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DSGE-ModelsCalibration and Introduction to Dynare
Dr. Andrea Beccarini Willi Mutschler, M.Sc.
Institute of Econometrics and Economic [email protected]
Summer 2012
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 1 / 24
Calibration and Introduction to Dynare
1 Overview Estimation-Methods
2 CalibrationHints for calibrating a model
3 Exercise 2: Calibration of a RBC-model with monopolistic competition
4 Calibration - Pros & Cons
5 Exercise 3: A simple RBC model - Practicing Dynare
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 2 / 24
Overview Estimation-Methods
Econometrically, a DSGE-Model is a state-space model of which onehas to determine the parameters.
Three concepts:1 Calibration: The parameters are set in such a way, that they closely
correspond to some theoretical moment or stylized fact of data.2 Methods of limited information or weak econometric interpretation:
Minimize the distance between theoretical and empirical moments, i.e.General-Method-of-Moments or Indirect Inference.
3 Methods of full information or strong econometric interpretation:The goal is an exact characterization of observed data, i.e.Maximum-Likelihood or bayesian methods.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 3 / 24
Calibration
Goal: To answer a specific quantitative research question using astructural model.
Construct and parameterize the model such, that it corresponds tocertain properties of the true economy.
Use steady-state-characteristics for choosing the parameters inaccordance with observed data.
Often: stable long-run averages (wages, working-hours, interest rates,inflation, consumption-shares, government-spending-ratios, etc.).
You can use micro-studies as well, however, one has to be carefulabout the aggregation!
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 4 / 24
CalibrationHints for calibrating a model
Use long-term averages of interest rates, inflation, average growth ofproductivity, etc. for steady-state values.
BUT: Weil (1989) shows, that in models with representative agentsthere is an overestimation of steady-state interest rates (risk-free ratepuzzle). It is possible that you get absurd constellation of parameters,like a discount-factor of β > 1.
Usual mark-up on prices is around 1.15 (Corsetti et al (2012)).
Intertemporal elasticity of substitution 1/σ somewhere between σ = 1and σ = 3 (King, Plosser and Rebelo (1988), Rotemberg andWoodford (1992), Lucas (2003)).
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 5 / 24
CalibrationHints for calibrating a model
Rigidity of prices: For an average price adjustment of 12-15 monthssee Keen and Wang (2007).
Coefficients of monetary policy: Often Taylor-Rule, you can use therelative coefficients to put more emphasize/weight on the stability ofprices or on smoothing the business cycle.
Parameters of stochastic processes: Often persistent, smallstandard-deviations, otherwise you get high oscillations. You can alsoestimate the production function via OLS (Solow-residual).
How to choose shocks: Look at similar studies: Christiano,Eichenbaum and Evans (2005), Smets and Wouters (2003), etc..
Ultimately: Try & Error!
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 6 / 24
Exercise 2:Calibration of a RBC-model with monopolistic competition
Households maximize expected utility over consumption ct and leisureft = 1− lt , where lt denotes labor:
Et
∞∑t=0
β[log(ct) + ψlog(1− lt)],
taking account of the following budget constraint:
ct + kt+1 = wt lt + rtkt︸ ︷︷ ︸=yt
+(1− δ)kt .
kt ist the capital stock of the economy, wt the real wage, rt the realinterest rate and δ the rate of depreciation. Further, investment is given by:
it = yt − ct .
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 7 / 24
Exercise 2:Calibration of a RBC-model with monopolistic competition
In the market for intermediate goods there is monopolistic competition,whereas perfect competition applies to the market for final goods. Theproduction-function of a firm i ∈ [0; 1], that sells intermediate goods, isgiven by:
yit = Atkαit l
1−αit , 0 < α < 1,
log(At) = ρlog(At−1) + εt , εt ∼ N(0, σ2),
with At denoting the level of technology. Firms cannot influence the realwage wt or the real interest rate rt . However, they have market power overtheir price pit for their good yit . The intermediate goods are combined intoa final good by a Dixit/Stiglitz-type aggregator:
yt =
(∫ 1
0(yit)
εε−1
) ε−1ε
,
with ε being the elasticity of substitution.Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 8 / 24
Exercise 2:Calibration of a RBC-model with monopolistic competition
(a) Show that, the structural form of the DSGE-model is given by thefollowing equations and interpret these.
1
ct= βEt
[1
ct+1(1 + rt+1 − δ)
](1)
wt = ψct
1− lt(2)
yt = ct + it (3)
yt = Atkαt l
1−αt (4)
wt = (1− α)ytlt
ε− 1
ε(5)
rt = αytkt
ε− 1
ε(6)
it = kt+1 − (1− δ)kt (7)
log(At) = ρlog(At−1) + εt (8)
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 9 / 24
Exercise 2:Calibration of a RBC-model with monopolistic competition
(b) What are the parameters of the model?
(c) Write a mod-file for the model and calibrate the vector of parametersµ. Simulate the model for 1000 periods with Dynare. Save the middle100 observations of ct , yt , it ,wt and rt into an Excel-file as well as intoa mat-file. Plot the path of consumption.
(d) Reformulate the structural equations such that variables are expressedas percentage deviations from steady-state:xt = e log(xt)−log(x)+log(x) = xe xt . Write a mod-file for this model.What has changed?
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 10 / 24
CalibrationPros
Calibration is commonly used in the literature. It gives a firstimpression, a flavor of the strengths and weaknesses of a model.
A good calibration can provide a valuable and precise image of data.
Using different calibrations, one can asses interesting implications ofdifferent policies:
How does the economy react, if the central bank focuses more onsmoothing the business cycle than on price stability?What happens to consumption, if the households have a strongintertemporal elasticity of substitution? What if it is low?
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 11 / 24
CalibrationCons
This Ad-hoc-approach is at the center of criticism of DSGE-models.
There is no statistical foundation, it is based upon subjective views,assessments and opinions.
Many parameter, such as those of the exogenous processes, leaveroom for different values and interpretations (intertemporal elasticityof substitution, monetary and fiscal parameters, coefficients of rigidity,standard deviations, etc.).
Prescott (1986, S. 10) regarding RBC-models:
The models constructed within this theoretical framework are necessarilyhighly abstract. Consequently, they are necessarily false, and statisticalhypothesis testing will reject them. This does not imply, however, thatnothing can be learned from such a quantitative theoretical exercise.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 12 / 24
Exercise 3: A simple RBC model
Consider the following model of an economy.
Representative agent preferences
U =∞∑t=1
(1
1 + ρ
)t−1Et
[log (Ct)−
L1+γt
1 + γ
].
The household supplies labor and rents capital to the corporatesector.
Lt is labor services.ρ ∈ (0,∞) is the rate of time preference.γ ∈ (0,∞) is a labor supply parameter.Ct is consumption.wt is the real wage.rt is the real rental rate.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 13 / 24
Exercise 3: A simple RBC model
The household faces the sequence of budget constraints
Kt = Kt−1 (1− δ) + wtLt + rtKt−1 − Ct ,
whereKt is capital at the end of period.δ ∈ (0, 1) is the rate of depreciation.
The production function is given by the expression
Yt = AtKαt−1((1 + g)t Lt
)1−α,
where g ∈ (0,∞) is the growth rate and α and β are parameters.
At is a technology shock that follows the process
At = Aλt−1 exp (et) ,
where et is an i.i.d. zero mean normally distributed error withstandard deviation σ and λ ∈ (0, 1) is a parameter.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 14 / 24
Exercise 3: A simple RBC modelThe household problem
Lagrangian
L = maxCt ,Lt ,Kt
∞∑t=1
(1
1 + ρ
)t−1Et
[log (Ct)−
L1+γt
1 + γ
− µt (Kt − Kt−1 (1− δ)− wtLt − rtKt−1 + Ct)].
First order conditions
∂L
∂Ct=
(1
1 + ρ
)t−1( 1
Ct− µt
)= 0,
∂L
∂Lt=
(1
1 + ρ
)t−1(Lγt − µtwt) = 0,
∂L
∂Kt= −
(1
1 + ρ
)t−1µt +
(1
1 + ρ
)t
Et (µt+1(1− δ + rt+1)) = 0.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 15 / 24
Exercise 3: A simple RBC modelFirst order conditions
Eliminating the Lagrange multiplier, one obtains
Lγt =wt
Ct,
1
Ct=
1
1 + ρEt
(1
Ct+1(rt+1 + 1− δ)
).
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 16 / 24
Exercise 3: A simple RBC modelThe firm problem
maxLt ,Kt−1
AtKαt−1((1 + g)t Lt
)1−α − rtKt−1 − wtLt .
First order conditions:
rt = αAtKα−1t−1
((1 + g)t Lt
)1−α,
wt = (1− α)AtKαt−1((1 + g)t
)1−αL−αt .
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 17 / 24
Exercise 3: A simple RBC modelGoods market equilibrium
Kt + Ct = Kt−1(1− δ) + AtKαt−1((1 + g)t Lt
)1−α︸ ︷︷ ︸wtLt+rtKt
.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 18 / 24
Exercise 3: A simple RBC modelDynamic Equilibrium
1
Ct=
1
1 + ρEt
(1
Ct+1(rt+1 + 1− δ)
),
Lγt =wt
Ct,
rt = αAtKα−1t−1
((1 + g)t Lt
)1−α,
wt = (1− α)AtKαt−1((1 + g)t
)1−αL−αt ,
Kt + Ct = Kt−1(1− δ) + AtKαt−1((1 + g)t Lt
)1−α,
log(At) = λlog(At−1) + et .
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 19 / 24
Exercise 3: A simple RBC modelExistence of a balanced growth path
Good markets equilibrium for each period t:
Kt + Ct = Kt−1(1− δ) + AtKαt−1((1 + g)t Lt
)1−α.
So, there must exist growth rates gc and gk such that
(1 + gk)tK1 + (1 + gc)tC1 =
(1 + gk)t
1 + gkK1(1− δ) + At
((1 + gk)t
1 + gkK1
)α ((1 + g)t Lt
)1−α
⇔ K1 +
(1 + gc1 + gk
)t
C1 =
K1
1 + gk︸ ︷︷ ︸K0
(1− δ) + At
(K1
1 + gk
)α(( 1 + g
1 + gk
)t
Lt
)1−α
.
This is only valid, if gc = gk = g .Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 20 / 24
Exercise 3: A simple RBC modelStationarized model
Let’s define
Ct = Ct/(1 + g)t ,
Kt = Kt/(1 + g)t ,
wt = wt/(1 + g)t .
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 21 / 24
Exercise 3: A simple RBC modelStationarized model (continued)
1
Ct(1 + g)t=
1
1 + ρEt
(1
Ct+1(1 + g)(1 + g)t(rt+1 + 1− δ)
),
Lγt =wt(1 + g)t
Ct(1 + g)t,
rt = αAt
(Kt−1
(1 + g)t
1 + g
)α−1 ((1 + g)tLt
)1−α,
wt(1 + g)t = (1− α)At
(Kt−1
(1 + g)t
1 + g
)α ((1 + g)t
)1−αL−αt ,(
Kt + Ct
)(1 + g)t = Kt−1
(1 + g)t
1 + g(1− δ)
+ At
(Kt−1
(1 + g)t
1 + g
)α ((1 + g)tLt
)1−α.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 22 / 24
Exercise 3: A simple RBC modelStationarized model (continued)
1
Ct
=1
1 + ρEt
(1
Ct+1(1 + g)(rt+1 + 1− δ)
),
Lγt =wt
Ct
,
rt = αAt
(Kt−11 + g
)α−1L1−αt ,
wt = (1− α)At
(Kt−11 + g
)αL−αt ,
Kt + Ct =Kt−11 + g
(1− δ) + At
(Kt−11 + g
)αL1−αt ,
log(At) = λlog(At−1) + et .
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 23 / 24
Exercise 3: A simple RBC modelPracticing Dynare
(a) Write a mod-File for this simple RBC-model and use for calibration:α = 0.33, δ = 0.1, ρ = 0.03, λ = 0.97, γ = 0, g = 0.015.Use initval with these values:C = 1,K = 3, L = 0.9,w = 1, r = 0.15,A = 1.
(b) Show that the steady-state implies:
A = 1, r = (1 + g)(1 + δ) + δ − 1
L =
(1− α
rα − δ − g
)( rα
), K = (1 + g)
( rα
) 1α−1
L
C = (1− δ)K
1 + g+
(K
1 + g
)αL1−α − K , w = C
(c) Use this analytical solution for the mod-file, i.e. usesteady state model instead of initval. Dynare creates a steady-statem-file. Have a look at it.
Willi Mutschler (Institute of Econometrics) DSGE-Models Summer 2012 24 / 24