 = 0, α23 = 0 the
likelihood ratio test statistics was 2.66, and then the null-hypothesis
could not be rejected.

To summarize our findings in this subsection among Tokyo andHong Kong financial markets, we have found that the simultaneouseﬀects of two markets are significant while the eﬀects of past events
are rather small.

36

Figure 6-1 : Relative Power Contributions

6.3

A Further Empirical Analysis

We also use the spectral decomposition and the relative power contributions as we explained in Section 3. Three figures of US, UK and HK
are given as Figure 6-1, 6-2 and 6-3. In the first two decompositions
we assume that there are no co-jumps while in the last one we do haveco-jumps terms. We have adopted the cases when cij (x) = x because
the resulting models have the minimum AIC.
For the relationship between Tokyo-NY financial markets, the self
contribution play major contribution while there is some contribution
from NY to Tokyo in the low frequency, which corresponds to thelong-run relation. On the other hand, for the relationship between
Tokyo-HK financial markets, the instantaneous contribution plays amajor contribution in all frequencies as well as the self contribution.
This aspect reflects the fact that we have used the SHPP models.
37

Figure 6-2 : Relative Power Contributions

Figure 6-3 : Relative Power Contributions

38

7

Conclusions

In this paper we developed a new method of econometric analysis
of multivariate time series of events and proposed the simultaneous
Hawkes-type point process modeling. Unlike some existing literatures,
we develop and use the new statistical models for simultaneous sudden
and large events and delayed events occurred explicitly. By using thesimultaneous multivariate Hawkes-type point process approach andthe SHPP models, we have investigated the Granger-causality andthe instantaneous Granger causality on diﬀerent financial markets andeconomies and developed the non-causality tests.
By applying the non-causality tests for both the Granger noncausality (GNC) and the Granger instantaneous non-causality (GINC),
we have found the important relations among major financial markets
and several empirical findings. In Tokyo-NY financial markets, there
is a strong one way direction in causation while in Tokyo-HK financial
markets the simultaneous eﬀects are dominant.
There are several questions remained to be answered. First, although we have used the Hawkes-type marked point processes, there
can be many possible non-linear point processes and Kurisu (2016)
has found one way to justify the use ot SHPP models. In economic
and financial econometrics it is standard to handle the discrete observations of time series such as year, month, weak, day, an hour and aminute. Thus we need a coherent way of investigating abrapt or sudden events and we are proposing one way to deal with discrete time39

series events in this paper. In this respect, it should be interesting
to investigate the robustness of our empirical results further. Second,the choice of threshold parameter is an important one, which is related
to the relevance of the generalized Pareto distribution (GPD) in thestatistical extreme value theory in our empirical analysis. Since wehave used a simple threshold parameter, apparently we need a moreconvincing statistical theory on the choice of threshold. Finally, whend > 2 there can be many parameters to be estimated and often the estimated parameters would be statistically not significant. This aspect
is important when we have the possibility of co-jumps. Hence there
should be some ways to handle this problem.
These questions are currently under investigation and we shall report our progresses in another occasion.

References[1] Ait-Sahalia, Y. and J. Jacod (2014), High-Frequency Financial
Econometrics, Princeton University Press.
[2] Ait-Sahalia, Y., Cacho-Diaz,J., and Laeven, L. (2015) ”Modeling financial contagion using mutually exciting jump processes,”
Journal of Financial Economics, 117, 585-606[3] Bacry, E., Mastromatteo, I. and Muzy, J-F. (2015) ”Hawkes
processes in Finance,” Market Microstructure and Liquidity, 1,
1550005, World Scientific.
40[4] Bartrett, M.S. (1963) “The Spectral Analysis of Point Processes,”
Journal of Royal Statistical Society (B), Vol.25-2, 264-296.
[5] Daley, D.J. and D. Vere-Jones (2003), An Introduction to the
Theory of Point Processes, Volume I, Volume II, 2nd Edition,
Springer.
[6] Embrechts, P., Liniger, T., Lin, L. (2011), ”Multivariate Hawkes
processes: an application to financial data,” Journal of Applied
Probability, Special Volume, 48A, 367-378.
[7] Florens, J-P. and D. Fougere (1996) “Noncausality in Continuous
Time,”Econometrica, 64, 1195-1212.
[8] Granger, C.W. (1969) “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods,” Econometrica, 37,
161-194.
[9] Grothe, O., V. Korniichuk and H. Mannera (2014)“Modeling multivariate extreme events using self-exciting point processes,” Journal of Econometrics, 182, 269-289.
[10] Hamao, Y, R. W. Masulis and V. Ng. (1990), ”Correlations inPrice Changes and Volatility across International Stock Markets,”
Review of Financial Studies, 3 (1990), 281-308[11] Hawkes, A. G. (1971) “Point Spectra of Some Mutually Exciting
Point Processes,” Journal of the Royal Statistical Society. Series
B, 33-3, 438-443.
41

[12] Ikeda, N. and S. Watanabe. (1989), Stochastic Diﬀerential Equations and Diﬀusion Processes, 2nd Edition, North-Holland.
[13] Kunitomo,N., A. Ehara and D. Kurisu (2017) ”A Causality Analysis of Financial Markets by Multivariate Hawkes-type Models,”(in Japanese), Journal of Japan Statistical Society, 46-2, 137171.
[14] Kurisu, D. (2016), ”Discretization of Self-Exciting Peaks Over
Threshold Models, Unpublished Manuscript.
[15] Ogata, Y. (1978) “The Asymptotic Behavior of Maximum Likelihood Estimators of Stationary Point Processes,” Anals of Institute of Statistical Mathematics, 30, 243-261.
[16] Ogata, Y. (2015) “Studies of Probabilistic Prediction of Earthquakes:

A Survey,” Statistical Mathematics, 63-11, 3-27 (inJapanese).
[17] Protter, P. (2003), Stochastic Integration and Diﬀerential Equations, Springer.
[18] Resnick, S. (2007), Heavy-Tail Phenomena, Springer.
[19] Solo, V. (2007) “Likelihood Functions for Multivariate Point Processes with Coincidences,” Proceedings of the 46th IEEE Conference on Decision and Control, 4245-4250.

42APPENDIX : Mathematical Details
In this Appendix, we give some mathematical details we have usedin the previous sections. In the statistical analysis of point processes,
Ogata (1978) derived the asymptotic properties of consistency andasymptotic normality of the maximum likelihood estimation for one
dimensional intensity models, which have been classical and often citedin the related studies. He obtained the results by using a martingale
central limit (CLT) theorem for point processes, which has not beenwell-known for econometricians, and also the asymptotic normality
holds under more general conditions often cited. Hence, we first discuss some properties of jump martingales with continuous time parameter. We omit the subscript n without any loss of generality inthis Appendix.
(i) A Martingale CLT
We present a general martingale CLT for one-dimensional point processes and then we can apply to our situation as an application.
Theorem A.1 : Let an F−adapted simple point process on R+ be
N and the F−(continuous)compensator be A. We assume that for
any T (> 0) there exists an F−adapted gT (t) and an F0 −adapted(positive) random variable η, which satisfy the following conditions.
∫(i) E[ T1 0T (gT (x))2 dA(x)] < ∞ ,(ii) For any δ (> 0),
(A.1)

1
T 1+δ

p

A(T ) −→ 0,
43

(iii) As T −→ ∞1∫Tp(gT (x))2 dA(x) −→ η 2 ,
T 0

(A.2)

(iv) For any c > 0, as c → ∞
(A.3)

E[

1∫T
p
(gT (x)I(|gT (x)| > c))2 dA(x)|F0 ] −→ 0 .
T 0

Then

1 ∫T
XT = √(A.4)
gT (x)[dN (x) − dA(x)]T 0converges to U η in the sense of F0 −(stable convergence sense), where
U is N (0, 1), which is independent of F0 .
Remark A-1 : The method of proof is basically a modification of
the one given by Daley=Vere-Jones (2008, Vol-II), as Theorem 14.5.I.
They derived a martingale CLT under a Lyapunov condition. Ourcondition includes the speed of compensator, which may be a reasonable condition.
√
Proof : For any real number y and fT (u) = (1/ T )gT (u), we define
(

)

∫ t

1 ∫t
ζT (t, y) = exp iy fT (u)[dN (u) − dA(u)] + y 2 [fT (u)]2 dA(u) .
0
0
2
(A.5)
By using Lemma A-1 below, when A(t) and N (t) are a continuous
process and a pure jump process, respectively, we can represent
)

(

(A.6)

∫ t
1
ζT (t, y) = exp ( y 2 [fT (u)]2 − iy fT (u))dA(u)
0
2

×

∏

∫ t

i

0

[1 + (exp(iy44fT (ti ) − 1)∆N (ti )] ,where ti are jump times. By using the transformation of jump process,we haveζT (t, y) − 1
=
=

∫ t

[

]

1
ζT (u−, y) y 2 [fT (u)]2 − iyfT (u)]dA(u) + [exp(iyfT (u)) − 1]dN (u)
0
2

∫ t

ζT (u−, y)(exp(iyfT (u)) − 1)(dN (u) − dA(u))
[
]
∫ t
1 2
2
+ ζT (u−, y) exp(iyfT (u)) − 1 − iyfT (u) + y [fT (u)] dA(u) .
020

We define the stopping time τ by τ = inf{t :∫T
2
0 [fT (u)] dA(u)≥η 2 }. Then for any F0 −measureable and essentially bounded randomvariable Z, we set t = T ∧ τ . By the martingale property we have[

E Z

∫ T ∧τ0]ζT (u−, y)(exp(iyfT (u)) − 1)(dN (u) − dA(u))|F0 = 0 .

Hence|E(ZζT (T ∧τ )|F0 ]−Z)| ≤ E[|Z|∫ T ∧τ0|ζ(u−, y)R(fT (u), y)|dA(u)|F0 ] ,where
1R(fT (u), y) = exp(iyfT (u)) − 1iyfT (u) + y 2 [fT (u)]2 .
2
For 0 < u < T ∧ τ we have1
1 2 ∫ T ∧τ[fT (u)]2 dA(u)) ≤ exp( y 2 η 2 ) .
|ζT (T ∧ τ )| ≤ exp( y022Also by the Taylor-expansion,|θy|3|R(fT (u), y)| ≤ y |fT (u)| I[|fT (u)| > cT ]+|fT (u)|3 I[|fT (u)| ≤ cT ]
3!
2

2

45

and then|E(ZζT (T ∧ τ )|F0 ] − Z|
≤ |Z|[y2∫ T ∧τ

0∫ T ∧τ

+|yθ|3|fT (u)|2 I[|fT (u)| > cT ]dA(u)0|fT (u)|3 I[|fT (u)| ≤ cT ]dA(u) ,where |θ| ≤ 1. Therefore the right-hand side multiplying exp(−1/2y 2 η 2 )is bounded by|E(Z[ρT eiyXT − e−1/2y η ])| ,
2 2

where
[

]

∫ T

1 2 ∫TρT = exp iy
fT (u)([dN (u) − dA(u)] − (η − [fT (u)]2 dA(u))+ .
T ∧τ
0
2
√
√
We set gT (u) = fT (u)/ T and c = cT / T . Then
∫ T ∧τ
0c3|fT (u)| I[|fT (u) ≤ cT ]dA(u) ≤ 3/2 A(T ∧ τ ) ,T3which converges to zero by our conditions. Here we haveζT (u−, y)e−y 2 η 2 /2

[
iyXT

= e

]
∫ T ∧τ
∫T
2 ∫T
2 2
iy 0
fT (u)(dN −dA)+ y2 0 fT (u)2 dA−iy 0 fT (u)(dN −dA)− y 2ηe= eiyXT ρT .
Because |ρT | ≤ 1 and ρ → 1, we find that E[Z(ρT − 1)eitXT )] → 0 andthen
E[Z exp(iyXT )] −→ E[Ze− 2 y η /2 ] .
1 2 2Then by the use of weak-convergence and stable convergence (Dalay=VereJones(2008), Jacod=Protter (2012)), we have that XT −→ X (F0 )stably). This means that for any bounded F0 −measurable random46

variable Z, E[ZeiyX ] = E[Ze−y η /2 ], which implies E[eiyXT /η |F0 ] =2 2e−y /2 .
2

Q.E.D.
We give the integration-by-parts formula, which has been known instochastic analysis (see Chapter II of Protter (2003), for instance).
Lemma A.1 : Let(A.7) G1 (t) =

∏

(∫

i

0

(1 + w(ti ))∆N (ti ), G2 (t) = expt)v(u)dA(u) ,where v(u) = (y 2 /2)[fT (u)]2 − iyfT (u) and w(ti ) = exp(iyfT (ti ) − 1).
Then by the integration-by-parts formula,
(A.8)
=
=

G1 (t)G2 (t) − G1 (0)G2 (0)
∫ t
0
∫ t
0

G1 (u)dG2 (u) +∫ t
0G2 (u)dG1 (t)G1 (u−)G2 (u)v(u)dA(u) +∑G2 (ti )G1 (ti −)w(ti )∆N (ti ) .

i

By using Theorem A.1, it is straightforward to obtain a martingale
convergence result under the same assumptions of Theorem A.1. Thatis, for any ϵ > 0 we have
(A.9)

YT =

1

∫ T

T 1/2+ϵ 0

pgT (x)[dN (x) − dA(x)] −→ 0 .

Thus, we do not need to use the Ergodic Theorem for stationary
stochastic processes, which was one of key arguments on the asymptotic results obtained by Ogata (1978).
It is also straightforward to extend Theorem A.1 to the multivariate
47

cases. Let N = (Ni ) be a p × 1 vector F−adapted simple point processes on R+ and A = (Ak ) are the F−(continuous)compensators. For
any T (> 0) we consider q × p F−adapted and predictable processesgT (t) = (gTij (t)) and a q × q F0 −adapted (positive-definite) randommatrix η = (ηij ), we assume the following conditions.
∫′(i) max1≤i,j≤q max1≤k≤p E[ T1 0T |gTik (t)||gTik (t)|dAk (t)] < ∞ ,′(ii) For any δ (> 0),
1(A.10)Tp

max Ak (T ) −→ 0,
1+δ
1≤k≤p

′

(iii) As T −→ ∞
p
1∫T ∑
p
gTik (t)gtjk (x)dAk (t) −→ ηij ,T 0 k=1(A.11)

where η = (ηij ) is a q × q non-negative definite matrix.
′(iv) For any c > 0, as c → ∞(A.12) max E[
1≤k≤p

1 ∫ T ·,kp
∥gT (t)∥2 I(∥gT· k (t)∥ > c)dAk (t)|F0 ] −→ 0 ,T 0′

where gT·k (t) = (gT1,k , · · · , gTp,k ) .
Then we have the result.
Theorem A.2 : For the proint processes N = (Ni ) and their com′′pensators A = (Ai ) stated, we assume the conditions (i) −(iv) . Thena q × 1 vector process(A.13)p
1 ∫T∑
gT·,k (t)[dNk (t) − dAk (t)]XT = √
T 0 i=1

converges to η 1/2 U in the sense of F0 −(stable convergence sense),
where U is Nq (0, Iq ), which is independent of F0 and we have used48

the notation η 1/2 η 1/2 = η.
(ii) A Wilks Property
We consider the parametric point process models for the case when
the intensity function is λi (s, θ) for the point processes Ni (s, θ) (i =1, · · · , p) over the observation period [0, T ]. We take θ = (θi ) ∈ Rr .
Then the log-likelihood function is given by(A.14)

LT (θ) =p∑

LiT (θ) ,

i=1

where
(A.15)

LiT (θ) =∫ T
0log λi (s, θ)dNi (s) −∫ T
0λi (s, θ)ds ,and its derivatives are given by(A.16)∂LiT (θ) ∫ T log λi (s, θ)=[dNi (s) − λi (s, θ)ds] ,
0
∂θ
∂θ

and
∫ T log λ (s, θ)′∂ 2 LiT (θ) ∫ T 1 ∂ 2 λii=[∂θ ]λi (s, λ)ds .
′′ [dNi (s)−λi (s, θ)ds]−
0 λi ∂θ∂θ
0∂θ∂θ∂θ(A.17)

Theorem A.3 : Let the log-likelihood function be LT (θ), the loglikelihood function under the true parameter vector θ0 be LT (θ0 ), andthe log-likelihood function under the maximum likelihood estimatorθ̂M L be LT (θ̂M L ). Then under the following regularity conditions as
T →∞(A.18)

d2{LT (θ̂M L ) − LT (θ0 )} → χ(r) ,49

where r is the dimension of θ = (θk ) and χ(r) is the χ2 −distribution
with degrees of freedom r. The conditions are
p ∫ T ∂ log λ ∂ log λ
1 ∑
p
i
i
[
]λi (s, θ)ds −→ I(θ0 ) > 0 (a positive definite matrix),′
T i=1 0∂θ∂θ
(A.19)p ∫ T ∂ log λ
1 ∑
i
w
√
(A.20)][dNi (s) − λi (s, θ)ds] −→ Nr (0, I(θ0 )) ,[∂θ
T i=1 0(A.21)
and(A.22)p ∫ T ∂ 2λ
1 ∑
p
i 1
[
[dNi (s) − λi (s, θ)ds] −→ 0 ,′]T i=1 0 ∂θ∂θ λi
p ∫ T ∂ log λ ∂ log λ
1 ∑
p
i
i
[
][dNi (s) − λi (s, θ)ds] −→ 0 ,′T i=1 0∂θ∂θwhere I(θ0 ) is the Fisher information matrix.

As Corollaries of Theorem A.2, it is straight-forward, but lengthy
and standard arguments, to give the formal proofs of Theorem 4.1
and Theorem 4.2 as the non-causality tests we have developed anddiscussed in Section 4.

As the final remark of Appendix we should point out again thatwhile Ogata (1978) has discussed a set of suﬃcient conditions for theconsistency and the asymptotic normality of the ML estimator in onedimensional self-exciting point processes, we have extended his results
significantly to the multivariate point processes under a set of weakerconditions. For instance, I(θ0 ) is not necessarily a constant matrix
and our conditions means the mixed Gaussiann distribution in our50formulation. Then the limiting χ2 property of the statistics is often
called the Wilks Property.

51