Computing first and second order approximations of DSGE models with DYNARE

Transcription

1 Computing first and second order approximations of DSGE models with DYNARE Michel Juillard CEPREMAP Computing first and second order approximations of DSGE models with DYNARE p. 1/33

2 DSGE models E t {f(y t+1,y t,y t 1,u t )} = 0 u t = σǫ t E(ǫ t ) = 0 E(ǫ t ǫ t) = Σ ǫ y : vector of endogenous variables u : vector of exogenous stochastic shocks σ : stochastic scale variable ǫ : auxiliary random variables Computing first and second order approximations of DSGE models with DYNARE p. 2/33

3 Remarks The exogenous shocks may appear only at the current period There is no deterministic exogenous variables Not all variables are necessarily present with a lead and a lag Generalization to leads and lags on more than one period Computing first and second order approximations of DSGE models with DYNARE p. 3/33

4 Solution function y t = g(y t 1,u t,σ) Then, y t+1 = g(y t, u t+1, σ) g(g(y t 1, u t, σ), u t+1, σ) F(y t 1, u t, u t+1, σ) = f(g(g(y t 1, u t, σ), u t+1, σ), g(y t 1, u t, σ), y t 1, u t ) E t {F(y t 1,u t,u t+1,σ)} = 0 Computing first and second order approximations of DSGE models with DYNARE p. 4/33

5 Steady state A deterministic steady state, ȳ, for the model satisfies f(ȳ,ȳ,ȳ, 0) = 0 A model can have several steady states, but only one of them will be used for approximation. Furthermore, ȳ = g(ȳ, 0, 0) Computing first and second order approximations of DSGE models with DYNARE p. 5/33

6 First order approximation Around ȳ: } E t {F (1) (y t 1,u t,u t+1,σ) = 0 = { ( E t f(ȳ,ȳ,ȳ, 0) + f y+ gy (g y ŷ + g u u + g σ σ) + g u u + g σ σ ) } +f y0 (g y ŷ + g u u + g σ σ) + f y ŷ + f u u with ŷ = y t 1 ȳ, u = u t, u = u t+1, f y+ = f y t+1, f y0 = f y t, f y = f y t 1, f u = f u t, g y = g y t 1, g u = g u t, g σ = g σ. Computing first and second order approximations of DSGE models with DYNARE p. 6/33

7 Taking the expectation } E t {F (1) (y t 1,u t,u t+1,σ) = f(ȳ,ȳ,ȳ, 0) + f y+ (g y (g y ŷ + g u u + g σ σ) + g σ σ) } +f y0 (g y ŷ + g u u + g σ σ) + f y ŷ + f u u = (f y+ g y g y + f y0 g y + f y )ŷ + (f y+ g y g u + f y0 g u + f u ) u +(f y+ g y g σ + f y0 g σ )σ = 0 Computing first and second order approximations of DSGE models with DYNARE p. 7/33

8 Recovering g y (f y+ g y g y + f y0 g y + f y )ŷ = 0 Structural state space representation: [ ] [ ] [ 0 f y+ I f y f y0 g y ŷ = I 0 0 I g y ] [ I g y ] ŷ or [ 0 f y+ I 0 ] [ y t ȳ y t+1 ȳ ] = [ f y0 0 I f y ] [ y t 1 ȳ y t ȳ ] Computing first and second order approximations of DSGE models with DYNARE p. 8/33

9 Structural state space representation Dx t+1 = Ex t with x t+1 = [ y t ȳ y t+1 ȳ ] x t = [ y t 1 ȳ y t ȳ ] There is an infinity of solutions but we want a unique stable one. Problem when D is singular. Computing first and second order approximations of DSGE models with DYNARE p. 9/33

10 Real generalized Schur decomposition Taking the real generalized Schur decomposition of the pencil < E,D >: D = QTZ E = QSZ with T, upper triangular, S quasi-upper triangular, Q Q = I and Z Z = I. Computing first and second order approximations of DSGE models with DYNARE p. 10/33

11 Generalized eigenvalues λ i solves λ i Dx i = Ex i For diagonal blocks on S of dimension 1 x 1: T ii 0: λ i = S ii T ii T ii = 0, S ii > 0: λ = + T ii = 0, S ii < 0: λ = T ii = 0, S ii = 0: λ C Computing first and second order approximations of DSGE models with DYNARE p. 11/33

12 Applying the decomposition [ D T 11 T 12 0 T 22 [ ] [ I g y ] g y ŷ = E Z 11 Z 12 Z 21 Z 22 = ] [ [ [ I g y I g y ] ] ŷ g y ŷ S 11 S 12 0 S 22 ] [ Z 11 Z 12 Z 21 Z 22 ] [ I g y ] ŷ Computing first and second order approximations of DSGE models with DYNARE p. 12/33

13 Selecting stable trajectory To exclude explosive trajectories, one imposes Z 21 + Z 22 g y = 0 g y = Z 1 22 Z 21 A unique stable trajectory exists if Z 22 is non-singular: there are as many roots larger than one in modulus as there are forward looking variables in the model (Blanchard and Kahn condition) and the rank condition is satisfied. Computing first and second order approximations of DSGE models with DYNARE p. 13/33

14 Recovering g u f y+ g y g u + f y0 g u + f u = 0 g u = (f y+ g y + f y0 ) 1 f u Computing first and second order approximations of DSGE models with DYNARE p. 14/33

15 Recovering g σ f y+ g y g σ + f y0 g σ = 0 g σ = 0 Yet another manifestation of the certainty equivalence property of first order approximation. Computing first and second order approximations of DSGE models with DYNARE p. 15/33

16 irst order approximated decision function y t = ȳ + g y ŷ + g u u E {y t } Σ y = ȳ = g y Σ y g y + σ 2 g u Σ ǫ g u The variance is solved for with an algorithm for Lyapunov equations. Computing first and second order approximations of DSGE models with DYNARE p. 16/33

17 econd order approximation of the model E t ÒF (2) Ó (y t 1, u t, u t+1, σ) = E t F (1) (y t 1, u t, u t+1, σ) +0.5 F y y (ŷ ŷ) + F uu (u u) + F u u (u u ) + F σσ σ 2 +F y u(ŷ u) + F y u (ŷ u ) + F y σŷσ + F uu (u u) + F uσ uσ + F u σu σ = E t F (1) (y t 1, u t, u t+1, σ) = F y y (ŷ ŷ) + F uu (u u) + F u u (σ2 Σǫ ) + F σσ σ 2 +F y u(ŷ u) + F y σŷσ + F uσ uσ Computing first and second order approximations of DSGE models with DYNARE p. 17/33

18 epresenting the second order derivatives The second order derivatives of a vector of multivariate functions is a three dimensional object. We use the following notation 2 F x x = 2 F 1 x 1 x 1 2 F 1 2 F 2 x 1 x 1 2 F 2. 2 F m x 1 x 1 x 1 x F 1 x 2 x F 1 x n x n x 1 x F 2 x 2 x F 2 x n x n F m x 1 x F m x 2 x F m x n x n Computing first and second order approximations of DSGE models with DYNARE p. 18/33

19 Composition of two functions Let y = g(s) f(y) = f(g(s)) then, 2 f s s = f y 2 g s s + 2 f y y ( g s g ) s Computing first and second order approximations of DSGE models with DYNARE p. 19/33

20 Recovering g yy F y y = f y+ (g yy (g y g y ) + g y g yy ) + f y0 g yy + B = 0 where B is a term that doesn t contain second order derivatives of g(). The equation can be rearranged: (f y+ g y + f y0 )g yy + f y+ g yy (g y g y ) = B This is a Sylvester type of equation and must be solved with an appropriate algorithm. Computing first and second order approximations of DSGE models with DYNARE p. 20/33

21 Recovering g yu F y u = f y+ (g yy (g y g u ) + g y g yu ) + f y0 g yu + B = 0 where B is a term that doesn t contain second order derivatives of g(). This is a standard linear problem: g yu = (f y+ g y + f y0 ) 1 (B + f y+ g yy (g y g u )) Computing first and second order approximations of DSGE models with DYNARE p. 21/33

22 Recovering g uu F uu = f y+ (g yy (g u g u ) + g y g uu ) + f y0 g uu + B = 0 where B is a term that doesn t contain second order derivatives of g(). This is a standard linear problem: g uu = (f y+ g y + f y0 ) 1 (B + f y+ g yy (g u g u )) Computing first and second order approximations of DSGE models with DYNARE p. 22/33

23 Recovering g yσ, g uσ F yσ F uσ = f y+ g y g yσ + f y0 g yσ = 0 = f y+ g y g uσ + f y0 g uσ = 0 as g σ = 0. Then g yσ = g uσ = 0 Computing first and second order approximations of DSGE models with DYNARE p. 23/33

24 Recovering g σσ F σσ + F u u Σ ǫ = f y+ (g σσ + g y g σσ ) + f y0 g σσ = 0 taking into account g σ = 0. This is a standard linear problem: +(f y+ y + (g u g u ) + f y+ g uu ) Σ ǫ g σσ = (f y+ (I + g y ) + f y0 ) 1 (f y+ y + (g u g u ) + f y+ g uu ) Σ ǫ Computing first and second order approximations of DSGE models with DYNARE p. 24/33

25 Second order decision functions y t = ȳ + 0.5g σσ σ 2 + g y ŷ + g u u + 0.5(g yy (ŷ ŷ) + g uu (u u)) + g yu (ŷ u) We can fix σ = 1. Second order accurate moments: Σ y = g y Σ y g y + σ 2 g u Σ ǫ g u E {y t } = (I g y ) 1 ( ȳ (g σσ + g yy Σy + g uu Σǫ )) Computing first and second order approximations of DSGE models with DYNARE p. 25/33

26 Stochastic versus deterministic SS Deterministic steady state: the point where the agents decide to stay, in the absence of shocks, and ignoring futur shocks. Stochastic steady state: the point where the agents decide to stay, in the absence of shocks, but taking into account the likelihood of futur shocks. It is possible to compute a second order approximation around the stochastic steady state. Computing first and second order approximations of DSGE models with DYNARE p. 26/33

27 Further issues Impulse response functions depend of state at time of shocks and history of future shocks. For large shocks second order approximation simulation may explode pruning algorithm (Sims) truncate normal distribution (Judd) Computing first and second order approximations of DSGE models with DYNARE p. 27/33

28 DYNARE commands Commands: check; shocks;... end; stoch_simul(options) variable list; Options: order = 1,[2] solve_algo = 0,1,[2] dr_algo = [0],1 irf = 0,...,[40],... noprint Computing first and second order approximations of DSGE models with DYNARE p. 28/33

29 Optimal Linear Regulator Consider, max {u} t=0 E 0 t=0 β t ( y tw 11 y t + 2y tw 12 u t + u tw 22 u t ) s.t. A + E t (y t+1 ) + A 0 y t + A y t 1 + Bu t + Ce t = 0 Lagrangian: L = E 1 β t 1[ y tw 11 y t + 2y tw 12 u t + u tw 22 u t t=1 +λ t (A +E t (y t+1 ) + A 0 y t + A y t 1 + Bu t + Ce t ) ] Computing first and second order approximations of DSGE models with DYNARE p. 29/33

30 First order conditions L y 1 = 2W 11 y 1 + 2W 12 u t + A 0 λ 1 + βa E 1 (λ 2 ) = 0 L y t = 2W 11 y t + 2W 12 u t + β 1 A +λ t 1 + A 0λ t + βa E t (λ t+1 ) t = 2,... = 0 L u t = 2W 12y t + 2W 22 u t + B λ t t = 1,... = 0 L λ t = A + E t (y t+1 ) + A 0 y t + A y t 1 + Bu t + Ce t = 0 One can write the first equation (for t = 1) as the second one (for t > 1) if and only if λ 0 = 0. Computing first and second order approximations of DSGE models with DYNARE p. 30/33

31 Augmented model 2W 11 y t + 2W 12 u t + β 1 A +λ t 1 + A 0λ t + βa E t (λ t+1 ) = 0 2W 12y t + 2W 22 u t + B λ t = 0 A + E t (y t+1 ) + A 0 y t + A y t 1 + Bu t + Ce t = 0 for y 0 given and λ 0 = 0. Computing first and second order approximations of DSGE models with DYNARE p. 31/33

32 Example: cgg_olr.mod var y inf r; varexo e_y e_inf; parameters delta sigma alpha kappa; delta = 0.44; kappa = 0.18; alpha = 0.48; sigma = -0.06; model(linear); y = delta * y(-1) + (1-delta) * y(+1) + sigma *(r - inf(+1)) + e_y; inf = alpha * inf(-1) + (1-alpha) * inf(+1) + kappa*y + e_inf; end; Computing first and second order approximations of DSGE models with DYNARE p. 32/33

33 Example: cgg_olr.mod (continued) shocks; var e_y; stderr 0.63; var e_inf; stderr 0.4; end; olr_inst r; optim_weights; y 1; inf 1; end; olr; Computing first and second order approximations of DSGE models with DYNARE p. 33/33