On the maximum and minimum response to an impulse in Svars e-Luis

Introduction
Notation
Projection
Calibration
Application
Conclusion
On the maximum and minimum response
to an impulse in Svars
Bulat Gafarov (PSU), Matthias Meier (UBonn), and José-Luis
Montiel-Olea (NYU)
September 23, 2015
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Introduction
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Introduction
? Structural VAR: Theoretical Restrictions imposed on a VAR.
(Sims [1980, 1986])
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
Σ = E[ηt ηt0 ]
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Introduction
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Projection
Calibration
Application
Conclusion
Introduction
? Structural VAR: Theoretical Restrictions imposed on a VAR.
(Sims [1980, 1986])
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
Σ = E[ηt ηt0 ]
? Goal: Ik,ij : (A1 , . . . Ap , Σ) 7→ IRFk,ij .
(response of variable i at horizon k to a ‘structural impulse’ j)
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Introduction
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Calibration
Application
Conclusion
Introduction
? Structural VAR: Theoretical Restrictions imposed on a VAR.
(Sims [1980, 1986])
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
Σ = E[ηt ηt0 ]
? Goal: Ik,ij : (A1 , . . . Ap , Σ) 7→ IRFk,ij .
(response of variable i at horizon k to a ‘structural impulse’ j)
? Restrictions =⇒ point identification or set identification.
(i.e., Ik,ij is one-to-one or one-to-many)
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Introduction
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Calibration
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Introduction
? Structural VAR: Theoretical Restrictions imposed on a VAR.
(Sims [1980, 1986])
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
Σ = E[ηt ηt0 ]
? Goal: Ik,ij : (A1 , . . . Ap , Σ) 7→ IRFk,ij .
(response of variable i at horizon k to a ‘structural impulse’ j)
? Restrictions =⇒ point identification or set identification.
(i.e., Ik,ij is one-to-one or one-to-many)
? We study SVARs that are set-identified with ±/0 restrictions.
(Faust [1998], Uhlig[2005])
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Introduction
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Projection
Calibration
Application
Conclusion
Introduction
? Structural VAR: Theoretical Restrictions imposed on a VAR.
(Sims [1980, 1986])
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
Σ = E[ηt ηt0 ]
? Goal: Ik,ij : (A1 , . . . Ap , Σ) 7→ IRFk,ij .
(response of variable i at horizon k to a ‘structural impulse’ j)
? Restrictions =⇒ point identification or set identification.
(i.e., Ik,ij is one-to-one or one-to-many)
? We study SVARs that are set-identified with ±/0 restrictions.
(Faust [1998], Uhlig[2005])
? Our context: ‘Prior-free’ inference in sign-restricted SVARs.
(Giacomini & Kitagawa [2015]-RBayes; MSG [2013]-M. Inequ.)
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This paper
? Uniformly consistent in level, frequentist confidence interval for:
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This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
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Introduction
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Calibration
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Conclusion
This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
2. The (i, j)-th structural impulse response function, up to k = h:
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
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Introduction
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Calibration
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Conclusion
This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
2. The (i, j)-th structural impulse response function, up to k = h:
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
? Strategy: focus on the bounds of the identified set for IRFk,ij .
(the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ))
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Introduction
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Projection
Calibration
Application
Conclusion
This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
2. The (i, j)-th structural impulse response function, up to k = h:
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
? Strategy: focus on the bounds of the identified set for IRFk,ij .
(the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ))
? Message: Frequentist inference is conceptually straightforward.
(and as general as Uhlig’s current Bayesian Approach)
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Introduction
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Projection
Calibration
Application
Conclusion
This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
2. The (i, j)-th structural impulse response function, up to k = h:
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
? Strategy: focus on the bounds of the identified set for IRFk,ij .
(the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ))
? Message: Frequentist inference is conceptually straightforward.
(and as general as Uhlig’s current Bayesian Approach)
Proposal: Projection Inference
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Introduction
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Projection
Calibration
Application
Conclusion
This paper
? Uniformly consistent in level, frequentist confidence interval for:
1. The k-th coefficient of the structural impulse-response fn:
IRFk,ij ∈ R
2. The (i, j)-th structural impulse response function, up to k = h:
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
? Strategy: focus on the bounds of the identified set for IRFk,ij .
(the maximum and minimum response, v k,ij (A, Σ), v k,ij (A, Σ))
? Message: Frequentist inference is conceptually straightforward.
(and as general as Uhlig’s current Bayesian Approach)
h
i
CS for θ ≡ (A, Σ) =⇒ min v k,ij (θ), max v k,ij (θ)
θ∈CS
θ∈CS
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Details
? maxθ∈CS v k,ij (θ) is a nonlinear mathematical program.
(with gradients that can be computed analytically)
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Details
? maxθ∈CS v k,ij (θ) is a nonlinear mathematical program.
(with gradients that can be computed analytically)
? Implementing projection requires us to solve such program.
(No numerical inversion of tests, no sampling from rotations)
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Details
? maxθ∈CS v k,ij (θ) is a nonlinear mathematical program.
(with gradients that can be computed analytically)
? Implementing projection requires us to solve such program.
(No numerical inversion of tests, no sampling from rotations)
? Projection is conservative (> 1-α), but can be ‘calibrated’.
(the degrees of freedom in CS for (A, Σ) can be adjusted)
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Introduction
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Projection
Calibration
Application
Conclusion
Details
? maxθ∈CS v k,ij (θ) is a nonlinear mathematical program.
(with gradients that can be computed analytically)
? Implementing projection requires us to solve such program.
(No numerical inversion of tests, no sampling from rotations)
? Projection is conservative (> 1-α), but can be ‘calibrated’.
(the degrees of freedom in CS for (A, Σ) can be adjusted)
? MC output provides a lower bound on the d.o.f. adjustment.
(this point is illustrated with economic applications)
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Introduction
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Projection
Calibration
Application
Conclusion
Details
? maxθ∈CS v k,ij (θ) is a nonlinear mathematical program.
(with gradients that can be computed analytically)
? Implementing projection requires us to solve such program.
(No numerical inversion of tests, no sampling from rotations)
? Projection is conservative (> 1-α), but can be ‘calibrated’.
(the degrees of freedom in CS for (A, Σ) can be adjusted)
? MC output provides a lower bound on the d.o.f. adjustment.
(this point is illustrated with economic applications)
? +/0 on only one shock, delta-method inference is available.
(we emphasized this in the previous version of the paper)
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Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
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Introduction
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Calibration
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Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
F-U: Numerical Bayes; BH: Analytic, Full-Bayesian Analysis.
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
5 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
MSG: Frequentist Bonferroni-M.I.; GK: Robust Bayesian
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
5 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
SVARs could be a nice application of these general papers!
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Brief Overview of Related Work
a) ‘Set Identified’ SVARs—Bayesian Approaches
Faust (1998); Uhlig (2005); Baumeister and Hamilton (2014);
b) ‘Set Identified’ SVARs—Non-Bayesian Approaches
Moon, Schorfheide, Granziera (2013); Giacomini, Kitagawa (2015)
c) Transformations of set-identified parameters
Bugni, Canay, Shi (2015); Kaido, Molinari, Stoye (2015)
SVARs could be a nice application of these general papers!
(Projection inference involved in MI models, but not in SVARs.)
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Introduction
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Projection
Calibration
Application
Conclusion
Outline
1. Notation and Main Assumptions
2. Projection Inference and Implementation in SVARs
3. ‘Calibrated’ Projection Inference: Idea
4. Illustrative Example: Unconventional MP
5. Conclusion
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Outline
1. Notation and Main Assumptions
2. Projection Inference and Implementation in SVARs
3. ‘Calibrated’ Projection Inference: Idea
4. Illustrative Example: Unconventional MP
5. Conclusion
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Outline
1. Notation and Main Assumptions
2. Projection Inference and Implementation in SVARs
3. ‘Calibrated’ Projection Inference: Idea
4. Illustrative Example: Unconventional MP
5. Conclusion
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Outline
1. Notation and Main Assumptions
2. Projection Inference and Implementation in SVARs
3. ‘Calibrated’ Projection Inference: Idea
4. Illustrative Example: Unconventional MP
5. Conclusion
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Outline
1. Notation and Main Assumptions
2. Projection Inference and Implementation in SVARs
3. ‘Calibrated’ Projection Inference: Idea
4. Illustrative Example: Unconventional MP
5. Conclusion
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Calibration
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1. Notation and Main Assumptions
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Gaussian SVAR(p)
? Vector Autoregression for the n-dimensional vector Yt :
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
A ≡ (A1 , A2 , . . . Ap )
and
Σ ≡ E[ηt ηt0 ].
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Gaussian SVAR(p)
? Vector Autoregression for the n-dimensional vector Yt :
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
A ≡ (A1 , A2 , . . . Ap )
and
Σ ≡ E[ηt ηt0 ].
θ = (vec(A)0 , vech(Σ)0 )0
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Gaussian SVAR(p)
? Vector Autoregression for the n-dimensional vector Yt :
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
A ≡ (A1 , A2 , . . . Ap )
and
Σ ≡ E[ηt ηt0 ].
? ‘Structural’ Model for the vector of Forecast Errors ηt :
ηt = Hεt ,
H ∈ Rn×n ,
εt ∼ Nn (0, In ),
i.i.d.
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Gaussian SVAR(p)
? Vector Autoregression for the n-dimensional vector Yt :
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
A ≡ (A1 , A2 , . . . Ap )
and
Σ ≡ HH 0
? ‘Structural’ Model for the vector of Forecast Errors ηt :
ηt = Hεt ,
H ∈ Rn×n ,
εt ∼ Nn (0, In ),
i.i.d.
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Structural IRF
? Structural Impulse Response Functions (variable i, shock j)
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Structural IRF
? Structural Impulse Response Functions (variable i, shock j)
∂Yi,t+k
· ≡ IRFk,ij (A, H) = ei0 Ck (A) Hj
| {z }
∂εjt
1×n
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Structural IRF
? Structural Impulse Response Functions (variable i, shock j)
∂Yi,t+k
· ≡ IRFk,ij (A, H) = ei0 Ck (A) Hj
| {z }
∂εjt
1×n
Linear combination of the j-th column of H.
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Calibration
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Conclusion
Structural IRF
? Structural Impulse Response Functions (variable i, shock j)
∂Yi,t+k
· ≡ IRFk,ij (A, H) = ei0 Ck (A) Hj
| {z }
∂εjt
1×n
Linear combination of the j-th column of H.
? Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A
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Introduction
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Calibration
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Conclusion
Structural IRF
? Structural Impulse Response Functions (variable i, shock j)
∂Yi,t+k
· ≡ IRFk,ij (A, H) = ei0 Ck (A) Hj
| {z }
∂εjt
1×n
(ei is the i-th column of In )
? Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A
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(A, Σ) 7→ IRFk,ij is one-to-many
? The reduced-form parameters
θ = (A, Σ)
are compatible with any
IRFk,ij ≡ ei0 Ck (A)Hj
such that
Hj
satisfies
HH 0 = Σ
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(A, Σ) 7→ IRFk,ij is one-to-many
? The reduced-form parameters
θ = (A, Σ)
are compatible with any
IRFk,ij ≡ ei0 Ck (A)Hj
such that
Hj
satisfies
HH 0 = Σ
? The set is usually refined with other restrictions on H
(the most common are zero/sign restrictions)
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Identified set for IRFk,ij
n
Ik,ij (A, Σ) ≡ v ∈ R|v = ei0 Ck (A)Hj ,
0
HH = Σ,
±/0
o
rest. on H
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Identified set for IRFk,ij
n
Ik,ij (A, Σ) ≡ v ∈ R|v = ei0 Ck (A)Hj ,
0
HH = Σ,
ei00 Ck 0 (A)Hj 0
o
≥0
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Identified set for IRFk,ij
n
Ik,ij (A, Σ) ≡ v ∈ R|v = ei0 Ck (A)Hj ,
0
HH = Σ,
ei00 (H 0 )−1 ej
o
≥0
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The maximum response in the population
v k,ij (A, Σ) ≡ max ei0 Ck (A)Hj
H∈Rn×n
subject to
HH 0 = Σ
and
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The maximum response in the population
v k,ij (A, Σ) ≡ max ei0 Ck (A)Hj
H∈Rn×n
subject to
HH 0 = Σ
and
zero restrictions:
Z 0 (A, Σ)Hj 0 = 0,
Z (A, Σ) ∈ Rn×mz
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The maximum response in the population
v k,ij (A, Σ) ≡ max ei0 Ck (A)Hj
H∈Rn×n
subject to
HH 0 = Σ
and
zero restrictions:
Z 0 (A, Σ)Hj 0 = 0,
Z (A, Σ) ∈ Rn×mz
sign restrictions:
S 0 (A, Σ)Hj 00 ≥ 0,
S(A, Σ) ∈ Rn×ms
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Obviously . . .
i
h
Ik,ij (A, Σ) ⊆ v k,ij (A, Σ) , v k,ij (A, Σ)
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And this is almost all we need to know to conduct inference on IRFk,ij
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2. Projection Inference and its implementation
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2.1 Main Idea
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What is it that we are looking for?
? Want to construct CST : Data → Subs(R) such that:
lim inf inf
inf
Pθ IRFk,ij ∈ CST ≥ 1 − α
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
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Introduction
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Calibration
Application
Conclusion
What is it that we are looking for?
? Want to construct CST : Data → Subs(R) such that:
lim inf inf
inf
Pθ IRFk,ij ∈ CST ≥ 1 − α
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? And we know that (uniformly over Θ):
√
d
T (θbT − θ) → Ndθ (0, Ω(θ))
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Introduction
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Calibration
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Conclusion
What is it that we are looking for?
? Want to construct CST : Data → Subs(R) such that:
lim inf inf
inf
Pθ IRFk,ij ∈ CST ≥ 1 − α
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? And we know that (uniformly over Θ):
√
d
T (θbT − θ) → Ndθ (0, Ω(θ))
? Thus, a CS is available for the reduced-form parameters:
n
o
b −1 (θbT −θ) ≤ χ2 (1−α)
CS(θbT ; 1−α) ≡ θ ∈ Θ|T (θbT −θ)0 Ω
dθ
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Introduction
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Calibration
Application
Conclusion
What is it that we are looking for?
? Want to construct CST : Data → Subs(R) such that:
lim inf inf
inf
Pθ IRFk,ij ∈ CST ≥ 1 − α
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? And we know that (uniformly over Θ):
√
d
T (θbT − θ) → Ndθ (0, Ω(θ))
? Thus, a CS is available for the reduced-form parameters:
n
o
b −1 (θbT −θ) ≤ χ2 (1−α)
CS(θbT ; 1−α) ≡ θ ∈ Θ|T (θbT −θ)0 Ω
dθ
? And this CS is uniformly valid of level (1 − α):
lim inf inf Pθ θ ∈ CS(θbT ; 1 − α) ≥ 1 − α
T →∞ θ∈Θ
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Projection Confidence Set for IRFk,ij
? A conceptually straightforward CS for IRFk,ij uses projection:
h
i
max
CSPT ≡
min
v k,ij (θ),
v k,ij (θ)
θ∈CS(θbT ,1−α)
θ∈CS(θbT ,1−α)
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Conclusion
Projection Confidence Set for IRFk,ij
? A conceptually straightforward CS for IRFk,ij uses projection:
h
i
max
CSPT ≡
min
v k,ij (θ),
v k,ij (θ)
θ∈CS(θbT ,1−α)
θ∈CS(θbT ,1−α)
? Build a confidence set for θ, and evaluate the max/min.
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Introduction
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Projection
Calibration
Application
Conclusion
Projection Confidence Set for IRFk,ij
? A conceptually straightforward CS for IRFk,ij uses projection:
h
i
max
CSPT ≡
min
v k,ij (θ),
v k,ij (θ)
θ∈CS(θbT ,1−α)
θ∈CS(θbT ,1−α)
? Build a confidence set for θ, and evaluate the max/min.
? Proof: Note that for any IRFk,ij ∈ Ik,ij (θ):
Pθ IRFk,ij ∈ CSPT
≥ Pθ v k,ij (θ), v k,ij (θ) ∈ CSPT
≥ Pθ θ ∈ CSPT (θbT , 1 − α)
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Introduction
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Calibration
Application
Conclusion
Projection Confidence Set for IRFk,ij
? A conceptually straightforward CS for IRFk,ij uses projection:
h
i
max
CSPT ≡
min
v k,ij (θ),
v k,ij (θ)
θ∈CS(θbT ,1−α)
θ∈CS(θbT ,1−α)
? Build a confidence set for θ, and evaluate the max/min.
? Proof: Note that for any IRFk,ij ∈ Ik,ij (θ):
Pθ IRFk,ij ∈ CSPT
≥ Pθ v k,ij (θ), v k,ij (θ) ∈ CSPT
≥ Pθ θ ∈ CSPT (θbT , 1 − α)
? =⇒ lim inf T →∞ inf θ∈Θ inf IRFk,ij ∈Ik,ij (θ) Pθ IRFk,ij ∈ CST
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Introduction
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Projection
Calibration
Application
Conclusion
Projection Confidence Set for IRFk,ij
? A conceptually straightforward CS for IRFk,ij uses projection:
h
i
max
CSPT ≡
min
v k,ij (θ),
v k,ij (θ)
θ∈CS(θbT ,1−α)
θ∈CS(θbT ,1−α)
? Build a confidence set for θ, and evaluate the max/min.
? Proof: Note that for any IRFk,ij ∈ Ik,ij (θ):
Pθ IRFk,ij ∈ CSPT
≥ Pθ v k,ij (θ), v k,ij (θ) ∈ CSPT
≥ Pθ θ ∈ CSPT (θbT , 1 − α)
? ≥ lim inf T →∞ inf θ∈Θ Pθ θ ∈ CSPT (θbT , 1 − α) = 1 − α
18 / 58
Introduction
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Projection
Calibration
Application
Conclusion
v k,ij (·), v k,ij (·) are only required to be measurable fns. of θ!
19 / 58
Introduction
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Projection
Calibration
Application
Conclusion
2.2 Implementation
20 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The upper end of the projection CS
max v k,ij (A, Σ)
A,Σ
subject to
21 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The upper end of the projection CS
max v k,ij (A, Σ)
A,Σ
subject to
n
o
b −1 (θbT − θ) ≤ χ2 (1 − α)
θ ∈ θ | T (θbT − θ)0 Ω
dθ
θ = (vec(A)0 , vech(Σ)0 )0 ∈ Rdθ
21 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Since the maximum response is given by
v k,ij (A, Σ) ≡ max ei0 Ck (A)Hj
H∈Rn×n
subject to
22 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Since the maximum response is given by
v k,ij (A, Σ) ≡ max ei0 Ck (A)Hj
H∈Rn×n
subject to
HH 0 = Σ
and
zero restrictions:
Z 0 (A, Σ)Hj 0 = 0,
Z (A, Σ) ∈ Rn×mz
sign restrictions:
S 0 (A, Σ)Hj 00 ≥ 0,
S(A, Σ) ∈ Rn×ms
22 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The upper end of projection CS is the value of a NLP
max
A,Σ,H∈Rn×n
ei0 Ck (A)Hj
subject to
HH 0 = Σ
and
zero restrictions:
Z 0 (A, Σ)Hj 0 = 0,
Z (A, Σ) ∈ Rn×mz
sign restrictions:
S 0 (A, Σ)Hj 00 ≥ 0,
S(A, Σ) ∈ Rn×ms
CS:
b −1 (θbT − θ) ≤ χ2 (1 − α)
T (θbT − θ)0 Ω
dθ
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Introduction
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Projection
Calibration
Application
Conclusion
We solve this program using fmincon
24 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
fmincon finds local minimum even when dθ is large, say 100.
(as long as the CS for θ contains only stationary matrices)
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Introduction
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Projection
Calibration
Application
Conclusion
2.3 A Simple (Small-Scale) Toy Example
26 / 58
Introduction
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Projection
Calibration
Application
Conclusion
Demand-Supply SVAR HB(2015): two shocks (dθ = 7)
? Effect of a structural shock on labor demand over wages/emp?
(2-SVAR, 1 lag (AIC, BIC), Q1-1970/Q2-2014: ∆ ln wt , ∆ ln et .)
SR1: demand shock increases wt and empt
SR2: supply shock increases wt , but decreases empt
? wt : comprnfb; et : payems (St. Louis Fed).
27 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
19 seconds vs. 107.11 seconds
(laptop @2.4GHz IntelCore i7)
28 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
19 seconds vs. 107.11 seconds
(laptop @2.4GHz IntelCore i7)
Point estimator: .01 seconds
29 / 58
Introduction
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Projection
Calibration
Application
Conclusion
Frequentist CS are larger than Bayesian Credible Sets (no surprise)
(Moon and Schorfheide [ECMA; 2012])
30 / 58
Introduction
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Projection
Calibration
Application
Conclusion
Are they unnecessarily large?
31 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
3. ‘Calibrated’ Projection
32 / 58
Introduction
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Projection
Calibration
Application
Conclusion
The well-known problem of Projection
? Projection inference could be conservative, in the sense that:
lim inf inf
inf
Pθ IRFk,ij ∈ CSPT > 1 − α.
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
33 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The well-known problem of Projection
? Projection inference could be conservative, in the sense that:
lim inf inf
inf
Pθ IRFk,ij ∈ CSPT > 1 − α.
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? In fact, when the max/min are differentiable CSPT is approx:
q
q
h
χ2dθ (1 − α)
χ2dθ (1 − α) i
b
b
b
√
√
v k,ij (θT ) −
σ
b , v k,ij (θT ) +
σ
T
T
33 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The well-known problem of Projection
? Projection inference could be conservative, in the sense that:
lim inf inf
inf
Pθ IRFk,ij ∈ CSPT > 1 − α.
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? In fact, when the max/min are differentiable CSPT is approx:
q
q
h
χ2dθ (1 − α)
χ2dθ (1 − α) i
b
b
b
√
√
v k,ij (θT ) −
σ
b , v k,ij (θT ) +
σ
T
T
? But in this case, we would like to use something like:
h
1.64
v k,ij (θbT ) − √ σ
b
T
1.64 bi
, v k,ij (θbT ) + √ σ
T
33 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The well-known problem of Projection
? Projection inference could be conservative, in the sense that:
lim inf inf
inf
Pθ IRFk,ij ∈ CSPT > 1 − α.
T →∞ θ∈Θ IRFk,ij ∈Ik,ij (θ)
? In fact, when the max/min are differentiable CSPT is approx:
q
q
h
χ2dθ (1 − α)
χ2dθ (1 − α) i
b
b
b
√
√
v k,ij (θT ) −
σ
b , v k,ij (θT ) +
σ
T
T
? But in this case, we would like to use something like:
h
1.64
v k,ij (θbT ) − √ σ
b
T
1.64 bi
, v k,ij (θbT ) + √ σ
T
? Thus, when dθ = 186: “14” times std. errors vs. “1.64”.
33 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Calibrated Projection: Thought Experiment
? Asymptotically θbT is approximately Ndθ (θ, Ω(θ)/T ).
34 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Calibrated Projection: Thought Experiment
? Asymptotically θbT is approximately Ndθ (θ, Ω(θ)/T ).
? Hence, we can use this limiting DGP to compute:
Ck (d) ≡ inf
inf
Pθ IRFk,ij ∈ CSPT (θbT , d)
θ∈Θ IRFk,ij ∈Ik,ij (θ)
34 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Calibrated Projection: Thought Experiment
? Asymptotically θbT is approximately Ndθ (θ, Ω(θ)/T ).
? Hence, we can use this limiting DGP to compute:
Ck (d) ≡ inf
inf
Pθ IRFk,ij ∈ CSPT (θbT , d)
θ∈Θ IRFk,ij ∈Ik,ij (θ)
? Which depends on the degrees of freedom used in CS for θ.
34 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Calibrated Projection: Thought Experiment
? Asymptotically θbT is approximately Ndθ (θ, Ω(θ)/T ).
? Hence, we can use this limiting DGP to compute:
Ck (d) ≡ inf
inf
Pθ IRFk,ij ∈ CSPT (θbT , d)
θ∈Θ IRFk,ij ∈Ik,ij (θ)
? Which depends on the degrees of freedom used in CS for θ.
? Brute Force: ‘Calibrate’ the parameter d until Ck (d) = 1 − α.
(theoretically, all we need to show is continuity of Ck (d))
34 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Calibrated Projection: Thought Experiment
? Asymptotically θbT is approximately Ndθ (θ, Ω(θ)/T ).
? Hence, we can use this limiting DGP to compute:
Ck (d) ≡ inf
inf
Pθ IRFk,ij ∈ CSPT (θbT , d)
θ∈Θ IRFk,ij ∈Ik,ij (θ)
? Which depends on the degrees of freedom used in CS for θ.
? Brute Force: ‘Calibrate’ the parameter d until Ck (d) = 1 − α.
(theoretically, all we need to show is continuity of Ck (d))
? Equivalent to setting different MC exercises over a grid for θ.
(making the minimum coverage probability over MCs 1 − α)
34 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Practically, projection can be calibrated over a grid of values of θ
35 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Find dk (ΘG ) such that:
inf
inf
θ∈ΘG IRFk,ij ∈Ik,ij (θ)
Pθ IRFk,ij ∈ CSPT (θbT , dk (ΘG )) = 1 − α.
36 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
The calibrated dk (ΘG ) is a lower bound for dk (Θ)
(the grid can include only one point; namely θbT )
37 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Lower Bound on dk (Θ)
WAGE
EMP
8
calibrated degrees of freedom
calibrated degrees of freedom
8
6
4
2
0
6
4
2
0
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Grid of 1 point (θbT ), only integer values dk allowed.
(Time: 1.5 hours, using 25 parallel cores)
38 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Dotted: Calibrated Projection corresponding to dk (θbT )
39 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Dotted: Calibrated Projection corresponding to dk (θbT )
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Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Calibration over Θ would give lines between dashed and dotted.
41 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Projection is conservative for IRFk,ij , yes . . .
42 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
but even if we were to eliminate ‘projection bias’ . . .
43 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
95% CS gives different conclusions than a 95% Credible Set.
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Is projection also conservative for the impulse-response function?
45 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Projection can also be calibrated to cover
(IRF0,ij , IRF1,ij , . . . IRFh,ij ) ∈ Rh+1
46 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Dotted: Calibrated Projection for (IRF0,ij , . . . IRF20,ij ); d(θbT ) = 3
47 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
WAGE
EMP
5
4
4
3
3
% change
% change
5
2
2
1
1
0
0
-1
-1
0
5
10
Months after shock
15
20
0
5
10
15
20
Months after shock
Dotted: Calibrated Projection for IRFk,ij ; dk (θbT )
48 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
4. Empirical Application
49 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
An ‘Unconventional’ Monetary Shock
? What is an ‘unconventional’ monetary shock?
(forward guidance, large-scale asset purchase program)
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Introduction
Notation
Projection
Calibration
Application
Conclusion
An ‘Unconventional’ Monetary Shock
? What is an ‘unconventional’ monetary shock?
(forward guidance, large-scale asset purchase program)
? Hard to define exactly, but easy to state minimal properties
50 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
An ‘Unconventional’ Monetary Shock
? What is an ‘unconventional’ monetary shock?
(forward guidance, large-scale asset purchase program)
? Hard to define exactly, but easy to state minimal properties
? At the very minimum: shock that ↓ long term rates . . .
(sign restriction on the contemporaneous response)
50 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
An ‘Unconventional’ Monetary Shock
? What is an ‘unconventional’ monetary shock?
(forward guidance, large-scale asset purchase program)
? Hard to define exactly, but easy to state minimal properties
? At the very minimum: shock that ↓ long term rates . . .
(sign restriction on the contemporaneous response)
? . . . but perhaps no effect over the fed funds rate.
(zero restriction on contemporaneous response)
50 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
An ‘Unconventional’ Monetary Shock
? What is an ‘unconventional’ monetary shock?
(forward guidance, large-scale asset purchase program)
? Hard to define exactly, but easy to state minimal properties
? At the very minimum: shock that ↓ long term rates . . .
(sign restriction on the contemporaneous response)
? . . . but perhaps no effect over the fed funds rate.
(zero restriction on contemporaneous response)
? Extra: No contractionary effect over inflation and output
50 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Monetary SVAR
4 variable SVAR(2) with monthly data (07-1979:12-2007):
1. Consumer Price Index, ∆ log
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Monetary SVAR
4 variable SVAR(2) with monthly data (07-1979:12-2007):
1. Consumer Price Index, ∆ log
2. Industrial Production Index, ∆ log
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Introduction
Notation
Projection
Calibration
Application
Conclusion
Monetary SVAR
4 variable SVAR(2) with monthly data (07-1979:12-2007):
1. Consumer Price Index, ∆ log
2. Industrial Production Index, ∆ log
3. 2 year government bond rate, ∆
51 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Monetary SVAR
4 variable SVAR(2) with monthly data (07-1979:12-2007):
1. Consumer Price Index, ∆ log
2. Industrial Production Index, ∆ log
3. 2 year government bond rate, ∆
4. Federal Funds rate, ∆
51 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
95% Projection (blue) vs. 95% Bayesian Credible Set (gray)
IP
CPI
10
4
6
% change
% change
8
4
2
0
2
0
-2
-2
-4
-4
5
10
15
20
0
5
10
Months after shock
Months after shock
GS2
FFR
0.4
0.4
0.2
0.2
% change
% change
0
0
-0.2
15
20
15
20
0
-0.2
-0.4
-0.4
0
5
10
Months after shock
15
20
0
5
10
Months after shock
28 seconds vs. 283 seconds
(laptop @2.4GHz IntelCore i7)
52 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Second Round of Quantititative Easing
Use the pre-crisis bands to bound the post QE2 responses
53 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Response of IP, CPI: 95% Uhlig’s Credible Set
108
103
107
102.5
106
102
105
101.5
104
101
103
100.5
102
100
101
100
Dec Jan
Feb Mar
Apr May Jun
Jul
Aug Sep Oct
Nov Dec Jan
Feb Mar
99.5
Dec Jan
Apr May
Feb Mar
Apr May Jun
Jul
Aug Sep Oct
Nov Dec Jan
Feb Mar
Apr May
The Credible Set ‘covers’ IP 7/9 (78%) and CPI 5/9 (56%) times
54 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Response of IP, CPI: 95% Projection CS
108
103.5
107
103
106
102.5
105
102
104
101.5
103
101
102
100.5
100
101
100
Dec Jan
Feb Mar
Apr May Jun
Jul
Aug Sep Oct
Nov Dec Jan
Feb Mar
99.5
Dec Jan
Apr May
Feb Mar
Apr May Jun
Jul
Aug Sep Oct
Nov Dec Jan
Feb Mar
Apr May
The Projection CS ‘covers’ IP 9/9 (100 %) and CPI 8/9 (89 %).
55 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
5. Conclusion
56 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Main Messages from our Revised Project
? Projection CS for IRFk,ij ∈ R and (IRF0,ij , . . . IRFh,ij ) ∈ Rh+1
(as general as Uhlig’s current Bayesian Approach)
57 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Main Messages from our Revised Project
? Projection CS for IRFk,ij ∈ R and (IRF0,ij , . . . IRFh,ij ) ∈ Rh+1
(as general as Uhlig’s current Bayesian Approach)
? Implementing projection requires evaluation of maxθ∈CS v k,ij (θ)
(value function of a nonlinear mathematical program)
57 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Main Messages from our Revised Project
? Projection CS for IRFk,ij ∈ R and (IRF0,ij , . . . IRFh,ij ) ∈ Rh+1
(as general as Uhlig’s current Bayesian Approach)
? Implementing projection requires evaluation of maxθ∈CS v k,ij (θ)
(value function of a nonlinear mathematical program)
? Projection is conservative, but—in theory—can be calibrated.
(propose a lower bound on the calibrated degrees of freedom)
57 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Main Messages from our Revised Project
? Projection CS for IRFk,ij ∈ R and (IRF0,ij , . . . IRFh,ij ) ∈ Rh+1
(as general as Uhlig’s current Bayesian Approach)
? Implementing projection requires evaluation of maxθ∈CS v k,ij (θ)
(value function of a nonlinear mathematical program)
? Projection is conservative, but—in theory—can be calibrated.
(propose a lower bound on the calibrated degrees of freedom)
? One structural shock: delta-method type inference is available
(contemporaneous restrictions, we establish unif. validity)
57 / 58
Introduction
Notation
Projection
Calibration
Application
Conclusion
Main Messages from our Revised Project
? Projection CS for IRFk,ij ∈ R and (IRF0,ij , . . . IRFh,ij ) ∈ Rh+1
(as general as Uhlig’s current Bayesian Approach)
? Implementing projection requires evaluation of maxθ∈CS v k,ij (θ)
(value function of a nonlinear mathematical program)
? Projection is conservative, but—in theory—can be calibrated.
(propose a lower bound on the calibrated degrees of freedom)
? One structural shock: delta-method type inference is available
(contemporaneous restrictions, we establish unif. validity)
? max/min differentiable and strictly set identified model:
Calibrated Projection ≈ Delta Method
57 / 58
Introduction
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Projection
Calibration
Application
Conclusion
Thanks!
58 / 58