Maximum trimmed likelihood estimation for discrete multivariate Vasicek processes

Fulle on, Thomas M.; Pokojo y, Michael; Anum, And ews T.; Nkum, Ebeneze
A icle
Maximum immed likelihood es ima ion o disc e e
mul i a ia e Vasicek p ocesses
Economies
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Fulle on, Thomas M.; Pokojo y, Michael; Anum, And ews T.; Nkum, Ebeneze
(2025) : Maximum immed likelihood es ima ion o disc e e mul i a ia e Vasicek p ocesses,
Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 3, pp. 1-28,
h ps://doi.o g/10.3390/economies13030068
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Recei ed: 14 Janua y 2025
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Published: 6 Ma ch 2025
Ci a ion: Fulle on, T. M., J ., Pokojo y,
M., Anum, A. T., & Nkum, E. (2025).
Maximum T immed Likelihood
Es ima ion o Disc e e Mul i a ia e
Vasicek P ocesses. Economies,13(3), 68.
h ps://doi.o g/10.3390/
economies13030068
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A icle
Maximum T immed Likelihood Es ima ion o Disc e e Mul i a ia e
Vasicek P ocesses
Thomas M. Fulle on, J . 1,* , Michael Pokojo y 2, And ews T. Anum 3and Ebeneze Nkum 4
1Depa men o Economics and Finance, The Uni e si y o Texas a El Paso, El Paso, TX 79968, USA
2Depa men o Ma hema ics and S a is ics, Old Dominion Uni e si y, No olk, VA 23529, USA;
[email p o ec ed]
3Depa men o Ma hema ical Sciences, The Uni e si y o Memphis, Memphis, TN 38152, USA;
[email p o ec ed]
4Cigna Heal hca e, Nash ille, TN 37228, USA; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : The mul i a ia e Vasicek model is commonly used o cap u e mean- e e ing
dynamics ypical o sho a es, asse p ice s ochas ic log- ola ili ies, e c. Repa ame iz-
ing he disc e ized p oblem as a VAR(1) model, he pa ame e s a e o en imes es ima ed
using he mul i a ia e leas squa es (MLS) me hod, which can be suscep ible o ou lie s.
To accoun o po en ial model iola ions, a maximum immed likelihood es ima ion
(MTLE) app oach is u ilized o de i e a sys em o nonlinea es ima ing equa ions, and an
i e a i e p ocedu e is de eloped o sol e he la e . In addi ion o obus ness, ou new
echnique allows o eliable eco e y o he long- e m mean, unlike exis ing me hodolo-
gies. A se o simula ion s udies ac oss mul iple dimensions, sample sizes and obus ness
con igu a ions a e pe o med. MTLE ou comes a e compa ed o hose o mul i a ia e leas
immed squa es (MLTS), MLE and MLS. Empi ical esul s sugges ha MTLE no only
main ains good ela i e e iciency o uncon amina ed da a bu signi ican ly imp o es
o e all es ima ion quali y in he p esence o da a i egula i ies. Addi ionally, eal da a
examples con aining daily log- ola ili ies o six common asse s (commodi ies and cu en-
cies) and US/Eu o sho a es a e also analyzed. The esul s indica e ha MTLE p o ides
an a ac i e ins umen o in e es a e o ecas ing, s ochas ic ola ili y modeling, isk
managemen and o he applica ions equi ing s a is ical obus ness in complex economic
and inancial en i onmen s.
Keywo ds: econome ic modeling; au o eg essi e models; imes se ies analysis; maximum
immed likelihood es ima ion; s a is ical obus ness; ou lie s
1. In oduc ion
Policymake s, cen al banks, inancial ins i u ions, e c., ely on econome ic modeling
o guide decision making. By u ilizing s a is ical echniques and economic heo y, many
econome ic models ha e been de eloped o unde s and, o example, how s ock ma ke
pe o mance ela es o mac oeconomic indica o s and o quan i y he ela ionships ha shape
ma ke beha io and p edic u u e economic ou comes. Many mac oeconomic and inancial
p ocesses a e empi ically obse ed o ollow he mul i a ia e Vasicek
model (Vasicek,1977)
.
This model is commonly applied o sho - e m in e es a es, log- ola ili ies and o he
mac oeconomic a iables ha exhibi endencies o e e owa d a long- e m a e age
o e ime. By inco po a ing mul iple a iables in o a uni ied amewo k, he mul i a ia e
Vasicek model p o ides a powe ul ool o analyzing he co-mo emen s o economic
indica o s (Campbell e al.,1996;Ego o e al.,2011;Lü kepohl,2005;Sims,1980).
Economies 2025,13, 68 h ps://doi.o g/10.3390/economies13030068
Economies 2025,13, 68 2 o 28
The mul i a ia e Vasicek model is cha ac e ized by h ee undamen al pa ame e s:
he long- e m mean ec o , he mean- e e sion speed ma ix and he squa ed ola ili y
ma ix. The long- e m mean ec o ep esen s he a e age le els a ound which he ec o ial
p ocess oscilla es, and is expec ed o e e o, in he long un unde app op ia e condi ions.
The mean- e e sion speed ma ix go e ns he speed a which he ec o ial p ocess mo es
owa d he long- e m mean; la ge absolu e eigen alues indica e highe speed and ice
e sa. The co a iance ma ix, on he o he hand, ep esen s he squa ed ola ili y in he
luc ua ions a ound he long- e m mean. While all model pa ame e s play an impo an ole
in s a is ical in e ence and economic decision making, inancial analys s and economis s a e
pa icula ly in e es ed in es ima ing he long- e m mean o a ious pu poses, including,
bu no limi ed o, isk managemen and hedging (Fabozzi & Mann,2011), o ecas ing
and planning, e c. The long- e m mean is a c ucial indica o in isk managemen and
hedging, as i p o ides a s able e e ence poin o assess expec ed e u ns, de ec new ends
and op imize s a egies o mi iga e po en ial losses o e ime. Wi h ega d o planning,
he long- e m mean p o ides a eliable benchma k o se ealis ic goals, make in o med
decisions and assess po en ial isks o e ex ended pe iods. A he same ime, es ima ing
he long- e m mean can be pa icula ly challenging due o iden i iabili y issues.
Fo mul i a ia e con inuous- ime models, when he da a a e only obse ed o e a
disc e e ime g id, he pa ame e s can be es ima ed using nume ous pa ame e es ima-
ion app oaches, including Bayesian me hods (E ake ,2001), mul i a ia e leas squa es
(MLS) and me hods based on momen s (Me on,1980). Mo eo e , simula ion-based ech-
niques (Gallan & Tauchen,1996;Gou ie oux e al.,1993) as well as nonpa ame ic ech-
niques (Aï -Sahalia,1995;S an on,1997) also exis . None heless, o ob ain accu a e es i-
ma es o he pa ame e s, simula ion-based me hods ypically equi e many eplica ions
and highe compu a ional ime, especially in highe dimensions. P io knowledge abou
he pa ame e s, which a e usually no a ailable, is addi ionally expec ed o Bayesian
app oaches. Momen -based es ima o s a e less applicable in p ac ice since hey hea ily ely
on heo e ical assump ions and a e suscep ible o ou lie s and hea y ails.
In s a is ical li e a u e, he disc e e Vasicek p ocess is ypically epa ame ized as a
VAR(1) model and calib a ed using he MLS me hod, whose es ima ion powe can be
signi ican ly comp omised in he p esence o ou lie s (Chang & Shi,2024;Hamil on,2020).
Financial ime se ies may be p one o ola ili y shocks o unusual ma ke e en s (B ockwell
& Da is,2002), leading o he in oduc ion o anomalous alues in he da a (Ji e al.,2020;
Pokojo y & Anum,2022;Spel a e al.,2023). These alues may emana e om a shi in he
inno a ion p ocess, a ec ing cu en obse a ions and subsequen ones, o may come om
obse a ion(s) whose own alue(s) a e con amina ed (Chang & Shi,2024). The p esence o
ou lie s in a da ase can cause many es ima o s o p oduce biased o ine icien pa ame e
es ima es. To accoun o ou lie s in mul i a ia e ime se ies, MM-es ima ion (Kud aszow
& Ma onna,2011;Yohai,1987) and mul i a ia e leas immed squa es (MLTS) (Agulló
e al.,2008;C oux & Joossens,2008) a e p oposed in he li e a u e as obus al e na i es
o he MLS app oach. A somewha lowe e iciency o he MLTS es ima o mo i a ed
he de elopmen o eweigh ed mul i a ia e leas immed squa es (RMLTS) (C oux &
Joossens,2008). MM-es ima o s, like mos i e a i e echniques, equi e a choice o ini ial
es ima es o “wa ms a s”. Imp ope speci ica ion o his inpu can a ec he obus ness
and a ine equi a iance o he es ima o . Addi ionally, when he mean- e e sion speed
ma ix is ill-condi ioned, he MLTS app oach ails o es ima e he long- e m mean, while he
ecen ly p oposed modi ied MLE (Pokojo y e al.,2024) app oach can be unduly in luenced
by ou lie s and o he model iola ions. Fu he mo e, Bayesian me hods (E ake ,2008) and
simula ion-based echniques (Gallan & Tauchen,2010) ha e been applied in econome ic
Economies 2025,13, 68 3 o 28
modeling in o de o mi iga e es ima ion biases in oduced by non-Gaussian shocks and
ma ke anomalies.
In his pape , we adop a maximum immed likelihood es ima ion (MTLE) app oach
ins ead o MM-es ima ion o Bayesian al e na i es. While MM-es ima o s a e appealing
as hey o e a balance be ween obus ness and e iciency, hey end o be mo e compu a-
ionally expensi e as hey ypically in ol e cos ly loa ing-poin ope a ions a ising om
compu ing anscenden al unc ions. Also, hey a e ulne able o nume ical ins abili y
in high-dimensional o nea -nons a iona y se ings (Boud e al.,2020). This limi a ion is
pa icula ly acu e in inancial ime se ies wi h clus e ed ou lie s (e.g., du ing ma ke c ises).
Bayesian p ocedu es end o in ol e e en highe compu a ional cos s. We he e o e used
he maximum immed likelihood es ima ion (MTLE) app oach o es ima e he pa ame-
e s while accoun ing o possible ou lie s and pu o h a sys em o nonlinea es ima ing
equa ions wi h an i e a i e p ocedu e o sol e he sys em. No only does ou o mula-
ion accoun o ou lie s, bu i also p o ides egula iza ion h ough he u iliza ion o he
Moo e–Pen ose pseudoin e se when he mean- e e sion speed ma ix is ill-condi ioned.
While being on pa wi h he s a e-o - he-a MLTS me hod a es ima ing he mean- e e sion
speed ma ix and he squa ed ola ili y ma ix, ou new es ima o is also able o e ec i ely
es ima e he long- e m mean o he Vasicek p ocess, especially o e la ge ime ho izons.
In addi ion o me hodological de elopmen s ha unde pin he p oposed MTLE es ima-
o , we also illus a e how his new econome ic ins umen can be used o acili a e obus
analysis o mul i a ia e ime-se ies da a. Focusing on sho - a e and s ochas ic ola ili y
modeling as wo common applica ion examples, we ha e e alua ed he disc e e Vasicek
model o a wide ange o syn he ic da ase s in Sec ion 3.1 co esponding o mul iple
hypo he ical ou lie con igu a ions and wo- eal wo ld da ase s in Sec ion 3.2, espec i ely.
In he con ex o sho - a e modeling and o ecas ing, ou indings a e consis en wi h hose
o Pokojo y e al. (2024) and clea ly demons a e ha ou MTLE no only can se e as a
obus and e icacious al e na i e o MLE o gene al model calib a ion and o ecas ing
pu poses bu signi ican ly ou pe o ms MLTS as i s sole obus compe i o a es ima ing
he long- e m mean. As epo ed by Chang and Shi (2024), adi ional es ima o s o VAR
models may signi ican ly o e es ima e some pa ame e s. Con e sely, obus VAR models
deli e mo e eliable esul s. In he con ex o s ochas ic ola ili y modeling, ou simula-
ion s udy and eal-wo ld example con i m his conclusion while addi ionally p o iding
compelling e idence o MTLE’s supe io i y o e all compe i o s conside ed in es ima ing
he long- e m mean om con amina ed da a. In conclusion, i espec i e o applica ion
domain, ou MTLE me hodology o e s an a ac i e econome ic ins umen o obus and
e icien es ima ion o key pa ame e s necessa y o s a is ical and econome ic in e ence,
o ecas ing and simula ing u u e dynamics, s a is ical p ocess moni o ing and change
poin de ec ion, isk managemen , de elopmen o in es men s a egies, e c.
Ou esul s align wi h and ex end p io wo k on obus es ima ion o mul i a ia e
ime se ies, pa icula ly in he con ex o sho - a e modeling. T adi ional obus me h-
ods, such as mul i a ia e leas immed squa es (MLTS) (Agulló e al.,2008;C oux &
Joossens,2008), ha e demons a ed success in es ima ing pa ame e s o VAR(1) models
unde con amina ion. Howe e , hese me hods we e no explici ly designed o he disc e e
mul i a ia e Vasicek model, whe e he long- e m mean plays a c i ical ole in inancial
applica ions such as isk managemen and po olio op imiza ion (Ego o e al.,2011;Va-
sicek,1977). In line wi h Chang and Shi (2024), who highligh ed he limi a ions o classical
VAR es ima o s in con amina ed se ings, ou simula ions con i m ha MLTS s uggles
o eliably es ima e long- e m mean when he mean- e e sion ma ix is ill-condi ioned o
he calib a ion ho izon is sho . This de iciency is pa icula ly p onounced in empi ical
en i onmen s, whe e small samples and low- equency da a wo sen nume ical ins abili y
Economies 2025,13, 68 4 o 28
(Pokojo y e al.,2024)
. Ou MTLE me hodology add esses his gap by in eg a ing MLTS’s
obus ness wi h a egula ized ixed-poin i e a ion. The imp o emen is consis en wi h
he indings in obus po olio op imiza ion (Ko n & Koziol,2006), whe e egula iza ion
enhances s abili y in ill-condi ioned sys ems. Resul s om ou analysis o US/Eu o sho
a es ex end he wo k o Pokojo y e al. (2024), who demons a ed he non- obus MLE’s
e icacy in uncon amina ed da a. Impo an ly, o he bes o ou knowledge, ou s udy is
he i s o sys ema ically alida e a obus es ima o ’s abili y o eco e long- e m mean in
ini e samples—a c ucial ad ancemen o applica ions like yield cu e modeling and s ess
es ing, whe e accu a e long- e m mean es ima es a e pa amoun
(Fabozzi & Mann,2011)
.
In summa y, while MLTS emains a gold s anda d o obus VAR es ima ion, ou MTLE
b idges a c i ical gap in Vasicek-speci ic applica ions, o e ing a ailo ed solu ion o sho -
a e models plagued by con amina ion and nume ical challenges due o ill-condi ioned
ma ices. This ad ancemen aligns wi h he b oade adi ion in inancial econome ics ha
p io i izes obus ness wi hou sac i icing in e p e abili y (Aï -Sahalia,1995;S an on,1997).
The emainde o his pape is o ganized as ollows. In Sec ion 2, we in oduce he
mul i a ia e Vasicek model, gi e a e iew o VAR p ocesses and discuss MLTS es ima ion
and p esen a disc e e-con inuous op imiza ion p oblem unde lying ou new MTLE es i-
ma o along wi h a sys em o nonlinea es ima ing equa ions and an i e a i e p ocedu e
o sol ing he la e . Sec ion 3.1 p o ocols he esul s o an ex ensi e simula ion s udy
pe o med in his pape o benchma k and compa e he pe o mance o MTLE wi h MLTS,
MLE and MLS a es ima ing mul i a ia e Vasicek pa ame e s. Fu he mo e, we analyze
wo eal-wo ld da ase s and discuss he esul s in Sec ion 3.2. Finally, Sec ion 4summa izes
he pape and p o ides concluding ema ks.
2. Ma e ials and Me hods
The mul i a ia e Vasicek model (Pla en & Rendek,2009) is gi en by he sys em o
s ochas ic di e en ial equa ions (SDEs)
dR =A(R∗−R )d +Σ1/2 dW o ∈[0, T],R0=R0, (1)
o a
p
- a ia e Ma ko ian di usion p ocess
(R ) ≥0
, whe e
R∗∈Rp
deno es he long- e m
mean ec o ,
A∈Rp×p
s ands o he mean- e e sion speed ma ix and
Σ∈Rp×p
is he
posi i e de ini e squa ed ola ili y ma ix, while
(W ) ≥0
is a
p
- a ia e s anda d Wiene
p ocess and he oo ma ix is de ined ia he spec al heo em, i.e.,
Σ1/2 =
p
∑
k=1
λ1/2
keke′
k
wi h
λk
and
ek
,
k=
1,
. . .
,
p
, deno ing he
k
- h eigen alue and eigen ec o o
Σ
, espec i ely.
When s udying low- equency p ocesses, e.g., daily sho a es,
Pokojo y e al. (2024)
conside a disc e e e sion o Equa ion
(1)
ob ained by applying he explici Eule –
Ma uyama disc e iza ion scheme on an equispaced ime g id wi h a cons an ime s ep
∆ =T
n. This esul s in he di e ence equa ion
1
∆ (R j+1−R j) = AR∗−R j+1
∆ Σ1/2(W j+1−W j),j=0, . . . , n−1 (2)
which can equi alen ly be exp essed as a ec o au o eg essi e model
R j+1= (∆ )AR∗+Ip×p−(∆ )AR j+ε j o j=0, . . . , n−1 (3)

Economies 2025,13, 68 5 o 28
wi h inno a ions ε j:=Σ1/2(W j+1−W j)sa is ying
ε j
i.i.d.
∼ N(0p,(∆ )Σ). (4)
See Sec ion 2.1 o de ails.
F om a o mal s andpoin , model iola ions o Equa ions
(3)
and
(4)
can be desc ibed
as a Hube - ype con amina ion
ε j|δ j,ξ j∼(1−δ j)ξ j+δ jG j
δ j
i.i.d.
∼Be noulli(ε),ξ j
i.i.d.
∼ N(0p,(∆ )Σ)(5)
whe e
ε∈[
0,
1
2)
is he popula ion-le el ac ion o ou lie s and
(G j)j=0,...,n−1
is some
unknown (po en ially nons a iona y and au oco ela ed) con amina ion p ocess.
Tu ning back o he uncon amina ed case, using ma hema ical induc ion and in oking
he law o la ge numbe s, he ollowing well-known la ge- ime asymp o ics esul holds. I
is in e es ing o obse e ha he limi in Equa ion
(7)
explains why
R∗
is e e ed o as he
long- e m mean.
Theo em 1 (La ge- ime asymp o ics).The disc e e Vasicek p ocess is explici ly gi en as
R j=Ip×p−(∆ )AjR0+Ip×p−Ip×p−(∆ )AjR∗+
j
∑
l=0Ip×p−(∆ )Aj−lε l.(6)
Assuming he ma ix
Ip×p−(∆ )A
in Equa ion
(2)
is s ic ly con ac ing, i.e., he (possibly
complex) eigen alues o Asa is y
1−(∆ )λ<1 o λ∈σ(A),
We addi ionally ha e
Eh1
T
n−1
∑
j=0
(∆ )R j−R∗2i=O(T−1/2)uni o mly in ∆ →0as T →∞
and, hus,
1
T
n−1
∑
j=0
(∆ )R j=R∗+OP(T−1/2)uni o mly in ∆ →0as T →∞(7)
whe e T = (∆ )n.
Gi en a (single) ba ch o da a
{R 0
,
. . .
,
R n−1}
om a con amina ed model, he goal
is o es ima e he model pa ame e s
R∗
,
A
and
Σ
while a aining a p esc ibed b eakdown
poin o
α∈[
0,
1
2)
. Empi ically, exis ing es ima o s o
R
end o ail e en i
A
is jus sligh ly
ill-condi ioned, which can be u he exace ba ed by he p esence o ou lie s. All es ima o s
based on VAR(1) pa ame iza ion, including he mul i a ia e leas immed squa es (MLTS)
es ima o (see Sec ion 2.1), ail o be applicable in his si ua ion.
Unlike exis ing me hods, ins ead o ea ing
R∗
as a meaningless nominal model
pa ame e and a emp ing o indi ec ly econs uc i om an es ima e o
AR∗
, we a he
le e age he la ge- ime con e gence o he ac ual long- e m mean in acco dance wi h
Equa ion
(7)
. I is impo an o emphasize ha he la e holds e en i no eliable es ima es
o
A
and/o
Σ
a e a ailable. In ligh o his ac , ou app oach (see Sec ion 2.2) le e ages an
Economies 2025,13, 68 6 o 28
in insic connec ion be ween likelihood maximiza ion and he la ge- ime beha io o he
p ocess
(R j) j
(c . Theo em A2 in Appendix A) o eco e he long- e m mean
R∗
. Unlike
con en ional s a is ical iden i iabili y, ou a gumen a ion has a deg ee o eminiscence wi h
bo h s uc u al iden i iabili y and de ec abili y o dynamical sys ems.
2.1. Mul i a ia e Leas T immed Squa es Es ima ion
The mul i a ia e leas immed squa es (MLTS) es ima o (Agulló e al.,2008) is a
obus al e na i e o he mul i a ia e leas squa es me hod o es ima ing he mul i a ia e
eg ession model. The MLTS es ima o is also applicable o es ima ing he pa ame e s
o
VAR(k)
models. Fo a
p
- a ia e disc e e- ime p ocess
(y j) j
wi h
j=j(∆ )
o some
∆ >0, he la e eads as
y j+1=β′
0+β′
1y j+. . . +β′
ky j−k+1+ε j o j=k−1, . . . , n−k(8)
whe e
k∈N
is he lag size,
β0∈Rp
is he in e cep pa ame e ,
β1
,
. . .
,
βk∈Rp×p
a e
he pa ial slopes, and
ε i.i.d.
∼ N(0p
,
Σ)
a e he e o s. Fo lag leng h 1, he
VAR(k=
1
)
model becomes
y j+1=β′
0+β′
1y j+ε j o j=0, 1, . . . , n−1 (9)
which can be exp essed in he o m o he mul i a ia e eg ession model
y j+1=β′x j+ε j(10)
whe e
x j= (
1,
y′
j)′∈Rp+1
and
β= (β′
0
,
β′
1)′∈R(p+1)×p
is he ma ix con aining he
eg ession coe icien s. No e ha gene al
VAR(k)
models can be educed o he Ma ko ian
case o VAR(1)by ex ending he phase space o include kp e ious p ocess alues.
Le ing
X= (x 0
,
. . .
,
x n−1)′∈Rn×(p+1)
and
Y= (y 1
,
. . .
,
y n)′∈Rn×p
deno e he
design ma ix and he esponse ma ix, espec i ely, he usual non- obus mul i a ia e leas
squa es es ima o s o βand Σ
ˆ
βMLS = (X′X)−1X′Y,ˆ
ΣMLS =1
n−p−1(Y−Xˆ
βMLS)′(Y−Xˆ
βMLS)
ollow.
Akin o he Minimum Co a iance De e minan (MCD) o Rousseeuw (1984), obus
MLTS es ima ion in ol es inding a subse o
h
obse a ions such ha he MLS i o
hese obse a ions minimizes he de e minan o he esidual co a iance. Fo an index se
I ⊂ {
0,1,
. . .
,
n−
1
}
wi h
|I| =h
, he MLS es ima o s based on a subsample indexed by
I
a e gi en as
ˆ
βMLS(I) = (X′
IXI)−1X′
IYI
ˆ
ΣMLS(I) = 1
h−p−1YI−XIˆ
βMLS(I)′YI−XIˆ
βMLS(I).
The MLTS es ima o s o βand Σa e hen exp essed as
ˆ
βMLTS =ˆ
βMLS(I∗),ˆ
ΣMLTS =cp,αˆ
ΣMLS(I∗)(11)
whe e
I∗=a gmin
I∈{0,1,...,n−1}
|I|=h
log de ˆ
ΣMLS(I)(12)
and cp,αis a consis ency ac o de ined in Equa ion (21) below.
Economies 2025,13, 68 7 o 28
The es ima o s
ˆ
βMLTS
and
ˆ
ΣMLTS
se e as a obus al e na i e o MLS es ima ion.
The MLTS es ima o s ha e been shown o be high-b eakdown and Fishe -consis en
(Agulló e al.,2008)
, while asymp o ic- and
√n
-consis ency a e only known o scala -
on- ec o LTS eg ession (Víšek,2006a,2006b;Zuo,2024).
The combina o ial op imiza ion p oblem in Equa ion
(12)
is ypically sol ed wi h
a sampling-based algo i hm ha in ol es i e a i e applica ion o he concen a ion
s ep
(C-s ep)
(Agulló e al.,2008;Rousseeuw & Van D iessen,1999). F om a heo e -
ical s andpoin ,
de (·)
o
logde (·)
can equi alen ly be used in Equa ion
(12)
; how-
e e , compu a ional implemen a ions ypically p e e he la e due o be e nume ical
s abili y p ope ies.
The mul i a ia e Vasicek model in Equa ion
(2)
can be exp essed as a
VAR(
1
)
-p ocess
R j+1=β′
0+β′
1R j+ε j(13)
whe e
β′
0= (∆ )AR∗
,
β′
1=Ip×p−(∆ )A
and
Co [ε j]≡Ξ= (∆ )Σ
. This epa ame iza-
ion is equi alen i and only i
A
is in e ible since he o iginal Vasicek pa ame e s can be
econs uc ed om he VAR(1)pa ame e s ia
R∗=1
∆ A−1β′
0,A=1
∆ (Ip×p−β′
1),Σ=1
∆ Ξ, (14)
which leads o eponymous MLTS es ima o s o Vasicek pa ame e s. Theo e ically, maxi-
mum likelihood es ima ion o he Vasicek model is asymp o ically equi alen wi h ha
o VAR(1) on he s eng h o he in a iance p inciple. In u n, maximum likelihood es i-
ma ion o he Vasicek model is hen also equi alen wi h he leas squa es es ima ion o
he
VAR(
1
)
model (up o he Bessel co ec ion ac o o he es ima e o
Σ
). Howe e , i
A
is e en sligh ly ill-condi ioned, Equa ion
(14)
ails o u nish a easonable es ima e o he
long- e m mean R∗e en in ela i ely la ge samples.
2.2. Maximum T immed Likelihood Es ima ion
Ins ead o epa ame izing he Vasicek model in Equa ion
(2)
as a
VAR(
1
)
model and
applying maximum likelihood es ima ion o he la e , simila o Pokojo y e al. (2024) in
he non- obus si ua ion, we pu o h a sys em o es ima ing equa ions o he maximum
immed likelihood es ima o (MTLE) and sol e i wi h a nume ical scheme based on
i e a i e applica ion o he C-s ep. Replacing he in e se o
A
wi h a sui ably unca ed
Moo e–Pen ose pseudoin e se, a nume ically s able es ima e o he long- e m mean
R∗
can be ob ained.
In he absence o ou lie s, Pokojo y e al. (2024) discuss maximum likelihood es i-
ma ion o he disc e e Vasicek model
(2)
. In oducing he log-likelihood unc ion (scaled
by ∆ )
ℓθ|R= (∆ )
n−1
∑
j=0
log φ∆R j−A(R∗−R j)∆ 0p,(∆ )Σ(15)
whe e
θ= (R∗
,
A
,
Σ)
,
R= (R 0
,
R 1
,
. . .
,
R n)′
,
∆R j:=R j+1−R j
and
φ(x|µ
,
Σ)
is he
p- a ia e Gaussian densi y, he maximum likelihood es ima o is de ined as
ˆ
θMLE ≡(ˆ
R∗
MLE,ˆ
AMLE,ˆ
ΣMLE) = a gmax
θ∈Θ
ℓθ|R≡a gmax
(R∗,A,Σ)∈Θ
ℓR∗,A,Σ|R(16)
wi h open pa ame e se Θ=(R∗,A,Σ)|R∗∈Rp,A egula , Σ∈Rp×p,Σ′=Σ,Σ≻0.
Fixing an in ege
h
and le ing
α:=
1
−h/n
, in lieu o he likelihood unc ion in
Equa ion (15), we conside he immed scaled log-likelihood unc ion
Economies 2025,13, 68 8 o 28
ℓIθ|R= (∆ )∑
j∈I
log φ∆R j−A(R∗−R j)∆ 0p,(∆ )Σ
=−(1−α)pT
2log 2π(∆ )−(1−α)T
2log |Σ|
−1
2∑
j∈I
d2∆R j−(∆ )A(R∗−R j)|0p,(∆ )Σ
(17)
o a bi a y I ⊂ {0, 1, . . . , n−1}wi h |I| =h, whe e
d2(x|µ,Σ) = (x−µ)′Σ−1(x−µ)(18)
is he usual squa ed Mahalanobis dis ance unc ion.
We de ine he maximum immed likelihood es ima o (MTLE) as
ˆ
R∗
MTLE :=ˆ
R∗
†,ˆ
AMTLE :=ˆ
A†,ˆ
ΣMTLE :=c2
p,αˆ
Σ† o α=1−h
n(19)
wi h
(I†,ˆ
θ†):=a gmax
θ∈Θ
I⊂{0,1,...,n−1},|I|=h
ℓI(θ|R)(20)
whe e he co a iance es ima e is adjus ed using he asymp o ic bias co ec ion ac o
c2
p,α=1−α
P{χ2
p+2≤χ2
p,1−α}. (21)
In his pape , we p ima ily ocus on he obus ness o ou es ima o . The e o e,
we solely conside un eweigh ed o “ aw” es ima o s. Howe e , he usual eweigh ing
app oach (c . Agulló e al. (2008); C oux and Joossens (2008) o eweigh ed MLTS) can
be adop ed o ob ain eweigh ed coun e pa s should one be in e es ed in inc easing he
s a is ical e iciency o he es ima o .
The double maximiza ion p oblem in Equa ion
(20)
is a mixed p og amming p oblem
in ol ing disc e e op imiza ion in
I
and con inuous op imiza ion in
θ
. Fixing
θ∈Θ
, he
op imum o
ℓ(·)
wi h espec o
I
is a ained a
I={i1
,
i2
,
. . .
,
ih}
, whe e
di1≤di2≤ ··· ≤
dina e he so ed squa ed Mahalanobis dis ances
d2
j=d2∆R j−(∆ )A(R∗−R j)|0p,(∆ )Σ.
Con e sely, ixing
I
, on he s eng h o (Pokojo y e al.,2024, Theo em 1), he op imum
o ℓ(·)wi h espec o θis a ained a a solu ion o he es ima ing equa ions
R∗=1
h∑
j∈I
R j+1
(1−α)TA−1∑
j∈I
(R j+1−R j), (22)
A=1
∆ ∑
j∈I
(R j+1−R j)(R∗−R j)′∑
j∈I
(R∗−R j)(R∗−R j)′−1
, (23)
Σ=1
(1−α)T∑
j∈I ∆R j−AR∗−R j(∆ )∆R j−AR∗−R j(∆ )′. (24)
Ins ead o he usual ma ix in e se, a sui able unca ed Moo e–Pen ose pseudoin e se
A−1=
p
∑
k=1
1[ϵ,∞)(σk/∥A∥)1
σk
uk ′
kwi h ∥A∥=max
k=1,...,kσk
Economies 2025,13, 68 15 o 28
he es ima es ob ained wi h MTLE a e e y s able i espec i e o he p esence o absence o
ex eme obse a ions. Thus, MTLE is bo h mo e obus and less biased. These obse a ions
a e consis en wi h Sec ion 3.1.1. When es ima ing
A
and
Σ
, non- obus es ima o s, MLE
and MLS, ha e smalle e o s compa ed o MLTS and MTLE, al hough he di e ence is no
e y la ge. These esul s highligh wha was obse ed ac oss he di e en con igu a ions
conside ed unde his simula ion se ing.
100 150 200 250 300 350
0
5
10
15
20
25
30
100 150 200 250 300 350
0.1
0.15
0.2
0.25
0.3
0.35
0.4
100 150 200 250 300 350
0
0.2
0.4
0.6
0.8
1
Figu e 3. Simula ed c
e alues o ε=0.20, ncp =25 and bdp =0.25.
Table 3. Simula ed c
e alues o ε=0.20, ncp =25 and bdp =0.25.
ˆ
R∗ˆ
Aˆ
Σ
TMTLE MLTS MLE MLS MTLE MLTS MLE MLS MTLE MLTS MLE MLS
60 0.39 62.41 2.15 17.06 0.3648 0.3684 0.3041 0.3042 0.9192 0.9501 0.4559 0.4559
90 0.41 405.71 1.33 5.04 0.2906 0.2918 0.2354 0.2354 0.6142 0.6232 0.3422 0.3422
120 0.42 14.48 0.97 0.99 0.2475 0.2487 0.2011 0.2011 0.4804 0.4832 0.2865 0.2865
150 0.42 24.23 0.83 0.84 0.2242 0.2248 0.1809 0.1809 0.4167 0.4187 0.2542 0.2542
180 0.42 18.34 0.76 0.76 0.2041 0.2048 0.1659 0.1659 0.3690 0.3698 0.2306 0.2306
210 0.43 8.59 0.71 0.71 0.1915 0.1922 0.1559 0.1559 0.3393 0.3401 0.2136 0.2136
240 0.43 6.66 0.67 0.67 0.1807 0.1810 0.1482 0.1482 0.3128 0.3141 0.2001 0.2001
270 0.43 4.02 0.64 0.65 0.1721 0.1724 0.1412 0.1412 0.2934 0.2944 0.1890 0.1890
300 0.43 8.96 0.62 0.62 0.1648 0.1648 0.1358 0.1358 0.2761 0.2767 0.1800 0.1800
330 0.43 5.18 0.61 0.61 0.1590 0.1592 0.1317 0.1317 0.2636 0.2644 0.1721 0.1721
360 0.43 1.40 0.59 0.59 0.1538 0.1538 0.1283 0.1283 0.2503 0.2510 0.1656 0.1656
100 150 200 250 300 350
0
5
10
15
20
25
30
100 150 200 250 300 350
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
100 150 200 250 300 350
0
0.5
1
1.5
Figu e 4. Simula ed c
e alues o ε=0.10, ncp =25 and bdp =0.35.
In summa y, when he e a e no ou lie s in he da a (
ε=
0), he simula ion esul s
sugges ha MTLE pe o ms on pa wi h he non- obus es ima o MLE and, o some ex en ,
MLS, while he ad an age o MTLE becomes mos p onounced o
ε>
0, especially when
es ima ing
R∗
. These obse a ions highligh he need o obus es ima o s, in pa icula
hose able o eliably es ima e he long- e m mean, as is he case wi h ou MTLE.

Economies 2025,13, 68 16 o 28
Table 4. Simula ed c
e alues o ε=0.10, ncp =25 and bdp =0.35.
ˆ
R∗ˆ
Aˆ
Σ
TMTLE MLTS MLE MLS MTLE MLTS MLE MLS MTLE MLTS MLE MLS
60 0.18 29.79 1.52 28.06 0.3966 0.4010 0.3037 0.3037 1.5352 1.5462 0.4546 0.4546
90 0.18 10.75 0.99 8.82 0.3156 0.3172 0.2342 0.2342 0.8691 0.8649 0.3420 0.3420
120 0.19 7.70 0.59 1.49 0.2688 0.2716 0.1997 0.1997 0.6275 0.6280 0.2852 0.2852
150 0.20 7.71 0.46 0.46 0.2429 0.2435 0.1788 0.1788 0.5286 0.5276 0.2532 0.2532
180 0.20 5.82 0.41 0.41 0.2211 0.2219 0.1639 0.1639 0.4571 0.4577 0.2296 0.2296
210 0.20 14.24 0.38 0.38 0.2065 0.2069 0.1532 0.1532 0.4173 0.4165 0.2121 0.2121
240 0.20 2.78 0.36 0.36 0.1944 0.1946 0.1449 0.1449 0.3820 0.3828 0.1985 0.1985
270 0.20 1.11 0.34 0.34 0.1853 0.1853 0.1386 0.1386 0.3559 0.3560 0.1872 0.1872
300 0.20 1.26 0.33 0.33 0.1766 0.1770 0.1333 0.1333 0.3336 0.3330 0.1786 0.1786
330 0.20 0.87 0.32 0.32 0.1699 0.1700 0.1287 0.1287 0.3170 0.3170 0.1708 0.1708
360 0.20 1.92 0.31 0.31 0.1641 0.1640 0.1252 0.1252 0.3007 0.3010 0.1634 0.1634
3.2. Examples
Two eal-wo ld da ase s a e s udied in his sec ion. In Sec ion 3.2.1, we analyze a
bi a ia e ime se ies o simul aneous US and EU sho a es o iginally s udied by Pokojo y
e al. (2024). The Vasicek model is calib a ed wi h each o he ou me hods, MTLE, MLTS,
MLS and MLE, on one yea ’s wo h o his o ical da a and used o pe o m a 3-mon h
o ecas . Sec ion 3.2.2 builds upon he wo k o Chang and Shi (2024), who analyze a da ase
composed o log- ola ili ies o six common asse s. While he la e pape used a VAR(1)
model in hei analyses, we a he use he disc e e- ime Vasicek model and show how
MTLE can o e majo ad an ages a es ima ing he long- e m mean pa ame e .
3.2.1. Daily T easu y and Eu o Pa Yields
This example combines wo publicly a ailable inancial da ase s, he Daily T easu y
Pa Yield Cu e Ra es and he Daily Eu o Pa Yield Cu e Ra es, o iginally p esen ed
by Pokojo y e al. (2024). These da ase s a e a ailable om he US Depa men o he
T easu y (US Depa men o he T easu y,2024) and he Eu opean Cen al Bank (Eu opean
Cen al Bank,2024), espec i ely. The Daily T easu y Pa Yield Cu e Ra es da ase con ains
daily es ima es o US T easu y secu i ies’ yield cu es, wi h ma u i ies anging om
1 mon h
o 30 yea s, spanning om Janua y 2023 o Decembe 2023. This yield cu e se es
as a key indica o o he ela ionship be ween yield and ma u i y o US T easu y secu i ies,
which a e ega ded as isk- ee in es men s. Simila ly, he Daily Eu o Pa Yield Cu e Ra es
da ase p o ides yield da a o Eu o-denomina ed go e nmen bonds, co e ing he same
ma u i y ange and ime pe iod. This da ase includes yields on seconda y ma ke - aded
bonds, wi h sepa a e cu es o AAA- a ed Eu o-a ea cen al go e nmen bonds and all
Eu o-a ea cen al go e nmen bonds. The da a a e upda ed on a ge business days, wi h
addi ional de ails on ze o-coupon, o wa d and Pa Yield cu es a ailable on he Eu opean
Cen al Bank’s websi e.
In ou analysis, we calib a ed a mul i a ia e disc e e Vasicek model on a bi a ia e ime
se ies cons uc ed om he Daily T easu y and AAA- a ed Eu o-a ea 3-mon h Pa Yield
da a, spanning om 1 Janua y 2023 o 31 Decembe 2023. The 3-mon h ma u i y was chosen
as a p oxy o he sho a e. As poin ed ou by Pokojo y e al. (2024), du ing his pe iod,
he FED Funds a e and he Eu opean Cen al Bank’s main e inancing a e emained
ela i ely s able, allowing us o apply he model wi hou adjus men s o po en ial change
poin s. We hen assessed he pe o mance o he maximum immed likelihood es ima o
(MTLE), mul i a ia e leas immed squa es (MLTS), maximum likelihood es ima o (MLE)
and mul i a ia e leas squa es (MLS) es ima o s using 3-mon h ma u i ies om Janua y
2024 o Ma ch 2024. Fo consis ency ac oss business days, holidays and weekends we e
Economies 2025,13, 68 17 o 28
impu ed h ough linea in e pola ion and ex apola ion. The MTLE and MLTS me hods
we e speci ically chosen o p o ec agains po en ial ou lie s due o i egula i ies, sudden
shi s and/o con inuous d i s in inancial da a.
In all ou panels o Figu e 5, his o ical da a a e displayed o he le o he e ical
do ed line, ma king he calib a ion pe iod (1 Janua y 2023–31 Decembe 2023). To he
igh side o he do ed line, o ecas ing esul s gene a ed based on each o he es ima o s
a e plo ed, displaying bo h he p edic ed mean and he 90% p edic ion in e als. These
p ojec ions se e o compa e each es ima o ’s capaci y o ob ain igh p edic ion bounds
while main aining a desi ed con idence le el. As can be seen, unlike hei non- obus com-
pe i o s, bo h MTLE and MLTS p oduced igh e and mo e upwa ds-poin ing con idence
egions due o hei abili y o il e ou ou lie s and change(s) in end. Using sho a es o
p ice bonds o a bi a y ma u i ies (Mamon,2004), one could simila ly compu e alue a
isk (VaR) o o he con en ional isk managemen me ics o hese bonds.
Figu e 5. His o ic US/EU 3-mon h a es (1 Janua y 2023–31 12 Decembe 2023) as well as o ecas ed
mean and 90% p ojec ion bands (1 Janua y 2024–31 Ma ch 2024).
Figu e 6displays con ou plo s o bi a ia e densi y unc ions o he o ecas ed sho
a e dis ibu ion on 31 Ma ch 2024 unde he Vasicek model calib a ed using each o he
ou es ima o s unde compa ison. The plo s indica e ha he dis ibu ion o he o ecas ed
sho a e appea s o be Gaussian, as Equa ion
(6)
in Theo em 1p edic s. The igu e also
sugges s ha he dis ibu ions a ising om MLS and MLE es ima o s a e mo e sp ead
ou , indica ing g ea e unce ain y in compa ison wi h he obus compe i o s (MTLE and
MLTS), which pe o m head- o-head.
5.2 5.4 5.6 5.8 6
3
3.2
3.4
3.6
3.8
4
4.2
4.4
5.2 5.4 5.6 5.8 6
3
3.2
3.4
3.6
3.8
4
4.2
4.4
5.2 5.4 5.6 5.8 6
3
3.2
3.4
3.6
3.8
4
4.2
4.4
5.2 5.4 5.6 5.8 6
3
3.2
3.4
3.6
3.8
4
4.2
4.4
Figu e 6. The con ou plo s o he p obabili y densi y unc ion o he o ecas ed sho a e
R
dis ibu ion on 31 Ma ch 2024.
Economies 2025,13, 68 18 o 28
To u he e alua e and benchma k model pe o mance, we compu ed sphe ed em-
pi ical esiduals based on pa ame e es ima es gi en in Table 5. The esiduals, a e being
deco ela ed and s anda dized, a e displayed in Figu e 7 o each es ima o accompanied by
95% p edic ion ci cles. Bo h MLE and MLS iden i y up o i e empi ical sphe ed esiduals
as ou lie s and appea o ha e an iden ical pe o mance. A compa able pe o mance o
hese wo non- obus es ima o s can also be seen om he es ima es o he pa ame e s
epo ed in Table 5. MTLE and MLTS, on he o he hand, labeled mo e sphe ed esiduals
(up o 14 poin s) as ou lie s. These wo es ima o s also appea o ha e simila pe o -
mance o his example, which is also con i med by Table 5. The la ge numbe o esiduals
lagged by hese me hods may indica e he exis ence o possible model inadequacies, e.g., a
hea ie - han-Gaussian ail, o could be caused by wha is known as “swamping” e ec s
(Jobe & Pokojo y,2015).
Table 5. Pa ame e es ima es using MTLE (
bdp =
0.2), MLTS (
bdp =
0.2), MLE and MLS es ima o s.
ˆ
R∗ˆ
Aˆ
Σ
MTLE 5.6890
4.0105 0.0171 −0.0050
−0.0050 0.0167 10−3·0.2526 −0.0396
−0.0396 0.6459 
MLTS 5.6706
3.9619 0.0170 −0.0049
−0.0049 0.0175 10−3·0.2548 −0.0427
−0.0427 0.6359 
MLE 5.5689
3.7141 0.0299 −0.0116
−0.0116 0.0233 10−3·0.9193 −0.0435
−0.0435 0.7551 
MLS 5.5705
3.7171 0.5269 −0.0116
−0.0116 0.0232 10−3·0.9193 −0.0435
−0.0435 0.7551 
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
Figu e 7. Sphe ed empi ical esiduals o MTLE (
bdp =
0.2), MLTS (
bdp =
0.2), MLE and MLS
es ima o s wi h espec i e 95% p edic ion ci cles.
Fo addi ional pe o mance assessmen , we epo empi ical oo mean squa ed
e o (RMSE)
[
MSE( j) = 1
10,000
10,000
∑
i=1ˆ
RUS
j,i−RUS
j,ob 2+ˆ
REU
j,i−REU
j,ob 2, (27)
in he le -hand panel and he mean absolu e pe cen age e o (MAPE)
MAPE( j) = 1
10,000
10,000
∑
i=1
ˆ
RUS
j,i−RUS
j,ob 
RUS
j,ob 
+ˆ
REU
j,i−REU
j,ob 
REU
j,ob 
, (28)
in he igh -hand panel o Figu e 8, calcula ed om a Mon e Ca lo simula ion o size
N=
10
,
000, whe e
R j,ob
ep esen s he obse ed US and EU a es and
ˆ
R j,i
deno es
o ecas ed a es om each simula ion un.
Economies 2025,13, 68 19 o 28
Jan Feb Ma Ap
2024
0
0.05
0.1
0.15
0.2
0.25
0.3
Jan Feb Ma Ap
2024
0
1
2
3
4
5
6
7
8
Figu e 8. Empi ical back es ing oo -MSE and MAPE using MTLE, MLTS, MLE and MLS es ima o s.
Bo h MTLE and MLTS es ima o s we e uned a a nominal b eakdown poin o
bdp =0.2
and used, alongside MLE and MLS, o compu e he p ojec ed US and EU a es.
The esul s om his back es ing analysis, which a e epo ed in Figu e 8, u he bu ess
he ad an age o he obus es ima o s o e he classical ones seen in he p e ious igu e.
MTLE and MLTS pe o m head- o-head and ou pe o m MLE and MLS, howe e , no
emendously in his example.
As b ie ly desc ibed in Sec ion 1, wi h obus ly es ima ed Vasicek pa ame e s a hand,
a wide a ie y o p ac ically ele an p oblems can be add essed. Fo example, s a ing
in a s eady s a e, a obus mul i a ia e Shewha
X
-cha (Mason & Young,2002) can be
cons uc ed o de ec shi s in he long- e m mean based on a Ho elling-like
signal s a is ic
T2
k= (R k−ˆ
R∗)′ˆ
Ψ−1(R k−ˆ
R∗)
wi h he asymp o ic co a iance ma ix
ˆ
Ψ= (∆ )
∞
∑
n=0Ip×p−(∆ )ˆ
Aˆ
ΣIp×p−(∆ )ˆ
A′n
= (∆ )Ip×p−Ip×p−(∆ )ˆ
Aˆ
ΣIp×p−(∆ )ˆ
A′−1
whe e
∆ >
0 is assumed su icien ly small so ha he la e geome ic se ies (c . Theo em 1)
con e ges wi h espec o a sui able
ˆ
Σ1/2
-weigh ed ope a o no m. The well-known S ahel–
Donoho es ima o has ecen ly been s udied in he con ex o i.i.d. da a by Raji e al.
(2021). Since
R
s a e au oco ela ed, a co a iance co ec ion due o G imshaw (2023) can
be adop ed o u he enhance he pe o mance o he cha . Fixing he in-con ol a e age
un leng h (IC ARL) a a desi ed le el, e.g.,
ICARL =
365 days, he cha would issue a
alse ala m e e y
ICARL
ime pe iod, e.g., 365 days, on a e age p o ided he long- e m
mean emains unchanged, while being sensi i e o ac ual shi s in he la e pa ame e . This
obus moni o ing scheme can be used o de ec c edi egimes o po olio ebalancing
o used as a model isk managemen ool, e c. Wi h MTLE ou pe o ming all compe i o s
a es ima ing
R∗
as demons a ed in his pape , we expec his cha o be supe io o i s
MLS-, MLE- and MLTS-based compe i o s.
Ano he p omising applica ion ield lies in op imal po olio alloca ion. Mul i a ia e
obus po olio op imiza ion ex ends he classical mean– a iance amewo k (Ma kowi z,
1952) by inco po a ing s a is ical obus ness in o po olio selec ion, pa icula ly o bond
po olios in luenced by mul i- ac o e m s uc u es (Ko n & Koziol,2006). In his app oach,
he expec ed po olio e u n is modeled as a ec o
R =w′µ
, whe e
w
is he weigh
Economies 2025,13, 68 20 o 28
alloca ion ac oss bonds and
µ
is he ec o o expec ed e u ns. Simila ly, he po olio
isk is cap u ed by a ull co a iance ma ix
Σ =w′Σ w
p ese ing he mul i-dimensional
dependencies among di e en bonds and in e es a e ac o s. A key inno a ion in his
obus amewo k is he inco po a ion o a obus es ima o , such as ou p oposed MTLE,
o es ima e he mul i a ia e Vasicek model pa ame e s, ensu ing ha ou lie con amina ion
and model iola ions do no dis o he op imiza ion p ocess. Unlike adi ional es ima ion
me hods ha a e sensi i e o anomalies in inancial da a, he obus app oach le e ages a
immed likelihood echnique o p o ide mo e eliable es ima es o he long- e m mean
and co a iance s uc u e. This ensu es ha he po olio op imiza ion emains s able and
e icien e en in he p esence o ex eme ma ke condi ions. The inal op imiza ion p oblem
seeks o maximize he obus Sha pe a io while accoun ing o hese obus ly es ima ed pa-
ame e s, leading o a mo e esilien and heo e ically g ounded bond
po olio alloca ion.
As a simple illus a ion, we ea he a o edesc ibed sho - e m US and EU bonds as a
mul i-cu ency cash accoun . A e a daily compounding (
∆ =
1 day) un il he ime pe iod
n= (∆ )n, he o iginally deposi ed USD 1 and EUR 1 will p oduce he e u ns
n
∏
l=01+ (∆ )RUS
l−1and n
∏
l=01+ (∆ )REU
l−1,
espec i ely. Fo simplici y, we assume he exchange a e is nea ly cons an (should his
assump ion be iola ed, he Vasicek model can be ecalib a ed on his o ical sho da a a e
adjus ing hem o a iable exchange a e) and cons uc a 90-day po olio (s a ing on
1 Janua y 2024) wi h a minimum e u n a e o 1.1% maximizing he Sha pe a e. Using
Equa ion
(3)
o simula e u u e e u ns, he mean ec o and co a iance ma ix o he
e u ns can be compu ed unde each o he ou choices o he es ima o s. Excluding
bo owing and sho -selling, he usual quad a ic p og amming p oblem
w′Σ w→min o e w∈[0,1]2such ha 1′
2w=1 and w′µ ≥0.011 (29)
can be easily sol ed o ob ain he op imal weigh s (Gold a b & Idnani,1983). Op imal po -
olio weigh s, es ima ed mean ec o and co a iance o e u ns as well as a e o e u ns a e
epo ed in Table 6. As can be seen, e en despi e he e y sho 90-day in es men ho izon,
he obus me hodologies (MTLE and MLTS) o e a a e supe io o non- obus coun e -
pa s, wi h ou p oposed MTLE ou pe o ming all compe i o s. Though beyond he scope
o his wo k, s udying inhe en ly iskie bonds o longe ma u i ies
(Ko n & Koziol,2006)
in lieu o a cash accoun , one would na u ally expec obus po olios o exhibi e en
s onge dominance.
Table 6. Ma kowi z-s yle op imal po olios consis ing o US T easu ies and 3-mon h Eu o bonds
based on e u ns ob ained om MTLE, MLTS, MLE and MLS es ima o s wi h a a ge a e o e u n
o 1.1% o e a 90-day ime pe iod s a ing on 1 Janua y 2024.
Es ima ed Mean Es ima ed Co a iance Op imal Op imal
Vec o o Re u ns µ Ma ix o Re u ns Σ US/EU Weigh s Ra e o Re u n
MTLE 0.0137
0.009410−7·0.1964 0.1338
0.1338 0.5286 0.5036
0.49641.1571%
MLTS 0.0137
0.009310−7·0.1991 0.1340
0.1340 0.5059 0.5036
0.49641.1550%
MLE 0.0137
0.009210−7·0.4588 0.3192
0.3192 0.5900 0.5037
0.49631.1476%
MLS 0.0137
0.009210−7·0.4548 0.3170
0.3170 0.5815 0.5037
0.49631.1472%

Economies 2025,13, 68 21 o 28
3.2.2. Commodi ies and Cu encies
This example aims o e isi he analysis o six common asse (commodi ies and
cu encies) p ices (exp essed in US dolla s), iz., he u u es o gold (XAU), sil e (XAG),
B en oil (BRE) and Wes Texas In e media e oil (WTI) and he cu encies o Swiss F ancs
(CHF) and Japanese Yen (JPY), anging om July 2017 o June 2020, o iginally pe o med
by Chang and Shi (2024). The e a e 771 eco ds o each asse . To demons a e how
he mul i a ia e Vasicek model can be applied o s ochas ic ola ili ies, we analyzed he
( ans o med) daily log- ola ili ies o he a o emen ioned asse p ices di ec ly p o ided
by he au ho s o Chang and Shi (2024). The daily ola ili y is calcula ed as he oo o
summa ion o squa ed hou ly close p ices (Chang & Shi,2024). Figu e 9plo s he daily
logged ola ili ies o e he en i e ange conside ed.
The sex i a ia e ime se ies appea s o be s a iona y om July 2017 un il he ea ly
mon hs o 2020, when he ola ili ies (i.e., ola ili ies-o -(log-) ola ili y) s a o ise h ough
he hi d mon h o 2020. The e is a s ong indica ion o a po en ial shi in he “long- e m”
mean o he mul i a ia e p ocess. The shi can p obably be a ibu ed o a spike in ola ili y
associa ed wi h he onse o he COVID-19 pandemic. One can also obse e om he plo s
ha he inc eased ola ili ies in all o he six ime se ies appea o e e o head down
owa d hei no mal le els. In gene al, he cu encies appea o ha e lowe ola ili ies han
he oil p ices.
While Chang and Shi (2024) ho oughly in es iga ed how obus es ima ion using
MLTS can imp o e es ima ion and in e ence o he mean- e e sion speed
A
and he
squa ed ola ili y ma ix
Σ
in he p esence o ou lie s, we solely ocus on long- e m mean
es ima ion in ou analyses as ou MTLE is known o pe o m head- o-head wi h MLTS
when applied o he o he wo pa ame e s. Thus, all ad an ages o MLTS epo ed by
Chang and Shi (2024) a e sha ed by ou MTLE.
Since no “g ound u h” alue is known o he long- e m mean
R∗
o ou ime se ies,
we chose o pe o m a di e en ype o expe imen . To his end, we c ea ed di e en
subse s o daily log- ola ili ies o each o he asse s and es ima ed
R∗
on each subse . The
subse s we e c ea ed in he same ashion mo ing a e ages a e compu ed, i.e., by “sliding”
ac oss he da ase based on some gi en window size
w
. Choosing
w=
50, we o med he
ollowing (n−w+1)con iguous subse s:
{x(1), . . . , x(w)},{x(2), . . . , x(w+1)},{x(3), . . . , x(w+2)}, . . . , {x(n−w+1), . . . , x(n)}.
Bo h obus es ima o s, MTLE and MLTS, uned wi h
bdp =
0.25 and non- obus MLE
and MLS es ima o s we e employed o es ima e
R∗
om each o he subse s. The esul ing
mo ing “long- e m means” o he log- ola ili ies o he asse s a e plo ed s. subse index
in Figu e 10. The plo s show ha MLTS can be seen o be o e ly ola ile and “exploded” on
many occasions. The leng hy upwa d and downwa d “candles” o spikes obse ed in he
plo s a e indica i e o occasions on which he MLTS ails as an adequa e s a is ical es ima o .
Ou MTLE, on he o he hand, exhibi s good s a is ical obus ness and mode a e a iabili y
compa ed o he o he es ima o s. Th ough he end o 2019 (up o subse index 595), MTLE
is essen ially ollowing he o he non- obus es ima o s, sugges ing only mino luc ua ions
in he i s wo momen s o he log- ola ili y p ocess. In con as , s a ing in ea ly 2020 (a e
subse index 595), MTLE s a s o exhibi mode a e oscilla ions, which could be indica i e
o possible changes in he “long- e m” mean and/o he squa ed ola ili y o he p ocess.
In summa y, MTLE is clea ly p e e ed o e MLTS. Addi ionally, some e idence exis s ha
MTLE should be chosen o e MLE and MLS s a ing in ea ly 2020 o be e cap u e a shi
in p ocess pa ame e s o his da ase .
Economies 2025,13, 68 22 o 28
Jan 2018 Jan 2019 Jan 2020
-2
-1
0
1
2
Jan 2018 Jan 2019 Jan 2020
-2
-1
0
1
2
3
Jan 2018 Jan 2019 Jan 2020
-1
0
1
2
3
4
Jan 2018 Jan 2019 Jan 2020
-1
0
1
2
3
4
5
6
Jan 2018 Jan 2019 Jan 2020
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Jan 2018 Jan 2019 Jan 2020
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Figu e 9. Daily logged ola ili ies: July 2017–June 2020.
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
0 100 200 300 400 500 600 700 800
-3
-2
-1
0
1
2
3
Figu e 10. Es ima es o R∗ o daily log- ola ili ies wi h w=50.
4. Conclusions
Modeling econome ic and inancial p ocesses is essen ial o analyzing and o ecas ing
dynamic in e ac ions be ween mul iple po en ially use ul economic and inancial a iables.
Economies 2025,13, 68 23 o 28
The mul i a ia e Vasicek model, o en imes epa ame ized as a VAR(1) model, is widely
used as a quan i a i e desc ip ion o he mean- e e ing dynamics ypical o sho a es,
s ochas ic log- ola ili ies and o he mac oeconomic p ocesses. Nume ous s udies ha e
emphasized he impo ance o s a is ical obus ness in econome ic modeling and why he
esul s ob ained wi h he usual MLS can be lawed in he p esence o ou lie s, hea y ails
and o he model iola ions.
Adop ing he maximum immed likelihood es ima ion (MTLE) app oach o accoun
o hese ype o model inadequacies, we de i ed a disc e e-con inuous op imiza ion
p oblem unde lying he MTLE es ima o and de eloped an i e a i e p ocedu e o sol e
o he maximum o he immed likelihood. We applied ou me hod o he mul i a ia e
Vasicek model o bo h syn he ic and eal-wo ld da ase s in he con ex o sho a es
and s ochas ic log- ola ili ies. Empi ical e idence in his s udy indica es ha MTLE, a
minimum, pe o ms head- o-head and o en imes ou pe o ms MLS and MLE on da ase s
wi h ou lie s and MLTS, especially a es ima ing he long- e m mean. Thus, his wo k
p o ides and ho oughly e alua es a new econome ic ool o es ima ing he pa ame e s
o he mul i a ia e Vasicek model. Ou p oposed me hodology is applicable unde b oad
ci cums ances and can be help ul o in es o s and analys s ac oss a wide a ie y o use
cases in po olio op imiza ion, isk managemen and o he ypes o decision making. Wi h
a p ima y ocus on obus ness ins ead o imp o ed s a is ical e iciency, un eweigh ed
“ aw” es ima o s we e compa ed in his s udy.
Ou indings gene ally con i m wha is known abou MLTS in e ms o obus calib a-
ion o VAR(1) models. Howe e , while MLTS and exis ing me hodologies a e success ul a
es ima ing he mean- e e sion speed ma ix
A
and co a iance
Σ
, we documen ed majo de-
iciencies o exis ing me hodologies a es ima ing he long- e m mean, especially in smalle
samples. Using MLTS as a “p ime ” es ima o o p oduce a wa m s a , we we e able o
alle ia e his de iciency by adop ing a egula ized immed EM- ype i e a ion o compu e
MTLE es ima es. This unde pins he obus na u e o MLTS mani es in i s o iginal abili y
o ind an ou lie - ee bulk se while explaining he ole o ou MTLE in imp o ing he
es ima ion accu acy o he long- e m mean in he p esence o ou lie s and ill-condi ioned
mean- e e sion speed ma ices.
In u u e wo k, a eweigh ed e sion o MTLE can be de eloped in a simila ashion
as i s MLTS coun e pa (Agulló e al.,2008;Chang & Shi,2024). O he po en ial ex en-
sions include high-dimensional ime se ies, missing da a impu a ion, s a is ical p ocess
moni o ing and change poin de ec ion, among o he p oblems.
Supplemen a y Ma e ials: The ollowing suppo ing in o ma ion can be downloaded a h ps://
www.mdpi.com/a icle/10.3390/economies13030068/s1 and h ps://gi hub.com/And ewsJunio /
Vasicek-MTLE (accessed on 25 Feb ua y 2025), Supplemen a y MATLAB
®
codes; supplemen a y
igu es and ables (PDF).
Au ho Con ibu ions: Concep ualiza ion, T.M.F.J., M.P. and A.T.A.; me hodology, M.P.; so wa e,
M.P. and A.T.A.; alida ion, M.P., A.T.A. and E.N.; o mal analysis, T.M.F.J., M.P. and A.T.A.; in-
es iga ion, T.M.F.J., M.P., A.T.A. and E.N.; esou ces, T.M.F.J.; da a cu a ion, A.T.A. and E.N.;
w i ing—o iginal d a p epa a ion, M.P., A.T.A. and E.N.; w i ing— e iew and edi ing, T.M.F.J., M.P.,
A.T.A. and E.N.; isualiza ion, A.T.A. and E.N.; supe ision, T.M.F.J. and M.P.; p ojec adminis a ion,
T.M.F.J. and M.P.; unding acquisi ion, T.M.F.J. and M.P. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding: This wo k was pa ially suppo ed by he Na ional Science Founda ion (DMS-2210929);
(ERC-ASPIRE-1941524); (DUE-2216396) and Depa men o Educa ion (Awa d #P116S210004). Ad-
di ional pa ial unding suppo was p o ided by El Paso Wa e (EPW 226621069A); Rio G ande
Council o Go e nmen s (COG 226621071A); RAIZ FCU (RAIZ 2024E); and UTEP Cen e o he
S udy o Wes e n Hemisphe ic T ade (BRMP 2024E).
Economies 2025,13, 68 24 o 28
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : The sho - a e da a epo ed in his wo k a e a ailable a h ps://
gi hub.com/And ewsJunio /Vasicek-MTLE (accessed on 25 Feb ua y 2025). The da ase con aining
he six asse s can be eques ed om he au ho s o he pape Chang and Shi (2024).
Con lic s o In e es : The au ho s decla e no con lic s o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses o in e p e a ion o da a; in he w i ing o he manusc ip o
in he decision o publish he esul s.
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
bdp B eakdown poin
CLT Cen al Limi Theo em
COVID Co ona i us disease
C-s ep Concen a ion s ep
EU Eu opean Union
LTS Leas immed squa es
MCD Minimum co a iance de e minan
MLE Maximum likelihood es ima o
MLS Mul i a ia e leas squa es
MLTS Mul i a ia e leas immed squa es
MM Modi ied maximum likelihood- ype
MSE Mean squa ed e o
MTLE Maximum immed likelihood es ima ion
ncp Non-cen ali y pa ame e
RMLTS Reweigh ed mul i a ia e leas immed squa es
RMSE Roo mean squa ed e o
SDE S ochas ic di e en ial equa ion
US Uni ed S a es
VAR Vec o au o eg ession
Appendix A. Asymp o ic Consis ency
Two asymp o ic consis ency esul s a e p esen ed in his Appendix. Theo em A1
desc ibes he la ge-sample asymp o ics o MTLE, assuming espec i e MLTS es ima o s a e
consis en . Exploi ing he la ge- ime beha io o he disc e e Vasicek p ocess, Theo em A2
gi es an al e na i e s a emen unde milde assump ions, solely elying on he la ge- ime
asymp o ics o he Vasicek p ocess. This o e s a heo e ical explana ion o he empi ically
obse ed e icacy o ou app oach (c . Sec ions 3.1 and 3.2).
To show he la e esul , assuming he ma ix
Ip×p−(∆ )A
is s ic ly con ac i e,
he idea is o s udy he es ima ing Equa ion (22)
R∗
MTLE =1
h∑
j∈I†
R j+1
(1−α)TA−1
MTLE ∑
j∈I†
(R j+1−R j)(A1)
om Sec ion 2.2 (wi h
A−1
deno ing he Moo e–Pen ose pseudoin e se). Indeed, a guing
heu is ically, i he la e e m can be shown o anish, which one would in ui i ely expec
in iew o asymp o ic s abili y, Theo em 1 eadily u nishes he con e gence o
R∗
MTLE
o
he (nominal and ac ual) long- e m mean R∗.