A Class of Time-Varying Parameter Structural VARs for Inference under Exact or Set Identiﬁcation

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A Class of Time-Varying Parameter Structural VARs for Inference under

Exact or Set Identification

Mark Bognanni¹

1Federal Reserve Bank of Cleveland

January 27, 2018 Norges Bank

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Boring Fed disclaimer

The views expressed in this presentation are not necessarily those of the Federal Reserve Bank of Cleveland or the Board of Governors of the Federal Reserve System or its staff.

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To fix ideas

y⁰_t

(1×n)

A

(n×n)

=y⁰_t−1F₁

(n×n)

+· · ·+y⁰_t−pF_p+ c

(1×n)

+ ε⁰_t

(1×n)

, ε_t ∼N(0,I_n)

Define

x_t ≡[y⁰_t−1, . . . ,y⁰_t−p,1]⁰ and F≡[F⁰₁, . . . ,F⁰_p,c⁰]⁰ Write

y_t⁰A=x⁰_tF+ε⁰_t Want to infer (A,F) because they

• represent equilibrium relationships between variables

• determine response ofyt to the mutually orthogonal

“structural” shocks in ε_t But (A,F) don’t come for free.

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The identification problem

Rewriting the SVAR

y_t⁰ =x⁰_tFA⁻¹+ε⁰_tA⁻¹, Likelihood fory_t

p(y_t|A,F,y_t−p:t−1) =Npdf(y_t|x⁰_tFA⁻¹

| {z }

µ

,(AA⁰)⁻¹

| {z }

Σ

)

But consider the alternative parameter point (eA,F)e (eA,eF) = (AQ,FQ) forQ∈ O_n

µ=eFAe⁻¹ =FQ(AQ)⁻¹ =FQQ⁻¹A⁻¹ =FA⁻¹ Σ=AeAe⁰ = (AQ)(AQ)⁰ =AQQ⁰A⁰ =AA⁰ Hence, we cannot identify (A,F).

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The reduced-form VAR

We can identify

g(A,F) = (FA⁻¹,AA⁰) = (B,Σ)

y⁰_t =x_tB+u⁰_t, u_t ∼N(0,Σ) Key practical feature:

• Easy to estimate (B,Σ) Key drawback:

• (Σ,B) are not (A,F)

Most traditional approaches to estimating (A,F) construct a one-to-one mapping from (A,F) to (Σ,B).

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The literature since then

1 Set identification (with static VAR parameters):

• Canova and de Nicolo (2002)

• Uhlig (2005)

2 Coefficients that change (with exact identification)

• Cogley and Sargent (2005)

• Primiceri (2005)

• Sims and Zha (2006), Sims, Waggoner and Zha (2008) Not obvious how to coherently combine these approaches.

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A Motivating Example

• Based on Baumeister and Peersman (2013, AEJ Macro)

• y_t = [∆p_t^oil,∆q^oil_t ,∆GDP_t,∆p_t^CPI]⁰

• Identify time-varying IRFs of oil supply shocks Their method:

• Estimate Primiceri (2005) VAR-TVP-SV

• Reassemble into “reduced-form VAR” parameters t-by-t

• Find structural parameters satisfying sign-restrictions εôil,s_t <0⇒ ∆q_t+hôil <0<∆p_t+hôil for h= 0, ...,4

• RRWZ “algorithm” applied to “reduced-form” parameters t-by-t.

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“Reduced-form”

Primiceri (2005)

y⁰_t =vec(B_t)⁰(I_n⊗x_t) +ε⁰_tΞ_t∆⁻¹_t where

Ξ_t =







ξ_1,t 0 · · · 0 0 ξ_2,t . .. ... ... . .. ... 0 0 · · · 0 ξ_n,t







, ∆_t =







1 δ_12,t · · · δ_1n,t 0 1 . .. ...

... . .. ... δn−1n,t

0 · · · 0 1







and

Ξ_t =Ξt−1diag(exp(η_t)), η_t ∼N(0n×1,Σ_η)

δ_t =δt−1+ζ_t, ζ_t ∼N(0n(n−1)

2 ×1,Σ_ζ)

vec(B_t) = vec(Bt−1) +υ_t, υ_t ∼N(0mn×1,Σ_υ)

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• Supply shock causing

∆q^oil =−1%.

• “baseline” IRFs

• x-axis: time in quarters

• p^oil_t IRF:

contemporaneous response at eacht

• GDP_t and ∆p_t IRFs:

cumulative change over 4 quarters at eacht

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∆q^oil =−1%.

• Finding: oil demand has become

increasingly inelastic

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A Motivating Example

• y_t = [∆p_t^oil,∆q^oil_t ,∆GDP_t,∆p_t^CPI]⁰

• Identify time-varying IRFs of oil supply shocks The method:

• RRWZ “algorithm”

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A Motivating Example Revisited

• y_t = [∆p_tôil,∆qôil_t ,∆GDP_t,∆p_t^CPI]⁰ y_t = [∆p_t^CPI,∆GDP_t,∆qôil_t ,∆p_tôil]⁰

• Identify time-varying IRFs of oil supply shocks The method:

• RRWZ “algorithm”

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∆q^oil =−1%.

• IRFs under alternative variable ordering

• Time-variation in IRFs is gone!

• Would have been a different paper!

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Takeaway from the exercise

• Not that Baumeister Peersman are “wrong.”

(Indeed, I will find something similar them).

But

• Methodologically, the BP method is deeply problematic.

• The “reduced-form” can be sensitive to variable ordering.

• Spills over into any inference based on the “reduced-form”

Key resulting shortcomings:

1 Results driven as much by an unacknowledged modeling choice (variable ordering) as by the explicit identifying assumptions.

2 n! different candidate reduced-forms.

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Examining the posterior I

LetS_t = (A_t,F_t) and S_t∗Q_t = (A_tQ_t,F_tQ_t) p(φ,S_0:T|y_1:T)∝p(φ,S₀)

| {z }

prior

p(S_1:T|φ,S₀)

| {z }

density of theS_1:T sequence under the model’s law of motion

p(y_1:T|φ,S₀,S_1:T)

| {z }

data density givenS_0:T

where

p(y_1:T|φ,S₀,S_1:T)=

T

Y

t=1

p(y_t|yt−p:t−1,S_t)

=

T

Y

t=1

p_N(y_t| x⁰_tF_tA⁻¹_t

| {z }

x⁰_tFtQtQ⁻¹_t A⁻¹_t

, (A_tA⁰_t)⁻¹

| {z }

(AtQtQ⁰_tA⁰_t)⁻¹

)

⇒In each t, S_t∗Q_t gives same evaluation of this term as S_t.

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Examining the posterior II

p(φ,S_0:T|y_1:T)∝p(φ,S₀)

| {z }

prior

p(S_1:T|φ,S₀)

| {z }

density of theS_1:T sequence under the model’s law of motion

p(y_1:T|φ,S₀,S_1:T)

| {z }

data density givenS_0:T

where

p(S_1:T|φ,S₀)=

T

Y

t=1

p(S_t|φ,S_t−1)

(This is the tricky part.)

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This paper

• Let’s try something else.

• I define a class of models with laws of motion for S_t such that:

1 whole sequences of S_0:T have densities invariant to orthogonal rotations

2 yield a shared reduced-form Key benefits

• Time-varying parameter model amenable to identification driven by RRWZ conditions/algorithms.

• (Also, more straightforward to estimate.)

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This paper

• Let’s try something else.

• I define a class of models with laws of motion for S_t such that:

1 whole sequences of S_0:T have densities invariant to orthogonal rotations

2 yield a shared reduced-form Key benefits

• Time-varying parameter model amenable to identification driven by RRWZ conditions/algorithms.

• (Also, more straightforward to estimate.)

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Outline

1 A new SVAR with dynamic parameters

2 Reduced-form Representation

3 Structural Inference Revisited

4 Revisiting the time-varying oil demand elasticity

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Extending the SVAR

y⁰_tA_t =x⁰_tF_t+ε⁰_t, ε_t ∼N(0,I_n) Law of motion and stochastic processes for (A_t,F_t):

(A_t,F_t)∼p(A_t−1,F_t−1,φ)

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Extending the SVAR

y⁰_tA_t =x⁰_tF_t+ε⁰_t, ε_t ∼N(0,I_n) Law of motion for (A_t,F_t):

A_t =β^−1/2At−1Ω_t F_t =F_t−1A⁻¹_t−1A_t+Θ_t . Shocks:

Ω_t =L_th(Γ_t)R_t , Γ_t ∼B_n(β/(2(1−β)),1/2) Θ_t ∼MN_m,n(0,W,I_n)

where

β∈[(n−1)/n,1]

L_t,R_t ∈ O_n

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Detour: alternate form of SVAR

y⁰_t =x⁰_tBt+ε⁰_tQ⁰_th(Ht)⁻¹, εt ∼N(0,In) Law of motion for (A_t,F_t) = (B_t,H_t,Q_t) :

h(H_t)Q_t =β^−1/2h(Ht−1)Qt−1Ω_t B_th(H_t)Q_t =B_t−1h(H_t−1)Q_t+Θ_t

Qt =p(Qt|Bt,Ht) Shocks:

Ω_t =h(Γ_t) Γ_t ∼B_n(β/(2(1−β)),1/2) Θ_t ∼MN_m,n(0,W,I_n)

where

β ∈[(n−1)/n,1]

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Some notation

A Dynamic SVAR (call it DSVAR) denoted:

S_0:T^U (L_1:T,R_1:T) and let

φ= (β,W)

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Key result

Theorem (Theorem 1)

LetS_0:T^U (L_1:T,R_1:T) have prior p(φ,S₀) for which p(φ,S₀) =p(φ,S₀∗P) for anyP∈ O_n.

For anyQ_0:T such that each Q_t ∈ O_n, the model

S_0:T^U (eL_1:T,Re_1:T)defined by (eL_t,Re_t) = (Q⁰_t−1L_t,R_tQ_t) is such that, for every pointS_0:T, the point Se_0:T =S_0:T ∗Q_0:T satisfies

p(φ,S_0:T|y_1:T,S_0:T^U (L_1:T,R_1:T))

=p

φ,Se_0:T|y_1:T,S_0:T^U (eL_1:T,Re_1:T) .

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Theorem 1: restatement and implications

For

1 any realization of the data,

2 any dynamic structural VAR,

3 and anyQ_1:T

there exists an alternative model with the “same posterior” as the original model, but with each point rotated byQ_1:T.

• Set of equivalent models does not depend on y1:T

• ⇒All structural models in the class are observationally equivalent.

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Outline

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Reduced-form VAR with TVP-SV

Define (H_t,B_t) = g(S_t) = (A_tA⁰_t,F_tA⁻¹_t ) y⁰_t =x⁰_tB_t+u⁰_t, u_t ∼N(0,H⁻¹_t ) Laws of motion for (B_t,H_t):

H_t = 1

βh(Ht−1)⁰Γ_th(Ht−1) Bt =Bt−1+Vt

distributions of shocks (Γ_t,V_t)

Γ_t ∼Beta_n(β/(2(1−β)),1/2) Vt ∼MNm,n(0,W,H⁻¹_t )

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A short history of the reduced-form

The reduced-form model is “a known quantity.”

• Uhlig (1994, 1997) – the stochastic volatility part

• Mike West and coauthors – “dynamic linear model with discounted Wishart stochastic volatility,” (DLM-DWSV)

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Why does this work?

Suppose I’ve estimated the reduced-formH_0:T.

Shocks rationalizing movement fromA_t−1 to A_t satisfy, βA⁻¹_t−1 Ht

|{z}

AtA⁰_t

A⁻¹_t−1⁰ =Γt

Suppose instead my identification scheme said that int −1, Ae_t−1 =A_t−1Q_t−1.

Shocks rationalizing movement to H_t:

βA⁻¹_t−1H_tA⁻¹_t−1⁰ =Qt−1Γ_tQ⁰_t−1 = ˜Γ_t Critical thing: Γ_t and ˜Γ_t have the same density!

A property of the multivariate Beta distribution:

Srivastava (2003) Corollary 4.1,

p(Γ_t) = p(Q_tΓ_tQ⁰_t)

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Estimation of reduced-form

Need to characterize

p(β,W,B_0:T,H_0:T|y_1:T).

• Can’t characterize it analytically.

• Can construct an MCMC algorithm.

Gibbs Sampler

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• Block 2. p(β,B_0:T,H_0:T|y_1:T,W)

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Gibbs sampler: block 1

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• Super easy.

• If prior isW∼IW(Ψ0, ν0),

W|y1:T, β,B0:T,H0:T ∼IW( ¯Ψ,ν)¯ where

Ψ¯ =Ψ(y_1:T,B_0:T,H_0:T) +Ψ₀

¯

ν =Tn+ν₀

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Gibbs sampler: block 2

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• Factor joint density as p(β,B_0:T,H_0:T|y_1:T,W)

=p(β|y1:T,W)

| {z }

Block 2a

·p(B0:T,H0:T|y1:T, β,W)

| {z }

Block 2b

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Gibbs sampler: block 2

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• 2a. p(β|y_1:T,W)

• 2b. p(B_0:T,H_0:T|y_1:T, β,W)

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Gibbs sampler: block 2a

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• 2a. p(β|y_1:T,W)

• 2b. p(B_0:T,H_0:T|y_1:T, β,W)

Random-walk Metropolis-Hastings,

• “Propose” aβ^∗ ∼q(β^∗|β⁽ⁱ⁻¹⁾) =Npdf(β⁽ⁱ⁻¹⁾, σ²_β)

• Setβ^∗ =β⁽ⁱ⁾ with probability

α(β^∗|y_1:T,W) = min (

∝p(β^∗,W⁽ⁱ⁾)·p(y1:T|β^∗,W⁽ⁱ⁾)

z }| {

p β^∗,W⁽ⁱ⁾|y1:T

p(β⁽ⁱ⁻¹⁾,W⁽ⁱ⁾|y_1:T) , 1

)

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Gibbs sampler: block 2a

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• 2a. p(β|y_1:T,W)

• 2b. p(B_0:T,H_0:T|y_1:T, β,W)

Evaluatingα(β^∗|y_1:T,W) requires pointwise evaluation of p(y_1:T|β^∗,W⁽ⁱ⁾)

= Z

(H0:T,B0:T)

p(y_1:T|β^∗,W⁽ⁱ⁾,H_0:T,B_0:T)p(H_0:T,B_0:T)d(H_0:T,B_0:T)

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Block 2a: evaluating p(y

_1:T

|β

^∗

, W

⁽ⁱ⁾

)

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Block 2b: simulation smoother

• Block 1. p(W|y_1:T, β,B_0:T,H_0:T)

• 2a. p(β|y_1:T,W)

• 2b. p(B_0:T,H_0:T|y_1:T, β,W)

Analogous to Kalman smoother.

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Outline

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From reduced-form back to structural

Given

1 restriction regions R_t for each t

2 and posterior samples {H⁽ⁱ⁾_0:T,B⁽ⁱ⁾_0:T,φ⁽ⁱ⁾}^Nsim_i=1 one can

1 construct a sequence of arbitrary (A⁽ⁱ_0:T⁾ ,F⁽ⁱ⁾_0:T) consistent with (H⁽ⁱ⁾_0:T,B⁽ⁱ⁾_0:T) period-by-period

2 t-by-t, find Q⁽ⁱ_t⁾∈ O_n such that (A⁽ⁱ⁾_t Q⁽ⁱ⁾_t ,F⁽ⁱ⁾_t Q⁽ⁱ⁾_t )∈ R_t.

3 Set (eA⁽ⁱ_t⁾,Fe⁽ⁱ⁾_t ) = (A⁽ⁱ⁾_t Q⁽ⁱ_t⁾,F⁽ⁱ⁾_t Q⁽ⁱ⁾_t ) Note,Q⁽ⁱ⁾_t can be constructed via:

• Algorithm 1 of RRWZ (exact id), or

• Algorithm 2 of RRWZ (set id)

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Outline

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Prior vs. Posterior: β

0.0 0.2 0.4 β 0.6 0.8 1.0

0102030405060

Prior Posterior

Density

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∆q^oil =−1%.

• IRFs under alternative variable ordering

• Results frommy model.

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Concluding Remarks

Main contributions:

1 Developed a new class of SVAR with time-varying parameters amenable to a variety of identification methods.

• All models in the class have the same reduced-form representation.

2 Developed an MCMC algorithm for the fully-Bayesian estimation of the reduced-form model.

3 Applied to set identification of a time-varying object of interest about the effect of oil supply shocks.

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Appendix

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Outline

5 More on the Density of latent states

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Dynamic parameters

Now suppose

(A_t,F_t)∼p(φ,At−1,Ft−1) We lose everything.

1 No easy “reduced-form” to estimate or analyze.

2 (Without part 1 who cares?).

But most importantly, the same basic approach isn’t on the table anymore. Why?

Lack of observational equivalence between alternative rotated sequences ofstructural parameters.

Some notatation before we go on:

S_t = (A_t,F_t) S_t∗Q_t = (A_tQ_t,F_tQ_t)

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Multivariate Beta

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A Class of Time-Varying Parameter Structural VARs for Inference under Exact or Set Identiﬁcation