INTRODUCTION TO DYNARE

Transcription

1 INTRODUCTION TO DYNARE ICE 2005 Mchel Jullard CEPREMAP, Pars Scences Economcs, Unversty Pars 8 INTRODUCTION TO DYNARE p. 1/1

2 Acknowledgments DYNARE started at CEPREMAP n DYNARE development: S. Adjeman, O. Kamenk Bult on work of: R. Boucekkne, F. Collard, J.P. Laffargue, M. Ratto, F. Schorfhede, C. Sms, R. Wouters Publc doman software: cygwn, gnumex, lapack, styxbox, asamn, asa. INTRODUCTION TO DYNARE p. 2/1

3 DYNARE 1. computes the steady state of the model 2. computes the soluton of determnstc models (arbtrary accuracy) 3. computes frst and second order approxmaton to soluton of stochastc models 4. estmates (maxmum lkelhood or Bayesan approach) parameters of DSGE models INTRODUCTION TO DYNARE p. 3/1

4 Soluton of determnstc models based on work of Laffargue, Boucekkne and myself approxmaton: mpose return to equlbrum n fnte tme nstead of asymptotcally computes the trajectory of the varables numercally uses a Newton type method usefull to study full mplcatons of non lneartes INTRODUCTION TO DYNARE p. 4/1

5 An example The effect of a change n tax rate n a model wth monopolstc competton (adapted from Harault, Langot and Porter, 2001) W t = ln c t + η ln(1 h t ) + βw t+1 c t + t = Āk α t 1 h1 α t t = k t (1 δ)k t = βe t (z t δ) c t c t+1 η 1 h t = w t α 1 c t α (1 α) kt 1 h t kt 1 = (1 + µ)(1 + τ t )z t h t α = (1 + µ)(1 + τ t )w t INTRODUCTION TO DYNARE p. 5/1

6 DYNARE mplementaton Preambule var Welf w c h k z; varexo tau; parameters beta delta alpha mu eta rho Abar; delta = 0.025; eta = 2; mu = 0.1; alpha = 0.36; rho = 0.95; beta = 0.988; Abar = 1; INTRODUCTION TO DYNARE p. 6/1

7 DYNARE mplementaton Model model; Welf = log(c)+eta*log(1-h)+beta*welf(+1); c+ = Abar*k(-1)ˆalpha*hˆ(1-alpha); = k - (1-delta)*k(-1); 1/c = beta*(1/c(+1))*(z(+1)+1-delta); eta/(1-h) = w/c; alpha*(k(-1)/h)ˆ(alpha-1) = (1+mu)*(1+tau)*z; (1-alpha)*(k(-1)/h)ˆalpha = (1+mu)*(1+tau)*w; end; INTRODUCTION TO DYNARE p. 7/1

8 DYNARE mplementaton Intalzaton ntval; Welf = -100; w = 0.5; c = 0.6; h = 0.3; = 0.4; k = 3; z = 0.1; tau = 0; end; steady; INTRODUCTION TO DYNARE p. 8/1

9 DYNARE mplementaton Intalzaton endval; Welf = -100; w = 0.5; c = 0.6; h = 0.3; = 0.4; k = 3; z = 0.1; tau = -mu/(1+mu); end; steady; INTRODUCTION TO DYNARE p. 9/1

10 DYNARE mplementaton Computaton smul(perods=300); dsample 0 50; rplot Welf; rplot k; rplot c; rplot h; INTRODUCTION TO DYNARE p. 10/1

11 Stochastc models: Frst order approxmato In a a stochastc framework, the unknowns are the decson functons. For a large class of DSGE models, DYNARE computes approxmated decson rules and transton equatons of the form y t = ȳ + Aŷ t 1 + Bu t wth ŷ t = y t ȳ. Method proposed by Klen (2000) and Sms (2002). DYNARE computes also theoretcal moments and IRFs. INTRODUCTION TO DYNARE p. 11/1

12 Example In the prevous example, one ntroduces stochastc productvty accordng to ln A t = (1 ρ) ln Ā + ρ ln A t 1 + e t and one consders the case of no tax. New nstructons: shocks; var e; stderr 0.072; end; stoch_smul(order=1) Welf h c w z; INTRODUCTION TO DYNARE p. 12/1

13 Second order approxmaton Two features: decson rules and transton functons are 2nd order polynomals departure from certanty equvalence: the varance of future shocks matters Decson rules and transton equatons of the form y t = ȳ+aŷ t 1 +Bu t +0.5 ( ŷ t 1Cŷ t 1 + u tdu t ) +ŷ t 1 Fu t + (Σ u ) Method suggested by K. Judd, developped by C. Sms (2002), S. Schmtt-Grohe and M. Urbe (2003), F. Collard and M. Jullard (2000). INTRODUCTION TO DYNARE p. 13/1

14 p. 1/1 A k-order Perturbaton Approach to Solve Complete Market RBC Models July 2004 prepared for the conference Computatonal Methods for Dynamc Stochastc Economc Models, SITE Mchel Jullard CEPREMAP and Unversty Pars 8

15 p. 2/1 Solvng DSGE models Let s consder the followng model: E t (f(y t 1,y t,y t+1,u t )) = 0 wth u t = σε t, E(ε t ) = 0, E The soluton takes the form: ( ) [ε t ] β 1...[ε t ] β k = [Σ] β 1...β k y t = g(y t 1,u t,σ)

16 p. 3/1 The perturbaton method Computes a Taylor expanson for g() from the coeffcents of the Taylor expanson of f(). The Taylor expanson s generaly computed around the determnstc equlbrum of the model: f(ȳ,ȳ,ȳ, 0, 0) = 0

17 p. 4/1 The state varables The state varables are y t 1, and u t. Then, y t+1 = g(y t,u t+1,σ) = g(g(y t 1,u t,σ),u t+1,σ) and ( F(y t 1,u t,σ,u t+1 ) = f y t 1,g(y t 1,u t,σ), ) g(g(y t 1,u t,σ),u t+1,σ),u t ) F(y t 1,u t,σ) = E t (F(y t 1,u t,σ,u t+1 ) = 0

18 p. 5/1 The frst order approxmaton Frst order Taylor expanson of the structural model: F (1) (y t 1,u t,σ) = E t {f(ȳ,ȳ,ȳ, 0, 0) + f y ŷ +f y (g y ŷ + g u u + g σ σ) +f y+ g y (g y ŷ + g u u + g σ σ) } + f y+ g u u + f y+ g σ σ + f u u = 0 where ŷ = y t 1 ȳ, u = u t, u = u t+1. The partal dervatves are taken at the determnstc equlbrum and aren t stochastc.

19 p. 6/1 for ths to hold... (f y + f y g y + f y+ g y g y ) ŷ = 0 (f y g u + f y+ g y g u + f u )u = 0 (f y g σ + f y+ g y g σ + f y+ g σ )σ = 0

20 p. 7/1 k order approxmaton Let s wrte s = [y t 1,u t ] s = [ȳ, 0] ŝ = s s u = u t u = u t+1

21 p. 8/1 Tensor notaton j F s α1... s αj = [ F s j ]α 1...α j and n α 1... n α j j F ŝ α1...ŝ αj = [ F ] s s α1... s j αj α 1...α j [ŝ] α 1...[ŝ] α j

22 p. 9/1 Taylor expanson of the model F (p) (s, σ, u ) = F ( s, 0, 0) + p j=1 1 j! F s j [ŝ] α 1... [ŝ] α j α 1...α j j 1 k=1 j 1 k=1 j 1 k=1 j 2 F s k σ j k [ŝ] α 1... [ŝ] α k σ j k α 1...α k F s ku j k [ŝ] α 1... [ŝ] α k β u 1 β... u j k α 1...α k β 1...β j k F u kσ u β 1 β j k... u k σ j k β 1...β k j 1 k=1 m=k+1 F s ku m kσj m [ŝ] α 1... [ŝ] α k β u 1 β... u m k σ j m α 1...α k β 1...β m k + F σ σ j j + F u u β 1 β1...βj j... u β j

23 p. 10/1 Remndng E ( ) u = σε [u] β 1...[u] β k = σ k [Σ] β 1...β k

24 Takng the expectaton E F (p) (s, σ) = F ( s, 0, 0) + p j=1 1 j! F s j [ŝ] α 1... [ŝ] α j α 1...α j j 1 k=1 j 1 k=1 j 1 k=1 j 2 F s k σ j k [ŝ] α 1... [ŝ] α k σ j k α 1...α k F s ku j k [ŝ] α 1... [ŝ] α k [Σ] β 1...β j k σ j k α 1...α k β 1...β j k F u kσ j k [Σ] β 1...β k σ j β 1...β k j 1 k=1 m=k+1 + F σ j σ j + F s ku m kσj m [ŝ] α 1... [ŝ] α k [Σ] β 1...β m k σ j k α 1...α k β 1...β m F u j β1...βj [Σ] β 1...β j σ j = 0. p. 11/1

25 p. 12/1 Constrants on the partal dervatves (1) (2) (3) [ F s ]α j 1...α j = 0 [ F s k σ ]α j k 1 + [ ] F...α k s k u j k [Σ] β 1...β j k α 1...α k β 1...β j k + j 1 [ ] m=k+1 F [Σ] β 1...β m k = 0 s k u m k σ j m α 1...α k β 1...β [ ] [ ] m F σ + F j u j [Σ] β 1...β j β 1...β j + j 1 [ ] k=1 F [Σ] β 1...β k = 0 u k σ j k β 1...β k j = 1,...,p k = 1,...,j 1

26 p. 13/1 F as a composton of functons Let s defne z as y t 1 y t z = y t+1 = z(y,u,σ,u ) = u t y g(y,u,σ) g(g(y,u,σ),u,σ) u and F(y,u,σ,u ) = f ( z(y,u,σ,u ) )

27 p. 14/1 kth order dervatves of a composton Faa D Bruno formula: f y = f(z(s)), then [ f s j] α 1...α j = j l=1 [ f z l]β 1...β l c M l,j l m=1 [z s N(cm)] β m αc m where M l,j s the set of all parttons of the set of j ndces wth l classes and N(c m ) s the cardnalty of class c m. Note that M 1,j = {1,...,j} and M j,j = {{1}, {2},..., {j}}. Good news: The hghest order dervatves appear only once and are multpled by frst order dervatves.

28 p. 15/1 Recoverng g y j Back nto matrx notaton. The partal dervatves unfold along the columns. F y j = f y+ (g y j ) j g y + g y g y j + f y g y j + D = 0 k=1 where D s a term dependng on partal dervatves of g() of order lower than j and therefor already computed. Ths requres solvng the generalzed Sylvester equaton (f y+ g y + f y )g y j + f y+ g y j usng Kamenk s algorthm. j k=1 g y = D,

29 p. 16/1 Recoverng other terms n g s j F s j = f y+ (g y j j g s + g y g s j ) + f y g s j + D = 0 k=1 where D s a term dependng on partal dervatves of g() of order lower than j and therefor already computed. Ths requres solvng the lnear system (f y+ g y + f y )g s j = D f y+ g y j j k=1 g s,

30 p. 17/1 Recoverng g y k σ j k Must be solved n decreasng order of k. F yk σ j k = f y+ ( g yk σ j k ) k g y + g y g yk σ + f j y g y j + D + E = 0 l=1 where D s a term not dependng on g yk σ j k, but on g y r σ j r, for r > k and = [ F ] s k u j k [E] α 1...α k + j 1 m=k+1 α 1...α k β 1...β j k [Σ] β 1...β j k [ F s k u m k σ j m ] α 1...α k β 1...β m [Σ] β 1...β m k

31 p. 18/1 Recoverng g y k σ j k (contnued) Ths requres solvng the generalzed Sylvester equaton (f y+ g y + f y )g yk σ j k + f y+g yk σ j k k l=1 g y = D E.

32 p. 19/1 Recoverng g σ j F σ j = f y+ g y g σ j + f y+ g σ j + f y g σ j + D + E = 0 where D s a term not dependng on g σ j and [E] β 1...β j = [ F u j ] β 1...β j [Σ] β 1...β j + j 1 k=1 Ths requres solvng the lnear system (f y+ g y + f y+ + f y )g σ j = D E [ F u k σ j k ] β 1...β k [Σ] β 1...β k