inference in bayesian proxy-svars · this is a restricted ngn density over the structural...

Introduction Methodology Application I Application II Conclusion

Inference in Bayesian Proxy-SVARs∗

Jonas Arias1 Juan F. Rubio-Ramırez2,3,4 Daniel F. Waggoner3

1Federal Reserve Bank of Philadelphia

2Emory University

3Federal Reserve Bank of Atlanta

4BBVA Research

March 25, 2019

∗The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta, theFederal Reserve Bank of Philadelphia, or the Federal Reserve System.

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Introduction

• Proxy-SVARs aim to identify one or more structural shocksby augmenting SVARs with instruments which must be cor-related with the shocks of interest and uncorrelated with allother shocks

• Proxy-SVARs have become influential in empirical macro

See e.g. Stock and Watson (12), Montiel-Olea, Stock and Watson

(12), Mertens and Ravn (13), Mertens and Montiel-Olea (18)

• Most studies work in the frequentist paradigm. Exceptions are

Bahaj (14), Drautzburg (16), and Caldara and Herbst (16)

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Introduction

• This paper makes two novel methodological contributions

Efficient algorithms to independently draw from the family ofNGN over the structural parameterization of a proxy-SVAR

Proxies may not be enough to identity a proxy-SVAR

Set identification

. Combine external instruments with sign and zero restrictions

. Restrictions unrelated to the proxies can be added

• We revisit Mertens and Montiel-Olea (QJE ’18)

Some conclusions change when using less restrictive assumptions

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Setup and Notation

• Consider the SVAR

y′tA0 =

p∑`=1

y′t−`A` + c + ε′t for 1 ≤ t ≤ T

• y ′t = [y ′t m′t ] is a vector of endogenous variables and instruments

Let the n × 1 vector yt denote the endogenous variables

Let the k × 1 vector mt denote the instruments

• A` is a (n + k)× (n + k) matrix of parameters for 0 ≤ ` ≤ p

• A0 invertible, c is a 1× (n + k) vector of parameters

• ε′t = [ε′t ν′t ], where εt is n × 1 and νt is k × 1

• εt is conditionally standard normal

• p is the lag length, and T is the sample size

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Setup and Notation

• The SVAR can be written as

y′tA0 = x′tA+ + ε′t for 1 ≤ t ≤ T

where x ′t = [y ′t−1 · · · y ′t−p 1] and A′+ = [A′1 · · · A′p c ′]

• We will begin by imposing two types of restrictions on (A0, A+)

Block restrictions

Exogeneity restrictions

• We turn to this next

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Block Restrictions

• We assume that yt evolves according to

y ′tA0 = x ′tA+ + ε′t ,

• This implies that

A` =

[A` Γ`,1

0k,n Γ`,2

], for 0 ≤ ` ≤ p

where Γ`,1 is n × k and Γ`,2 is k × k

• We call these zero restrictions on A` the block restrictions

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A Proxy-SVAR

• SVAR + block restrictions (bc) = Proxy-SVAR

• (A0, A+) + bc = Proxy-SVAR structural parameters

• The proxy-SVAR structural parameters (A0, A+) and (A0, A+)are observationally equivalent if and only if A0 = A0Q andA+ = A+Q, for some matrix Q ∈ Q ⊂ O(n), where O(n) isthe set of n × n orthogonal matrices and Q is defined by Q =Q ∈ O(n)|Q = diag(Q1,Q2),Q1 ∈ O(n), and Q2 ∈ O(k)

• A proxy-SVAR is not identified

If (A0, A+) are proxy-SVAR structural parameters then(A0Q, A+Q) for Q ∈ Q are observationally equivalentproxy-SVAR structural parameters

• Proxy-SVARs are identified using exogeneity restrictions

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Exogeneity Restrictions (I/II)

• We assume that mt are proxies for the last k shocks in εt andthat they are uncorrelated with the first n − k shocks in εt

E[mtε′t ] = [0k×(n−k) Vk×k ]

• If (A0, A+) are proxy-SVAR structural parameters(A−10

)′=

[ (A−10

)′0n,k

−(A−10 Γ0,1Γ

−10,2

)′ (Γ−10,2

)′]

• Multiplying the proxy-SVAR by A−10 , the last k equations are

m′t = x ′tA+

[−A−10 Γ0,1Γ

−10,2

Γ−10,2

]− ε′tA−10 Γ0,1Γ

−10,2 + ν ′tΓ

−10,2

• Thus

E[mtε′t ] = −

(A−10 Γ0,1Γ

−10,2

)′8/49


Exogeneity Restrictions (II/II)

• Hence

(A−10

)′=

(A−10

)′0n,k

E[mtε′t ](Γ−10,2

)−1 =

[ (A−10

)′0n,k

[0k×(n−k) Vk×k ](Γ−10,2

)′]

• Exogeneity restrictions: the lower left-hand k × (n − k) block

of(A−10

)′must be zero

• We also need Vk×k to be full rank

Following the literate we refer to this as the relevance condition

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Our Prior

• The proxy-SVAR structural parameters have a prior density pro-portional to NGN(ν,Φ,Ψ,Ω)(A0, A+), where

NGN(ν,Φ,Ψ,Ω)(A0, A+) ∝ | det(A0)|ν−ne−12vec(A0)′Φ vec(A0)

e−12(vec(A+)−Ψ vec(A0))′Ω−1(vec(A+)−Ψ vec(A0))

• This is a restricted NGN density over the structural represen-tation of the proxy-SVAR because it is a normal-generalized-normal density over Rn2+mn conditional on the block restric-tions, where m = pn + 1

• Conjugate family of priors commonly used in the literature

Sims and Zha (1998)

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Our Posterior

• Since our prior is conjugate, the posterior is also a restrictedNGN distribution over proxy-SVAR structural parameters andit is proportional to

NGN(ν, Φ, Ψ, Ω)

where

ν = T + ν

Ω = (In ⊗ X ′X + Ω−1)−1

Ψ = Ω(In ⊗ X ′Y + Ω−1Ψ)

Φ = In ⊗ Y ′Y + Φ + Ψ′Ω−1Ψ− Ψ′Ω−1Ψ

Y = [y1 · · · yT ]′ and X = [x1 · · · xT ]′

• One may choose another prior/posterior . . . not necessarily anissue

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Simulation

• ARRW showed how to independently draw from the restrictednormal-generalized-normal posterior distribution over the struc-tural representation for the case of a SVAR identified with signand zero restrictions

• One would like to use ARRW algorithm. But, the techniques ofthat paper cannot be directly applied here

• Example: n = 3, k = 1 and p = 1

Maximum number of zero restrictions using ARRW = 6

Number of proxy-SVAR restrictions ARRW = 8

• Even so, the techniques can be adapted for proxy-SVARs

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Simulation• The key insight in ARRW

Draw the reduced-form parameters from NIW of choice

Draw from set of orthogonal matrices such that zerorestrictions hold

Map the draws to the structural parameterization and takeappropriate care of the volume elements

IS to draw structural parameters from a NGN of choice

• This paper

Draw the triangular-block parameters from a NGN of choice

Draw from set of orthogonal matrices such that exogeneityrestrictions hold

Map the draws to the structural parameterization of theProxy-SVAR and take appropriate care of the volume elements

IS to draw structural parameters from a NGN of choice

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Orthogonal-Triangular-Block Parameterization (I/III)

• This parameterization is characterized by four matrices

1. An (n + k)× (n + k) upper-triangular matrix Λ0

2. A (p(n + k) + 1)× (n + k) matrix Λ+

. Λ′+ = [Λ′1 · · · Λ′p d ′], where Λi is (n + k)× (n + k) with thelower left-hand k × n block equal to zero, and d is 1× (n + k)

3. An n × n orthogonal matrix Q1

4. A k × k orthogonal matrix Q2

• (Λ0, Λ+,Q1,Q2) maps to structural parameters satisfying theblock restrictions by

(Λ0, Λ+,Q1,Q2)f−→ (Λ0 diag(Q1,Q2)︸︷︷︸

A0

, Λ+ diag(Q1,Q2)︸︷︷︸A+

),

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Orthogonal-Triangular-Block Parameterization (II/III)

• The mapping f has an inverse

(A0, A+)f −1

−→ (A0P︸︷︷︸Λ0

, A+P︸︷︷︸Λ+

, P ′1︸︷︷︸Q1

, P ′2︸︷︷︸Q2

).

where

P is such that A−10 = PR is the QR-decomposition of A−10

The lower left-hand k × n block of A−10 is zero, and hence

. P = diag(P1,P2), where P1 is n × n and P2 is k × k

Λ0 is upper triangular because A0P = R−1

The lower left-hand k × n block of each Λi is zero because

. P is block diagonal

. The lower left-hand k × n block of Ai is zero

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Orthogonal-Triangular-Block Parameterization (III/III)• We can write the proxy-SVAR as

y ′tΛ0 = x ′tΛ+ + u′t ,

where u′t = ε′t diag(Q1,Q2)′ are conditionally normal

• Our ultimate goal is drawing the proxy-SVAR structuralparameters from a NGN posterior of choice

1. We draw orthogonal-triangular-block parameters from aposterior distribution

2. We map them to the structural parameterization

3. We calculate the volume element of the mapping

4. Note that (1)-(3) imply we will be drawing from a posteriorover the structural parameterization of the proxy-SVAR, butnot the NGN posterior of choice

5. Because we know the volume element we can use an importantsampling to draw from the NGN posterior of choice

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The Algorithm

• We independently draw triangular-block parameters(

Λ0, Λ+

)from a NGN posterior distribution

• Given a draw of(

Λ0, Λ+

)the exogeneity restrictions are

linear restrictions on the columns of the orthogonal matrix Q1

Thus, we use ideas in ARRW to independently draw orthogonalmatrices (Q1,Q2) such that the exogeneity restrictions hold

• Then, we use f to map (Λ0, Λ+,Q1,Q2) into (A0, A+)

• These draws are not from the NGN posterior of choice

But, we will be able to numerically compute the densityassociated with the implied distribution

• We can use those draws as an intermediate step in animportance sampler to draw from the desired distribution

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Independent Draws of the Triangular-Block Parameters

• We use the Gibbs sampler of Waggoner and Zha (’03) to indepen-dently draw from a restricted normal-generalized-normal pos-terior distribution over the triangular-block parameters charac-terized by NGN(ν, Φ, Ψ, Ω)

• The triangular and block restrictions on (Λ0, Λ+) satisfy theconditions to use the sampler

• Since Λ0 is upper triangular, the Gibbs draws are independent

• Often, it suffices to chose (ν, Φ, Ψ, Ω) to be equal to (ν, Φ, Ψ, Ω)

However, sometimes this can lead to small effective samplesizes in our sampler. Hence, a more tailored choice of(ν, Φ, Ψ, Ω) is needed in such cases

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Exogeneity Restrictions Linear over Q1

• Let J = [0k×n Ik ], the exogeneity restrictions are of the form

J(A−10 )′en,j = 0k×1 for 1 ≤ j ≤ n − k,

• This is equivalent to

J(Λ−10 )′ diag(Q1,Q2)en,j = J(Λ−10 )′L′︸︷︷︸G(Λ0)

Q1en,j = 0k×1

for 1 ≤ j ≤ n − k , where L = [ In 0n,k ]

• We will denote the number of exogeneity restrictions on thej th column of Q1 by zj , which is k if 1 ≤ j ≤ n− k and is zeroif n − k < j ≤ n

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Algorithm 11. Draw (Λ0, Λ+) independently from the restricted NGN(ν, Φ, Ψ, Ω)

distribution

2. For 1 ≤ j ≤ n, draw x1,j ∈ Rn+1−j−zj from a N (0, 1), set w1,j = x1,j/ ‖ x1,j ‖3. For 1 ≤ j ≤ k, draw x2,j ∈ Rk+1−j from a N (0, 1), set w2,j = x2,j/ ‖ x2,j ‖

4. Define Q1 = [q1,1 · · · q1,n] recursively by q1,j = K1,jw1,j for any matrixK1,j whose columns form an orthonormal basis for the null space of the(j − 1 + zj)× n matrix

M1,j =

[q1,1 · · · q1,j−1 G(Λ0)′

]′for 1 ≤ j ≤ n − k[

q1,1 · · · q1,j−1

]′for n − k + 1 ≤ j ≤ n

5. Define Q2 = [q2,1 · · · q2,k ] recursively by q2,j = K2,jw2,j for any matrixK2,j whose columns form an orthonormal basis for the null space of the(j − 1)× k matrix

M2,j =[q2,1 · · · q2,j−1

]′for 1 ≤ j ≤ k.

6. Set (A0, A+) = f (Λ0, Λ+,Q1,Q2).

7. Return to Step 1 until the required number of draws has been obtained.

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Density Implied by Algorithm 1 (I/II)

• Step 1 independently draws (Λ0, Λ+) from a restricted NGN

• Step 2 draws w1,j from the uniform distribution on the unitsphere in Rn+1−j−zj

• Step 3 draws w2,j from the uniform distribution on the unitsphere in Rk+1−j

• The density implied by Steps 1-3 is ∝ a restricted NGN

• Step 4 and 5 map (Λ0, Λ+,w1,1, · · · ,w1,n,w2,1, · · · ,w2,k) to(Λ0, Λ+,Q1,Q2)

• Step 6 maps (Λ0, Λ+,Q1,Q2) into (A0, A+) using f

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Density Implied by Algorithm 1 (II/II)

• The composite mapping implied by Steps 1 to 6 together withTheorem 3 in ARRW will be used to compute the densityimplied the Algorithm

• Let Z denotes the set of all proxy-SVAR structural parametersthat satisfy the exogeneity restrictions

• By Theorem 3 in ARRW, the posterior density overproxy-SVAR structural parameters subject to the exogeneityrestrictions implied by the algorithm is proportional to

NGN(ν,Φ,Ψ,Ω)(Λ0, Λ+)v(gf −1)|Z (A0, A+),

where (Λ0, Λ+,Q1,Q2) = f −1(A0, A+)

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Algorithm 2

1. Use Algorithm 1 to independently draw (A0, A+)

2. Set its importance weight to

NGN(ν,Φ,Ψ,Ω)(A0, A+)

NGN(ν,Φ,Ψ,Ω)(Λ0, Λ+)v(gf−1)|Z (A0, A+),

where (Λ0, Λ+,Q1,Q2) = f −1(A0, A+) and Z denotes the set ofall proxy-SVAR structural parameters that satisfy the exogeneityrestrictions

3. Return to Step 1 until the required number of draws has beenobtained

4. Re-sample with replacement using the importance weights

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The Need of Additional Restrictions

• Exogeneity restrictions separate the structural shocks into twoblocks

• If we only use the exogeneity restrictions, we have an identifi-cation problem within the set of the structural shocks that arecorrelated with the proxies unless k = 1

• The same problem occurs within the set of the structural shocksthat are not correlated with the proxies

• Let (A0, A+) and (A0, A+) be proxy-SVAR parameters thatsatisfy the exogeneity restrictions and the relevance condition,then (A0, A+) and (A0, A+) are obs equivalent iff there existsa matrix Q ∈ X ⊂ Q ⊂ O(n) such that A0 = A0Q and A+ =A+Q, where X is defined by X = Q ∈ Q|Q = diag(Q3,Q4,Q5),Q3 ∈ O(n − k) ,Q4 ∈ O(k), and Q5 ∈ O(k)

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Additional Zero Restrictions (I/II)

• Let Zj be a zj × r matrix of full row rank

• Assume that the zero restrictions in the proxy-SVAR structuralparameterization are ZjF (A0, A+)en,j = 0zj ,1 for 1 ≤ j ≤ n

• The zero restrictions in the orthogonal-triangular-blockparameterization are

ZjF (f (Λ0, Λ+,Q1,Q2))en,j= ZjF (f (Λ0, Λ+, In, Ik)) diag(Q1,Q2)en,j

= 0zj ,1 for 1 ≤ j ≤ n

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Additional Zero Restrictions (II/II)

• The zero restrictions can be written as

ZjF (f (Λ0, Λ+, In, Ik))[I ′n 0′k,n

]′︸︷︷︸G1,j (Λ0,Λ+)

Q1en,j = 0zj ,1 for 1 ≤ j ≤ n

• Again, linear restrictions on Q1

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Algorithm 3

1. Draw (Λ0, Λ+) independently from the restricted NGN(ν, Φ, Ψ, Ω)

2. For 1 ≤ j ≤ n, draw x1,j ∈ Rn+1−j−zj−zj from a N (0, 1), set w1,j = x1,j/ ‖ x1,j ‖3. For 1 ≤ j ≤ k, draw x2,j ∈ Rk+1−j−zn+j from a N (0, 1), set w2,j = x2,j/ ‖ x2,j ‖

4. Define Q1 = [q1,1 · · · q1,n] recursively by q1,j = K1,jw1,j for any matrixK1,j whose columns form an orthonormal basis for the null space of the(j − 1 + zj + zj)× n matrix

M1,j =

[q1,1 · · · q1,j−1 G(Λ0)′ G1,j(Λ0, Λ+)′

]′for 1 ≤ j ≤ n − k[

q1,1 · · · q1,j−1 G1,j(Λ0, Λ+)′]′

for n − k + 1 ≤ j ≤ n

5. Define Q2 = [q2,1 · · · q2,k ] recursively by q2,j = K2,jw2,j for any matrixK2,j whose columns form an orthonormal basis for the null space of the(j − 1 + zn+j)× k matrix

M2,j =[q2,1 · · · q2,j−1

]′for 1 ≤ j ≤ k.

6. Set (A0, A+) = f (Λ0, Λ+,Q1,Q2)

7. Return to Step 1 until the required number of draws has been obtained

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Density Implied by Algorithm 3

• We proceed as when computing the density implied byAlgorithm 1

• Let X denote the set of all proxy-SVAR structural parametersthat satisfy the proxy and the zero additional restrictions

• The density over the structural parameterization of theproxy-SVAR subject to the proxy and the zero additionalrestrictions implied by the algorithm is proportional to

NGN(ν,Φ,Ψ,Ω)(Λ0, Λ+)v(gf −1)|X (A0, A+),

where (Λ0, Λ+,Q1,Q2) = f −1(A0, A+)

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Additional Sign and Zero Restrictions

• Let Sj be an sj × r matrix of full row rank

• Sj defines the additional sign restrictions on the j th structuralshock. In particular,

SjF (A0, A+)en,j > 0sj ,1

• F continuous

• At least one parameter value satisfy the sign restrictions

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Algorithm 4

1. Use Algorithm 3 to independently draw (A0, A+).

2. If (A0, A+) satisfies the sign restrictions, set its importance weightto

NGN(ν,Φ,Ψ,Ω)(A0, A+)

NGN(ν,Φ,Ψ,Ω)(Λ0, Λ+)v(gf−1)|X (A0, A+),

where (Λ0, Λ+,Q1,Q2) = f −1(A0, A+) where X denotes the set ofall proxy-SVAR structural parameters that satisfy the proxy and theadditional zero restrictions.

3. Return to Step 1 until the required number of draws has beenobtained.

4. Re-sample with replacement using the importance weights.

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The Dynamic Effects of TFP Shocks• The proxy-SVAR features 5 endogenous variables and 2

proxies. Quarterly data: 1947Q2-2015Q4

• The endogenous variables are

1. Real GDP growth2. Employment growth3. Inflation4. Real consumption (non-durables and services) growth5. Real investment (equipment + durables consumption) growth

• The proxies are

1. A proxy for consumption TFP shocks2. A proxy for investment TFP shocks

• Prior: NGN(ν,Φ,Ψ,Ω)(A0, A+). We set ν = n, Φ = 0n,n,

Ψ = 0mn,n2 and Ω−1 = 0mn,mn. And, we set ν = ν, Φ = Φ,

Φ = Φ and Ω−1 = Ω−1.

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The Dynamic Effects of TFP ShocksIdentification Restrictions

• The relevance condition and the exogeneity restrictions are

E[mtε

TFP′t

]= V 6= 02×2 and E

[mtε

O′t

]= 02×3

where m′t = [mC ,t mI ,t ] are the proxies for the consumptionand investment TFP shocks.

• We also impose the following sign restrictionsE [mC ,tεC ,t ] > 0, E [mI ,tεI ,t ] > 0, E [mC ,tεC ,t ] > E [mC ,tεI ,t ]and E [mI ,tεI ,t ] > E [mI ,tεC ,t ] on the entries of V .

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The Dynamic Effects of TFP Shocks

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The Dynamic Effects of Personal Tax Rates ShocksRoadmap

1. Replicate Mertens and Montiel-Olea (’18) using our algorithmwhen there is only one proxy

2. Replicate Mertens and Montiel-Olea (’18) with both AMTR andATR proxies assuming AMTR does not respond to ATR shock

3. Combine Mertens and Montiel-Olea (’18) instruments with signrestrictions

4. Compare effects of income specific marginal tax rate shocksusing Mertens and Montiel-Olea (’18) instruments andidentification scheme versus sign restrictions.

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Specification to Study AMTR Shocks (I/II)

• The proxy-SVAR features 9 endogenous variables, 2exogenous variables, and 1 proxy. Yearly data: 1946-2012


1. The negative of log net of-tax-rate2. Log of reported income level3. Log of real GDP per tax unit4. The unemployment rate5. Log of real stock market index6. Inflation7. The federal funds rate8. Log of real government spending per tax unit9. Change in log real federal government debt per tax unit

35/49


Specification to Study AMTR Shocks (II/II)

• Exogenous variables: Dummy variables for 1949 and 2008

• Proxy: Collection of instances of variation in marginal taxrates considered to be contemporaneously exogenous changesin AMTR

• Prior: NGN(ν,Φ,Ψ,Ω)(A0, A+). We set ν = n, Φ = 0n,n,Ψ = 0mn,n2 and Ω−1 = 0mn,mn

• We normalize the sign of the AMTR shock by assuming thatthe IRF of the AMTR is negative in response to a negativeAMTR shock, thus we use Algorithm 2 to obtain an effectivesample size of 10,000

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AMTR shock (rate cut)One proxy for AMTR shock. Replicates Figure 5 from Mertens and Montiel-Olea (’18)

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Specification to Study AMTR and ATR Shocks (I/II)

• The proxy-SVAR features 10 endogenous variables, 2exogenous variables, and 2 proxies. Yearly data: 1946-2012


1. The negative of log net of-tax-rate2. Log of reported income level3. Log of real GDP per tax unit4. The unemployment rate5. Log of real stock market index6. Inflation7. The federal funds rate8. Log of real government spending per tax unit9. Change in log real federal government debt per tax unit

10. Log of ATR

38/49


Specification to Study AMTR and ATR Shocks (II/II)

• Proxies: Mertens and Montiel-Olea (’18) identify AMTR and ATRshocks using two proxies. Analogously to the case of theAMTR proxy, the new proxy (which we call the ATR proxy) isa collection of instances of variation in average tax rates thatthe authors reasonably consider to be contemporaneouslyexogenous changes in the ATR

• Mertens and Montiel-Olea (’18) introduce an additional zerorestriction

1. e′2A0e1 = 0 (AMTR ordered first)2. e′1A0e2 = 0 (ATR ordered first)

• If AMTR ordered first, this means AMTR cannot reactcontemporaneously to the ATR

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Effects of AMTR and ATR shocksReplicates Panels (A) and (B) in Figure 10 of Mertens and Montiel-Olea (’18)

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Effects of AMTR and ATR shocks

• These figures clearly justify sentences like these one

• There is, on the other hand, no evidence for any effect onincomes when average tax rates decline but marginal rates donot (See Mertens and Montiel-Olea (’18) page 2.)

• The main finding is that, in sharp contrast to the results formarginal tax rate changes after controlling for average taxrates, there is no evidence that income responds strongly toaverage tax rate changes once marginal rate changes arecontrolled for. The point estimates are in fact slightlynegative, although they are not statistically significant at anyhorizon. (See Mertens and Montiel-Olea (’18) page 35.)

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Effects of AMTR and ATR shocksIdentification

• Proxies are only correlated with AMTR and ATR shocks

• To separate the shocks, we use sign restrictions

AMTR proxy is positively correlated with AMTR shock

ATR proxy is positively correlated with ATR shock

Covariance between AMTR shock and proxy larger thancovariance between ATR shock and AMTR proxy

Covariance between ATR shock and proxy is larger thancovariance between AMTR shock and ATR proxy

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Effects of AMTR and ATR shocksMertens and Montiel-Olea (’18) without Additional Exclusion Restriction

43/49


Specification to Study Income Specific AMTR Shocks(I/II)

• The proxy-SVAR features 11 endogenous variables, 4exogenous variables, and 2 proxies. Yearly data: 1946-2012


1. The negative of log net of-tax-rate for the top 1%2. The negative of log net of-tax-rate for the bottom 99%3. Log of reported income level for the top 1%4. Log of reported income level for the bottom 99%5. Log of real GDP per tax unit6. The unemployment rate7. Log of real stock market index8. Inflation9. The federal funds rate

10. Log of real government spending per tax unit11. Change in log real federal government debt per tax unit

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Specification to Study Income Specific AMTR Shocks(II/II)

• Exogenous variables: Dummy variables for 1949 and 2008 aswell as a linear and a quadratic trend to capture long trends inincome inequality

• Proxies: Newly built disaggregated measures of exogenousvariation in tax rates across the income distribution as proxiesfor top and bottom marginal tax rate shocks

• Mertens and Montiel-Olea (’18) identification

1. C21 : Proxy restriction + e′2A0e1 = 0 (AMTR Bottom 99%ordered first)

2. C12 : Proxy restriction + e′1A0e2 = 0 (AMTR Bottom 99%ordered first)

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Effects of income specific MTR shocksIRFs to a one std dev MTR shock of the Top 1% (Gray) and Bottom 99% (Red)

Replicates Figures 11 and 12 from Mertens and Montiel-Olea (’18)46/49



Replicates Figures 11 and 12 from Mertens and Montiel-Olea (’18)47/49



Mertens and Montiel-Olea (’18) without Additional Exclusion Restriction48/49


Conclusions

In this paper we develop:

• An efficient algorithm for Bayesian inference

• The algorithm gives independent posterior draws of thestructural parameters

• Can be efficiently applied with priors from the NGN family,which is an often used conjugate family of priors

• Perhaps most importantly the external instruments can easilybe combined with sign and zero restrictions

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inference in bayesian proxy-svars · this is a restricted ngn density over the structural...

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