# Point Process Models for Multivariate High-Frequency Irregularly

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2 2 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data Cosider a K dimesioal multivariate poit process. Let N k t deote the coutig process associated with the k-th poit process which is simply the umber of evets which have occured by time t. Let F t deote the filtratio of the pooled process N t of K poit processes cosistig of the set t k <t k 1 <t k 2 < <t k i < deotig the history of arrival times of each evet type associated with the k=1 K poit processes. At time t, the most recet arrival time will be deoted t k k Nt. A process is said to be simple if o poits occur at the same time, thas, there are o zero-legth duratios. The coutig process ca be represeted as a sum of Heaviside step fuctios θt)= { t< 1 t N t k = ti k t θt k ) 1) The coditioal itesity fuctio gives the coditioal probability per uit time that a evet of type k occurs i the exstat. For small values of t we have λ k t F t )= lim t k rn t+ t N k t > F t ) t 2) so that λ k k t F t ) t=en t+ t N k t F t )+o t) 3) k EN t+ t N k t ) λ k t F t ) t)=o t) 4) ad 4) will be ucorrelated with the past of F t as t. Next cosider lim t s 1 s ) t k Ns+j t = lim N s k N s1 t =N k s N k s1 ) k ) s ) k N s +j 1) t λ k s +j t F t ) t s 1 s ) t s 1 λ k t F t )dt λ k j t F t ) t 5) which will be ucorrelated with F s, thas s 1 E λ k t F t )dt) =N k k s N s1 6) s The itegrated itesity fuctio is kow as the compesator, or more precisely, the F t -compesator ad will be deoted by Λ k s 1 s,s 1 )= λ k t F t )dt 7) s Let x k = t k k i 1 deote the time iterval, or duratio, betwee the i-th ad i 1)-th arrival times. The F t -coditioal survivor fuctio for the k-th process is give by Let S k x i k )= k k >x i k F ti 1+τ) 8) Ẽ k i = λ k t F t )dt=λ k 1, ) 1 the provided the survivor fuctio is absolutely cotiuous with respect to Lebesgue measurewhich is a assumptio that eeds to be verified, usually by graphical tests) we have S k x k i )=e t λ k t F t )dt k i 1 =e Ẽ i 9)

3 ,, 3 ad Ẽ Nt) is a i.i.d. expoetial radom variable with uit mea ad variace. Sice E Ẽ Nt) ) =1 the radom variable k E Nt) =1 Ẽ Nt) 1) has zero mea ad uit variace. ositive values of E Nt) idicate that the path of coditioal itesity fuctio λ k t F t ) uder-predicted the umber of evets i the time iterval ad egative values of E Nt) idicate that λ k t F t ) over-predicted the umber of evets i the iterval. I this way, 8) ca be iterpreated as a geeralized residual. The backwards recurrece time give by icreases liearly with jumps back to at each ew poit. U k) t)=t t N k t) 11) Stochastic Itegrals. The stochastic Stieltjes itegral[1, 2.1][8, 2.2] of a measurable process, havig either locally bouded or oegative sample paths, Xt) with respect to N k exists ad for each t we have Xs)dN k s = θt t k i )Xt k i ) 12),t] i The Expoetial Autoregressive Coditioal DuratioEACD) Model. Lettig p i be the family of coditioal probability desity fuctios for arrival time, the log likelihood of the expoetial) ACD model ca be expressed i terms of the coditioal desities or itesities as [11] ll{ } i= ) = logp i t,, 1 ) i= ) t = logλ i 1,t,, 1 ) λu,t,,t Nu )du i=1 = i=1 = i=1 logλ i 1,t,, 1 ) λu,t,,t Nu )du 1 ) logλ i 1,t,, 1 ) Ẽ i t = lλt)dnt t t t λt)dt We will see that λ ca be parameterized i terms of t ) 13) λt N t,t 1,,t Nt )=ω+ i=1 N t π i t Nt+1 i t Nt i) 14) so that the impact of a duratio betwee successive evets depeds upo the umber of iterveig evets. Let x i = 1 be the iterval betwee cosecutive arrival times; the x i is a sequece of duratios or waitig times. The coditioal desity of x i give its pass the give directly by Ex i x i 1,,x 1 )=ψ i x i 1,,x 1 ;θ)=ψ i 15) The the ACD models are those that cosist of the assumptio x i =ψ i ε i 16) where ε i is idepedetly ad idetically distributed with desity pε; φ) where θ ad φ are variatio free. ACD processes are limited to the uivariate settig but later we will see that this model ca be combied with a Hawkes process i a multivariate framework. [6] The coditioal itesity of a ACD model ca be expressed i geeral as λt N t,t 1,,t Nt )=λ t tnt ψ Nt+1 ) 1 ψ Nt+1 17)

4 4 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data where λ t) is a determiistic baselie hazard, so that the past history iflueces the coditioal itesity by both a multiplicative effect ad a shif the baselie hazard. This is called a accelerated failure time model sice pasformatio iflueces the rate at which time passes. The simplest model is the expoetial ACD which assumes that the duratios are coditioally expoetial so that the baselie hazard λ t)=1 ad the coditioal itesity is λt x Nt,,x 1 )= 1 ψ Nt +1 18) The compesator for cosecutive evets of the ACD model i the case of costat baselie itesity λ t)=1 is simply Ẽ i =Λ k 1, ) = λt x i,,x 1 )dt t i 1 1 = dt 1 ψ Nt +1 1 = dt ψ i 1 = 1 ψ i = x i ψ i where x i = 1. A geeral model without limited memory is referred to as ACDm,q) where m ad q refer to the order of the lags so that there are m+q+1) parameters. m ψ i =ω+ q α j x i j + 19) ψ i j 2) ω 1 q j=q where ω,α j, ad ψ i = for i=1 maxm,q) so the coditioal itesity is the writte 1 λt x Nt,,x 1 )= ω+ m α j x q 21) Nt+1 j+ ψ Nt+1 j The log-likelihood for the ACDm,q) model is the writte i terms of the duratios x i = 1 ) ll{x i } i=1,, ) = lλ i 1,t,, 1 ) Ẽ i i=1 = ) Sxi ) l ψ i i=1 ) e Ẽi ψ i = l i=1 = i=1 = i=1 l e xi ψ i ψ i 1 l ψ i ) x i ψ i 22) A ACD process is statioary if m α j + q i=1 i=1 <1 23)

5 ,, 5 i which case the ucoditioal mea exists ad is give by ω µ=e[x i ]= 1 m i=1 α j + q i=1 ) 24) The goodess of fit ca be checked by testig that residuals Ẽ i have mea ad variace equal to 1 ad o autocorrelatio The Weibull-ACD Model. The WACDWeibull-ACD) model exteds the EACD model by assumig a Weibull distributio for the residuals ε i i 16) istead of a expoetial. We have the itesity give by ad log-likelihood by 1+ λt x Nt,,x 1 )= Γ 1 γ ψ Nt+1 ) ) ll{x i } i=1,, ) = ) γ 1+ l +γl Γ 1 γ x i ψ i i=1 γ t t Nt ) γ 1 γ 25) x i ) 1+ Γ 1 γ ψ i x i γ 26) The goodess of fit ca be checked by testig that the mea of Ẽ i is equal to 1 ad graphically checkig whas kow as a weibull plot. If is a good fit, the empirical curve will be ear the straight lie. I the example show below, the weibull does better tha the expoetial bus still ot a great fit Weibull robability lot.1 robability Data Figure 1. Weibull plot for WACD1,1) model fit to SY INET o The Hawkes rocess.

6 6 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data Liear Self-Excitig rocesses. A uivariate) liear self-excitig coutig) process N s oe that ca be expressed as [15][7][14][3] t λt) =λ t)κ+ νt s)dn s =λ t)κ+ 27) νt ) <t where λ t) is a determiistic base itesity, see 77), ν: R + R + expresses the positive ifluece of past evets o the curret value of the itesity process, ad κ takes the place of the λ costa the refereced papers. The expoetial) Hawkes process of order is a liear selfexcitig process defied by the expoetial kerel so that the itesity is writte as νt)= α j e βjt 28) λt) =λ t)κ+ t N t 1 α j e βjt s) dn s =λ t)κ+ α j e βjt ti) i= =λ t)κ+ N t 1 α j e βjt ti) =λ t)κ+ =λ t)κ+ i= N t 1 α j e βjt tk) i= α j B j N t ) 29) where B j i) is give recursively by i 1 A uivariate Hawkes process is statioary if B j i) = 3) k= =1+B j i 1))e βjt ti) e βjt tk) αj If a Hawkes process is statioary the the ucoditioal mea is <1 31) λ µ=e[λt)] = 1 νt)dt = λ 1 λ = 1 α j α j e βjt dt 32)

7 ,, 7 For cosecutive evets, we have the compesator 7) Λ 1, ) = λt)dt 1 = λ t)+ α j B j N t ) dt 1 = λ s)ds+ i 1 αj e 1 t k) e βjti tk)) 1 k= = λ s)ds+ αj ) 1 e 1) A j i 1) 1 where there is the recursio A j i) = e βjti tk) t k i 1 e βjti tk) = k= 33) 34) =1+e βjti ti 1) A j i 1) with A j )=. If λ t)=λ the 33) simplifies to Λ 1, ) = 1 )λ + k= = 1 )λ + i 1 αj αj e 1 t k) e βjti tk)) 1 e 1) ) A j i 1) 35) Similiarly, aother parameterizatio is give by Λ 1, ) = κλ s)ds+ αj ) 1 e 1) A j i 1) 1 =κ λ s)ds+ αj ) 1 e 1) A j i 1) 1 36) =κλ 1, )+ αj 1 e 1) ) A j i 1) where κ scales the predetermied baselie itesity λ s). I this parameterizatio the itesity is also scaled by κ λt) =κλ t)+ α j B j N t ) 37) this allows to precompute the determiistic part of the compesator Λ 1, )= 1 λ s)ds The Hawkes1) Model. The simplest case occurs whe the baselie itesity λ t) is costat ad =1 where we have λt)=λ + αe βt ti) 38) <t

8 8 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data which has the ucoditioal mea E[λt)]= λ 1 α β 39) Maximum Likelihood Estimatio. The log-likelihood of a simple poit process is writte as T llnt) t [,T] ) = =T 1 λs))ds+ T λs)ds + T T lλs)dn s lλs)dn s 4) which i the case of the Hawkes model of order ca be explicitly writte [13] as ll{ } i=1 ) =T Λ,T)+ i=1 =T + i=1 lλ ) lλ ) Λ 1, ) =T Λ,T)+ lλ ) i=1 =T Λ,T)+ l κλ )+ i=1 =T Λ,T)+ i=1 i 1 k=1 l κλ )+ α j R j i) α j e βjti tk) 41) =T + i=1 T κλ s)ds αj i=1 l κλ )+ α j R j i) ) 1 e t ) where T =t ad we have the recursio[12] i 1 e βjti tk) R j i) = k=1 =e βjti ti 1) 1+R j i 1)) 42) If we have costat baselie itesity λ t)=1 the the log-likelihood ca be writte ll{ } i=1 ) =T κt i=1 + i=1 αj l λ + α j R j i) ) 1 e t ) 43) Note that was ecessary to shift each by t 1 so that t 1 = ad t =T. Also ote that T is just a additive costat which does ot vary with the parameters so for the purposes of estimatio ca be removed from the equatio The Hawkes rocess i Quatum Theory.

9 ,, 9 The Hawkes process arises i quatum theory by cosiderig feedback via cotiuous measuremets where the quatum aalog of a self-excitig poit process is a source of irreversibility whose stregth is cotrolled by the rate of detectios from that source. [16] The Hawkes rocess Havig a Weibull Kerel. The expoetial kerel of the Hawkes process ca be replaced with that of a Weibull kerel. [1, 6.3] Recall that the itesity is defied by 27) t λt) =λ t)κ+ νt s)dn s =λ t)κ+ νt ) <t 44) where the expoetial kerel νt)= so the Weibull-Hawkes itesity is writte is λt) =λ t)κ+ α j e βjs replaced by the Weibull kerel νt)= ) ) ) κj κj t κj 1 t ω α j e j 45) ω j ω j =λ t)κ+ i= t N t 1 α j κj ω j α j κj ω j ) t s ω j ) t ti ω j ) ) κj κj 1 e βj t s ω j dn s ) κj 1 e t ω j ) κj 46) 1.5. Combiig the ACD ad Hawkes Models. The ACD ad Hawkes models ca be combied to provide a model for itraday volatility. [2] Let λt) =λ t)+ 1 t + νt s)dn s 47) ψ Nt where λ t) is the determistic baselie itesity77) ad where the ACD2) pars m ψ i =ω+ q α j x i j + ψ i j 48) ad the Hawkes part has the expoetial kerel28) νt)= γ j e ϕjt 49) so that t νt s)dn s = t γ j e ϕjt s) dn s N t νt t k ) = k= N t = k= = N t γ j k= = γ j B j N t ) γ j e ϕjt tk) e ϕjt tk) 5)

10 1 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data where we have replaced α=γ ad β=ϕ i the Hawkes part so that the parameter ames do ot coflict with the ACD part where α ad β are also used as parameter ames. The Hawkes part of the itesity has a recursive structure similiar to that of the compesator. Let where B j )=. The we have i 1 B j i) = 51) k= =1+B j i 1))e ϕjt ti) e ϕjt tk) 1 λt) =λ t)+ ω+ m α j x q + γ j B j N t ) 52) Nt j+ ψ Nt j The log-likelihood for this hybrid model ca be writte as ll{ } i=1,.., ) = i=1 lλ ) 1 ) λt)dt = lλ ) Λ 1, )) i=1 53) = i=1 lλti ) Ẽ i ) By direct calculatio, combiig 19) ad 33), ad lettig x i = 1 we have the compesator Ẽ i =Λ 1, ) = λt)dt t i 1 = λ t)+ 1 1 ψ Nt +1 = x i + λ t)+ ψ i 1 ti = λ t)dt+ x i + ψ ti 1 i k= = 1 + t t νt s)dn s )dt νt s)dn s )dt i 1 γj e ϕjti 1 tk) e ϕjti tk)) ϕ j λ t)dt+ x i + γj 1 e ϕjxi )A j i 1) ψ i ϕ j 54) where ψ i is defied by 48) ad is give by 34) so that 53) ca be wriitte as A j i)=1+e ϕjxi A j i 1) 55) ll{ } i=,.., ) = lλ ) Ẽ i ) i=1 = lλti ) x i + γj 1 e ϕjxi )A j i 1) ψ i ϕ j i=1 = l 1 i 1 + γ ψ j e ϕjti tk) x i + γj 1 e ϕjxi )A i ψ i ϕ j i 1) j i=1 k= = l 1 + γ ψ j B j i) x i + γj 1 e ϕjxi )A i ψ i ϕ j i 1) j i= Multivariate Hawkes Models. Let M N ad { m )} m=1,,m be a M-dimesioal poit process. The associated coutig process will be deoted N t =N t 1,,N t M ). A multivariate Hawkes process[7][5][9] is defied with 56)

11 ,, 11 itesities λ m t),m=1 M give by M λ m t) =λ m t)κ m + =1 =λ m t)κ m + =1 =λ m t)κ m + =1 =λ m t)κ m + =1 =λ m t)κ m + =1 =λ m t)κ m + =1 t M t k <t M M M M α m, j e β m, j t s) dn s α j m, α m, j e β m, j t t k ) t k <t α j m, t k <t N t 1 α j m, k= α j m, B j m, N t ) e β m, j t t k ) e β m, j t t k ) e β m, j t t k ) where i this parameterizatio κ is a vector which scales the baselie itesities, i this case, specified by piecewise polyomial splies 77). We ca write B j m, i) recursively 57) B j m, i) = i 1 k= e β m, j t t k ) =1+B j m, i 1))e m, t ) I the simplest versio with =1 ad λ m t)=1 costat we have λ m t) =κ m + =1 =κ m + =1 M t M M =κ m + =1 N t 1 α m, e βm, t s) dn s α m, e βm, t t k ) k= N t 1 e βm, t t k ) α m, k= 58) 59) M =κ m + =1 Rewritig 59) i vectorial otio, we have λt)=κ+ where α m, B 1 m, N t ) t Gt s)dn s 6) Gt)=α m, e βm, t s) ) m,=1 M 61) Assumig statioarity gives E[λt)] = µ a costat vector ad thus κ µ = I Gu)du κ = I αm, ) β m, = κ I Γ A sufficiet coditio for a multivariate Hawkes process to be statioary is that the spectral radius of the brachig matrix Γ= Gs)ds= αm, β m, 63) 62)

12 12 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data be strictly less tha 1. The spectral radius of the matrix G is defied as where SG) deotes the set of eigevalues of G. ρg)= max a 64) a SG) The Compesator. The compesator of the m-th coordiate of a multivariate Hawkes process betwee two cosecutive evets t m i 1 ad t m i of type m is give by Λ m t m i 1,t m i ) = m m 1 M + =1 M + =1 λ m s)ds t m k < 1 t m i 1 t m k < m, α j β m,[e β m, j t m i 1 t ) k e β m, j t m i t ) k ] j m, α j β m,[1 e β m, j t m i t ) k ] j To save a cosiderable amout of computatioal complexity, ote that we have the recursio A m, j i) = e β m, j t m i t k ) t m k < =e β m, j t m i 1 m ) A j m, i 1)+ t m i 1 t m k < 65) e β 66) m, j t m i t k ) ad rewrite 65) as Λ m t m i 1,t m i ) =κ m ti m λ s)ds+ m m m 1 1 =κ m ti m λ m s)ds m 1 M m, α j m, β j m + =1 =κ m + =1 tm i 1 M [ λ m s)ds ti m M =1 t k <s α j m, e m, s tk ) ds 1 e β m, j ti m ti 1 m ) ) A m, j i 1)+ m, αj m, β 1 e β m, j m ti ti 1 m ) ) j t k <tm i 1 e β m, j m ti 1 t k ) + t m i 1 t m k < t m i 1 t k <tm i ] 1 e βjm, ti m ) tk ) 1 e m, m ti t ) k ) 67) where we have the iitial coditios A j m, )= Log-Likelihood. The log-likelihood of the multivariate Hawkes process ca be computed as the sum of the loglikelihoods for each coordiate. Let where each term is defied by ll m { })= M ll{ } i=1,,nt )= m=1 which i this case ca be writte as ll m { }) =T Λ m,t)+ N T z m i l m λ )κ m + i=1 =1 N T m =T Λ m,t)+ i=1 T ll m { }) 68) 1 λ m T s))ds+ lλ m m s)dn s 69) l m λ t m i )κ m + =1 M α m, j e βjm, t k ) t k < M t m k < α j m, e m, m t k ) 7)

15 ,, 15 1 by desig ad for a oisso process the mea ad variace are equal. The ext thig to check is that the Λ series is ot autocorrelated. κ α 1 β 1 ll{ }) E[λt)] E[Λ] Var[Λ] Table 1. arameters ad statistics for model fitted to data without diural adjustmets κ α 1 β 1 ll{ }) E[λt)] E[Λ] Var[Λ] Table 2. arameters ad statistics for model fitted to data with diural adjustmets

16 16 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data Autocorrelatio of Λ for =1 Autocorrelatio of Λ for =2.8.8 Sample Autocorrelatio Sample Autocorrelatio Autocorrelatio of Λ for =3 Autocorrelatio of Λ for =4.8.8 Sample Autocorrelatio Sample Autocorrelatio Autocorrelatio of Λ for =5 Autocorrelatio of Λ for =6.8.8 Sample Autocorrelatio Sample Autocorrelatio Figure 3. Autocorrelatios of Λ 1, ) for =1 6 without diural adjustmets

17 ,, 17 Autocorrelatio of Λ for =1 Autocorrelatio of Λ for =2.8.8 Sample Autocorrelatio Sample Autocorrelatio Autocorrelatio of Λ for =3 Autocorrelatio of Λ for =4.8.8 Sample Autocorrelatio Sample Autocorrelatio Autocorrelatio of Λ for =5 Autocorrelatio of Λ for =6.8.8 Sample Autocorrelatio Sample Autocorrelatio Figure 4. Autocorrelatios of Λ 1, ) for =1 6 with diural adjustmets As ca be see by visually ispectig the autocorrelatios, all of the residual series are prettymuch acceptable *without* diural adjustmets except for = 1 with still had sigificat leftover autocorrelatio. Stragely, it seems thaclusio of the diural adjustmet sigificatly worses the model fi early all cases. I am tempted to suspect somethig wrog with the code.

18 18 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data x 1 4 Figure 5. rice history for SY traded o INET o Oct 22d, Figure 6. x i = 1 i blue ad {Λ 1, ): =1} i gree

19 ,, Figure 7. x i = 1 i blue ad {Λ 1, ): =6} i gree

21 ,, SY Trades o BATS Buys Λ BATS Sells Λ INET Buys Λ INET Sells Λ ARCA Trades Λ x 1 4 Figure 9. We say trades for ARCA because the type set from the data broker is Ukow, idiciatig thas ukow whether is a buyer or seller iitiated trade. We have the followig parameter estimates where large values of α >.1) are highlighted i bold. α= λ= β= ) 8) 81) with a log-likelihood score of Multivariate SY Data for

22 22 oit rocess Models for Multivariate High-Frequecy Irregularly Spaced Data Cosider the same symbol, SY, as a 5-dimesioal Hawkes process as i 3.1.3, for a differet day, o , estimated with order = 2 for a total of 15 parameters. α j coefficiets that are >.1 are highlighted i bold. The parameters listed below resulted i a log-likelihood value of A iterestig patter emerges i the β coefficiets where it takes o some approximate stair-step patter ragig from 2 to 22. This might be idicitative of some fixedfrequecy algorithms operatig across the differet exchages at approximate 1-secod itervals. λ= α 1= α 2= β 1 = β 2 = Bibliography 82) 83) 84) [1] C.G. Bowsher. Modellig security market evets i cotiuous time: itesity based, multivariate poit process models. Joural of Ecoometrics, 1412): , 27. [2] Y. Cai, B. Kim, M. Leduc, K. Szczegot, Y. Yixiao ad M. Zamfir. A model for itraday volatility.,, 27. [3] V. Chavez-Demouli ad JA McGill. High-frequecy fiacial data modelig usig hawkes processes. Joural of Bakig & Fiace,, 212. [4] JE Deis ad D.J. Woods. Optimizatio o microcomputers: the elder-mead simplex algorithm. New Computig Eviromets: Microcomputers i Large-Scale Computig, : , [5]. Embrechts, T. Liiger ad L. Li. Multivariate hawkes processes: a applicatio to fiacial data. Joural of Applied robability, 48: , 211. [6] R.F. Egle ad J.R. Russell. Autoregressive coditioal duratio: a ew model for irregularly spaced trasactio data. Ecoometrica, : , [7] A.G. Hawkes. Spectra of some self-excitig ad mutually excitig poit processes. Biometrika, 581):83 9, [8] A. Karr. oit processes ad their statistical iferece, volume 7. CRC, [9] T.J. Liiger. Multivariate Hawkes rocesses. hd thesis, Swiss Federal Istitute Of Techology Zurich, 29. [1] F. Loreze. Aalysis of Order Clusterig Usig High Frequecy Data: A oit rocess Approach. hd thesis, Tilburg School of Ecoomics ad Maagemet, Fiace Departmet, August 212. [11] Y. Ogata. The asymptotic behaviour of maximum likelihood estimators for statioary poit processes. Aals of the Istitute of Statistical Mathematics, 31): , [12] Y. Ogata. O lewis simulatio method for poit processes. Iformatio Theory, IEEE Trasactios o, 271):23 31, [13] T. Ozaki. Maximum likelihood estimatio of hawkes self-excitig poit processes. Aals of the Istitute of Statistical Mathematics, 311): , [14] H. Shek. Modelig high frequecy market order dyamics usig self-excited poit process. Available at SSRN ,, ) 86)

23 ,, 23 [15] Ioae Mui Toke. A itroductio to hawkes processes with applicatios to fiace., [16] HM Wisema. Quatum theory of cotiuous feedback. hysical Review A, 493):2133, 1994.

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