Ambiguous Business Cycles Cosmin Ilut

∗

Martin Schneider ∗ Duke

∗∗

Univ.

∗∗ Stanford

Univ.

2nd BU/Boston Fed Conference on Macro-Finance Linkages, 2011

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

1 / 26

Motivation

Role of uncertainty shocks in business cycles? usually: uncertainty = risk ⇒ agents confident in probability assessments ⇒ shocks to uncertainty = shocks to volatility this paper: uncertainty = risk + ambiguity (Knightian uncertainty) ⇒ allows for lack of confidence in prob assessments ⇒ shocks to uncertainty can be shocks to confidence

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

2 / 26

Overview Standard business cycle model with ambiguity aversion recursive multiple priors preferences ambiguity about mean aggregate productivity ⇒ 1st order effects of uncertainty I I

Methodology I I I

study uncertainty shocks with 1st order approximation simple estimation strategy based on linearization motivate & bound set of priors by concern w/ nonstationarity

Properties I I

ambiguity shocks work like “unrealized” news shocks with bias. in medium scale DSGE model estimated on US data, ambiguity shocks F F

generate comovement and account for > imply countercyclical asset premia.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

1 2

of fluctuations in Y , C , I , H.

BU/Boston Fed, 2011

3 / 26

Literature 1

Multiple Priors Utility: Gilboa-Schmeidler (1989), Epstein and Wang (1994), Epstein and Schneider (2003).

2

Business cycles with preference for robustness: Hansen, Sargent and Tallarini (1999), Cagetti, Hansen, Sargent and Williams (2002), Smith and Bidder (2011).

3

Signals: Beaudry and Portier (2006), Christiano, Ilut, Motto and Rostagno (2008), Schmitt-Grohe and Uribe (2008), Jaimovich and Rebelo (2009), Lorenzoni (2009), Blanchard, L’Huillier and Lorenzoni (2010), Barsky and Sims (2010, 2011).

4

Risk shocks: Justiniano and Primiceri (2007), Fernandez-Villaverde and Rubio-Ramirez (2007), Bloom (2009), Bloom, Floetotto and Jaimovich (2009), Christiano, Motto and Rostagno (2010), Gourio (2010), Arellano, Bai and Kehoe (2010), Basu and Bundick (2011)

5

Shocks to ‘non-fundamentals’: Farmer (2009), Angeletos and La’O (2011), Martin and Ventura (2011).

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

4 / 26

Ambiguity aversion & preferences S = state space I I

one element s ∈ S realized every period histories s t ∈ S t

Consumption streams C = (Ct (s t )) Recursive multiple-priors utility (Epstein and Schneider (2003)) Ut C ; s t = u Ct s t + β min E p Ut+1 C ; s t+1 , p∈Pt (s t )

Primitives: I I

felicity u (possibly over multiple goods) & discount factor β one-step-ahead belief sets Pt (s t ) – size captures (lack of) confidence

Why this functional form? I I

preference for known odds over unknown odds (Ellsberg Paradox) formally, weaken Independence Axiom

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

5 / 26

A stylized business cycle model with ambiguity Representative agent with recursive multiple priors utility. Felicity from consumption, hours u(Ct , Nt ) =

Ct1−γ − βNt 1−γ

Output Yt produced by Yt = Zt Nt−1 Labor chosen one period in advance Belief sets that enter utility I I

I

specify ambiguity about exogenous productivity beliefs about endogenous variables derived from “structural knowledge” of economy true TFP process: iid lognormal with E [Zt ] = 1, var (log Zt ) = σz2

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

6 / 26

Belief set: time variation in ambiguity Agents experience changes in confidence, described by process at ⇒ Representation of one-step-ahead belief set Pt 1 log Zt+1 = µt − σz2 + σz εz,t+1 2 µt ∈ [−at , at ] Examples for evolution of (lack of) confidence at 1

Linear, homoskedastic law of motion at = (1 − ρa ) ¯a + ρa at−1 + εa,t

2

Interpretation: intangible information affects confidence Feedback from realized volatility p at = 2ησz,t Interpretation: observed turbulence lowers confidence Follows if Pt is ”constant entropy” ball around true DGP: µt ∈ [−at , at ] ⇔ Rt =

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

µ2t 2 ≤η 2σz,t BU/Boston Fed, 2011

7 / 26

Social planner problem

Bellman equation

0

µ

0

0

0

u(ZN, N ) + β min E V (N , Z , a ) V (N, Z , a) = max 0 N

µ∈[−a,a]

Worst-case belief: future technology is low µ∗ = −a ⇒ planner acts as if bad times ahead Interpretation: precautionary behavior First order effects of ambiguity.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

8 / 26

Characterizing equilibrium Two Steps 1

2

Solve planner problem under worst case belief µ∗ = −a Optimal hours from FOC h −γ 0 i 1 = E −a β Z 0 N 0 Z Characterize variables under true shock process (in logs) 1 nt = − (1/γ − 1) (at + γσz2 ) 2 yt+1 = zt+1 + nt

⇒ Worst case belief reflected in action nt , but not in shock realization zt+1

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

9 / 26

Properties of equilibrium Dynamics yt+1 = zt+1 + nt 1 nt = − (1/γ − 1) (at + γσz2 ) 2 at = (1 − ρa ) ¯a + ρa at−1 + εa,t I I 1 2

I I

first order effects of uncertainty on output, even as σz2 → 0 if substitution effect is strong enough (1/γ > 1) : loss of confidence generates a recession increase in confidence leads to an expansion TFP is not unusual in either cases hours do not forecast TFP if cov (at , zt+1 ) = 0

Asset prices reflect time varying ambiguity premia I

price of 1-step-ahead consumption claim =

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

Et Ct+1 Rtf

exp −at − γσz2 BU/Boston Fed, 2011

10 / 26

Comparison of shocks nt = − (1/γ − 1) (at + 12 γσz2 )

Ambiguity shocks I I I

loss of confidence generates recession if 1/γ > 1 hours do not forecast TFP: regress zt+1 on nt to get slope 0 time varying ambiguity premium on consumption claim = at

News & noise shocks I I I I

I I I

signal about productivity st = zt+1 + σs s,t with π := σz2 /(σz2 + σs2 ) bad signal (news or noise) generates recession if 1/γ > 1 hours forecast TFP: regress zt+1 on nt to get slope γ/ (1 − γ) constant risk premium on consumption claim

Volatility shocks I

nt = (1/γ − 1) πst − 12 γ (1 − π) σz2

2 nt = − (1/γ − 1) 12 γσz,t

2 volatility process σz,t = vart (zt+1 ), mean adjusts so Et Zt+1 = 1 volatility increase generates recession if 1/γ > 1 hours do not forecast TFP (but forecast turbulence) 2 time varying risk premium on consumption claim = γσz,t

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

11 / 26

General framework: Rep. agent & Markov uncertainty Notation (as before s = exogenous state) I I I

X = endogenous states (e.g. capital) A = agent actions (e.g. consumption, investment) Y = other endogenous variables (e.g. prices)

Recursive equilibrium I

Functions for actions A, other endog vars Y , value V s.t., for all (X , s):

V (X , s) =

max

u (c (A)) + β min E p V (X 0 , s 0 )

A∈B(Y ,X ,s) 0

p∈P(s) 0

s.t. X = T (X , A, Y , s, s ) I I

endog var determination: G (A, Y , X , s) = 0 true exogenous Markov state process p ∗ (s) ∈ P (s)

Analysis again in 2 steps I I

find recursive equilibrium characterize variables under “true” state process

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

12 / 26

Characterizing equilibrium: a guess-and-verify approach basic idea 1 2 3 4

guess the worst case belief p 0 find recursive equilibrium under expected utility & belief p 0 compute value function under worst case belief, say V 0 verify that the guess p 0 indeed achieves the minimum

“essentially linear” economies & productivity shocks I I

environment T , B, G s.t. 1st order approx. ok under expected utility ambiguity is about mean of innovations to s

simplification in essentially linear case I I

I

in step 1, guess that worst case mean is linear in state variables step 2 uses loglinear approximation around “zero risk” steady state (sets risk to zero, but retains worst case mean) step 4 then checks monotonicity of value function

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

13 / 26

An estimated DSGE model with ambiguity Similar to CEE (2005), SW (2007) 1

Intermediate goods producers I

Price setting monopolist; competitive in the factor markets F

2

Final goods producers. I

3

Combines intermediate goods to produce a homogenous good.

Households: ambiguity-averse I I

Own capital stock, consume, monopolistically supply specialized labor investment adjustment costs, internal habit in consumption. F

4

efficiency of investment and price of investment shocks.

“Employment agencies” I

5

mark-up shocks.

aggregate specialized labor into homogenous labor.

Government I

Taylor-type interest rule: reacts to inflation, output gap and growth. F

government spending and monetary policy shocks.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

14 / 26

Technology The intermediate good j is produced using the function: α Yj,t = Zt Kj,t (t Lj,t )1−α − Ft

Final goods: Z

1

Yj,t

Yt =

1 λf ,t

λf ,t dj

0

Capital accumulation: It ¯ ¯ It . Kt+1 = (1 − δ)Kt + 1 − S ζt It−1 Beliefs about technology: log Zt+1 = ρz log Zt + µt + σu εz,t+1 µt ∈ [−at , at ] at − ¯a = ρa (at−1 − ¯a) + σa εa,t C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

15 / 26

A bound on ambiguity Beliefs about technology log Zt+1 = ρz log Zt + µt + σu εz,t+1 µt ∈ [−at , at ] at − ¯a = ρa (at−1 − ¯a) + σa εa,t I I I I

agents view innovations as risky (σu εz,t+1 ) and ambiguous (µt ) they know empirical moments of true {µ∗t }, but not the exact sequence empirical moments say log Zt+1 − ρz log Zt ∼ i.i.N 0, σz2 ; σz2 > σu2 agents respond to uncertainty about µ∗t as if minimizing over [−at , at ]

Constrain at to lie in a maximal interval [−amax , amax ] I I

require that “boundary” beliefs ±amax imply “good enough” forecasts: there exists σu2 s.t. for every potential true DGP {µ∗t }, amax or −amax is best forecasting rule at least α of the time

I

amax is best forecasting rule at date t if true mean µ∗t > amax

I

for example, α = 5% implies amax = 2σz = bound used in estimation

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

16 / 26

Estimation Linearization → estimation using standard Kalman filter methods. Data: US 1984Q1-2010Q1: Output, consumption, investment, price of investment growth, hours, FFR, inflation. Law of motion for at is estimated Estimates: I

productivity dynamics ρz = 0.95, σz = 0.0045

I

confidence dynamics ¯a = 0.0043, ρa = 0.96, σa = 0.00041

I

other parameters broadly consistent with previous studies.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

17 / 26

Role of ambiguity in fluctuations

Variance decompositions: business cycle frequency Shock/Var. Ambiguity Stationary techn. Efficiency of invest. Stochastic Growth Price mark-up I I

Output 27 11 33 7 12

Consumption 51 13 7 7 12

Investment 14 9 53 6 12

Hours 30 2 32 13 13

any other shocks < 5% for above variables. long-run theoretical decomposition: εa,t about 50% of fluctuations.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

18 / 26

technology

ambiguity

GDP

1 0.5

8

−0.5

0

6

−1

−0.5

4 −1.5

−1

2 10

20

30

40

10

Consumption

20

30

40

10

Investment

20

30

40

Hours worked −0.4

−0.6

−1

−0.6

−0.8

−0.8 −2

−1 −1.2

−1 −1.2

−3

−1.4 20

30

40

10

ex−post excess return 0.02

6

0

4

−0.02

0

30

40

20

30

C.Ilut, M. Schneider (Duke, Stanford)

40

10

20

30

20

30

40

Net real interest rate

−0.04 10

10

price of capital

8

2

20

Annualized, percent

10 percent deviation from ss

percent deviation from ss

percent deviation from ss

Dynamics: loss of confidence

40

Ambiguous Business Cycles

1.4 1.3 1.2 10

20

30

40

BU/Boston Fed, 2011

19 / 26

Estimated ambiguity path −3

x 10

2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

1985

1990

C.Ilut, M. Schneider (Duke, Stanford)

1995

2000

Ambiguous Business Cycles

2005

2010

BU/Boston Fed, 2011

20 / 26

Historical shock decomposition Output Growth

Consumption Growth

0.02

0.02

0

0

−0.02

−0.02

−0.04

−0.04 Data (year over year)

−0.06 1985

Model implied: only ambiguity shock

1990

1995

2000

2005

−0.06 1985

1990

Investment Growth 0.4

0

0.2

−0.05

0

−0.1

−0.2

−0.15

−0.4

−0.2

−0.6 1990

1995

C.Ilut, M. Schneider (Duke, Stanford)

2000

2000

2005

Log Hours

0.05

−0.25 1985

1995

2005

−0.8 1985

Ambiguous Business Cycles

1990

1995

2000

2005

BU/Boston Fed, 2011

21 / 26

Welfare cost of fluctuations through ambiguity Setting σz = 0 vs estimate =⇒ ¯a = 0 vs estimate Welfare: V ≡ Value function under ”zero risk steady state” (with estimated ¯a) I

Welfare cost of fluctuations, as % of CSS (¯a = 0), due to:

1

ambiguity: λambig = V − V SS (¯a = 0) (1 − β)β −1 = 13%

2

risk (known probability distributions): λrisk = Vσσ (1 − β)β −1 = 0.01% F

Vσσ : effect of fluctuations in εz,t+1 in a second order approx. of V (.).

Other vars: Output, Capital, Consumption, Hours lower by 15% C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

22 / 26

Conclusion Standard business cycle model with ambiguity aversion: I

recursive multiple priors preferences.

I

ambiguity about mean productivity.

I

discipline from modeling concern with nonstationarity

With ambiguity, uncertainty shocks have 1st order effects: I

can apply standard linearization techniques for solution and estimation

I

work like “unrealized” news shocks with bias

I

potentially large role in business cycle

Next I

characterize further essentially linear settings

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

23 / 26

Parametrization Recall at − ¯a = ρa (at−1 − ¯a) + σa εa,t Restrictions: process at s.t. 1

at is positive: ¯a − m p

2

σa 1 − ρ2a

≥0

at is bounded by the discipline of the non-stationary argument ¯a + m p

σa 1 − ρ2a

≤ 2σz

Scale ¯a = nσz , n ∈ [0, 1] Directly estimate n, ρa .

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

24 / 26

Solution method: Steps Find deterministic “distorted” steady state xo based on −¯a zo = exp 1 − ρz Linearize around distorted SS: Find A, B : xt − xo = A(xt−1 − xo ) + BP(st−1 − so ) + B(Ξt − Ξo ) ∗ ∗ ρ 0 0 st−1 st b zt−1 + Ξt zt = 0 ρz −1 st = b at−1 at 0 0 ρa Equilibrium: I

True DGP dynamics: et−1 b b zt = E zt + at−1

I

Endogenous variables: b t − Ξo ) xt − xo = A(xt−1 − xo ) + BP(st−1 − so ) + B(Ξ b t = at−1 /σz Ξ 0(n−1)×1

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

25 / 26

Ellsberg Paradox

bet on black from risky urn bet on black from ambiguous urn bet on white from risky urn bet on white from ambiguous urn expected utility cannot capture choices, but minp∈P E p [u (c)] can! Ambiguity

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

26 / 26

∗

Martin Schneider ∗ Duke

∗∗

Univ.

∗∗ Stanford

Univ.

2nd BU/Boston Fed Conference on Macro-Finance Linkages, 2011

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

1 / 26

Motivation

Role of uncertainty shocks in business cycles? usually: uncertainty = risk ⇒ agents confident in probability assessments ⇒ shocks to uncertainty = shocks to volatility this paper: uncertainty = risk + ambiguity (Knightian uncertainty) ⇒ allows for lack of confidence in prob assessments ⇒ shocks to uncertainty can be shocks to confidence

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

2 / 26

Overview Standard business cycle model with ambiguity aversion recursive multiple priors preferences ambiguity about mean aggregate productivity ⇒ 1st order effects of uncertainty I I

Methodology I I I

study uncertainty shocks with 1st order approximation simple estimation strategy based on linearization motivate & bound set of priors by concern w/ nonstationarity

Properties I I

ambiguity shocks work like “unrealized” news shocks with bias. in medium scale DSGE model estimated on US data, ambiguity shocks F F

generate comovement and account for > imply countercyclical asset premia.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

1 2

of fluctuations in Y , C , I , H.

BU/Boston Fed, 2011

3 / 26

Literature 1

Multiple Priors Utility: Gilboa-Schmeidler (1989), Epstein and Wang (1994), Epstein and Schneider (2003).

2

Business cycles with preference for robustness: Hansen, Sargent and Tallarini (1999), Cagetti, Hansen, Sargent and Williams (2002), Smith and Bidder (2011).

3

Signals: Beaudry and Portier (2006), Christiano, Ilut, Motto and Rostagno (2008), Schmitt-Grohe and Uribe (2008), Jaimovich and Rebelo (2009), Lorenzoni (2009), Blanchard, L’Huillier and Lorenzoni (2010), Barsky and Sims (2010, 2011).

4

Risk shocks: Justiniano and Primiceri (2007), Fernandez-Villaverde and Rubio-Ramirez (2007), Bloom (2009), Bloom, Floetotto and Jaimovich (2009), Christiano, Motto and Rostagno (2010), Gourio (2010), Arellano, Bai and Kehoe (2010), Basu and Bundick (2011)

5

Shocks to ‘non-fundamentals’: Farmer (2009), Angeletos and La’O (2011), Martin and Ventura (2011).

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

4 / 26

Ambiguity aversion & preferences S = state space I I

one element s ∈ S realized every period histories s t ∈ S t

Consumption streams C = (Ct (s t )) Recursive multiple-priors utility (Epstein and Schneider (2003)) Ut C ; s t = u Ct s t + β min E p Ut+1 C ; s t+1 , p∈Pt (s t )

Primitives: I I

felicity u (possibly over multiple goods) & discount factor β one-step-ahead belief sets Pt (s t ) – size captures (lack of) confidence

Why this functional form? I I

preference for known odds over unknown odds (Ellsberg Paradox) formally, weaken Independence Axiom

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

5 / 26

A stylized business cycle model with ambiguity Representative agent with recursive multiple priors utility. Felicity from consumption, hours u(Ct , Nt ) =

Ct1−γ − βNt 1−γ

Output Yt produced by Yt = Zt Nt−1 Labor chosen one period in advance Belief sets that enter utility I I

I

specify ambiguity about exogenous productivity beliefs about endogenous variables derived from “structural knowledge” of economy true TFP process: iid lognormal with E [Zt ] = 1, var (log Zt ) = σz2

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

6 / 26

Belief set: time variation in ambiguity Agents experience changes in confidence, described by process at ⇒ Representation of one-step-ahead belief set Pt 1 log Zt+1 = µt − σz2 + σz εz,t+1 2 µt ∈ [−at , at ] Examples for evolution of (lack of) confidence at 1

Linear, homoskedastic law of motion at = (1 − ρa ) ¯a + ρa at−1 + εa,t

2

Interpretation: intangible information affects confidence Feedback from realized volatility p at = 2ησz,t Interpretation: observed turbulence lowers confidence Follows if Pt is ”constant entropy” ball around true DGP: µt ∈ [−at , at ] ⇔ Rt =

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

µ2t 2 ≤η 2σz,t BU/Boston Fed, 2011

7 / 26

Social planner problem

Bellman equation

0

µ

0

0

0

u(ZN, N ) + β min E V (N , Z , a ) V (N, Z , a) = max 0 N

µ∈[−a,a]

Worst-case belief: future technology is low µ∗ = −a ⇒ planner acts as if bad times ahead Interpretation: precautionary behavior First order effects of ambiguity.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

8 / 26

Characterizing equilibrium Two Steps 1

2

Solve planner problem under worst case belief µ∗ = −a Optimal hours from FOC h −γ 0 i 1 = E −a β Z 0 N 0 Z Characterize variables under true shock process (in logs) 1 nt = − (1/γ − 1) (at + γσz2 ) 2 yt+1 = zt+1 + nt

⇒ Worst case belief reflected in action nt , but not in shock realization zt+1

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

9 / 26

Properties of equilibrium Dynamics yt+1 = zt+1 + nt 1 nt = − (1/γ − 1) (at + γσz2 ) 2 at = (1 − ρa ) ¯a + ρa at−1 + εa,t I I 1 2

I I

first order effects of uncertainty on output, even as σz2 → 0 if substitution effect is strong enough (1/γ > 1) : loss of confidence generates a recession increase in confidence leads to an expansion TFP is not unusual in either cases hours do not forecast TFP if cov (at , zt+1 ) = 0

Asset prices reflect time varying ambiguity premia I

price of 1-step-ahead consumption claim =

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

Et Ct+1 Rtf

exp −at − γσz2 BU/Boston Fed, 2011

10 / 26

Comparison of shocks nt = − (1/γ − 1) (at + 12 γσz2 )

Ambiguity shocks I I I

loss of confidence generates recession if 1/γ > 1 hours do not forecast TFP: regress zt+1 on nt to get slope 0 time varying ambiguity premium on consumption claim = at

News & noise shocks I I I I

I I I

signal about productivity st = zt+1 + σs s,t with π := σz2 /(σz2 + σs2 ) bad signal (news or noise) generates recession if 1/γ > 1 hours forecast TFP: regress zt+1 on nt to get slope γ/ (1 − γ) constant risk premium on consumption claim

Volatility shocks I

nt = (1/γ − 1) πst − 12 γ (1 − π) σz2

2 nt = − (1/γ − 1) 12 γσz,t

2 volatility process σz,t = vart (zt+1 ), mean adjusts so Et Zt+1 = 1 volatility increase generates recession if 1/γ > 1 hours do not forecast TFP (but forecast turbulence) 2 time varying risk premium on consumption claim = γσz,t

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

11 / 26

General framework: Rep. agent & Markov uncertainty Notation (as before s = exogenous state) I I I

X = endogenous states (e.g. capital) A = agent actions (e.g. consumption, investment) Y = other endogenous variables (e.g. prices)

Recursive equilibrium I

Functions for actions A, other endog vars Y , value V s.t., for all (X , s):

V (X , s) =

max

u (c (A)) + β min E p V (X 0 , s 0 )

A∈B(Y ,X ,s) 0

p∈P(s) 0

s.t. X = T (X , A, Y , s, s ) I I

endog var determination: G (A, Y , X , s) = 0 true exogenous Markov state process p ∗ (s) ∈ P (s)

Analysis again in 2 steps I I

find recursive equilibrium characterize variables under “true” state process

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

12 / 26

Characterizing equilibrium: a guess-and-verify approach basic idea 1 2 3 4

guess the worst case belief p 0 find recursive equilibrium under expected utility & belief p 0 compute value function under worst case belief, say V 0 verify that the guess p 0 indeed achieves the minimum

“essentially linear” economies & productivity shocks I I

environment T , B, G s.t. 1st order approx. ok under expected utility ambiguity is about mean of innovations to s

simplification in essentially linear case I I

I

in step 1, guess that worst case mean is linear in state variables step 2 uses loglinear approximation around “zero risk” steady state (sets risk to zero, but retains worst case mean) step 4 then checks monotonicity of value function

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

13 / 26

An estimated DSGE model with ambiguity Similar to CEE (2005), SW (2007) 1

Intermediate goods producers I

Price setting monopolist; competitive in the factor markets F

2

Final goods producers. I

3

Combines intermediate goods to produce a homogenous good.

Households: ambiguity-averse I I

Own capital stock, consume, monopolistically supply specialized labor investment adjustment costs, internal habit in consumption. F

4

efficiency of investment and price of investment shocks.

“Employment agencies” I

5

mark-up shocks.

aggregate specialized labor into homogenous labor.

Government I

Taylor-type interest rule: reacts to inflation, output gap and growth. F

government spending and monetary policy shocks.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

14 / 26

Technology The intermediate good j is produced using the function: α Yj,t = Zt Kj,t (t Lj,t )1−α − Ft

Final goods: Z

1

Yj,t

Yt =

1 λf ,t

λf ,t dj

0

Capital accumulation: It ¯ ¯ It . Kt+1 = (1 − δ)Kt + 1 − S ζt It−1 Beliefs about technology: log Zt+1 = ρz log Zt + µt + σu εz,t+1 µt ∈ [−at , at ] at − ¯a = ρa (at−1 − ¯a) + σa εa,t C.Ilut, M. Schneider (Duke, Stanford)

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A bound on ambiguity Beliefs about technology log Zt+1 = ρz log Zt + µt + σu εz,t+1 µt ∈ [−at , at ] at − ¯a = ρa (at−1 − ¯a) + σa εa,t I I I I

agents view innovations as risky (σu εz,t+1 ) and ambiguous (µt ) they know empirical moments of true {µ∗t }, but not the exact sequence empirical moments say log Zt+1 − ρz log Zt ∼ i.i.N 0, σz2 ; σz2 > σu2 agents respond to uncertainty about µ∗t as if minimizing over [−at , at ]

Constrain at to lie in a maximal interval [−amax , amax ] I I

require that “boundary” beliefs ±amax imply “good enough” forecasts: there exists σu2 s.t. for every potential true DGP {µ∗t }, amax or −amax is best forecasting rule at least α of the time

I

amax is best forecasting rule at date t if true mean µ∗t > amax

I

for example, α = 5% implies amax = 2σz = bound used in estimation

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

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Estimation Linearization → estimation using standard Kalman filter methods. Data: US 1984Q1-2010Q1: Output, consumption, investment, price of investment growth, hours, FFR, inflation. Law of motion for at is estimated Estimates: I

productivity dynamics ρz = 0.95, σz = 0.0045

I

confidence dynamics ¯a = 0.0043, ρa = 0.96, σa = 0.00041

I

other parameters broadly consistent with previous studies.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

17 / 26

Role of ambiguity in fluctuations

Variance decompositions: business cycle frequency Shock/Var. Ambiguity Stationary techn. Efficiency of invest. Stochastic Growth Price mark-up I I

Output 27 11 33 7 12

Consumption 51 13 7 7 12

Investment 14 9 53 6 12

Hours 30 2 32 13 13

any other shocks < 5% for above variables. long-run theoretical decomposition: εa,t about 50% of fluctuations.

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

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technology

ambiguity

GDP

1 0.5

8

−0.5

0

6

−1

−0.5

4 −1.5

−1

2 10

20

30

40

10

Consumption

20

30

40

10

Investment

20

30

40

Hours worked −0.4

−0.6

−1

−0.6

−0.8

−0.8 −2

−1 −1.2

−1 −1.2

−3

−1.4 20

30

40

10

ex−post excess return 0.02

6

0

4

−0.02

0

30

40

20

30

C.Ilut, M. Schneider (Duke, Stanford)

40

10

20

30

20

30

40

Net real interest rate

−0.04 10

10

price of capital

8

2

20

Annualized, percent

10 percent deviation from ss

percent deviation from ss

percent deviation from ss

Dynamics: loss of confidence

40

Ambiguous Business Cycles

1.4 1.3 1.2 10

20

30

40

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Estimated ambiguity path −3

x 10

2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

1985

1990

C.Ilut, M. Schneider (Duke, Stanford)

1995

2000

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2005

2010

BU/Boston Fed, 2011

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Historical shock decomposition Output Growth

Consumption Growth

0.02

0.02

0

0

−0.02

−0.02

−0.04

−0.04 Data (year over year)

−0.06 1985

Model implied: only ambiguity shock

1990

1995

2000

2005

−0.06 1985

1990

Investment Growth 0.4

0

0.2

−0.05

0

−0.1

−0.2

−0.15

−0.4

−0.2

−0.6 1990

1995

C.Ilut, M. Schneider (Duke, Stanford)

2000

2000

2005

Log Hours

0.05

−0.25 1985

1995

2005

−0.8 1985

Ambiguous Business Cycles

1990

1995

2000

2005

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Welfare cost of fluctuations through ambiguity Setting σz = 0 vs estimate =⇒ ¯a = 0 vs estimate Welfare: V ≡ Value function under ”zero risk steady state” (with estimated ¯a) I

Welfare cost of fluctuations, as % of CSS (¯a = 0), due to:

1

ambiguity: λambig = V − V SS (¯a = 0) (1 − β)β −1 = 13%

2

risk (known probability distributions): λrisk = Vσσ (1 − β)β −1 = 0.01% F

Vσσ : effect of fluctuations in εz,t+1 in a second order approx. of V (.).

Other vars: Output, Capital, Consumption, Hours lower by 15% C.Ilut, M. Schneider (Duke, Stanford)

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BU/Boston Fed, 2011

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Conclusion Standard business cycle model with ambiguity aversion: I

recursive multiple priors preferences.

I

ambiguity about mean productivity.

I

discipline from modeling concern with nonstationarity

With ambiguity, uncertainty shocks have 1st order effects: I

can apply standard linearization techniques for solution and estimation

I

work like “unrealized” news shocks with bias

I

potentially large role in business cycle

Next I

characterize further essentially linear settings

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

23 / 26

Parametrization Recall at − ¯a = ρa (at−1 − ¯a) + σa εa,t Restrictions: process at s.t. 1

at is positive: ¯a − m p

2

σa 1 − ρ2a

≥0

at is bounded by the discipline of the non-stationary argument ¯a + m p

σa 1 − ρ2a

≤ 2σz

Scale ¯a = nσz , n ∈ [0, 1] Directly estimate n, ρa .

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

24 / 26

Solution method: Steps Find deterministic “distorted” steady state xo based on −¯a zo = exp 1 − ρz Linearize around distorted SS: Find A, B : xt − xo = A(xt−1 − xo ) + BP(st−1 − so ) + B(Ξt − Ξo ) ∗ ∗ ρ 0 0 st−1 st b zt−1 + Ξt zt = 0 ρz −1 st = b at−1 at 0 0 ρa Equilibrium: I

True DGP dynamics: et−1 b b zt = E zt + at−1

I

Endogenous variables: b t − Ξo ) xt − xo = A(xt−1 − xo ) + BP(st−1 − so ) + B(Ξ b t = at−1 /σz Ξ 0(n−1)×1

C.Ilut, M. Schneider (Duke, Stanford)

Ambiguous Business Cycles

BU/Boston Fed, 2011

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Ellsberg Paradox

bet on black from risky urn bet on black from ambiguous urn bet on white from risky urn bet on white from ambiguous urn expected utility cannot capture choices, but minp∈P E p [u (c)] can! Ambiguity

C.Ilut, M. Schneider (Duke, Stanford)

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