DSGE Methods - Wirtschaftswissenschaftliche Fakultät · Willi Mutschler (Institute of Econometrics) DSGE Methods Summer 2014 19 / 30. Identi cation of DSGE-models Qu and Tkachenko’s

DSGE MethodsIdentification of DSGE models

Willi Mutschler, M.Sc.

Institute of Econometrics and Economic StatisticsUniversity of Munster

[email protected]

Summer 2014

Willi Mutschler (Institute of Econometrics) DSGE Methods Summer 2014 1 / 30

Identification

Identification problem

Distinct parameter values do not lead to distinct probabilitydistribution of data

Source of identification influences findings

Lack of identification leads to wrong conclusions from calibration andestimation

Identification of DSGE models

concerned with two mappingsuniqueness of solution

→ from the deep parameters to the reduced-form parameters

uniqueness of probability distribution

→ from the solution to observable data

Bayesian approach

circumvents badly shaped likelihood by using tight priorscomparison of prior and posterior can be misleading


Identification

DSGE context: formal identification criteria via

(i) the autocovariogram (Iskrev 2010)(ii) the spectral density (Komunjer and Ng 2011 & Qu and Tkachenko

2012)(iii) Bayesian indicators (Koop, Pesaran, and Smith 2012)


Identification

Basic idea

Question of uniqueness of a solution, i.e. injectivity of functions

Formally, given an objective function f(θ) a sufficient condition for θ0

being globally identified is given by

f(θ1) = f(θ0)⇒ θ1 = θ0 for any θ1 ∈ Θ

If this is only true in an open neighborhood of θ0, the identification ofθ0 is local

Population moments (mean, variance, autocovariance, spectrum) arefunctions of data

⇒ Check, whether mapping from θ to population moments is unique

Conditions are in general only sufficient, unless data is generated froma Gaussian distribution (very common for DSGE)


IdentificationApproaches to time series analysis

Time series analysis: try to explain regular behavior/patterns

Time domain approach

Idea: Regression of the present on the past

Objective: Identify bunch of parameters

Fundamental representation via autocovariances or Wold decomposition

Frequency domain approach

Idea: Regression of the present on periodic sines and cosines, i.e. decompositioninto regular components

Regularity: periodic variations expressed as Fourier frequencies, driven by sines andcosines

Objective: Identify dominant frequencies in a series

Fundamental representation via spectral density


IdentificationApproaches to time series analysis

ω = 6 100 A2 = 13

Time

0 20 40 60 80 100

−10

−5

05

10

ω = 10 100 A2 = 41

Time

x2

0 20 40 60 80 100

−10

−5

05

10

ω = 40 100 A2 = 85

0 20 40 60 80 100

−10

−5

05

10

sumx

0 20 40 60 80 100

−15

−5

515


Identification of DSGE-models

Iskrev’s approach


IdentificationIskrev’s approach

Iskrev (2010)’s approach:

Idea: Check whether derivative of the mean and the predictedautocovariogram of observables w.r.t deep parameters has full rank→ Time domain approach

Collect for t = 0, 1, . . . ,T − 1 all elements in a vector

m(θ,T ) :=(µ′d vech(Σd)′ vec(Σd(1))′ . . . vec(Σd(T − 1))′

)′If m(θ, q) is a continuously differentiable function of θ, then θ0 islocally identifiable if the Jacobian

M(q) :=∂m(θ0, q)

∂θ′

has full column rank at θ0 for q ≤ T , i.e. equal to nθ.



Iskrev (2010)’s necessary condition:

Stack all elements of the mean and the solution matrices that dependon θ into a vector τ :

τ(θ) :=(µ′d vec(hx)′ vec(gx)′ vech(ηxη

′x)′ vech(ηdη

′d)′)′

and consider the factorization

M(q) =∂m(θ, q)

∂τ(θ)′∂τ(θ)

∂θ′.

θ0 is locally identifiable if the rank of J := ∂τ(θ0)∂θ′ at θ0 is equal to nθ.

This condition is, however, only necessary, because τ may beunidentifiable.



Implementation and interpretation:

Compute derivatives analytically or numerically

If θ is identifiable, M(q) and is likely to have full rank for q muchsmaller than T

Rank deficiency: Evaluating the nullspace one can pinpointproblematic parameters



Komunjer and Ng’s approach


IdentificationKomunjer and Ng’s approach

Based upon results from control theory for minimal systems(observable input and output sequences):

minimality and invertibility are enough to characterize observationalequivalence of spectral densities

Consider minimal DSGE model, i.e. dynamics are entirely driven bythe smallest possible dimension of the state vector (and shocks)

Derive restrictions implied by equivalent spectral densities withoutcomputing any autocovariances or the spectral density

→ (Time and) Frequency domain approach



Consider minimal DSGE model

x2,t = x2 + A(x2,t−1 − x2) + Bεt ,

dt = d + C(x2,t−1 − x2) + Dεt .

i.e. smallest possible dimension nx2 of the state vector such thatdynamics are entirely driven by x2,t and εt .

(i) Controllability: For any initial state, it is always possible to design aninput sequence that puts the system in the desired final state, i.e. the

matrix[B AB . . . Anx2

−1B]

has full row rank,

(ii) Observability: Given the evolution of the input it is always possible toreconstruct the initial state by observing the evolution of the output,

i.e. the matrix[C′ A′C′ . . . Anx2

−1′ C′]′

has full column rank.



Since dt is weakly stationary and εt is either white noise or iid:

dt = d + Det +∞∑j=1

CAj−1Bεt−j = d + He(L−1; θ)εt ,

where L is the lag-operator. For z ∈ C the transfer function(z-transform) is

Hε(z ; θ) := D +∞∑j=1

CAj−1Bz−j = D + C[Inx2− Az

]−1B.



Collect all hyperparameters of the state space solution into a vector

Λ(θ) :=(vec(A)′, vec(B)′, vec(C)′, vec(D)′, vech(Σε)

′)′

For all z ∈ C the spectral density matrix of dt is defined as

Ωd (z; θ) := Σd +∞∑j=1

Σd (j)z−j +∞∑j=1

Σd (−j)z−j = Hε(z; θ)Σε(θ)Hε(z−1; θ)′.

Observational equivalence of θ0 and θ1 is equivalent to ∀z ∈ C :

Hε(z; Λ(θ1)) · Σε(θ1) · Hε(z−1; Λ(θ1))′ = Hε(z; Λ(θ0)) · Σε(θ0) · Hε(z−1; Λ(θ0))′

implies θ0 = θ1.Equivalent spectral densities arise if

(i) for given Σε(θ), each transfer function Hε(z ; Λ(θ)) is potentially

obtained from a multitude of quadruples (A, B, C, D),

(ii) there are many pairs Hε(z ; Λ(θ)) and Σε(θ) that jointly generate thesame spectral density.



Komunjer and Ng (2011) show that this is equivalent to the existenceof a nx2 × nx2 similarity transformation matrix T and a nε × nε fullcolumn rank matrix U = Lε(θ0)VLε(θ1)−1 such that

A(θ1) = T A(θ0)T−1, B(θ1) = T B(θ0)U, C(θ1) = C(θ0)T−1

D(θ1) = D(θ0)U, Σε(θ1) = U−1Σε(θ0)U−1′ ,

with Lε being the Cholesky decomposition of Σε(θ) = LεL′ε and V a

constant matrix such that VV ′ = I

⇒ Now we have mappings!



Define a continuously differentiable mapping

δ : Θ× Rn2x2 × Rn2

ε → RnΛ as

δ(θ,T ,U) :=

vec(T A(θ)T−1)

vec(T B(θ)U)

vec(C(θ)T−1)

vec(D(θ0)U)

vech(U−1Σε(θ0)U−1′)

.

θ is now locally identifiable from the spectral density (or equivalentlyautocovariances) of dt at a point θ0 if and only if δS(θ,T ,U) islocally injective at (θ0, Inx2

, Inε)

Sufficient condition: matrix of partial derivatives of δS(θ,T ,U) hasfull column rank at (θ0, Inx2

, Inε)



∆(θ0) :=(∂δ(θ0,Inx2

,Inε )

∂θ′∂δ(θ0,Inx2

,Inε )

∂vec(T )′∂δ(θ0,Inx2

,Inε )

∂vec(U)′

)

=

∂vec(A)∂θ′ A′ ⊗ Inx2

− Inx2⊗ A 0n2

x2×n2ε

∂vec(B)∂θ′ B′ ⊗ Inx2

Inε ⊗ B∂vec(C)∂θ′ −Inx2

⊗ C 0ndnx2×n2ε

∂vec(D)∂θ′ 0ndnx2

×n2x2

Inε ⊗ D∂vech(Σe )∂θ′ 0(nε(nε+1)/2)×n2

x2−2Ξnε [Σe ⊗ Inε ]

=:(

∆Λ(θ0) ∆T (θ0) ∆U(θ0))

with Ξnε being the left-inverse of the n2ε + nε(nε + 1)/2 duplication matrix Gnε for vech(Σe)

Order and rank condition

Order (necessary): nθ + n2x2

+ n2ε ≤ nS

Λ := (nx2 + nd)(nx2 + nε) + nε(nε + 1)/2

Rank (necessary and sufficient): rank(∆s(θ0)) = nθ + n2x2

+ n2ε



Implementation and interpretation

Analytical or numerical derivatives

Derive minimal state vector: Check observability and controllability

Remove the entries from all solution matrices and its derivativescorresponding to redundant state variables

Model diagnostics via other submatrices

Nullspace can be used to pinpoint problematic parameters



Qu and Tkachenko’s approach


IdentificationQu and Tkachenko’s approach

Idea: Check whether derivative of the spectrum of observables w.r.tthe deep parameters has full rank → Frequency domain approach

Assume that dt is covariance-stationary:

dt = d +∞∑j=0

Dgxhjxσηxεt−j + ηdεt = d + Hε(L; θ)εt

with Hε(L; θ) = Dgx (Inx − hxL)−1 σηx + ηd

Using the Fourier transformation the spectral density matrix Ωd is

Ωd(ω, θ) =1

2πHε(e

−iω; θ) · Σε(θ) · Hε(e−iω; θ)∗, ω ∈ [−π;π],

with ∗ denoting the conjugate transpose of a complex valued matrix



The test focuses on

G(θ0)︸︷︷︸nθ×nθ

=

∫ π

−π

(∂vec(Ωd (ω; θ0))′

∂θ′

)′ (∂vec(Ω(ω; θ0))′

∂θ′

)dω +

∂µd (θ0)′

∂θ

∂µd (θ0)

∂θ′.

θ is locally identifiable at a point θ0 from the mean and spectrum ofdt if and only if G (θ0) is nonsingular, that is its rank is equal to nθ.




Derivative of Ωd(ω; θ0) can be calculated analytically or numerically,but numerically approximate integral

Full rank G or G : identification via spectrum or mean and spectrum

Rank deficiency: check whether the diagonal elements of G or G arenonzero


Exercise: Identification of An andSchorfheide (2007)


Exercise: Identification of An and Schorfheide (2007)

Consider the An and Schorfheide (2007) model

1 Write a mod file for the log-linearized model.

2 Run and interpret Dynare’s identification command.

3 Harder: Implement the other criteria based on ranks using numericalderivatives. Compare the results.


Bayesian identification criteria


Bayesian identification criteriaKoop, Pesaran, and Smith (2013)’s approach

Koop, Pesaran, and Smith (2013, JBES)’s approach

Indicates weak identification using Bayesian simulation techniques(Bayesian-learning-rate-indicator)

Rate at which the posterior precision of a given parameter getsupdated with the sample size T using simulated data

For not identified or weakly identified parameters this rate is slowerthan T


Bayesian identification criteriaKoop, Pesaran, and Smith (2013)’s approach


Simulate data, divide into growing subsamples

Calculate variance or precision via

Hessian (analytically or numerically) at posterior modeMCMC variance of posterior distribution

Precision divided by sample size will go to

zero for weakly identified parametersa constant for identified parameters


Comparison


Comparison

Methods differ due to different perspectives and assumptions:

Iskrev: time domain, very general assumptions (moments), computederivative of autocovariogramQu/Tkachenko: frequency domain, very general assumptions (VMArepresentation), compute derivative of spectrumKomunjer/Ng: (time and) frequency domain, strictest assumptions(minimality and left-invertibility), conditions without computingautocovariances or spectral densityKPS: Bayesian simulation approach

Computational issues and numerical errors

If feasible: Use analytical derivatives rather than numericalIsk: lag length, QT: subintervals for integral, KN: Find minimalrepresentation, KPS: Filtering, SpeedFor all: numerical instability of solution algorithm, size of matrices

Methods include different model diagnostics


Documents

DSGE Methods - Wirtschaftswissenschaftliche Fakultät · Willi Mutschler (Institute of Econometrics) DSGE Methods Summer 2014 19 / 30. Identi cation of DSGE-models Qu and Tkachenko’s