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3.1. CONTINUOUSSTATEMARKOVCHAINS
281
TheoreticalResults Analogoustothefinitecase,givenastochastickernelpandcorresponding
Markovoperatorasdefinedin(3.10),adensityy
onSiscalledstationaryforPifitisafixedpoint
oftheoperatorP
Inotherwords,
y
(y)=
Z
p(x,y)y
(x)dx,
8y2S
(3.15)
Aswiththefinitecase,ify
isstationaryforP,andthedistributionofX
0
isy
,then,inviewof
(3.11),X
t
willhavethissamedistributionforallt
Hencey
isthestochasticequivalentofasteadystate
Inthefinitecase,welearnedthatatleastonestationarydistributionexists,althoughtheremaybe
many
Whenthestatespaceisinfinite,thesituationismorecomplicated
Evenexistencecanfailveryeasily
Forexample,therandomwalkmodelhasnostationarydensity(see,e.g.,EDTC,p.210)
However,therearewell-knownconditionsunderwhichastationarydensityy
exists
Withadditionalconditions,wecanalsogetauniquestationarydensity(y2Dandy=yP =)
y=y
),andalsoglobalconvergenceinthesensethat
8y2D, yP
t
!y
as t!¥
(3.16)
Thiscombinationofexistence,uniquenessandglobalconvergenceinthesenseof(3.16)isoften
referredtoasglobalstability
Underverysimilarconditions,wegetergodicity,whichmeansthat
1
n
n
å
t=1
h(X
t
)!
Z
h(x)y
(x)dx asn!¥
(3.17)
forany(measurable)functionh:S!Rsuchthattheright-handsideisfinite
Notethattheconvergencein(3.17)doesnotdependonthedistribution(orvalue)ofX
0
Thisisactuallyveryimportantforsimulation—itmeanswecanlearnabouty
(i.e.,approximate
therighthandsideof(3.17)viathelefthandside)withoutrequiringanyspecialknowledgeabout
whattodowithX
0
Sowhataretheseconditionswerequiretogetglobalstabilityandergodicity?
Inessence,itmustbethecasethat
1. Probabilitymassdoesnotdriftofftothe“edges”ofthestatespace
2. Sufficient“mixing”obtains
Foronesuchsetofconditionsseetheorem8.2.14ofEDTC
Inaddition
• [SLP89]containsaclassic(butslightlyoutdated)treatmentofthesetopics
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.1. CONTINUOUSSTATEMARKOVCHAINS
282
• Fromthemathematicalliterature,[LM94]and[MT09]giveoutstandingindepthtreatments
• Section8.1.2ofEDTCprovidesdetailedintuition,andsection8.3givesadditionalreferences
• EDTC,section11.3.4providesaspecifictreatmentforthegrowthmodelweconsideredin
thislecture
AnExampleofStability Asstatedabove,thegrowthmodeltreatedhereisstableundermildcon-
ditionsontheprimitives
• SeeEDTC,section11.3.4formoredetails
Wecanseethisstabilityinaction—inparticular,theconvergencein(3.16)—bysimulatingthe
pathofdensitiesfromvariousinitialconditions
Hereissuchafigure
Allsequencesareconvergingtowardsthesamelimit,regardlessoftheirinitialcondition
Thedetailsregardinginitialconditionsandsoonaregiveninthisexercise,whereyouareaskedto
replicatethefigure
ComputingStationaryDensities Intheprecedingfigure,eachsequenceofdensitiesisconverg-
ingtowardstheuniquestationarydensityy
Evenfromthisfigurewecangetafairideawhaty
lookslike,andwhereitsmassislocated
However,thereisamuchmoredirectwaytoestimatethestationarydensity,anditinvolvesonly
aslightmodificationofthelookaheadestimator
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.1. CONTINUOUSSTATEMARKOVCHAINS
283
Let’ssaythatwehaveamodeloftheform(3.3)thatisstableandergodic
Letpbethecorrespondingstochastickernel,asgivenin(3.7)
Toapproximatethestationarydensityy
,wecansimplygeneratealongtimeseriesX
0
,X
1
,...,X
n
andestimatey
via
y
n
(y)=
1
n
n
å
t=1
p(X
t
,y)
(3.18)
Thisisessentiallythesameasthelookaheadestimator(3.13),exceptthatnowtheobservations
wegenerateareasingletimeseries,ratherthanacrosssection
Thejustificationfor(3.18)isthat,withprobabilityoneasn!¥,
1
n
n
å
t=1
p(X
t
,y)!
Z
p(x,y)y
(x)dx=y
(y)
wheretheconvergenceisby(3.17)andtheequalityontherightisby(3.15)
Therighthandsideisexactlywhatwewanttocompute
Ontopofthisasymptoticresult,itturnsoutthattherateofconvergenceforthelookaheadesti-
matorisverygood
Thefirstexercisehelpsillustratethispoint
Exercises
Exercise1 Considerthesimplethresholdautoregressivemodel
X
t+1
=qjX
t
j+(1 q
2
)
1/2
x
t+1
where fx
t
g
IID
 N(0,1)
(3.19)
Thisisoneofthoserarenonlinearstochasticmodelswhereananalyticalexpressionforthesta-
tionarydensityisavailable
Inparticular,providedthatjqj<1,thereisauniquestationarydensityy
givenby
y
(y)=2f(y)F
qy
(1 q2)1/2
(3.20)
HerefisthestandardnormaldensityandFisthestandardnormalcdf
Asanexercise,computethelookaheadestimateofy
,asdefinedin(3.18),andcompareitwith
y
in(3.20)toseewhethertheyareindeedcloseforlargen
Indoingso,setq=0.8andn=500
Thenextfigureshowstheresultofsuchacomputation
Theadditionaldensity(blackline)isanonparametrickerneldensityestimate,addedtothesolu-
tionforillustration
(Youcantrytoreplicateitbeforelookingatthesolutionifyouwantto)
Asyoucansee,thelookaheadestimatorisamuchtighterfitthanthekerneldensityestimator
Ifyourepeatthesimulationyouwillseethatthisisconsistentlythecase
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.1. CONTINUOUSSTATEMARKOVCHAINS
284
Exercise2 Replicatethefigureonglobalconvergenceshownabove
Thedensitiescomefromthestochasticgrowthmodeltreatedatthestartofthelecture
Beginwiththecodefoundinstationary_densities/stochasticgrowth.py
Usethesameparameters
Forthefourinitialdistributions,usethebetadistributionandshifttherandomdrawsasshown
below
psi_0 Beta(5.05.0# Initial distribution
1000
# .... more setup
for i=1:4
# .... some code
rand_draws (rand(psi_0, n) .+ 2.5i) ./ 2
Exercise3 Acommonwaytocomparedistributionsvisuallyiswithboxplots
Toillustrate,let’sgeneratethreeartificialdatasetsandcomparethemwithaboxplot
using PyPlot
500
500
randn(n) # N(0, 1)
exp(x) # Map p x x to lognormal
randn(n) 2.0 # N(2, 1)
randn(n) 4.0 # N(4, 1)
fig, ax subplots()
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.1. CONTINUOUSSTATEMARKOVCHAINS
285
ax[:boxplot]([x y z])
ax[:set_xticks]((123))
ax[:set_ylim](-214)
ax[:set_xticklabels]((L"$X$"L"$Y$"L"$Z$"), fontsize=16)
plt.show()
Thethreedatasetsare
fX
1
,...,X
n
gLN(0,1)fY
1
,...,Y
n
gN(2,1), and fZ
1
,...,Z
n
gN(4,1),
Thefigurelooksasfollows
Eachdatasetisrepresentedbyabox,wherethetopandbottomoftheboxarethethirdandfirst
quartilesofthedata,andtheredlineinthecenteristhemedian
Theboxesgivesomeindicationasto
• thelocationofprobabilitymassforeachsample
• whetherthedistributionisright-skewed(asisthelognormaldistribution),etc
Nowlet’sputtheseideastouseinasimulation
Considerthethresholdautoregressivemodelin(3.19)
WeknowthatthedistributionofX
t
willconvergeto(3.20)wheneverjqj<1
Let’sobservethisconvergencefromdifferentinitialconditionsusingboxplots
Inparticular,theexerciseistogenerateJboxplotfigures,oneforeachinitialconditionX
0
in
initial_conditions linspace(80, J)
ForeachX
0
inthisset,
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.2. THELUCASASSETPRICINGMODEL
286
1. Generatektimeseriesoflengthn,eachstartingatX
0
andobeying(3.19)
2. Createaboxplotrepresentingndistributions,wherethet-thdistributionshowsthekobser-
vationsofX
t
Useq=0.9,n=20,k=5000,J=8
Solutions
Solutionnotebook
Appendix
Here’stheproofof(3.6)
LetF
U
andF
V
bethecumulativedistributionsofUandVrespectively
BythedefinitionofV,wehaveF
V
(v)=Pfa+bUvg=PfU(v a)/bg
Inotherwords,F
V
(v)=F
U
((v a)/b)
Differentiatingwithrespecttovyields(3.6)
3.2 TheLucasAssetPricingModel
Contents
• TheLucasAssetPricingModel
– Overview
– TheLucasModel
– Exercises
– Solutions
Overview
Asstatedinanearlierlecture,anassetisaclaimonastreamofprospectivepayments
Whatisthecorrectpricetopayforsuchaclaim?
TheelegantassetpricingmodelofLucas[Luc78]attemptstoanswerthisquestioninanequilib-
riumsettingwithriskaverseagents
WhilewementionedsomeconsequencesofLucas’modelearlier,itisnowtimetoworkthrough
themodelmorecarefully,andtrytounderstandwherethefundamentalassetpricingequation
comesfrom
AsidebenefitofstudyingLucas’modelisthatitprovidesabeautifulillustrationofmodelbuild-
ingingeneralandequilibriumpricingincompetitivemodelsinparticular
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.2. THELUCASASSETPRICINGMODEL
287
TheLucasModel
Lucasstudiedapureexchangeeconomywitharepresentativeconsumer(orhousehold),where
• Pureexchangemeansthatallendowmentsareexogenous
• Representativeconsumermeansthateither
– thereisasingleconsumer(sometimesalsoreferredtoasahousehold),or
– allconsumershaveidenticalendowmentsandpreferences
Eitherway,theassumptionofarepresentativeagentmeansthatpricesadjusttoeradicatedesires
totrade
Thismakesitveryeasytocomputecompetitiveequilibriumprices
BasicSetup Let’sreviewthesetup
Assets Thereisasingle“productiveunit”thatcostlesslygeneratesasequenceofconsumption
goodsfy
t
g
¥
t=0
Anotherwaytoviewfy
t
g¥
t=0
isasaconsumptionendowmentforthiseconomy
WewillassumethatthisendowmentisMarkovian,followingtheexogenousprocess
y
t+1
=G(y
t
,x
t+1
)
Herefx
t
gisaniidshocksequencewithknowndistributionfandy
t
0
Anassetisaclaimonallorpartofthisendowmentstream
Theconsumptiongoodsfy
t
g¥
t=0
arenonstorable, soholdingassetsistheonlywaytotransfer
wealthintothefuture
Forthepurposesofintuition,it’scommontothinkoftheproductiveunitasa“tree”thatproduces
fruit
Basedonthisidea,a“Lucastree”isaclaimontheconsumptionendowment
Consumers Arepresentativeconsumerranksconsumptionstreamsfc
t
gaccordingtothetime
separableutilityfunctional
¥
å
t=0
b
t
u(c
t
)
(3.21)
Here
• b2(0,1)isafixeddiscountfactor
• uisastrictlyincreasing,strictlyconcave,continuouslydifferentiableperiodutilityfunction
isamathematicalexpectation
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.2. THELUCASASSETPRICINGMODEL
288
PricingaLucasTree Whatisanappropriatepriceforaclaimontheconsumptionendowment?
We’llpriceanexdividendclaim,meaningthat
• thesellerretainsthisperiod’sdividend
• thebuyerpaysp
t
todaytopurchaseaclaimon
– y
t+1
and
– therighttoselltheclaimtomorrowatpricep
t+1
Sincethisisacompetitivemodel,thefirststepistopindownconsumerbehavior,takingpricesas
given
Nextwe’llimposeequilibriumconstraintsandtrytobackoutprices
Intheconsumerproblem,theconsumer’scontrolvariableisthesharep
t
oftheclaimheldineach
period
Thus,theconsumerproblemistomaximize(3.21)subjectto
c
t
+p
t+1
p
t
p
t
y
t
+p
t
p
t
alongwithc
t
0and0p
t
1ateacht
Thedecisiontoholdsharep
t
isactuallymadeattime1
Butthisvalueisinheritedasastatevariableattimet,whichexplainsthechoiceofsubscript
Thedynamicprogram
Wecanwritetheconsumerproblemasadynamicprogrammingproblem
Ourfirstobservationisthatpricesdependoncurrent information, andcurrentinformationis
reallyjusttheendowmentprocessupuntilthecurrentperiod
InfacttheendowmentprocessisMarkovian,sothattheonlyrelevantinformationisthecurrent
statey2R
+
(droppingthetimesubscript)
Thisleadsustoguessanequilibriumwherepriceisafunctionpofy
Remarksonthesolutionmethod
• Sincethisisacompetitive(read:pricetaking)model,theconsumerwilltakethisfunctionp
asgiven
• Inthiswaywedetermineconsumerbehaviorgivenpandthenuseequilibriumconditions
torecoverp
• Thisisthestandardwaytosolvecompetitiveequilibrummodels
Usingtheassumptionthatpriceisagivenfunctionpofy,wewritethevaluefunctionandcon-
straintas
v(p,y)=max
c,p0
u(c)+b
Z
v(p
0
,G(y,z))f(dz)
subjectto
c+p
0
p(y)py+pp(y)
(3.22)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.2. THELUCASASSETPRICINGMODEL
289
Wecaninvokethefactthatutilityisincreasingtoclaimequalityin(3.22)andhenceeliminatethe
constraint,obtaining
v(p,y)=max
p0
u[p(y+p(y)) p
0
p(y)]+b
Z
v(p
0
,G(y,z))f(dz)
(3.23)
Thesolutiontothisdynamicprogrammingproblemisanoptimalpolicyexpressingeitherp
0
orc
asafunctionofthestate(p,y)
• Eachonedeterminestheother,sincec(p,y)=p(y+p(y)) p
0(
p,y)p(y)
Nextsteps Whatweneedtodonowisdetermineequilibriumprices
Itseemsthattoobtainthese,wewillhaveto
1. Solvethistwodimensionaldynamicprogrammingproblemfortheoptimalpolicy
2. Imposeequilibriumconstraints
3. Solveoutforthepricefunctionp(y)directly
However,asLucasshowed,thereisarelatedbutmorestraightforwardwaytodothis
Equilibriumconstraints Sincetheconsumptiongoodisnotstorable,inequilibriumwemust
havec
t
=y
t
forallt
Inaddition,sincethereisonerepresentativeconsumer(alternatively,sinceallconsumersareiden-
tical),thereshouldbenotradeinequilibrium
Inparticular,therepresentativeconsumerownsthewholetreeineveryperiod,sop
t
=1forallt
Pricesmustadjusttosatisfythesetwoconstraints
Theequilibriumpricefunction Nowobservethatthefirst orderconditionfor(3.23)canbe
writtenas
u
0
(c)p(y)=b
Z
v
0
1
(p
0
,G(y,z))f(dz)
wherev
0
1
isthederivativeofvwithrespecttoitsfirstargument
Toobtainv
0
1
wecansimplydifferentiatetherighthandsideof(3.23)withrespecttop,yielding
v
0
1
(p,y)=u
0
(c)(y+p(y))
Nextweimposetheequilibriumconstraintswhilecombiningthelasttwoequationstoget
p(y)=b
Z
u
0[
G(y,z)]
u
0(
y)
[G(y,z)+p(G(y,z))]f(dz)
(3.24)
Insequentialratherthanfunctionalnotation,wecanalsowritethisas
p
t
=
t
b
u
0
(c
t+1
)
u0(c
t
)
(c
t+1
+p
t+1
)
(3.25)
Thisisthefamousconsumption-basedassetpricingequation
Beforediscussingitfurtherwewanttosolveoutforprices
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.2. THELUCASASSETPRICINGMODEL
290
SolvingtheModel Equation(3.24)isafunctionalequationintheunknownfunctionp
Thesolutionisanequilibriumpricefunctionp
Let’slookathowtoobtainit
Settinguptheproblem
Insteadofsolvingforitdirectlywe’llfollowLucas’indirectapproach,
firstsetting
f(y):=u
0
(y)p(y)
(3.26)
sothat(3.24)becomes
f(y)=h(y)+b
Z
f[G(y,z)]f(dz)
(3.27)
Hereh(y):=b
R
u
0
[G(y,z)]G(y,z)f(dz)isafunctionthatdependsonlyontheprimitives
Equation(3.27)isafunctionalequationinf
Theplanistosolveoutforfandconvertbacktopvia(3.26)
Tosolve(3.27)we’lluseastandardmethod:convertittoafixedpointproblem
FirstweintroducetheoperatorTmappingfintoTfasdefinedby
(Tf)(y)=h(y)+b
Z
f[G(y,z)]f(dz)
(3.28)
Thereasonwedothisisthatasolutionto(3.27)nowcorrespondstoafunction f
satisfying
(Tf
)(y)=f
(y)forally
Inotherwords,asolutionisafixedpointofT
Thismeansthatwecanusefixedpointtheorytoobtainandcomputethesolution
Alittlefixedpointtheory Letcb
+
bethesetofcontinuousboundedfunctionsf:
+
!
+
Wenowshowthat
1. Thasexactlyonefixedpoint f
incb
+
2. Forany2cb
+
,thesequenceT
k
fconvergesuniformlytof
(Note: Ifyoufindthemathematicsheavygoingyoucantake1–2asgivenandskiptothenext
section)
RecalltheBanachcontractionmappingtheorem
Ittellsusthatthepreviousstatementswillbetrueifwecanfindana<1suchthat
kTf Tgkakf gk,
8f,g2cb
+
(3.29)
Herekhk:=sup
x2
+
jh(x)j
Toseethat(3.29)isvalid,pickanyf,g2cb
+
andanyy2
+
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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