Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics...

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Temporal Resolution of Uncertaintyand Recursive Models of Ambiguity Aversion

Tomasz StrzaleckiHarvard University

Preference for Earlier Resolution of Uncertainty

• instrumental value of information Spence Zeckhauser (1972)

• intrinsic value of information Kreps Porteus (1978)

• hidden actions Ergin Sarver (2012)

Aversion to Persistence/Long Term Risk

Duffie Epstein (1992)

Ambiguity

Kenes (1921), Ellsberg (1961)

Ambiguity

Main Message: modeling tradeoffs

The way we choose to model ambiguity aversion will impact:

• preference for earlier resolution of uncertainty

• aversion to long-run risk,

so the model ties these dimensions of preference together

Similar to the modeling tradeoff in Epstein Zin (1989), where

• risk aversion

• intertemporal elasticity of substitution

• preference for earlier resolution of uncertainty and aversion tolong-run risk

are tied together

Preferences

Vt = u(ct)+βE (Vt+1) Discounted Expected Utility

Vt = W(u(ct), E (Vt+1)

)Kreps–Porteus, Epstein–Zin

Vt = u(ct)+βI (Vt+1) Discounted Ambiguity Aversion

Preferences

this % is indifferent to timing and to long run risks

Preferences

this % is

iff W (u, ·) is

convex

concave

linear

Preferences

• here the operator I replaces expectation

• captures ambiguity aversion

• question: how does PERU depend on I

Setting

T = 0, 1, . . . ,T — discrete time, T <∞

(S ,Σ) — shocks, measurable space

Ω = ST — states of nature

X — consequences, convex subset of a real vector space

h = (h0, h1, . . . , hT ) — consumption plan, ht : S t → X

Consumption Plan

IID uncertainty with EU

Vt(st , h) = u(ht(st)

((st , st+1), h

)dp(st+1 |st)

((st , st+1), h

)dp(st+1)

• what is IID is the underlying state process st

• consumption can have correlation

((st , st+1), h

)dp(st+1 |st)

((st , st+1), h

)dp(st+1)

((st , st+1), h

)dp(st+1 |st)

((st , st+1), h

)dp(st+1)

((st , st+1), h

)dp(st+1 |st)

((st , st+1), h

)dp(st+1)

generalize beyond EU

((st , st+1), h

)dp(st+1)

)+ βI

((st , ·), h

))where I : RS → R

I constant over time—IID Ambiguity (Epstein and Schneider, 2003)

((st , st+1), h

)dp(st+1)

)+ βI

((st , ·), h

((st , st+1), h

)dp(st+1)

)+ βI

((st , ·), h

Uncertainty Averse Preferences

I : RS → R is:

- continuous (supnorm)- monotonic (∀s∈S ξ(s) ≥ ζ(s)⇒ I (ξ) ≥ I (ζ))- normalized (∀r∈R I (r) = r)- quasiconcave

→ Uncertainty Aversion (Schmeidler, 1989)

Axiomatic foundations: Cerreia-Vioglio, Maccheroni, Marinacci,and Montrucchio (2011)

special cases

1. Maxmin expected utility: I (ξ) = minp∈C∫ξ dp

2. Variational: I (ξ) = minp∈∆(Σ)

[ ∫ξ dp + c(p)

]3. Multiplier: I (ξ) = minp∈∆(Σ)

[ ∫ξ dp + θR(p‖q)

]4. Confidence: I (ξ) = minp∈∆(Σ)|ϕ(p)≥α

∫ξ dp

]5. Second order expected utility: I (ξ) = φ−1

( ∫φ(ξ) dp

)6. Smooth ambiguity: I (ξ) = φ−1

( ∫∆(Σ)

φ(∫ξ dp) dµ(p)

special cases

1. Maxmin expected utility: Gilboa and Schmeidler (1989)

2. Variational: Marinacci, Maccheroni, and Rustichini (2006)

3. Multiplier: Hansen and Sargent (2001)

4. Confidence: Chateauneuf and Faro (2009)

5. Second order expected utility: Neilson (1993)

6. Smooth ambiguity: Klibanoff, Marinacci, and Muhkerhi (2005)

relations between them

• MEU ⊂ Variational

• MEU ⊂ Confidence

• Variational ∩ Confidence = MEU

• Multiplier ⊂ Variational

• SOEU ∩ Variational = Multiplier

• SOEU ⊂ KMM

two key properties

• I (ξ + k) = I (ξ) + k for all ξ ∈ RS , k ∈ R

– shift invariance, Constant Absolute Ambiguity Aversion(Variational)

• I (βξ) = βI (ξ) for all ξ ∈ RS , β ∈ (0, 1)

– scale invariance, Constant Relative Ambiguity Aversion(Confidence)

• so MEU has both

Discounted Uncertainty Averse Preferences

define by backward induction

VT (sT , h) := u(hT (sT ))

Vt(st , h) := u(ht(st)

)+ βI

((st , ·), h

)); t = 0, . . . ,T−1

so Dynamically Consistent:

• Sarin and Wakker (1998)

• Epstein and Schneider (2003)

• Marinacci, Maccheroni, and Rustichini (2006)

• Klibanoff, Marinacci, and Muhkerhi (2009)

(without a fixed filtration, well known problems with DC)

Applications to macroeconomics and finance

Dow and Werlang (1992), Epstein and Wang (1994), Chen andEpstein (2002), Epstein and Schneider (2007, 2008), Drechsler(2009), Ilut (2009), Mandler (2013), Condie Ganguli (2014),

Maenhout (2004), Karantounias, Hansen, and Sargent (2009),Kleshchelski and Vincent (2009), Barillas, Hansen, and Sargent(2009)

Ju and Miao (2012), Chen, Ju, and Miao (2009), Hansen (2007),Collard, Mukerji, Sheppard, and Tallon (2011), Benigno andNistico (2009)

Results

Main Message

What we assume about I will have impact on Preference for EarlierResolution of Uncertainty.

Theorem 1. A family of discounted uncertainty averse preferencessatisfies indifference to timing of resolution of uncertainty if andonly if I (ξ) = minp∈C

∫ξ dp.

• indifference to timing ⇒ shift-invariance and scale-invariance

• shift-invariance and scale-invariance ⇒ MEU

we have:

∃β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)

need to show:

∀β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)

(details in the paper)

we have:

∃β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)

need to show:

∀β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)

(details in the paper)

comments

• in some sense this argument could be used to axiomatize therecursive multiple priors model

• a related paper by Kochov (2012) axiomatizes MEU using astrong version of Stationarity, which has a flavor of IERU

Comparison to Risk

• Chew Epstein (1989) show IERU ⇒ EU

• Grant Kajii Polak (2000) show (rank-dependent orbetweenness) + PERU ⇒ EU

• so dispensing with objective probability makes more room

Theorem 2. A family of discounted variational preferences,I (ξ) = minp∈∆(Σ)

∫ξ dp + c(p), always satisfies preference toward

earlier resolution of uncertainty.

Theorem 3. A family of discounted confidence preferences,I (ξ) = minp∈∆(Σ)|ϕ(p)≥α

1ϕ(p)

∫ξ dp, displays a preference for

earlier resolution of uncertainty if and only if I (ξ) = minp∈C∫ξ dp.

Theorem 4. A family of discounted second order expected utilitypreferences I (ξ) = φ−1

( ∫φ(ξ) dp

)with φ concave, strictly

increasing and twice differentiable displays a preference for earlierresolution of uncertainty iff Condition 1 holds.

Condition 1. There exists a real number A > 0 such that−φ′′(x)φ′(x) ∈ [βA,A] for all x ∈ R

(this condition means that the curvature of φ doesn’t vary toomuch)

Theorem 5. A family of discounted smooth ambiguity preferencesI (ξ) = φ−1

( ∫∆(Σ) φ(

∫ξ dp) dµ(p)

)with φ concave, strictly

increasing and twice differentiable displays a preference for earlierresolution of uncertainty if Condition 1 holds

Only if under additional assumption on the support of µ.

Persistence

Theorem 6. A family of discounted variational preferences,I (ξ) = minp∈∆(Σ)

∫ξ dp + c(p), always satisfies preference for iid.

A family of discounted confidence preferences,I (ξ) = minp∈∆(Σ)|ϕ(p)≥α

1ϕ(p)

∫ξ dp, always satisfies preference

for iid.

In both cases, indifference to iid is satisfied if and only ifI (ξ) = minp∈C

∫ξ dp.

(I do not know how to extend this result to all of I )

Aggregator

(Vt+1((st , ·), h)

Vt(st , h) = W

(ht(st), I

(Vt+1((st , ·), h)

))where W : X × R→ R

Aggregator

(Vt+1((st , ·), h)

)Vt(st , h) = W

(ht(st), I

(Vt+1((st , ·), h)

))where W : X × R→ R

Recursive Uncertainty Averse Preferences

VT (sT , h) = v(hT (sT ))

Vt(st , h) = W

(ht(st), I

((st , ·), h

))); t = 0, . . . ,T−1

Question

(IMEU ,W disc)

Question

(IMEU ,W disc)

(I ,W disc)

Question

(IMEU ,W disc)

(I ,W disc)

(IMEU ,W )

Question

(IMEU ,W disc)

(I ,W disc)

(IMEU ,W )

Are they the same?

Question

(IMEU ,W disc)

(I ,W disc)

(IMEU ,W )

Same iff I (ξ) = minp∈C φ−1( ∫

φ(ξ) dp)

relation between the two models

• both W and I induce PERU

• but save for the case above, they are different %

• do they fit the data in a different way?

• W —workhorse model of macrofinance (Epstein–Zin)

• what does I add?

recent work: Epstein, Farhi, Strzalecki (2014)

• suppose you are endowed with a consumption process ht

• for what π ∈ (0, 1) are you indifferent between

[ht , gradual resolution] ∼ [(1− π)ht , early resolution]

• π ∈ (20%, 40%) for workhorse models in finance usingEpstein–Zin preferences (Bansal and Yaron 2004; Barro, 2009)

• how high is this number for models of ambiguity?

Conclusion:

interdependence of ambiguity and timing

MEU—only case of indifference

Questions:

theoretical: is this it? can we disentangle more?

empirical: how to measure this?

Thank you

Barillas, F., L. Hansen, and T. Sargent (2009): “Doubtsor variability?” Journal of Economic Theory, 144, 2388–2418.

Benigno, P. and S. Nistico (2009): “International PortfolioAllocation under Model Uncertainty,” NBER Working Papers14734, National Bureau of Economic Research, Inc.

Chen, H., N. Ju, and J. Miao (2009): “Dynamic AssetAllocation with Ambiguous Return Predictability,” mimeo.

Chen, Z. and L. Epstein (2002): “Ambiguity, Risk, and AssetReturns in Continuous Time,” Econometrica, 70, 1403–1443.

Collard, F., S. Mukerji, K. Sheppard, and J. M.Tallon (2011): “Ambiguity and the historical equitypremium,” mimeo.

Drechsler, I. (2009): “Uncertainty, Time-Varying Fear, andAsset Prices,” mimeo.

Epstein, L. G. and M. Schneider (2003): “IID:independently and indistinguishably distributed,” Journal ofEconomic Theory, 113, 32–50.

Epstein, L. G. and T. Wang (1994): “Intertemporal AssetPricing under Knightian Uncertainty,” Econometrica, 62,283–322.

Hansen, L. P. (2007): “Beliefs, Doubts and Learning: ValuingEconomic Risk,” American Economic Review, 97, 1–30.

Ilut, C. (2009): “Ambiguity Aversion: Implications for theUncovered Interest Rate Parity Puzzle,” mimeo.

Ju, N. and J. Miao (2012): “Ambiguity, Learning, and AssetReturns,” Econometrica, 80, 559–591.

Karantounias, A. G., L. P. Hansen, and T. J. Sargent(2009): “Managing expectations and fiscal policy,” mimeo.

Kleshchelski, I. and N. Vincent (2009): “RobustEquilibrium Yield Curves,” mimeo.

Maenhout, P. (2004): “Robust Portfolio Rules and AssetPricing,” Review of Financial Studies, 17, 951–983.

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