computational methods in macro1

8/12/2019 Computational Methods in Macro1

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Methodsforsolvingnonlinear,

rationalexpectationsbusinesscyclemodels

A1dayworskhop fortheBathBristolExeterDoctoralTrainingCentre,byTonyYates

UniversityofBristol+CentreforMacroeconomics

21May

21

14


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1.

Introduction:

milestonesin

history,purpose,curriculumforself

study,plan

for

today


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Purpose

WorkofSamuelson,Bewley,Brock,Lucas,

Kydland,Prescott

and

others

Emphasisandlaterdominanceofmicrofounded,oftenrationalexpectationsmodelsinmacro

Appliesto

models

of

growth

as

much

as

business

cycles.

Recentcolonisationoffinance,development,

politicaleconomy.

Alsorichmicroliteraturetoo[eg Pakes...]


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Purpose(2)

Dominantparadigmhasgeneratednew

industriesin

applied

econometrics

Methodsforestimatingmodels

Searchforfactstoconfrontwithmodels

Purposeistobeginaprocessofgivingyou

accesstothesevastliteratures


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Someofthegreatdebatesemploying

computationalmethods

for

macro

Causesofbusinesscycles. Weretheyrealornot?

The

great

inflation,

great

moderation,

great

contractions. Thecostsofbusinesscycles.

Optimalmonetaryandfiscalpolicy,incinstitutionaldesign.

Causeofandoptimalresponsetochangesinconsumption

andwealth

distribution

Financeasasourceandpropagatorofbusinesscycles,andofmisallocation

Labourmarketinstitutions(eg benefits),welfareandthe

businesscycle

R&Dandgrowth; financeandgrowth


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Applicationsofrecursive,numerical

methods

IO: optimalentryexit,pricing.

Labour:

searchproblems

when

decision

is

acceptreject.

DP^2: optimalgovernmentunemploymentcompensationpolicywhenagentssolveanaccept

reject

search

problem

in

the

labour

market.

Monetarypolicyandthezerobound[Imworking

onthis

now].

Optimalredistributivetaxationwithaggregateandidiosyncraticuncertainty


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Plan=f(speed) Quasilinear,perfectforesightmethod. [DSGEatzerolowerbound]

Nonlinear,perfectforesightusingNewtonRaphson [deterministicRBC]

Parameterisedexpectations[stochasticRBC]

AprojectionmethoddeployingChebyshev polynomical functionapproximation. [DeterministicRBC]

Dynamicprogramming

with

value

and

policy

function

iteration

[DeterministicandstochasticRBC]

DynamicprogrammingusingcollocationandChebyshevPolynomials[DeterministicRBC]

HeterogeneousagentmethodsusingKrusselSmith[DeterministicRBC]

DP^2 someremarksonly.


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Thevehicle:

RBC

model

(Initially)representativeagent,consumption

savingsinvestment

output

problem,

stochastic,

neutraltechnologicalprogress.

Usefulbuildingblockformany,moremodernand

realisticapplications

in

growth,

finance,

monetaryeconomics.

Manynumericalmethodstextbooksexplainusing

this

as

an

example. Buttorepeat:methodsmuchmorewidely

applicable.


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Minimaltoolsforappliedmacro

theory

Dynamicoptimisation tosolvetheproblemsof

thefirms

and

consumers

in

your

model

Functionapproximationusingsumsofpolynomials

Numericaloptimisation

Matrixalgebra

Numericalderivatives,numericalintegrals

Markovchains

Dynamicprogramming


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Desirableforappliedmacro,minimal

fortheoretical

macro

Realanalysis studyofexistenceand

convergencetheorems

on

which

dp and

fa rely

MostminormodificationsofRBCmodelswillbe

suchthatrequiredconditionsmet.

Manymajormodificationseg heterogeneousagents leaveyouinterritorywherethereisno

possibilityofprovingexistenceoruniqueness,so

youhavetorelyonexperimentation,homotopy.


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Notcovering

today:

Peturbation methodsforsolvingRBC/DSGE

models Thoughwearedeployingsimilarideatosolve

rootfinding problemsournonlinearmodelposes

Noweasy[maybetooeasy!]todothisinDynare,

softwarewritten

for

use

in

Matlab.

Thesearelocalmethods. Adequateunless: Largeshocks

Occasionally

binding

constraints

or

other

things

inducingkinksinpolicyfunctions

Choicesetforagentsdiscontinuous


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Goodresources

Stokey andLucas(1989) regularityconditions,

convergence

theorems

for

tools

to

work. Judd(1993) excellentdiscussionofnumerical

methodsforNLDSGEmodels

Heer andMaussner (2005) greathowtobook,including

some

good

details

on

tools

Adda andCooper(2003) niceexplanationsofdynamicprogramming,simulatedmethodof

moments.

Aread

in

the

bath

tour

of

macro.

MirandaandFackler (2002) includespowerfultoolboxofcode.


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Goodresources

(2)

Wouter denHaans lecturenoteson

projectionmethods,

+many

others

DenHaan andMarcet (1990)on

parameterisedexpectationsmethodis

excellentandverytransparent.

Wouter denHaan manuscriptonPEA

Christiano andFisher

(2000)on

unity

of

PEA

andprojectionmethods.


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Software

Pencilandpaperindispensible,butinsufficient!

Matlab,Gauss,

Python

for

development,

computation,simulation

LowerlevellanguagelikeFortran,C++for

computationallyintensivetasks

Mathematica,Maplefordebuggingpenciland

paper

calculation

of

derivatives

[perhaps

integratedintoyourMatlab orotherprograms]


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Companioncurriculumonempirical

macro

Timeserieseconometrics; VARs,random

processes,

Markov

processes,

ergodicity Kalman Filter

Filtering. Dualitywithoptimallinearregulator.

Computingthelikelihoodof[thess formofa]

DSGE/RBCmodel

MarkovChainMonteCarlomethods Characterisingnumericallyposteriordensities

Particlefiltering

Forwhenyoudonthaveanalyticalexpressionsforthelikelihood


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2.

Piecewiselinear,

perfect

foresightsolutionofanonlinearRE

NK

model[TakenfromBrendon,Paustian and

Yates,Thepitfallsofspeedlimit

interestrate

rules

at

the

ZLB.]


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Warmup: piecewiselinearREE

solutionmethod

Perfectforesight,nonlinearREE

Jung,Teranishi and

Watanabe

(2005),

Eggertson andWoodford(2003)

Policy

rules

in

NK

model

with

the

zero

lower

boundtointerestrates

GuessperiodatwhichZLBbinds;solve

resultant,2part

linear

RE

system,

verify

whetherwehaveanREE


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AnalmostlinearNewKeynesianRE

model

t 11

ytEtt1

y

t

Ety

t1

1

RtEt

t1

Rt max

ty

y

ty

y

ty

t1 ,RL


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WhyZLB

is

of

interest

RatesattheZLBinJapanfor20years,andin

UK,US

for

3.5

years

Woodfordsforwardguidancepolicycopied

(?)bycentralbanksisoptimalpolicyatthe

ZLB

Fiscalmultiplier,andbenefitsoffiscalpolicy,

heavilydependent

on

ZLB

Otherapplicationsofpiecewiselineartoo.


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Calibrationof

simple

NK

model

elasticity of intertemporal substitution 1

discount rate 0.99

Calvo hazard parameter 0.67

inverse Frisch elasticity of labour supply 2

weight on inflation in policy rule 1.5

y weight on output in policy rule 0

y weight on change in output in p.r. 2

t max

ty

y

ty

y

ty

t1,RL


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Solutionalgorithm

t

y tRt

a1

a2

a3

y t1 , t 1

1.GuessZLBbindsatt=0,butnotthereafter.

2.

Conventional

RE

solvers

giveus:

3.Nowsolveforinitial

period,substitutingout

expectationsusing

2.:

0 k

y

0 a1y

0

y 0 a2

y 0

1

RL a1y 0

4. Giveninitialvalues,use2.tosolve

recursivelyfort=1onwards...

R0shadow RLRt

shadow

RL, t 1

5. Verify:


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Dynamicsunder

self

fulfilling

recession

5 10 15 20

-0.04

-0.03

-0.02

-0.01

Output

5 10 15 20

-0.04

-0.03

-0.02

-0.01

Inflation

5 10 15 20

-0.04

-0.03

-0.02

-0.01

Labour

5 10 15 20-10

-8

-6

-4

-2

x 10-3 Poli cy Rate

Self-fulfilling crisis: a simple New Keynesian model

Inflationandoutput>4%

fromss

RatesattheZLB


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Realrate

under

self

fulfilling

recession

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18x 10

-3

The real interest rate during the crisis episode

Realrate

high

in

period

2,

sustainingforecastoflow

inflationandoutput.


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Generallessons

Solvingnonlinearproblemsbyrecastingasa

setof

linear

problem

solving

steps.

Guessandverify.

NB:

this

algorithm

is

a

stepping

stone

to

solvingforoptimalpolicyatthezlb.


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Whyperfect

foresight?

Hopelesslyunrealistic?

Maybe

ok

for

some

questions,

eg transition

fromoneSStoanother,inresponseto

preannouncedpolicy.

Usefulbenchmark.

Steppingstonetolearningothermethods.

Steppingstone

to

coding

other

methods,

once

theyarelearned.


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3.Solving

aperfect

foresight,

finite

horizonversionofthegrowthmodel

using

Newton

Raphson methods


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Afinite

horizon

yeoman

farmer

model

UC0 , . . .CT t0

T

Ct

1/

, , 1

fKt Kt, 0, 1

Kt1 Ct fKt,

0 Ct,

0 Kt1 , t 0, . . .T

Example1.1.1,Heer andMaussner,p9.


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FONCsfor

the

yeoman

farmer

model

Kt1 Kt Ct, t 0,1...T

CtCt1

1

Kt11 1, t 0,1...T 1

Eulerequationdoesnot

holdfor

period

T;

no

intertemporal choicehere,

justeateverything.

Ct Kt Kt1

Kt Kt1

Kt1 Kt2

1

Kt11 1

Kt1

Kt2Kt

Kt1

1

Kt11 0

UsebudgetconstrainttosuboutforCtermsinEulerequation.


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TheTdimensionalnonlinearequation

system

K1 K2

K0 K1

1

K11 0,

K2 K3

K1 K2

1

K21 0,. .

KT

KT1 KT

1KT

1 0

Thisis asetofTnonlinearequationsinTunknowns,whichwe

aregoingtosolveusingamodifiedNewtonRaphson method,

usefulinmanymany contexts.

Thisalso,likeourwarmupexample,convertssolofnonlinear

equationintosequentialsolvingoflinearapproximations.


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Unidimensional

Newton

Raphson

,s. t.fx 0

g0 x fx 0 fx 0 xx 0

1 , s. t.g0 x 1 0

1 x 0 fx0

fx0

Thisisourultimategoal,

formalised

Weapproximateflinearlyaround

someinitialpointx_0,usingfirst2

termsof

Taylors

theorem.

Thenwesolveforthex_1that

makestheapproximantg(x_1)=0


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Theiterative,unidimension NR

method

s1 x s fxs

fxs

Ifwe

iterate

on

this

equation,

then

the

estimatesslowlyconvergeontherootsof

theoriginalsystem.

Cangetproblemthatnewguesseslie

outsidethe

domain

of

the

function,

so

we

modifybyplacingboundsonthe

allowableguesses.


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Graphicalillustrationofuni

dimensionalNewton

Raphson

Source: Heer andMaussner,Chapter8: tools


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Multidimensional

Newton

Raphson

0

0

. . .

0

f1 x 1 , . . .x n

f2 x 1 , . . .x n

. . .

fnx 1 , . . .x n

0 fx

f1 K1

K2K0

K1

1

K11 0 K3 0.K4 . . . 0.KT

f2 . . .

. . .

fnT 0 K1 0 K2 . . . KT

KT1 KT

1

KT1

Definingthematrix

function

f

via

matrix

representationofour

systemofequations

Inour

deterministic

yeomanfarmer

model,the

fs

willlooklike

this

Defining the Jacobian and populating


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DefiningtheJacobian,andpopulating

it

using

the

deterministic

yeoman

farmereg

Jx

f11 f2

1 . . . fn1

f12 f2

2 . . . fn2

. . . . . . . . . . . .

f1n f2

n . . . fnn

,wherefji

fix

dxj

Ratherthanaderivative,

wenow

work

with

aJacobian,amatrixof

partialderivatives.

f11

df1

dx 1 df1

dK1 d

dK1

K1 K2

K0 K1

1

K11

K1 K2 11K0

K1 2 1K1 K2 K1

1 1K12

f2

1

K0

K

111

K

1

K

2

. . .

fn1 0,n 3. . .T

Definingthefirstrow mostentriesintheJacobian matrixwillbezeros.


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Fromlinearapproximationtoour

system,to

iterative

approx to

the

roots

gx fx0 Jx0 dx,dx xx0

x1 x0Jx0 1 fx

0

xs1 xsJxs1 fx

s

Linearapproximationto

themultivariatesystem

Solutiontog(x)=0at

the

point

x_0

Solutiontog(x_s+1)=0

atthepointx_s; and

iterationsonthis

convergeglobally

on

thesolutionx_*tothe

nonlinearsystem,with

somemodifications.


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General,multivariate,modifiedNR

algorithm

1. Initialise: choose x0 x,x, i 0

2. (i) ComputeJxi,

2. (ii) solvefxi Jx

ix

i1x i 0

2. (iii) ifxi1 x,x,choose 0, 1, s.t. xi1 xix i1xi x,x

2. (iv) set xi1 xi1

3. check convergence:stop if fxi1

, else set i i 1and go to 2

Modificationchecksifnewguesslieswithindomainoffunction.

NotethisisHMalgorithm8.5.1,edition1.


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Remarks Manysolutionmethodsboildowntosolving

nonlinearoptimisation

problems

like

this

Manyalternativemethodsandrefinements.

Notesimilaritieswithbefore; converting

problemofsolvingnonlinearproblemintoone

offindingeverbettersolutionstothelinear

approximationsto

the

nonlinear

problem.


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Alternative

methods

and

refinements

Usingnumericalderivativesbasedon

differences,in

this

NR

method

Usingasecantmethodexplicitlybasedondifferences.

Stochasticmodificiations to

prevent

getting

stuckatalocal. [simulatedannealing]

Manyprecodedfunctionstouse,butithelps

toknow

basic

method

to

be

able

to

use

them

effectively


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PS

on

perfect

foresight

and

NR Weusedthismethodinourpaperonthezero

boundtoo.

Insteadoflinearising theNKmodel,andthenworkingwiththezeroboundusingaguess

andverify

method

Formulatefull,nonlinearNKmodel,includingZLB,solvingsysteminvolvingfinitenumberof

periods,one

equation

for

each

period,

as

you

justsawfortheRBCmodel.


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WhatabouttheinfinitehorizonRBC

model?

Samemethodcanbeusedtosolveinfinite

horizonmethod,

on

assumption

that

we

get

veryclosetosteadystateinsomefinite

numberofperiods.


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N li f f d


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Nonlinearsystemforforward

iteration

0 K1

K2K0

K1K11

0 K2

K3K1

K2K2

1

. . .

0 KT

K

KT1 KT

KT1

Nowwe

have

log

utility,

and

discounting,soslightdifferenceinRHS.

NotebeforefinalKwas0; now

K_star=steadystate

ThisisTequationsinTunknowns.

K_0isgiven.

Iterationsneeded:

if

K_T

too

far

from

ss,thenneedtolengthenT.


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4.

Solvingthe

growth

model

using

theparameterisedexpectations

algorithm

S l i RBC d l i


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SolvingRBCmodelusing

parameterisedexpectations

DenHaan andMarcet (1989)

Analogywith

learning,

connection

with

Marcets

workwithSargent onpropertiesoflearning

algorithms

Lotsof

problems

with

it,

not

that

widely

used

now

Butverysimpletoconceiveandprogram,

illustratesthe

problem

nicely,

and

an

introduction

intogeneralclassofprojectionmethods


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The

RBC

or

growth

model

c t,ktt0

max Ett0

tct

1 1

1

subject to :

ctkt1 ztkt 1kt

lnzt lnzt1 et,et N0, 1

ct ckt,zt,kt1 kkt,ztSolutionisapairoffunctionslinkingthechoicevariables(consumption

andcapitalforusetomorrow)tothestatevariables[shockz,andcapital

today]


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ct

Etct1

kt11 1

ctkt1 ztkt 1kt

SolutionhastosatisfytheEulerequation[inmorecomplexmodels,

thefull

set

of

FONCs]

and

the

resource

constraint.

Inthespecialcaseoflogutility[gamma=1]andfulldepreciation

[delta=1],thesesolutionscanbecalculatedanalyticallyandare

knowntobe:

ct 1ztkt,kt1 ztkt

Choosing the approximant:


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Choosingtheapproximant:

expectationsin

the

Euler

equation

gkt,zt ea1 ln kta 2 lnzt

ct

Etct1

kt11

1

WeapproximatetheconditionalexpectationontheRHSofthe

Eulerequationabovewithapolynomialinthestatevariables.

NB1. maybedesirabletousehigherorderpolynomial

2.maybeachoiceaboutwhichfunctiontoapproximate[as

here],ormorethanonefunctionyouhavetoapproximate.Eg may

bemore

than

one

agent

forming

expectations!


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PEA

algorithm

1. Draw a long sequence of values for shockse tto generate the exogenous technology shock process z t. (Long100,000?)

2. Choose a starting value fora1 ,a2and a starting value fork0 , perhaps the steady state.

3. Simulate the model by:

3.1 computingc0

ea 1ln k0a 2lnz0

3.2 solving fork1from the resource constraint, thus: k1 z0k0 1k0c0

3.3. push time forward one period and return to 3.1

4. Compute residuals by:4.1 for each t, lettingc t

true ct1

kt11 1,ct

app ea1ln kta2lnzt

4.1Rt cttrue c t

app

5. Updatea1 ,a2by choosing them to mint0

T

R2 [nonlinear least squares]

6. Check convergence. If converged, stop, else return to 3.


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Problems

with

PEA Multicollinearity whenyoutrytogetmore

accurateapproximationsbyincludinghigherordertermsinthepolynomial.

Eg: Thesquareoftheshockiscorrelatedwiththeshockitself.

Solutionaccurate

only

close

to

steady

state,

since

simulationslivemostlythere.

Potentialforinstabilityintheupdatingalgorithm

Analogywith

intertemporal learning

models,

and

projectionfacilitiesforsolvingtheproblem.


54/152

Resolving

PEA

convergence

problems

Homotopy: usesolutionstoproblemsyoucan

solveto

inform

starting

values

to

problems

youhaventyetsolved

Projectionfacilitiesanddampedupdating:

slowdown

updating

process

in

PEA

algorithm.

PEA step with a projection facility or


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PEAstepwithaprojectionfacility,or

dampedupdating

5. 1 : a1j1p

,a2,j1p

arg mint0

T

R2

5. 2 : a i,j1 ai,j 1ai,j1p

Tradeoff:

Tooslowupdatingrisksgettingstuckawayfromthesolution.

Too

fast

updating

risks

inducing

instability

in

the

algorithm

and

not

findingthesolution.

Example of (not very useful) homotopy


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Exampleof(notveryuseful)homotopy

algorithm

f, fObjective: solve

RBCmodel

definedby

Initialise parameters:

RBC parameters0 1,0 1

PEA parameters:

i) solve model analytically, givingc t0 zt,kt,kt10 zt,ktii) simulate model giving sequencesc0 ,k0

iii) initialise values ofa10 anda2

0 in c t

ea 1 ln kta 2 lnzt that minimise the gap between ctsimulated in ii) andc t

Counter: set i 1

Main loop:1. Set i, i i1 , i1 f,f0 ,0

5. Solve forc t1 zt,kt,kt1

1 zt,ktusing PEA, initialising (a1 ,a2 a1i1 anda2

i1 , saving on convergencea1i ,a2

i

6. If (, f,f, set i i 1and go to 1, else you are done.


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Remarks Analogybetweenintratemporalprocessof

iterativealgorithm

to

find

best

approximate

expectationsfunction

andintertemporallearningwithagents

updatingexpectations

functions

as

data

becomesavailableeachperiod.

Similarity

not

just

superficial;

maths

of

convergenceorlackofithasconnections.

Inter and intra temporal learning


59/152

Interandintratemporallearning

analogy Consumersstartwithanexpectationsfunction.

Take

decisions.

Datarealised.

Agentscomputesurprise. Newfunction

updatedas

function

of

surprise,

and

imprecision

inestimates.

Gaindampsupdating.

Undersomeconditions,convergestoREE.


60/152

5.

Solving

RBC

model

using

collocation,andChebyshev

polynomials


61/152

Overview

of

this

section Preliminaryonapproximationusing

Chebyshev polynomials SolvingRBCmodelsusingCheb Polysto

approximatetheexpectationsfunction.

Iteratingon

the

Bellman

Equation

using

Cheb

Polystoapproximatethevaluefunction.


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5.1Function

approximation

and

Chebyshev polynomials


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Intro

remarks Functionapproximationisatechniquewidely

encounteredintheory,econometrics,numerical

analysis. Weencountereditalreadyinusinglinear

approximationsinNewtonRaphson tofindroots

of

nonlinear

equation

system. Regressionisfunctionapproximation.

Fourierseriesapproximationusedtocomputethespectrum.

Here: wewillapproximatetheexpectationsfunctioninconsumersFOC.

Typicalfunctionapproximation


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yp pp

problem,and

solution

x Ca,b

fx

i1

n

i ix

0 x,1 x, . . .

x fx

i

1

i ix, ix xi

x fx 0 1x 2x

2 . . .pxp

Wewanttoapproximatesomefwhichis

continuousoverthea,b interval

Wedoitbytakingsomeweighted

combinationofbasisfunctions,eg afamilyof

polynomials

Forexample,wecanrepresentany

continuousfunctionEXACTLYwith

thisinfinite

sum

of

polynomials

Whichinpracticemeansusingthe

firstpterms.


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Chebyshev Polynomials

Tix cosiarccosx

Ti1 x 2xTixTi1 x

T0 x cos0.arccosx 1

T1 x cos1.arccosx x

T2 x 2xT1 xT0 x 2x2

1T3 x 2xT2 xT1 x 4x 3 3x

GeneralexpressionfortheC.P.oforderi.

Theycanbedefinedrecursively.

HerearethefirstfourCPs.

Graph of 1st 3 Chebyshev polynomials


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Graphof1st 3Chebyshev polynomials

Source: Heer+Maussner,ch 8,p434


67/152

Domains

of

the

CP

and

your

f(x)

Xz 2zba

a bba

,z a,b

Zx x1ba2

a,x 1, 1

CPsonlydefinedon1,1. X(z)

convertsvaluesdefinedona,b

intovalues

defined

on

1,1.

Z(x)doesthereverse.

fx : a,b

gx fZx :1, 1

z; i0

n

iTiXzThiswillbeourapproximating

function.

TheCPswilltakeasaninput

transformationsoftheoriginal

pointsinourfunction.

Digression: derivingtheEuler

equation in the deterministic RBC


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equationinthedeterministicRBC

model

maxc0,c1...

U t0

tc t1

11 , 0, 1, 0

subject tokt1 c t kt 1kt, 0, 1

0 ct

0 kt1k0given

Nonlogarithic preferences,andthepresenceofonlypartial

depreciation,will

make

an

analytical

solution

impossible.

ButasafirststepwehavetoderivetheEulerequationanyway;

thoughthatstillleavesthetaskofsolvingforc(k),thepolicyfunction.

Forminganddifferentiatingthe


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Lagrangian inthe

RBC

model

Lagrangian fortheconsumers

problem.

dL

dct 0 t1

ct

1 t

0 tct t

ct

t

Whichwedifferentiatewrt

todaysconsumption

andtomorrowscapital

L t0

t c

t

1

1

1 tkt1 c tkt 1kt


70/152

Solving for the steady state for capital


71/152

Solvingforthesteadystateforcapital

0 k1 1 c

c

1

1 k1 1/

k1 1

1

k1 11

k1

11

k

11

1

1

StartwiththeEulerequation.

Dropthe

time

subscripts,

since

in

thesteadystateallvariablesare

constant.

Thensolveforkstarintermsof

themodels

primitive

parameters.

Thisvalueisgoingtohelpus

definegoodboundsforcapital

whensolvingthedynamicRBC

model.

Bounding the state space


72/152

Boundingthestatespace

X k,k

k 1. 5k,k 0. 5k

Setboundsexperimentally,bracketingthesteadystatesolutionfor

capital.

The residual function


73/152

Theresidualfunction

minkk

R,k

2

dk

Likeallprojectionmethods,westartwiththe

objectiveofminimisingtheintegralofa

residualfunction,evaluatedacrossthestatespace,asafunctionoftheparams that

definetheapproximatingfunction.

S kk

2L

l1

LR,k

kl

2 1kl

2

Thek_tilde_ls aremappedintok_up,k_down.

Thisintegralisgoingtobeapproximatedusingasum,evaluatedatpoints

correspondingtothezerosoftheChebyshev Polynomial. Anexampleof

quadrature,

a

form

of

numerical

integration.

Remarks


74/152

Remarks

NotethereareTWOapproximationsgoingon.

Weare

approximating

the

policy

function

c(k)

usingChebyshev polynomials

And,inordertofindthebestsuch

approximation. weareapproximatingtheintegraloftheresidual

functiondefinedoverthecapitalspacewitha

sum,using

quadrature.

Computingtheresidualfunctionfor


75/152

somearbitary k_0

R,k0

c 1c 0

1 k11 1

Ideaisthatifwehavegotthegammasrightintheapproximating

function,ie ifwehaveagoodapproximationtothepolicyfunction.

thentheEulerEquationshouldbeclosetoholding. Ifitdoesthenthe

LHS=0.

Butthenweneedanalgorithmforcomputingc_hat_0,c_hat_1and

k_1.

Computing the residual function


76/152

Computingtheresidualfunction

R,k0 c 1

c 0

1k1

1 1

1.computec 0

c,k0

2.computek1 k0 1k0c 0

3.computec 1

c,k1

c,k0 j0p jTjkk0

TheEEbasedresidual

function

we

are

trying

to

compute.

Givenc_hat_0(k_0),wegetk_1

fromtheresourceconstraint.

Thenweevaluatec_hat_1(k_1).

UsingtheCPformulahere.Wherek_tilde(k_0)mapsthe

latterintothe1,1intervalover

whichtheCPisdefined.

Not quite done!


77/152

Notquitedone!

Wehavetominimisethatapproximateintegralof

the[functionofthe]residualfunction,overthe

capitalspace.

.Usingsomeminimisationroutine!

Wehave

seen

elements

of

how

to

do

this

when

weusedNRmethodstofindthezeros ofa

nonlinearequationsystem.

Canwrite

afunction

that

computes

the

sum,

then

usefminsearch,orcsminwel tominimiseit.

Remarks


78/152

e a s

Theoutputofthisprocedureisafunctionthat

specifies

what

cis,

given

some

inherited

level

ofk.

Rememberthiswasthedeterministicgrowth

model.

Ifwewantshocks,wethenhaveatwo

dimensionalstatespace,andneedtodoCPin

twodimensions.


79/152

6.Intro

to

dynamic

programming,

valueandpolicyfunctioniteration

Sargent+L: Imperialismofrecursive


80/152

methods DPveryuseful.

Detailsof

conditions

under

which

its

lessons

hold,

andnumericalmethodstooperationaliseitwork,

arehard. Avoidedhere.

Implementationin

simple

settings

easy!

Essentialifeg agentschoicesarediscontinuous,

whenLagrangian methodsbreakdown.

[accept/reject,enter

or

not,

exit

or

not,

bus

or

train]

OverviewofDynamicProgramming


81/152

section 1.Deterministic,finiteelementdynamic

programming. 2. Stochasic,finiteelementDP

3. Continuousstatemethodsusing

collocation.

Alongtheway: MarkovChainapproximations

to

continuous

random

processes. FewwordsaboutDP^2


82/152

6.1

Dynamicprogramming:

generalsetting

Generalsetting


83/152

Chooseut t0 to

maxt0

trx t,u t

s.t.x t1 gx t,ut;x 0 Rn given

Chooseasequenceofcontrol

settings(u),

such

that

adiscountedsumofreturns (r)is

maximised,subjecttolawof

motionforthestate(x)

Wecouldbethinkingofaconsumer,orafirm,[and!/]oralarge

agentlikeapolicymakersettingpolicysubjecttolawsofmotion

generatedbythesolutionstothesmallagentsproblem

[aggregated].

MusingsonhandV


84/152

ut hx t

x t1 gx t,ut

Vx 0 maxu ss0 t0

trx t,u t

Dynamicprogramming

turns

problem

into

searchforV[valuefunction]andh[policy

function]suchthatifweiterateonthese2

equations thetermonRHSofRHSis

maximised.

maxurx,u Vx, x gx,u

IfweknewV,wecouldcompute

theRHSfordifferentusand

thereforefind

h.

Ofcourse,wedont.

TheBellmanequation


85/152

Vx maxurx,u Vgx,u

Vx rx,hx Vgx,hx

V(todaysstate)andV(tomorrows)arelinkedbytheBellman

Equationabove. Remembergisthetransitionlawthat

producestomorrowsstatefromtoday.

Thepolicyfunctionhisafunctionthatsatisfiesthemax

operater above,in

which

case

if

we

substitute

in

for

u,

we

cangetridofthemaxoperator.

Cakeeatingproblem


86/152

Canthaveyourcakeandeatit. Rather,cant

eat

your

cake

and

have

it! HowquicklyshouldIeatmycake,giventhatit

rots,thatIdontwanttogohungry,but

tomorrowmay

never

come.

BellmanEquation: supposethatfrom

tomorrow,Ihaveanoptimalcakeeatingplan.

Howmuch

should

Ieat

now?

Contractionmappingyieldedby


87/152

[iterationson]

the

Bellman

equation

Vj1 maxurx,uVjx

Undersomeconditions,startingfromanyguessofV_0,evenV_0=zeros,

iteratingontheBellmanequationwillconvergetogivethevalue

function. Thisiscalledvaluefunctioniteration. Asabiproductofthe

maxon

the

RHS

it

produces

the

policy

function

h.

AswewillseewecanalsoiterateonthePolicyfunction. Policyfunction

iteration.


88/152

6.2

Deterministic,analytical

DP

inthegrowthmodel

DeterministicVFIinanRBCmodelwe

l l i ll


89/152

cansolve

analytically

max

t0

tuc

s.t. ctkt1 yt1kt

yt fkt

k0given

u lnc

1

yt Ak

Deterministic:

noteno

expectations

term,andnorandomproductivity

proess.

CobbDouglasproduction.

Logutility.

Fulldepreciationofcapitalk.

DeterministicVFIintheRBCmodel


90/152

V0 k 0,k

V1 k maxc,k lnc . 0

ck Ak

V1 k lnAk lnA lnk

WestartwithanyoldguessatV,

V_0

WeplugitintotheBellman

Equation.

AstheBEinstructs,wemaximiseit

wrt tochoice

variables

c,k

Thisgivesusanewguessatthe

valuefunctionV_1. [Nowquite

differentfromV_0.

Werepeattheprocessoverand

overuntilconvergence.

2nd iterationontheBellmanEquation


91/152

V2k maxc,k

lnc lnA lnk

maxc,k lnAk

k

lnA

lnk

1Ak k

k 0

Ak k 1

k

Ak k1 1

k Ak

1 1

Ak

1

c Ak Ak

1

c Ak

1

WehavesubstitutedinourV_1

guess.

Nowwehavetomaximisethis

expressionwrt c,k

Evaluatedatthesemaximised

values,we

have

our

next

guess

at

thevaluefunction,V_2

Asabiproduct,wehavetheguess

atthepolicyfunctions(expressions

forcand

k

in

terms

of

k

Completingthe2nd iterationonthe

B ll E ti


92/152

BellmanEquation

V2 k ln Ak

1 lnA ln

Ak

1

ln A1

lnA ln A

1 1 lnk

ThisisourV_2guessoncewehavecompletedthe

maximisation,[hence

no

max

operator

now,

since

this

is

done!].

NotethatitisquitedifferentfromV_1

Inacomputer

program,

we

would

use

the

difference

betweeneachiterationsguesstodecidewhetherwe

shouldkeepgoingornot.

A3rd iteration?!


93/152

V3 k maxc,k

lnc V2 k

maxc,k

lnAk k ln A1

lnA ln A1

1 lnk

Continuing,we

would

:

differentiatewrt k,

solve

for

c,

k

that

maximise;

pluginthemaximisedvalues,toproduceV_3. ThensubstituteintheBE

againtogetanexpressionforV_4.Andsoon.

Finalexpressionforvalueandpolicy

functions


94/152

functions

Vk 11 lnA1 1

lnA 1

lnk

ck 1Ak

kk Ak

Wewouldneedininitely manyiterationstoconverge,butuseofthealgebra

ofgeometric

series

that

emerge

can

produce

these

expressions.

ThereisalsoaguessandverifywaytosolveforV,butdoingitthisway

illustratessomeoftheaspectsofnumericaliterationsontheBE.

RemarksonVFI


95/152

Onlyhaveconvergenceundercertainconditions. [SeeStokey andLucas].

Caniterateonthepolicyfunctioninstead,undermorerestrictiveconditions.

Can

use

combination

of

the

two

to

speed

computation.

Ingeneralwonthaveanalyticalexpressions

forderivatives

we

used

at

each

step

[and

of

coursenoanalyticalexpressionsforV,c,k]


96/152

6.3

Policyfunction

iteration


97/152


98/152


99/152

6.4 Numerical,deterministic

dynamicprogramming

in

the

growth

model


100/152

Alooptocomputethefirstnew

iterationontheBellmanEquationin

Matlab or similar


101/152

Matlab orsimilar

V1 ki maxucj V0 kj1.set k k1 kl y Akl

2.evaluateucj V0 kjfor each possiblecj,kj

3.choose highest [in this casec Ak,k 0], assign this number toV1 1

4.go back to1.and repeat for next value ofk,k2 kl 1/dku kl. . .

Subsequentloopsarethesame withexceptionthatoncewehave

dispensed

with

V=0the

V

is

going

to

influence

the

search

for

the

maximising

valuesofk,c.

Andwewillkeepgoinguntilwehavemeasuredanddecidedon

convergence.

Stoppingcriterionforavaluefunction

iteration loop


102/152

iterationloop

stop ifmaxVdiff

VkVk1

Aftereachnewiteration,wewillcomparetheoldandnewVselementby

element,and

compute

the

maximum

difference,

and

stop

provided

the

absolutevalueofthisdifferenceislessthanaprespecifiedtolerancevalue.

Thisisdonebyusingawhileabs(max(vdiff))


103/152

Themaximisationstepcanbeverytimeconsuming. Computerhastoenumerateand

checkeach

candidate

Value

for

each

policy

choice.

Convergencegreatlyspeededupbygoodstarting

values.

Orevenmissingoutthemaxstep.[Accelerator].

Syntaxofloopscanbeimportant:vectorising

RemarksonnumericalDP


104/152

Alternativestoppingcriteriaiswhenthepolicy

functions[vector(inourexample)ofpointers]

doesnotchange.

Thatis,ifyouarentinterestedinVitself.

Simulatingthe

model

just

requires

the

pointers.

Theeg weworkedoutallowsustotrace

trajectoryfrominitialk,tosteadystate.


105/152

6.5

Stochasticdynamic

programminginthegrowthmodel

Stochasticdynamicprogramming


106/152

Sofarwehaveassumedtherearenoshocks.

SowecantsolveandsimulatetheclassicRBC

model,drivenbytechnologyshocks.

Todothiswehavetoenlargethestatespace

toinclude

adimension

for

the

shocks.

SoourV=V(k,A)

Finiteelementapproximationofacontinuous

randomprocessusingMarkovchains.

DigressiononMarkovchains


107/152

Someeconomiccontextsbestdescribedwith

finiteelementrandomprocesses[regimes?]

Continuousrandomprocesseswell

approximatedbyMarkovchains[Tauchen

(1986)]

Markovchains


108/152

robx t1 |x t,x t1 , . . .x tk probx t1 |x t Markovproperty

0

Z

P

3objectsneededtodefineaMarkovchain. Initial

probabilties; vectorstoringpossiblevaluesforthe

chain; matrixdefiningtransitionprobabilities

0 Pk Computingprobabilitiesatt+k

Ergodic,orstationarydistributionofa

Markov chain


109/152

k k1

P

P IP 0

p ij 0i,j

p ijk 0i,j;Pk P P. . . .P [ktimes], somek 1

Wecaniterateonthisequationto

findthestationarydistribution

..whichhappenstosolvethis

equation.

Suchadistributionexists,andisindependentofpi_0ifeitherof

thesetwoconditionsholds. Basicallyrequirescantrandomlyget

stuckinoneofthestates.

DefiningaMarkovchainfor

technology


110/152

gy

2

1

0. 5

P

0. 3 0. 3 0. 4

0. 8 0. 1 0. 1

0.2 0.75 0.05

0 0.1,0.1.0.8

Technologycantakethree values:

high,mediumandlow.

Prob ofgoingfromhightolowis0.4;

Prob ofstaying

in

medium

if

you

start

in

mediumis0.1.

Theinitialperiodprobabilitiesofhigh,

mediumandlow.


111/152

StochasticversionoftheBellman

equation


112/152

qVj1 maxurx,uEVj

x |x

Vj1 k,A maxc,k lnc EVjk,A |A

Vj1 k,Am maxc,k lnc i1

3

pmiVjk,A i

|Am

Ourgeneralcasemodifiedto

includeshocks.

OurRBCcasewithshocks.

Expandingthe

expectations

operator

in

our

caseofadiscretestateMarkovchainusedto

approximatetheshocks.

Thenewtwodimensionalvalue

functionwithrandom,3state

technology


113/152

gy

V0 k,A

V0 k kl,A A1 V0 k k

l,A A2 V0 k kl,A A3

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

V0 k ku ,A A1 V0 k k

u ,A A2 V0 k ku ,A A3

Valuefunctionisnowamatrixstoringvaluescorrespondingtoinherited

outcomesforkfromk_l tok_h,andineachofthesecases,forinherited

productivitylevels

from

A(high=1_

to

A(low=3).

FillingoneelementoftheBellman

Equationwitha3stateMarkovchain

for technology


114/152

gy

Vj1 kl,A3 maxc,k

lnc 0. 2Vjk

,A

A1|A A3 0.75Vjk,A A2|A A3

0.05Vjk,A A3|A A3

RemarksonapproximatingAR(1)for

technology with a Markov chain Before you implement your value function iteration


115/152

Beforeyouimplementyourvaluefunctioniteration,

youwillhavetoapproximatetechnologywiththeMC.

Originalmethod

due

to

Tauchen (1986).

Themorestates,themoreaccuratetheapproximation.

Inthelimit,theapproximantcanbemadeequaltothe

continuous

counterpart. But,withmoreelements,themoretimeconsuming

willbetheVFIforagivencapitalgridsize.

Or,toholdcomputingtimeconstant,youwillhavetomake

do

with

acoarser

grid

for

capital.

6.6Dynamicprogrammingsquared

P li k d i i ti l l t


116/152

Policymakerdevisingoptimalunemployment

compensation; agentssolvingaccept/reject

searchproblem.

Nestedvaluefunctions. EachagentsVFisa

functionoftheothers.

Theseproblemscanbesolved. LQexamplesof

optimalpolicyinoptimisingRBC/NKmodels.

Solved

iteratively.

Guess

VF

for

one

agent;

iterateontheotheragentsBE. Thenswap.


117/152

6.6Solvinggrowthmodelusing

Chebyshev collocationto

solve

the

Bellmanequation.

Basicstrategy

A i t k [i thi l ]


118/152

Approximateunknown[inthiscasevalue]

functionwithfinitecombinationofnknown

basisfunctions,

coefficients

on

which

to

be

determined

Requirethis

approximant

to

satisfy

the

functionalequation[inthiscasetheBellman

Equation]atnprescribedpointsofthe

domain,known

as

the

collocation

nodes.

Functionalequationsandcollocation:

remarks


119/152

Vk,a maxk uc EVk,a

Thisisafunctionalequation.Regularequationproblem givesusaknownfunction,anasksustofind

avalue

such

that

some

condition

is

met,

eg find

x,

such

that

f(x)=0

Afunctionalequationproblemasksustofindanunknownfunction

[hereV(.)]suchthatsomeconditionismet[herethecondition

stipulatedintheBellmanEquation]

Collocationhereconvertsfunctionalequationproblemintoaregular

equationproblem.

Fromthefunctionalequationtothe

collocationequation


120/152

Vk j1

n

cjjk

ApproximateVwiththeweightedsumofChebyshev Polynomials

Then

substitute

in

on

both

sides

of

the

Bellman

Equation.

Nowwehavethecollocationequation,aregularbutnonlinearequationsystem.

j1

n

cjjki maxk uk E

j1

ncjjk

, i 1. . .n

Thecollocationequation


121/152

c vc,

ij jki

v ic maxkK

uki,k Ej1

n

cjj,kik

Collocationequation

Collocationmatrix

Conditionalvaluefunction

ataparticular

capital

value

k_i

2waystosolvethecollocation

equation1


122/152

c 1vc

cs1 1vcs

cvc 0,

cs1 cs v cs1 cs vcs

Writeinfixedpointform,theniterate

Or,poseasarootfindingproblem,andupdateusingNewtonsmethod.

Herev()istheJacobian ofthevaluefunctionataparticularvaluefork

NBthisisasystem,notjustoneequation. Oneeq foreverynode.


123/152

7.

Heterogeneousagents

Whyheterogeneousagentmodels

Some RBC like models have borrowers and


124/152

SomeRBClikemodelshaveborrowersand

lenders,consumersandentrepreneurs.

Butmaynotbeenoughforsomeproblems.

Cantstudydynamicsofincomeandwealth

distribution,nor

their

impacts.

Existenceofrepresentativeagentrulesout

interestingproblemslikebehaviourviz

uninsurableidiosyncratic

risk.

Heterogeneitystepbystep

Startwithnoaggregateuncertainty,but


125/152

gg g y,uninsurableidiosyncraticrisk.

Basicmethod

is

to

takeaggregatepricesasgiven

solveagentsdecisionproblemusingstochasticDP

Simulateindividual

Cs

to

get

distribution

Computeaggregatequantities

Checktoseeifmarketclearedatassumedprices; if

not,

find

price

that

does

clear

market,

then

repeat.

Moveontoaggregateuncertainty.

Choices

Checkingmarketclearing.


126/152

g g

Solvingindividualagentsdecisionproblem

[wehave

seen

some

of

these]

FiniteelementDP?

ContinuousstateDPviacollocation?

Computingthestationarydistributionofassetholdings

Montecarlo simulation

Approximatingdistributionfunction.

Historyofmethods[tobecompleted]

Huggett (1993)Pureexchangeeconomy;


127/152

gg ( ) g y;endowments,noproduction,noaggregate

uncertainty.

Idiosyncraticendowment

risk.

Aiyagari (1994)

Krussell Smith(1998); aggregateuncertainty.Means

of

distributions

are

sufficient

statistics.

Earlyresultsonexistenceanduniquenessforsimplecases. Generallynotavailableforlater,more

realistic

models.

Interestingrecentpapers

Heathcote et al (2009); Guvenen. Surveys.


128/152

Heathcote etal(2009); Guvenen. Surveys.

MackayandReiss(2012).Countercyclicaltax

policyinstickypricehetagentmodel.

Reiter (2006)Computationofhetagentmodel

usingfunction

approximation

to

model

the

distribution.

Ahetagentmodel


129/152

maxE0 t0

tuct

uct ct

1

1, 0

w tif e

btif u

| probt1 |t

Pu.u Pue

Peu Pee

Agentsmaximisediscountedstreamof

utilityfrom

consumption

Wagesifemployed,benefitsifnot.

Employmentstatusis

exogenous,and

Markov,withknown

transitionlaw


130/152

at1 1 1rtat1w tct, e 1 1rtatbtct, u

Statecontingent

budget

constraint.

Return

on

assets

and

wages

taxed

atratetau.

Eulerequationforconsumption.

u c t

Etu

ct1 11rt1

Firmsandproduction

Competitivefirmsowned


131/152

Yt FKt,Nt KtNt

1, 0. 1

rt NtKt

1

w t 1 KtNt

byhouseholds,maximise

profits,subjecttothis

technology.

Inequilibrium,

factors

are

paid

their

marginalproducts


132/152

Objectscomprisingstaionary eqm


133/152

V,a,c,a,a ,a

fe,a,fu,a

w, r

K,N,T,B

Valueandpolicyfunctionsforagents

Timeinvariantdensityfunctionsforemploymentstatus

Constantfactor

prices

wages

and

interest

rates

Constantcapital,labourinput,taxesandbenefits

Suchthat..

Aggregatequantitiesobtainedby

summing

across

agents


134/152

K e,u

amin

af,ada

N amin

fe,ada

C e,u

amin

c,af,ada

T wNrK

B 1 Nb


135/152

Basicstylisedstepstocomputethe

stationary

distribution Computationofindividualpolicyfunctions,


136/152

givenaggregateqsandps

Thisstep

we

have

already

done,

2different

ways

Computationofdistributiongivenindividual

policy

functions.Thisisthenewstepthatyouhaventseen.

Stepsfor

computing

the

stationary

eqm ofthehetagentmodelinmore

detail


137/152

1. Compute stationary employmentN

2. Make initial guesses at Kand

3. Compute the factor prices wandr

4. Compute household decision rulesc,a,a ,a

5. ComputeFa ,stationary distribution of assets for emp and unemp

6. ComputeKandTthat solve aggregate consistency7. Computethat balances the budget

8. UpdateKandif necessary and go to 2.

1. Computingstationaryemployment


138/152

t pue1Nt1 peeNt1

Todaysemploymentisyesterdays,timesthechancethattheystayin

employment,plus

yesterdays

unemployed

times

the

chance

they

leave

unemployment.

GivenanyinitialN_0wecansimplyiterateonthisequationtoproduce

thestationaryN

WecanalsocomputestationaryNanalytically.

5. Computingstationarydistribution

Threemethods


139/152

Montecarlo simulationFunctionapproximation

Discretisationofthedensityfunction

5. Computingstationarydn using

Monte

Carlo

simulation


140/152

1. Choose sample sizeN[not to be confused withNfor employment]x 10k

2. Initialise asset holdingsa0i and employment status0

i .

3. Using the policy function already computed, computea i i,aifor each of theNagents.

4. Use random number generator to draw i for each agent.

5. Compute summary moments of distribution of asset holdings, ega,a

6. Iterate until moments converge.

Remarks

Noguaranteethatconvergentsolutionexists,

if it d th t it i i


141/152

or,ifitdoes,thatitisunique

Noguarantee

that

even

if

you

have

aunique

stationarydistribution,thatthisalgorithm,

even

if

correctly

coded,

will

find

it. Muchtrialanderrorneeded!

KrusselSmith

Nextlogicalstepistoincludeaggregateuncertainty


142/152

uncertainty.

Recallwe

had

only

idiosyncratic

uncertainty.

Thisisfornexttime.

Thatpapercontainedmethodologicalinsightthat

agents

could

make

do

with

just

mean

of

momentsofdistributions.

Contains

within

it

the

hint

that

heterogeneity

doesntmatter.


143/152

Recapping

on

all

the

different

methods

Recap:

quasilinear,

perfect

foresight

methodtosolvethezerobound

problem Guesswhenthenonlinearitystopsbinding,

and the model is linear


144/152

andthemodelislinear.

Solvethe

linear

model.

Usethatlinearsolutiontosolvebackwardsfortheinitialperiod,whenthemodelisALSO

linear,and

with

1less

equation,

and

1fewer

unknowns.

Verifythattheguessholdsbycheckingthemax

policy

rule

implies

ZLB

binding.

Recap: perfectforesight,nonlinear

Newton

Raphson method RBCexample.

Solveforsteadystate. Problem: tosolvefortrajectory


145/152

y j ygivensomeinitialvalue.

Derivethe

FOCs;

substitute

out

for

C

using

the

resourceconstraint.

Formulatesystemofnonlinearequations,oneforeach

unknown

period. SolveusingmultidimensionalNR.

Linearise aroundapoint,thenfindtherootofthelinearsystem. Usethatroot[provideditfallsinthe

domain]as

the

next

point

around

which

to

approximate.

Recap: Parameterisedexpectations

algorithm RBCexample.

Derive Euler equation Substitute in


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DeriveEulerequation. Substitutein

approximatefunction,

involving

shocks

and

states.

Drawalargetimeseriesofshocks.

Choosenew

parameters

in

forecast

function

tominimisegapbetweensimulatedforecastsandwhatconsumptionactuallyturnsouttobe

in

the

simulation.

Recap: ProjectionusingChebyshev

Polynomials Anotherexampleoffunctionapproximation.

Approximate at the Chebyshev nodes gaining


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ApproximateattheChebyshev nodes,gaining

globalaccuracy

for

agiven

computational

time.

ApproximatepolicyfunctionsusingCPs. Define

residualfunctionusingtheEulerEquation.

Minimisetheintegralofthisresidualfunction,

approximatingthatwithasumofresidualsatthe

nodes.

Recap: deterministicdynamic

programming FormulatetheBellmanequation: recursive

definition of the value function


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definitionofthevaluefunction.

Magic:

startingfrom

any

degenerate

guess,

iterateontheBellmanequationtogetthe

solution. Discretisethestate[inthiscase,justcapital]

space.

Essentialwhen

choices

are

discontinuous.

Recap: stochasticdynamic

programming

RBC: approximatetechnologyshockwith

finitestate Markov chain


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finite stateMarkovchain.

Nowexpectation

of

value

function

next

period

isprobabilityweightedsumofdifferent

choices

under

the

different

outcomes

for

the

shock.

Recap: usingChebyshev collocationto

solvetheBellmanequation

ApproximatethevaluefunctionwithCP.

Alternative to discretising the capital/shock


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Alternativetodiscretisingthecapital/shock

space.

Recap: Aiyagari

Guessrealinterestratethatclearsthecapital

market.


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market.

Atthat

real

rate,

solve

the

individuals

problemusingdynamicprogramming.

Sumup

individual

capital

demand,

and

see

whatinterestrateclearsthemarket.

Repeatuntilconvergence.

Recap: KrusselSmith


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computational methods in macro1

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