Introduction to DSGE modelling. Nicola Viegi. University of Pretoria

Transcription

1 Inroducion o DSGE modelling Nicola Viegi Universi of reoria

2 Dnamic Sochasic General Equilibrium

3 Dnamic - expecaions

4 Sochasic Impulses ropagaion Flucuaions

5 General equilibrium Monear auhori Firms Households

6 Households Maximise presen discouned value of expeced uili from no unil infinie fuure, subjec o budge consrain Households characerised b uili maximisaion consumpion smoohing

7 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

8 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

9 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

10 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

11 Firms Full soluion αω x ( βe ) ( ω)( βω) Knon as he Ne Kenesian hillips curve Knon as he forard-looking hillips curve

12 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

13 Monear auhori Ses he ineres rae Simples case is simple rule Ineres rae reacs o inflaion, ih shocks i δ v

14 Baseline DSGE model Monear auhori i δ v Firms Households αω x ( β E ) ( ω)( βω) U '( C ) βe U '( C i )

15 Households To simplifing assumpions: CRRA uili funcion U ( C ) σ C σ U '( C ) C σ No capial C Y

16 Log-linearised DSGE model Monear auhori i δ v Firms Households αω x ( E ) ( ω)( βω) x ( E x σ i E )

17 Calibraion α ω β σ

18 Full DSGE model αω βω ω κ δ κ β σ ) )( ( ) ( v i x E E i E x x

19 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

20 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

21 Alernaive sae-space form E X A A X A B v A B E X AX Bv

22 ariioning of model Bv A E X backard-looking variables predeermined variables forard-looking variables conrol variables

23 Jordan decomposiion of A E A Bv A Λ eigenvecors diagonal marix of eigenvalues

24 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

25 Too man sable roos muliple soluions equilibrium pah no unique need alernaive echniques

26 Too man unsable roos no soluion all pahs are explosive ransversali condiions violaed

27 Blanchard-Kahn saisfied one soluion equilibrium pah is unique ssem has saddle pah sabili

28 Rearrangemen of Jordan form Λ Bv E Λ Bv E R

29 ariion of model Λ Λ Λ Λ Rv E 2 R R R sable unsable

30 Transformed problem ~ ~ E ~ ~ ~ ~ Λ Λ v R R E 2 2 ~ ~ ~ ~ Λ Λ v R R E

31 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

32 Soluion sraeg Solve unsable ransformed equaion Solve sable ransformed equaion Translae back ino original problem ~ ~

33 Soluion of unsable equaion Solve unsable equaion forard o ime j E ~ ~ j ( ) j Λ 2 As Λ 2 >, onl sable soluion is ~ ~ Forard-looking (conrol) variables are funcion of backard-looking (predeermined) variables

34 Soluion of sable equaion Solve sable equaion forard o ime j E ~ ~ j ( ) j Λ As Λ <, no problems ih insabili ~ ~ ( )

35 Soluion of sable equaion ~ ~ Λ Rv ~ ( ) ~ ( ) ( ( ) 22 Λ 2 ) ( R v ) Fuure backard-looking (predeermined) variables are funcion of curren backardlooking (predeermined) variables

36 Full soluion 22 2 ( ( ) 2 Λ ) ( R v ) All variables are funcion of backard-looking (predeermined) variables: recursive srucure

37 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

38 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

39 Blanchard-Khan condiions Require one sable roo and o unsable roos ariion model according o v x

40 More complex models Frisch-Slusk paradigm Impulses ropagaion Flucuaions

41 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 ε ε ε υ υ υ υ υ υ υ υ υ ρ ρ ρ ρ ρ ρ ρ ρ ρ

42 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 p p h h βγ γ

43 Simulaion possibiliies Slised facs Impulse response funcions Forecas error variance decomposiion

44 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 )

45 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

46 Brue force mehod Calculae simulaed loss for each combinaion Bes (opimal) coefficiens are hose saisfing B-K condiions and leading o smalles simulaed loss

47 Grid search δ x 2 For each poin check B-K condiions 2 δ Find loes loss amongs poins saisfing B-K condiion