Policy evaluation and uncertainty about the effects of oil prices on economic activity
Abstract
This paper addresses the issue of policy evaluation in a context in which policymakers are uncertain about the effects of oil prices on economic performance. I consider models of the economy inspired by Solow (1980), Blanchard and Gali (2007), Kim and Loungani (1992) and Hamilton (1983, 2005), which incorporate different assumptions on the channels through which oil prices have an impact on economic activity. I first study the characteristics of the model space and I analyze the likelihood of the different specifications. I show that the existence of plausible alternative representations of the economy forces the policymaker to face the problem of model uncertainty. Then, I use the Bayesian approach proposed by Brock, Durlauf and West (2003, 2007) and the minimax approach developed by Hansen and Sargent (2008) to integrate this form of uncertainty into policy evaluation. I find that, in the environment under analysis, the standard Taylor rule is outperformed under a number of criteria by alternative simple rules in which policymakers introduce persistence in the policy instrument and respond to changes in the real price of oil.
Full text
Policy e alua ion and unce ain y abou he e¤ec s o oil
p ices on economic ac i i y
F ancesca Rondinay
Ins i u e o Economic Analysis, CSIC and Ba celona GSE
No embe 2010
Abs ac
This pape add esses he issue o policy e alua ion in a con ex in which policymake s
a e unce ain abou he e¤ec s o oil p ices on economic pe o mance. I conside models
o he economy inspi ed by Solow (1980), Blancha d and Gali (2007), Kim and Loun-
gani (1992) and Hamil on (1983, 2005), which inco po a e di¤e en assump ions on he
channels h ough which oil p ices ha e an impac on economic ac i i y. I … s s udy he
cha ac e is ics o he model space and I analyze he likelihood o he di¤e en speci…ca-
ions. I show ha he exis ence o plausible al e na i e ep esen a ions o he economy
o ces he policymake o ace he p oblem o model unce ain y. Then, I use he Bayesian
app oach p oposed by B ock, Du lau and Wes (2003, 2007) and he minimax app oach
de eloped by Hansen and Sa gen (2008) o in eg a e his o m o unce ain y in o policy
e alua ion. I …nd ha , in he en i onmen unde analysis, he s anda d Taylo ule is ou -
pe o med unde a numbe o c i e ia by al e na i e simple ules in which policymake s
in oduce pe sis ence in he policy ins umen and espond o changes in he eal p ice o
oil.
JEL Classi…ca ion: C52, E52, E58
Keywo ds: model unce ain y, obus policy, Bayesian model a e aging, minimax, oil
p ices.
Con ac s: Campus UAB, 08193 Bella e a, Ba celona, Spain; el: +34 935 806 612, ax: +34 935 801 452,
email: [email p o ec ed]
yI am especially g a e ul o S e en Du lau o his guidance and cons an encou agemen . I am indeb ed o
William B ock, Giacomo Rondina and Emmanuele Bobbio o aluable discussions and sugges ions. I ha e also
bene… ed om commen s om Kenne h Wes , Noah Williams, Fede ico Diez, Nona i Bisonyabu and semina
pa icipan s a he Bank o England, Riksbank, Pomona College, Indiana Uni e si y, Uni e sidade No a, BIS,
EPFL-UNIL, Dallas Fed, Vassa College and IAE-CSIC. All e o s emain my own. Financial suppo om
he Go e nmen o Ca alonia and he Spanish Minis y o Science and Inno a ion (P og ama Ope a i o FSE
2007-2013) is g a e ully acknowledged.
1
1 In oduc ion
This pape in es iga es issues ela ed o he e alua ion o mone a y policy in he p esence o
model unce ain y. In pa icula , he analysis ocuses on en i onmen s in which he policymake
is unce ain abou he mechanism h ough which oil p ices a¤ec economic a iables. In his
con ex , his wo k aims o p esen a wide ange o measu es, based on a numbe o di¤e en
app oaches, ha can suppo policymake s’decision ac i i y by p o iding in o ma ion on he
sensi i i y o di¤e en policy ules o model unce ain y.
In ecen yea s, he li e a u e in mac oeconomics has de o ed la ge a en ion o he p oblem
o model unce ain y in economic policy. In pa icula , his issue has ecei ed inc easing in e es ,
among economis s as well as policymake s, when applied o mone a y policy.1Some ele an
con ibu ions in his a ea a e ep esen ed by B ock, Du lau and Wes (2003, 2007), Cogley
and Sa gen (2005), Giannoni (2007), Hansen and Sa gen (2001a, 2001b, 2008). These wo ks
de elop heo e ical amewo ks o policy design and e alua ion in unce ain en i onmen s and
p o ide applica ions o di¤e en o ms o unce ain y ha commonly a ise in mone a y policy
decisions.2
This pape applies some o he echniques de eloped in he li e a u e on model unce ain y
o a con ex in which he policymake is unce ain abou he e¤ec s o oil p ices on he economy.
Despi e he numbe o con ibu ions s udying he esponse o economic a iables o oil p ice
shocks, he e is s ill much deba e abou he mechanisms h ough which oil p ices a e belie ed
o ha e an impac on economic ac i i y. This deba e o igina es om he ac ha oil p ices
can indeed a¤ec he economy in se e al ways. Changes in oil p ices di ec ly a¤ec he cos s
o p oduc ion ( anspo a ion and hea ing, o ins ance) as well as he p ice o goods made
wi h pe oleum p oduc s. Mo eo e , oil p ice inc eases a e likely o inc ease he gene al p ice
le el, which can educe employmen i wages a e igid. Finally, oil p ice shocks can also lead o
ealloca ion o labo and capi al be ween sec o s o he economy, and induce g ea e unce ain y
abou he u u e, which migh educe pu chases o la ge- icke consump ion and in es men
goods. The di¤e en con ibu ions in his a ea o en disag ee on which o hese ac o s should
be ega ded as he main channel h ough which oil p ices a¤ec ou pu and o he economic
a iables.
The lack o consensus on he p edominan mechanism h ough which oil p ices a¤ec he
economy leads o di¤e en iews abou he abili y o mone a y policy o con as he e¤ec s o
oil p ice shocks. This gene a es a subs an ial disag eemen o e he way mone a y policy should
1On he policymaking side, see Dow (2004) o a desc ip ion o he me hodological app oach ha he Bank
o England and he ECB ha e aken in esponse o he p oblem o model unce ain y.
2Fo ins ance, B ock, Du lau and Wes (2007) p esen an example based on he unce ain y on he way he
public o ms expec a ions on u u e economic a iables, while in Cogley and Sa gen (2005) policymake s a e
unce ain abou he speci…ca ion o Phillips cu e o be adop ed o policy decisions.
2
op imally espond o changes in oil p ices.3Hence, his is a con ex in which he applica ion
o he echniques de eloped in li e a u e on model unce ain y seems o be qui e na u al, and
a he same ime essen ial o sound policymaking.
In his pape I conside he p oblem o a policymake who wan s o explo e possible cou ses
o ac ion o be unde aken in esponse o a change in oil p ices. He is unce ain abou he way
oil p ices a¤ec he economy, and he is pa icula ly in e es ed in in es iga ing he sensi i i y o
his policy decisions o his o m o unce ain y. This wo k p o ides an analysis o he ex en o
which mone a y policies and hei consequences a e model dependen , and s udies he policy
ecommenda ions o Bayesian and non-Bayesian c i e ia. The main …nding is ha , in he
desc ibed en i onmen , he s anda d Taylo (1993) ule is ou pe o med by al e na i e simple
ules in which he policymake in oduces pe sis ence in he policy ins umen , and esponds
o changes in he eal p ice o oil.
The con ibu ion o his wo k o he exis ing li e a u e is wo old. Fi s , I p o ide an analysis
o he likelihood o h ee main amewo ks ha ha e been p oposed o explain he e¤ec s o
oil p ices on economic a iables. Fo each o hese amewo ks, I s udy he consequences o
he implemen a ion o al e na i e simple policy ules, and I in es iga e he ex en o which
he op imal esponse o a change in he p ice o oil is model dependen . Second, I p esen an
applica ion o a ange o echniques de eloped in he model unce ain y li e a u e o he speci…c
o m o unce ain y unde analysis in his pape .
This s udy is ela ed o he li e a u e on policy design and e alua ion in unce ain en i-
onmen s. In ecen yea s, wo majo di ec ions o wo k ha e eme ged in his a ea. The … s
one is ep esen ed by he con ibu ions o Hansen and Sa gen (2001a, 2001b, 2008). In his
app oach, unce ain y is de…ned o e speci…ca ions ha lie wi hin some dis ance om a base-
line amewo k, and p e e ences a e assumed o ollow a minimax ule wi h espec o model
unce ain y.4A second di ec ion is ep esen ed by he con ibu ions o B ock, Du lau and Wes
(2003, 2007). In his app oach, he model space includes speci…ca ions ha a e no close o
each o he acco ding o some me ic, and model unce ain y is in oduced in he policy decision
p ocess h ough he echnique o Bayesian model a e aging.5Recen ly, B ock, Du lau , Nason
and Rondina (2007) ha e p oposed ways o in oducing he minimax app oach due o Hansen
and Sa gen o con ex s in which he elemen s o he model space do no necessa ily lie wi hin
3An example o his disag eemen is he deba e be ween Be nanke, Ge le and Wa son (1997, 2004) and
Hamil on and He e a (2004) abou he ole o mone a y policy in he economic down u ns ollowing he oil
p ice shocks episodes o he pos wa pe iod.
4In mo e de ail, he decision make is assumed o minimizes while na u e maximize losses o e he se o
models in he model space. Applica ions o his app oach o mone a y policy can be ound in Giannoni (2007),
Ona ski and S ock (2002) and B ock and Du lau (2004).
5The wo ks o Cogley and Sa gen (2005) and Cogley e al. (2010) a e examples o applica ions o his
app oach o he analysis o mone a y policy.
3
some dis ance om a baseline model speci…ca ion.
This pape is me hodologically based on B ock, Du lau and Wes (2003, 2007) ( om now
BDW, 2003, 2007) and B ock, Du lau , Nason and Rondina (2007) ( om now BDNR). The
decision o ollow hese app oaches was mo i a ed by he ac ha he unce ain y o e he
mechanisms h ough which oil p ices a¤ec economic pe o mance is la gely non-local. The
desc ip ion o he model space in sec ions 4 will p o ide mo e e idence abou his s a emen .
In addi ion, BDW (2007) and BDNR (2007) in oduce policy e alua ion echniques ha mo e
beyond s anda d model a e aging me hods, and ha a e use ul in p o iding a mo e ex ensi e
and comp ehensi e policy analysis. Mo e speci…cally, BDW (2007) p opose a ange o measu es
and isual ools ha supply he policymake wi h mo e in o ma ion han a simple summa y
s a is ic in which model dependence has been in eg a ed ou . On he o he hand, BDNR
in oduce applica ions o policy e alua ion o non-Bayesian app oaches based on he minimax
and minimax eg e c i e ia, which ha e he ad an age o no equi ing any p e ious knowledge
o he cha ac e is ics o he model space.
This wo k is also ela ed o he la ge li e a u e s udying he impac o oil p ices on economic
ac i i y. This pape does no in end o ake a posi ion in he deba e o e he di¤e en models
p oposed o explain he e¤ec s o a change in oil p ices on economic pe o mance. Ra he , I
show ha di¤e en amewo ks, based on di¤e en channels o ansmission o oil p ice shocks
in o he economy, a e plausible al e na i e app oxima ions o he ue da a gene a ing p ocess.
Finally, his pape is ela ed o he li e a u e in es iga ing he esponse o mone a y policy o
changes in oil p ices. Recen con ibu ions ha e ocused on he ole o mone a y policy in he
down u ns ollowing he la ge oil p ice shocks o he pos wa pe iod (Be nanke, Ge le and
Wa son 1997, 2004; Hamil on and He e a, 2004; Leduc and Sill, 2004), and on i s con ibu ion
o he milde eac ion o economic a iables o oil p ice shocks since he mid 1980s (Blancha d
and Gali, 2007; He e a and Pesa en o, 2009; Cla k and Te y, 2010). This wo k p o ides some
addi ional insigh s in his a ea by explici ly analyzing he ex en o which he consequences o
he mone a y policy esponse o a change in oil p ices depend on he model o he economy
unde conside a ion.
The emainde o he pape is o ganized as ollows. Sec ion 2 summa izes he echniques
ha I will use o inco po a e model unce ain y in o policy e alua ion. Sec ion 3 illus a es he
main mechanisms ha ha e been p oposed o model he e¤ec s o oil p ices in he economy.
Sec ion 4 cha ac e izes he model unce ain y p oblem, de…nes he model space and s udies
i s basic p ope ies. Sec ion 5 epo s he esul s o he policy e alua ion exe cise. Sec ion 6
concludes.
4
2 Policy e alua ion unde model unce ain y
In his sec ion, I summa ize he echniques de eloped by BDW (2003, 2007) and BDNR o
accoun o model unce ain y in he e alua ion o al e na i e economic policies.6These a e
he echniques ha will be employed in he exe cise in sec ion 5.
2.1 Gene al F amewo k
The cen al idea o he app oach p oposed by BDW (2003, 2007) is ha model unce ain y
should be conside ed as a componen o policy e alua ion. This idea has wo implica ions.
The … s one is ha model unce ain y should no be esol ed p io o he e alua ion o a
policy ule h ough he selec ion o a speci…c model o he economy. The second one is ha
policy e alua ion should explici ly accoun o he lack o comple e in o ma ion abou he ue
da a-gene a ing p ocess.
Conside he p oblem o a policymake who is in e es ed in e alua ing he e¤ec o a pol-
icy ule pon an ou come . Typically, his policy will be s udied based on he condi ional
p obabili y measu e:
(jm; p; m)(1)
whe e mdeno es a model and mis a ec o o pa ame e s ha indexes he model. I he
model mis known, he a ailable da a dcan be used o es ima e he ec o o pa ame e s m.
In his case, (1) can be ew i en as:
(jm; p; d)(2)
The app oach o policy e alua ion in unce ain en i onmen s p oposed by BDW (2003, 2007)
en ails compu ing he p obabili y measu e (jd; p) om (2) by ea ing model unce ain y
as any o he o m o unce ain y a¤ec ing . This can be done by elimina ing he condi ioning
on min (2). Le Mbe he space o possible da a-gene a ing p ocesses, hen we ha e:
(jd; p) = X
M
(jm; p; d)(mjd)(3)
whe e (mjd)is he pos e io p obabili y o model mgi en da a d. By Bayes’ ule, his
measu e can be cha ac e ized as ollows:
(mjd)/(djm)(m)(4)
6This sec ion only p o ides a b ie explana ion o he echniques ha I will use in sec ion 5 o he pape .
Fo a mo e ho ough desc ip ion o hese me hods, see BDW (2003, 2007) and BDNR.
5
whe e (djm)is he likelihood o he da a gi en model mand (m)is he p io p obabili y
assigned o model m.7
Le now conside a policymake ha e alua es policies acco ding o he expec ed losses gen-
e a ed by a loss unc ion l(). The p e ious discussion implies ha he measu e inco po a ing
model unce ain y in o he analysis is:
E(l()jd; p) = Z
l()(jp; d)d (5)
The empi ical pa o his pape will in ol e compu a ion o expec ed losses o his o m, gi en
a s anda d loss unc ion ha will be de…ned in sec ion 4.
The model a e aging app oach has some a ac i e p ope ies, … s and o emos he ac
ha i allows o he assessmen and compa ison o policies wi hou condi ioning on a gi en
elemen o he model space. Howe e , i s implemen a ion p esen s se e al issues, mainly ela ed
o he de…ni ion o he model space Mand o he speci…ca ion o he p io p obabili ies o i s
elemen s. See BDW (2003, 2007) o a mo e exhaus i e discussion o he implemen a ion issues
o his app oach.
2.2 Ou come dispe sion and ac ion dispe sion
In addi ion o he model a e aging app oach, BDW (2007) p opose addi ional ways o commu-
nica ing in o ma ion abou he e¤ec s o di¤e en policies in an en i onmen cha ac e ized by
model unce ain y. The in oduc ion o hese addi ional s a is ics is mo i a ed by se e al con-
side a ions. Fi s , he policymake migh wan o in es iga e aspec s o he condi ional densi y
(jm; p; d) ha a e los in he a e aging p ocess. Second, he migh be conce ned abou he
beha io o his condi ional densi y only in some speci…c models a he han o he s. Thi d, he
migh be in e es ed in knowing which policies ha e an ou come ha is ela i ely mo e s able
ac oss he di¤e en speci…ca ions composing he model space.
Fo all o hese easons, i could be use ul o en ich he policy e alua ion exe cise by including
addi ional measu es ha a e able o o¤e a b oade pic u e o he e¤ec s o a policy unde
al e na i e ep esen a ions o he economy. BDW (2007) in oduce wo measu es ha p o ide
a cha ac e iza ion o he ex en o which mone a y policies and hei consequences a e model
dependen . These measu es a e ou come dispe sion and ac ion dispe sion.
Ou come dispe sion measu es he a ia ion in loss ha occu s when di¤e en models a e
conside ed, gi en a …xed policy ule. In o he wo ds, his measu e desc ibes how he losses
associa ed wi h a speci…c policy ule a e model dependen , hus p o iding in o ma ion on he
7See BDW (2007) o an in e es ing discussion o some in e p e a ions o he ole o model unce ain y in
policy e alua ion ha can be in e ed om his de i a ion.
6
obus ness o he selec ed policy ule o e di¤e en models. Ac ion dispe sion, on he o he
hand, measu es how he op imal policy di¤e s ac oss al e na i e models. A dis inc op imal
policy can be compu ed o any gi en model, so ha a ange o di¤e en policies can be ob ained
om he elemen s o a model space. The analysis o ac ion dispe sion p o ides in o ma ion on
he sensi i i y o he op imal policy ule o model choice.
2.3 Minimax and minimax eg e
In addi ion o he ou come dispe sion and ac ion dispe sion measu es, I will also conside non-
Bayesian app oaches based on he minimax and minimax eg e c i e ia. These app oaches
a e based on he idea ha policymake s migh be in e es ed in ob aining in o ma ion abou
policy ules ha a e no op imal, bu ha wo k well in some o he di ec ions o aspec s o
he policy analysis. In pa icula , hese c i e ia add ess a conce n o con olling he maximum
losses ha can be incu ed unde al e na i e policies in an en i onmen cha ac e ized by model
unce ain y.
The minimax app oach has been la gely used by Hansen and Sa gen (2001a, 2001b, 2008) as
he basis o obus ness analysis in mac oeconomics. In he policy e alua ion exe cise pe o med
in sec ion 5, I will ollow BDNR and de…ne he minimax policy choice as he one sol ing:
min
p2Pmax
m2ME(l()jp; d; m)(6)
Because i always assumes he wo s possible scena io in assessing al e na i e policies, he
minimax c i e ia has been c i icized o being ex emely conse a i e. To a oid his issue, he
li e a u e has in oduced he concep o minimax eg e , which is based on he ela i e ( a he
han absolu e) loss associa ed wi h a gi en policy. Following again BDNR, he minimax eg e
policy ule will be ob ained as he solu ion o he ollowing p oblem:
min
p2Pmax
m2MR(p; d; m)(7)
whe e R(p; d; m)is he eg e unc ion de…ned as:
R(p; d; m) = E(l()jp; d; m)min
p2PE(l()jp; d; m)(8)
Gi en a model, he eg e unc ion measu es he loss su¤e ed by a policy ela i e o he loss
unde he op imal policy o ha speci…c model. The de…ni ion o he eg e unc ion illus a es
how his c i e ion is able o a oid he p oblems associa ed wi h models ha compo ela i ely
high losses ega dless o he choice o he policy ule.
7
BDNR o¤e a mo e comp ehensi e exposi ion o he p ope ies o he minimax and minimax
eg e c i e ia and desc ibe some applica ions ha ha e been p oposed in he li e a u e.
3 Modeling he e¤ec s o oil p ices on he economy
This sec ion p o ides a b ie e iew o he mos ele an con ibu ions on he e¤ec s o oil p ices
on economic ac i i y.8
The li e a u e in economics has p oposed many di¤e en mechanisms h ough which oil
p ices can a¤ec economic pe o mance. Some ea ly s udies, such as Solow (1980) and Pindyck
(1980) ocus on he demand-side e¤ec s o changes in oil p ices. In hese amewo ks, he di ec
and immedia e consequence o a change in oil p ices is a change in he o e all p ice le el, which
in u n has an e¤ec on employmen and o he eal a iables due o he Keynesian assump ion
o igid wages. Thus, wage igidi y is he main channel h ough which oil p ice a ia ions a¤ec
ou pu in hese models. A simila explana ion has been p oposed by Blancha d and Gali (2007),
which assume p ice igidi ies in addi ion o wage igidi ies.
A second s and o li e a u e conside s he supply-side e¤ec s o changes in oil p ices. These
wo ks a e usually based on a p oduc ion unc ion in which ene gy is one o he inpu s, so ha
an exogenous change in he p ice o oil a¤ec s ou pu di ec ly by changing p oduc i i y, and
employmen h ough a change in he wage le el. Some con ibu ions based on his mechanism
a e Rasche and Ta om (1977) and Kim and Loungani (1992). This way o explaining he
e¤ec o oil p ices on ou pu seems o be qui e na u al in he con ex o a s anda d neoclassical
economic model. O he con ibu ions ha e conside ed depa u es om he s anda d neoclassical
amewo k ha a e able o explain addi ional indi ec e¤ec s o an oil p ice shock on ou pu .
Fo ins ance, Finn (2000) ocuses on he impac o changing capaci y u iliza ion a es, while
Ro embe g and Wood o d (1996) conside a model cha ac e ized by impe ec compe i ion, in
which addi ional e¤ec s on ou pu o igina e om changes in business ma kups.
Finally, one las g oup o con ibu ions has ocused on he e¤ec s o oil p ice shocks on
sho - un economic pe o mance as he consequence o alloca i e dis u bances. Some examples
o his li e a u e a e Be nanke (1983) and Hamil on (1988). These s udies ha e he ele an
ea u e o sugges ing a nonlinea ela ion be ween oil p ices and ou pu . A ise in oil p ices will
dec ease demand o some goods, bu possibly inc ease i o o he s. As a consequence, i i is
cos ly o ealloca e labo o capi al be ween sec o s, hen an oil shock will be con ac iona y
in he sho un. Howe e , an oil p ice dec ease would equi e he same ype o ealloca i e
p ocess, and o his eason i could be con ac iona y as well in he sho un.
8Ex ensi e e iews o he di¤e en mechanisms ha ha e been p oposed o explain he impac o oil p ices
on he economy a e p o ided by Mo k (1994), Hamil on (2005), Segal (2007) and Kilian (2008).
8
4 Model Unce ain y
I conside he p oblem o a policymake who wan s o in es iga e possible policy esponses o
changes in he p ice o oil. He knows ha many di¤e en mechanisms ha e been p oposed in
he economic li e a u e o explain he e¤ec s o oil p ices on economic ac i i y. In pa icula ,
he belie es ha he ue model o he economy migh be one o he ollowing h ee amewo ks:
Solow (1980) ( om now on deno ed as S), in which he mos ele an e¤ec o a change
in oil p ices is a change in he o e all p ice le el, which in u n a¤ec s employmen and
eal a iables due o he assump ion o nominal wage igidi ies. The e o e, in his model
he main channel h ough which oil p ices ha e an impac on ou pu is nominal wage
igidi ies.
Blancha d and Gali (2007) ( om now on deno ed as BG), in which he cen al e¤ec o a
change in oil p ices is a change in he o e all p ice le el, which in u n a¤ec s employmen
and eal a iables due o he assump ion o p ice and eal wage igidi ies. This is a new
Keynesian ype o model, and p ice igidi ies a e in oduced in he economy h ough he
assump ion o Cal o p icing. In his amewo k, he channel h ough which oil p ices
ha e an impac on economic ac i i y is eal wage and p ice igidi ies.
Kim and Loungani (1992) and Hamil on (2005) ( om now on deno ed as H), in which
changes in he p ice o oil a¤ec ou pu di ec ly by changing p oduc i i y and ha e an
impac on employmen h ough a change in he wage le el. This is a s anda d neoclassical
ype o model, cha ac e ized by pe ec compe i ion and ‡exible p ices and wages.
Gi en his belie s on he possible ue da a gene a ing p ocess, he policymake conside s
h ee di¤e en app oxima ing amewo ks ha inco po a e he main ea u es o each one o
hese ep esen a ions o he economy. These amewo ks a e in he spi i o he empi ical
li e a u e on mone a y policy, along he lines o King, S ock and Wa son (1995), Rudebusch
and S ensson (1999), Cogley and Sa gen (2005) and P imice i (2006). Each amewo k consis s
o wo equa ions, one o he ou pu gap and one o he in‡a ion a e, and includes he ollowing
a iables: he ou pu gap (y ), co e CPI in‡a ion ( ), he in e es a e (i ), which is he policy
ins umen , and eal oil p ice changes (s ).9
The Sapp oxima ing model is desc ibed by he ollowing equa ions:
y =S
y(L)y 1+S
(L) [ 1E 2( 1)] + S
s(L)s 1+!S
y; (9)
=S
(L) 1+S
y(L)y 1+S
i(L)i 1+S
s(L)s 1+!S
; (10)
9The use o co e CPI in‡a ion ollows Blancha d and Gali (2007) and Cla k and Te y (2010).
9
abili ies ha e been escaled so ha hey add up o 1 ac oss he model space. Thus, able 2
can be in e p e ed as he p obabili y ha he ue da a gene a ing p ocess ollows he Solow,
Blancha d-Gali o Hamil on heo y on he p edominan channels h ough which oil p ices a e
assumed o a¤ec he economy. In his sense, i is clea ha he da a a o s he Solow heo y,
since his class o models inco po a es 85:85% o he pos e io p obabili y. Howe e , he pos e-
io s a ached o he Hamil on and Blancha d-Gali heo ies, while conside ably lowe ela i e
o he Solow class, a e s ill la gely di¤e en om ze o. In addi ion, …gu e 1 also shows ha a
ew speci…ca ions, belonging o di¤e en classes o models, exhibi pos e io p obabili ies ha
a e ac ually compa able wi h each o he . Fo hese easons, a policymake conce ned abou
model unce ain y should no disca d any o hese heo y as he possible ue ep esen a ion o
he economy, bu a he look o a policy ule ha is able o pe o m ela i ely well in all o
hem.
F om …gu e 1, i is e iden ha each class o models is cha ac e ized by an hand ul o
speci…ca ions ha ha e highe pos e io p obabili ies, and a la ge numbe o hem ha , on he
con a y, ha e nea ze o pos e io s. Gi en he la ge numbe o models in M, policymake s migh
wan o es ic he model space and ocus only on hose speci…ca ions ha o¤e a plausible
ep esen a ion o he economy. In his choice, decision make s ace a adeo¤ be ween allowing
o a su¢ cien ly la ge deg ee o model unce ain y, and making he policy e alua ion exe cise
cumbe some and possibly e en no in o ma i e.13 He e, I ollow BDW (2007) in he p ocedu e
used o es ic he analysis o a smalle model space.14 This p ocedu e en ails compu ing he
ela i e pos e io o a model wi hin a class, de…ned as:
Pm=(mjd)
P
m2C
(mjd)=b
Lm
P
m2Cb
Lm
(20)
whe e b
Lmis he BIC-adjus ed likelihood o model m, and Cis equal o MS; MBH o MH
depending on he class unde conside a ion. The second equali y ollows om he ac ha in
his se up pos e io p obabili ies a e p opo ional o BIC-adjus ed likelihoods and ha , wi hin
each class o models C, all models ha e he same p io . In wo ds, his o mula escales he
pos e io p obabili ies so ha hey add up o one wi hin each class o models. The measu e
ob ained om (20) is hen used o iden i y he models ha ha e he highes ela i e pos e io
p obabili ies wi hin each class. In his wo k, hese models will be de…ned as hose o which
Pmis a leas 1=100 = 1% o he model wi h he highes Pmin he class. The policy e alua ion
13Many o he speci…ca ions wi h nea ze o pos e io s a e e y uns able, and exhibi in…ni e losses unde a
wide ange o policies. Fo his eason, hey migh domina e he policy e alua ion exe cise, despi e he ac ha
hei pos e io p obabili y is essen ially ze o.
14This app oach is based on he "Occam’s window" echnique o iginally p oposed by Madigan and Ra e y
(1994).
16
exe cise de eloped in he nex sec ion will ocus on his subse o model speci…ca ions.15
Table 3 - Rela i e pos e io p obabili y Pm
MSMBG MH
(1) Minimum Pm2:02 1024 7:34 1026 2:47 1022
(2) Q1 Pm1:53 1015 2:16 1018 3:89 1014
(3) Median Pm6:40 1012 9:08 1012 1:74 1010
(4) Q3 Pm3:92 1098:58 1078:59 108
(5) Maximum Pm0:2137 0:1378 0:1849
(6) No. models wi h Pm>(max Pm)=100 55 87 56
(7) Sum o Pmmodels wi h Pm>(max Pm)=100 0:8581 0:8827 0:9274
(8) Sum o Pm o models in op qua ile 1:0000 0:9998 1:0000
(9) Sum o Pm o models in bo om 3 qua iles 3:25 1061:97 1041:89 105
(10) Sum o Pm111
(11) No. models 20;480 5;120 5;120
No e: The ela i e pos e io p obabili y Pmis de…ned by (20). The sum o Pm o each class o models equals
one by cons uc ion.
Table 3 p o ides some summa y s a is ics on he dis ibu ion o he ela i e pos e io p ob-
abili ies o each class o models. This able clea ly shows ha , in each class, a es ic ed
numbe o speci…ca ions co e almos he en i e pos e io p obabili y o he class. Indeed, he
… s h ee qua iles only con ain speci…ca ions wi h ela i e pos e io s ha a e essen ially ze o,
while he sum o Pm o he … s qua ile is nea ly one in all classes. The numbe o speci…ca-
ions o which Pmis a leas 1% o he model wi h he highes Pm; epo ed in line (6), is e y
small ela i e o he size o each class, bu hese ew speci…ca ions s ill co e a e y high ela i e
pos e io , as shown in line (7). Fo he policy e alua ion exe cise in he nex sec ion, he model
space Mand he classes o models MS,MBG and MHa e ede…ned o inco po a e only he
models wi h he highes ela i e pos e io p obabili y. The e o e, he new model space includes
198 speci…ca ions, while MS,MBG and MHa e composed o 55,87 and 56 models espec i ely.
A mo e de ailed desc ip ion o he model speci…ca ions used in he policy e alua ion exe cise,
and he de…ni ion o he new model space and classes o models a e p o ided in Appendix 1.
15The ac o ha is used in BDW (2007) o de…ne he se o models wi h high pos e io p obabili y is 1/20.
The eason why I se a lowe h eshold is ha , in his con ex , a la ge numbe o models, wi h pos e io
p obabili y di¤e en om ze o as a g oup, do no ge cap u ed by he 1/20 h eshold. Since I will use he
subse o models wi h high pos e io p obabili ies o he policy e alua ion exe cise in he nex sec ion, he
lowe h eshold o 1/100 allows me o ha e a g oup o models ha p o ide a be e ep esen a ion o he o iginal
model space M:
17
Table 4 - Pa ame e es ima es o he models wi h he highes pos e io p obabili y
(A)Ou pu equa ion
y1y2y312i1i2s1R2DW s:e:
Smodel 1:183
(0:007) 0:024
(0:016)
0:221
(0:006)
0:030
(0:002) 0:162
(0:002)
n:a: n:a: 0:0014
(0:000)
0:90 1:92 0:49
BG model 1:132
(0:006) 0:028
(0:014)
0:206
(0:006)
n:a: n:a: 0:088
(0:002) 0:152
(0:002)
0:0014
(0:000)
0:90 1:98 0:50
Hmodel 1:187
(0:007) 0:107
(0:017)
0:186
(0:007)
n:a: n:a: n:a: n:a: 0:0011
(0:000)
0:89 2:03 0:54
(B)In‡a ion equa ion
y1y21234i1i2i3i4R2DW s:e:
Smodel 0:137
(0:004) n:a: 0:290
(0:008) 0:073
(0:008) 0:349
(0:007) 0:235
(0:008) 0:438
(0:009) 0:461
(0:014)
0:379
(0:015) 0:398
(0:010)
0:78 1:80 1:83
BG model 0:622
(0:029) 0:392
(0:027)
0:637
(0:006)
0:590
(0:006)
n:a: n:a: n:a: n:a: n:a: n:a: 0:39 2:01 2:21
Hmodel 0:137
(0:004) n:a: 0:290
(0:008) 0:073
(0:008) 0:349
(0:007) 0:235
(0:008) 0:438
(0:009) 0:461
(0:014)
0:379
(0:015) 0:398
(0:010)
0:78 1:80 1:83
No es: 1. Panel (A) p esen s he es ima ed coe¢ cien s o equa ions (9), (11) and (13) and Panel (B) he es ima ed coe¢ cien s o equa ions (10), (12)
and (14) o he speci…ca ion wi h he highes pos e io p obabili y in each class o models. Cons an e ms we e included in all he eg essions, bu
a e no epo ed o cla i y o exposi ion.
2. In Panel (A), ou pu gap is he dependen a iable, yj is he coe¢ cien on ou pu gap a lag j,ij and j a e he lag jcoe¢ cien s on he annual
eal in e es a e and unan icipa ed in‡a ion espec i ely, and sj is he coe¢ cien on eal oil p ice changes a lag j. In Panel (B), o he Sand H
speci…ca ions, in‡a ion is he dependen a iable, yj is he coe¢ cien on ou pu gap a lag j,j is he coe¢ cien on in‡a ion a lag j, and ij is
he coe¢ cien s on he annual nominal in e es a e a lag j. Fo he BG speci…ca ion, he change in in‡a ion is he dependen a iable, yj is he
coe¢ cien on ou pu gap a lag jand j is he coe¢ cien on he change in in‡a ion a lag j.
3. The sample is composed o qua e ly da a om 1973:I o 2008:II, o a o al o 142 obse a ions. In‡a ion is he annualized change in co e CPI; he
ou pu gap is he di¤e ence be ween eal GDP and he CBO es ima e o po en ial GDP, bo h in lags; he in e es a e is he a e age annual Fede al
unds a e; eal oil p ice changes a e he annualized change in he eal p ice o oil, compu ed as he di¤e ence be ween he log o he nominal p ice o
oil and he log o co e CPI. Addi ional in o ma ion on he da a used in he es ima ions is p o ided in Appendix 1.
18
Finally, able 4 epo s he es ima ed coe¢ cien s o he speci…ca ion wi h he highes
pos e io p obabili y in each class o models. As I men ioned be o e, pos e io p obabili ies
a e p opo ional o model speci…c BIC-adjus ed likelihoods. I ollows ha he speci…ca ions
p esen ed in able 4co espond o hose ha would ha e been selec ed wi hin each class using
BIC as he selec ion c i e ion. No ice ha in hese speci…ca ions eal oil p ice changes en e in
he ou pu equa ion wi h only one lag, and hey do no en e in he in‡a ion equa ion. Howe e ,
he subspace o models wi h high pos e io p obabili ies used in he policy analysis includes
speci…ca ions wi h a highe numbe o lags o he oil measu e in bo h equa ions. Again, see
Appendix 1 o u he de ails on he elemen s o he es ic ed model space.
4.5 Simple ules
This wo k aims o compa e he pe o mance o al e na i e policy ules in an en i onmen
cha ac e ized by unce ain y on he way oil p ices a¤ec economic a iables. Thus, a e ha ing
desc ibed he space o models unde conside a ion, he second s ep is de…ning he se o policies
o be e alua ed. As p e iously men ioned, I assume ha policymake s only conside simple
policy ules in he o m o (18).
The … s ule included in he se o policies unde analysis is he one o iginally p oposed by
Taylo (1993) ( om now on deno ed as OT ule):
i = 1:5 + 0:5y (21)
This policy ule is widely used in he li e a u e and was likely also implemen ed in p ac ice,
so i will be conside ed as a benchma k. In addi ion o he OT ule, policymake s migh wan
o s udy he pe o mance o policies ha a e o some ex en op imal unde he heo ies hey
ega d as possibly gene a ing he da a. To ob ain hese policy ules, I ollowed BDW (2007)
and used he speci…ca ion wi h he highes pos e io p obabili y in each class o models. Mo e
speci…cally, I compu ed hese ules by pe o ming a g id sea ch o he pa ame e s g,gy,giand
gsin (18) ha minimize he condi ional expec ed loss:
b
Rm= a (1jd; p; m) + y a (y1jd; p; m) + i a (i1jd; p; m)(22)
o each o he h ee models desc ibed in able 4. I es ic ed his sea ch o ules in which he
long un e¤ec o ou pu and co e CPI in‡a ion on he nominal in e es a e is he same as
in he Taylo ule.16 No es ic ions we e imposed on he coe¢ cien on eal oil p ice changes,
gs. In o he wo ds, I assumed ha he mone a y au ho i y wan s o e alua e he pe o mance
16Mo e speci…cally, I pe o med a g id sea ch only on alues o g,gyand gi ha sa is y: g=(1 gi) = 1:5
and gy=(1 gi) = 0:5.
19
o he Taylo ule ela i e o al e na i e simple ules which di¤e om he o iginal Taylo ule
only in e ms o in e es a e smoo hing and he (possible) esponse o oil p ices. This exe cise
p o ides a clea pic u e o he impac ha eac ing o changes in he eal p ice o oil has
on policymake s’losses, and seems o be he mos app op ia e in a con ex cha ac e ized by
unce ain on he way in which oil p ices a¤ec economic a iables.17
The simple policy ules ob ained om he desc ibed p ocedu e, deno ed as S ule, BG ule
and H ule, a e epo ed in able 5.
Table 5 - Policy space: he simple policy ules
S ule BG ule H ule
gy0.1995 0.4635 0.2670
g0.5985 1.3905 0.8010
gi0.6010 0.0730 0.4660
gs-0.0071 -0.0194 -0.0040
Exp. loss 31.688 22.706 19.771
Long un
eg1.5 1.5 1.5
egy0.5 0.5 0.5
egs-0.0178 -0.0209 -0.0075
No es: 1. Simple ules in he o m desc ibed by (18). These ules we e ob ained by g id sea ch o he coe -
…cien s in (18) ha minimize (22) unde he es ic ions g=(1 gi) = 1:5and gy=(1 gi) = 0:5 o he
speci…ca ion wi h he highes pos e io p obabili y in each class o models.
2. The long un e¤ec o y; and son he nominal in e es a e is de…ned as: egk=gk=(1 gi); k =y; ; s:
The simple ules epo ed in able 5o¤e some ele an insigh s on he di¤e ences in he
op imal policy esponse o oil p ices in each o he h ee heo ies unde conside a ion. In
pa icula , we can compa e he sho un and long un e¤ec s o oil p ices on he nominal
in e es a e ha hese h ee policies imply. As expec ed, he BG ule ecommends he s onges
esponse o changes in he eal p ice o oil, bo h in he sho un and in he long un. Indeed, he
Blancha d-Gali heo y assumes ha he economy is cha ac e ized by a numbe o igidi ies ha
17In a p e ious e sion o he pape , I was compa ing he o iginal Taylo ule o he op imal simple ules
ob ained by minimizing (22) wi h no es ic ions on he alues o he coe¢ cien s gyand g. Howe e , I ound
ha exe cise o be less in o ma i e han he one pe o med he e. Indeed, he di¤e ences in pe o mance be ween
he al e na i e simple ules and he o iginal Taylo ule we e la gely d i en by hei di¤e en esponse o ou pu
and in‡a ion, and i was di¢ cul o disce n he ole o he eac ion o changes in he eal p ice o oil. He e, his
is no he case, because he long un esponse o he ou pu gap and co e CPI in‡a ion is se o be equal in all
he ules conside ed in he policy e alua ion exe cise.
20
ha e he po en ial o ampli y he impac o oil p ices on he a iables o in e es o policymake s.
A he same ime, in his heo y he policy ins umen can a¤ec he ou pu gap di ec ly so ha ,
by esponding o changes in he eal p ice o oil, policymake s a e able o con as he e¤ec s o
his a iable on he eal economy. On he o he hand, in he Solow heo y mone a y policy has
an impac on he ou pu gap only indi ec ly h ough unan icipa ed in‡a ion. None heless, he
policy sugges ed by his heo y s ill implies a ela i ely la ge eac ion o he nominal in e es
a e o oil p ices, especially in he long un. Finally, in he Hamil on heo y policymake s canno
modi y he ou pu gap wi h hei policy choices. Fo his eason, he in e es a e esponse o
he oil a iable is much smalle , and di ec ed o con as i s e¤ec s on co e CPI in‡a ion only.
Figu e 2 - Impulse esponses: simple ules and no ac ion
No e: Response o he ou pu gap, co e CPI in‡a ion and he Fede al unds a e o a 10% inc ease in he eal
p ice o oil. The … s columns epo s he ou pu gap, he second column co e CPI in‡a ion and he las column
he Fede al unds a e. Each ow ep esen s a di¤e en model and ela i e policy ule. In each panel, he
esponse o he a iable o in e es unde he selec ed policy ule (con inuous line) is compa ed o he esponse
when no ac ion is unde aken by he policymake , i.e. when he coe¢ cien s in (18) a e all se equal o ze o
(dashed line).
Fo a be e unde s anding o he implica ions o he simple ules desc ibed in able 5,
I s udied he policy esponse o a 10% inc ease in he eal p ice o oil ha each o hem
21
ecommends. Mo e speci…cally, I in es iga ed he esponse o ou pu , co e CPI in‡a ion and he
Fede al Funds a e in each o he models desc ibed in able 4, when he policymake implemen s
he espec i e op imal simple ule. In each model, he impac o he policy esponse o he
change in oil p ices is compa ed wi h he pa e n o he a iables o in e es when no ac ion
is unde aken by he policymake , i.e. when he coe¢ cien s on he policy ule in (18) a e all
se equal o ze o. This exe cise p o ides u he e idence abou he ac ha he abili y o
mone a y policy o con as an oil p ice shock is model dependen . The esul s o his exe cise
a e epo ed in …gu e 2; some u he analysis o he policy esponses implied by each o he
ules desc ibed in able 5 is p o ided in Appendix 2.
A ew hings can be obse ed om …gu e 2. Fi s , as discussed he ecommended esponse
o oil p ices is s onge in he Solow and Blancha d-Gali heo ies ela i e o he Hamil on
heo y. Second, in he Blancha d-Gali model i policymake s do no espond o he change
in he eal p ice o oil, bo h he ou pu gap and co e CPI in‡a ion quickly di e ge owa ds
in…ni e nega i e alues. Thus, in his model policymake s mus eac o changes in oil p ices
o p ese e he s abili y o he a iables o in e es . Thi d, …gu e 2 shows ha he abili y o
policymake s o con as he e¤ec s o a change in he eal p ice o oil on he ou pu gap is
qui e di¤e en depending on he model o he economy unde conside a ion. Fo his eason,
he exe cise epo ed in his …gu e p o ides some addi ional insigh s on he deba e be ween
Be nanke e al. (1997, 2004) and Hamil on and He e a (2004) o e he ole o mone a y policy
in he declines in ou pu ha ollowed mos o he oil p ice shocks o he pos wa pe iod. While
Be nanke e al. (1997, 2004) sugges ha he economic down u ns would ha e been milde i
he policymake had adop ed a less con ac iona y policy a e an oil p ice shock, Hamil on
and He e a (2004) a gue ha ou pu would ha e dec eased no ma e wha policy had been
implemen ed. Figu e 2 epo s an impulse esponse exe cise ha is e y simila o hose s udied
by Be nanke e al. (1997, 2004) and Hamil on and He e a (2004), and he panels in he … s
column o his …gu e a e ac ually consis en wi h he esul s o hese con ibu ions. In mo e
de ail, i he ue model o he economy is he BG model, hen …gu e 2 shows ha policymake s
can success ully educe he down all in ou pu caused by an oil p ice shock by implemen ing
an expansiona y policy ule. This conclusion suppo s he posi ion o Be nanke e al. (1997,
2004). On he o he hand, i he ue da a gene a ing p ocess is ei he he Hmodel o he S
model, hen policymake s a e no able o a oid he dec ease in ou pu caused by a change in oil
p ices, and a mo e expansiona y policy ule b ings no bene… s o he eal economy, which is he
opinion exp essed by Hamil on and He e a (2004). Thus, his exe cise p o ides e idence ha
bo h posi ions can be co ec , depending on which heo y is ega ded as he one gene a ing he
da a.
The simple ules epo ed in able 5 ha e been selec ed o minimize losses in a speci…c model
22
belonging o one o he h ee heo ies unde conside a ion. Howe e , hei pe o mance in he
o he speci…ca ions included in hei same class o in he o he classes o models is no ob ious,
and policymake s migh be in e es ed in e alua ing whe he he adop ion o one o hem o¤e s
ad an ages ela i e o he implemen a ion o he OT ule. This exe cise is ca ied ou in he
nex sec ion, using he measu es ha ha e been p e iously desc ibed in sec ion 2.
5 Policy E alua ion
In a con ex in which he mone a y au ho i y does no know whe he he ue model o he
economy belongs o he MS; MBG o MHclass, wha a e he consequences o adop ing a speci…c
policy ule? Wha ules a e mo e obus ac oss he di¤e en speci…ca ions? These ques ions
will be in es iga ed in his sec ion.
A la ge pa o he policy e alua ion exe cise pe o med in his sec ion is based on he s udy
o expec ed losses condi ional on a gi en model speci…ca ion and policy ule, as de…ned in (22).
In addi ion, he Bayesian po ion o he analysis equi es he compu a ion o expec ed losses
o he di¤e en classes o models and o he en i e model space. As in BDW (2007) and
Cogley e al. (2009), hese will be ob ained by aking a weigh ed a e age o he model speci…c
condi ional expec ed losses, using pos e io p obabili ies as weigh s. Thus, he expec ed loss
ac oss he en i e model space when model unce ain y is inco po a ed in o he analysis will be
de…ned as:
b
R=X
m2Mb
Rm(mjd)(23)
Using he same app oach, he expec ed loss o each class o models will be compu ed as:
b
RC=X
m2Cb
Rm(mjd) = P
m2Cb
Rmb
Lm
P
m2Cb
Lm
(24)
whe e again b
Lmis he BIC adjus ed likelihood o model m, and he second equali y ollows
om he ac ha wi hin each class o models all speci…ca ions ha e he same p io p obabili y.
5.1 Ou come dispe sion
Ou come dispe sion measu es he a ia ion in loss ha occu s when conside ing he e¤ec s o
he same policy ule in di¤e en model speci…ca ions. Table 6 epo s he p ope ies o he
dis ibu ion o losses o each class o models unde each o he ou policy ules included in o
he analysis (OT ule, S ule, BG ule and H ule). Table 7 p o ides a desc ip ion o he same
23
dis ibu ion ac oss he whole model space. Finally, …gu e 3 o¤e s a isual ep esen a ion o he
in o ma ion p esen ed in hese wo ables.
Table 6 - Dis ibu ion o model losses unde each o he policy ules
Class o models MSMBG MH
Policy ule OT S BG H OT S BG H OT S BG H
(1) Mean 42.92 39.48 40.88 40.31 68.46 46.87 40.67 57.29 32.90 30.86 32.49 30.54
(2) S . de ia ion 11.45 10.72 9.97 11.34 36.23 23.31 20.64 29.80 11.51 10.31 9.20 11.16
(3) Minimum 26.21 25.54 28.57 24.59 27.42 22.60 21.20 26.22 19.07 19.90 22.22 18.15
(4) Q1 34.30 31.00 33.37 31.54 40.90 27.90 24.92 34.22 25.11 22.92 25.66 22.21
(5) Median 41.26 36.62 38.51 38.10 49.46 35.34 33.01 43.75 30.08 28.31 29.05 28.39
(6) Q3 49.40 44.92 44.74 47.10 96.23 62.15 52.92 80.40 37.78 34.36 35.23 34.56
(7) Maximum 78.55 75.67 75.53 77.02 157.41 127.63 130.24 132.26 67.20 64.32 63.72 65.59
(8) P. w. a e age 38.70 35.31 36.97 36.07 57.56 36.80 31.87 46.43 26.85 25.30 27.19 24.71
(9) N. o models 55 87 56
No es: 1. Dis ibu ion o model speci…c losses o each class o models unde he Taylo ule and he h ee
simple ules desc ibed in able 5: he S ule, he BG ule and he H ule.
2. Rows (1) - (7) epo basic s a is ics o he dis ibu ion o losses o each class o models unde each policy
ule. Row (8) epo s he pos e io weigh ed a e age loss, compu ed using (24).
3. The composi ion o each class o models is desc ibed in Appendix 1.
In he en i onmen unde analysis, he simple ules desc ibed in able 5 pe o m be e
han he OT ule in e ms o he … s and second momen s o he dis ibu ion o losses ha
hey gene a e. In pa icula , able 6 shows ha while he OT ule implies highe and mo e
dispe se expec ed losses in all classes o models, i s pe o mance is signi…can ly wo se han he
o he ules in he BG class. Among he h ee simple ules, he H ule deli e s a highe mean
and s anda d de ia ion o losses han he Sand BG ules in he BG class, while all o hem
imply simila losses in he o he wo classes. The conside ably lowe s anda d de ia ion o
expec ed losses ha can be a ained by adop ing he So BG ule should be a cha ac e is ic
o pa icula in e es o policymake s in an en i onmen cha ac e ized by unce ain y on he
model ha gene a es he da a.
24
Table 7 - Dis ibu ion o losses ac oss he model space
OT ule S ule BG ule H ule
(1) Mean 51.308 40.287 38.414 45.009
(2) S anda d de ia ion 29.866 18.521 15.840 24.289
(3) Minimum 19.068 19.900 21.198 18.152
(4) Q1 32.592 27.894 26.692 29.511
(5) Median 41.377 33.323 33.948 36.224
(6) Q3 55.738 48.493 44.579 51.304
(7) Maximum 157.413 127.634 130.243 132.259
(8) Pos e io weigh ed a e age 37.866 34.164 35.647 35.030
(9) N. o models 198 198 198 198
No es: 1. Dis ibu ion o model speci…c losses ac oss he en i e model space unde he Taylo ule, he S ule,
he BG ule and he H ule.
2. Rows (1) - (7) epo basic s a is ics o he dis ibu ion o losses unde each policy ule. Row (8) epo s he
pos e io weigh ed a e age loss, compu ed using (23).
3. The composi ion o he model space is desc ibed in Appendix 1.
Figu e 3 - Ou come dispe sion o each policy ule
No es: 1. Model speci…c expec ed losses unde he o iginal Taylo (OT) ule, de…ned in (21), and he S,BG
and H ules desc ibed in able 5.
2. The summa y s a is ics o he dis ibu ion o losses in each class o models a e epo ed in able 6. The
summa y s a is ics o he dis ibu ion o losses ac oss he model space a e epo ed in able 7.
3. Models om 1 o 55 belong o he Sclass, om 56 o 142 o he BG class, and om 143 o 198 o he H
class. Addi ional in o ma ion on he model numbe s is p o ided in Appendix 1.
25
he OT ule and he S,BG and H ules desc ibed in able 5. As p e iously discussed, he
non-Bayesian minimax and minimax eg e app oaches do no ake in o accoun he models’
pos e io p obabili ies. The e o e, in his po ion o he policy e alua ion an equal weigh is
a ached o all he speci…ca ions included in he model space.
Table 11 - Minimax analysis
(1) All models (2) MS(3) MBG (4) MH
N. o models 198 55 87 56
Max Loss
Taylo ule 157.41 78.55 157.41 67.20
S ule 127.63 75.67 127.63 64.32
BG ule 130.24 75.53 130.24 63.72
H ule 132.26 77.02 132.26 65.59
Minimax S ule BG ule S ule BG ule
No es: 1. Robus policy ule ecommended by he minimax c i e ion o each class o models and o he en i e
model space. The minimax c i e ion is de…ned by (6).
2. The OT ule is de…ned by (21) and he S,BG and H ules a e de…ned by (18), wi h coe¢ cien alues as
epo ed in able 5.
3. The composi ion o each class o models is desc ibed in Appendix 1.
Table 12 - Minimax eg e analysis
(1) All models (2) MS(3) MBG (4) MH
N. o models 198 55 87 56
Max Reg e
Taylo ule 83.22 8.25 83.22 9.53
S ule 57.03 0.95 57.03 3.79
BG ule 59.65 5.52 59.65 10.25
H ule 57.71 4.00 57.71 5.70
Minimax Reg e S ule S ule S ule S ule
No es: 1. Robus policy ule ecommended by he minimax eg e c i e ion o each class o models and o he
en i e model space. The minimax eg e c i e ion is de…ned by (7).
2. The OT ule is de…ned by (21) and he S,BG and H ules a e de…ned by (18), wi h coe¢ cien alues as
epo ed in able 5.
3. The composi ion o each class o models is desc ibed in Appendix 1.
Table 11 epo s he esul o he minimax analysis o each class o models and o he en i e
model space. Ac oss he 198 speci…ca ions composing he model space, he policy ule ha
32
minimizes he maximum possible loss is he S ule. This is also he case i we only conside he
Blancha d-Gali class o models, while in he o he wo classes he BG ule deli e s a (sligh ly)
lowe maximum loss. In all se s o models, he OT ule induces he highes maximum loss.
Table 12 epo s he policy ecommenda ions o he minimax eg e c i e ion. Fo each
model, eg e is de…ned as he di¤e ence be ween he loss su¤e ed by a policy and he loss
unde he op imal policy o ha speci…c model. Thus, ela i e o he minimax c i e ion, his
measu e is able o educe he dominance o hose speci…ca ions ha en ail ela i ely high losses
ega dless o he selec ed policy. Table 12 shows ha he policy minimizing he maximum
eg e , in each class o models and in he en i e model space, is he S ule. Again, in all se s o
models, he OT ule deli e s he highes maximum eg e .
Fo he space o model speci…ca ions conside ed in his wo k, ables 11 and 12 show ha
he minimax and he minimax eg e c i e ia bo h ecommend he same policy, ha is he S
ule. This policy is also he one ha gene a es he lowes pos e io weigh ed a e age loss ac oss
he model space, as epo ed in able 7. Thus, among he policy ules conside ed in he policy
e alua ion exe cise, he Bayesian model a e aging app oach and he non-Bayesian minimax
and minimax eg e c i e ia ag ee on he choice o he obus policy unde model unce ain y.
Mo eo e , unde all measu es he leas ecommended policy is always he o iginal Taylo ule.
5.4 An al e na i e model space
In he baseline scena io, he de…ni ion o he model space was cen ed on he h ee di¤e en
heo ies ha policymake s belie e as possibly gene a ing he da a. As a consequence, he
speci…ca ions included in he es ic ed model space used o he ou come dispe sion, ac ion
dispe sion, minimax and minimax eg e analysis we e hose wi h he highes pos e io p ob-
abili ies wi hin each class o models. In his sec ion, I in es iga e whe he he esul s o he
policy e alua ion exe cise would be di¤e en unde an al e na i e de…ni ion o he model space
ha pu s less emphasis on he heo y om which each model speci…ca ion o igina es.
The model space conside ed in his sec ion was de…ned using he ollowing p ocedu e. S a -
ing om he ini ial se o 30;720 speci…ca ions, I a ached he same ini ial weigh o all o hem
by assuming a uni o m p io o 1=30720. Then, I selec ed all he models wi h pos e io p ob-
abili y o a leas 1=200 = 0:5% o he model wi h he highes pos e io in he en i e model
space.19 In his way, only speci…ca ions wi h high pos e io p obabili y in absolu e (and no in
ela i e) e ms we e included in he es ic ed model space used o he policy e alua ion exe -
cise. This p ocedu e selec ed a o al o 96 models, co e ing 89:27% o he pos e io p obabili y.
19I dec eased he h eshold ela i e o he baseline scena io o include an o e all pos e io p obabili y compa-
able wi h hose epo ed in able 3 o he di¤e en classes o models. In any case, he same exe cised pe o med
wi h he h eshold o 1% deli e s e y simila esul s.
33
O hese, 89 we e pa o he o iginal Solow class o models, 1o he Blancha d-Gali class, and
6o he Hamil on class. The highe p io a ached o he Solow speci…ca ions ela i e o he
baseline case is e‡ec ed in he composi ion o he es ic ed model space, which is almos en-
i ely cons i u ed o models belonging o his class. The speci…ca ion wi h he highes pos e io
p obabili y in his al e na i e de…ni ion o he model space co esponds o he speci…ca ion
wi h he highes pos e io in he Solow class, so i s es ima ed coe¢ cien s we e al eady epo ed
in able 4.
Table 13 - Dis ibu ion o model losses in he al e na i e model space
OT ule S ule BG ule H ule
(1) Mean 42.406 39.065 40.508 39.776
(2) S anda d de ia ion 12.516 11.754 11.0945 12.395
(3) Minimum 21.352 20.344 22.549 19.459
(4) Q1 34.006 30.279 33.281 30.535
(5) Median 40.205 36.521 38.340 37.291
(6) Q3 49.283 45.397 45.701 47.012
(7) Maximum 79.226 75.669 75.527 77.020
(8) Pos e io weigh ed a e age 38.729 35.379 37.033 36.101
(9) N. o models 96 96 96 96
No es: 1. Dis ibu ion o model losses unde di¤e en policy ules. The OT ule is de…ned by (21) and he S,
BG and H ules a e de…ned by (18), wi h coe¢ cien alues as epo ed in able 5.
2. Rows (1) - (7) epo basic s a is ics o he dis ibu ion o losses unde each policy ule. Row (8) epo s he
pos e io weigh ed a e age loss, compu ed using (23).
3. The models space is composed o 96 models, 89 om he Solow class, 1 om he Blancha d-Gali class, and
6 om he Hamil on class. These models we e selec ed using he p ocedu e desc ibed in he main ex .
Table 13 p o ides some summa y s a is ics o he dis ibu ion o losses ac oss he new model
space o he policy ules ha we e s udied in he o iginal analysis. The S ule co esponds
o he policy ecommended by he speci…ca ion wi h he highes pos e io in his new model
space. In addi ion, he Blancha d-Gali and Solow speci…ca ions desc ibed in able 4, which we e
used o compu e he BG and H ules, a e s ill pa o he model space, e en in his al e na i e
de…ni ion. Fo his eason, as well as o compa ison pu poses, he policy e alua ion exe cise
was pe o med using he same se o policies conside ed in he baseline model scena io.
The pe o mance o he OT ule in e ms o he … s wo momen s o he dis ibu ion o losses
ac oss he model space is conside ably imp o ed in his case. This esul was somehow expec ed,
since his policy ule o igina es high losses pa icula ly in he Blancha d-Gali class o models,
34
which is g ea ly unde ep esen ed he e compa ed o he baseline scena io (1speci…ca ion ins ead
o 87). In addi ion, he ma ginal p esence o Blancha d-Gali speci…ca ions o which, as shown
in …gu e 3, losses exhibi a gene al endency o be mo e ola ile, induces a educ ion in he
s anda d de ia ion o losses unde all policy ules. None heless, he di¤e ences in pe o mance
in e ms o pos e io weigh ed a e age loss a e almos he same as hose epo ed in able 7.
This happens because he Solow speci…ca ions domina e he model space in e ms o pos e io
p obabili ies, in his exe cise as well as in he baseline case. The e o e, he di¤e ences in he
expec ed losses gene a ed by he selec ed policies in he wo scena ios almos disappea when
hese a e weigh ed using he models’pos e io s.
Figu e 5 - Model losses o each policy ela i e o he Taylo ule in he al e na i e model space
No es: 1. Each panel epo s he a io be ween he loss gene a ed by one o he simple policy ules desc ibed
in able 5 and he loss gene a ed by he o iginal Taylo ule, o each speci…ca ion in he model space.
2. Model numbe s a e as ollows: speci…ca ions om 1 o 89 belong o he Solow class o models, speci…ca ion
90 belongs o he Blancha d-Gali class, and speci…ca ions om 91 o 96 belong o he Hamil on class. These
models ha e been selec ed using he p ocedu e desc ibed in he main ex .
Finally, …gu e 5 p o ides some addi ional in o ma ion abou he losses gene a ed by he
S,BG and Hpolicy ules ela i e o he OT ule. This …gu e shows ha , while o some
elemen s o he model space he OT ule is able o ou pe o m he BG ule in e ms o model
speci…c losses, his is almos ne e he case when his ule is compa ed o he Sand Hpolicies.
In all, om able 13 and …gu e 5we can conclude ha e en i he OT ule is conside ably
35
mo e compa able o he o he policies in his di¤e en model space, i s ill p oduces he highes
a e age losses, ei he non-weigh ed o weigh ed using he models’pos e io p obabili ies.
In his al e na i e de…ni ion o he model space, based on he pos e io weigh ed a e age
losses epo ed in able 13 a Bayesian policymake would selec he S ule. On he o he hand,
he minimax c i e ion would sugges he BG ule, while he minimax eg e app oach would
ecommend again he S ule. Rela i e o he baseline scena io, only he minimax c i e ion
selec s a di¤e en policy ule. Howe e , i is clea om line (7) in able 13 ha he lead o
he BG policy ule o e he S ule is minimal, since he maximum loss gene a ed by hese wo
policies is ac ually almos he same. The e o e, he conclusion ha in his en i onmen a ious
app oaches poin o he S ule as he obus policy choice unde model unce ain y can be
ega ded as alid e en unde he di¤e en de…ni ion o model space conside ed in his sec ion.
Fu he mo e, all he echniques s ill ega d he o iginal Taylo ule as he leas obus among
he se o policies unde s udy.
6 Concluding ema ks
In his pape , I analyzed he p oblem o a policymake ha is unce ain abou he mechanisms
h ough which oil p ices a¤ec economic ac i i y. I conduc ed an empi ical s udy o he likeli-
hood o h ee al e na i e heo ies ha ha e been p oposed o explain he e¤ec s o oil p ices on
he economy, and I p esen ed a policy e alua ion exe cise encompassing a ange o echniques
ha ha e been de eloped in he model unce ain y li e a u e. In his en i onmen , I ound ha
acco ding o a numbe o Bayesian and non-Bayesian measu es, he o iginal Taylo ule pe -
o ms wo se han a se o al e na i e simple ules in which policymake s in oduce pe sis ence
in he nominal in e es a e and espond o changes in he eal p ice o oil. In pa icula , I
showed ha allowing he policy ule o eac o oil p ices is impo an o con olling he mean
and ola ili y o expec ed losses ac oss he di¤e en speci…ca ions conside ed in he analysis.
This esul was no ob ious. Since he di¤e en elemen s o he model space ecommend a
con as ing op imal esponse o oil p ices (nega i e in some cases, posi i e in o he s) i could
ha e been possible as well ha a policy ule no eac ing o changes in he eal p ice o oil
pe o med be e han ano he one imposing a esponse in one speci…c di ec ion.
I belie e ha his wo k could be ex ended in a ew di¤e en di ec ions. Fi s , he policy
analysis could be en iched o accoun o he lack o consensus on he way oil p ices should
be measu ed. Indeed, while a pa o he li e a u e ocused on eal oil p ices, in le els o
di¤e ences (see, o ins ance, Blancha d and Gali, 2007 and He e a and Pesa en o, 2009),
o he con ibu ions in oduced al e na i e measu es o nominal oil p ice changes (see, among
he o he s, Hamil on, 2003 and Ca allo and Wu, 2009). This issue could be inco po a ed in he
36
amewo k p oposed in his pape by simply conside ing he unce ain y on he way oil p ices
should be de…ned as an addi ional o m o unce ain y cha ac e izing he model space.
A second ex ension could be he inclusion o models ha ocus on alloca i e dis u bances
as he channel h ough which oil p ices a¤ec economic ac i i y (see o ins ance Be nanke,
1983 o Hamil on, 1988). As explained by Hamil on (2005), i his is ac ually he mechanism
h ough which oil p ices a¤ec he economy, hen he e is no eason o expec a linea ela ion
be ween oil p ices and GDP. An oil p ice inc ease would dec ease demand o some goods and
possibly inc ease demand o o he s, and i would c ea e incen i es o households o pos pone
hei in es men ac i i y. Howe e , an oil p ice dec ease would ha e he same e¤ec on he
economy, so ha bo h an oil p ice inc ease and an oil p ice dec ease could be con ac iona y in
he sho un. Fo his eason, i migh be wo hy o hink abou possible ways o including his
addi ional channel o ansmission o he e¤ec s o oil p ices in he policy e alua ion exe cise.
Finally, a las ex ension could be in he di ec ion o in es iga ing he ole o expec a ions
in his en i onmen . As a … s s ep, he assump ion o backwa d looking expec a ions could be
eplaced by he use o su ey da a on expec ed in‡a ion. In addi ion, i migh be in e es ing
o in oduce unce ain y on he way expec a ions a e o med, in a way simila o BDW (2007).
37
Appendix 1
Da a desc ip ion and model labeling
Da a desc ip ion
The a iables used in he main ex a e he ollowing:
y is he ou pu gap, compu ed as he di¤e ence be ween eal GDP and he CBO es ima e
o po en ial GDP, bo h exp essed in logs.
is he annualized di¤e ence in log co e CPI, whe e co e CPI is he "CPI o all u ban
consume s: all i ems less ene gy p oduc s".
s is he annualized change in he eal p ice o oil. The eal p ice o oil is de…ned as he
di¤e ence be ween he nominal p ice o oil and co e CPI, bo h exp essed in logs. The
nominal p ice o oil is he Wes Texas In e media e spo oil p ice, while co e CPI is he
same used o compu e .
i is he a e age Fede al Funds a e.
The da a is qua e ly and includes obse a ions om 1973 : I o 2008 : II, wi h da a om
1971 : I o 1972 : IV used o p o ide lags. All he da a was ob ained om he Fede al Rese e
Bank o S . Louis web si e.
The compu a ion o he expec ed losses de…ned by (22) equi es ha policymake s know
he alue o he pa ame e s in he p ocess o he eal p ice o oil. As explained in he main
ex , Rondina (2010) es ima es hese pa ame e s using a MCMC algo i hm. Gi en he esul s
o his ela ed wo k, I se = 0:91; 2
o= 42(220) and 2
"= 42(1:9) :This implies ha 2
=
42(441:9) = 84:092:
Model labeling
The ull model space includes 30;720 models, 20;480 o MSand 5;120 o MBG and MH:The
numbe ing o he models is o ganized as ollows:
models om 1 o 20;480 a e he Sclass o models;
models om 20;481 o 25;600 a e he BG class o models;
models om 25;601 o 30;720 a e he Hclass o models.
38
The elemen s o each class o models di¤e in e ms o he a iables and he numbe o lags
o each a iable included in he ou pu and in‡a ion equa ions, as speci…ed in able 1. In each
class o models, he o de in which he lags change is he ollowing:
1. lags o s in he in‡a ion equa ion;
2. lags o i in he in‡a ion equa ion ( o he Sand Hmodels);
3. lags o in he in‡a ion equa ion;
4. lags o y in he in‡a ion equa ion;
5. lags o s in he ou pu equa ion;
6. lags o he eal in e es a e (i 1E 1( )) in he ou pu equa ion ( o he BG models);
7. lags o unan icipa ed in‡a ion ( E 1( )) in he ou pu equa ion ( o he Smodels);
8. lags o yin he ou pu equa ion.
In he policy e alua ion exe cise, I only conside a subse o he ini ial model space, composed
o 55 models o he Sclass, 87 models o he BG class and 56 models o he Hclass o a
o al o 198 model speci…ca ions. The p ocess used o selec hese models was explained in he
main ex . The numbe ing o he elemen s in his es ic ed model space is as ollows:
models om 1 o 55 a e he Sclass o models;
models om 56 o 142 a e he BG class o models;
models om 143 o 198 a e he Hclass o models.
Speci…cally, he lag composi ion o each o hese models is desc ibed in he nex ables.
39
Table 14 - Model speci…ca ions in he es ic ed Solow class
Table 15 - Model speci…ca ions in he es ic ed Blancha d-Gali class
40
Table 16 - Model speci…ca ions in he es ic ed Hamil on class
41
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