© 2019 Published by VŠB-TU Os a a. All igh s ese ed. ER-CEREI, Volume 22: 45–56 (2019).
ISSN 1212-3951 (P in ), 1805-9481 (Online) doi: 10.7327/ce ei.2019.06.02
Modelling o Ra ing Downg ades Based on
Mul iple Failu e-Time Da a
Ma ina NOVOTNÁ
*
Depa men o Finance, Facul y o Economics, VŠB – Technical Uni e si y, Sokolská řída 33, 70200, Os a a,
Czech Republic
Abs ac
This a icle aims o de elop a ing models based on su i al analysis me hods. The ocus is on he use o he Cox
p opo ional haza ds model o analyse he ime o an e en de ined as a a ing downg ade and o examine he e ec
o selec ed inancial a iables on he a ing. Two di e en app oaches a e used o es ima e he models depending
on whe he we a e conside ing one o mul iple e en s o a subjec . The esul s show ha he p obabili y o a a ing
downg ade is a ec ed by annual changes in inancial a iables. Fu he mo e, he applica ion indica es ha he s udy
o mul iple ailu e- ime da a leads o a mo e sui able model based on he s a is ical signi icance o he es ima ed
coe icien s and he goodness o i . O e all, he main indings sugges ha i is mo e app op ia e o use mul iple
ailu e- ime analysis, which co esponds be e o a gi en p oblem and allows he use o all he a ailable da a, o
modelling a ing downg ades.
Keywo ds
Cox model; c edi isk; haza d unc ion; a ing; su i al analysis
JEL Classi ica ion: C14, G24, G32
*
[email p o ec ed]
This pape was suppo ed by he Ope a ional P og amme o Compe i i eness unde P ojec CZ.1.07/2.3.00/20.0296 and P ojec
SGS SP2019/132.
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
46
Modelling o Ra ing Downg ades Based on
Mul iple Failu e-Time Da a
Ma ina NOVOTNÁ
1. In oduc ion
Ra ing analysis is a cu en opic as i is ela ed o as-
sessing he c edibili y o he deb o o issue o deb se-
cu i ies. Some esea ch on bond a ing da es back o he
1950s, o example he s udies by Hickman (1958) and
Fishe (1959). Reg ession analysis became one o he
mos used me hods o es ima e a ings in his pe iod. An
al e na i e app oach o p edic ing bond a ings is he
mul iple disc iminan analysis in oduced, o example,
by Pinches and Mingo (1973), Ang and Pa el (1975),
Al man and Ka z (1976) and Belkaoui (1980). Subse-
quen esea ch compa ed pa icula s a is ical me hods;
o ins ance, Kaplan and U wi z (1979) compa e o -
de ed p obi analysis wi h o dina y leas squa e eg es-
sion and Wingle and Wa s (1980) compa e o de ed
p obi analysis wi h mul iple disc iminan analysis. Re-
cen s udies come om he heo e ical amewo k men-
ioned abo e and ex end he s a is ical me hods o new
non-conse a i e app oaches, such as neu al ne wo ks
(Du a and Shekha , 1988; Su kan and Single on,
1990). Waagepe e sen (2010) assesses he ela ionship
be ween quan i a i e models and expe a ing e alua-
ion. In a mo e ecen s udy, Al man, Saba o and Wil-
son (2010) ocus on he impo ance o non- inancial in-
o ma ion wi hin isk managemen .
In ecen yea s, esea ch on a ing p edic ion has
shi ed mainly o he applica ion o machine lea ning
me hods. Hsu, Chen and Chen (2018) p opose a model
based on he a i icial bee colony app oach and suppo
he ec o machine echnique. The au ho s ind ha
hei bio-inspi ed compu ing mechanism p o ides im-
p o ed p edic ion accu acy compa ed wi h o he s a is-
ical me hods. Golbayani, Flo escu and Cha e jee
(2020) compa e neu al ne wo ks, suppo ec o ma-
chines and decision ees, inding ha decision ee-
based models achie e he bes pe o mance. In hei e-
sea ch, hey apply con en ional accu acy measu es and
in oduce he so-called no ch dis ance app oach, which
is sui able o compa ing he pe o mances o a ious
machine lea ning me hods. As i u ns ou , all he me h-
ods men ioned abo e a e app op ia e o a ing p edic-
ion. The indi idual models di e mainly in hei me h-
odology, he a iables used and hei abili y o p edic
a ings. The e o e, esea ch in his a ea is cu en ly o-
cused p ima ily on imp o ing he p edic i e accu acy
o he models. Fo example, Wang and Ku (2021) de-
elop he pa allel a i icial neu al ne wo ks model,
which c ea es se e al independen a i icial neu al ne -
wo ks. As he au ho s sugges , hei app oach ob ains
compe i i e esul s compa ed wi h con en ional a i i-
cial in elligence echniques.
The cu en esea ch shows ha con en ional ap-
p oaches and newe me hods based on a i icial in elli-
gence a e widely used o model c edi a ings. The huge
ad an age o hese models is hei p ac ical applicabil-
i y and he possibili y o use hem o po en ial a ing
e isions. In addi ion, esea ch s udies show ha mod-
els' p edic i e powe is su icien and compa able o
o he commonly used me hods in aluing his o ical
da a based on a e ages o g ow h a es. Fo example,
Jones, Johns one and Wilson (2015) examine he p e-
dic i e pe o mance o bina y classi ie s using a la ge
sample o in e na ional c edi a ings. They apply con-
en ional echniques (logi and p obi eg ession and
linea disc iminan analysis) and ully nonlinea classi-
ie s (neu al ne wo ks, suppo ec o machines, gen-
e al boos ing, AdaBoos and andom o es s). The au-
ho s conclude ha , al hough he newe classi ie s ou -
pe o m he olde ones, simple classi ie s can be iable
al e na i es o mo e sophis ica ed app oaches, pa icu-
la ly i in e p e abili y is an impo an objec i e o p e-
dic i e models.
An al e na i e way o assess a ings, which is used
in his a icle, is based on su i al analysis. Fo exam-
ple, Glennon and Nig o (2005) use a disc e e- ime haz-
a d amewo k o measu ing he de aul isk o small
business loans. Roa, Ga cía and Bonilla (2009) p opose
a su i al analysis me hodology o analyse alling a -
ing du a ion. They es mac oeconomic a iables o
p edic his e en in selec ed coun ies and ind di e -
ences be ween de eloped and eme ging economies.
Zhang and Thomas (2012) compa e linea eg ession
and su i al analysis o modelling eco e y a es in
u he esea ch. The au ho s ind ha linea eg ession
is be e o eco e y a e modelling; howe e , hey
sugges some adjus men s and addi ional alida ion.
O e all, su i al analysis me hods a e ypically used
o modelling a ing o c edi ansi ions and ime se ies
a ing pa e ns (e.g. Pa nes, 2007; Figlewski, F ydman
and Liang, 2012; Louis, Van Lae e and Baesens, 2013;
M.No o ná – Modelling o a ing downg ade based on mul iple ailu e- ime da a
47
Leow and C ook, 2014). Thus, we can model he a ing
beha iou o e ime and measu e, o example, he
p obabili y o a ce ain change in he a ing depending
on ime and o he ele an a iables. The e o e, su -
i al analysis allows us o gain a be e unde s anding
o he da a and hei dynamics o e ime. In addi ion, i
is a me hod used by a ing agencies o es ima e de aul
a es, which a e egula ly published and used by ana-
lys s and esea che s in he inancial ma ke .
I is e iden ha he cu en esea ch deals wi h
sco ing and a ing models and ha many s udies on his
opic ha e been published. S ill, academic esea ch
pays mo e a en ion o sco ing han a ing models. Sco -
ing models a e ypically used in p edic ing c edi de-
aul o co po a e bank up cy. Va ious modi ica ions o
hese models ha e a wide applica ion in co po a e i-
nance when assessing a company's inancial heal h, es-
pecially in he banking sec o , in which he p obabili y
o loan epaymen de aul is analysed. Ra ing models
wo k simila ly as hei pu pose is o assess he a ing o
a deb ins umen o issue . These models a e also im-
po an in co po a e inance, especially in analysing
deb secu i ies and he in es men p ocess. Thei appli-
ca ion is i al when a ce i ied a ing is no a ailable.
These models can also play a key ole in aluing bonds
ha a e no aded in he public ma ke . In hese cases,
i is especially necessa y o conside he isk o de aul
ca e ully, and his is when he models can be used.
The mo i a ion o ou esea ch is he insu icien
a en ion paid o his issue and he e o o unde s and
he a ing beha iou in selec ed coun ies o e ime.
The aim o his pape is o model he p obabili y o a
a ing downg ade using he me hod o su i al analy-
sis. The ocus is on assessing he impac o inancial
a iables on nega i e a ing changes. In his s udy, we
apply he Cox p opo ional haza ds model o iden i y
he economic a iables wi h g ea po en ial o signal a
de e io a ion in c edi a ings. The su i al models a e
es ima ed using wo app oaches. Fi s , we examine he
ime o he i s a ing downg ade, igno ing addi ional
e en s. Nex , we make use o all he a ailable da a and
accoun o mul iple ailu e- ype da a. This p ocedu e
allows us o compa e he wo app oaches and make ec-
ommenda ions. The es ima ed models a e designed o
e alua e he indi idual a ing o an en i y, secu i y o
o he deb ins umen . The p ac ical applica ion con-
sis s mainly o he alua ion o deb secu i ies and he
ela ed es ima ion o he isk p emium and he cos o
capi al.
The s uc u e o his a icle is as ollows. Fi s , he
main concep s and me hods o su i al analysis a e de-
sc ibed in Chap e 2. A en ion is paid especially o he
Cox p opo ional haza ds model and he s udy o he
mul iple- ime da a used in he applica ion. Then, in
Chap e 3, he da a and a iables en e ing he model a e
desc ibed. Subsequen ly, he models a e es ima ed and
he main esul s a e in e p e ed. Finally, he o e all
indings a e summa ized in Chap e 4.
2. Desc ip ion o he Me hodology
Su i al analysis is a s a is ical me hod used o analyse
he p obabili y o an e en occu ing as a unc ion o
ime. This me hod is p ima ily used in he na u al sci-
ences, in which, o example, he likelihood o pa ien
su i al om he momen o diagnosis, ini ia ion o
ea men and so on is examined. In economic and e-
la ed ields, his me hod is applied mainly o he analy-
sis o bank up cy o de aul . In ou s udy, su i al anal-
ysis is used o a ing modelling. This chap e is de-
o ed o explaining he me hodology used, including
he e minology and main p inciples.
Su i al analysis should be used o analyse da a in
which he ime un il he e en is o in e es . The e-
sponse a iable, he ime un il ha e en , is ypically
called he ailu e ime, su i al o e en ime (Ha ell,
2010). Su i al analysis allows he esponse o be in-
comple ely de e mined o some subjec s; o example,
we canno ollow all he obse a ions in he da ase .
This me hod is based on he mechanism o censo ing
when censo ed and uncenso ed obse a ions a e de-
ined. Fo example, Hosme e al. (2008, p. 18) desc ibe
a censo ed obse a ion as an incomple e alue due o
andom ac o s o each subjec . In he ollowing ex ,
we ocus on he undamen al heo e ical backg ound,
which can be supplemen ed by a a ie y o ele an li -
e a u e, o example Gou ie oux and Jasiak (2007),
Tabachnik and Fidell (2007), Hosme e al. (2008),
Cle es e al. (2010), Ha ell (2010), Roys on and Lam-
be (2011) and Klein e al. (2014).
2.1 Main Concep s o Su i al Analysis
A key issue in su i al analysis is o es ima e he like-
lihood ha subjec s will su i e a ce ain leng h o
ime. The likelihood o su i al o a ce ain poin in
ime is condi ioned by he ac ha he subjec has su -
i ed he p e ious leng h o ime. The o al p obabili y
o su i al is hen gi en by he p oduc o hese indi-
idual p obabili ies, o example
1 2 3
( ) ... ,
S p p p p=
(1)
whe e
1 2 3... ,
p p p p
is he condi ional p obabili y o
su i ing ime
a e ha ing su i ed ime
1 −
. The
p
can be exp essed as
()
1,
n d d
pnn
−
= = −
(2)
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
48
whe e
n
is he numbe o subjec s ali e a he s a o
he in e al ending a ime
1 +
and
d
is he numbe o
subjec s ailing in he sho ime in e al jus a e
.
Thus, equa ion (1) can be ew i en as
( ) 1 .
d
S n

=−



(3)
The successi e o e all su i al p obabili ies,
(1), (2),..., ( )S S S
, a e e e ed o as he Kaplan–Meie
(K-M) o p oduc -limi su i al es ima es. A g aphical
ep esen a ion o
()S
as a unc ion o ime is he K-
M es ima e o he su i al cu e, which is plo ed as a
s ep unc ion. The K-M es ima e desc ibes he ime o a
gi en e en based on all he a ailable da a.
Klein e al. (2014) de ine he dis ibu ion unc ion
o he su i al ime, commonly called he ailu e unc-
ion, as he p obabili y o ailu e up o ime ,
( ) P ( ),F T =
(4)
whe e T is a non-nega i e andom a iable deno ing he
ime o a ailu e e en and F( ) e e s o he cumula i e
dis ibu ion. Fo p ac ical easons, i is o en mo e ap-
p op ia e o use a complemen a y unc ion in su i al
analysis, he su i al unc ion S( ), which is he p oba-
bili y o su i ing beyond ime ,
( ) 1 ( ) P ( ).S F T = − = 
(5)
Using he su i al unc ion, we can es ima e he p ob-
abili y o no ailu e e en occu ing p io o . The den-
si y unc ion ( ) can be ob ained bo h om S( ) and
om F( ):


()
( ) 1 ( ) ( ).
dF d
S S
d d 
= = − = −
(6)
To assess how he isk o a pa icula ou come a -
ies wi h ime, we use he haza d a e
()h
. Acco ding
o Ha ell (2010), he haza d a ime
is ela ed o he
p obabili y ha he e en will occu in a small in e al
a ound
, gi en ha he e en has no occu ed be o e
ime
. Cle es e al. (2010) explain he haza d a e as
he condi ional ailu e a e o he in ensi y unc ion. As
hey emphasize, he haza d a e ep esen s he ins an a-
neous a e o ailu e wi h
1/
uni s:
0
P ( ()
( ) lim .
()
T T
h S
→
+    
==

(7)
The haza d unc ion can ange om ze o (no isk) o
in ini y ( he ce ain y o ailu e a ha ins an ) and can
be dec easing, inc easing o cons an ; al e na i ely, i
can e en ake on o he shapes.
Depending on he assump ions abou he ailu e-
ime dis ibu ion, he e a e a ious me hods o su i al
analysis. Cle es e al. (2010) speci y h ee app oaches
and ele an models as ollows:
• Nonpa ame ic models (Kaplan–Meie and
Nelson–Aalen);
• semipa ame ic models (Cox p opo ional haz-
a ds model); and
• pa ame ic models (e.g. exponen ial, Weibull,
logno mal, log-logis ic, gamma and Gom-
pe z).
While pa ame ic models equi e assump ions abou he
dis ibu ion o ailu e imes, semipa ame ic models a e
pa ame ic in he sense ha he e ec o he co a ia es
is assumed o ake a ce ain o m. In his case, no pa a-
me ic o m o he su i al unc ion is speci ied, ye he
e ec s o co a ia es a e pa ame ized o modi y he
baseline su i o unc ion. Thus, compa ed wi h he
p e ious app oaches, nonpa ame ic models do no e-
qui e any assump ions abou he dis ibu ion o ailu e
imes.
Nonpa ame ic me hods es ima e he p obabili y o
su i al pas a ce ain ime o compa e su i al expe i-
ences o di e en g oups (Cle es e al., 2010). The
common cha ac e is ic o nonpa ame ic models is ha
hey do no make any assump ions abou he dis ibu-
ion o ailu e imes o he way in which co a ia es
change he su i al expe ience. The Kaplan–Meie es-
ima o o he su i o ship unc ion a ime can ake
he o m o he ollowing equa ion:
()
ˆ( ) ,
i
ii
i
nd
S n

−
=
(8)
whe e
i
n
is he numbe a isk o dying (company ail-
u e) a
()i
,
i
d
e e s o he obse ed numbe o ailu es
and
ˆ( ) 1S =
i
()
.
i

I we assume ha he ime a i-
able is absolu ely con inuous, hen he su i al unc ion
may be exp essed as
()
( ) ,
H
S e−
=
(9)
whe e H( ), he cumula i e haza d unc ion, can be
w i en as
( ) ln( ( )).H S =−
(10)
Aalen, Nelson and Al shule p opose he indica o H( ),
which is e e ed o as he Nelson–Aalen es ima o
(Hosme e al., 2008). The Nelson–Aalen es ima o o
H( ) is gi en by
()
ˆ( ) .
i
i
i
d
H n

=
(11)
M.No o ná – Modelling o a ing downg ade based on mul iple ailu e- ime da a
49
Semipa ame ic models do no assume any pa a-
me ic o m o he su i al unc ions. The e ec s o he
co a ia es a e pa ame ized o modi y he baseline su -
i al unc ion. Acco ding o Hosme e al. (2008), he
eg ession model o he haza d unc ion can be ex-
p essed as
0
( , , ) ( ) ( , ),h x h x

=
(12)
whe e
0()h
cha ac e izes he change in he haza d
unc ion as a unc ion o he su i al ime and
( , ) x

desc ibes how he haza d unc ion changes as a unc-
ion o he subjec co a ia es. The model makes no as-
sump ions abou he shape o he haza d o e ime.
Howe e , he gene al shape o he haza d is he same
o e e yone. One subjec 's haza d is a mul iplica i e
eplica o ano he 's, which is cons an . The quan i ies
es ima ed om he model a e haza d a ios, which
measu e he ex en o which a co a ia e inc eases o de-
c eases he a e o a pa icula e en .
Pa ame ic models a e used when he dis ibu ion o
he su i al ime has a known pa ame ic o m. They
gene ally p o ide smoo h es ima es o he haza d and
su i al unc ions o any combina ion o co a ia e al-
ues. Acco ding o Hosme e al. (2008), using hese
models may ha e he ollowing ad an ages. Full maxi-
mum likelihood may be used o es ima e he pa ame-
e s, he es ima ed coe icien s o hei ans o ma ions
can p o ide clinically meaning ul es ima es o e ec s,
i ed alues om he model can p o ide es ima es o
su i al ime and esiduals can be compu ed as di e -
ences be ween obse ed and p edic ed alues o he
ime.
2.2 Cox P opo ional Haza d Model
The Cox p opo ional haza d model is a semipa ame ic
model o su i al analysis. The e ec o he co a ia es
is assumed o ake a ce ain o m compa ed wi h he
nonpa ame ic app oach. In his case, no pa ame ic
o m o he su i al unc ion is speci ied, ye he e ec s
o he co a ia es a e pa ame ized o modi y he base-
line su i o unc ion. In gene al, he baseline su i al
unc ion is he unc ion o which all he co a ia es a e
equal o ze o in a ce ain way. The haza d unc ion (9)
is he p oduc o wo pa s, which cha ac e izes how he
haza d unc ion changes as a unc ion o he su i al
ime, and he unc ion
( , ) x

=
desc ibes how he
haza d unc ion changes as a unc ion o he subjec co-
a ia es.
I ollows om he model ha :
• The unc ions mus be chosen such ha
( , , ) 0,h x


•
0()h
is he haza d unc ion when
( , ) 1, x

=
•
0()h
is e e ed o as he baseline haza d
unc ion when he unc ion
( , ) x

is pa a-
me ized such ha
( 0, ) 1. x

==
Thus, he baseline haza d unc ion can be seen as a
gene aliza ion o he in e cep o cons an e m ound in
pa ame ic eg ession models. We do no make any as-
sump ions abou
0()h
, howe e , a he cos o a loss o
e iciency. Al hough he model makes no assump ions
abou he shape o he haza d o e ime, he gene al
shape is assumed o be he same o e e yone.
The a io o he haza d unc ions o wo subjec s
wi h co a ia e alues deno ed
0
x
and
1
x
is:
1
10
0
01 1
10
0 0 0
( , , )
( , , ) ,o
( , , )
( ) ( , ) ( , )
( , , ) .
( ) ( , ) ( , )
h x
HR x x h x
h x x
HR x x h x x




=
==
(13)
As we can see in (13), he haza d a io (HR) depends
only on he unc ion
( , ). x

This model was o iginally p oposed in 1972 by
Cox, who sugges ed using
( , ) exp( ) x x

=
o p ac-
ical easons. Then, he haza d unc ion can be ex-
p essed as:
0
( , , ) ( ) ,
x
h x h e


=
(14)
and he haza d a io is
10
()
10
( , , ) .
x x x
HR x x e

−
=
(15)
This model is he mos -used semipa ame ic model,
called he Cox model, he Cox p opo ional haza ds
model o he p opo ional haza ds model. The e m p o-
po ional haza ds (PHs) e e s o he ac ha he haza d
unc ions a e mul iplica i ely ela ed; hus, hei HR is
cons an o e ime (Hosme e al., 2008, p. 70). In o he
wo ds, we assume ha he co a ia es mul iplica i ely
shi he baseline haza d unc ion. Then, one subjec 's
haza d is a mul iplica i e eplica o ano he 's (Cle es e
al., 2010). Besides he assump ion o p opo ional haz-
a ds, o he pa ame iza ions can be used, o example
addi i e models. These pa ame iza ion app oaches a e
desc ibed in he ele an li e a u e (Hosme e al., 2008;
Klein e al., 2014).
2.3 Mul iple Failu e-Time Da a
Cle es (2000) desc ibes mul iple ailu e- ime da a as
da a in which any o wo o mo e e en s ( ailu es) occu
o he same subjec o om iden ical e en s occu ing
o ela ed subjec s. The ypical ea u e is ha he ailu e
imes a e co ela ed wi hin a clus e (subjec o g oup),

Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
50
iola ing he independence o ailu e imes assump ion
equi ed in adi ional su i al analysis. As he au ho
poin s ou , ailu e e en s should be classi ied acco ding
o whe he hey ha e a na u al o de and ecu ences o
he same ype o e en s. The e en s a e supposed o be
o de ed when he second e en canno occu be o e he
i s e en . On he con a y, uno de ed e en s can hap-
pen in any sequence.
The e a e mo e app oaches o examining mul iple
ailu e- ime da a. The i s me hod conside s he ime
o he i s e en , igno ing addi ional ailu es. Howe e ,
his means ha we do no make use o all he a ailable
da a. The second me hod is based on analysing he
a ailable da a while accoun ing o he lack o inde-
pendence o he ailu e imes. Cle es (2000) sugges s
co esponding p ocedu es o es ima ing hese models
using he Cox p opo ional haza d model. Unde he
p opo ional haza d assump ion, he haza d unc ion
(14) o he i h clus e o he k h ailu e ype is as ol-
lows:
,
0
( , ) ( ) ,
i
Z
k ki
h Z h e

=
(16)
whe e
ki
Z
is a p- ec o o possibly ime-dependen co-
a ia es o he i h clus e o he k h ailu e ype. While
we p esume in equa ion (16) ha he baseline haza d
unc ion is equal o e e y ailu e ype, he baseline
haza d unc ion is allowed o di e by ailu e ype in
he ollowing o mula:
,
0
( , ) ( ) .
i
Z
k ki k
h Z h e

=
(17)
As Cle es (2000) sugges s, he maximum likelihood es-
ima es o models (16) and (17) a e ob ained om
Cox's pa ial likelihood unc ion
()L

, assuming inde-
pendence o ailu e imes.
Conce ning he analysis o mul iple ailu e- ime
da a, Cle es (2000) emphasizes he need o de e mine
whe he hey a e o de ed o uno de ed da a and selec a
sui able me hod o es ima ing models acco dingly. In
he case o uno de ed imes, which is he case o a ing
analysis, i is i s necessa y o de e mine whe he he
e en s a e o he same o di e en ypes. Simila ly, a
decision is equi ed on whe he he baseline haza d is
he same o di e en o all e en ailu es. In any case,
i is necessa y o implemen he me hods in which he
da a a e co ec ly s uc u ed, including iden i ying in-
di idual ailu e e en s.
1
MORE a ings classi y companies simila ly as a ing agen-
cies (Bu eau an Dijk Elec onic Publishing, 2008). MORE
a ings a e calcula ed using a unique model ha e e ences he
company's inancial da a o c ea e an indica ion o he com-
pany's inancial isk le el.
3. Modelling o Ra ing Downg ades
The aim o his s udy is o examine he ela ionship be-
ween ime and co po a e a ing downg ades, including
he in luence o he a iables used. In his s udy, he
e en , o ailu e in e ms o su i al analysis, is de ined
as a a ing downg ade. As he a ing can be downg aded
mo e han once du ing he pe iod unde ou obse a-
ion, mul iple ailu e- ime analysis app oaches should
be used. Thus, su i al analysis models a e es ima ed
using he Cox p opo ional haza d model o mul iple
ailu e- ime da a. Fu he mo e, we aim o model he a -
ing downg ade depending on he ime and he annual
changes in inancial a iables. The e o e, we use yea ly
changes in he a ing as he dependen a iable and
yea ly changes in he inancial co a ia es as he inde-
penden a iables in ou model. Speci ically, he de-
penden a iable is a a ing downg ade. This choice is
qui e na u al because i is c ucial o de ec a po en ial
de e io a ion in in es men quali y, which inc eases he
c edi isk.
This s udy is ocused on he analysis o co po a e
c edi a ings om eigh coun ies in Cen al and Eas -
e n Eu ope (CEE): he Czech Republic, Es onia, Hun-
ga y, La ia, Li huania, Poland, Slo akia and Slo enia.
As global a ing agencies do no a e many companies
in hese coun ies, he models a e es ima ed based on
he Mul i-Objec i e Ra ing E alua ion (MORE).
1
The
MORE me hodology, as pa o he issuance o c edi
a ings, complies wi h Regula ion (EC) N. 1060/2009
o he Eu opean Pa liamen and he Council o 16 Sep-
embe 2009 ( he C edi Ra ing Agencies Regula ion).
The e o e, wi h e ec om 10 July 2015, i is egis e ed
as a c edi a ing agency in acco dance wi h his egu-
la ion.
2
The da a sample con ains eco ds o 1249 com-
panies (2002–2007).
We ollow a ing assessmen s and a ious inancial
a iables on an annual basis. Thus, we ha e a o al o
7494 obse a ions. Wi hou speci ica ions o ela ed
subjec s based on id, he o e all da a consis o 6245
obse a ions, 705 (single) and 870 (mul iple) e en s de-
ined as a ing downg ades and 18,735 (single) and
15,358 (mul iple) o he o al analysis ime. In bo h
cases, we use 2,436 obse a ions. Since we assume ha
he obse a ions o each company may be co ela ed,
we adjus o his by clus e ing as ollows. The compa-
nies a e in e p e ed as he same sampling uni s by using
he id() iden i ie , allowing us o speci y ela ed sub-
jec s. Thus, we analyse he da a based on 737 clus e s.
2
MORE a ing by he i s Fin ech Ra ing Agency – S-Peek,
online access: h ps://www.s-peek.com/en/mo e- a ing (17
June 2019).
M.No o ná – Modelling o a ing downg ade based on mul iple ailu e- ime da a
51
Fo each company in ou da a sample, we ollow he
a ing assessmen and a ious inancial a iables annu-
ally (Table 1-1). The independen co a ia es in he
model a e he annual pe cen age changes in he ele an
inancial indica o s, e e ed o as co a ia es in he a-
ble. We conside all a ing downg ades as he same
e en ype and do no dis inguish he downg ade size.
Ne e heless, he a ing changes by mo e han one de-
g ee in a ma ginal numbe o cases, so we do no ake
his ac in o accoun .
Table 1–1 Desc ip ion o inancial a iables
Financial indica o
Co a ia e
Mean
To al asse s
ag
22.25
Re u n on asse s
oag
86.70
Re u n on equi y
oeg
66.24
EBITDA o o al deb
ebi da g
27.57
Equi y o o al
asse s
eq ag
12.33
Cash low
c g
55.62
In e es co e age
in co g
149.13
In his s udy, we es ima e su i al models using wo
app oaches. Fi s , we examine he ime o he i s
e en , igno ing addi ional downg ades. Nex , we make
use o all he a ailable da a and accoun o mul iple
ailu e- ype da a. This p ocedu e allows us o compa e
he wo app oaches and make ecommenda ions.
3.1 Es ima ion o Ra ing Su i al Models
The models will be de eloped based on he abo e-de-
sc ibed app oaches. Since we conside single and mul-
iple ailu e- ime da a, we es ima e wo models, e-
e ed o as single and mul iple. Fi s , we pe o m a
simple su i al analysis, in which we conside only he
i s a ing downg ade, igno ing addi ional ones o
each company. Thus, he da a consis o 256 de ined
e en s (downg ades) and 8253 o al analysis ime da a.
The ime is measu ed in yea s a e he ime o o igin,
se as he yea 2002.
The g aph o he Kaplan–Meie es ima e o he su -
i al unc ion is shown in Figu e 1-1. Nex , we es ima e
he su i al model based on mul iple ailu e- ime e en
da a. Using his p ocedu e, we will conside any a ing
downg ades o he en i y. The e o e, since nume ous
e en s can occu o each subjec , i is a mul iple ail-
u e- ime analysis o he same ype. The ime un il he
e en is measu ed as he ime since he las e en o
each subjec . The da a consis o 331 downg ades and
6732 o he o al analysis ime da a. The Kaplan–Mei
es ima e o he su i al unc ion o mul iple ailu e-
ime da a is shown in Figu e 2-1. In bo h cases, he su -
i al cu e has a descending, s epped shape. By com-
pa ing Figu es 1-1 and 2-1, we can see ha he p oba-
bili y o su i al, ha is, a s able o imp o ed a ing, is
highe in he case o single da a. I is a consequence o
he assump ions used in he su i al analysis as, in his
case, only one e en is allowed o each subjec .
Figu e 1–1 Kaplan–Meie su i al es ima e (single)
Figu e 2–1 Kaplan–Meie su i al es ima e (mul iple)
The esul ing models a e summa ized in Table 2-1.
We can see he es ima ed coe icien s o he independ-
en a iables (coe .), including he s anda d e o , in
he b acke s. Fo comple eness, he haza d a ios (HRs)
a e also lis ed in he able. I we compa e he wo mod-
els, we can see ha he es ima ed coe icien s do no
di e signi ican ly. Thus, he e ec o he a iables
used on he p obabili y o su i al, ha is, a s able o
upg aded a ing, is simila . All he coe icien s a e s a-
is ically signi ican a he 0.05 le el in he mul iple
model.
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
52
Table 2–1 Es ima ed coe icien s o he Cox models
Indep.
a iable
Coe .
Single
HR
Single
Coe .
Mul iple
HR
Mul iple
ag
0.0083*
(0.002)
1.0083
0.0084*
(0.002)
1.0084
oag
-0.0027*
(0.001)
0.9973
-0.0034*
(0.001)
0.9966
oeg
0.0030
(0.002)
1.0030
0.0038*
(0.001)
1.0038
ebi da g
-0.0079*
(0.004)
0.9921
-0.0083*
(0.003)
0.9918
eq ag
-0.0153*
(0.005)
0.9848
-0.0184*
(0.004)
0.9817
c g
-0.0117*
(0.003)
0.9884
-0.0142*
(0.003)
0.9859
in co g
-0.0062*
(0.002)
0.9940
-.0063*
(0.002)
0.9938
* Signi ican a he 0.05 le el; s anda d e o s adjus ed o 737
clus e s.
The es ima ed coe icien s can be used o in e p e
he e ec o indi idual a iables, bu i is mo e app o-
p ia e o use he haza d a es. Fo example, an inc ease
in ag by one uni ( he annual change in o al asse s by
1%) inc eases he haza d o a a ing downg ade by
0.84%. Con e sely, i he co a ia e eq ag inc eases by
one uni , he haza d dec eases by 1.83%. F om he
o e all esul s, we can see ha an annual pe cen age
change o one uni in he a iables oa, ebi da, eq a, c
and in co educes he haza d o he a ing downg ade.
In con as , he haza d is inc eased by changes in he
a iables a and oe.
To assess he in luence o he a iables on he haz-
a d, we de e mine he haza d o he so-called a e age
company (H1); ha is, he alues o he a iables a e
equal o hei mean alues. Subsequen ly, we compa e
his haza d wi h he baseline haza d, when he alues o
all he a iables a e equal o ze o (H0). The g aphical
ep esen a ion is p esen ed in Figu es 3-1 and 4-1. In
bo h g aphs, we can see ha H1 lies below he baseline
haza d, H0. I ollows om he ac ha he baseline
haza d co esponds o a si ua ion in which all he co-
a ia es a e ze o. Howe e , a ze o annual change in he
inancial indica o s means ha he isk o a a ing
downg ade mus be highe han in he a e age compa-
ny's yea ly changes.
Figu e 3–1 Baseline and a e age haza d (single)
Figu e 4–1 Baseline and a e age haza d (mul iple)
3.2 Resul s and In e p e a ion
When using he Cox model, i is ad isable o e i y ha
he haza d unc ions a e mul iplica i ely ela ed. The e-
o e, we use he plo o he es ima ed haza d on a log
scale using he ke nel smoo he o es he p opo ional
haza ds assump ion. Since he lines in Figu e 1 (Appen-
dix) seem o be pa allel, we conclude ha he p opo -
ionali y assump ion in bo h models is no iola ed.
The es o he p opo ional haza ds speci ica ion is
based on he Schoen eld esiduals a e i ing he
model. I is used o es he independence be ween e-
siduals and ime. The es esul s in Table 3-1 sugges
ha he haza d assump ion is no p opo ional o h ee
a iables: oag, oeg and in co g. Howe e , based on
he global es , we ind no e idence ha ou speci ica-
ion iola es he p opo ional haza ds assump ion in
bo h models.
M.No o ná – Modelling o a ing downg ade based on mul iple ailu e- ime da a
53
Table 3–1 Tes o PH assump ions
Indep.
a iable
Single
ho
(Chi2)
Mul iple
ho
(Chi2)
ag
-0.0862
(1.53)
-0.0724
(1.58)
oag
-0.0935
(6.44) *
-0.0800
(4.95) *
oeg
0.0954
(6.48) *
0.0836
(5.46) *
ebi da g
-0.0148
(0.13)
-0.0087
(0.05)
eq ag
0.0873
(2.65)
0.0556
(1.46)
c g
0.0817
(2.57)
0.0803
(3.34)
in co g
-0.0704
(7.23) *
-0.0683
(7.16) *
Global
(11.64)
(10.59)
* Signi ican a he 0.05 le el; s anda d e o s adjus ed o 737
clus e s.
The explained a ia ion in he es ima ed models is
measu ed using he adjus ed index o de e mina ion R2
(Roys on, 2006). The alue o R2 equals 0.4689 (SE =
0.104) o he single model and 0.4973 (SE = 0.0231)
o he mul iple model. The g ea es con ibu ion o he
explained a ia ion is made by he co a ia es in co g,
c g, eq ag and ag based on a compa ison o he models
wi h di e en numbe s o p edic o s. De ailed esul s
a e p esen ed in he Appendix (Table 1), which con ains
he da a o models wi h a dec easing numbe o co a i-
a es. The las ow includes a model con aining only one
a iable, in co g.
The o e all model i is e alua ed using Cox–Snell
esiduals. Figu es 5-1 and 6-1 show he Nelson–Aalen
cumula i e haza d es ima o plo s o he Cox–Snell e-
siduals o bo h models. We can see some a iabili y
a ound he 45° line, pa icula ly in he igh -hand ail.
Cle es e al. (2010) a gue ha some a iabili y is ex-
pec ed due o he educed e ec i e sample caused by
p io ailu es and censo ing. Compa ing Figu es 7-1
and 8-1, we can conclude ha he mul iple model i s
he da a be e han he single model.
O e all, he mul iple ailu e- ime da a analysis
leads o a mo e sui able model based on he s a is ical
signi icance o he es ima ed coe icien s and goodness
o i . On he o he hand, i should be no ed ha he wo
su i al models a e e y simila based on he es ima ed
coe icien s and explained a ia ion. Howe e , wi h e-
spec o he alida ion and in e p e a ion o bo h mod-
els, we should use a mul iple ailu e- ime analysis.
Figu e 5-1 Cumula i e haza d o Cox–Snell esiduals (sin-
gle)
Figu e 6-1 Cumula i e haza d o Cox–Snell esiduals (mul-
iple)
4. Conclusion
This s udy aimed o de elop a ing models using su -
i al analysis me hods. Speci ically, we applied he
Cox p opo ional haza ds model o analyse he su i al
ime un il he e en . In ou case, we ocused on using
su i al analysis o model he ime o a a ing down-
g ade. As pa o he analysis, we examined he e ec
o inancial a iables on he p obabili y o a nega i e
annual a ing change.
Two di e en app oaches we e used o es ima e he
models, depending on whe he we we e conside ing
only one o mo e e en s, de ined as a ing downg ades,
o one company. The single model was de i ed on he
assump ion ha he e en can occu only once o each
subjec . The mul iple model, on he o he hand, as-
sumed ha he e en can occu epea edly. Due o hese
di e en assump ions, he inpu da a and hei s uc u e
also had o be adjus ed.