A Bayesian Analysis for Negative Binomial INGARCH Models with Minimal Prior Information

A Bayesian Analysis o Nega i e Binomial INGARCH
Models wi h Minimal P io In o ma ion
Xiaoyin Wang
Yunwei Cui
Depa men o Ma hema ics, Towson Uni e si y
Abs ac
This s udy p esen s a Bayesian amewo k o es ima ing Nega i e Binomial In ege - alued
Gene alized Au o eg essi e Condi ional He e oskedas ici y (NB-INGARCH) models, designed o
add ess o e -dispe sed coun ime se ies da a. Minimally in o ma i e p io s such as Di ichle p io
and hie a chical p io a e employed o ensu e he lexibili y oge he wi h he model cons ain s.
Nume ical in e ence is ca ied ou ia Ma ko Chain Mon e Ca lo (MCMC) me hods. Applica ions
o empi ical da a demons a e ha he p oposed Bayesian app oach deli e s mo e obus and
in e p e able es ima es compa ed o equen is me hods.
Keywo ds: Bayesian es ima ion, Ma ko Chain Mon e Ca lo, NB-INGARCH model, Di ichle dis-
ibu ion, ime se ies analysis, minimal-in o ma i e p io s.
1 In oduc ion
Coun ime se ies ha e b oadly used in he a ious ields such as c iminology, inance, public heal h,
and ma ke ing. Rececenly, i was applied o a s udy analyzing so d ink sales (Jia e al., 2023) and
COVID-19 mo ali y da a (Palme e al., 2021). Ea ly wo k in coun ime se ies modeling la gely
ocused on condi ional Poisson dis ibu ions (Da is e al., 2000; Fe land e al., 2006; Weiß, 2008;
Fokianos e al., 2009, 2020; Kong and Lund, 2025). Howe e , he Poisson assump ion o equal
mean and a iance o en ails in p ac ice, pa icula ly o o e -dispe sed da a. Empi ical s udies ha e
shown ha non-Poisson al e na i es, such as he Nega i e Binomial dis ibu ion, o en cap u e his
a iabili y mo e e ec i ely (see Da is and Wu, 2009; Zhu, 2011; Ch is ou and Fokianos, 2014; Da is
and Liu, 2016; Ahmad and F ancq, 2016). Ano he no able sho coming o exis ing esea ch is ha he
es ima ion o he dispe sion pa ame e has ecei ed limi ed a en ion, especially in classical se ings.
Zhu (2012) emphasized he need o u he esea ch and sugges ed momen -based app oaches,
while Da is and Liu (2016) in oduced a wo-s age p o ile likelihood me hod: i s es ima ing he
pa ame e s 𝜶,𝜷, 𝛿 as unc ions o he dispe sion pa ame e 𝜙, ollowed by op imizing he dispe sion
pa ame e using he p o ile likelihood. None heless, hese wo ks did no p o ide a comp ehensi e
in e ence o he dispe sion pa ame e .
Bayesian me hods ha e been widely applied in con inuous GARCH models (Bauwens and Lu-
b ano, 1998; Aus´
ın and Galeano, 2007) and Poisson GARCH models (A dia, 2008, 2009; A dia and
Hooge heide, 2010). Mo e ecen ly, Chen e al. (2021) demons a ed ha Bayesian analysis wi h
nai e p io s shows Nega i e Binomial INGARCH models ou pe o m Poisson-based al e na i es. A
key s eng h o he Bayesian amewo k is i s abili y o join ly es ima e dispe sion and model pa-
ame e s wi hin a uni ied p ocess. The choice o p io dis ibu ion is c i ical in Bayes in e ence. The
gamma p io has been commonly used o dispe sion (Chen and Lee, 2017; Chen and Kham hong,
2020; Chen e al., 2021; Chu and Yu, 2023), ye he selec ion o he hype pa ame e s is no explici ly
s a ed, and o en elies on subjec i e judgmen .
1
To add ess hese limi a ions, his pape de elops Bayesian es ima ion o he NB-INGARCH
p ocess unde non-in o ma i e p io s, allowing in e ence o ely p ima ily on obse ed da a. Fo
u he e iews o coun ime se ies models, see Fokianos (2012); Da is e al. (2021); Liu e al. (2023).
We begin by p o iding an o e iew o NB-INGARCH models and hei dynamic speci ica ions in
Sec ion 2. Following his, we in oduce Bayesian es ima ion echniques, elabo a ing on he selec ion
o p io dis ibu ions o model pa ame e s and he de i a ion o he join pos e io dis ibu ion in he
Sec ion 3. Ob aining he closed- o m exp ession o he pos e io dis ibu ion o model pa ame e s is
challenging, and Ma ko Chain Mon e Ca lo (MCMC) me hods can be applied o gene a e nume ical
esul s. To demons a e he p ac ical applica ion o Bayesian es ima ion o NB-INGARCH models,
Sec ion 4 p esen s an empi ical example, on he ansac ion da a o E icsson B s ock. Since i is widely
used in li e a u e, his da ase se es as a s anda dized benchma k o compa ing and e alua ing ou
esea ch indings. Finally, we p esen conclusions and ema ks in Sec ion 5.
2 The NB-INGARCH(p, q) Models
This sec ion p esen s he NB-INGARCH model, which p o ides a lexible amewo k o o e -
dispe sed coun ime se ies. The INGARCH p ocess speci ies coun s 𝑌𝑡wi h a ime- a ying mean
𝜆𝑡e ol ing as
𝜆𝑡=𝛿+
𝑝
∑︁
𝑖=1
𝛼𝑖𝜆𝑡−𝑖+
𝑞
∑︁
𝑗=1
𝛽𝑗𝑌𝑡−𝑗,(1)
whe e 𝛿 > 0 is he baseline le el, 𝛼𝑖, 𝛽𝑗≥0 a e coe icien s o lagged means and coun s, and 𝑝, 𝑞
deno e he model o de s.
The equa ion (1) desc ibes ha he dynamics o 𝜆𝑡depends on he pa ame e s 𝜶=𝛼1,· · · , 𝛼𝑝,
𝜷=𝛽1,· · · , 𝛽𝑞, and 𝛿, as well as he lagged alue o (𝜆𝑡−1,· · · , 𝜆𝑡−𝑝)and (𝑌𝑡−1,· · · ,𝑌𝑡−𝑞). The
s a e pa ame e s 𝜶is o en e e ed o as he pe sis ence pa ame e , and 𝜷is he a iabili y pa ame e ,
while 𝛿se s he long- e m baseline.
Condi ioned on 𝜆𝑡,𝑌𝑡 ollows a Nega i e Binomial dis ibu ion wi h mean 𝜆and dispe sion 𝜙:
𝑃(𝑋=𝑘)=𝑘+𝜙−1
𝑘 𝜙
𝜙+𝜆𝜙𝜆
𝜙+𝜆𝑘
,
wi h a iance 𝜆+𝜆2/𝜙. This mean-dispe sion pa ame e iza ion is especially use ul o modeling he
e olu ion o coun s h ough he condi ional mean. Da is and Liu (2016) show ha in o de o ensu e
he exis ence o a unique s ic ly s a iona y and e godic solu ion o he ime se ies {𝑌𝑡, 𝜆𝑡}wi h
𝐸(𝑌𝑡)<∞and 𝐸(𝜆𝑡)<∞, i equi es Í𝑖𝛼𝑖+Í𝑗𝛽𝑗<1.
The likelihood o obse a ions 𝒀=(𝑌1, . . . , 𝑌𝑇)is
𝐿(𝜙, 𝜶,𝜷, 𝛿 |𝒀)=
𝑇
Ö
𝑡=1𝑦𝑡+𝜙−1
𝑦𝑡 𝜙
𝜙+𝜆𝑡𝜙𝜆𝑡
𝜙+𝜆𝑡𝑦𝑡
.
3 Bayesian Es ima ion
3.1 P io Speci ica ion
Choosing p io s is a c i ical ye challenging ask in Bayesian analysis. The selec ion o he p io
dis ibu ion in Bayesian s a is ics ypically depends on he s udy’s pa icula objec i es and he
cha ac e is ics o he a ailable da a. P io s can be in o ma i e o non-in o ma i e depending on
he ex en o p io knowledge. In he absence o p io in o ma ion, we adop non-in o ma i e o
hie a chical p io s o allow da a-d i en in e ence.
While he dispe sion pa ame e 𝜙in he Nega i e Binomial dis ibu ion can heo e ically be
ea ed as a con inuous a iable, i is o en modeled using a gamma p io due o i s lexibili y and
conjugacy p ope ies (Chen and Lee, 2017; Chen and Kham hong, 2020; Chen e al., 2021; Chu and
Yu, 2023). Howe e , he p ocess o selec ing hype pa ame e s o he gamma p io lacks o malized
2
guidance and ends o ely hea ily on subjec i e judgmen . In p ac ical applica ions, 𝜙is equen ly
in e p e ed as an in ege - alued pa ame e , which aligns wi h he disc e e na u e o coun da a.
Al hough he Poisson dis ibu ion is a na u al choice o modeling such da a, i does no o e he
same adap abili y as he gamma dis ibu ion in exp essing a wide ange o p io belie s abou 𝜙. To
in eg a e he lexibili y o he gamma dis ibu ion and he na u al alignmen o he Poisson wi h coun
da a, we cons uc a hie a chical Poisson-gamma p io o 𝜙as below:
𝜙∼Poisson(𝜌), 𝜌 ∼Gamma(𝜏, 𝜂),
wi h hype pa ame e s 𝜏and 𝜂chosen o yield ague dis ibu ions, allowing i o a y eely ac oss a
b oad ange o plausible alues.
No ably, as he hype pa ame e s 𝜏and 𝜂o he gamma p io app oach ze o, he p io dis ibu ion
o 𝜙becomes nea ly la , indica ing ha all easible alues o 𝜙a e assigned app oxima ely equal
p obabili y. The esul ing ma ginal p io is
𝜋(𝜙) ∝ Γ(𝜙+𝜏)
(𝜂+1)𝜙+𝜏+1𝜙!.
As 𝜏, 𝜂 →0, he ma ginal p io o 𝜙becomes in e sely p opo ional o 𝜙, i.e., he p io densi y
beha es like 1/𝜙. In e es ingly, his o m o in e se p opo ionali y co esponds o he Je eys p io
o he dispe sion pa ame e in he Nega i e Binomial dis ibu ion.
Fo he s a e pa ame e s (𝜶and 𝜷), uni o m dis ibu ions wi h s a iona i y cons ain s a e used
as p io s as discussed in Chen and Lee (2017); Chen and Kham hong (2020); Chen e al. (2021),
whe e he indi idual componen s 𝛼𝑖and 𝛽𝑗a e ea ed as independen pa ame e s. None heless,
he cons ain s imposed by he model a e no embedded di ec ly in he p io dis ibu ions bu a e
ins ead en o ced du ing he compu a ional phase o in e ence. We adop a Di ichle dis ibu ion o e
he (𝑝+𝑞)-simplex o en o ce he s a iona i y cons ain Í𝑖𝛼𝑖+Í𝑗𝛽𝑗<1. This dis ibu ion is
equi alen o say ha (𝜶,𝜷)join ly ollow a mul i a ia e uni o m dis ibu ion o e he open s anda d
(𝑝+𝑞)-simplex, and ha 𝛼𝑖and 𝛽𝑗(whe e 𝑖=1,· · · , 𝑝 and 𝑗=1,· · · , 𝑞) each ha e a ma ginal
dis ibu ion o 𝐵𝑒𝑡𝑎(1, 𝑝 +𝑞).
Fo he posi i e baseline pa ame e 𝛿, p e ious s udies such as Chen and Lee (2017); Chen and
Kham hong (2020); Chen e al. (2021) ha e adop ed a uni o m (o la ) p io o e he posi i e eal line
(0,∞). While his choice is s aigh o wa d and non-in o ma i e, i has been equen ly c i icized o
being imp ope , as i does no in eg a e o a ini e alue. To add ess his conce n, we ins ead adop a
hie a chical unca ed no mal p io , which ensu es p ope ness while e aining lexibili y in modeling
unce ain y abou 𝛿.
𝛿∼𝑁(𝜇, 𝜎2)𝐼(0,∞), 𝜇 ∼𝑁(0,105), 𝜎2∼In -Gamma(0.1,0.1),
wi h hype pa ame e s se o p oduce di use dis ibu ions.
3.2 Pos e io Dis ibu ion
Le 𝜋(𝜙),𝜋(𝛿), and 𝜋(𝜶,𝜷)deno e he p io densi ies o 𝜙,𝛿, and (𝜶,𝜷), espec i ely. Assuming
independence, he join pos e io is
𝑝(𝜙, 𝜶,𝜷, 𝛿 |𝒀) ∝ 𝐿(𝒀|𝛿, 𝜶,𝜷)𝜋(𝛿)𝜋(𝜶,𝜷)𝜋(𝜙).(2)
In heo y, a closed- o m join pos e io can be ob ained by no malizing equa ion (2). Howe e ,
due o he unc ion’s complexi y, in eg a ing o e all pa ame e s is o en imp ac ical, so MCMC
me hods such as Gibbs sampling, implemen ed ia JAGS, a e commonly used o emedy his ype
o di icul ies.
In Bayesian analysis, he pos e io mean is a common choice o poin es ima ion o he pa ame e ,
and is known as he Bayes es ima o ; whe eas he highes pos e io densi y (HPD) c edible in e al
is o en p e e ed o in e al es ima ion, especially o asymme ic o mul imodal pos e io s. This
in e al is cha ac e ized as he sho es in e al con aining a speci ied p opo ion o he pos e io
3
p obabili y densi y. I ep esen s he mos c edible ange o alues o he pa ame e gi en he
obse ed da a and he p io dis ibu ion. S a iona y pa ame e s such as he mean 𝜇and s anda d
de ia ion 𝜎a e essen ial o cha ac e izing he long- e m beha io o a ime se ies. Le e aging he
pos e io dis ibu ion enables mo e accu a e es ima ion o hese quan i ies, o e ing deepe insigh s
in o he unde lying dynamics.
4 Empi ical Example
We illus a e he ad an ages o he p oposed Bayesian app oach using E icsson B s ock ansac ion
da a. The se ies o ansac ions pe minu e on July 2, 2002 (460 obse a ions) has been widely
s udied in Da is and Liu (2016); Fokianos e al. (2009); Fokianos and Neumann (2013); Doukhan
and Kengne (2015); Diop and Kengne (2017) and se es as a benchma k. Figu e 1 shows he
ime se ies and au oco ela ion unc ion (ACF), indica ing signi ican empo al dependence and
o e dispe sion (mean = 9.91, a iance = 32.84).
We i a NB-INGARCH(1,1) model:
𝑌𝑡| F𝑡−1∼𝑁𝐵(𝜆𝑡, 𝜙),(3)
𝑃(𝑌𝑡=𝑦𝑡| F𝑡−1)=𝑦𝑡+𝜙−1
𝑦𝑡 𝜙
𝜙+𝜆𝑡𝜙𝜆𝑡
𝜙+𝜆𝑡𝑦𝑡
,(4)
𝜆𝑡=𝛿+𝛼𝜆𝑡−1+𝛽𝑌𝑡−1,(5)
wi h 𝜙 > 0, 𝛼, 𝛽 ≥0, 𝛿 > 0, and F𝑡−1=𝜎{𝑌𝑡−1,𝑌𝑡−2, . . . }. The model is s a iona y i 0 < 𝛼 +𝛽 < 1
(Da is and Liu, 2016).
As p e iously discussed in Sec ion 3, we assign a Poisson-Gamma p io o 𝜙wi h hype pa-
ame e s 𝜏=𝜂=0.0001, a Di ichle (1,1,1) p io o (𝛼, 𝛽), and a hie a chical unca ed no mal
p io o 𝛿. Pos e io in e ence is conduc ed ia Ma ko Chain Mon e Ca lo (MCMC) using JAGS
in R. Ini ializa ion, bu n-in, and sampling leng hs a e au oma ically managed by unc ions wi hin
he JAGS package, which i e a i ely ex ends he simula ion un il con e gence is achie ed o all
moni o ed pa ame e s (Gelman and Rubin, 1992). Addi ional i e a ions a e pe o med o compensa e
any au oco ela ion. Con e gence diagnos ics con i m adequa e mixing o he chains (Figu e 2).
The pos e io dis ibu ions (Figu e 3) show ha 𝜙peaks nea 8, 𝛼exhibi s le skewness, while he
emaining pa ame e s a e igh -skewed. Due o hese asymme ies, highes pos e io densi y (HPD)
in e als a e p e e ed o in e ence (Gelman e al., 2013; Da idson-Pilon, 2015).
A summa y o he pa ame e es ima es is lis ed in Table 1.
Pa . Mean SD Lowe 95 Uppe 95 Median
𝜙7.89 0.97 6.00 10.00 8
𝛼0.83 0.04 0.76 0.90 1
𝛽0.14 0.03 0.09 0.19 0
𝛿0.28 0.16 0.01 0.58 0
Table 1: E icsson S ock: Bayesian Es ima es Summa y S a is ics
One ad an age o he Bayesian app oach is i s e ec i eness in es ima ing he dispe sion pa ame e
𝜙in he NB-INGARCH model. Due o he di icul y associa ed wi h he likelihood base es ima ion
me hod and he sepe a ion na u e o he pa ame e s, adi ional me hods o en ix 𝜙and es ima e only
he s a e pa ame e s, 𝛼,𝛽, and 𝛿. The Bayesian app oach allows join in e ence o all pa ame e s in
a single p ocedu e. Fo he da ase conside ed, se ing 𝜙=8 is common (e.g., Da is and Wu, 2009;
Da is and Liu, 2016), and he Bayesian es ima e aligns wi h his alue.
Table 2 p esen s MLEs o 𝛼,𝛽, and 𝛿, which a e simila o he Bayesian es ima es. No ably, he
classical 95% con idence in e al o 𝛿has a nega i e lowe bound, iola ing i s posi i i y cons ain ,
whe eas he Bayesian HPD c edible in e al espec s pa ame e cons ain s.
4
Pa . Mean SD Lowe 95 Uppe 95
𝛼0.84 0.03 0.78 0.91
𝛽0.13 0.03 0.07 0.18
𝛿0.27 0.14 −0.01 0.54
Table 2: E icsson S ock: Summa y S a is ics o Classical MLE
Fo he NB-INGARCH(1,1) model, he s a iona y mean 𝜇and s anda d de ia ion 𝜎a e
𝜇=𝛿
1−𝛼−𝛽, 𝜎 =
u
u
𝜇+𝜇2
𝜙+1+1
𝜙𝛽2𝜇(1+𝜇
𝜙)
(1− (𝛼+𝛽)2−𝛽2
𝜙)
.
The MLEs o he s a iona y pa ame e s 𝜇and 𝜎2a e
ˆ𝜇=
ˆ
𝛿
1−ˆ𝛼−ˆ
𝛽
=9,and ˆ𝜎=
u
u
u
ˆ𝜇+ˆ𝜇2
ˆ
𝜙+ (1+1
ˆ
𝜙)
ˆ
𝛽2ˆ𝜇(1+ˆ𝜇
ˆ
𝜙)
(1− ( ˆ𝛼+ˆ
𝛽)2−ˆ
𝛽2
ˆ
𝜙)
=5.05.
Es ima ing s anda d e o o he MLE is challenging due o he complex asymp o ic p ope ies o
he model. In con as , he Bayesian app oach o e s a comp ehensi e assessmen o unce ain y,
p o iding bo h s anda d e o s and c edible in e als di ec ly (Bauwens and Lub ano, 1998; Aus´
ın
and Galeano, 2007; A dia, 2008). Table 3 epo s he Bayesian poin and in e al es ima es o 𝜇and
𝜎.
Pa . Mean SD Lowe 95 Uppe 95 Median
𝜇10.68 1.63 7.96 14.15 10
𝜎6.64 3.21 4.38 10.73 6
Table 3: E icsson S ock: Bayesian Es ima es o S a iona y Pa ame e s
5 Conclusion
This pape in es iga es he NB-INGARCH(𝑝, 𝑞) model, a coun ime se ies model cha ac e ized
by a condi ional Nega i e Binomial dis ibu ion, making i pa icula ly sui able o o e -dispe sed
coun da a. Classical pa ame e es ima ion poses challenges due o he complexi y o he likelihood
unc ion. To add ess his, we ad oca e o Bayesian es ima ion using minimally in o ma i e p io s,
which help mi iga e p io in luence on he join pos e io and p o ide a obus and uni ied app oach
o in e ence (Bauwens and Lub ano, 1998; Aus´
ın and Galeano, 2007; A dia, 2008).
Unlike classical wo-s age app oaches, Bayesian me hods allow o he join es ima ion o all
model pa ame e s, including he dispe sion pa ame e 𝜙. The use o hie a chical p io s enhances
lexibili y, and Di ichle p io s cap u es he dependency among he pa ame e s. MCMC-based nu-
me ical esul s show ha Bayesian highes pos e io densi y (HPD) in e als a e mo e eliable han
classical con idence in e als, as hey espec model cons ain s.
Fu he mo e, he Bayesian amewo k acili a es comp ehensi e es ima ion o s a iona y pa am-
e e s, such as he mean and a iance, by p o iding bo h poin es ima es and c edible in e als,
which o e s deepe insigh s in o he unde lying s uc u e o he ime se ies and s eng hens he
in e p e abili y o model ou pu s.
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Time Se ies Plo
Time
Numbe o T ansac ions
0 100 200 300 400
0 10 30
0 5 10 15 20 25
0.0 0.4 0.8
Lag
ACF
Au o−Co ela ion Func ion Plo
Figu e 1: Time Se ies and ACF Plo s o E icsson B du ing July 2nd, 2002
8
Figu e 2: E icsson S ock: MCMC Con e gence Diagnos ic Plo s
9