# Introduction to Dynare First order approximation by itlpw9937

VIEWS: 0 PAGES: 30

• pg 1
```									 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
◮   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:
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
′

¯
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
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;