Temporal Resolution of Uncertainty and Recursive Models of Ambiguity Aversion. Tomasz Strzalecki Harvard University

Transcription

1 Temporal Resolution of Uncertainty and Recursive Models of Ambiguity Aversion Tomasz Strzalecki Harvard University

2 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)

3 Aversion to Persistence/Long Term Risk Duffie Epstein (1992)

4 Ambiguity Kenes (1921), Ellsberg (1961)

5 Ambiguity Kenes (1921), Ellsberg (1961)

6 Ambiguity Kenes (1921), Ellsberg (1961)

7 Ambiguity Kenes (1921), Ellsberg (1961)

8 Ambiguity Kenes (1921), Ellsberg (1961)

9 Ambiguity Kenes (1921), Ellsberg (1961)

10 Ambiguity Kenes (1921), Ellsberg (1961)

11 Ambiguity Kenes (1921), Ellsberg (1961)

12 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

13 Similar to the modeling tradeoff in Epstein Zin (1989), where risk aversion intertemporal elasticity of substitution preference for earlier resolution of uncertainty and aversion to long-run risk are tied together

14 Preferences V t = u(c t )+βe(v t+1 ) Discounted Expected Utility

15 Preferences V t = u(c t )+βe(v t+1 ) V t = W ( u(c t ), E(V t+1 ) ) Discounted Expected Utility Kreps Porteus, Epstein Zin

16 Preferences V t = u(c t )+βe(v t+1 ) V t = W ( u(c t ), E(V t+1 ) ) V t = u(c t )+βi (V t+1 ) Discounted Expected Utility Kreps Porteus, Epstein Zin Discounted Ambiguity Aversion

17 Preferences V t = u(c t )+βe(v t+1 ) Discounted Expected Utility this is indifferent to timing and to long run risks

18 Preferences V t = W ( u(c t ), E(V t+1 ) ) Kreps Porteus, Epstein Zin this is PERU PLRU iff W (u, ) is IERU convex concave linear

19 Preferences V t = u(c t )+βi (V t+1 ) Discounted Ambiguity Aversion here the operator I replaces expectation captures ambiguity aversion question: how does PERU depend on I

20 Setting

21 Setting T = {0, 1,..., T } discrete time, T < (S, Σ) shocks, measurable space Ω = S T states of nature X consequences, convex subset of a real vector space h = (h 0, h 1,..., h T ) consumption plan, h t : S t X

22 Consumption Plan

23 IID uncertainty with EU V t (s t, h) = u ( h t (s t ) ) ( + β V t+1 (s t, s t+1 ), h ) dp(s t+1 s t ) S

24 IID uncertainty with EU V t (s t, h) = u ( h t (s t ) ) + β V t (s t, h) = u ( h t (s t ) ) + β S S V t+1 ( (s t, s t+1 ), h ) dp(s t+1 s t ) V t+1 ( (s t, s t+1 ), h ) dp(s t+1 )

25 IID uncertainty with EU V t (s t, h) = u ( h t (s t ) ) + β V t (s t, h) = u ( h t (s t ) ) + β S S V t+1 ( (s t, s t+1 ), h ) dp(s t+1 s t ) V t+1 ( (s t, s t+1 ), h ) dp(s t+1 ) what is IID is the underlying state process s t

26 IID uncertainty with EU V t (s t, h) = u ( h t (s t ) ) + β V t (s t, h) = u ( h t (s t ) ) + β S S V t+1 ( (s t, s t+1 ), h ) dp(s t+1 s t ) V t+1 ( (s t, s t+1 ), h ) dp(s t+1 ) what is IID is the underlying state process s t consumption can have correlation

27 generalize beyond EU V t (s t, h) = u ( h t (s t ) ) ( + β V t+1 (s t, s t+1 ), h ) dp(s t+1 ) S

28 generalize beyond EU V t (s t, h) = u ( h t (s t ) ) ( + β V t+1 (s t, s t+1 ), h ) dp(s t+1 ) S V t (s t, h) = u ( h t (s t ) ) ( + βi (V t+1 (s t, ), h )) where I : R S R

29 generalize beyond EU V t (s t, h) = u ( h t (s t ) ) ( + β V t+1 (s t, s t+1 ), h ) dp(s t+1 ) S V t (s t, h) = u ( h t (s t ) ) ( + βi (V t+1 (s t, ), h )) where I : R S R I constant over time IID Ambiguity (Epstein and Schneider, 2003)

30 Uncertainty Averse Preferences I : R S 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)

31 special cases 1. Maxmin expected utility: I (ξ) = min p C ξ dp 2. Variational: I (ξ) = min p (Σ) [ ξ dp + c(p) ] 3. Multiplier: I (ξ) = min p (Σ) [ ξ dp + θr(p q) ] 4. Confidence: I (ξ) = min {p (Σ) ϕ(p) α} [ 1 ϕ(p) 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) )

32 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)

33 relations between them MEU Variational MEU Confidence Variational Confidence = MEU Multiplier Variational SOEU Variational = Multiplier SOEU KMM

34 two key properties I (ξ + k) = I (ξ) + k for all ξ R S, k R shift invariance, Constant Absolute Ambiguity Aversion (Variational) I (βξ) = βi (ξ) for all ξ R S, β (0, 1) scale invariance, Constant Relative Ambiguity Aversion (Confidence) so MEU has both

35 Discounted Uncertainty Averse Preferences define by backward induction V T (s T, h) := u ( h T (s T )) V t (s t, h) := u ( h t (s t ) ) + βi ( V t+1 ( (s t, ), 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)

36 Applications to macroeconomics and finance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (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 and Nisticò (2009)

37 Results

38 Main Message What we assume about I will have impact on Preference for Earlier Resolution of Uncertainty.

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42 Theorem 1. A family of discounted uncertainty averse preferences satisfies indifference to timing of resolution of uncertainty if and only if I (ξ) = min p C ξ dp.

43 Proof indifference to timing shift-invariance and scale-invariance shift-invariance and scale-invariance MEU

44 Proof

45 Proof

46 Proof

47 Proof

48 Proof

49 Proof

50 Proof

51 Proof

52 Proof we have: β (0,1) x R ξ R S x + βi (ξ) = I (x + βξ)

53 Proof we have: β (0,1) x R ξ R S x + βi (ξ) = I (x + βξ) need to show: β (0,1) x R ξ R S x + βi (ξ) = I (x + βξ) (details in the paper)

54 comments in some sense this argument could be used to axiomatize the recursive multiple priors model a related paper by Kochov (2012) axiomatizes MEU using a strong version of Stationarity, which has a flavor of IERU

55 Comparison to Risk Chew Epstein (1989) show IERU EU Grant Kajii Polak (2000) show (rank-dependent or betweenness) + PERU EU so dispensing with objective probability makes more room

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59 Theorem 2. A family of discounted variational preferences, I (ξ) = min p (Σ) ξ dp + c(p), always satisfies preference toward earlier resolution of uncertainty.

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64 Theorem 3. A family of discounted confidence preferences, 1 I (ξ) = min {p (Σ) ϕ(p) α} ϕ(p) ξ dp, displays a preference for earlier resolution of uncertainty if and only if I (ξ) = min p C ξ dp.

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67 Theorem 4. A family of discounted second order expected utility preferences I (ξ) = φ 1( φ(ξ) dp ) with φ concave, strictly increasing and twice differentiable displays a preference for earlier resolution of uncertainty iff Condition 1 holds. Condition 1. There exists a real number A > 0 such that [βa, A] for all x R φ (x) φ (x) (this condition means that the curvature of φ doesn t vary too much)

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72 Theorem 5. A family of discounted smooth ambiguity preferences I (ξ) = φ 1( (Σ) φ( ξ dp) dµ(p) ) with φ concave, strictly increasing and twice differentiable displays a preference for earlier resolution of uncertainty if Condition 1 holds Only if under additional assumption on the support of µ.

73 Persistence

74 Persistence Theorem 6. A family of discounted variational preferences, I (ξ) = min p (Σ) ξ dp + c(p), always satisfies preference for iid. A family of discounted confidence preferences, 1 I (ξ) = min {p (Σ) ϕ(p) α} ϕ(p) ξ dp, always satisfies preference for iid. In both cases, indifference to iid is satisfied if and only if I (ξ) = min p C ξ dp. (I do not know how to extend this result to all of I )

75 Aggregator

76 Aggregator V t (s t, h) = u ( h t (s t ) ) +βi ( V t+1 ((s t, ), h) )

77 Aggregator V t (s t, h) = u ( h t (s t ) ) +βi ( V t+1 ((s t, ), h) ) V t (s t, h) = W (h t (s t ), I ( V t+1 ((s t, ), h) )) where W : X R R

78 Recursive Uncertainty Averse Preferences V T (s T, h) = v ( h T (s T )) ( ( V t (s t, h) = W h t (s t ( ), I V t+1 (s t, ), h ))) ; t = 0,..., T 1

79 (I MEU, W disc ) Question

80 Question (I, W disc ) (I MEU, W disc )

81 Question (I, W disc ) (I MEU, W disc ) (I MEU, W )

82 Question (I, W disc ) (I MEU, W disc ) Are they the same? (I MEU, W )

83 Question (I, W disc ) (I MEU, W disc ) Same iff I (ξ) = min p C φ 1 ( φ(ξ) dp ) (I MEU, W )

84 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?

85 recent work: Epstein, Farhi, Strzalecki (2014) suppose you are endowed with a consumption process h t for what π (0, 1) are you indifferent between [h t, gradual resolution] [(1 π)h t, early resolution] π (20%, 40%) for workhorse models in finance using Epstein Zin preferences (Bansal and Yaron 2004; Barro, 2009) how high is this number for models of ambiguity?

86 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?

87 Thank you

88 Barillas, F., L. Hansen, and T. Sargent (2009): Doubts or variability? Journal of Economic Theory, 144, Benigno, P. and S. Nisticò (2009): International Portfolio Allocation under Model Uncertainty, NBER Working Papers 14734, National Bureau of Economic Research, Inc. Chen, H., N. Ju, and J. Miao (2009): Dynamic Asset Allocation with Ambiguous Return Predictability, mimeo. Chen, Z. and L. Epstein (2002): Ambiguity, Risk, and Asset Returns in Continuous Time, Econometrica, 70, Collard, F., S. Mukerji, K. Sheppard, and J. M. Tallon (2011): Ambiguity and the historical equity premium, mimeo. Drechsler, I. (2009): Uncertainty, Time-Varying Fear, and Asset Prices, mimeo. Epstein, L. G. and M. Schneider (2003): IID: independently and indistinguishably distributed, Journal of Economic Theory, 113, Epstein, L. G. and T. Wang (1994): Intertemporal Asset Pricing under Knightian Uncertainty, Econometrica, 62,