A Class of Time-Varying Parameter Structural VARs for Inference - PowerPoint PPT Presentation

A Class of Time-Varying Parameter Structural VARs for Inference under Exact or Set Identification Mark Bognanni 1 1 Federal Reserve Bank of Cleveland January 27, 2018 Norges Bank

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.

To fix ideas y ′ = y ′ + · · · + y ′ + ε ′ A t − 1 F 1 t − p F p + c , ε t ∼ N ( 0 , I n ) t t (1 × n ) ( n × n ) ( n × n ) (1 × n ) (1 × n ) Define x t ≡ [ y ′ t − 1 , . . . , y ′ t − p , 1] ′ F ≡ [ F ′ 1 , . . . , F ′ p , c ′ ] ′ and Write y ′ t A = x ′ t F + ε ′ t Want to infer ( A , F ) because they • represent equilibrium relationships between variables • determine response of y t to the mutually orthogonal “structural” shocks in ε t But ( A , F ) don’t come for free.

The identification problem Rewriting the SVAR t FA − 1 + ε ′ t A − 1 , y ′ t = x ′ Likelihood for y t p ( y t | A , F , y t − p : t − 1 ) = Npdf ( y t | x ′ t FA − 1 , ( AA ′ ) − 1 ) � �� � � �� � µ Σ But consider the alternative parameter point ( � A , � F ) ( � A , � F ) = ( AQ , FQ ) for Q ∈ O n A − 1 = FQ ( AQ ) − 1 = FQQ − 1 A − 1 = FA − 1 µ = � F � A ′ = ( AQ )( AQ ) ′ = AQQ ′ A ′ = AA ′ Σ = � A � Hence, we cannot identify ( A , F ).

The reduced-form VAR We can identify g ( A , F ) = ( FA − 1 , AA ′ ) = ( B , Σ ) y ′ t = x t B + u ′ 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 ).

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.

A Motivating Example • Based on Baumeister and Peersman (2013, AEJ Macro) • y t = [∆ p oil t , ∆ q oil t , ∆ GDP t , ∆ p CPI ] ′ t • 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 ε oil , s < 0 ⇒ ∆ q oil t + h < 0 < ∆ p oil for h = 0 , ..., 4 t t + h • RRWZ “algorithm” applied to “reduced-form” parameters t -by- t .

“Reduced-form” Primiceri (2005) y ′ t = vec ( B t ) ′ ( I n ⊗ x t ) + ε ′ t Ξ t ∆ − 1 t where     ξ 1 , t 0 · · · 0 1 δ 12 , t · · · δ 1 n , t . . ... ...     . . 0 ξ 2 , t . 0 1 .     Ξ t =  , ∆ t =     . . ... ... ... ...  .  .  . 0 . δ n − 1 n , t 0 · · · 0 ξ n , t 0 · · · 0 1 and Ξ t = Ξ t − 1 diag (exp( η t )) , η t ∼ N ( 0 n × 1 , Σ η ) δ t = δ t − 1 + ζ t , ζ t ∼ N ( 0 n ( n − 1) × 1 , Σ ζ ) 2 vec ( B t ) = vec ( B t − 1 ) + υ t , υ t ∼ N ( 0 mn × 1 , Σ υ )

• Supply shock causing ∆ q oil = − 1%. • “baseline” IRFs • x-axis: time in quarters • p oil IRF: t contemporaneous response at each t • GDP t and ∆ p t IRFs: cumulative change over 4 quarters at each t

• Supply shock causing ∆ q oil = − 1%. • “baseline” IRFs • Finding: oil demand has become increasingly inelastic

A Motivating Example • Based on Baumeister and Peersman (2013, AEJ Macro) • y t = [∆ p oil t , ∆ q oil t , ∆ GDP t , ∆ p CPI ] ′ t • Identify time-varying IRFs of oil supply shocks The method: • Estimate Primiceri (2005) VAR-TVP-SV • Reassemble into “reduced-form VAR” parameters t -by- t • Find structural parameters satisfying sign-restrictions ε oil , s < 0 ⇒ ∆ q oil t + h < 0 < ∆ p oil for h = 0 , ..., 4 t t + h • RRWZ “algorithm”

A Motivating Example Revisited • Based on Baumeister and Peersman (2013, AEJ Macro) • y t = [∆ p oil t , ∆ q oil t , ∆ GDP t , ∆ p CPI ] ′ t t ] ′ y t = [∆ p CPI , ∆ GDP t , ∆ q oil t , ∆ p oil t • Identify time-varying IRFs of oil supply shocks The method: • Estimate Primiceri (2005) VAR-TVP-SV • Reassemble into “reduced-form VAR” parameters t -by- t • Find structural parameters satisfying sign-restrictions ε oil , s < 0 ⇒ ∆ q oil t + h < 0 < ∆ p oil for h = 0 , ..., 4 t t + h • RRWZ “algorithm”

• Supply shock causing ∆ q oil = − 1%. • “baseline” IRFs • IRFs under alternative variable ordering • Time-variation in IRFs is gone! • Would have been a different paper!

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.

Examining the posterior I Let S t = ( A t , F t ) and S t ∗ Q t = ( A t Q t , F t Q t ) p ( φ , S 0: T | y 1: T ) ∝ p ( φ , S 0 ) p ( S 1: T | φ , S 0 ) p ( y 1: T | φ , S 0 , S 1: T ) � �� � � �� � � �� � prior density of the S 1: T data density given S 0: T sequence under the model’s law of motion where T � p ( y 1: T | φ , S 0 , S 1: T ) = p ( y t | y t − p : t − 1 , S t ) t =1 � T p N ( y t | x ′ t F t A − 1 ( A t A ′ t ) − 1 = , ) t � �� � � �� � t =1 t F t Q t Q − 1 A − 1 t ) − 1 ( A t Q t Q ′ t A ′ x ′ t t ⇒ In each t , S t ∗ Q t gives same evaluation of this term as S t .

Examining the posterior II p ( φ , S 0: T | y 1: T ) ∝ p ( φ , S 0 ) p ( S 1: T | φ , S 0 ) p ( y 1: T | φ , S 0 , S 1: T ) � �� � � �� � � �� � prior density of the S 1: T data density given S 0: T sequence under the model’s law of motion where � T p ( S 1: T | φ , S 0 ) = p ( S t | φ , S t − 1 ) t =1 (This is the tricky part.)

This paper • Let’s try something else.

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.)

Outline 1 A new SVAR with dynamic parameters 2 Reduced-form Representation 3 Structural Inference Revisited 4 Revisiting the time-varying oil demand elasticity

Extending the SVAR y ′ t A t = x ′ t F t + ε ′ 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 )

Extending the SVAR y ′ t A t = x ′ t F t + ε ′ t , ε t ∼ N ( 0 , I n ) Law of motion for ( A t , F t ): A t = β − 1 / 2 A t − 1 Ω t F t = F t − 1 A − 1 t − 1 A t + Θ t . Shocks: Ω t = L t h ( 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

Detour: alternate form of SVAR t h ( H t ) − 1 , y ′ t = x ′ t B t + ε ′ t Q ′ ε t ∼ N ( 0 , I n ) Law of motion for ( A t , F t ) = ( B t , H t , Q t ) : h ( H t ) Q t = β − 1 / 2 h ( H t − 1 ) Q t − 1 Ω t B t h ( H t ) Q t = B t − 1 h ( H t − 1 ) Q t + Θ t Q t = p ( Q t | B t , H t ) 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]

Some notation A Dynamic SVAR (call it DSVAR) denoted: S U 0: T ( L 1: T , R 1: T ) and let φ = ( β, W )

Key result Theorem (Theorem 1) Let S U 0: T ( L 1: T , R 1: T ) have prior p ( φ , S 0 ) for which p ( φ , S 0 ) = p ( φ , S 0 ∗ P ) for any P ∈ O n . For any Q 0: T such that each Q t ∈ O n , the model 0: T ( � L 1: T , � R 1: T ) defined by ( � L t , � R t ) = ( Q ′ S U t − 1 L t , R t Q t ) is such that, for every point S 0: T , the point � S 0: T = S 0: T ∗ Q 0: T satisfies p ( φ , S 0: T | y 1: T , S U 0: T ( L 1: T , R 1: T )) � � φ , � 0: T ( � L 1: T , � S 0: T | y 1: T , S U = p R 1: T ) .

Theorem 1: restatement and implications For 1 any realization of the data, 2 any dynamic structural VAR, 3 and any Q 1: T there exists an alternative model with the “same posterior” as the original model, but with each point rotated by Q 1: T . • Set of equivalent models does not depend on y 1: T • ⇒ All structural models in the class are observationally equivalent.

Outline 1 A new SVAR with dynamic parameters 2 Reduced-form Representation 3 Structural Inference Revisited 4 Revisiting the time-varying oil demand elasticity