identiﬁcation analysis of dsge models with dynare · pdf fileidentiﬁcation with dynare...

Identification with DYNARE

Identification Analysis of DSGE models withDYNARE

FP7 Funded, Project MONFISPOL Grant no.: 225149.

Marco RattoEuropean Commission, Joint Research Centre

with the contribution of Nikolai IskrevBank of Portugal

MONFISPOL Final Conference, Frankfurt


Outline1 Introduction

2 DSGE modelsMoments

3 IdentificationLocal identification

4 DYNARE ImplementationIdentification strengthAnalyzing identification patternsMain features of the softwareDYNARE procedureNotes

5 Possible extensions

6 Examples


Intro

Introduction

In developing the identification software, we took into considerationthe most recent developments in the computational tools foranalyzing identification in DSGE models. A growing interest isbeing addressed to identification issues in economic modeling(Canova and Sala, 2009; Komunjer and Ng, 2011a,b; Iskrev,2010b). The identification toolbox includes the new efficientmethod for derivatives computation presented in Ratto and Iskrev(2010a,b) and the identification tests proposed by Iskrev.


DSGE Models

DSGE Models: Structural model and reduced form

A DSGE model is summarized by a system g of m non-linearequations:

Et

(

g(zt , zt+1, zt−1,ut |θ))

= 0 (1)

Most studies use linear approximations of the original models:

Γ0(θ)zt = Γ1(θ) Et zt+1 + Γ2(θ)zt−1 + Γ3(θ)ut (2)

where zt = zt − z∗. The elements of the matrices Γ0, Γ1, Γ2 andΓ3 are functions of θ.


DSGE Models

Assuming that a unique solution exists, it can be cast in thefollowing form

zt = A(θ)zt−1 + B(θ)ut (3)

Some of the variables in zt are not observed, so the transitionequation (3) is complemented by a measurement equation

xt = Czt +Dut + νt (4)

τ collects the non-constant elements ofz∗, A, and Ω,i.e. τ := [τ ′

z , τ′

A, τ′

Ω]′.


DSGE Models

Moments

Theoretical first and second moments

The unconditional first and second moments of xt are given by

E xt := µx = s (5)

cov(xt+i , x′

t) := Σx(i) =

CΣz(0)C′ if i = 0

CAiΣz(0)C′ if i > 0

(6)

where Σz(0) := E ztz′

t solves the matrix equation

Σz(0) = AΣz(0)A′ +Ω (7)


DSGE Models

Moments

Denote the observed data with XT := [x′1, . . . , x′

T ]′, and let ΣT be

its covariance matrix, i.e.

ΣT := EXTX′

T

=

Σx(0), Σx(1)′, . . . , Σx(T − 1)′

Σx(1), Σx(0), . . . , Σx(T − 2)′

. . . . . . . . . . . .Σx(T − 1), Σx(T − 2), . . . , Σx(0)

(8)


DSGE Models

Moments

We define mT := [µ′,σ′

T ]′, where

σT := [vech(Σx(0))′, vec(Σx(1))

′, ..., vec(Σx(T − 1))′]′

mT is a function of θ. If either ut is Gaussian, or there are nodistributional assumptions about the structural shocks, themodel-implied restrictions on mT contain all information that canbe used for the estimation of θ. The identifiability of θ depends onwhether that information is sufficient or not.


Identification

Global identification: the Gaussian case

Theorem

Suppose that the data XT is generated by the model (3)-(4) withparameter vector θ0. Then θ0 is globally identified if

mT (θ) = mT (θ0) ⇔ θ = θ0 (9)

for any θ ∈ Θ. If (9) is true only for values θ in an open

neighborhood of θ0, the identification of θ0 is local.


Identification

Local identification

Local identification: The rank condition

Theorem

Suppose that mT is a continuously differentiable function of θ.

Then θ0 is locally identifiable if the Jacobian matrix J(q) :=∂mq

∂θ′

has a full column rank at θ0 for q ≤ T. This condition is both

necessary and sufficient when q = T if ut is normally distributed.


Identification

Local identification

Given the chain rule

J(T ) =∂mT

∂τ ′

∂τ

∂θ′(10)

another necessary condition discussed in Iskrev (2010b):

Corollary

The point θ0 is locally identifiable only if the rank of J2 =∂τ∂θ′ at

θ0 is equal to k.

The condition is necessary because the distribution of XT dependson θ only through τ , irrespectively of the distribution of ut . It isnot sufficient since, unless all state variables are observed, τ maybe unidentifiable.


DYNARE Implementation

DYNARE implementation I

The identifiability of each draw θj is then established using thenecessary and sufficient conditions discussed by Iskrev (2010b):

Finding that matrix J2 is rank deficient at θj implies that thisparticular point in Θ is unidentifiable in the model.

Finding that J2 has full rank but J(T ) does not, means thatθj cannot be identified given the set of observed variables andthe number of observations.

we also analyze the derivatives of the LRE form of the model(JΓ = ∂γ

∂θ′ ), to check for ‘trivial’ non-identification problem,like two parameters always entering as a product in Γimatrices;

if θ is identified check J(q) with q < T . According toTheorem 2 this is sufficient for identification;



Tracking singularities

Whenever some of the matrices J2, J(T ) or JΓ is rank deficient,the code tries to diagnose the subset of parameters responsible forthe rank deficiency:

1 if there are columns of zeros in the J(·) matrix, the associatedparameter is printed on the MATLAB command window;

2 compute pairwise- and multi-correlation coefficients for eachcolumn of the J(·) matrix: if there are parameters withcorrelation coefficients equal to unity, these are printed on theMATLAB command window;

3 take the Singular Values Decomposition (SVD) of J(·) andtrack the eigenvectors associated to the zero singular values.



Identification strength

Identification strengthA measure of identification strength is introduced, following thework of Iskrev (2010a) and Andrle (2010), based on theinformation matrix

the strength of identification for parameter θi :

si =√

θ2i /(IT (θ)−1)(i ,i) (11)

a sort of a priori ‘t-test’ for θi ;

alternative normalization using the prior standard deviationσ(θi ):

spriori = σ(θi )/

√

(IT (θ)−1)(i ,i) (12)

∆priori = σ(θi ) ·

√

IT (θ)(i ,i) (13)



Identification strength

‘sensitivity’ and ‘correlation’ components (Iskrev, 2010a),

std(θi ) ≥√

IT (θ)−1 =1√∆i

1√

1− ρ2i

(14)

i.e. weak identification may be due to ∆i ≈ 0 or ρi ≈ 1;

the sensitivity component is defined as

∆i =√

θ2i · IT (θ)(i ,i) (15)

The identification toolbox shows, after the check of rankconditions, the plots of the strength of identification and of thesensitivity component for all estimated parameters.



Analyzing identification patterns

Analyzing identification patterns

1 Andrle (2010): the identification patterns are shown by takingthe singular value decomposition of IT (θ) or of the J(q)matrix and displaying the eigenvectors corresponding to thesmallest (or highest) singular values;

2 Iskrev (2010b): check which group of one, two or moreparameters is most capable to mimic (replace) the effect ofeach parameter. A brute force search is done for each columnof J(q)(j) to detect the group of columns J(q)(I+j), having thehighest explanatory power for J(q)(j) by a linear regression.



Features

Main features of the software

The new DYNARE keyword identification triggers the routinesdeveloped at JRC. This option has two modes of operation; pointidentification check (default) and Monte Carlo mode.



Features

Point identification check

With a list of estimated parameters:

with prior definition: the program performs the localidentification checks for the estimated parameters at the priormean (prior mode, posterior mean and posterior mode are alsoalternative options);

for ML estimation (no prior definition), local identificationchecks are performed for the estimated parameters at theactual or initial value declared for estimation (ML value is alsopossible);

No list of estimated parameters:

the program computes the local identification checks for allthe model parameter values declared in the DYNARE modelfile.



Features

Monte Carlo exploration

When information about prior distribution is provided, a fullMonte Carlo analysis is also possible.

For a number of parameter sets sampled from priordistributions, the local identification analysis is performed inturn. This provides a ‘global’ prior exploration of localidentification properties of DSGE models.

This is a ‘glocal’ procedure.

This Monte Carlo mode can also be linked to the the globalsensitivity analysis toolbox, also available in the officialDYNARE as of version 4.3.



Features

The optional Monte Carlo implementation I

a sample from Θ is made of many randomly drawn pointsfrom Θ′, where Θ ∈ Θ′ discarding values of θ that do notimply a unique solution.

The set Θ′ contains all values of θ that are theoreticallyplausible, and may be constructed by specifying a lower andan upper bound for each element of θ or by directly specifyingprior distributions.

After specifying a distribution for θ with support on Θ′, onecan obtain points from Θ by drawing from Θ′ and removingdraws for which the model is either indetermined or does nothave a solution. Conditions for existence and uniqueness areautomatically checked by DYNARE.



Features

The optional Monte Carlo implementation II

The checks for local identification are then performed for eachparameter set in turn, obtaining a Monte Carlo sample ofidentification features for the model under analysis.



Features

The Advanced option (point)

analysis of the LRE form and the reduced form;

identification patterns (Iskrev, 2010b; Andrle, 2010).



Features

The Advanced option (Monte Carlo)

analysis of the condition number of the Jacobians J(q), J2and JΓ and detection of the parameters that mostly drivelarge condition numbers (i.e. weaker identification);

analysis of the identification patters across the Monte Carlosample;

detailed point-estimate (identification strength and collinearityanalysis) of the parameters set having the smallest/largestcondition number;

when some singularity (rank condition failure) is detected forsome elements of the Monte Carlo sample, detailedpoint-estimates are performed for such critical points.



Features

Test routines

A library of test routines is also provided in the official DYNAREtest folder. Such tests implement some of the examples describedin the present document.

Kim (2003) : the DYNARE routines for this example are placedin the folderdynare_root/tests/identification/kim;

An and Schorfheide (2007) : the DYNARE routines for thisexample are placed indynare_root/tests/identification/as2007;



DYNARE procedure

DYNARE procedure

A new syntax is available in DYNARE. The simple keywordidentification(<options>=<values>);

triggers the point local identification checks, performed at the priormean. Prior definitions and the list of observed values are needed,using the standard DYNARE syntax for setting-up an estimation.



DYNARE procedure

DYNARE procedure: Options I

parameter set = prior mode | prior mean |posterior mode | posterior mean | posterior median.Specify the parameter set to use. Default: prior_mean.

prior_mc = INTEGER sets the number of Monte Carlo draws(default = 1); prior_mc=1 triggers the default pointidentification analysis; prior_mc>1 triggers the Monte Carlomode;

prior_range = INTEGER triggers uniform sample within therange implied by the prior specifications (when prior_mc>1).Default: 0

load_ident_files = 0, triggers a new analysis, whileload_ident_files = 1, loads and displays a previouslyperformed analysis (default = 0);



DYNARE procedure

DYNARE procedure: Options II

ar = <integer> (default = 3), triggers the value for q incomputing J(q);

useautocorr: this option triggers J(q) in the form ofauto-covariances and cross-covariances (useautocorr = 0),or in the form of auto-correlations and cross-correlations(useautocorr = 1). The latter form normalizes all mq

entries in [−1, 1] (default = 0).

advanced = INTEGER triggers standard or advancedidentification analysis (default = 0).

max_dim_cova_group = INTEGER In the brute force search(performed when advanced=1) this option sets the maximumdimension of groups of parameters that best reproduce thebehavior of each single model parameter. Default: 2



DYNARE procedure

DYNARE procedure: Options III

periods = INTEGER triggers the length of the stochasticsimulation to compute the analytic Hessian. Default: 300

periods = INTEGER When the analytic Hessian is notavailable (i.e. with missing values or diffuse Kalman filter orunivariate Kalman filter), this triggers the length of stochasticsimulation to compute Simulated Moments Uncertainty.Default: 300

replic = INTEGER When the analytic Hessian is notavailable, this triggers the number of replicas to computeSimulated Moments Uncertainty. Default: 100.



DYNARE procedure

DYNARE procedure: Options IV

gsa_sample_file = INTEGER If equal to 0, do not usesample file. If equal to 1, triggers GSA prior sample. If equalto 2, triggers GSA Monte-Carlo sample (i.e. loads a samplecorresponding to pprior=0 and ppost=0 in thedynare_sensitivity options). Default: 0

gsa_sample_file = FILENAME Uses the provided path to aspecific user defined sample file. Default: 0



Notes

Notes I

DYNARE symbolic preprocessor interprets and implementsthe model definitions as expressed in the DYNARE file;

It will not reflect all parameter definitions which may behidden in the <>_steadystate.m file.

The # syntax should be used in the model block of theDYNARE file, instead.


Possible extensions

Possible extensions IKomunjer and Ng (2011a):

use structural properties of the canonical solution of DSGEmodels and restrictions implied by observational equivalenceto derive rank conditions that do not require knowledge ofinfinite autocovariances or Markov parameters;

like Iskrev, their conditions do not depend on data but also donot depend on the estimator (i.e. do not rely on Gaussianassumptions about shocks) nor on T (i.e. apply to T → ∞);


Possible extensions

Possible extensions II

useful framing in terms of controllability/observability andminimal state space representation (control systemengineering);

no big practical advance w.r.t. the present approach foridentification purposes, but useful additional diagnostics toanalyse DSGE models.

use numerical derivatives.


Possible extensions

Possible extensions I

Koop et al. (2011) suggest two Bayesian identification indicators:

is a subset of params is known to be identified, the marginalposterior is compared to the posterior expectation of the priorconditionasl on the identified parameters;

considers the rate at which posterior precision gets updated asthe sample size T is increased: for unidentified parameters,rate of update is smaller than T .


Examples

Examples


Examples

Kim (2003)

Kim (2003)

This paper demonstrated a functional equivalence between twotypes of adjustment cost specifications, coexisting inmacroeconomic models with investment: intertemporal adjustmentcosts which involve a nonlinear substitution between capital andinvestment in capital accumulation, and multisectoral costs whichare captured by a nonlinear transformation between consumptionand investment.We reproduce results of Kim (2003), worked out analytically,applying the DYNARE procedure on the non-linear form of themodel.


Examples

Kim (2003)

The representative agent maximizes

∞∑

t=0

βt logCt (16)

subject to a national income identity and a capital accumulationequation:

(1− s)( Ct

1− s

)1+θ+ s

( It

s

)1+θ= (AtK

αt )

1+θ (17)

Kt+1 =

[

δ

(

It

δ

)1−φ

+ (1− δ)K 1−φt

]1

1−φ

(18)

where s = βδα∆ , ∆ = 1− β + βδ.


Examples

Kim (2003)

φ(≥ 0) is the inverse of the elasticity of substitution betweenIt and Kt (represents the size of intertemporal adjustmentcosts)

θ(≥ 0) is the inverse of the elasticity of transformationbetween consumption and investment (called the multisectoraladjustment cost parameter). θ .

the two adjustment cost parameter only enter through an‘overall’ adjustment cost parameter Φ = φ+θ

1+θ , thus implyingthat they cannot be identified separately.


Examples

Kim (2003)

On the effect of the number of states.

Assume to have an additional equation for the Lagrange multiplier

λt =(1−s)θ

(1+θ)C(1+θ)t

, with λt entering the Euler equation.

We still assume that only Ct and It can be observed.


Examples

An Schorfheide (2007)

An and Schorfheide (2007)

yt = Et [yt+1] + gt − Et [gt+1]− 1/τ · (Rt − Et [πt+1]− Et [zt+1])(19)

πt = βEt [πt+1] + κ(yt − gt) (20)

Rt = ρRRt−1 + (1− ρR)ψ1πt + (1− ρR)ψ2(∆yt + zt) + εR,t

(21)

gt = ρggt−1 + εg ,t (22)

zt = ρzzt−1 + εz ,t (23)

where yt is GDP in efficiency units, πt is inflation rate, Rt isinterest rate, gt is government consumption and zt is change intechnology.


Examples


The model is completed with three observation equations forquarterly GDP growth rate (YGRt), annualized quarterly inflationrates (INFt) and annualized nominal interest rates (INTt):

YGRt = γQ + 100 ∗ (yt − yt−1 + zt) (24)

INFLt = πA + 400πt (25)

INTt = πA + rA + 4γQ + 400Rt (26)

where β = 11+rA/400

.


Examples


Bibliography I

Sungbae An and Frank Schorfheide. Bayesian analysis of DSGEmodels. Econometric Reviews, 26(2-4):113–172, 2007.DOI:10.1080/07474930701220071.

Michal Andrle. A note on identification patterns in DSGE models(august 11, 2010). ECB Working Paper 1235, 2010. Availableat SSRN: http://ssrn.com/abstract=1656963.

Fabio Canova and Luca Sala. Back to square one: identificationissues in DSGE models. Journal of Monetary Economics, 56(4),May 2009.

Nikolay Iskrev. Evaluating the strenght of identification in DSGEmodels. an a priori approach. unpublished manuscript, 2010a.

Nikolay Iskrev. Local identification in DSGE models. Journal ofMonetary Economics, 57:189–202, 2010b.


Examples


Bibliography II

Jinill Kim. Functional equivalence between intertemporal andmultisectoral investment adjustment costs. Journal of Economic

Dynamics and Control, 27(4):533–549, February 2003. URLhttp://ideas.repec.org/a/eee/dyncon/v27y2003i4p533-549.htm

Ivana Komunjer and Serena Ng. Dynamic identification of DSGEmodels. Econometrica, 2011a. accepted.

Ivana Komunjer and Serena Ng. Global identification in nonlinearmodels with moment restrictions. Econometric Theory, 2011b.accepted.

Gary Koop, Hashem Pesaran, and Ron Smith. On identification ofDSGE models. mimeo, 2011.

http://ideas.repec.org/a/eee/dyncon/v27y2003i4p533-549.html


Examples


Bibliography III

M. Ratto and N. Iskrev. Computational advances in analyzingidentification of DSGE models. 6th DYNARE Conference, June3-4, 2010, Gustavelund, Tuusula, Finland, 2010a. Bank ofFinland, DSGE-net and Dynare Project at CEPREMAP.

M. Ratto and N. Iskrev. Identification toolbox for DYNARE. 1stMONFISPOL conference, London, 4-5 November 2010, 2010b.The London Metropolitan University and the European Researchproject (FP7-SSH) MONFISPOL.

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