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Introduction to Dynare First order approximation Michel Juillard November 18, 2009 DYNARE 1. computes the solution of deterministic models (arbitrary accuracy) 2. computes ﬁrst and second order approximation to solution of stochastic models 3. estimates (maximum likelihood or Bayesian approach) parameters of DSGE models 4. computes optimal policy. 5. performs global sensitivity analysis of a model (thanks to Marco Ratto’s tools) The general problem Deterministic, perfect foresight, case: f (yt+1 , yt , yt−1 , ut ) = 0 y : vector of endogenous variables u : vector of exogenous shocks Solution of deterministic models ◮ based on work of Laffargue, Boucekkine and myself ◮ recently much accelerated by Mihoubi ◮ approximation: impose return to equilibrium in ﬁnite time instead of asymptotically ◮ computes the trajectory of the variables numerically ◮ uses a Newton–type method ◮ usefull to study full implications of non–linearities The general problem Stochastic case: Et {f (yt+1 , yt , yt−1 , ut )} = 0 y : vector of endogenous variables u : vector of exogenous shocks E (ut ) = 0 E (ut ut′ ) = Σu ′ E (ut uτ ) = 0 t =τ Stochastic models In a a stochastic framework, the unknowns are the decision functions: yt = g(yt−1 , ut ) For a large class of DSGE models, DYNARE computes approximated decision rules and transition equations by a perturbation method. First order approximation yt ˆ = g (yt−1 , ut ) ¯ ˆ = y + Ayt−1 + But ˆ ¯ with yt = yt − y . Method proposed by Klein (2000) and Sims (2002). DYNARE computes also theoretical moments and IRFs. Second order approximation Two features: ◮ decision rules and transition functions are 2nd order polynomials ◮ departure from certainty equivalence: the variance of future shocks matters Decision rules and transition equations of the form ¯ ˆ ˆ′ ˆ ˆ′ yt = y +Ayt−1 +But +0.5 yt−1 C yt−1 + ut′ Dut + yt−1 Fut +∆ (Σu ) Method suggested by K. Judd, developped by C. Sims (2002), S. Schmitt-Grohe and M. Uribe (2003), F. Collard and M. Juillard (2000). Estimation DYNARE estimates the structural parameters of a model based on a linear approximation. Et {f (yt+1 , yt , yt−1 , ut ; θ)} = 0 Estimation steps: 1. computes the steady state 2. linearizes the model 3. solves the linearized model 4. computes the log–likelihood via the Kalman ﬁlter 5. ﬁnds the maximum of the likelihood or posterior mode 6. simulates posterior distribution with Metropolis algorithm 7. computes various statistics on the basis of the posterior distribution 8. computes smoothed values of unobserved variables 9. computes forecasts and conﬁdence intervals Optimal policy ∞ max β τ −t U (yτ , zτ , uτ ) zt τ =t s.t. Et {f (yτ +1 , yτ , yτ −1 , zτ , uτ )} = 0 where U() is the objective function of the public authority and zτ a set of policy instruments. Computation of ﬁrst order approximation ◮ Perturbation approach: recovering a Taylor expansion of the solution function from a Taylor expansion of the original model. ◮ A ﬁrst order approximation is nothing else than a standard solution thru linearization. ◮ A ﬁrst order approximation in terms of the logarithm of the variables provides standard log-linearization. General model Reparametrization for perturbation approach: Et {f (yt+1 , yt , yt−1 , ut )} = 0 E (ut ) = 0 E (ut ut′ ) = Σǫ ′ E (ut uτ ) = 0 y : vector of endogenous variables u : vector of exogenous stochastic shocks 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 (2nd order approximation requires a more complicated algorithm) Solution function yt = g(yt−1 , ut , σ) where σ is the stochastic scale of the model. If σ = 0, the model is deterministic. For σ > 0, the model is stochastic. Then, yt+1 = g(yt , ut+1 , σ) = g(g(yt−1 , ut , σ), ut+1 , σ) F (yt−1 , ut , ut+1 , σ) = f (g(g(yt−1 , ut , σ), ut+1 , σ), g(yt−1 , ut , σ), yt−1 , ut ) Et {F (yt−1 , ut , ut+1 , σ)} = 0 Stochastic scale ut+1 = σǫt+1 E (ǫt ) = 0 E (ǫt ǫ′ ) = Σǫ t E (ǫt ǫ′ ) = 0 τ then Et ut+1 ut+1 = σ 2 Σǫ ′ Steady state ¯ A deterministic steady state, y , for the model satisﬁes ¯ ¯ ¯ f (y , y , y , 0) = 0 A model can have several steady states, but only one of them will be used for approximation. Furthermore, ¯ ¯ y = g(y , 0, 0) First order approximation ¯ Taylor expansion around y with respect to yt−1 , ut and σ: Et F (1) (yt−1 , ut , σǫt+1 , σ) = ¯ ¯ ¯ ˆ Et f (y , y , y , 0) + fy+ gy (gy y + gu u + gσ σ) + gu σǫ′ + gσ σ ˆ ˆ +fy0 (gy y + gu u + gσ σ) + fy− y + fu u = 0 ˆ ¯ ∂f ∂f with y = yt−1 − y , u = ut , ǫ′ = ǫt+1 , fy+ = ∂yt+1 , fy0 = ∂yt , ∂f ∂f ∂g ∂g ∂g fy− = ∂yt−1 , fu = ∂ut , gy = ∂yt−1 , gu = ∂ut , gσ = ∂σ . Taking the expectation Et F (1) (yt−1 , ut , σǫt+1 , σ) = ¯ ¯ ¯ ˆ f (y , y , y , 0) + fy+ (gy (gy y + gu u + gσ σ) + gσ σ) ˆ ˆ +fy0 (gy y + gu u + gσ σ) + fy− y + fu u = ˆ fy+ gy gy + fy0 gy + fy− y + (fy+ gy gu + fy0 gu + fu ) u + (fy+ gy gσ + fy0 gσ ) σ = 0 Recovering gy ˆ fy+ gy gy + fy0 gy + fy− y = 0 Structural state space representation: 0 fy+ I −fy− −fy0 I ˆ gy y = ˆ y I 0 gy 0 I gy or 0 fy+ ¯ yt − y −fy− −fy0 yt−1 − y¯ = I 0 yt+1 − y¯ 0 I ¯ yt − y Structural state space representation Dxt+1 = Ext with ¯ yt − y yt−1 − y¯ xt+1 = xt = yt+1 − y¯ ¯ yt − y ◮ There are multiple solutions but we want a unique stable one. ◮ Problem when D is singular. 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. Generalized eigenvalues λi solves λi Dxi = Exi For diagonal blocks on S of dimension 1 x 1: Sii ◮ Tii = 0: λi = Tii ◮ Tii = 0, Sii > 0: λ = +∞ ◮ Tii = 0, Sii < 0: λ = −∞ ◮ Tii = 0, Sii = 0: λ ∈ C Applying the decomposition I I D ˆ gy y = E ˆ y gy gy T11 T12 Z11 Z12 I ˆ gy y 0 T22 Z21 Z22 gy S11 S12 Z11 Z12 I = ˆ y 0 S22 Z21 Z22 gy Selecting stable trajectory To exclude explosive trajectories, one imposes Z21 + Z22 gy = 0 −1 gy = −Z22 Z21 A unique stable trajectory exists if Z22 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. Recovering gu fy+ gy gu + fy0 gu + fu = 0 gu = − (fy+ gy + fy0 )−1 fu Recovering gσ fy+ gy gσ + fy0 gσ = 0 gσ = 0 Yet another manifestation of the certainty equivalence property of ﬁrst order approximation. First order approximated decision functions ¯ ˆ yt = y + gy y + gu u ¯ E {yt } = y Σy = gy Σy gy + σ 2 gu Σǫ gu ′ ′ The variance is solved for with an algorithm for Lyapunov equations. A simple example A simple monetary model yt = Et δyt−1 + (1 − δ)yt+1 + σ(rt − πt+1 ) + ey t πt = Et {απt−1 + (1 − α)πt+1 + κyt + eπt } rt = γ1 πt + γ2 yt where yt is the output gap, πt , the inﬂation rate, rt , the nominal short term interest rate. All variables are in relative distance to the steady state. simple.mod var y pie r; varexo e_y e_pie; parameters delta sigma alpha kappa gamma1 gamma2; delta = 0.44; kappa = 0.18; alpha = 0.48; sigma = -0.06; gamma1 = 1.5; gamma2 = 0.5; A simple example (continued) model(linear); y = delta*y(-1)+(1-delta)*y(+1) +sigma *(r-pie(+1))+e_y; pie = alpha*pie(-1)+(1-alpha)*pie(+1) +kappa*y+e_pie; r = gamma1*pie+gamma2*y; end; shocks; var e_y; stderr 0.63; var e_pie; stderr 0.4; end; steady; check; stoch_simul;