Session: Using DSGE models with Dynare
Aurélien Poissonnier1
1 Insee-European Commission
Applied Macroeconometrics - ENSAE
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Dynare
”Dynare is a software platform for handling a wide class of
economic models, in particular dynamic stochastic general
equilibrium (DSGE)”.
Developed by a team of researcher and hosted at Cepremap.
Runs on Matlab, Octave, C++ and currently being developed for
Julia.
Has become a must in macro-economists’ toolbox.
Stéphane Adjemian et al. (Apr. 2011). Dynare: Reference Manual
Version 4. Dynare Working Papers 1. CEPREMAP
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Main blocks
How to declare a standard DSGE model in Dynare.
Remember from Lecture 4 and (Villemot, 2011), the general
formula for a DSGE model:
Et f (yt+1 , yt , yt−1 , vt ) = 0 (1)
Quite logically then, you should declare:
the endogenous variables (y )
the exogenous variables (v )
the model (f), i.e.
−→ the parameters
−→ the equations
Et is implicit in Dynare and solved under rational expectations
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Main blocks
Therefore a code includes:
var list of vars ;
varexo list of shocks ;
parameters list of parameters ; (possibly followed by
calibrations beta =0.99; )
model; list of equations end ;
More options
Dynare allows for a lot of useful options. Variables and parameters
can be assigned long names or Latex names. So can equations.
You can use calibrated parameters to calibrate new ones (e.g.
beta = 0.99; gamma = beta^ 2;) The model can be declared
linear (to avoid the computation of its linearization)...
Check the reference manual for more (Adjemian et al., 2011).
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Timing convention
When writing the equations of a model be aware that:
shocks economic decisions
exot endot
t −1 t t +1
This timing convention implies in particular that stock variables are
stocks at the end of the period.
e.g. for capital, it implies that output (Yt ) depends on capital at
t − 1, a part of which will be used as investment (It ) to increase
capital (Kt ).
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Steady state
steady;
This command computes the steady state of your model, i.e
returns the value of ȳ such that
f (ȳ , ȳ , ȳ , 0) = 0 (2)
Test
In lecture 4 we exemplified this command on a small
neo-Keynesian model.
Let’s try on a small RBC model (Collard, 2001). (example2.mod)
The solution is stored in oo .steady state.
Specifying the initial value to search for the steady state of a non
linear model may help the algorithm ( initval; ...). NB: this
command finds different uses in different contexts.
Check the reference manual (Adjemian et al., 2011, chap 4.10).
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Checking the rank condition
check;
This command computes the eigenvalues of the (linearized)
system, which is key to solve the model under rational expectations
(see Lecture 4).
The solution is stored in oo .dr.eigval.
Dynare automatically checks for the rank condition.
Test
In lecture 4 we exemplified this command on a small
neo-Keynesian model as well.
Let’s also try on a small RBC model (Collard, 2001).
(example2.mod)
Check the reference manual (Adjemian et al., 2011, chap 4.11).
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Simulated and theoretical moments
To compute simulated or theoretical moments and other statistics
linked to the stochastic part of the model, we need to specify the
exogenous variables (v ). This is done with a ”shocks” block.
shocks;
var an exovar name = a value for the variance ;
...
end;
One can also specify correlations between exogenous variables. By
assumption, exogenous variables follow normal distributions with
zero mean. Check the reference manual (Adjemian et al., 2011,
chap.4.8).
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Simulated and theoretical moments
stoch simul(irf=0);
This command computes many interesting results, among which
the moments of the endogenous variables.
stoch simul(irf=0, periods=0); computes the theoretical
moments
stoch simul(irf=0, periods=1000); computes the
empirical moments on 900 simulated periods (see the option drop
for why 900 and not 1000)
Test
smallneoK.mod or example2.mod
Many options are available, such as a first, second or third order
approximation of the model. Check the reference manual
(Adjemian et al., 2011, chap.4.13).
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Variance decomposition
stoch simul(irf=0);
stoch simul(irf=0); also returns the variance decomposition
(see also Lecture 3 on SVAR).
Closely related is the conditional variance decomposition (the
variance of the forecast error k steps ahead). It can be returned
using stoch simul(irf=0,
conditional variance decomposition = [1:3]); for 1 to 3
steps ahead forecast errors.
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Policy function
stoch simul(irf=0);
stoch simul(irf=0); also returns the policy function (or
decision rule).
Remember from Lecture 4 and (Villemot, 2011), this is solving the
model for unobserved expectations using the rational expectation
assumption:
−
yt = g (yt−1 , vt ) (3)
This takes the form of a linear model (matrices) for a first order
approximation of the model. Check the reference manual
(Adjemian et al., 2011, chap.4.13.3, 4 & 5) to see under what
form Dynare returns the solution to a second and third order
approximation.
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Stochastic simulations
Deterministic simulations
Estimation of models
Optimal policy
Wrapping-up
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Plan
Simulating models
Stochastic simulations
Deterministic simulations
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IRF
stoch simul;
As you probably guessed stoch simul; (withouth the option
irf=0) returns the impulse response functions of the endogenous
variables to shocks on the exogenous variables.
The outputs are stored in oo .irfs.endo exo.
Graphs are printed (and saved).
You can adjust the list of (endogenous and exogenous) variables to
be plotted.
Once again, check the reference manual (Adjemian et al., 2011,
chap.4.13.1).
Test
Let’s try on example2.mod and NK baseline.mod for a change.
(Note the existence of a file NK baseline steadystate.m)
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Plan
Simulating models
Stochastic simulations
Deterministic simulations
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Deterministic simulations
The above exercises are related to solving a DSGE model under
rational expectations and putting it under the form of a
constrained structural VAR. As anticipated in Lecture 4, this allows
to perform exercises similar to those with VAR (e.g. IRF, variance
decomposition).
Instead, one could work with
E@t f (yt+1 , yt , yt−1 , vt ) = 0
@ (4)
that is assuming perfect foresight of the sequence of v (and not
rational expectations).
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Deterministic simulations
The above exercises are related to solving a DSGE model under
rational expectations and putting it under the form of a
constrained structural VAR. As anticipated in Lecture 4, this allows
to perform exercises similar to those with VAR (e.g. IRF, variance
decomposition).
Instead, one could work with
E@t f (yt+1 , yt , yt−1 , vt ) = 0
@ (4)
that is assuming perfect foresight of the sequence of v (and not
rational expectations).
Before that, which of the above exercises is not just a rational
expectation exercise?
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Declaring shocks differently
A completely different perspective: you don’t calibrate standard
deviations but values of the shocks v in specific periods.
Dynare stacks the equations f (yt+1 , yt , yt−1 , vt ) = 0 at the
different periods and solve for y (non linear problem).
You should give a starting and ending value for y .
initval ;, endval ;, steady ;, histval ;
resid ; may also be useful.
Check the reference manual (Adjemian et al., 2011, chap.4.8) to
be sure about what you ask Dynare to do as these commands can
have different behaviors in different contexts!
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Declaring shocks differently
You declare shocks such as
shock ;
var x ;
periods 1:3 ;
values 1 ;
end ;
Check the reference manual (Adjemian et al., 2011, chap.4.7) to
be sure you understand the difference between shock declaration in
the stochastic and deterministic context.
You run the simulation
simul; in a condensed way
or perfect foresight setup; & perfect foresight solver;
Results are stored in oo .endo simul & oo .exo simul
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Deterministic simulations
Test
Let’s try on ramst.mod.
Exercise
On NK baseline.mod, code a simulation for an economy where
the central banker would change the inflation target (e.g. to move
away from the risk of ZLB).
NB: You may want to change a parameter into an exovar (e.g. to
simulate a transition between two steady states linked to two
calibrations). There is a special command for this (change type)
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison 24 / 43
Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Observed variables
varobs;
To estimate a model you need to declare which endogenous
variables are observed with varobs a list of endovars ;.
Thanks to the Kalman filter, you can estimate the model without
observing all the data.
You cannot have more observed variables than exogenous variables
(stochastic singularity).
You point Dynare to the observed data with
estimation(datafile=...);
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Estimation
estimation(...);
This is a central command of Dynare.
It allows in particular for MLE and Bayesian estimations of models.
We will see two examples fd2000.mod and (Smets and Wouters,
2003) (see also Lecture 6).
Check the reference manual (Adjemian et al., 2011, chap.4.14). It
includes 28 pages of options.
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Prior distribution
estimated params;
For a Bayesian estimation you can declare your priors about
estimated parameters.
You can specify them choosing from various distributions.
If you don’t specify any prior, you must calibrate the non estimated
parameters.
Test
Let’s have a look in fd2000.mod.
For a maximum likelihood estimation, you can force the estimated
parameters within chosen intervals with
astimated params bounds ;
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Posterior distribution
Metropolis-Hastings
On top of the estimated mode (and an approx of the distribution
around it), you can estimate the posterior distribution with a
specific algorithm (Monte Carlo Markov Chain -
Metropolis-Hastings, see Lecture 6).
The estimation command has a series of specific options
mh replic, mh nblocks, mh jscale, once again check the
reference manual (Adjemian et al., 2011, chap.4.14)
Test
Let’s estimate the posterior distribution of the parameters on
simulated data with fd2000.mod.
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Posterior distribution
Metropolis-Hastings
On top of the estimated mode (and an approx of the distribution
around it), you can estimate the posterior distribution with a
specific algorithm (Monte Carlo Markov Chain -
Metropolis-Hastings, see Lecture 6).
The estimation command has a series of specific options
mh replic, mh nblocks, mh jscale, once again check the
reference manual (Adjemian et al., 2011, chap.4.14)
Test
Let’s estimate the posterior distribution of the parameters on
simulated data with fd2000.mod.
Discuss the identification of the parameters.
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Shock decomposition
Test
Let’s run fd2000.mod with a shock decomposition command.
Check the reference manual (Adjemian et al., 2011, chap.4.16) for
more options.
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Smets and Wouters, 2007
Test
Let’s run the Smets and Wouters model estimation for the US.
Commented in the author’s code (updated by Johannes Pfeifer)
are many other commands to replicate their results.
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Plan
Estimation of models
Observed variables
Estimation
Prior distribution
Posterior distribution
Shock decomposition
Replication code for (Smets and Wouters, 2007)
Model Comparison
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Model Comparison
model comparison
In a Bayesian setup, a comparison of 2 models on the same dataset
is done through the posterior odds ratio (Lecture 6).
Dynare allows you to do that with the command
model comparison (Adjemian et al., 2011, chap.4.15).
Exercise
Using the Smets and Wouter model, compare the baseline model
to the same model estimated without habit formation.
Check the reference manual (Adjemian et al., 2011, chap.4.16) for
more options.
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Optimal simple rule
osr;
You can search for the optimal values of (policy) parameters such
that they minimize a quadradic function of the endogenous
variables.
(Galı́, 2015, Chapter 6) shows that the welfare loss function in our
smallneoK.mod example is a function of the variances of output
gap, price inflation and wage inflation.
Test
Let’s use GaliChap6.mod to find the coefficients of the Taylor rule
such that the loss function is minimized.
We use osr params to chose the parameters with respect to which
we optimize.
osr weights to set the weights in the quadratic function.
osr; to run the command.
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Plan
Introduction: what is Dynare
Main building blocks in a dynare code
Elementary outputs
Simulating models
Estimation of models
Optimal policy
Wrapping-up
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Applications with Dynare
What we have covered
Main code blocks, solving a model under rational expectations,
stochastic vs. deterministic simulations, estimation, optimal policy
rule
Also available
Sensitivity analysis, forecasting, Ramsey policy, markov switching
SBVAR, DSGE-VAR, macro-processing language, time-series
manipulation, Dynare and Latex, Dynare and Matlab, Dynare++
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References I
Adjemian, Stéphane et al. (Apr. 2011). Dynare: Reference Manual
Version 4. Dynare Working Papers 1. CEPREMAP.
Collard, Fabrice (2001). “Stochastic simulations with DYNARE. A
practical guide. (Adapted for Dynare 4.1, 2009)”. In: pp. 1 –8.
Galı́, Jordi (2015). Monetary policy, inflation, and the business
cycle: an introduction to the new Keynesian framework and its
applications. Princeton University Press.
Smets, Frank and Raf Wouters (2003). “An Estimated Dynamic
Stochastic General Equilibrium Model of the Euro Area”. In:
Journal of the European Economic Association 1.5,
pp. 1123–1175.
Smets, Frank and Rafael Wouters (2007). “Shocks and Frictions in
US Business Cycles: A Bayesian DSGE Approach”. In: American
Economic Review 97.3, pp. 586–606.
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References II
Villemot, Sébastien (Apr. 2011). Solving rational expectations
models at first order: what Dynare does. Dynare Working Papers
2. CEPREMAP.
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