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 1

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x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 4

subsampling rate k = 2 x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 5

subsampling rate k = 2 Can we s%ll learn the structure? x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 6

subsampling rate k = 3 x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 7

subsampling rate k = 3 Can we s%ll learn the structure? x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 8

k = (1 , 2) subsampling rate Mixed Frequency (MF) x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 9

k = (2 , 3) subsampling rate Mixed Frequency + Subsampling x 31 x 41 x 11 x 21 x 71 x 51 x 61 x 22 x 32 x 72 x 12 x 42 x 52 x 62 %me 10

Causes of Subsampling and Mixed Frequencies • 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. 11

Previous Work • Gong et al. 2015 study subsampled VAR models with independent errors. x t = A x t − 1 + e t x 31 x 11 x 71 x 51 x 32 x 72 x 12 x 52 12

Previous Work • Gong et al. 2015 study subsampled VAR models with independent errors. x t = A x t − 1 + e t • We extend their framework to deal with mixed subsampling frequencies and correlated errors. x 11 x 41 x 71 x 32 x 72 x 12 x 52 13

Structural Vector Autoregressive Model (SVAR) x 11 x 41 x 71 x 32 x 72 x 12 x 52 14 14

Structural Vector Autoregressive Model (SVAR) x t = A x t − 1 + C e t x 11 x 41 x 71 x 32 x 72 x 12 x 52 15 15

Structural Vector Autoregressive Model (SVAR) Instantaneous errors, ‘shocks’ x t = A x t − 1 + C e t transi%on matrix structural matrix x 11 x 41 x 71 x 32 x 72 x 12 x 52 16

Structural Vector Autoregressive Model (SVAR) Instantaneous errors, ‘shocks’ x t = A x t − 1 + C e t transi%on matrix structural matrix x 11 x 41 x 71 C 21 A 21 x 32 x 72 x 12 x 52 17

Subsampled/MF SVAR x t = A x t − 1 + C e t x 11 x 41 x 71 x 32 x 72 x 12 x 52 18

Subsampled/MF SVAR x t = A x t − 1 + C e t ˜ ˜ ˜ ˜ ˜ x 1 x 2 x 3 x 4 x 5 x 31 x 11 x 11 x 41 x 71 x 71 x 32 x 32 x 72 x 72 x 12 x 12 x 52 x 52 19 19

Subsampled/MF SVAR x t = A x t − 1 + C e t ˜ X = ˜ ˜ ˜ ˜ ˜ x 5 x 1 x 2 x 3 x 4 x 31 x 11 x 11 x 41 x 71 x 71 x 32 x 32 x 72 x 72 x 12 x 12 x 52 x 52 20 20

Subsampled SVAR process • When subsampling at same rate: x t = A k ˜ x t − 1 + L ˜ ˜ e t C , AC , . . . , A k − 1 C � � L = • not iden%fiable from first two moments of ! • Implies not iden%fiable if Gaussian

Subsampled SVAR process • When subsampling at same rate: x t = A k ˜ x t − 1 + L ˜ ˜ e t C , AC , . . . , A k − 1 C � � L = ˜ • not iden%fiable from first two moments of ! X ( A , C ) • Implies not iden%fiable if Gaussian. e t

Non-Gaussian SVAR x t = A x t − 1 + C e t A1 independent of e it e jt e it ∼ p e i non-Gaussian A2 e t p e i 6 = p e j 8 i, j A3 C 23

Non-Gaussian SVAR x t = A x t − 1 + C e t A1 independent of e it e jt e it ∼ p e i non-Gaussian A2 e t p e i 6 = p e j 8 i, j A3 No restric%ons iden%fiable C (Lanne et al 2015) C Perm. to lower triangular DAG and its ordering associated w/ iden%fiable. C (Hyvarninen et al 2010, 2013 and Peters et al 2013 ) 24

Subsampled/MF Non-Gaussian SVAR • Non-Gaussian subsampled/MF ( A , C , e, k ) parameteriza%on • Iden%fiability in this seing is a unique map between ( A , C , e, k ) Distribu%on of ˜ X • Proof technique: • Show that if two parameteriza%ons ( A , C , e, k ) ˜ and lead to same distribu%on of ( A 0 , C 0 , e 0 , k ) X then . ( A , C ) = ( A 0 , C 0 ) 25

Iden%fiability for Subsampled/MF SVAR ˜ Theorem : Suppose is generated according to subsampled/MF X SVAR process and also admits another ( A , C , e, k ) representa-on . Assume A1-3 hold and ( A 0 , C 0 , e 0 , k ) || A || 2 < 1 C = C 0 P where is a permuta-on matrix with P 1 and -1 entries. → C = C 0 If lower triangular C If asymmetric → ( A , C ) = ( A 0 , C 0 ) p e i and full rank C 26

Iden%fiability for Subsampled/MF SVAR ˜ Theorem : Suppose is generated according to subsampled/MF X SVAR process and also admits another ( A , C , e, k ) representa-on . Assume A1-3 hold and ( A 0 , C 0 , e 0 , k ) || A || 2 < 1 C = C 0 P where is a permuta-on matrix with P 1 and -1 entries. → C = C 0 If lower triangular C If asymmetric → ( A , C ) = ( A 0 , C 0 ) p e i and full rank C Corollary : If the instantaneous interac-ons follow a DAG structure, may be permuted to lower triangular, DAG C ˜ structure and ordering iden-fiable from . X 27

Bayesian Es%ma%on: Gibbs Sampler Non-Gaussianity: • Model the as a mixture of Gaussians with 2 e it components. • Introduce auxiliary binary variables such that: z it e it = z it w 1 it + (1 − z it ) w 2 it w 1 it ∼ N ( µ 1 i , τ 1 w 2 it ∼ N ( µ 2 i , τ 2 i ) i ) Subsampling: • Treat the unobserved data in as missing . Sampler X will impute missing values. 28

Bayesian Es%ma%on: Gibbs Sampler Place conjugate priors on all parameters Θ = ( A , C , µ, τ , π ) Gibbs sampler steps: 1. Jointly impute missing data: ( X , Z ) ∼ p ( X , Z | ˜ X , Θ ) z 1 x 1 x 2 z 2 2. Standard condi%onal Gibbs updates for all . Θ 1. Sample as in Wozniak et al 2015. C 29

Bayesian Es%ma%on: Gibbs Sampler • Simula%on with subsampling rate k = 2. • C = I, T = 403 30 80 Density 20 40 10 p ( A | ˜ X ) ≈ 0 0 0.90 0.94 0.98 0.155 0.165 0.175 25 60 15 40 20 5 0 0 − 0.38 − 0.34 − 0.30 0.950 0.960 0.970 0.980 30

Conclusion • 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 31