Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
On the maximum and minimum response
to an impulse in Svars
Pepe Montiel Olea (NYU) and Bulat Gafarov (PSU)
September 2, 2014
1 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
The maximum (or minimum) response of yt+k to a structural
shock ǫt .
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
The maximum (or minimum) response of yt+k to a structural
shock ǫt .
⋆ Examples:
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
The maximum (or minimum) response of yt+k to a structural
shock ǫt .
⋆ Examples:
i) What is the maximum effect of a σ-structural uncertainty
shock over household spending?
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
The maximum (or minimum) response of yt+k to a structural
shock ǫt .
⋆ Examples:
i) What is the maximum effect of a σ-structural uncertainty
shock over household spending?
ii) What is the maximum effect of a σ-structural monetary shock
over yt+k , πt+k or rt+k ?
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Plan for Today
→ Estimate and conduct frequentist inference on:
The maximum (or minimum) response of yt+k to a structural
shock ǫt .
⋆ Examples:
i) What is the maximum effect of a σ-structural uncertainty
shock over household spending?
ii) What is the maximum effect of a σ-structural monetary shock
over yt+k , πt+k or rt+k ?
iii) What is the maximum effect of a σ-structural oil shock over
yt+k , πt+k or Pt+k ?
2 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Messages of this Talk
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
3 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Messages of this Talk
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
2. Efficient delta-method inference available for these objects.
(max and min depend nicely on reduced-form VAR parameters)
3 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Messages of this Talk
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
2. Efficient delta-method inference available for these objects.
(max and min depend nicely on reduced-form VAR parameters)
3. Our inference procedure can accommodate alternative sources
of identification
(sign restrictions or external instruments).
3 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Related Work
a) ‘Partially Identified’ SVARs
Faust (1998); Uhlig (2005); Moon, Schorfheide, Granziera (2013)
b) Value functions of Stochastic Programs
Faccio-Ishizuka (1990); Shapiro (1991); Santos and Fang (2014)
4 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Related Work
a) ‘Partially Identified’ SVARs
Faust (1998); Uhlig (2005); Moon, Schorfheide, Granziera (2013)
Min/Max will give efficient estimators of the ‘identified set’.
b) Value functions of Stochastic Programs
Faccio-Ishizuka (1990); Shapiro (1991); Santos and Fang (2014)
4 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Related Work
a) ‘Partially Identified’ SVARs
Faust (1998); Uhlig (2005); Moon, Schorfheide, Granziera (2013)
b) Value functions of Stochastic Programs
Faccio-Ishizuka (1990); Shapiro (1991); Santos and Fang (2014)
4 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Related Work
a) ‘Partially Identified’ SVARs
Faust (1998); Uhlig (2005); Moon, Schorfheide, Granziera (2013)
b) Value functions of Stochastic Programs
Faccio-Ishizuka (1990); Shapiro (1991); Santos and Fang (2014)
Min/Max will be cont. and directionally differentiable functions.
4 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Outline
1. Notation
2. Identification and Estimation
3. Illustrative Example
4. Sign-Restricted SVARs
5. Frequentist Inference
5 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Outline
1. Notation
2. Identification and Estimation
3. Illustrative Example
4. Sign-Restricted SVARs
5. Frequentist Inference
5 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Outline
1. Notation
2. Identification and Estimation
3. Illustrative Example
4. Sign-Restricted SVARs
5. Frequentist Inference
5 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Outline
1. Notation
2. Identification and Estimation
3. Illustrative Example
4. Sign-Restricted SVARs
5. Frequentist Inference
5 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Outline
1. Notation
2. Identification and Estimation
3. Illustrative Example
4. Sign-Restricted SVARs
5. Frequentist Inference
5 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
1. Notation
6 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
SVAR(p)
⋆ Vector Autoregression for the n-dimensional vector Yt :
7 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
SVAR(p)
⋆ Vector Autoregression for the n-dimensional vector Yt :
Yt = A1 Yt−1 + . . . Ap Yt−p + ηt ,
7 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
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 ηt′ ].
7 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
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 ηt′ ].
⋆ Structural Model for the vector of Forecast Errors ηt :
7 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
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 ηt′ ].
⋆ Structural Model for the vector of Forecast Errors ηt :
ηt = Hεt ,
H ∈ Rn×n .
7 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Orthogonal SVAR(p)
⋆ Key Assumption of the Structural Model: Orthogonality
E[εt ε′t ] = diag(σ12 , σ22 , . . . σn2 ) ≡ D
8 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Orthogonal SVAR(p)
⋆ Key Assumption of the Structural Model: Orthogonality
E[εt ε′t ] = diag(σ12 , σ22 , . . . σn2 ) ≡ D
⋆ (ηt = Hεt ) + (Orthogonality) =⇒
8 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Orthogonal SVAR(p)
⋆ Key Assumption of the Structural Model: Orthogonality
E[εt ε′t ] = diag(σ12 , σ22 , . . . σn2 ) ≡ D
⋆ (ηt = Hεt ) + (Orthogonality) =⇒
i
i
h
h
Σ ≡ E ηt ηt′ = E Hεt ε′t H ′ = HDH ′
8 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Orthogonal SVAR(p)
⋆ Key Assumption of the Structural Model: Orthogonality
E[εt ε′t ] = diag(σ12 , σ22 , . . . σn2 ) ≡ D
⋆ (ηt = Hεt ) + (Orthogonality) =⇒
i
i
h
h
Σ ≡ E ηt ηt′ = E Hεt ε′t H ′ = HDH ′
⋆ The ‘structural’ parameter HD 1/2 is restricted.
8 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Quick Observation
Orthogonal SVAR(p) are partially identified
(no need of sign restrictions)
9 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A.
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A.
⋆ Structural Impulse Response Functions (variable i , shock j)
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A.
⋆ Structural Impulse Response Functions (variable i , shock j)
∂Yi ,t+k
· σj ≡ IRFk,ij (A, Σ) = ei′ Ck (A)Hej σj
∂εjt
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Structural Impulse Response Function
⋆ Moving-Average Representation of the SVAR
Yt =
∞
X
Ck (A)Hεt−k ,
k=0
Ck (A) is a nonlinear transformation of A.
⋆ Structural Impulse Response Functions (variable i , shock j)
∂Yi ,t+k
· σj ≡ IRFk,ij (A, Σ) = ei′ Ck (A)Hej σj
∂εjt
ei is the i -th column of In . So Hej is Hj , the j-th column of H.
10 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
2. Identification and Estimation
11 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Message of this Section:
The minimum and maximum response are ‘point-identified’
(even if the structural IRF is not)
12 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
(A, Σ)
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
(A, Σ)
are compatible with any
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
(A, Σ)
are compatible with any
IRFk,ij ≡ ei′ Ck (A)Hj σj
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
(A, Σ)
are compatible with any
IRFk,ij ≡ ei′ Ck (A)Hj σj
such that
Hj σj
satisfies HDH ′ = Σ
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The structural IRFk,ij is partially identified
⋆ The reduced-form parameters
(A, Σ)
are compatible with any
IRFk,ij ≡ ei′ Ck (A)Hj σj
such that
Hj σj
satisfies HDH ′ = Σ
⋆ However, . . .
13 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The minimum response in the population is identified
⋆ Consider the mathematical program
min ei′ Ck (A)Hj σj
Hj σj ∈Rn
14 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The minimum response in the population is identified
⋆ Consider the mathematical program
min ei′ Ck (A)Hj σj
Hj σj ∈Rn
subject to the quadratic equality constraint
14 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The minimum response in the population is identified
⋆ Consider the mathematical program
min ei′ Ck (A)Hj σj
Hj σj ∈Rn
subject to the quadratic equality constraint
HDH ′ = Σ
14 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The minimum response in the population is identified
⋆ Consider the mathematical program
min ei′ Ck (A)Hj σj
Hj σj ∈Rn
subject to the quadratic equality constraint
HDH ′ = Σ
⋆ (A, Σ) maps to the unique value of the program above:
14 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
The minimum response in the population is identified
⋆ Consider the mathematical program
min ei′ Ck (A)Hj σj
Hj σj ∈Rn
subject to the quadratic equality constraint
HDH ′ = Σ
⋆ (A, Σ) maps to the unique value of the program above:
v k (A, Σ) ≡ −
q
ei′ Ck (A)ΣCk (A)′ ei
14 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
b−A
A
b −Σ
Σ
!
d
→ Nd (0, Ω).
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
b−A
A
b −Σ
Σ
!
d
→ Nd (0, Ω).
(no unit-roots, no singular covariance matrices)
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
b−A
A
b −Σ
Σ
!
d
→ Nd (0, Ω).
(no unit-roots, no singular covariance matrices)
⋆ We will consider the plug-in estimator:
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
b−A
A
b −Σ
Σ
!
d
→ Nd (0, Ω).
(no unit-roots, no singular covariance matrices)
⋆ We will consider the plug-in estimator:
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Estimation and Inference
b Σ)
b such that
⋆ Assumption 0: There exist estimators (A,
b−A
A
b −Σ
Σ
!
d
→ Nd (0, Ω).
(no unit-roots, no singular covariance matrices)
⋆ We will consider the plug-in estimator:
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
⋆ We provide conditions for consistency and study asy. dist.
15 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
q
b Σ)
b = − e ′ Ck (A)
b ΣC
b k (A)
b ′ ei ,
v k (A,
i
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
q
b Σ)
b = − e ′ Ck (A)
b ΣC
b k (A)
b ′ ei ,
v k (A,
i
is continuously differentiable for every (A, Σ), except, when
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
q
b Σ)
b = − e ′ Ck (A)
b ΣC
b k (A)
b ′ ei ,
v k (A,
i
is continuously differentiable for every (A, Σ), except, when
Ck (A)′ ei = 0.
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
q
b Σ)
b = − e ′ Ck (A)
b ΣC
b k (A)
b ′ ei ,
v k (A,
i
is continuously differentiable for every (A, Σ), except, when
Ck (A)′ ei = 0.
⋆ The function, however, is everywhere directionally differentiable
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Main Idea for Asymptotic Distribution
⋆ The function
q
b Σ)
b = − e ′ Ck (A)
b ΣC
b k (A)
b ′ ei ,
v k (A,
i
is continuously differentiable for every (A, Σ), except, when
Ck (A)′ ei = 0.
⋆ The function, however, is everywhere directionally differentiable
⋆ Adjusted delta-method inference is asymptotically valid.
(details later)
16 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
3. Example
17 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Effects of Uncertainty on Household Spending
⋆ Bivariate SVAR(12) with monthly data (01-1963: 05-2014)
18 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Effects of Uncertainty on Household Spending
⋆ Bivariate SVAR(12) with monthly data (01-1963: 05-2014)
New one family houses sold (HSN1f-FRED)
18 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Effects of Uncertainty on Household Spending
⋆ Bivariate SVAR(12) with monthly data (01-1963: 05-2014)
New one family houses sold (HSN1f-FRED)
S&P 100 Volatility Index (VXO-Bloom’s Website)
18 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Effects of Uncertainty on Household Spending
⋆ Bivariate SVAR(12) with monthly data (01-1963: 05-2014)
New one family houses sold (HSN1f-FRED)
S&P 100 Volatility Index (VXO-Bloom’s Website)
⋆ Logarithm of both series is HP-filtered (λ = 129, 600)
18 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Effects of Uncertainty on Household Spending
⋆ Bivariate SVAR(12) with monthly data (01-1963: 05-2014)
New one family houses sold (HSN1f-FRED)
S&P 100 Volatility Index (VXO-Bloom’s Website)
⋆ Logarithm of both series is HP-filtered (λ = 129, 600)
⋆ See Knotek and Khan (2011, FRB Kansas) for details
18 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Maximum and Minimum Response of New O-F Houses sold
Response of HSN1f to a Structural Uncertainty Shock
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
19 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Cholesky Estimate of Structural IRF (Uncertainty First)
Response of HSN1f to a Structural Uncertainty Shock
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
20 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
4. Sign-Restricted SVARs
21 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Two Modifications to the Basic Idea
1. Min/Max response subject to sign restrictions on the IRFs.
(mathematical program with linear inequality constraints)
22 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Two Modifications to the Basic Idea
1. Min/Max response subject to sign restrictions on the IRFs.
(mathematical program with linear inequality constraints)
2. Min/Max Forecast Error Variance Decomposition.
(quadratic objective function)
22 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
need not have a closed form solution.
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
need not have a closed form solution.
⋆ Hence, the plug-in solution v k will be numerical
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
need not have a closed form solution.
⋆ Hence, the plug-in solution v k will be numerical
⋆ Standard Bayesian vs. Frequentist Approach:
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Additional Complication: Estimation
⋆ The mathematical program
b j σj
min ei′ Ck (A)H
Hj σj ∈Rn
b
subject to HDH ′ = Σ
and S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
need not have a closed form solution.
⋆ Hence, the plug-in solution v k will be numerical
⋆ Standard Bayesian vs. Frequentist Approach:
grid search vs. nonlinear programming
23 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Maximum and Minimum Response of New O-F Houses sold
Response of HSN1f to a Structural Uncertainty Shock
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
24 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Maximum and Minimum Response s.t. (∂vxot /∂ǫt,unc ) > 0
Response of HSN1f to a Structural Uncertainty Shock (h11>0)
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
25 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
5. Frequentist Inference
26 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Inference
⋆ We are solving the mathematical program
b Σ)
b ≡ min e ′ Ck (A)H
b j σj
v k (A,
n i
Hj σj ∈R
subject to
b
HDH ′ = Σ
27 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Inference
⋆ We are solving the mathematical program
b Σ)
b ≡ min e ′ Ck (A)H
b j σj
v k (A,
n i
Hj σj ∈R
subject to
b
HDH ′ = Σ
and subject to S inequality constraints on the IRFs:
27 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Inference
⋆ We are solving the mathematical program
b Σ)
b ≡ min e ′ Ck (A)H
b j σj
v k (A,
n i
Hj σj ∈R
subject to
b
HDH ′ = Σ
and subject to S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
27 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Inference
⋆ We are solving the mathematical program
b Σ)
b ≡ min e ′ Ck (A)H
b j σj
v k (A,
n i
Hj σj ∈R
subject to
b
HDH ′ = Σ
and subject to S inequality constraints on the IRFs:
b j σj ≥ 0 m = 1, . . . S
ei′m Ckm (A)H
⋆ How do we construct 1-α% confidence bands?
27 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
A d
→ Z ∼ Nd (0, Ω)
Σ
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
A d
→ Z ∼ Nd (0, Ω)
Σ
b Σ)
b ...
and we are interested in v k (A,
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
A d
→ Z ∼ Nd (0, Ω)
Σ
b Σ)
b ...
and we are interested in v k (A,
⋆ Can’t we just apply the delta-method to get the limit of
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
A d
→ Z ∼ Nd (0, Ω)
Σ
b Σ)
b ...
and we are interested in v k (A,
⋆ Can’t we just apply the delta-method to get the limit of
√ b Σ)
b − v (A, Σ) ?
T v k (A,
k
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method
⋆ Since we have assumed
A d
→ Z ∼ Nd (0, Ω)
Σ
b Σ)
b ...
and we are interested in v k (A,
⋆ Can’t we just apply the delta-method to get the limit of
√ b Σ)
b − v (A, Σ) ?
T v k (A,
k
⋆ Question: Is v k (·) differentiable in (A, Σ)?
28 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
⋆ Sign-restricted problems satisfy the Mangasarian-Fromowitz CQ
(Lagrange Multipliers Exist)
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
⋆ Sign-restricted problems satisfy the Mangasarian-Fromowitz CQ
(Lagrange Multipliers Exist)
⋆ Result: If sign-restricted problems satisfy the LICQ and have
unique minimizer or maximizer:
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
⋆ Sign-restricted problems satisfy the Mangasarian-Fromowitz CQ
(Lagrange Multipliers Exist)
⋆ Result: If sign-restricted problems satisfy the LICQ and have
unique minimizer or maximizer:
a) Lagrange Multipliers are unique
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
⋆ Sign-restricted problems satisfy the Mangasarian-Fromowitz CQ
(Lagrange Multipliers Exist)
⋆ Result: If sign-restricted problems satisfy the LICQ and have
unique minimizer or maximizer:
a) Lagrange Multipliers are unique
b) Value function is fully differentiable:
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Envelope Theorem (Bonnans-Shapiro, Faccio-Ishizuka)
⋆ Sign-restricted problems have a Lipschitz Cont. Value Fn
(Enough for Consistency)
⋆ Sign-restricted problems satisfy the Mangasarian-Fromowitz CQ
(Lagrange Multipliers Exist)
⋆ Result: If sign-restricted problems satisfy the LICQ and have
unique minimizer or maximizer:
a) Lagrange Multipliers are unique
b) Value function is fully differentiable:
v ′k (A, Σ) = ∇(A,Σ) L(Hj∗ (A, Σ), λ∗ (A, Σ)),
L : Lagrangian,
29 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method Confidence Sets for IRFs(no sign restriction)
Response of HSN1f to a Structural Uncertainty Shock
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
30 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method Confidence Sets for IRFs (sign restriction)
Response of HSN1f to a Structural Uncertainty Shock
8
6
Percent Deviation from trend
4
2
0
−2
−4
−6
−8
0
5
10
15
20
25
Months
30
35
40
45
50
31 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method Confidence Sets for IRFs(no sign restriction)
Response of VXO to a Structural Uncertainty Shock
20
15
Percent Deviation from trend
10
5
0
−5
−10
−15
−20
0
5
10
15
20
25
Months
30
35
40
45
50
32 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Delta-Method Confidence Sets for IRFs (sign restriction)
Response of VXO to a Structural Uncertainty Shock
20
15
Percent Deviation from trend
10
5
0
−5
−10
−15
−20
0
5
10
15
20
25
Months
30
35
40
45
50
33 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Other things we have done
⋆ Results for Monetary and Oil SVARs
(Identification power of sign restrictions/external instruments)
34 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Other things we have done
⋆ Results for Monetary and Oil SVARs
(Identification power of sign restrictions/external instruments)
⋆ Monte-Carlo exercises to analyze coverage
(we also analyze a Lagrangian Bootstrap)
34 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Other things we have done
⋆ Results for Monetary and Oil SVARs
(Identification power of sign restrictions/external instruments)
⋆ Monte-Carlo exercises to analyze coverage
(we also analyze a Lagrangian Bootstrap)
⋆ Comparison with Moon, Schorfheide, Granziera (2013)
(our procedure is efficient at points of full differentiability)
34 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Other things we have done
⋆ Results for Monetary and Oil SVARs
(Identification power of sign restrictions/external instruments)
⋆ Monte-Carlo exercises to analyze coverage
(we also analyze a Lagrangian Bootstrap)
⋆ Comparison with Moon, Schorfheide, Granziera (2013)
(our procedure is efficient at points of full differentiability)
⋆ Relax full differentiability: directional differentiability
34 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Conclusion: Frequentist Approach to SVARs
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
35 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Conclusion: Frequentist Approach to SVARs
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
35 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Conclusion: Frequentist Approach to SVARs
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
2. Efficient delta-method inference available for these objects.
(max and min are diff. w.r.t. reduced-form VAR parameters)
35 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Conclusion: Frequentist Approach to SVARs
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
2. Efficient delta-method inference available for these objects.
(max and min are diff. w.r.t. reduced-form VAR parameters)
√ d
b Σ)
b − v (A, Σ) →
T v k (A,
vk′ (A, Σ)Z .
k
35 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Conclusion: Frequentist Approach to SVARs
1. The maximum and minimum are ‘point-identified’ in SVARs
(even if impulse response functions are not).
q
b
b
b ΣC
b k (A)
b ′ ei ,
v (A, Σ) = − ei′ Ck (A)
2. Efficient delta-method inference available for these objects.
(max and min are diff. w.r.t. reduced-form VAR parameters)
√ d
b Σ)
b − v (A, Σ) →
T v k (A,
vk′ (A, Σ)Z .
k
3. Can accommodate alternative sources of identification
(sign restrictions or external instruments).
35 / 36
Introduction
Notation
Identification and Estimation
Example
Sign Restrictions
Inference
Thanks!
36 / 36