Discoun ing he dis an u u e: How much does model
selec ion affec he ce ain y equi alen a e?
Ben G oom∗Phoebe Koundou i†Eka e ini Panopoulou‡
Theologos Pan elidis§
Decembe 20, 2004
Abs ac
E alua ing in es men s wi h long- e m consequences using discoun a es ha decline wi h
he ime ho izon, (Declining Discoun Ra es o DDRs) means ha u u e wel a e changes a e
o g ea e consequence in p esen alue e ms. Recen wo k in his a ea has u ned owa ds
ope a ionalising he heo y and es ablishing a schedule o DDRs o use in cos bene i analysis.
Using US da a we make he ollowing poin s conce ning his ansi ion: i) model selec ion has
impo an implica ions o ope a ionalising a heo y o DDRs ha depends upon unce ain y;
ii) misspeci ica ion es ing na u ally leads o employing models ha accoun o changes in he
in e es a e gene a ing mechanism. Las ly, we p o ide an analysis o he policy implica ions o
DDRsin hecon ex o clima echange o heUS and show ha he use o a s a e space model
can inc ease alua ions by 150% compa ed o con en ional cons an discoun ing.
JEL classi ica ion: C13, C53, Q2, Q4
Keywo ds: long- un discoun ing, in e es a e o ecas ing, s a e-space models, egime-swi ching
models, clima e change policy.
∗Depa men o Economics, Uni e si y College London.
†Depa men o Economics, Uni e si y o Reading, UK and Depa men o Economics, Uni e si y
College London, UK.
‡Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, G eece and Depa men o
Economics, Na ional Uni e si y o I eland Maynoo h. Co espondence o: Eka e ini Panopoulou, Depa -
men o Economics, Na ional Uni e si y o I eland Maynoo h, Co.Kilda e, Republic o I eland. E-mail:
[email protected]. Tel: 00353 1 7083793. Fax: 00353 1 7083934.
§Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, G eece.
Acknowledgemen s: We a e g a e ul o Ch is ian Gollie , Came on Hepbu n, Dimi ios Mallia opulos,
Da id Pea ce, Niki as Pi is, pa icipan s in he 2003 Royal Economic Socie y Con e ence, he 2004
Applied En i onmen al Economics Con e ence Royal Socie y, he Fi s Hispanic Po uguese Cong ess o
En i onmen al and Na u al Resou ce Economics, he 13 h Annual EAERE Con e ence, and semina pa -
icipan s a he Uni e si y College London and Reading Uni e si y o help ul commen s and sugges ions.
Panopoulou and Pan elidis hank he EU o inancial suppo unde he “PYTHAGORAS: Funding o
esea ch g oups in he Uni e si y o Pi aeus” h ough he G eek Minis y o Na ional Educa ion and
Religious Affai s.
1
1 In oduc ion
Thed ama iceffec s o con en ional exponen ial discoun ing on p esen alues o cos s and
bene i s ha acc ue in he dis an u u e along wi h he issues o in e gene a ional equi y
ha a ise a e well documen ed (see e.g. Po ney and Weyan 1999, Pea ce e al. 2003).
The eme gence o a long- e m policy a ena con aining issues as di e se as clima e change,
nuclea build and decommission, biodi e si y conse a ion, g oundwa e pollu ion, and
he use o social Cos Bene i Analysis (CBA) o guide decision-make s in his a ena has
b ough he discussion o long- un discoun ing o he o e. Discoun a es ha decline
wi h he ime ho izon (Declining Discoun Ra es o DDRs) ha e o en been ou ed as
an app op ia e esolu ion o wha Pigou (1932) desc ibed as he ‘de ec i e elescopic
acul y’ o con en ional discoun ing, and he e has been much discussion abou he mo al
and heo e ical jus i ica ion o such a s a egy (see e.g. Dyb ig e al. 1996, Sozou
1998, Wei zman 1998, 2001, Po ney and Weyan 1999, Gollie 2002a). O pa icula
in e es a e he declining ye socially efficien discoun a es esul ing om he analysis
o Wei zman (1998, 2004) and Gollie (2002a, 2002b, 2004) bo h o which appea o offe
a heo e ical pa h h ough he ‘da k jungles o he second bes ’ (Baumol 1968) and he
in e gene a ional equi y-efficiency ade-offcon ained he ein.
I hese heo e ical solu ions offe e en a pa ial esolu ion o he p oblems o con-
en ional discoun ing hen i is clea ly impo an ha hey can be ope a ionalised and a
schedule o DDRs can be de e mined. In he case o Gollie (2002a) and Wei zman (1998)
i is unce ain y ha d i es DDRs, wi h ega d o u u e g ow h o consump ion and he
discoun a e espec i ely, hus he ques ion o implemen a ion is one o cha ac e ising
he unce ain y o hese p imals in some cohe en way. Howe e , o hese wo app oaches
i is Wei zman (1998) ha has p o en o be mo e amenable o implemen a ion mainly
because he in o ma ional equi emen s s op a he cha ac e isa ion o unce ain y, and
do no ex end o speci ic a ibu es o u u e gene a ions’ isk p e e ences as would be
una oidable in he case o Gollie (2002a, 2002b).1
1Wei zman (1998) assumes isk neu al agen s o exposi ion, bu his ep esen s a special case o
his gene al poin . Fo ealis ic scena ios, de e mina ion o DDRs a la Gollie (2002a, 2002b) equi es
knowledge o he 4 h and 5 h de i a i es o u ili y unc ions, some hing ha he admi s is e y a om
2
Wei zman’s Ce ain y Equi alen Discoun Ra e (CER) is de i ed om he expec ed
discoun ac o and is he e o e a summa y s a is ic o he dis ibu ion o he discoun a e.
The le el and beha iou o e ime o his s a is ic is clea ly dependen upon he manne
in which unce ain y is cha ac e ised and he wo applica ions ha exis ha e aken diffe -
en app oaches s emming om diffe en in e p e a ions o unce ain y. Wei zman (2001)
de ines unce ain y by he cu en lack o consensus on he app op ia e discoun a e o
he e y long e m. His su ey o p o essional economis s esul s in a Gamma p obabil-
i y dis ibu ion o he discoun a e which leads o he so-called ‘Gamma discoun ing’
app oach, a e sion o which can also be seen in Sozou (1998). Apa om unce ain y
his model has pe sis ence in-buil , he assump ion being ha each indi idual discoun s
he u u e a hei p e e ed cons an a e, ha is each o he esponses ha make up he
p obabili y dis ibu ion emain cons an o e ime.
Mo e ecen ly, Newell and Pize (2003) (N&P, hence o h) sugges ha while we a e
ela i ely ce ain abou he cu en le el o discoun a es, he e is conside able unce -
ain y in u u e. F om his s andpoin hey assume ha he pas is in o ma i e abou he
u u e and cha ac e ise in e es a e unce ain y by he pa ame e unce ain y ypically
ound in any econome ic model. They choose o desc ibe he beha iou o he US long-
e m eal in e es a e wi h a educed- o m model. Thei model is he di ec analogue o
he Vasicek (1977) model o he e m s uc u e o in e es a es in he sense ha only
he condi ional mean equa ion is speci ied and he condi ional a iance is held cons an .
In his espec , he au ho s ge a wo king de ini ion o he CER based upon an econo-
me ic model and es ima ion o he CER schedule comes om a o ecas ing simula ion.
Wei zman (2004) goes one s ep u he and builds a “s a is ical op imal g ow h model” by
combining a neoclassical economic model o op imal g ow h unde unce ain y wi h a ully
in eg a ed Bayesian s a is ical model o es ima ing, upda ing and p edic ing he ou come
o his unce ain y. His model is able o p oduce pe sis en unce ain y in he in e es a e
and as a esul DDRs s emming mainly om he unce ain y o e u u e echnological
p og ess. F om a diffe en poin o iew, mainly d i en by he exis ing inance li e a u e
being accomplished.
3
on he e m s uc u e o in e es a es, Gollie (2004) eaches simila conclusions. He,
speci ically, inds ha a posi i ely co ela ed g ow h p ocess leads o a dec easing yield
cu e in he case o a p uden ep esen a i e agen due o inc eased unce ain y o he
dis an u u e. He also links his model wi h second o de s ochas ic co ela ion and as a
esul o he Cox, Inge soll and Ross model (1985) (CIR, hence o h) o he inance li e a-
u e, in oducing he analogue o he e oscedas ici y in his p ocess o he in e es a e. In
wo simula ion expe imen s, one including disc e e jumps in he g ow h o consump ion
and he o he pa ame e unce ain y, he p o ides e idence o DDRs and sugges s ha he
discoun a e should be as low as 1% o pe iods exceeding 400 yea s.
The a o emen ioned s udies b ing o ligh some in e es ing issues conce ning he cha -
ac e isa ion o he u u e pa h o in e es a es. I is mainly pe sis ence combined wi h
unce ain y ha leads o decline in discoun a es o e ime. In he heo e ical s udies o
Gollie and Wei zman, pe sis ence is gene a ed by he economy i sel , while in N&P, he
exis ence o pe sis ence is an empi ical ques ion and i is he deg ee o pe sis ence in he
se ies ha de e mines he a e o decline o he CER. In pa icula , N&P speci y a simple
AR(p) model o in e es a e unce ain y, which limi s he cha ac e isa ion o unce ain y
o a p ocess in which he dis ibu ion o he pe manen and empo a y s ochas ic com-
ponen s is cons an o all ime. Such a p ocess gua an ees declining CERs, bu i akes
in o accoun only he e olu ion o he mean o he p ocess. As al eady men ioned hei
model is a disc e e ime e sion o he Vasicek (1977) con inuous- ime model in which
he d i o he p ocess is linea and mean- e e ing, while he diffusion unc ion is held
cons an . Since he seminal con ibu ion o Vasicek (1977), an immense li e a u e on he
e m s uc u e o in e es a es has p oduced in e es ing insigh s as o wha d i es efficien
discoun a es. The basic ex ensions mainly come om he speci ica ion o he a iance o
he p ocess, namely he diffusion unc ion. Fo example, CIR model he diffusion unc ion
as a linea unc ion o he le el o he in e es a e, while Chan e al. (1992) allow he
diffusion unc ion o be any powe unc ion o he le el o he in e es a e. Howe e , he
a o emen ioned one- ac o models display ime-homogenei y, i.e. hei pa ame e s emain
cons an o e ime. I is easonable o expec ha he ins an aneous e u n and ola il-
4
i y slowly e ol e o e ime. In his espec , a ious effo s ha e been made o p oduce
ime-dependen models, such hose o Ho and Lee (1986), Black e al. (1990), Hull and
Whi e (1990) and Black and Ka asinski (1991). These models speci y bo h he d i and
he diffusion p ocess o he ins an aneous s ochas ic a e ia ime- a ying unc ions o he
le el o in e es a es.
The empi ical issues s emming om he en i onmen al li e a u e on declining discoun
a es along wi h he de elopmen o an econome ic model, e sa ile enough o ep oduce
he empi ical egula i ies ypically encoun e ed in in e es a e da a a e he main con-
ce n o his pape and we build upon he ollowing poin s. Fi s ly, i is clea ha i we
belie e ha he pas is in o ma i e abou he u u e, i is impo an o cha ac e ise he
pas as accu a ely as possible. Indeed, he selec ion o he econome ic model is o con-
side able momen in ope a ionalising a heo y o DDRs ha depends upon unce ain y
and de ines he CER in s a is ical e ms. Each speci ica ion diffe s in he assump ions
made conce ning he ime se ies p ocess, hence he o ecas s o he in e es a e and he
a ibu es o he esul ing schedule o he CER will diffe acco dingly. Secondly, he p e-
sc ip ion o CBA will diffe ma kedly depending upon he empi ical schedule o discoun
a es employed, pa icula ly o p ojec s wi h a long ime ho izon such as clima e change
p e en ion. Mo eo e , model selec ion is also an empi ical ques ion. Typical misspec-
i ica ion es ing and compa isons among a ious econome ic models based upon hei
ou -o -sample o ecas ing pe o mance should guide model selec ion o he p ac i ione .
We e isi hese issues o US in e es a e da a and show ha misspeci ica ion es ing
gene a es a na u al p og ession away om he simple AR(p) speci ica ion owa ds models
which accoun o second-o de dependence and explici ly conside changes in he ime
se ies p ocess o e ime. We employ, o compa ison pu poses, he same da a se o he
US in e es a es wi h N&P and show he policy implica ions o in e es a e unce ain y
and model selec ion in he alue o ca bon damages o seques a ion.
The pape is o ganised as ollows. In Sec ion 2, we in oduce he heo y o he CER
offe ed by Wei zman (1998), ou me hodology o model selec ion and he econome ic
models employed o eplica e he s ochas ic na u e o US in e es a es. The esul s o
5

he es ima ion and he simula ions a e p esen ed in Sec ion 3. Sec ion 4 d aws policy
implica ions o model selec ion in he case o he alue o ca bon mi iga ion and Sec ion
5 concludes he pape .
2 F om Theo y o P ac ice
2.1 The Ce ain y Equi alen Discoun Fac o and Ra e
Discoun ing u u e consequences in pe iod back o he p esen is ypically calcula ed
using he discoun ac o P ,whe e P =exp(−
P
i=1
i).When is s ochas ic, he expec ed
discoun ed alue o a dolla deli e ed a e yea s is:
E(P )=EÃexp(−
X
i=1
i)!(1)
Following Wei zman (1998) we de ine (1) as he ce ain y equi alen discoun ac o ,and
he co esponding ce ain y-equi alen o wa d a e o discoun ing be ween adjacen pe-
iods a ime as equal o he a e o change o he expec ed discoun ac o :
E(P )
E(P +1)−1=e (2)
whe e e is he o wa d a e om pe iod o pe iod +1a ime in he u u e, o
he ma ginal discoun a e. Gollie (2002a) shows ha he ce ain y equi alen a e is
he socially efficien discoun a e in a isk neu al wo ld − isk neu al agen s a e only
conce ned wi h he expec ed alue o he discoun ac o a he han highe o de momen s
−by showing ha an a bi age exis s i his is no he case.2In effec his ep esen s
he economic heo y unde lying Wei zman’s de ini ion, howe e he beha iou o ˜ o e
ime is dependen upon he na u e o he unce ain y su ounding he discoun a e.
Wei zman (1998) and N&P show ha e as de ined in (2) is a declining unc ion o ime
p o ided ha he e is sufficien pe sis ence in he se ies o e ime.3This makes i clea
2S ic ly, Gollie deals wi h he a e age ce ain y equi alen a e, howe e he same a gumen s hold as
→∞. His p oo ollows Dyb ig e al. (1996).
3Wei zman (1998) gi es a p oo o a gene al bu ime in a ian dis ibu ion unc ion o h .Wei zman
(2001) es ima es his dis ibu ion empi ically as a Gamma dis ibu ion. Pea ce e al. (2003) p o ide a
6
ha ope a ionalising his heo y is an empi ical ques ion, equi ing he de e mina ion o
hes ochas icna u eo e .
2.2 Pa ame e isa ion o Real In e es Ra es
N&P employed a simula ion me hod o o ecas discoun a es in he dis an u u e,
which was p ope ly designed o accoun o unce ain y in he u u e pa h o in e es
a es and was mainly based on he es ima ion esul s o wo econome ic models, namely
an au o eg essi e Mean-Re e ing (MR) model and a Random Walk (RW) model. They
es ima ed he ollowing AR(p)model o :
=η+e (3)
e =
p
X
i=1
aie −i+ξ
whe e ξ ∼N(0,σ2
ξ),η ∼N¡η,σ2
η¢and
p
P
i=1
ai<1 o he MR model, while
p
P
i=1
ai=1 o
he RW model. The au ho s p o e ha in he case o an AR(1) model, he CER akes
he ollowing o m:
e =η− σ2
η−σ2
ξ (ρ, )(4)
whe e ηis he uncondi ional mean discoun a e, ρis he au o eg essi e coefficien ,
(ρ, )=1−ρ2−2log(ρ)ρ +1(1+ρ−ρ +1)
2(1−ρ)3(1+ρ) o MR and (ρ, )= 1
12(1 + 6 +6 2) o RW. I
is s aigh o wa d o see ha (4) is a declining unc ion o (See N&P o de ails).
This model, al hough simple, is success ul in cap u ing he basic ea u es o he un-
de lying Da a Gene a ion P ocess (DGP) which lead o DDRs, namely pe sis ence and
unce ain y. Howe e , gi en he abundance o models al eady designed o cap u e he
dynamics o he in e es a e da a ei he in disc e e o con inuous ime, i is ha d o
belie e ha simply modelling he mean o such a p ocess is an adequa e pa ame e isa ion
o eali y. As ea ly as 1985, CIR in oduce second-o de dependence in he s ochas ic
p ocess o he in e es a e by le ing he condi ional a iance a y wi h he le el o he
nume ical example o he decline o he ce ain y equi alen discoun a e o a uni o m dis ibu ion.
7
in e es a e.4The simple disc e ised diffusion model mo i a ed by he CIR model is he
GARCH (1,1) model, in which he condi ional a iance depends on i s own lag as well as
he lag o squa ed inno a ions. Howe e , when i ing a GARCH model o in e es a es,
one o en inds ha he pa ame e es ima es imply ha he condi ional a iance p ocess
is ei he in eg a ed o explosi e. Engle e al. (1987, 1990), Hong (1988), Ha ey (1993)
and Kees e al. (1997) documen such a beha iou mainly o he US sho e m in e es
a es. In such cases, p ope s a is ical es ing usually canno ejec he hypo hesis ha
he condi ional a iance o he p ocess ollows an in eg a ed GARCH p ocess (IGARCH).
In ou s udy, we employ he AR(p)-GARCH(l, m)model o accoun o bo h mean and
ola ili y effec s in he US in e es a e p ocess. Speci ically ou model is as ollows:
=η+e
e =
p
X
i=1
aie −i+ξ
ξ =h1/2
z (5)
h =c+
m
X
i=1
βiξ2
−i+
l
X
i=1
γih −i
whe e h is he condi ional ola ili y o ξ (gi en all a ailable in o ma ion a ime −1)and
z ∼IIDN(0,1). In he case ha
m
P
i=1
βi+
l
P
i=1
γi=1,we ha e an AR(p)-IGARCH(l, m)
model.
Bo h he AR(p)and AR(p)-GARCH(l, m)models assume ha he pa ame e s
d i ing he s ochas ic p ocess a e cons an o e he sample pe iod, i.e. hey a e ime-
homogenous. This is likely o be an un ealis ic assump ion o a pe iod o 200 yea s
and ce ainly o o ecas ing he CER o e he long- e m policy ho izon in hand which,
ollowing N&P, ex ends o 400 yea s. I is well known ha he beha iou o in e es
a es is s ongly affec ed by he economic cycles as well as shocks des abilising hem, i.e.
pe iods o economic c isis. Fo example, in he US, du ing he pe iod 1979 h ough 1982,
he Fede al Rese e Bank (FED) s opped i s usual p ac ice o a ge ing in e es a es and
decided o use non-bo owed ese es as a a ge ins umen o mone a y policy. As a
4Chan e al. (1992) ex end he CIR model o include any powe unc ion o he diffusion unc ion.
8
esul , he ola ili y o US in e es a es inc eased d ama ically du ing ha pe iod. O he
pe iods o high ola ili y o he US in e es a es we e he OPEC oil c isis (1973-1975),
he Oc obe 1987 s ock ma ke c ash and wa s in ol ing he US. Such u bulen pe iods
a e likely o induce pe sis ence in ola ili y, which is o en an a i ac o he changes in
he economic mechanism gene a ing he in e es a e (see G ay 1996). Lamou ex and
Las apes (1990) show ha any s uc u al shi in he uncondi ional a iance is likely o
lead o un eliable es ima es o he GARCH pa ame e s such ha hey imply oo much
pe sis ence in ola ili y. In his sense, egime shi s a e mis aken o pe iods o ola ili y
clus e ing. Consequen ly, s udies in he e m s uc u e li e a u e ha e modelled disc e e
egime shi s in he spo in e es a e p ocess (Hamil on 1988, Das 1994, G ay 1996 and
Naik and Lee 1997). These models ypically posi a spo in e es a e p ocess ha can
shi andomly be ween wo o mo e egimes ( o example a low-mean and a high-mean
egime). The diffusion and d i unc ions a e kep he same bu he speci ic pa ame e
alues a e diffe en in each egime. This makes he p ocess ime-he e ogeneous. Each
egime inco po a es a diffe en speed o mean- e e sion o a diffe en long- un mean and
adiffe en uncondi ional a iance. Speci ically, in ou s udy we conside he ollowing
Regime-Swi ching (RS) model wi h wo s a es:
=ηk+e (6)
e =
p
X
i=1
ak
ie −i+ξ
whe e ξ ∼IIDN(0,σ2
k),k=1,2 o he i s and second egime, espec i ely. A any
pa icula poin in ime he e is unce ain y as o which egime we a e in. The p obabili y
o being in each egime a ime is speci ied as a Ma ko 1 p ocess, i.e. i depends only
on he egime a ime −1.We de ine he p obabili y ha he p ocess emains a he
i s egimeasP, while he p obabili y ha he p ocess emains a he second egime is
Q. The ma ix o he ansi ion p obabili ies is assumed o be cons an .5
5We de ine he ollowing ma ix o ansi ion p obabili ies:
P ob(R =1|R −1=1)=P, P ob(R =2|R −1=2)=Q
P ob(R =2|R −1=1)=1−P, P ob(R =1|R −1=2)=1−Q
9
CER inc eases sligh ly due o some o e shoo ing du ing he i s 40 yea s. Excep o
his o e shoo ing, he RS model egains i s quick declining pa h o he es o he pe iod
eaching a a e o 0.7% a e 400 yea s. The highes e minal a e is p oduced by he SS
model, which p ojec s a a e o 1.6%, ollowed by MR a 1.4%.
{INSERT TABLE 2 HERE: 2: CERs}
In summa y, he o ecas s o he al e na i e models diffe subs an ially. In his e-
spec , we need o e alua e he models wi h espec o hei p edic i e abili y. Typical
misspeci ica ion es ing has shown ha a cons an coefficien model may no be able o
ully cap u e he dynamics o he US in e es a es o e he pe iod examined. Along
his line o easoning, we sugges ed wo ime- a ying coefficien models (RS and SS), one
accommoda ing ab up changes and he o he allowing o a g adual change o e ime
in he gene a ing mechanism o he in e es a es. These wo models seem eminen ly
p e e able o he cons an coefficien models. In he ollowing subsec ion, we pe o m an
ou -o -sample o ecas exe cise o selec among he a ious models.
3.4 Model Selec ion
E alua ing he ou -o -sample o ecas ing pe o mance o he models unde conside a ion
o he long un is impossible due o limi a ion o da a, as o wa d a es exis o a
maximum pe iod o 30 yea s. Howe e , we a emp o disc imina e be ween hese models
on he g ounds o hei o ecas ing pe o mance o e a 30-yea ho izon using a ailable eal
da a. We speci ically make use o annual o wa d a es sugges ed by he e m s uc u e o
he in la ion-indexed US go e nmen bonds. Then, we calcula e he commonly-used Mean
Squa e Fo ecas E o (MSFE) and judge he models by his c i e ion. Al e na i ely,
we calcula e ou modi ied MSFE c i e ia by inco po a ing ou ke nels15 which weigh
obse a ions by hei ele an p oximi y o he p esen . The esul s a e p esen ed in
Table 3.
{INSERT TABLE 3: A e age MSFEs}
In e es ingly, he a ious speci ica ions o he MSFE c i e ion unanimously ank he
15The Ba le (B), he Pa zen(P), he Quad a ic-Spec al (QS) and he Tukey-Hanning (TK) ke nels
a e he weigh ing unc ions used in ou e alua ion.
16

SS model i s ollowed by he RS model in mos o he cases. The AR-IGARCH model
anks hi d ollowed by MR and hen RW.
In sum, i we selec a model on he basis o i s abili y o cha ac e ise he pas and
i s accu acy conce ning o ecas s o he u u e, we a e inclined o accep he SS model as
he bes model (among he es ima ed models) o desc ibe he US eal in e es a es. Ou
second bes choice would be he RS model.
4 Policy Implica ions o Model Selec ion
The o egoing has es ablished he impo ance o model selec ion in de e mining a schedule
o declining discoun a es o use in CBA. The diffe ences ha a ise om al e na i e
speci ica ions o he ime se ies p ocess ha e been e ealed and a me hod o selec ing one
model o e ano he has been p oposed. In his sec ion we highligh he policy implica ions
o declining discoun a es and he impac o model misspeci ica ion by conside ing he
same case s udy as N&P, ha is, clima e change and he alue o ca bon seques a ion.16
We es ablish he p esen alue o he emo al o 1 on o ca bon om he a mosphe e, and
hence he p esen alue o he bene i s o he a oidance o clima e change damages o each
o he speci ied models. To unde s and wha ollows i is impo an o be amilia wi h
he p o ile o bene i s esul ing om he emo al o 1 on o ca bon om he a mosphe e.
We use he es ima es aken om he DICE model o No dhaus and Boye (2000) shown
in Figu e 3 (Appendix B). Table 4 shows he p esen alue pe on o ca bon emissions
when e alua ed using he schedule o discoun a es associa ed wi h each o he models
desc ibed in Sec ion 3.2.
{INSERT TABLE 4 HERE}
The RS model gi es he lowe alua ions ollowed by he con en ional 4% discoun ing.
In e es ingly, he SS model gi es he highe alua ion ollowedby heRWmodel. Fo
example, he p esen alue o ca bon emissions educ ion is o e 150 % la ge in he case o
he SS model compa ed o he case o cons an discoun ing a 4 %. On he o he hand, he
p esen alue o he emo al o 1 on o ca bon emissions om he a mosphe e inc eases
16See N&P o he assump ions conce ning he modeling o ca bon emissions damages.
17
by only 12 % based on he MR’s o ecas s compa ed o he cons an a e discoun ing
app oach.
The p eceding discussion has a gued ha he RS and SS models a e o be p e e ed
o e he o he s since hey allow o changes in he in e es a e gene a ing p ocess and
ha e desi able p ope ies. F om he policy pe spec i e we ha e es ablished ha bo h
hese models p o ide well speci ied ep esen a ions o he in e es a e se ies. Howe e ,
he RS model p o ides oughly equi alen alues o ca bon o he cons an discoun ing
a e alues ( he e is a 9% diffe ence), while he SS model p oduces alues ha a e up o
150% highe han hose o he cons an a e.
The dispa i y be ween he RS and he SS models, and he p oximi y o he ca bon
alues gene a ed by he o me o hose gene a ed by con en ional cons an discoun ing
ep esen s a clea signal o he policy ele ance o model selec ion in de e mining he
CER. I is c ucial om a policy pe spec i e o make a clea judgmen as o which o
he wo models (RS and SS) is mos app op ia e o he case in hand. Ou o ecas ing
exe cise e eals ha he SS model is p e e able o he RS model due o i s lowe MSFE
o he 30-yea ho izon. Hence in he con ex o SS he ca bon alues a e inc eased by
150% compa ed o con en ional discoun ing and 40% compa ed o N&P’s app oach. In
sho , in he US con ex , he selec ion o econome ic models on he basis o o ecas ing
pe o mance, and he p e e ed schedule o discoun a es makes clima e change p e en ion
a mo e desi able in es men .
5 Conclusions
In esponse o he need o app aise p ojec s o e e y long ime ho izons, a numbe o
heo e ical discussions ha e a isen conce ning he app op ia eness o discoun a es ha
all wi h he ime ho izon conside ed. Such Declining Discoun Ra es (DDRs) would add
g ea e weigh o he cos s and bene i s ha acc ue o u u e gene a ions and he eby a
leas pa ially add ess he issue o in e -gene a ional equi y ha so o en bese s he long
e m policy a ena.
Wei zman’s (Wei zman 1998) heo e ical jus i ica ion o DDRs depends upon un-
18
ce ain y o he discoun a e and he e o e he ope a ionalising o his heo y is highly
dependen upon he manne in which one in e p e s and cha ac e ises unce ain y. Wei z-
man (2001) sugges ed ha i was he lack o consensus abou he co ec discoun a e
o employ in he a dis an u u e ha was he sou ce o unce ain y and his es ima ed
Gamma dis ibu ion p o ided he means o ope a ionalising his heo y and de e mining
he declining Ce ain y Equi alen Ra e (CER). Newell and Pize (2003) (N&P) ook
an al e na i e iew, accoun ing o he unce ain y h ough an econome ic o ecas ing
app oach.
This pape builds on N&P’s app oach in de e mining DDRs and i makes he ollowing
poin s conce ning he model selec ion and he use o DDRs in gene al. Fi s ly, N&P’s
app oach is p edica ed upon he assump ion ha he pas is in o ma i e abou he u u e
and he e o e cha ac e izing unce ain y in he pas can assis us in o ecas ing he u u e
and de e mining he pa h o CERs. We ha e a gued ha i one subsc ibes o his iew i
is impo an o cha ac e ise he pas as well as possible by co ec ly speci ying he model
o he ime se ies p ocess. This is pa icula ly so when dealing wi h leng hy ime ho izons
whe e he accu acy o o ecas s is impo an . Indeed he selec ion o he econome ic
model is o conside able momen in ope a ionalising a heo y o DDRs ha depends upon
unce ain y, because econome ic models con ain diffe en assump ions conce ning he
p obabili y dis ibu ion o he objec o in e es . We ha e shown ha when modelling
he US in e es a e da a, he econome ic model should allow o changes o e ime in
he da a gene a ing p ocess and ha s a e space and egime swi ching models a e likely
o be app op ia e.
Ou es ima ions, simula ions and case s udy bea ou his asse ion. The pa h o he
CER diffe s conside ably om one model o ano he and he e o e each places a diffe en
weigh upon he u u e. The policy implica ions o hese es ima es is e ealed in he
con ex o a case s udy ha calcula es he p esen alue o ca bon emissions educ ion.
The u ilisa ion o a s a e space model o es ima e he discoun ac o s esul s in an inc ease
o 150% in he p esen alue o ca bon emissions educ ion compa ed o a cons an a e
discoun ing app oach.
19
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22
Table 1. Ce ain y Equi alen Discoun Fac o s
Model 4% Mean Random AR Regime S a e
Yea Cons an Re e ing Walk IGARCH Swi ching Space
10.96154 0.96154 0.96154 0.96154 0.96154 0.96154
20 0.45639 0.45906 0.46177 0.45876 0.45390 0.56424
40 0.20829 0.21661 0.22917 0.21250 0.19576 0.33136
60 0.09506 0.10471 0.12480 0.10062 0.08458 0.20296
80 0.04338 0.05150 0.07777 0.04894 0.03700 0.12889
100 0.01980 0.02567 0.05082 0.02455 0.01647 0.08408
150 0.00279 0.00476 0.02333 0.00529 0.00238 0.03132
200 0.00039 0.00095 0.01830 0.00178 0.00041 0.01255
250 0.00006 0.00022 0.01119 0.00104 0.00010 0.00526
300 0.00001 0.00006 0.00890 0.00086 0.00003 0.00227
350 0.00000 0.00002 0.00715 0.00080 0.00002 0.00100
400 0.00000 0.00001 0.00669 0.00078 0.00001 0.00044
Table 2. Ce ain y Equi alen Discoun Ra es
Model Mean Random AR Regime S a e
Yea Re e ing Walk IGARCH Swi ching Space
14.00 4.00 4.00 4.00 4.00
20 3.91 3.85 3.96 4.22 2.79
40 3.76 3.46 3.88 4.31 2.59
60 3.65 3.08 3.74 4.26 2.38
80 3.58 2.60 3.60 4.18 2.23
100 3.51 2.17 3.42 4.09 2.10
150 3.36 1.39 2.75 3.79 1.91
200 3.16 0.94 1.62 3.31 1.79
250 2.87 0.75 0.65 2.46 1.72
300 2.43 0.56 0.23 1.83 1.67
350 1.87 0.43 0.09 0.95 1.64
400 1.41 0.34 0.04 0.70 1.61
23
Table 3. A e age MSFEs
Model Mean Random AR Regime S a e
C i e ion Re e ing Walk IGARCH Swi ching Space
AMSFE 2.058 2.171 2.102 2.323 1.832
AMSFE (B) 1.692 1.724 1.692 1.687 1.499
AMSFE (P) 1.725 1.746 1.720 1.683 1.426
AMSFE (QS) 0.842 0.870 0.848 0.879 0.760
AMSFE (TH) 1.769 1.797 1.765 1.738 1.550
No es: The weigh ing unc ions a e as ollows: Ba le (B), Pa zen(P), Quad a ic-
Spec al (QS) and Tukey-Hanning (TK).
Table 4. Value o Ca bon Damages
Ca bon Values Rela i e o Rela i e o Rela i e o
Model ($/ c) Cons an Ra e Mean Re e ing Random Walk
Regime-Swi ching 5.22 -9.0% -18.8% -49.4%
Cons an (4.0%) 5.74 –-10.7% -44.4%
AR-IGARCH 6.37 11.0% -0.9% -38.3%
Mean Re e ing 6.43 12.0% –-37.7%
Random Walk 10.32 79.8% 60.5% –
S a e Space 14.44 151.6% 124.6% 39.9%
24
Appendix A: Tables
Table A.1: Uni Roo Tes s
Tes Lags /Bandwid h -s a . 5% c i ical alue Decision
ADF 13 -2.314 -2.877 non-s a iona y
Phillips-Pe on 12 -2.016 -2.876 non-s a iona y
DF-GLS 13 -0.473 -1.942 s a iona y
ERS Poin -Op imal 12 19.733 3.170 non-s a iona y
Ng-Pe on 12 -0.824 -8.100 non-s a iona y
KPSS 15 1.158 0.463 non-s a iona y
No es: SIC is employed o de e mine he lag-leng h o he se ies. The ke nel sum-o -
co a iances es ima o wi h Pa zen weigh s is used, while he bandwid h is de e mined based on
he Newey-Wes bandwid h selec ion me hod.
Table A.2: Es ima ion Resul s
PanelA:AR(3)-IGARCH(1,1)model
Coefficien Es ima e S d. E o -s a .
n1.330 0.104 12.811
a11.951 0.085 23.033
a2-1.322 0.156 -8.472
a30.355 0.080 4.441
c0.000 0.000 3.236
β10.442 0.092 4.805
Panel B: Regime Swi ching model
Coefficien Es ima e S d. E o -s a .
n11.189 0.128 9.327
a1
11.589 0.078 20.36
a1
2-0.660 0.086 -7.630
n21.714 0.238 7.206
a2
11.787 0.050 35.55
a2
2-0.800 0.049 -16.395
σ2
10.004 0.001 5.651
σ2
20.000 0.000 6.070
P0.867 0.058 14.934
Q0.917 0.035 25.976
Panel C: S a e Space model
Coefficien Es ima e S d. E o -s a .
n0.510 0.082 6.185
n10.990 0.002 494.9
ln(σ2
e)-9.158 1.324 -6.917
ln(σ2
u)-6.730 0.144 -46.63
25