DSGE-Models - uni- · PDF fileDSGE-Models Calibration and Introduction to Dynare Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics

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


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.


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!


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)).


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!


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 .



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


(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)



(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?


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?


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.


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.


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.


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.


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)

).


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 .


Exercise 3: A simple RBC modelGoods market equilibrium

Kt + Ct = Kt−1(1− δ) + AtKαt−1((1 + g)t Lt

)1−α︸︷︷︸wtLt+rtKt

.


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 ,


)1−α,

log(At) = λlog(At−1) + et .


Exercise 3: A simple RBC modelExistence of a balanced growth path

Good markets equilibrium for each period t:


)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 .


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.


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 .


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.


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DSGE-Models - uni- · PDF fileDSGE-Models Calibration and Introduction to Dynare Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics