Session : DSGE models and rational expectations · 2019-03-11 · Plan of part III of the course \Rational Expectation and DSGE models"- Lectures 4,5,6,7 Lecture 4 (This lecture)

$Page 1: Session : DSGE models and rational expectations · 2019-03-11 · Plan of part III of the course \Rational Expectation and DSGE models"- Lectures 4,5,6,7 Lecture 4 (This lecture)$
Session : DSGE models and rational expectations

Aurelien Poissonnier1

1Insee-European Commission

Applied Macroeconometrics - ENSAE

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Applied Macroeconometrics Part III: DSGE models

DynamicStochastic

GeneralEquilibrium

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Plan of part III of the course “Rational Expectation andDSGE models”- Lectures 4,5,6,7

Lecture 4 (This lecture)

Introduction to DSGE models (examples, linearizing)

Solving a DSGE model (i.e. solving for unobservedexpectations)

Lecture 5-6 (HLB) Estimation of DSGE modelsLecture 7 (AP) Simulation and use of DSGE models with Dynare.

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Plan

Rational expectations

DSGE models

Solving DSGE models

Wrapping-up

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Rational expectation models

Rational expectation models (e.g. DSGE) differ from

Traditional structural models, ”Cowles Commission”-type(taught by P.O. Beffy in Lecture 1 and 2)

Structural VAR models (Lecture 3)

Common point: dynamic time-series modelsKey differences:

RE models make expectations explicit

RE models are more structural than “structural VARs” (whichimpose only few restrictions)

RE models are in general derived from micro foundations.

Proeminent examples: DSGE modelsNB: expectations are (most often) non-observed variables.→ rational expectations; help circumvent this

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Expectations in macro models

• Expectations are everywhere in macro!

• Consumption-saving decision:U ′(ct) = β(1 + rt)EtU

′(ct+1) (ct consumption, rt real interestrate)

• asset yields; in the risk-neutral case: rt = Etp+1−ptpt

+ dtpt

(rt riskless rate, dt dividend, pt stock price)

• money demand (Cagan) ln(Mt/Pt) = −αEt(∆Pt+1/Pt) + εt

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The rational expectation assumption

Notation: EtYt+1 expectation of Yt+1 formulated at date t

Rational Expectations :

EtYt+1 = E (Yt+1|It),

E (.|It) mathematical expectation conditional on It , information setavailable at date t,

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Rational Expectation Hypothesis: a discussion

• Justifications:- RE expectations are ”model-consistent” : no reason to assumeeconomic agents have less knowledge than the econometricianabout the structure of the model- Lucas critique (Lucas, 1976) : traditional reduced form modelsare non invariant to an economic policy intervention

• Remarks:- RE is a strong assumption- rational expectations 6= perfect expectations- RE can be tested (direct tests, using expectations data likesurveys – or indirect tests)

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Some alternatives to rational expectations

• Adaptatives expectations

EtYt+1 = λYt + (1− λ)Et−1Yt ,

• Naive expectations, an extreme particular case: EtYt+1 = Yt

(case λ = 1).

• Many others: rational learning, limited rationality...

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Some alternatives to rational expectations (continued)

Expectations data that can be used for direct tests of the REHEtYt+1 Such data are either directly observed or built.Examples:

quantitative or qualitative answers to business survey (inFrance surveys of consumers or firms by Insee)

expectations data inferred from financial market data (e.g.inflation swap)

Some limits:

assumptions underlying quantification

directly observed expectations are most often forecast of somespecific agents (e.g. professional forecasters like OECDexperts or Consensus Forecast)

in practice, expectations are most often unoserved

These alternatives are not developed in this lecture10 / 66

Rational expectations in DSGE

EtYt+1 = E (Yt+1|It),

We will assume for E (•|It) that agents know the model structureand past data when forming expectations.

Expectations will be internally consistent

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Plan


DSGE modelsA simple RBC modelThe simplest neo-Keynesian model(large) DSGE models in public administrations andinternational institutionsLinearizing

Solving DSGE models

Wrapping-up

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This course is not about deriving DSGE models

For detailed material on the derivation of such models, see

2A Macroeconomie 2: fluctuations with Franck Malherbet

3A Monetary Economics with Olivier Loisel

3A Structural macroeconomics with Edouard Challe

appendix slides for this class from last year by BenoıtCampagne (Smets and Wouters, 2003)

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This course focuses on linear models

• linear models ... or linearized around the stationary equilibrium

• Alternative to linearization:- value function iteration,- second-order (or higher order) approximationsThese approaches will not be detailed here.See (Canova, 2011), (DeJong and Dave, 2011), (Juillard andOcaktan, 2008).NB: in some cases you may lose relevant information by linearizing(Linde and Trabandt, 2018)

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What is this course about?

This course will show in the next session how linearized DSGEmodels can be put under the form of a restricted SVAR.

Before that we will describe DSGE models as macroeconomic tools.

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Plan


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A simple RBC model

Keynes-Ramsey’s rules:1

ct= βEt

[1

ct+1(1 + rt+1)

]Hours worked: Ht = 1− φ ct

wt

Production function: yt = Atkαt H

1−αt

Real interest rate: rt = αytkt− δ

Real wage: wt = (1− α)ytHt

Investment: it = γkt+1 − (1δ)kt

Market clearing: yt = ct + it

Productivity shock: log(At) = ηlog(At−1 + (1− η)log(A) + εt

cf. Franck Malherbet’s class on Fluctuations (or King and Rebelo,1999)

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Plan


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The simplest neo-Keynesian model

yt = Et (yt+1)− 1

σ(it − Et (πt+1)− rnt ) (1)

πt = βEt (πt+1) + κyt (2)

it = φpπt + φy yt + εit (3)

cf. Olivier Loisel’s class on Monetary economics (or Jordi Galı(2015). Monetary policy, inflation, and the business cycle: anintroduction to the new Keynesian framework and its applications.Chap. 3)

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RBC-neoK controversy

There is a bit of a dogmatic controversy between both types ofmodels...

Don’t get involved

Keep a scientific approach: the model does or does not fit thedata, channel X or Y is or is not important in explaining macrofluctuations...You should not care whether Prof. W or Z had the right intuitiondecades ago!

Anyway, the truth will (dis)agree partially with both so benuanced.

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Plan


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The Smets-Wouters/Christiano Eichenbaum and Evanscore

The most popular mid-size model is (Smets and Wouters, 2003;Christiano, Eichenbaum, and Evans, 2005) (see Campagne’s annexor Challe’s class).

Closed economy

Capital and labour

Nominal rigidities (monetary policy has a role) and realrigidities

Exogenous fiscal policy

12 equations - 10 shocks - 7 observables

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DSGE in policy institutions

DSGE models are very popular in central banks and othereconomic policy institutions. The most popular mid-size model isSmets-Wouters/CEE (see Campagne’s annex or Challe’s class).

IMF-GIMF (Kumhof et al., 2010)

IMF-GEM (Bayoumi et al., 2004)

European Commission-Quest (Ratto, Roeger, and Veld, 2009)

ECB-NAWM (Warne, Coenen, and Christoffel, 2008)

ECB-EAGLE (Gomes, Jacquinot, and Pisani, 2012)

Fed-Sigma (Erceg, Guerrieri, and Gust, 2006)

DG Tresor-Omega3 (Carton and Guyon, 2007)

Insee-Meleze (Campagne and Poissonnier, 2016)

...

There are libraries of models macromodelbase.com and privatecollections (J. Pfeifer)

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https://ec.europa.eu/info/business-economy-euro/economic-and-fiscal-policy-coordination/economic-research/macroeconomic-models_en

http://www.macromodelbase.com/

https://github.com/johannespfeifer/dsge_mod

Main blocks in a DSGE model

Firms

−→ Production function−→ Balancing factors of production−→ Set prices / marginal cost−→ Banking sector (optional)

Households

−→ Consume−→ Save-Invest−→ Supply labour−→ Negotiate wages / marginal productivity−→ Heterogenous agents (optional)

Policy (fiscal and monetary, optional)

Laws of nature

−→ Market clearing−→ Capital dynamics−→ Other countries and trade (optional)

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What would you want to include in a DSGE model?

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A general form

A generalized DSGE model (non linear)

Et f (yt+1, yt , yt−1, vt) = 0 (4)

with . . . endogenous variables, . . . exogenous variables, and fcontaining . . . equations for ldots endogenous variables.

In general only a subset y+t+1 (y−t−1) of the . . . . . . . . . variables are

forward (reps. backward) looking.

The endogenous variables need to be solved for, as a function ofthe predetermined variables and the shocks: yt = g(. . . , . . .) withg the ”. . . . . . . . .”.

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Plan


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Why linear?

The simplest RBC and neo-Keynesian model are ”naturally” linearwhen taken in logs.

For large DSGE models this cannot be true.

We perform a first order Taylor development around a SteadyState.

You need to think twice about trends and the definition ofequilibrium in your model.

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(Standard) Notations

Xt a model variable, X its steady state, Xt = Xt−XX

the deviationfrom the steady state.

In a linearized model you assume that X � 1 (e.g. 5% at most) sothat X 2 ' 0 (is negligible).

Xt = Xt−XX' log(Xt)− log(X ) (but I recommend that you do not

use the log definition of X , cf. pen& paper). You will see why in aminute.

Y = f (X ) =⇒ Y = f ′(X )X

YX

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Pen and paper

Let’s take 2 simple equations:

Yt = Kαt (AtLt)

1−α (5)

and the growth rate

dYt =Yt − Yt−1

Yt−1≈ log

(Yt

Yt−1

)(6)

Let’s assume that At = A0eat is a deterministic trend. We denote

yt = Yt/At and kt = Kt/At the stationary component of outputand capital. Lt is stationary.Compute a log-linearisation of the 2 equations above

with yt = log(yt/y)

with yt = yt−yy

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Pen and paper - Solution

The first equation gives the same result in both cases:

yt = kαt L1−αt ; y = kαL1−α ; yt = αkt + (1− α)Lt (7)

The second does not:

dYt ≈ log

(Yt

Yt−1

)= log(yt)− log(yt−1) + a = yt − yt−1 + a (8)

dYt =Yt − Yt−1

Yt−1=

ytea − yt−1

yt−1=

ea(1 + yt)

(1 + yt−1)− 1 (9)

≈ea(1 + yt − yt−1)− 1 = ea(yt − yt−1) + (ea − 1)

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Linearized outcome

The generalized DSGE model (non linear)

Et f (yt+1, yt , yt−1, vt) = 0 (10)

can then be turned into a matrix expression (linear)

AYt = BEtYt+1 + CYt−1 + DVt (11)

see (Villemot, 2011) in the Dynare documentation to understandhow it’s done automatically.

Note for the future, linearized model makes an implicit certaintyequivalence assumption.

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Plan


DSGE models

Solving DSGE models

Wrapping-up

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Solving rational expectation models

• A benchmark (simple) example:

yt = βEtyt+1 + γxt + εt

• ”Forward ”solution :

yt = βEt(βEt+1yt+2 + γxt+1 + εt+1) + γxt + εt

yt = γ∑T

i=0βiEtxt+i + βT+1Etyt+T+1 + εt

• The ”fundamental” solution (in the case β < 1) :

yt = γ

∞∑i=0

βiEtxt+i + εt

• An interpretation: Discounted Present Value

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Computing a closed form

Let’s assume the model gives us a backward dynamic for xte.g. AR(1) xt = ρxt−1 + vtProperty:

Etxt+i = ρixt

Implied reduced form:

yt = γ1

(1− βρ)xt + εt

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Models with a lagged endogenous variable

Model specification

yt = bEtyt+1 + ayt−1 + cxt + εt

xt = ρxt−1 + ut

Examples :• new “hybrid” Phillips curve;• consumption Euler equation with habit formation in utility,• models with adjustment costs,...

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Example of a resolution method:the undetermined coefficient approach

”Guess” the form of the solution:

yt = ϕyt−1 + αxt + εt

This implies

Etyt+1 = ϕyt + αEtxt+1 = ϕ(ϕyt−1 + αxt + εt) + αρxt

= ϕ2yt−1 + α(ϕ+ ρ)xt + ϕεt

Substitute this in yt = bEtyt+1 + ayt−1 + cxt + εt

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This yields a second degree equation for ϕ:

ϕ = bϕ2 + a

And α = c + bα(ϕ+ ρ)We choose the root such that ϕ < 1

It results that ϕ = 1−√

1−4ba2b , and α = c

1−b(ϕ+ρ)

yt = (1−√

1− 4ba

2b)yt−1 + (

c

1− b(ϕ+ ρ))xt + (

1

1− bϕ)εt

Shortcoming: this approach is not systematic

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The multivariate case

• A general formulation for a multivariate model:

AYt = BEtYt+1 + CYt−1 + DXt

Xt = ΦXt−1 + Vt

where A,B,C,D, Φ are matrices, Yt , a vector of endogenousvariables, Xt , a vector of exogenous variables (e.g. shocks), Vt

i.i.d. inovation• Other general formulations are possible:

AYt = BEtYt+1 + DXt

Xt = ΦXt−1 + Vt

The two formulations are equivalent up to a redifinition of vectorYt .

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• In general, no analytical solution is available

Various procedures that produce numerical soultions:• (Uhlig, 1995): undetermined coefficient approach• Using matrices decompositions (Jordan, Schur,QZ...) to provideforward solutions: (Blanchard and Kahn, 1980), (Klein, 2000),(Sims, 2002)

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Sketch of the undetermined coefficient approach (Uhlig,1995).

• Canonical formulation of a multivariate model

FEtYt+1 + GYt + HYt−1 + MXt = 0

Xt = NXt−1 + Vt

where F ,G ,H ,M ,N are matrices, Xt , is a vector of endogenousvariables”Guess” the solution:

Yt = PYt−1 + QXt

P and Q fulfill:

FP2 + GP + H = 0 (12)

(FQ + L)N + (FP + G )Q + M = 0 (13)

→ Numerical solution of quadratic equation (12)41 / 66

The Blanchard and Kahn (1980) method for theMultivariate case

• Formulation of the model

EtYt+1 = AYt + CXt

Xt = ΦXt−1 + Vt

→ Distinguish in Yt between variablesy1t predetermined (n1 ) andy2t non-predetermined (≡ no initial condition ) .

→ y1t with size n1, y2t with size n2

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Note: unicity/stability of the solution

Blanchard and Kahn (1980) Conditions :

In the above model:Let n the number of eigenvalues of matrix A that are larger than 1If n = number of non -predetetermined variables (ie n2) : then thestable soution is unique (saddle-path solution)If n > n2( number of non-predetermined variables) : instability(unit-root, explosive dynamics for the variables)If n < n2 number of non-predetermined variables : multiplicity ofsolutions

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Example: the simple model

yt = βEtyt+1 + γxt + εt

xt = ρxt−1 + vt

with xt predetermined variable, yt non predetermined variable,Yt = (xt , yt)

′

EtYt+1 = AYt + Vt

with A =

[ρ 0−γ/β 1/β

]Eigenvalues of A are ρ, (1/β).

Unicity and stability if −1 < ρ < 1,−1 < β < 1

If β > 1, ”rational stationary bubbles ” are possible (multiplicity ofsolutions)Example of a bubble: bt = (1/β)bt−1 + et

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Blanchard and Kahn (1980) method

• Formulation of the model

EtYt+1 = AYt + CXt

Xt = ΦXt−1 + Vt

• We seek a solution that has the form :

y2t = Hy1t + NXt

y1t = My1t−1 + LXt−1

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Blanchard and Kahn (1980) method

The system writes:[y1t+1

Ety2t+1

]= A

[y1t

y2t

]+ CXt

Perform a Jordan decomposition of A :

A = Λ−1JΛ

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J is a bloc-diagonal matrix containing the eigenvalues of A

J =

[J1 00 J2

]where J1 collects eigenvalues smaller than 1and J2 collects eigenvalues larger than 1Note J is diagonal if eigenvalues are distinct,otherwise J contains both ”0”’s and ”1”’s on the line above thediagonal.

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Implementing Blanchard and Kahn (1980)The system can be re written as:

Λ

[y1t+1

Ety2t+1

]= JΛ

[y1t

y2t

]+ Λ

[C1

C2

]Xt

We define auxiliary variables[y1t

y2t

]=

[Λ11 Λ12

Λ21 Λ22

] [y1t

y2t

]Then we can write a “decoupled” system.[

y1t+1

Et y2t+1

]=

[J1 00 J2

] [y1t

y2t

]+

[D1

D2

]Xt

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The sub-system associated with eigenvalues larger than one can besolved forward :

Et y2t+1 = J2y2t + D2Xt

hencey2t = J−1

2 Et y2t+1 − J−12 D2Xt

Solving:

y2t = −∞∑k=0

J−1−k2 D2EtXt+k

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Using the properties of the forcing process Xt :VAR(1) EtXt+k = ΦkXt

y2t = −∞∑k=0

J−1−k2 D2ΦkXt

An explicit form for this sum (using properties of the Kroneckerproduct):

y2t = −J−12 (I − Φ′ ⊗ J−1

2 )D2Xt

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Expression as a fonction of the predetermined variablesy2t = Λ−1

22 y2t − Λ−122 Λ21y1t

Soy2t = −Λ−1

22 J−12 (I − Φ′ ⊗ J−1

2 )D2Xt − Λ−122 Λ21y1t

This expression is indeed of the form

y2t = Hy1t + NXt

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Last step (!)Solving for predetermined variables y1t

y1t+1 = A11y1t + A12y2t + D1Xt

where A =

[A11 A12

A21 A22

]Replacing y2t with its solved form, the process for y1t is indeed ofthe form:

y1t = My1t−1 + LXt−1

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Extensions and alternatives to Blanchard-Kahn method

• (Sims, 2002), (Klein, 2000): use of Schur and QZ decompositionBEtYt+1 = AYt + CXt

Xt = ΦXt−1 + Vt

→ In a general model B may be non invertible(if B invertible, back to B-K case with A = B−1A)→ Sims method does not require specification ofpredetermined/non-predetermined variables→ The various procedures are available in the form of Matlab,Gauss,... routines→ The Sims/Klein method is implemented in the Dynare toolbox

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Other procedures

• (Anderson and Moore, 1985) : multiple leads and lags ofendogenous variable

Yt =J∑

j=1AjEtYt+j +

I∑i=1

BiYt−i + Vt

AIM Algorithm (Federal Reserve) with Matlab, GAUSS• DYNARE allows the user to write multiple leads and lags

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Plan


DSGE models

Solving DSGE models

Wrapping-up

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Bottom line of this lecture

• Start with a non linear model

Et f (yt+1, yt , yt−1, vt) = 0

• Linearize it into

AYt = BEtYt+1 + CYt−1 + DVt

• Obtain the reduced form

Yt = MYt−1 + Dηt

You obtain a constrained VAR model • In practice done bycomputer routines (eg Dynare)• From there you can do the same as with a SVAR• + you can derive normative results

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What are A,B,C,D numerically?

Calibration

Associated to RBC models.

Main principle of calibration: set values of parameters(including shocks standard deviations) Then compare modelpredictions with second moments of the data

Cf. DeJong and Dave (2011) chap.6

This approach is not strictly speaking an econometric one.

It can be viewed as a particular case, and less formalized version ofthese approaches: Minimum Distance Estimation, BayesianApproach

Next Lectures with HLB, you will discuss the estimation ofsuch models

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Prepare for the next Lectures

Lecture 5-6: have a look at (Christiano, Eichenbaum, andEvans, 2005)

Lecture 7: Install Dynare + Matlab/Octave (if not alreadydone)

try the examples provided by the dynare team

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https://www.dynare.org/

References I

Anderson, Gary and George Moore (1985). “A linear algebraicprocedure for solving linear perfect foresight models”. In:Economics Letters 17.3, pp. 247–252.

Bayoumi, Tamim et al. (Nov. 2004). GEM: A New InternationalMacroeconomic Model. IMF Occasional Papers 239.International Monetary Fund.

Blanchard, Olivier Jean and Charles M Kahn (1980). “TheSolution of Linear Difference Models under RationalExpectations”. In: Econometrica 48.5, pp. 1305–1311.

Campagne, B. and A. Poissonnier (2016). MELEZE: A DSGEmodel for France within the Euro Area. Documents de Travailde l’Insee - INSEE Working Papers g2016-05. Institut Nationalde la Statistique et des Etudes Economiques.

Canova, Fabio (2011). Methods for applied macroeconomicresearch. Princeton university press Princeton, NJ.

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References II

Carton, Benjamin and Thibault Guyon (2007). Divergences deproductivite en union monetaire Presentation du modeleOmega3. Tech. rep. Technical Report 2007/08, DirectionGenerale du Tresor et de la Politique Economique.

Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans(2005). “Nominal Rigidities and the Dynamic Effects of a Shockto Monetary Policy”. In: Journal of Political Economy 113.1,pp. 1–45.

DeJong, David N and Chetan Dave (2011). Structuralmacroeconometrics. Princeton University Press.

Erceg, Christopher J., Luca Guerrieri, and Christopher Gust(2006). “SIGMA: A New Open Economy Model for PolicyAnalysis”. In: International Journal of Central Banking 2.1.

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References III

Galı, Jordi (2015). Monetary policy, inflation, and the businesscycle: an introduction to the new Keynesian framework and itsapplications. Princeton University Press.

Gomes, S., P. Jacquinot, and M. Pisani (2012). “The EAGLE. Amodel for policy analysis of macroeconomic interdependence inthe euro area”. In: Economic Modelling 29.5, pp. 1686–1714.

Juillard, Michel and Tarik Ocaktan (2008). “Methodes desimulation des modeles stochastiques d’equilibre general”. In:Economie & Prevision 0.2, pp. 115–126.

King, Robert G. and Sergio T. Rebelo (1999). “Resuscitating realbusiness cycles”. In: Handbook of Macroeconomics. Ed. byJ. B. Taylor and M. Woodford. Vol. 1. Handbook ofMacroeconomics. Elsevier. Chap. 14, pp. 927–1007.

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References IV

Klein, Paul (2000). “Using the generalized Schur form to solve amultivariate linear rational expectations model”. In: Journal ofEconomic Dynamics and Control 24.10, pp. 1405–1423.

Kumhof, Michael et al. (2010). “The global integrated monetaryand fiscal model (GIMF)- Theoretical structure”. In: IMFWorking Paper.

Linde, Jesper and Mathias Trabandt (2018). “Should we uselinearized models to calculate fiscal multipliers?” In: Journal ofApplied Econometrics 33.7, pp. 937–965.

Lucas, Robert Jr (1976). “Econometric policy evaluation: Acritique”. In: Carnegie-Rochester Conference Series on PublicPolicy 1.1, pp. 19–46.

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References V

Ratto, Marco, Werner Roeger, and Jan in ’t Veld (2009). “QUESTIII: An estimated open-economy DSGE model of the euro areawith fiscal and monetary policy”. In: Economic Modelling 26.1,pp. 222–233.

Sims, Christopher A (2002). “Solving Linear Rational ExpectationsModels”. In: Computational Economics 20.1-2, pp. 1–20.

Smets, Frank and Raf Wouters (2003). “An Estimated DynamicStochastic General Equilibrium Model of the Euro Area”. In:Journal of the European Economic Association 1.5,pp. 1123–1175.

Uhlig, Harald (1995). A toolkit for analyzing nonlinear dynamicstochastic models easily. Discussion Paper / Institute forEmpirical Macroeconomics 101. Federal Reserve Bank ofMinneapolis.

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References VI

Villemot, Sebastien (Apr. 2011). Solving rational expectationsmodels at first order: what Dynare does. Dynare Working Papers2. CEPREMAP.

Warne, Anders, Gunter Coenen, and Kai Christoffel (Oct. 2008).The new area-wide model of the euro area: a micro-foundedopen-economy model for forecasting and policy analysis.Working Paper Series 944. European Central Bank.

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An alternative solution method :Factorisation

Rely on ”Forward ” operator:

Fyt = Etyt+1

Benchmark equation writes:

yt − bFyt − aLyt = cxt + εt

henceP(F )Lyt = −(1/b)(cxt + εt)

where P(F ) = F 2 − (1/b)F + (a/b)P(F ) can be factored P(F ) = (F − ϕ1)(F − ϕ2) with ϕ1 < 1

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Dividing by polynomial (F − ϕ2)→ Solving ”forward” for the root ϕ2 > 1

(1− ϕ1L)yt =1

bϕ2

1

1− (1/ϕ2)F(βxt + εt)

where ϕ1 = 1−√

1−4ba2b (note ϕ1 = ϕ, same as the undetermined

coefficient approach)and

yt = ϕ1yt−1 − 1bϕ2

∞∑k=0

(1/ϕ2)kEt(cxt+k + εt+k)

Using Et(xt+k) = ρhxt , we find (as in the UC approach):α = c

1−b(ϕ1+ρ)

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