Inroducion o DSGE modelling Nicola Viegi Universi of reoria
Dnamic Sochasic General Equilibrium
Dnamic - expecaions
Sochasic Impulses ropagaion Flucuaions
General equilibrium Monear auhori Firms Households
Households Maximise presen discouned value of expeced uili from no unil infinie fuure, subjec o budge consrain Households characerised b uili maximisaion consumpion smoohing
Households ) '( ) '( i C U E C U β General soluion for sochasic -horizon case Knon as he dnamic IS curve Knon as he Euler equaion for consumpion
Households - inuiion U '( C ) βe U '( C i ) i U (C ) C Higher ineres raes reduce consumpion E U (C ) C Higher expeced fuure inflaion increases consumpion
Firms Maximise presen discouned value of expeced profi from no unil infinie fuure, subjec o demand curve, nominal price rigidi and labour suppl curve. Firms characerised b profi maximisaion subjec o nominal price rigidi
Nominal price rigidi Calvo model of price rigidi roporion of firms able o change heir price in a period roporion of firms unable o change heir price in a period ω ω
Firms Full soluion αω x ( βe ) ( ω)( βω) Knon as he Ne Kenesian hillips curve Knon as he forard-looking hillips curve
Firms - inuiion αω x ( βe ) ( ω)( βω) ( -βe ) < x < Inflaion expeced o rise in fuure, firms se high prices no, choking suppl E p i x Higher expeced fuure inflaion chokes suppl
Monear auhori Ses he ineres rae Simples case is simple rule Ineres rae reacs o inflaion, ih shocks i δ v
Baseline DSGE model Monear auhori i δ v Firms Households αω x ( β E ) ( ω)( βω) U '( C ) βe U '( C i )
Households To simplifing assumpions: CRRA uili funcion U ( C ) σ C σ U '( C ) C σ No capial C Y
Log-linearised DSGE model Monear auhori i δ v Firms Households αω x ( E ) ( ω)( βω) x ( E x σ i E )
Calibraion α ω β σ.5.75.99 2
Full DSGE model αω βω ω κ δ κ β σ ) )( ( ) ( v i x E E i E x x
Alernaive represenaion x E v x E E x κ β σ δ σ σ v x E E x σ κ δ σ β σ
Sae-space form Generalised sae-space form A E X A X B v Models of his form (generalised linear raional expecaions models) can be solved relaivel easil b compuer
Alernaive sae-space form E X A A X A B v A B E X AX Bv
ariioning of model Bv A E X backard-looking variables predeermined variables forard-looking variables conrol variables
Jordan decomposiion of A E A Bv A Λ eigenvecors diagonal marix of eigenvalues
Blanchard-Kahn condiion The soluion of he raional expecaions model is unique if he number of unsable eigenvecors of he ssem is exacl equal o he number of forard-looking variables. i.e., number of eigenvalues in Λ greaer han in magniude mus be equal o number of forard-looking variables
Too man sable roos muliple soluions equilibrium pah no unique need alernaive echniques
Too man unsable roos no soluion all pahs are explosive ransversali condiions violaed
Blanchard-Kahn saisfied one soluion equilibrium pah is unique ssem has saddle pah sabili
Rearrangemen of Jordan form Λ Bv E Λ Bv E R
ariion of model Λ Λ Λ 2 22 2 2 Λ Rv E 2 R R R sable unsable
Transformed problem ~ ~ E ~ ~ ~ ~ 22 2 2 2 22 2 2 2 22 2 2 Λ Λ v R R E 2 2 ~ ~ ~ ~ Λ Λ v R R E
Decoupled equaions 2 2 ~ ~ ~ ~ Λ Λ v R R E ~ ~ Λ v R 2 2 ~ ~ Λ v R E Decoupled equaions can be solved separael sable unsable
Soluion sraeg Solve unsable ransformed equaion Solve sable ransformed equaion Translae back ino original problem ~ ~
Soluion of unsable equaion Solve unsable equaion forard o ime j E ~ ~ j ( ) j Λ 2 As Λ 2 >, onl sable soluion is ~ ~ 2 22 22 2 Forard-looking (conrol) variables are funcion of backard-looking (predeermined) variables
Soluion of sable equaion Solve sable equaion forard o ime j E ~ ~ j ( ) j Λ As Λ <, no problems ih insabili ~ 22 2 2 ~ ( 2 22 2 )
Soluion of sable equaion ~ ~ Λ Rv ~ ( 2 22 2 ) ~ ( 2 22 2 ) ( ( 2 22 2 2 ) 22 Λ 2 ) ( R v 2 22 2 ) Fuure backard-looking (predeermined) variables are funcion of curren backardlooking (predeermined) variables
Full soluion 22 2 ( ( 2 22 2 2 22 ) 2 Λ ) ( R v 2 22 2 ) All variables are funcion of backard-looking (predeermined) variables: recursive srucure
Baseline DSGE model v x E E x σ κ δ σ β σ v v ε ρ Sae space form To make model more ineresing, assume polic shocks v follo an AR() process
Ne sae-space form v x v E E x v ε κ δ σ σ ρ β σ x, One backard-looking variable To forard-looking variables
Blanchard-Khan condiions Require one sable roo and o unsable roos ariion model according o v x
More complex models Frisch-Slusk paradigm Impulses ropagaion Flucuaions
Shocks ma be correlaed Impulses Can add exra shocks o he model v i u x E g E i E x x δ κ β σ ) ( g u v g u v g u v 33 32 3 23 22 2 3 2 33 32 3 23 22 2 3 2 ε ε ε υ υ υ υ υ υ υ υ υ ρ ρ ρ ρ ρ ρ ρ ρ ρ
ropagaion Add lags o mach dnamics of daa (Del Negro-Schorfeide, Smes-Wouers) x v x i δ δ Talor rule p p p x E E i E x h x h h x ) ( κ βγ β βγ γ σ.29.35 p p h h βγ γ
Simulaion possibiliies Slised facs Impulse response funcions Forecas error variance decomposiion
Opimised Talor rule Wha are bes values for parameers in Talor rule? i δ δ x v x Inroduce an (ad hoc) objecive funcion for polic min i i 2 2 β ( x i λ x λ i 2 )
Brue force approach Tr all possible combinaions of Talor rule parameers Check heher Blanchard-Kahn condiions are saisfied for each combinaion For each combinaion saisfing B-K condiion, simulae and calculae variances
Brue force mehod Calculae simulaed loss for each combinaion Bes (opimal) coefficiens are hose saisfing B-K condiions and leading o smalles simulaed loss
Grid search δ x 2 For each poin check B-K condiions 2 δ Find loes loss amongs poins saisfing B-K condiion