ModelingFinancialIntradayJumpTailContagionwithHigh ...consideration on emerging markets. It is well...

Research ArticleModeling Financial Intraday Jump Tail Contagion with HighFrequency Data Using Mutually Exciting Hawkes Process

Chao Yu 1 Jianxin Bi1 and Xujie Zhao2

1School of Statistics University of International Business and Economics Beijing China2School of International Trade and Economics University of International Business and Economics Beijing China

Correspondence should be addressed to Chao Yu chaoyuuibeeducn

Received 17 November 2019 Revised 23 January 2020 Accepted 4 February 2020 Published 20 May 2020

Academic Editor Ricardo Lopez-Ruiz

Copyright copy 2020ChaoYu et alis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Financial extreme jumps in asset price may propagate across stock markets and lead to the market-wide crashes which severelythreatens the stability of the financial system In order to analyzing the contagion features of jump tail risk this paper proposes amutually exciting contagion model based on Hawkes process with intraday high frequency data We use a simple two-stage methodthat first extracts the jump component nonparametrically from the high frequency data and thenmodels the intraday jump tail usingmutually exciting Hawkes process Moreover we take both the occurrence time andmagnitude of jump into account inmodeling theconditional intensity of Hawkes processe proposedmethod is applied to the five-minute high frequency data of the Chinese stockmarket e empirical results show that for the two main Chinese stock markets only background intensity is significant in theShanghai stock market while mutually exciting effect is significant in the Shenzhen stock market Both the location and size of jumpin the Shanghai stock market have significant stimulation to the next occurrences of jump in the Shenzhen stock market Fur-thermore the proposed model performs very well in predicting the future jump tail events

1 Introduction

It is well recognized that the financial asset returns are notnormally distributed but instead exhibit more slowlydecaying and asymmetric tails Numerous studies have shownthat these fatter tails may be attributable to stochastic volatilityandor occasionally large absolute price changes calledldquojumpsrdquo in the underlying asset price process With theavailability of reliable financial high frequency data over thelast two decades many closer research studies on the dy-namics of financial asset prices have documented the presenceof jumps see [1ndash6] While both components can account forthe extreme tail behavior they have different mechanisms andfurther have very different implications on pricing and riskmanagement as explored by Bollerslev and Todorov [7]

In contrast to the numerous studies on tail risk resultingfrom stochastic volatility there is less work to study the jumptail risk However the recent financial crisis has furtherspurred the interest of studying the jump tail events Bol-lerslev et al [8] first used the extreme value theory to study the

tail distribution of jumps and the dependence of jumps withhigh frequency data eir research reveals a strong degree oftail dependence between the market-wide jumps and thesystematic jumps in individual stocks Aıt-Sahalia et al [9]also pointed out that the jump occurred in one market maypropagate over time and spread to other markets as welleyfirst used the Hawkes process to model the self-exciting andmutually exciting features of jumps and then established aclass of jump contagion asset pricemodele proposed jumpcontagion model is then applied to the problems of derivativepricing [10] and portfolio investment [11] In addition manystudies also consider the self-exciting features of jump in theproblems of options pricing [12] and volatility risk premiummodeling [13] However these econometric analysis of jumptransmission is often conducted using low sampling fre-quencies such as daily or lower As many researchers pointedout the real price paths of many financial assets change fast atthe microstructure level especially in periods of financialcrisis and moreover the tail-type jump events tend to occurmore frequently at the intraday frequency rather than daily or

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 7940647 10 pageshttpsdoiorg10115520207940647

lower frequencies [14ndash16] Hence combined with themethods on jumps with high frequency data this paperproposes a simple two-stage method to model the jumpcontagion at the intraday frequency We first use the non-parametric method to identify the intraday jump series withhigh frequency data and then combine the mutually excitingHawkes process and peaks-over-threshold (POT) approach toconstruct a mutually exciting jump tail contagion model Byusing the identified jump series at high frequency we canfocus on studying its self-exciting and mutually excitingfeatures at the intraday frequency

In comparison with the numerous research studies relatedto jump risk on developed financial markets there is limitedconsideration on emerging markets It is well known thatstock returns in emerging markets usually exhibit differentcharacteristics such as higher volatility fatter tails and moresudden shifts erefore jumps in emerging markets may bemore frequent and the ways in which they transmit amongdifferent markets may differ from developed markets Withthe enormous growth in the past decades the Chinese stockmarket has become a more and more influential emergingmarket among the world stock markets Hence the proposedmethod is applied to the Chinese stock market We explorethe contagion mechanism of the jump tail between the twostock markets in Chinarsquos mainland the Shanghai stockmarket and Shenzhen stock market based on the proposedbivariate mutually exciting jump contagion model

Our research makes three differences from the existingliterature First by employing Hawkes process what we reallycare about is not the continuous risk caused by volatility butthe contagion features of jump tails or to say extreme jumprisks Second instead of extending the existing asset pricemodels with mutually exciting jump component we use asimple nonparametric method to separate jump componentfirst and then focus on modeling the jump tail contagionwhich allows us to study the contagion feature of jump at theintraday frequency ird in the modeling of the conditionalintensity of Hawkes process we consider the stimulationeffects of both the jump magnitude and occurrence time

e rest of this paper is organized as follows Section 2presents the extraction method of jump in asset price withhigh frequency data and proposes the mutually excitingjump tail contagion model e maximum likelihood esti-mation and goodness of fit of the model are further dis-cussed Section 3 presents the empirical analysis Section 4concludes the paper

2 Mutually Exciting Jump TailContagion Model

21 Extraction of Jump with High Frequency DataAssume that the efficient logarithmic price p

(j)t of the jth

asset defined on a filtered probability space (Ω F (Ft)tge0P) evolves as

dp(j)t b

(j)t dt + σ(j)

t dWt + dJ(j)t j 1 2 M (1)

where W (Wt) is an F-adapted standard Brownianmotion e drift b(j) (b

(j)t ) and the volatility σ(j) (σ(j)

t )

are progressively measurable processes which guarantee that(1) has a unique strong solution which are adapted andright continuous with left limits (cadlag) processes J(j)

(J(j)t ) is a pure jump processFrom the theory of high frequency data analysis we can

extract the jump component by consistently estimating thequadratic variation (QV) and the integrated volatility (IV) ofprice process Considering that there are T trading days forsimplicity assume that on the time horizon [0 T] theobservations are equally spaced On each trading day thereare totally n discrete observations of log-return Hence therealized volatility (RV) and the bipower variation (BV) onday t are given as follows

RV(j)t 1113944

(tminus1)n+n

i(tminus1)n+1Δn

i p(j)

11138681113868111386811138681113868

111386811138681113868111386811138682

BV(j)t

π2

1113944

(tminus1)n+n

i(tminus1)n+2Δn

i p(j)

11138681113868111386811138681113868

11138681113868111386811138681113868 Δniminus1p

(j)11138681113868111386811138681113868

11138681113868111386811138681113868 t 1 2 T

(2)

where Δni p(j) represents the log-return of the jth asset on the

time interval [(i minus 1)n in] ie

Δni p

(j) p

(j)

(in) minus p(j)

((iminus1)n) (3)

Barndorff-Neilsen and Shephard [1 2] demonstratedthat RV and BV converge to quadratic variation (QV) andintegrated volatility (IV) respectively when the time in-terval is small enough

To eliminate the intraday effects of high frequencyvolatility we adopt theTOD statistics proposed by Bollerslevet al [8]

TOD(j)

i n1113936

Tt1 Δ

nitp(j)

11138681113868111386811138681113868

111386811138681113868111386811138682I Δn

itp(j)

11138681113868111386811138681113868

11138681113868111386811138681113868le τ

BV(j)t and RV

(j)t

1113969

nminusω1113874 1113875

1113936nTs1 Δ

ns p(j)

111386811138681113868111386811138681113868111386811138682I Δn

s p(j)1113868111386811138681113868

1113868111386811138681113868le τ

BV(j)

[sn] and RV(j)

[sn]

1113969

nminusω1113874 1113875

it (t minus 1)n + i i 1 2 n

(4)where I(middot) is an indicator function and τ and ω are constantssatisfying τ gt 0ω isin (0 05) TOD

(j)i actually reflects the

intensity of the intraday effect of the ith interval TOD(j)i

larger than 1 indicates that there is a strong intraday effectand vice versa

Next we can use the threshold method of Mancini [17]combined with the intraday effect of volatility to identify theintervals where jump occurred on each trading day elocations of intervals containing the jump component onday t for asset j denoted by I

(j)t is estimated by

1113954I(j)

t i isin 12 n Δnitp

(j)11138681113868111386811138681113868

11138681113868111386811138681113868gtα(j)it

nminusω

it (t minus1)n + i1113882 1113883

t 12 T

(5)

2 Discrete Dynamics in Nature and Society

where

α(j)it

τ

BV(j)t and RV

(j)t1113872 1113873 times TOD

(j)i

1113969

i 1 2 n

(6)

In the empirical analysis in Section 3 we set τ 25 andω 049en the jump component can be estimated as thereturns on the intervals where jumps occurred because ifthere occurs a jump the jump will dominate the returnHence the identified jump series on day t for asset j can beobtained by

1113954J(j)

itn Δnitp

(j) it (t minus 1)n + i i isin 1113954I

(j)

t1113882 1113883 t 1 2 T

(7)

where 1113954J(j)

itndenotes the estimated jump at occurrence time

itn en itn it (t minus 1)n + i i isin 1113954I(j)

t is the arrival timesof jumps on day t

With the identified jump series on each day we can putthe results of all the trading days over [0 T] together in thechronological order to obtain one series of jump We denoteit by J

(j)

t(j)

k

where t(j)

k is the time of jump and J(j)

t(j)

k

is the jumpsize at time t

(j)

k for asset j In the following we will focus onmodeling the contagion behavior of these jump events basedon the Hawkes process and extreme value theory

22 Bivariate Mutually Exciting Jump Contagion ModelHere we focus on modeling the contagion behavior of theleft jump tail since we usually care about the events thatlead to extreme losses We combine the peaks-over-threshold (POT) approach in the extreme value theory(EVT) and the mutually exciting Hawkes process to con-struct the contagion model e Hawkes process is acounting process that models a sequence of lsquoarrivalsrsquo ofevents over time where each arrival excites the process inthe sense that the chance of a subsequent arrival is in-creased for some period after the initial arrival HenceHawkes process is often used to model the clustering ofevents Ogata [18] first introduced self-exciting Hawkesprocess in studying the earthquake occurrence ere areincreasing applications of Hawkes process in finance forinstance modeling of the risk [19 20] modeling of theduration between trades [21] or the arrival process of buyand sell orders [22] Aıt-Sahalia et al [9] use Hawkesprocess to capture the contagion of jumps in differentregions of the world In their modeling of jump contagionthe factor of interest that affects the conditional intensity ofjump is the occurrence times of earlier events However inthe financial high-frequency context it is natural to also letintensity depend on the magnitudes Hence in this paperwe consider that the conditional intensity could be affectedby both the magnitudes and occurrence times of previousjumps

Consider the bivariate situation Assume that J(j)

t(j)

k

j 1 2 is the identified negative jump series of two assetsfor all the trading days over the fixed time interval [0 T]according to the approach proposed in Section 21 Set u1 gt 0

and u2 gt 0 which are the threshold values of these two jumpseries respectively If |J

(j)

t(j)

k

|ge uj j 1 2 then extremejump occurs Define the excess of extreme jump occurring att(j)

k over the threshold uj as X(j)

k |J(j)

t(j)

k

| minus uj Following thenotation from [19 20] let T

(j)

k1113966 1113967kisinZ denote the series of

occurrence times of extreme jumps and X(j)

k1113966 1113967kisinZ the

corresponding series of magnitudes of excess jumps en(T

(j)

k X(j)

k )kisinZ forms two marked point processes whereT

(j)

k defines the arrival times and X(j)

k the correspondingmarks for the jth asset LetHt be the sigma algebra generatedby the two processes that is the entire history of the marksand their occurrence up to but not including the time tAssume that there are total n(1) and n(2) extreme jumps overinterval [0 T] for two assets Use (t(1)

k x(1)k ) k 1 2 n(1)

to denote the observed sequence of the first marked pointprocess over the period [0 T] and (t(2)

k x(2)k ) k 1 2

n(2) the observed sequence of the second marked pointprocess en according to the theory of marked pointprocess (see [23] for more details) assume that the condi-tional ground intensity for the jth marked point process iethe conditional intensity of the jth marginal point process oflocations of extreme jumps is given as

λj t |Ht( 1113857 λj + 11139442

m11113944

kt(m)

kltt

θjmgjm t minus t(m)k x

(m)k1113872 1113873 j 12

(8)where λj gt 0 and θjm gt 0 λj is the background intensity andgjm(t minus t

(m)k x

(m)k ) is the exciting function which shows how

the jump events before the time t affect the intensity of thejumps occurring at time t Furthermore we assume that

gjm t minus t(m)k x

(m)k1113872 1113873 exp δjmx

(m)k minus ηjm t minus t

(m)k1113872 11138731113872 1113873 (9)

where δjm gt 0 and ηjm gt 0 is exciting function indicatesthat the impact of extreme jump on the intensity of theprocess is driven exponentially by the magnitude of theexcess jump and decreases exponentially in relation to thedistance from the jump event en the conditional groundintensity is given by

λj t Ht

11138681113868111386811138681113872 1113873 λj + θj1 1113944

kt(1)

kltt

exp δj1x(1)k minus ηj1 t minus t

(1)k1113872 11138731113960 1113961

+ θj2 1113944

kt(2)

kltt


(2)k1113872 11138731113960 1113961

(10)

In this model we believe that the conditional intensity ofextreme jumprsquos arrival for asset j can be decomposed intotwo parts the background intensity part λj and the responsepart which is triggered by the self-exciting and cross-excitingeffects of previous extreme jumps from the asset itself andthe other asset respectively Specifically if an extreme jumpoccurs in the first asset its own intensity of extreme jumpswould increase immediately by θ11exp(δ11x

(1)k ) and then

decay exponentially with the parameter η11 And the in-tensity of extreme jumps in the second asset would increase

Discrete Dynamics in Nature and Society 3

instantaneously by θ21exp(δ21x(1)k ) and then decay expo-

nentially with the parameter η21 Likewise if one extremejump occurs in the second asset its own intensity of extremejumps would rise by θ22exp(δ22x

(2)k ) and decay exponen-

tially with the parameter η22 while the intensity in the firstasset would increase by θ12exp(δ12x

(2)k ) and then decay

exponentially with the parameter η12 erefore we can

analyze the contagion behavior of the extreme jump risksbased on this bivariate mutually exciting Hawkes process

23 Maximum Likelihood Estimation Assuming that themark X

(j)

k1113966 1113967 is independent of time T(j)

k1113966 1113967 and the past of themarked point process the log-likelihood function of thebivariate marked point process (T

(j)

k X(j)

k ) is given by

log L 11139442

j11113944

kt(j)

kleT1113864 1113865

log λj t(j)

k1113872 1113873 minus 1113946T

0λj(t)dt + 1113944

kt(j)

kleT1113864 1113865

logfj x(j)

k1113872 1113873⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠ (11)

See the details in the study by Embrechts et al [24] andChavez-Demoulin et al [20] Let

log L1 1113944

kt(1)

kleT

log λ1 t(1)k1113872 1113873 minus 1113946

T

0λ1(t)dt

log L2 1113944

kt(2)

kleT

log λ2 t(2)k1113872 1113873 minus 1113946

T

0λ2(t)dt

log L3 1113944

kt(1)

kleT

log f1 x(1)k1113872 1113873

log L4 1113944

kt(2)

kleT

log f2 x(2)k1113872 1113873

(12)

where for j 1 2

1113946T

0λj(t)dt λjT + θj1 1113944

kt(1)

kleT

1113946T

t(1)

k


(1)k1113872 11138731113872 1113873dt

+ θj2 1113944

kt(2)

kleT

1113946T

t(2)

k


(2)k1113872 11138731113872 1113873dt

λjT +θj1

ηj11113944

kt(1)

kleT

exp δj1x(1)k1113872 1113873

times 1 minus exp minusηj1 T minus t(1)k1113872 11138731113872 11138731113960 1113961

+θj2

ηj21113944

kt(2)

kleT

exp δj2x(2)k1113872 1113873


(13)

fj(x) is the probability density of X(j)

k According to theextreme value theory if the threshold uj is large enough thedistribution of X

(j)

k can be approximated by generalizedPareto distribution (GPD) Hence we can set

fj x ξj βj1113872 1113873

1βj

1 + ξj

x

βj

1113888 1113889

minus 1ξjminus 1

if ξj ne 0

1βj

exp minusx

βj

1113888 1113889 if ξj 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)where βj gt 0 and ξj are the scale and shape parameters of thegeneralized Pareto distribution respectively

en the total log-likelihood function of the model canbe written as a summation of logL1 logL2 logL3 and logL4at is

log L log L1 + log L2 + log L3 + log L4 (15)

Hence we can get the maximum likelihood estimation ofthe parameters for generalized Pareto distribution and theconditional ground intensity of Hawkes process separatelyIn particular log L1 and log L2 can be used to estimate theparameters of the intensity and log L3 and log L4 can beused to estimate the parameters of the generalized Paretodistribution

24 Goodness of Fit With the observations (t(j)

k x(j)

k )k isin 1 2 n(j)1113864 1113865 j 1 2 of the two marked point pro-cesses define

t(j)lowastk 1113946

t(j)

k

0λj(s)ds (16)

Based on model (10) formula (16) can be given as

t(j)lowastk λjt

(j)

k +θj1

ηj11113944

mt(1)m let

(j)

k1113864 1113865

exp δj1x(1)m1113872 1113873

times 1 minus exp minusηj1 t(j)

k minus t(1)m1113872 11138731113872 11138731113960 1113961

+θj2

ηj21113944

mt(2)m let

(j)

k1113864 1113865

exp δj2x(2)m1113872 1113873


k minus t(2)m1113872 11138731113872 11138731113960 1113961

(17)


According to the residual analysis of point process thesequence t

(j)lowast1 t

(j)lowast2 t

(j)lowastn(j)1113966 1113967 forms a Poisson process with

unit rate us the interarrival times τ1τ21113864

τn(j)minus1 t(j)lowast2 minust

(j)lowast1 t

(j)lowast3 minust

(j)lowast2 t

(j)lowastn(j) minust

(j)lowastn(j)minus11113966 1113967 should

satisfy τi simiidexp(1) erefore it is feasible to use QQ plotusing the exponential distribution to see how well the Hawkesprocess fits the observations Alternatively we can use theKolmogorovndashSmirnov test to test whether τi follows expo-nential distribution

3 Empirical Analysis

31 Estimation of Mutually Exciting Jump Tail ContagionModel We collect the tick-by-tick transaction data of theShanghai (SH) composite index in the Shanghai stockmarket and Shenzhen (SZ) component index in Shenzhenstock market from the Chinese RESSET high-frequencydatabasee sample starts on January 4 2006 and ends onDecember 31 2013 ere are totally 1932 valid tradingdays after excluding the holidays and the trading days withsuccessive missing values over intervals with length of tenminutes or above en we sample the tick-by-tick data atfive-minute frequency by assigning the last observation ineach five-minute interval as the price at five-minute fre-quency Finally we calculate the five-minute log-returnswith five-minute prices for these two indexese followinganalysis is carried out by using these five-minute returndata

Firstly we extract the jump components from the five-minute returns by using the threshold method proposed inSection 21 Also we are interested in the negative jumpsen we use the mean excess function to find an appro-priate threshold which can make the jump tails subject tothe generalized Pareto distribution Figure 1 shows themean excess functions of the SH composite index and SZcomponent index respectively From the figure we canobtain that both mean excess functions tend to be positivelylinear when the threshold is over 03 erefore we select03 as the threshold with which we can identify the extremejumps and obtain the excesses of extreme jumps for the twoindexes

Table 1 reports some summary statistics about the ex-treme jumps over threshold identified in the SH compositeindex and SZ component index respectively

With the identified extreme jump times and their marksie magnitudes of the excesses of extreme jumps for thesetwo indexes we next estimate the bivariate mutually excitingjump contagion model by the maximum likelihood esti-mation method Table 2 reports the estimation results of twogeneralized Pareto distributions

Figure 2 gives the QQ-plots of two fitted generalizedPareto distributions for the SH composite index and SZcomponent index respectively which show that the negativejump tails of both indexes are well fitted by the generalizedPareto distribution

Table 3 reports the estimation results of parameters inthe conditional ground intensity of Hawkes process

Moreover we use the bootstrap method to obtain thestandard errors of parameter estimation by Monte Carlosimulation In particular we first simulate 1000 paths of thebivariate Hawkes process with the estimated values ofparameters and then run the parameter estimation pro-cedure with these 1000 bootstrap samples to get 1000 es-timates of the parameters en we can compute thestandard error of each parameter with these 1000 estimatedvalues From the results of intensity process in the Shanghaistock market we can see that only background intensity issignificant which shows that there are no significant self-exciting and mutually exciting effects in the Shanghai stockmarket

However from the results of the Shenzhen stock marketwe can see that λ2 δ21 and η21 are significant at 5 level andθ21 is significant at 10 level which shows that the jumpintensity in the Shenzhen stock market are mainly affectedby the background factors and jumps occurred in theShanghai stock market as well Both the locations and sizesof jumps occurred in the Shanghai stock market have sig-nificant stimulations to the extreme jumps in the Shenzhenstock market ese results can be attributed to the differentcharacteristics of the two stock markets e Shanghai stockmarket mainly consists of large-cap stocks such as financialstocks or blue chips and the companies listed in Shanghaistock exchange are often the leading enterprises in their ownindustry while the main components of the Shenzhen stockmarket are medium and small-cap stocks and growth en-terprises market board erefore the performance ofShenzhen component index tends to follow the changes inthe Shanghai composite index

With the estimation results the simulated paths of theconditional ground intensities for the two indexes are givenin Figure 3 e figure shows that the intensity of the SZcomponent index is more volatile than the SH compositeindex since there exists cross-exciting effect in the Shenzhenstock market apart from the influence of common factorsFigure 4 presents the QQ-plots of two fitted Hawkes processfor the SH composite index and SZ component index re-spectively From the figure it can be easily seen that thesample quantiles and the theoretical quantiles of exponentialdistribution almost lie in a line which indicates that theobservations are well fitted by the model

32 Prediction Here we consider the problem of predictingthe future jump events out of sample Since our model isbased on Hawkes point process it is impossible to use it topredict the exact locations of future jumps However similarto the forecasting of earthquake we can forecast theprobability of the future jumprsquos arrivals within a time periodSuch kind of forecasting can be implemented by simulatingthe bivariate Hawkes process with the estimated intensityfunction repeatedly and then calculate the frequency of thearrivals within a given time interval en we can calculatethe frequencies of the arrivals within different time periodsand finally get the empirical distribution of times for futurejump arrivals For instance we can obtain the empiricaldistribution of days within which the next jump occurs or


the next two jumps and other situations occur Hence withthe estimated results in Section 31 we simulate 2000 pathsof our model over the next month after December 31 2013

ie January in 2014 We choose the following one month asthe time interval of the simulation and prediction becausethis time period is long enough for the future occurrences of

reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)

Figure 1 Mean excess functions for the (a) SH composite index and (b) SZ component index

Table 1 Summary statistics of extreme jumps in the SH composite index and SZ component index

Count Mean Std dev Minimum MaximumSH 946 08006 05947 03008 62527SZ 948 08595 05801 03010 46786

Table 2 Estimation results for generalized Pareto distribution

Scale β Shape ξSH 04257lowastlowast (00210) 01496lowastlowast (00374)SZ 05434lowastlowast (00253) 00286 (00334)Note e values in parenthesis are the results of standard errors lowastlowast e parameter is significant at 5 level

Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)

Figure 2 QQ-plots of generalized Pareto distribution fitting for the (a) SH composite index and (b) SZ component index


jump events which makes the prediction procedure de-scribed above and the checking of its performance feasibleTable 4 reports the frequencies of the first extreme jump

occurring on the following days for the SH composite indexand SZ component index based on simulation Figure 5shows the histograms

Table 3 Estimation results for ground intensity process

Parameter Value Std err Parameter Value Std errλ1 00102lowastlowast 00002 λ2 00101lowastlowast 00002θ11 00003 00012 θ21 00025lowast 00015δ11 01281lowastlowast 00387 δ21 00962lowastlowast 00367η11 03631lowastlowast 00635 η21 04728lowastlowast 00541θ12 48eminus 7 00015 θ22 25eminus 10 00012δ12 00579 00435 δ22 02697lowastlowast 00544η12 06962lowastlowast 00496 η22 05179lowastlowast 00567Note e results of standard errors are obtained by 1000 bootstrap simulations lowast and lowastlowast denote that the parameters are significant at 10 and 5 levelrespectively

Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)

Figure 3 Paths of ground intensity processes of the (a) SH composite index and (b) SZ component index

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)

Figure 4 QQ-plots of Hawkes process fitting for the (a) SH composite index and (b) SZ component index


From Table 4 we can obtain that for the SH compositeindex the frequency that the next negative extreme jumpoccurs within one day after December 31 2013 is 03790within two days 06195 (sum of the frequencies of the firsttwo days) and within three days 07545 (sum of thefrequencies of the first three days) As for the SZ com-ponent index the corresponding frequencies are 0374006060 and 07550 respectively In summary our pre-diction results show that in both stock markets the fre-quency that the next negative extreme jump occurs withinthree days is above 075 which implies that the next jumpoccurs within following three days with a relatively largeprobability

To check the performance of our prediction we use theintraday five-minute high frequency data from January 22014 to January 31 2014 and separate the negative extremejumps for the two indexes to make a comparison with ourprediction results We find that in the Shanghai stockmarket the first jump occurs at 10 10 am on January 2 andin the Shenzhen stock market the first jump occurs at 09 35am on January 3 Overall the arrivals of the first jump areobserved within two days for the two stock markets whichverifies our above prediction

In order to make the idea of the prediction more clearwe proceed to report the frequency tables and histograms ofthe second extreme jump occurrence on following days forthe SH composite index and SZ component index in Table 5and Figure 6 respectively

From the results in Table 5 we can obtain that thefrequencies that first two jumps occur in the SH compositeindex within 3 days and within 6 days are 04415 and 0791respectively And the frequencies that first two jumps occurin the SZ component index within 3 days and within 6 daysare 0444 and 0798 respectively From the results ofidentified jumps with real data we find that the second jumpoccurs at 9 35 am on Januray 3 and 10 00 am on January3 respectively for the SH composite index and SZ com-ponent index which means that the arrivals of the next twojumps are observed within two days for the two indexes eresults are compatible with our prediction

Next we adopt the procedure described above repeatedlyand use the rolling window method to make prediction overthe months from February to December in 2014 e per-formances of total 11 predictions with rolling window arereported in Table 6 for the SH composite index and inTable 7 for the SZ component index In both tables wereport the frequencies of the first jump occurring within 3days and 7 days over the forecasting interval and theidentified dates and times of the first extreme jump oc-currence by using the actual data for comparison FromTables 6 and 7 we can see that for all the rolling windowpredictions the frequencies of the first extreme jump oc-currence in 3 days are around 75 a relatively largeprobability and the frequencies in 7 days are around 97 avery high probability Furthermore the jump arrivals withrelatively large probability in every rolling window

Table 4 Frequencies of the first extreme jump occurrence on following days

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Above 7 daysSH 03790 02405 01350 00865 00555 00365 00300 00370SZ 03740 02320 01490 00935 00605 00410 00180 00320Note e values in the table are the frequencies of the fist extreme jump occurring on the following days For instance for the SH composite index 03790 isthe frequency of the first extreme jump occurring on the first day and 02405 is the frequency of the first extreme jump occurring on the second day over theone-month prediction window

Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)

Figure 5 Histogram of the next jump occurrence


prediction are almost observed In particular the next jumpoccurs within 3 days over prediction window for all thepredictions except that in the sixth rolling prediction thenext jump for the SZ component index occurs in next 7 daysin the seventh rolling prediction the next jump for the SHcomposite index occurs in next 4 days and in the eighthrolling prediction the next jumps for both indexes occurs innext 6 days e results further demonstrate our modelrsquospower

4 Conclusions

In this paper we focus on modeling the contagion feature ofthe intraday jump tail with financial high frequency data Weuse a two-stage method that first extracts the intraday jumpnonparametrically with high frequency data and then con-structs the mutually exciting jump tail contagion model basedon Hawkes process and peaks-over-threshold approach In themodeling of the conditional intensity of the Hawkes process

Table 6 Performances of the rolling window prediction of the SH composite index

Prediction no 1 2 3 4 5 6 7 8 9 10 11Dates Feb 7 Mar 4 Apr 3 May 7 Jun 5 Jul 3 Aug 6 Sep 9 Oct 9 Nov 3 Dec 3Times 09 35 09 35 14 00 09 35 09 35 09 35 09 35 09 50 11 00 14 05 13 20In 3 days 07630 07665 07715 07605 07855 07440 07635 07640 07415 07605 07575In 7 days 09700 09730 09710 09645 09705 09620 09715 09640 09575 09645 09625Notee rows named ldquoDatesrdquo and ldquoTimesrdquo in the table are the identified dates and times of the first jump on each prediction window and the rows named ldquoIn3 daysrdquo and ldquoIn 7 daysrdquo are the results of the cumulative frequencies of the first jump occurring within 3 days and 7 days over each prediction window

Table 7 Performances of the rolling window prediction of the SZ component index

Prediction no 1 2 3 4 5 6 7 8 9 10 11Dates Feb 7 Mar 4 Apr 3 May 5 Jun 4 Jul 9 Aug 5 Sep 9 Oct 9 Nov 3 Dec 3Times 09 35 09 35 14 00 09 50 09 35 14 05 10 20 09 50 11 00 14 05 13 20In 3 days 07630 07665 07550 07655 07500 07525 07635 07565 07675 07460 07840In 7 days 09650 09660 09615 09630 09620 09590 09635 09600 09715 09595 09675Note e meaning of the results in the table is the same as the ones in Table 6

Table 5 Frequencies of the second extreme jump occurrence on following days

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Above 9 daysSH 00865 01775 01775 01420 01245 00830 00535 00520 00365 00670SZ 00990 01635 01815 01520 01120 00900 00620 00465 00245 00690Note e values in the table are frequencies of the second extreme jump occurring on the following days For instance for the SH composite index 00865 isthe frequency of the second extreme jump occurring on the first day and 01775 is the frequency of the second extreme jump occurring on the second day overthe one-month prediction window

Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)

Figure 6 Histogram of the second jump occurrence


we take into account the effects of both occurrence time andexcess jump magnitude We then discuss the maximumlikelihood estimation and the goodness of fit of the modelFinally we apply the proposed method to the real high fre-quency data in the Chinese stock market We first separate thenegative jump series from the five-minute high frequency dataof the Shanghai composite index and Shenzhen componentindex which are the two main market indexes in China andthen apply the proposed bivariate mutually exciting jumpcontagion model to the identified jump series e empiricalresults show that mutually exciting feature is significant in theShenzhen stock market while both self-exciting and mutuallyexciting features are not significant in the Shanghai stockmarket In particular the intensity of extreme jump occurrencein the Shanghai stock market is only significantly affected bythe background factors while the intensity in the Shenzhenstock market is significantly affected by the background factorsand the extreme jump events in the Shanghai stock market aswell Both the occurrence times and magnitudes of jumpsoccurred in the Shanghai stock market have significantstimulations to the extreme jumps in the Shenzhen stockmarket Furthermore the prediction results of the future jumpevents based on simulation demonstrate our modelrsquos power

Data Availability

e data that support the findings of this study are availablefrom the corresponding author upon request

Conflicts of Interest

Jianxin Birsquos current address is as follows School of Eco-nomics Xiamen University Xiamen Fujian PRChina eauthors declare that they have no conflicts of interest

Acknowledgments

is research was supported in part by the National NaturalScience Foundation of China (NSFC) (71601048 and 11501104)National Social Science Found of China (NSSFC) (17CJY052)and the Fundamental Research Funds for the Central Uni-versities in UIBE (CXTD9-07 CXTD10-10 and 13QD09)

References

[1] O E Barndorff-Nielsen and N Shephard ldquoPower andbipower variation with stochastic volatility and jumpsrdquoJournal of Financial Econometrics vol 2 no 1 pp 1ndash37 2004

[2] O E Barndorff-Nielsen and N Shephard ldquoEconometrics oftesting for jumps in financial economics using bipowervariationrdquo Journal of Financial Econometrics vol 4 no 1pp 1ndash30 2006

[3] X Huang and G Tauchen ldquoe relative contribution of jumpsto total price variancerdquo Journal of Financial Econometricsvol 3 no 4 pp 456ndash499 2005

[4] Y Aıt-Sahalia and J Jacod ldquoTesting for jumps in a discretelyobserved processrdquo lte Annals of Statistics vol 37 no 1pp 184ndash222 2009

[5] S S Lee and J Hannig ldquoDetecting jumps from Levy jumpdiffusion processesrdquo Journal of Financial Economics vol 96no 2 pp 271ndash290 2010

[6] S S Lee and P A Mykland ldquoJumps in equilibrium prices andmarket microstructure noiserdquo Journal of Econometricsvol 168 no 2 pp 396ndash406 2012

[7] T Bollerslev and V Todorov ldquoTails fears and risk premiardquolte Journal of Finance vol 66 no 6 pp 2165ndash2211 2011

[8] T Bollerslev V Todorov and S Z Li ldquoJump tails extremedependencies and the distribution of stock returnsrdquo Journalof Econometrics vol 172 no 2 pp 307ndash324 2013

[9] Y Aıt-Sahalia J Cacho-Diaz and R J A Laeven ldquoModelingfinancial contagion using mutually exciting jump processesrdquoJournal of Financial Economics vol 117 no 3 pp 585ndash6062015

[10] Y Aıt-Sahalia R J A Laeven and L Pelizzon ldquoMutualexcitation in eurozone sovereign CDSrdquo Journal of Econo-metrics vol 183 no 2 pp 151ndash167 2014

[11] Y Aıt-Sahalia and T R Hurd ldquoPortfolio choice in marketswith contagionrdquo Journal of Financial Econometrics vol 14no 1 pp 1ndash28 2015

[12] A Fulop J Li and J Yu ldquoSelf-exciting jumps learning andasset pricing implicationsrdquo Review of Financial Studiesvol 28 no 3 pp 876ndash912 2015

[13] K Chen and S H Poon ldquoVariance swap premium understochastic volatility and self-exciting jumpsrdquo e Universityof Manchester Manchester England Manchester BusinessSchool Working Paper No 634 2013

[14] Y Aıt-Sahalia and J Jacod ldquoAnalyzing the spectrum of assetreturns jump and volatility components in high frequencydatardquo Journal of Economic Literature vol 50 no 4pp 1007ndash1050 2012

[15] Y Aıt-Sahalia and D Xiu ldquoIncreased correlation among assetclasses are volatility or jumps to blame or bothrdquo Journal ofEconometrics vol 194 no 2 pp 205ndash219 2016

[16] M Dungey D Erdemlioglu M Matei and X Yang ldquoTestingfor mutually exciting jumps and financial flights in highfrequency datardquo Journal of Econometrics vol 202 no 1pp 18ndash44 2018

[17] C Mancini ldquoNon-parametric threshold estimation formodels with stochastic diffusion coefficient and jumpsrdquoScandinavian Journal of Statistics vol 36 no 2 pp 270ndash2962009

[18] Y Ogata ldquoStatistical models for earthquake occurrences andresidual analysis for point processesrdquo Journal of the AmericanStatistical Association vol 83 no 401 pp 9ndash27 1988

[19] V Chavez-Demoulin A C Davison and A J McNeilldquoEstimating value-at-risk a point process approachrdquo Quan-titative Finance vol 5 no 2 pp 227ndash234 2005

[20] V Chavez-Demoulin and J A McGill ldquoHigh-frequency fi-nancial data modeling using Hawkes processesrdquo Journal ofBanking amp Finance vol 36 no 12 pp 3415ndash3426 2012

[21] L Bauwens and N Hautsch Handbook of Financial TimeSeries Modelling Financial High Frequency Data Using PointProcesses Springer Berlin Germany 2009

[22] E Bacry S Delattre M Hoffmann and J F Muzy ldquoMod-elling microstructure noise with mutually exciting pointprocessesrdquo Quantitative Finance vol 13 no 1 pp 65ndash772013

[23] D Daley and D Vere-Jones An Introduction to the lteory ofPoint Processes Volume I Elementary lteory and MethodsSpringer New York NY USA 2nd edition 2003

[24] P Embrechts T Liniger and L Lin ldquoMultivariate Hawkesprocesses an application to financial datardquo Journal of AppliedProbability vol 48 no A pp 367ndash378 2011


lower frequencies [14ndash16] Hence combined with themethods on jumps with high frequency data this paperproposes a simple two-stage method to model the jumpcontagion at the intraday frequency We first use the non-parametric method to identify the intraday jump series withhigh frequency data and then combine the mutually excitingHawkes process and peaks-over-threshold (POT) approach toconstruct a mutually exciting jump tail contagion model Byusing the identified jump series at high frequency we canfocus on studying its self-exciting and mutually excitingfeatures at the intraday frequency

In comparison with the numerous research studies relatedto jump risk on developed financial markets there is limitedconsideration on emerging markets It is well known thatstock returns in emerging markets usually exhibit differentcharacteristics such as higher volatility fatter tails and moresudden shifts erefore jumps in emerging markets may bemore frequent and the ways in which they transmit amongdifferent markets may differ from developed markets Withthe enormous growth in the past decades the Chinese stockmarket has become a more and more influential emergingmarket among the world stock markets Hence the proposedmethod is applied to the Chinese stock market We explorethe contagion mechanism of the jump tail between the twostock markets in Chinarsquos mainland the Shanghai stockmarket and Shenzhen stock market based on the proposedbivariate mutually exciting jump contagion model

Our research makes three differences from the existingliterature First by employing Hawkes process what we reallycare about is not the continuous risk caused by volatility butthe contagion features of jump tails or to say extreme jumprisks Second instead of extending the existing asset pricemodels with mutually exciting jump component we use asimple nonparametric method to separate jump componentfirst and then focus on modeling the jump tail contagionwhich allows us to study the contagion feature of jump at theintraday frequency ird in the modeling of the conditionalintensity of Hawkes process we consider the stimulationeffects of both the jump magnitude and occurrence time

e rest of this paper is organized as follows Section 2presents the extraction method of jump in asset price withhigh frequency data and proposes the mutually excitingjump tail contagion model e maximum likelihood esti-mation and goodness of fit of the model are further dis-cussed Section 3 presents the empirical analysis Section 4concludes the paper

2 Mutually Exciting Jump TailContagion Model

21 Extraction of Jump with High Frequency DataAssume that the efficient logarithmic price p

(j)t of the jth

asset defined on a filtered probability space (Ω F (Ft)tge0P) evolves as

dp(j)t b

(j)t dt + σ(j)

t dWt + dJ(j)t j 1 2 M (1)

where W (Wt) is an F-adapted standard Brownianmotion e drift b(j) (b

(j)t ) and the volatility σ(j) (σ(j)

t )

are progressively measurable processes which guarantee that(1) has a unique strong solution which are adapted andright continuous with left limits (cadlag) processes J(j)

(J(j)t ) is a pure jump processFrom the theory of high frequency data analysis we can

extract the jump component by consistently estimating thequadratic variation (QV) and the integrated volatility (IV) ofprice process Considering that there are T trading days forsimplicity assume that on the time horizon [0 T] theobservations are equally spaced On each trading day thereare totally n discrete observations of log-return Hence therealized volatility (RV) and the bipower variation (BV) onday t are given as follows

RV(j)t 1113944

(tminus1)n+n

i(tminus1)n+1Δn

i p(j)

11138681113868111386811138681113868

111386811138681113868111386811138682

BV(j)t

π2

1113944

(tminus1)n+n

i(tminus1)n+2Δn

i p(j)

11138681113868111386811138681113868

11138681113868111386811138681113868 Δniminus1p

(j)11138681113868111386811138681113868

11138681113868111386811138681113868 t 1 2 T

(2)

where Δni p(j) represents the log-return of the jth asset on the

time interval [(i minus 1)n in] ie

Δni p

(j) p

(j)

(in) minus p(j)

((iminus1)n) (3)

Barndorff-Neilsen and Shephard [1 2] demonstratedthat RV and BV converge to quadratic variation (QV) andintegrated volatility (IV) respectively when the time in-terval is small enough

To eliminate the intraday effects of high frequencyvolatility we adopt theTOD statistics proposed by Bollerslevet al [8]

TOD(j)

i n1113936

Tt1 Δ

nitp(j)

11138681113868111386811138681113868

111386811138681113868111386811138682I Δn

itp(j)

11138681113868111386811138681113868

11138681113868111386811138681113868le τ

BV(j)t and RV

(j)t

1113969

nminusω1113874 1113875

1113936nTs1 Δ

ns p(j)

111386811138681113868111386811138681113868111386811138682I Δn

s p(j)1113868111386811138681113868

1113868111386811138681113868le τ

BV(j)

[sn] and RV(j)

[sn]

1113969

nminusω1113874 1113875

it (t minus 1)n + i i 1 2 n

(4)where I(middot) is an indicator function and τ and ω are constantssatisfying τ gt 0ω isin (0 05) TOD

(j)i actually reflects the

intensity of the intraday effect of the ith interval TOD(j)i

larger than 1 indicates that there is a strong intraday effectand vice versa

Next we can use the threshold method of Mancini [17]combined with the intraday effect of volatility to identify theintervals where jump occurred on each trading day elocations of intervals containing the jump component onday t for asset j denoted by I

(j)t is estimated by

1113954I(j)

t i isin 12 n Δnitp

(j)11138681113868111386811138681113868

11138681113868111386811138681113868gtα(j)it

nminusω

it (t minus1)n + i1113882 1113883

t 12 T

(5)


where

α(j)it

τ

BV(j)t and RV

(j)t1113872 1113873 times TOD

(j)i

1113969

i 1 2 n

(6)


1113954J(j)

itn Δnitp


(j)

t1113882 1113883 t 1 2 T

(7)

where 1113954J(j)





(j)

t(j)

k

where t(j)


t(j)

k


(j)




t(j)

k



(j)

t(j)

k



k |J(j)

t(j)

k


(j)



k1113966 1113967kisinZ the


(j)

k X(j)


(j)



k x(1)k ) k 1 2 n(1)


k x(2)k ) k 1 2


λj t |Ht( 1113857 λj + 11139442

m11113944

kt(m)

kltt


(m)k1113872 1113873 j 12


(m)k x



gjm t minus t(m)k x

(m)k1113872 1113873 exp δjmx


(m)k1113872 11138731113872 1113873 (9)


λj t Ht

11138681113868111386811138681113872 1113873 λj + θj1 1113944

kt(1)

kltt


(1)k1113872 11138731113960 1113961

+ θj2 1113944

kt(2)

kltt


(2)k1113872 11138731113960 1113961

(10)


(1)k ) and then











(j)



(j)

k X(j)

k ) is given by

log L 11139442

j11113944

kt(j)

kleT1113864 1113865

log λj t(j)

k1113872 1113873 minus 1113946T

0λj(t)dt + 1113944

kt(j)

kleT1113864 1113865

logfj x(j)

k1113872 1113873⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠ (11)


log L1 1113944

kt(1)

kleT

log λ1 t(1)k1113872 1113873 minus 1113946

T

0λ1(t)dt

log L2 1113944

kt(2)

kleT

log λ2 t(2)k1113872 1113873 minus 1113946

T

0λ2(t)dt

log L3 1113944

kt(1)

kleT

log f1 x(1)k1113872 1113873

log L4 1113944

kt(2)

kleT

log f2 x(2)k1113872 1113873

(12)

where for j 1 2

1113946T

0λj(t)dt λjT + θj1 1113944

kt(1)

kleT

1113946T

t(1)

k


(1)k1113872 11138731113872 1113873dt

+ θj2 1113944

kt(2)

kleT

1113946T

t(2)

k


(2)k1113872 11138731113872 1113873dt

λjT +θj1

ηj11113944

kt(1)

kleT

exp δj1x(1)k1113872 1113873


+θj2

ηj21113944

kt(2)

kleT

exp δj2x(2)k1113872 1113873


(13)



(j)


fj x ξj βj1113872 1113873

1βj

1 + ξj

x

βj

1113888 1113889

minus 1ξjminus 1

if ξj ne 0

1βj

exp minusx

βj

1113888 1113889 if ξj 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩






k x(j)


t(j)lowastk 1113946

t(j)

k

0λj(s)ds (16)


t(j)lowastk λjt

(j)

k +θj1

ηj11113944

mt(1)m let

(j)

k1113864 1113865

exp δj1x(1)m1113872 1113873


k minus t(1)m1113872 11138731113872 11138731113960 1113961

+θj2

ηj21113944

mt(2)m let

(j)

k1113864 1113865

exp δj2x(2)m1113872 1113873


k minus t(2)m1113872 11138731113872 11138731113960 1113961

(17)



(j)lowast1 t

(j)lowast2 t




(j)lowast1 t

(j)lowast3 minust

(j)lowast2 t


















reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)






Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)







Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References


























where

α(j)it

τ

BV(j)t and RV

(j)t1113872 1113873 times TOD

(j)i

1113969

i 1 2 n

(6)


1113954J(j)

itn Δnitp


(j)

t1113882 1113883 t 1 2 T

(7)

where 1113954J(j)





(j)

t(j)

k

where t(j)


t(j)

k


(j)




t(j)

k



(j)

t(j)

k



k |J(j)

t(j)

k


(j)



k1113966 1113967kisinZ the


(j)

k X(j)


(j)



k x(1)k ) k 1 2 n(1)


k x(2)k ) k 1 2


λj t |Ht( 1113857 λj + 11139442

m11113944

kt(m)

kltt


(m)k1113872 1113873 j 12


(m)k x



gjm t minus t(m)k x

(m)k1113872 1113873 exp δjmx


(m)k1113872 11138731113872 1113873 (9)


λj t Ht

11138681113868111386811138681113872 1113873 λj + θj1 1113944

kt(1)

kltt


(1)k1113872 11138731113960 1113961

+ θj2 1113944

kt(2)

kltt


(2)k1113872 11138731113960 1113961

(10)


(1)k ) and then











(j)



(j)

k X(j)

k ) is given by

log L 11139442

j11113944

kt(j)

kleT1113864 1113865

log λj t(j)

k1113872 1113873 minus 1113946T

0λj(t)dt + 1113944

kt(j)

kleT1113864 1113865

logfj x(j)

k1113872 1113873⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠ (11)


log L1 1113944

kt(1)

kleT

log λ1 t(1)k1113872 1113873 minus 1113946

T

0λ1(t)dt

log L2 1113944

kt(2)

kleT

log λ2 t(2)k1113872 1113873 minus 1113946

T

0λ2(t)dt

log L3 1113944

kt(1)

kleT

log f1 x(1)k1113872 1113873

log L4 1113944

kt(2)

kleT

log f2 x(2)k1113872 1113873

(12)

where for j 1 2

1113946T

0λj(t)dt λjT + θj1 1113944

kt(1)

kleT

1113946T

t(1)

k


(1)k1113872 11138731113872 1113873dt

+ θj2 1113944

kt(2)

kleT

1113946T

t(2)

k


(2)k1113872 11138731113872 1113873dt

λjT +θj1

ηj11113944

kt(1)

kleT

exp δj1x(1)k1113872 1113873


+θj2

ηj21113944

kt(2)

kleT

exp δj2x(2)k1113872 1113873


(13)



(j)


fj x ξj βj1113872 1113873

1βj

1 + ξj

x

βj

1113888 1113889

minus 1ξjminus 1

if ξj ne 0

1βj

exp minusx

βj

1113888 1113889 if ξj 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩






k x(j)


t(j)lowastk 1113946

t(j)

k

0λj(s)ds (16)


t(j)lowastk λjt

(j)

k +θj1

ηj11113944

mt(1)m let

(j)

k1113864 1113865

exp δj1x(1)m1113872 1113873


k minus t(1)m1113872 11138731113872 11138731113960 1113961

+θj2

ηj21113944

mt(2)m let

(j)

k1113864 1113865

exp δj2x(2)m1113872 1113873


k minus t(2)m1113872 11138731113872 11138731113960 1113961

(17)



(j)lowast1 t

(j)lowast2 t




(j)lowast1 t

(j)lowast3 minust

(j)lowast2 t


















reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)






Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)







Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References


































(j)



(j)

k X(j)

k ) is given by

log L 11139442

j11113944

kt(j)

kleT1113864 1113865

log λj t(j)

k1113872 1113873 minus 1113946T

0λj(t)dt + 1113944

kt(j)

kleT1113864 1113865

logfj x(j)

k1113872 1113873⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠ (11)


log L1 1113944

kt(1)

kleT

log λ1 t(1)k1113872 1113873 minus 1113946

T

0λ1(t)dt

log L2 1113944

kt(2)

kleT

log λ2 t(2)k1113872 1113873 minus 1113946

T

0λ2(t)dt

log L3 1113944

kt(1)

kleT

log f1 x(1)k1113872 1113873

log L4 1113944

kt(2)

kleT

log f2 x(2)k1113872 1113873

(12)

where for j 1 2

1113946T

0λj(t)dt λjT + θj1 1113944

kt(1)

kleT

1113946T

t(1)

k


(1)k1113872 11138731113872 1113873dt

+ θj2 1113944

kt(2)

kleT

1113946T

t(2)

k


(2)k1113872 11138731113872 1113873dt

λjT +θj1

ηj11113944

kt(1)

kleT

exp δj1x(1)k1113872 1113873


+θj2

ηj21113944

kt(2)

kleT

exp δj2x(2)k1113872 1113873


(13)



(j)


fj x ξj βj1113872 1113873

1βj

1 + ξj

x

βj

1113888 1113889

minus 1ξjminus 1

if ξj ne 0

1βj

exp minusx

βj

1113888 1113889 if ξj 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩






k x(j)


t(j)lowastk 1113946

t(j)

k

0λj(s)ds (16)


t(j)lowastk λjt

(j)

k +θj1

ηj11113944

mt(1)m let

(j)

k1113864 1113865

exp δj1x(1)m1113872 1113873


k minus t(1)m1113872 11138731113872 11138731113960 1113961

+θj2

ηj21113944

mt(2)m let

(j)

k1113864 1113865

exp δj2x(2)m1113872 1113873


k minus t(2)m1113872 11138731113872 11138731113960 1113961

(17)



(j)lowast1 t

(j)lowast2 t




(j)lowast1 t

(j)lowast3 minust

(j)lowast2 t


















reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)






Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)







Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References



























(j)lowast1 t

(j)lowast2 t




(j)lowast1 t

(j)lowast3 minust

(j)lowast2 t


















reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)






Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)







Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References




























reshold

Mea

n ex

cess

1 2 3 4

05

06

07

08

09

10

11

(a)

0 1 2 3 4Threshold

Mea

n ex

cess

03

04

05

06

07

08

09

(b)






Ordered data

GPD

qua

ntile

s ξ =

01

4961

3

1 2 3 4 5 6

0

2

4

6

8

10

12

14

(a)

Ordered data

GPD

qua

ntile

s ξ =

00

2864

744

1 2 3 4

0

2

4

6

8

(b)







Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References






























Time

Inte

nsity

Process 1

0 20000 40000 60000 80000

00102

00104

00106

(a)

Inte

nsity

Process 2

Time0 20000 40000 60000 80000

0010

0011

0012

0013

0014

(b)


Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 2 4 6

4

6

0

2

(a)

Ordered data

Expo

nent

ial d

istrib

utio

n qu

antil

es

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

(b)










Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References

































Time

Den

sity

2 4 6 8 10

00

01

02

03

04

05

06

(a)

Time2 4 6 8 10

Den

sity

00

01

02

03

04

05

06

(b)




4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References



























4 Conclusions








Time

Den

sity

2 4 6 8 10

000

005

010

015

020

025

(a)

Time2 4 6 8 10

Den

sity

000

005

010

015

020

025

(b)




Data Availability




Acknowledgments


References



























Data Availability




Acknowledgments


References


























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