Informational Fragility of Dynamic Rational Expectations Equilibria - PowerPoint PPT Presentation

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility of Dynamic Rational Expectations Equilibria Giacomo Rondina 1 Todd B. Walker 2 1 Department of Economics University of California, San Diego 2 Department of Economics Indiana University, Bloomington Expectations in Dynamic Macro San Francisco Fed, August 2013

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary What is this paper about? REE in dynamic economies fragility of REE to information set perturbation

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary What motivates this paper? dynamic models of incomplete/dispersed information increasingly popular Link what equilibria robust to small changes in information set of agent?

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary What do we do in this paper? eq. is Informationally Fragile if small perturbations in agents’ information set lead to divergence

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary What do we do in this paper? eq. is Informationally Fragile if small perturbations in agents’ information set lead to divergence linear univariate REE framework ⇒ full information eq. is fragile ⇒ incomplete information eq. is stable informational fragility and learning applications: news shocks, productivity shocks with nominal rigidites

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Model and Equilibrium

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Model and Equilibrium structural equation y t = κ E ( y t +1 | Ω t ) + ϕa t (1)

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Model and Equilibrium structural equation y t = κ E ( y t +1 | Ω t ) + ϕa t (1) information set of representative agent Ω t = U t ∨ M t ( y )

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Model and Equilibrium structural equation y t = κ E ( y t +1 | Ω t ) + ϕa t (1) information set of representative agent Ω t = U t ∨ M t ( y ) Definition Given the exogenous information specification { U t } , t ∈ N , a dynamic Rational Expectations Equilibrium is a stochastic process for { y t , t ∈ N } and a stochastic process for the information set { Ω t , t ∈ N } such that the equilibrium condition (1) holds.

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) (2)

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) (2) Adding AR component will not change results

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) (2) Adding AR component will not change results if θ > 1 , MA(1) lends itself to “news” shock a t = (1 − ς ) u t + ςu t − 1 , ς ∈ (0 , 1) E [ a t +1 | u t ] = ςu t

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) (2) Adding AR component will not change results if θ > 1 , MA(1) lends itself to “news” shock a t = (1 − ς ) u t + ςu t − 1 , ς ∈ (0 , 1) E [ a t +1 | u t ] = ςu t if θ > 1 , MA(1) can be equivalent to signal plus noise η t ∼ N (0 , σ 2 x t = u t + η t η ) ( x t − E ( u t | x t )) 2 � ( u t − E | θ | > 1 ( u t | a t )) 2 � � � E = E θ 2 = σ 2 u + σ 2 η σ 2 u

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u )

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) full information: U F t = { u t } y t = ϕ (1 + θκ ) u t + ϕθu t − 1 (3)

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) full information: U F t = { u t } y t = ϕ (1 + θκ ) u t + ϕθu t − 1 (3) Proposition Consider an arbitrary exogenous information set U t ⊆ { u t } . If | θ | ≤ 1 then (3) is always a dynamic REE independent of U t . 1 If | θ | > 1 then (3) is a dynamic REE if and only if u τ ∈ U t for some τ ≤ t . 2

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) full information: U F t = { u t } y t = ϕ (1 + θκ ) u t + ϕθu t − 1 (3) Proposition Consider an arbitrary exogenous information set U t ⊆ { u t } . If | θ | ≤ 1 then (3) is always a dynamic REE independent of U t . 1 If | θ | > 1 then (3) is a dynamic REE if and only if u τ ∈ U t for some τ ≤ t . 2 When | θ | > 1 equation (3) is REE when information is “ initialized ” at u 0 .

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Full Information Equilibrium exogenous stochastic process u t ∼ N (0 , σ 2 a t = u t + θu t − 1 , u ) full information: U F t = { u t } y t = ϕ (1 + θκ ) u t + ϕθu t − 1 (3) Proposition Consider an arbitrary exogenous information set U t ⊆ { u t } . If | θ | ≤ 1 then (3) is always a dynamic REE independent of U t . 1 If | θ | > 1 then (3) is a dynamic REE if and only if u τ ∈ U t for some τ ≤ t . 2 When | θ | > 1 equation (3) is REE when information is “ initialized ” at u 0 . If information is initialized in a neighborhood of u 0 , (3) is no longer REE

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t }

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t } equilibrium representation with observable variables, y t and a t = (1 + θL ) u t y t = − θy t − 1 + ϕ (1 + θκ ) a t + ϕθa t − 1 (4)

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t } equilibrium representation with observable variables, y t and a t = (1 + θL ) u t y t = − θy t − 1 + ϕ (1 + θκ ) a t + ϕθa t − 1 (4) consider equilibrium at t = 1 y 1 = − θy 0 + ϕθa 0 + ϕ (1 + θκ ) a 1

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t } equilibrium representation with observable variables, y t and a t = (1 + θL ) u t y t = − θy t − 1 + ϕ (1 + θκ ) a t + ϕθa t − 1 (4) consider equilibrium at t = 1 y 1 = − θy 0 + ϕθa 0 + ϕ (1 + θκ ) a 1 “initial” condition: span { a 0 , y 0 } = u 0

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t } equilibrium representation with observable variables, y t and a t = (1 + θL ) u t y t = − θy t − 1 + ϕ (1 + θκ ) a t + ϕθa t − 1 (4) consider equilibrium at t = 1 y 1 = − θy 0 + ϕθa 0 + ϕ (1 + θκ ) a 1 “initial” condition: span { a 0 , y 0 } = u 0 perturbed initial condition ˆ u 0 = u 0 + ε

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Informational Fragility recall information set of representative agent Ω t = U t ∨ M t ( y ) M t ( y ) = { y t , a t } equilibrium representation with observable variables, y t and a t = (1 + θL ) u t y t = − θy t − 1 + ϕ (1 + θκ ) a t + ϕθa t − 1 (4) consider equilibrium at t = 1 y 1 = − θy 0 + ϕθa 0 + ϕ (1 + θκ ) a 1 “initial” condition: span { a 0 , y 0 } = u 0 perturbed initial condition ˆ u 0 = u 0 + ε dynamic path of perturbed system for t → ∞ y t = ϕ (1 + θκ ) u t + ϕθu t − 1 + θ t ε (5) ⇒ when | θ | > 1 REE fragile to initial information set perturbation!

Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary Equilibrium with Confounding Dynamics