Iden%fiability of Subsampled/Mixed- Frequency Structural VAR models - - PowerPoint PPT Presentation

Iden%fiability of Subsampled/Mixed- Frequency Structural VAR models

Alex Tank University of Washington Joint work with Emily Fox and Ali Shojaie

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

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subsampling rate k = 2

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

Can we s%ll learn the structure?

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subsampling rate k = 2

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

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subsampling rate k = 3

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

Can we s%ll learn the structure?

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subsampling rate k = 3

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

Mixed Frequency (MF)

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k = (1, 2)

subsampling rate

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x11 x21 x31 x41 x51 x61 x71 x12 x22 x32 x42 x52 x62 x72

Mixed Frequency + Subsampling

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k = (2, 3)

subsampling rate

• Costly data collec%on:
• GDP
• Housing prices
• Other econometric indicators.
• Biomarker health indicators.
• Technological limita%ons:
• fMRI/EEG all sample neural ac%vity at fixed rates.

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Causes of Subsampling and Mixed Frequencies

Previous Work

• Gong et al. 2015 study subsampled VAR models with

independent errors.

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xt = Axt−1 + et x11 x31 x51 x71 x12 x32 x52 x72

Previous Work

• Gong et al. 2015 study subsampled VAR models with

independent errors.

• We extend their framework to deal with mixed subsampling

frequencies and correlated errors.

xt = Axt−1 + et

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x11 x71 x12 x32 x52 x72 x41

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x11 x71 x12 x32 x52 x72 x41 Structural Vector Autoregressive Model (SVAR)

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x11 x71 x12 x32 x52 x72 x41 xt = Axt−1 + Cet Structural Vector Autoregressive Model (SVAR)

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x11 x71 x12 x32 x52 x72 x41

transi%on matrix structural matrix

xt = Axt−1 + Cet Structural Vector Autoregressive Model (SVAR)

Instantaneous errors, ‘shocks’

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x11 x71 x12 x32 x52 x72 x41

C21 A21

transi%on matrix structural matrix Instantaneous errors, ‘shocks’

xt = Axt−1 + Cet Structural Vector Autoregressive Model (SVAR)

Subsampled/MF SVAR xt = Axt−1 + Cet

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x11 x71 x12 x32 x52 x72 x41

x11 x31 x71 x12 x32 x52 x72 xt = Axt−1 + Cet ˜ x1 ˜ x2 ˜ x3 ˜ x4

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x11 x71 x12 x32 x52 x72 x41 ˜ x5 Subsampled/MF SVAR

xt = Axt−1 + Cet ˜ X = x11 x31 x71 x12 x32 x52 x72 ˜ x1 ˜ x2 ˜ x3 ˜ x4

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x11 x71 x12 x32 x52 x72 x41 ˜ x5 Subsampled/MF SVAR

• When subsampling at same rate:
• not iden%fiable from first two moments of !
• Implies not iden%fiable if Gaussian

Subsampled SVAR process ˜ xt = Ak˜ xt−1 + L˜ et

L =

• C, AC, . . . , Ak−1C

• When subsampling at same rate:
• not iden%fiable from first two moments of !
• Implies not iden%fiable if Gaussian.

Subsampled SVAR process ˜ xt = Ak˜ xt−1 + L˜ et

˜ X

(A, C)

L =

• C, AC, . . . , Ak−1C
• et

Non-Gaussian SVAR xt = Axt−1 + Cet eit ejt

independent of

eit ∼ pei non-Gaussian

pei 6= pej 8i, j

et C

A1 A2 A3

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Non-Gaussian SVAR xt = Axt−1 + Cet eit ejt

independent of

eit ∼ pei non-Gaussian

pei 6= pej 8i, j

et C

No restric%ons iden%fiable

(Lanne et al 2015)

• Perm. to lower triangular DAG and its
• rdering associated w/ iden%fiable.

(Hyvarninen et al 2010, 2013 and Peters et al 2013 )

C C

A1 A2 A3

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Subsampled/MF Non-Gaussian SVAR

(A, C, e, k)

• Non-Gaussian subsampled/MF

parameteriza%on

• Iden%fiability in this seing is a unique map between
• Proof technique:
• Show that if two parameteriza%ons

and lead to same distribu%on of then . Distribu%on of ˜

X

(A, C, e, k)

(A, C, e, k)

(A0, C0, e0, k)

(A, C) = (A0, C0)

˜ X

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where is a permuta-on matrix with 1 and -1 entries. Theorem: Suppose is generated according to subsampled/MF SVAR process and also admits another representa-on . Assume A1-3 hold and

(A, C, e, k)

˜ X

(A0, C0, e0, k)

||A||2 < 1

C = C0P

P

If asymmetric and full rank

pei

C

→ (A, C) = (A0, C0)

If lower triangular

→ C = C0

C

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Iden%fiability for Subsampled/MF SVAR

where is a permuta-on matrix with 1 and -1 entries. Theorem: Suppose is generated according to subsampled/MF SVAR process and also admits another representa-on . Assume A1-3 hold and

(A, C, e, k)

˜ X

(A0, C0, e0, k)

||A||2 < 1

C = C0P

P

If asymmetric and full rank

pei

C

→ (A, C) = (A0, C0)

If lower triangular

→ C = C0

C

Corollary: If the instantaneous interac-ons follow a DAG structure, may be permuted to lower triangular, DAG structure and ordering iden-fiable from .

˜ X

C

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Iden%fiability for Subsampled/MF SVAR

Non-Gaussianity:

• Model the as a mixture of Gaussians with 2

components.

• Introduce auxiliary binary variables such that:

eit

eit = zitw1

it + (1 − zit)w2 it

w1

it ∼ N(µ1 i , τ 1 i )

w2

it ∼ N(µ2 i , τ 2 i )

zit

Subsampling:

• Treat the unobserved data in as missing. Sampler

will impute missing values. X

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Bayesian Es%ma%on: Gibbs Sampler

Place conjugate priors on all parameters Gibbs sampler steps:

• 1. Jointly impute missing data:
• 2. Standard condi%onal Gibbs updates for all .
• 1. Sample as in Wozniak et al 2015.

(X, Z) ∼ p(X, Z| ˜ X, Θ)

Θ = (A, C, µ, τ, π)

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Θ

C

Bayesian Es%ma%on: Gibbs Sampler

x1

x2

z2 z1

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Bayesian Es%ma%on: Gibbs Sampler

• Simula%on with subsampling rate k = 2.
• C = I, T = 403

0.90 0.94 0.98 10 20 30 Density −0.38 −0.34 −0.30 5 15 25 0.950 0.960 0.970 0.980 20 40 60 0.155 0.165 0.175 40 80

p(A| ˜ X) ≈

• Extended iden%fiability results of Gong et al 2015 to

structural VAR models with mixed subsampling.

• Allows instantaneous covariance and interac%ons.
• May iden%fy the structural matrix and transi%on matrix

under non-Gaussian errors.

• Developed Gibbs sampler for inference.

References

• Gong et al. ‘Learning Temporal Causal Rela%ons from

Subsampled Time Series’ ICML 2015

• Wozniak et al. ‘Assessing Monetary Policy Models: Bayesian

Inference for Heteroskedas%c Structural VARs’ 2015

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Conclusion