Informational Fragility of Dynamic Rational Expectations Equilibria - - PowerPoint PPT Presentation

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Informational Fragility of Dynamic Rational Expectations Equilibria - - PowerPoint PPT Presentation

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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

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

  • f Dynamic Rational Expectations Equilibria

Giacomo Rondina1 Todd B. Walker2

1Department of Economics

University of California, San Diego

2Department of Economics

Indiana University, Bloomington

Expectations in Dynamic Macro San Francisco Fed, August 2013

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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

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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?

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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

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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

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Model and Equilibrium

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Model and Equilibrium

structural equation yt = κE (yt+1|Ωt) + ϕat (1)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Model and Equilibrium

structural equation yt = κE (yt+1|Ωt) + ϕat (1) information set of representative agent Ωt = Ut ∨ Mt (y)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Model and Equilibrium

structural equation yt = κE (yt+1|Ωt) + ϕat (1) information set of representative agent Ωt = Ut ∨ Mt (y) Definition Given the exogenous information specification {Ut}, t ∈ N, a dynamic Rational Expectations Equilibrium is a stochastic process for {yt, t ∈ N} and a stochastic process for the information set {Ωt, t ∈ N} such that the equilibrium condition (1) holds.

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

(2)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

(2) Adding AR component will not change results

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

(2) Adding AR component will not change results if θ > 1, MA(1) lends itself to “news” shock at = (1 − ς)ut + ςut−1, ς ∈ (0, 1) E[at+1|ut] = ςut

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

(2) Adding AR component will not change results if θ > 1, MA(1) lends itself to “news” shock at = (1 − ς)ut + ςut−1, ς ∈ (0, 1) E[at+1|ut] = ςut if θ > 1, MA(1) can be equivalent to signal plus noise xt = ut + ηt ηt ∼ N(0, σ2

η)

E

  • (xt − E(ut|xt))2

= E

  • (ut − E|θ|>1(ut|at))2

θ2 = σ2

u + σ2 η

σ2

u

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

full information: UF

t = {ut}

yt = ϕ(1 + θκ)ut + ϕθut−1 (3)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

full information: UF

t = {ut}

yt = ϕ(1 + θκ)ut + ϕθut−1 (3) Proposition Consider an arbitrary exogenous information set Ut ⊆ {ut}.

1

If |θ| ≤ 1 then (3) is always a dynamic REE independent of Ut.

2

If |θ| > 1 then (3) is a dynamic REE if and only if uτ ∈ Ut for some τ ≤ t.

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

full information: UF

t = {ut}

yt = ϕ(1 + θκ)ut + ϕθut−1 (3) Proposition Consider an arbitrary exogenous information set Ut ⊆ {ut}.

1

If |θ| ≤ 1 then (3) is always a dynamic REE independent of Ut.

2

If |θ| > 1 then (3) is a dynamic REE if and only if uτ ∈ Ut for some τ ≤ t. When |θ| > 1 equation (3) is REE when information is “initialized” at u0.

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Full Information Equilibrium

exogenous stochastic process at = ut + θut−1, ut ∼ N(0, σ2

u)

full information: UF

t = {ut}

yt = ϕ(1 + θκ)ut + ϕθut−1 (3) Proposition Consider an arbitrary exogenous information set Ut ⊆ {ut}.

1

If |θ| ≤ 1 then (3) is always a dynamic REE independent of Ut.

2

If |θ| > 1 then (3) is a dynamic REE if and only if uτ ∈ Ut for some τ ≤ t. When |θ| > 1 equation (3) is REE when information is “initialized” at u0. If information is initialized in a neighborhood of u0, (3) is no longer REE

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at}

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at} equilibrium representation with observable variables, yt and at = (1 + θL)ut yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (4)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at} equilibrium representation with observable variables, yt and at = (1 + θL)ut yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (4) consider equilibrium at t = 1 y1 = −θy0 + ϕθa0 + ϕ(1 + θκ)a1

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at} equilibrium representation with observable variables, yt and at = (1 + θL)ut yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (4) consider equilibrium at t = 1 y1 = −θy0 + ϕθa0 + ϕ(1 + θκ)a1 “initial” condition: span{a0, y0} = u0

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at} equilibrium representation with observable variables, yt and at = (1 + θL)ut yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (4) consider equilibrium at t = 1 y1 = −θy0 + ϕθa0 + ϕ(1 + θκ)a1 “initial” condition: span{a0, y0} = u0 perturbed initial condition ˆ u0 = u0 + ε

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility

recall information set of representative agent Ωt = Ut ∨ Mt (y) Mt (y) = {yt, at} equilibrium representation with observable variables, yt and at = (1 + θL)ut yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (4) consider equilibrium at t = 1 y1 = −θy0 + ϕθa0 + ϕ(1 + θκ)a1 “initial” condition: span{a0, y0} = u0 perturbed initial condition ˆ u0 = u0 + ε dynamic path of perturbed system for t → ∞ yt = ϕ(1 + θκ)ut + ϕθut−1 + θtε (5) ⇒ when |θ| > 1 REE fragile to initial information set perturbation!

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Equilibrium with Confounding Dynamics

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Equilibrium with Confounding Dynamics

structural equation yt = κE (yt+1|Ωt) + ϕat (6) information set of representative agent Ωt = Mt (y)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Equilibrium with Confounding Dynamics

structural equation yt = κE (yt+1|Ωt) + ϕat (6) information set of representative agent Ωt = Mt (y) Proposition Consider an exogenous information set Ut = {∅}. If |θ| > 1 the stochastic process yt = ϕ

  • θ + κ)˜

ut + ϕ˜ ut−1 where ˜ ut ≡ 1 + θL θ + L

  • ut.

is a dynamic REE of model (6).

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at
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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at

consider equilibrium at t = 1 y1 = 1 θ (ϕa0 − y0) + ϕ θ + κ θ

  • a1
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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at

consider equilibrium at t = 1 y1 = 1 θ (ϕa0 − y0) + ϕ θ + κ θ

  • a1

“initial” condition: span{a0, y0} = ˜ u0

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at

consider equilibrium at t = 1 y1 = 1 θ (ϕa0 − y0) + ϕ θ + κ θ

  • a1

“initial” condition: span{a0, y0} = ˜ u0 perturbed initial condition ˆ ˜ u0 = ˜ u0 + ε

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at

consider equilibrium at t = 1 y1 = 1 θ (ϕa0 − y0) + ϕ θ + κ θ

  • a1

“initial” condition: span{a0, y0} = ˜ u0 perturbed initial condition ˆ ˜ u0 = ˜ u0 + ε dynamic path of perturbed system for t → ∞ yt = ϕ(θ + κ)˜ ut + ϕ˜ ut−1 + 1 θt ε

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Stability

equilibrium representation with observable variables, yt and at yt = −1 θ yt−1 + ϕ θ

  • θ + κ + L
  • at

consider equilibrium at t = 1 y1 = 1 θ (ϕa0 − y0) + ϕ θ + κ θ

  • a1

“initial” condition: span{a0, y0} = ˜ u0 perturbed initial condition ˆ ˜ u0 = ˜ u0 + ε dynamic path of perturbed system for t → ∞ yt = ϕ(θ + κ)˜ ut + ϕ˜ ut−1 + 1 θt ε ⇒ since |θ| > 1, REE robust to initial information set perturbation

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Stability

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Stability

fragile equilibrium Ωt = {at ∨ yt ∨ ut} yt = −θyt−1 + ϕ(1 + θκ)at + ϕθat−1 (7) stable equilibrium Ωt = {at ∨ yt} yt = −1 θ yt−1 + ϕ θ (θ + κ)at + ϕ θ at−1

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

bivariate representation of dynamic system yt = κηet + ϕat et = yt − ηet−1 where et ≡ yt − E

  • yt|Ut−1 ∨ Mt−1(y)
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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

bivariate representation of dynamic system yt = κηet + ϕat et = yt − ηet−1 where et ≡ yt − E

  • yt|Ut−1 ∨ Mt−1(y)
  • least-squares learning algorithm

yt = κηt−1et + ϕat et = yt − ηt−1et−1 ηt =

1 t

t

s=1 yses−1 1 t

t

s=1 e2 s−1

,

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

Recursive form of learning algorithm et = ηt−1 1 − κηt−1 et−1 + ϕ 1 − κηt−1 (ut + θut−1) (8) ηt = ηt−1 + 1 t S−1

t−1etet−1

(9) St = St−1 + 1 t (e2

t − St−1) + 1

t2 −t t + 1(e2

t − St−1)

(10) where St is t estimate of variance-covariance matrix of forecast errors.

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

Recursive form of learning algorithm et = ηt−1 1 − κηt−1 et−1 + ϕ 1 − κηt−1 (ut + θut−1) (8) ηt = ηt−1 + 1 t S−1

t−1etet−1

(9) St = St−1 + 1 t (e2

t − St−1) + 1

t2 −t t + 1(e2

t − St−1)

(10) where St is t estimate of variance-covariance matrix of forecast errors. learning is about λ =

  • η

S

  • , ˆ

λ is (at, yt, ut); λ∗ is (at, yt)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

Recursive form of learning algorithm et = ηt−1 1 − κηt−1 et−1 + ϕ 1 − κηt−1 (ut + θut−1) (8) ηt = ηt−1 + 1 t S−1

t−1etet−1

(9) St = St−1 + 1 t (e2

t − St−1) + 1

t2 −t t + 1(e2

t − St−1)

(10) where St is t estimate of variance-covariance matrix of forecast errors. learning is about λ =

  • η

S

  • , ˆ

λ is (at, yt, ut); λ∗ is (at, yt) λ is learnable if dynamic system converges to λ as t → ∞ for λ0 in neighborhood of λ

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

fragile equilibrium (Ωt = at, yt, ut) yt = ϕ(1 + θκ)ut + ϕθut−1 stable equilibrium (Ωt = at, yt) yt = ϕ

  • θ + κ)˜

ut + ϕ˜ ut−1 where ˜ ut ≡ 1 + θL θ + L

  • ut.
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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

fragile equilibrium (Ωt = at, yt, ut) yt = ϕ(1 + θκ)ut + ϕθut−1 stable equilibrium (Ωt = at, yt) yt = ϕ

  • θ + κ)˜

ut + ϕ˜ ut−1 where ˜ ut ≡ 1 + θL θ + L

  • ut.

Proposition Suppose that |θ| > 1.

1

The fragile equilibrium is not least-squares learnable.

2

The stable equilibrium is always least-squares learnable.

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Informational Fragility and Learning

fragile equilibrium (Ωt = at, yt, ut) yt = ϕ(1 + θκ)ut + ϕθut−1 stable equilibrium (Ωt = at, yt) yt = ϕ

  • θ + κ)˜

ut + ϕ˜ ut−1 where ˜ ut ≡ 1 + θL θ + L

  • ut.

Proposition Suppose that |θ| > 1.

1

The fragile equilibrium is not least-squares learnable.

2

The stable equilibrium is always least-squares learnable. Proof: Eigenvalues of transition matrix G(λ) in learning algorithm: ⇒ fragile equilibrium (−θ, 0, 0) ⇒ stable equilibrium (−θ−1, 0, 0)

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Applications

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Applications

news shocks and the business cycle productivity shocks with nominal rigidities

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: News Shocks and the Business Cycle

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: News Shocks and the Business Cycle

linearized growth model with capital αβEt(kt+1) − (1 + α2β)kt + αkt−1 = αβEt(at+1) productivity shocks at = ut + θut−1, ut ∼ N(0, σ2

u)

when θ > 1 observing at or ut not equivalent “news shock” is observing ut, which improves Et(at+1)

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Application: News Shocks and the Business Cycle

news shocks equilibrium kt = αkt−1 + θut = θ (1 − αL)(1 + θL)at ⇒ informationally fragile equilibrium without news shocks kt = αkt−1 + ˜ ut = θ−1 (1 − αL)(1 + θ−1L)at ⇒ informationally stable

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Application: Productivity Shocks with Nominal Rigidities

linearized economy from Lorenzoni (AER, 2009) yt = E(yt+1|Ωt) −

  • it − E(πt+1|Ωt)
  • πt = α(yt − at) + βE(πt+1|Ωt)

it = φπt productivity shocks at = at−1 + ut + θut−1, ut ∼ N(0, σ2

u)

θ > 1 to capture dynamics of diffusion process (Canova (2003))

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: Productivity Shocks with Nominal Rigidities

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: Productivity Shocks with Nominal Rigidities

solution can be reduced to yt = κE

  • yt+1|Ωt
  • + (1 − κ)at,

κ ≡ 1 1 + αφ ∈ (0, 1)

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: Productivity Shocks with Nominal Rigidities

solution can be reduced to yt = κE

  • yt+1|Ωt
  • + (1 − κ)at,

κ ≡ 1 1 + αφ ∈ (0, 1) fragile equilibrium, ARMA(1,1) yt − yt−1 = (1 + θκ)ut + θ(1 − κ)ut−1

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Introduction Model and Equilibrium Fragility and Stability Learning Applications Summary

Application: Productivity Shocks with Nominal Rigidities

solution can be reduced to yt = κE

  • yt+1|Ωt
  • + (1 − κ)at,

κ ≡ 1 1 + αφ ∈ (0, 1) fragile equilibrium, ARMA(1,1) yt − yt−1 = (1 + θκ)ut + θ(1 − κ)ut−1 stable equilibrium, ARMA(2,2) yt − θ − 1 θ yt−1 − 1 θ yt−2 = θ + κ θ ut + 1 − κ θ + θ + κ

  • ut−1 + (1 − κ)ut−2
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Application: Productivity Shocks with Nominal Rigidities

impulse response of output and employment to innovation in productivity

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Summary

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Summary

multivariate extension in appendix

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Summary

multivariate extension in appendix notion of informational fragility in dynamic settings

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Summary

multivariate extension in appendix notion of informational fragility in dynamic settings characterized fragile and stable equilibria

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Summary

multivariate extension in appendix notion of informational fragility in dynamic settings characterized fragile and stable equilibria connection with learning

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Summary

multivariate extension in appendix notion of informational fragility in dynamic settings characterized fragile and stable equilibria connection with learning applications: news shocks and productivity shocks with nominal rigidities

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Literature on Incomplete/Dispersed Information and Macroeconomics

recent literature (non-exhaustive list) decentralized islands: Woodford (2003), Hellwig (2005), Lorenzoni (2009, 2010), Amador and Weill (2010), Angeletos and La’O (2009, 2011) rational inattention: Sims (2002), Mackowiak and Wiederholt (2009) sticky information: Mankiw and Reis (2002), Reis (2006)

  • lder literature

Lucas (1972, 1973, 1975) King (1982) Townsend (1983) underground classic Futia (Econometrica, 1981)

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