Aversion to Ambiguity and Model Misspecification in Dynamic

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Downloaded by guest on September 26, 2021 a Hansen Peter Lars environments in stochastic misspecification dynamic model and ambiguity to Aversion www.pnas.org/cgi/doi/10.1073/pnas.1811243115 uncertainty uncertainty. of presence limiting problems the control in time solving uncertainty. for continuous subjective equations imply Hamilton–Jacobi–Bellman to representations aversion recursive recursive ambiguity These construct smooth we and penal- ized for illustrations, allow that preferences these intertemporal of representations on time Drawing design, continuous By revealing our limits. have environments. environments illustrate time stochastic discrete We stylized these uncertainty. some incorporating on to to previ- approach responses approach generalizing tractable conceived By a disciplines. broadly propose many we valu- in analyses, a analysis ous are settings of dynamic tool Strzalecki) in Tomasz able and misspecification Maccheroni potential Fabio uncertainty by its reviewed subjective and 2018; to 29, aversion June accommodate review that for Preferences (sent 2018 27, July Hansen, Peter Lars by Contributed China and 100081, China; Beijing 610074, Economics, Chengdu and Economics, Finance and Finance of 60637; University IL Southwestern Chicago, Chicago, of University Business, yn hc,w itnus nasapwytecontributions joint the the of way entropy sharp relative the a before and in likelihood under- distinguish the an examples to from we using exposure the problems shock, changing two By lying misspecifica- these limits. solve prior some explore we about and 3, concern potential section a the In targets tion. features problem problem statistics. from second this familiar A density illustrate, predictive we the of As misspecification 3. back dating known ref. control the robust to underlies and prefer- first sensitive The variational risk 2. between the and connection 1 of refs. by cases axiomatized special ences are entropy when relative Both pose using we revealing penalizations. problems 2, be robustness section static to In alternative observation aversion. find two future of we forms a that various parameter over conceptualizing decomposition unknown distribution entropy an probability relative and joint a Specifi- present the uncertainty. we of 1, quantify section to in entropy cally, relative Shannon’s of the which limit. for the in equations, provide contribute aversions also (HJB) they counterparts Hamilton–Jacobi–Bellman but limiting right, time to own continuous meaningful their to in examples as interest These guidance of us. only interest not that ambigu- are to misspecification aversion same model of the and exam- forms some of ity some for implementation appealing Through mechanical not is a another. approach that to suggest we environment parame- ples, one rate discount from subjective ters and parameters risk-aversion guidance. little provide to treatments question axiomatic a existing environments, which alternative across parameters transport model to these how to is arises and that issue prefer- ambiguity, important An calibrate to misspecification. to aversion how including decide parameters, must ence maker decision a applications, however, In responses. uncertainty capture with to ways researchers alternative applied provide that different preferences rep- of justifies resentations on theory many axioms decision impose Axiomatic developed choices. to over theory preferences decision have in standard researchers is it approaches, While play. into I eateto cnmc,Uiest fCiao hcg,I 60637; IL Chicago, Chicago, of University Economics, of Department h usino o ocp ihsbetv netit comes uncertainty economics, subjective and with cope theory, to how decision of theory, question the control statistics, n ebidoraayi nmlil tp.W rtueaversion a use first We steps. multiple in analysis our build We transport to economics within practice common become has It | robustness a,b,c,1,2 | misspecification n inu Miao Jianjun and | ambiguity d eateto cnmc,Bso nvriy otn A02215; MA Boston, University, Boston Economics, of Department d,e,f,1 | risk b eateto ttsis nvriyo hcg,Ciao L60637; IL Chicago, Chicago, of University Statistics, of Department 1073/pnas.1811243115/-/DCSupplemental ulse nieAgs 8 2018. 28, August online Published at online information supporting contains article This under distributed 2 is article 1 BY-NC-ND). (CC 4.0 access License NonCommercial-NoDerivatives open This interest. of conflict no declare authors The University. Harvard T.S., and University; Bocconi F.M., Reviewers: paper. the wrote and tools, contributed reagents/analytic research, performed new research, designed J.M. and L.P.H. contributions: Author rpnybtenpoaiiymaue.W s tt ud our guide dis- to it of use measure We tractable measures. and probability attractive between an crepancy is entropy Relative Penalization HJB Entropy 1. provides it in uncertainty Moreover, confront forms. to 7. various maker time its and decision a discrete 6, allow the that 5, equations of uncertainty refs. counterpart alternative of a to specifications aversions provides of This specifica- forms components. time continuous distinct general with more in tion a Finally, propose limit. an we time 5, continuous including misspecifi- how section a environments, as model on emerges dynamic that and light environment alternative ambiguity shed across to to prior cation aversion calculations to aversion parameterize these an know to by use we motivated which We be 4, can uncertainty. sometimes ref. 5 from ref. specification from ambiguity parametrization smooth learn- convenient a of recursive explore assess- simple also We of the environments. sequence ing compounding a of of in misspecification for probabilities impact potential subjective the the and ambiguity, study risk, of example, we ments For 4, uncertainty environments. section of stochastic in calibration dynamic and in formulation preferences the into inputs economy. able limiting so the parameters in aversion even intact, scale remains to impact way their that a provide We distribution. owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To work.y this to equally contributed J.M. and L.P.H. f hne r rmsn ra o plcto fteetools. these of application climate for and areas finance, promising are macroeconomics, change, as such obscure, is Brown-ture struc- with uncertainty’s where Problems environments structures. information time ian revealing continuous have for representations implications subjective these of design, and misspecification By averse uncertainty. potential ambiguity the be to about for them concerned allow preferences that intertemporal makers recursive decision adopts analysis Our conceived. incor- broadly uncertainty, to to ways responses behavioral revealing porate and tractable suggest We of economics. and engineering, ways from statistics, in insights conceptualized alternative as on theory explore decision drawing by We challenges poten- models. these the in acknowledging assess mistakes or for uncertainty it tial finds the maker quantify decision to a settings, challenging economic dynamic many In Significance hn cnmc n aaeetAaey eta nvriyof University Central Academy, Management and Economics China h iista eepoei eto r,b ein valu- design, by are, 3 section in explore we that limits The PNAS | etme 1 2018 11, September . e nttt fCieeFnnilStudies, Financial Chinese of Institute | o.115 vol. www.pnas.org/lookup/suppl/doi:10. raieCmosAttribution- Commons Creative | o 37 no. c ot colof School Booth | 9163–9168

APPLIED ECONOMIC MATHEMATICS SCIENCES formulation of preferences for a decision maker with an aversion enforces a commitment to the baseline specifications. Since the to model ambiguity and concerns about misspecification. As a function U(y) depends only on y and not on θ, the solution precursor to formulating such preferences, we explore the rela- distorts only the predictive density tive entropy relations between priors, likelihoods, and predictive 1 densities in a static setting. exp − U (y) φˆ(y) φ∗(y) = κ We start with a prior distribution π for hypothetical parameter R exp − 1 U (y)φˆ(y)τ(dy) values θ over a set Θ and a likelihood λ that informs us of the Y κ density for possible outcomes y ∈ Y given θ (with respect to a and not the posterior π+ of θ given y. The resulting minimized measure τ). Let p denote the implied joint probability measure objective is over Y × Θ: Z 1 ˆ p(dy, dθ) = λ(y|θ)τ(dy)π(dθ). −κ log exp − U (y) φ(y)τ(dy). Y κ The corresponding predictive density for y that integrates over The exponential tilting toward outcomes y with lower utility θ is Z is familiar from the extensive literature applying relative entropy φ(y) = λ(y|θ)π(dθ). penalizations in control and estimation problems. In particular, Θ Problem 2.1 has the same solution as In our analysis, it is the predictive density that the decision maker Z cares about when taking actions, but he or she has uncertainty min U (y)φ(y)τ(dy) φ about subjective inputs into its construction. We introduce base- Y Z h i line counterparts with ˆ· and explore discrepancies relative to +κ log φ(y) − log φˆ(y) φ(y)τ(dy). these baselines. Y The relative entropy discrepancy of the joint distribution for outcomes and parameters is defined as This latter problem is the static counterpart to the dynamic recursive specification used extensively in robust control theory: ∗ Z dπ for instance, refs. 3, 8, 9, 10, and 11. While the construc- D (p | bp)= log (θ) π(dθ) tion of the predictive density φˆ embeds the reference prior Θ dπˆ " # πˆ, prior sensitivity is only confronted in an indirect way by Z Z λ(y|θ) Problem 2.1 through potential modifications in the predictive + λ(y|θ) log τ(dy)π(dθ), [1] Θ Y λˆ(y|θ) density. As an alternative, we target prior robustness by restricting λ = where we presume that p is in a set P of probability measures λ.ˆ This eliminates specification concerns about the likelihood. that are absolutely continuous with respect to the baseline bp. One justification for this omission is that uncertainty about how This representation gives us two potential contributions to rel- a prior weights alternative likelihoods already gives us a way to ative entropy: one coming from differences in the prior and the capture many forms of likelihood uncertainty. Let Π denote the other coming from differences in the likelihood. set of priors that are absolutely continuous with respect to πˆ and As we show in SI Appendix, there is an equivalent way to solve Problem 2.2. represent the same relative entropy in terms of predictive densities and posterior distributions, where the posterior distribution, Problem 2.2. + Z Z dπ π , is min U (θ)π(dθ) + κ log (θ) π(dθ), λ(y | θ) π∈Π dπˆ π+(dθ | y) = π(dθ) Θ φ (y) where + Z and πˆ is defined analogously. U (θ) ≡ U (y)λˆ(y|θ)τ(dy). Y 2. Relative Entropy and Robustness Problem Notice that U is constructed without likelihood uncertainty. In In this section, we pose and solve two problems that adjust for using the formulation to investigate prior sensitivity, we penalize robustness. These problems are the ingredients to preferences only the prior contribution to relative entropy. This formulation that capture a concern for robustness. While both use relative follows ref. 5. entropy to constrain the robustness assessment, one does so in a The solution to this problem gives the worst case prior more restricted way to target the robust choice of a prior. The solutions to both are special cases of the solution to a more exp − 1 U (θ) π∗(dθ) = κ πˆ(dθ), general problem that entails exponential tilting toward lower R exp − 1 U (θ)πˆ(dθ) expected utilities. Θ κ Consider the first robust evaluation of a y-dependent utility provided that the right-hand side integral is finite. The optimized U(y). objective is

Problem 2.1. Z 1 Z − κ log exp − U (θ) πˆ(dθ). [2] min U (y)dp + κD (p | p). Θ κ p∈P b The solution to this particular problem is recognizable as a A more primitive starting point would be to write Ue (a, y), smooth ambiguity objective and a special case of ref. 4.† where a is chosen before y is observed. We may then rank alternative actions using the corresponding solutions to this problem. Notably, we presume that the utility function does not depend on ∗Fabio Maccheroni reminded us that an alternative way to connect with this robust the unknown parameter. control theory literature is to impose a prior with a point mass on a single value of θ In this problem, κ > 0 is a parameter that penalizes the search and targeting only likelihood uncertainty. for robustness in the likelihood and prior. The κ = ∞ limit †Refs. 12 and 13 discuss earlier motivations for smooth ambiguity decision problems.

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APPLIED ECONOMIC MATHEMATICS SCIENCES environments in particular.§ Given that there are known ways and the conditional variance update is for ambiguity and misspecification aversion to have an impact 2 on preferences in discrete time, we find it most appealing to (qt ) qt+ − qt = 2 . [7] adopt parameterizations with more meaningful and revealing |ς| + qt continuous time limits. In what follows, partition the interval [0, 1] into subintervals Both equations have well-known continuous time limits. The of length 1/n, and let = 1/n. Let Fj be the sigma algebra limiting version of Eq. 6 for the conditional mean is generated by θ and {Y0, Y1, Y2, ... , Yj } for j = 0, 1, 2, ... , 2 n − 1. qt |ς| dm = d log Y − m dt + dt , t |ς|2 t t 2 A. Risk Aversion. We first consider a recursive construction of risk aversion over subintervals. Introduce a certainty equivalent and the limiting version of Eq. 7 is operator applied to a positive random variable X that is Ft+ 2 measurable: dqt (qt ) = − . dt |ς|2 1 1−γ 1−γ R,t (X ) ≡ E X | F t ,

where t = j and γ > 0 represents the risk-aversion parameter. C. Recursive Robust Priors. Analogous to ref. 2, deﬁne a certainty For γ = 1, R,t (X ) ≡ exp [E (log X | Ft )]. Note that equivalent robustness adjustment operator h γ 2i 1 log (Y ) = log Y + µ(θ) − |ς| ,t (X ) ≡ −κ() log E exp − X | G R,t t+ t 2 B κ() t

2 for > 0. The risk adjustment γ|ς| /2 scales with . for a random variable X that is Ft measurable. Here, we allow Introduce a recursive construction for the certainty equivalent the robustness parameter κ() > 0 to depend on . Motivated of the terminal consumption Y1, by the static model in sections 3 and 4, we define the recursive adjustment for robust priors for log utility over the terminal consumption Y1, Ut = R,t (Ut+), U1 = Y1 [5] log Bt = ,t [E (log Bt+ | F )], B1 = Y1, [8] for t = 0, , ... ,(n − 1). We show that B t for t = 0, , ... ,(n − 1). h γ i U = (Y ) = Y exp µ(θ) − |ς|2 We show in SI Appendix that 0 R1,0 1 0 2 log Bt = log Yt + (1 − t)mt + bt , for any value of . Notice that, in the recursive construction on the right-hand side of Eq. 5, we use the same risk-aversion where coefficient γ in the certainty equivalent operator for any . 2 2 2 2 This simple example shows why the common practice of hold- |ς| + (1 − t)qt |ς| bt+ − bt = qt 2 + . ing γ fixed across environments (in our case, indexed by ) 2κ() |ς| + qt 2 can be sensible, even as we shrink the exposure to risk by looking at small time intervals, due to the law of iterated From this formula, we see that the worst case prior/posterior expectations. for θ has a mean given by

2 B. Parameter Learning. Before investigating a recursive specifica- |ς| + (1 − t)qt mt − qt 2 . tion of smooth ambiguity, we remind readers of Bayesian updat- κ() |ς| + qt ing within this setting. Analogous to our previous discussion, we let Gj be the sigma algebra generated by {Y0, Y1, Y2 ... , Yj } If the robustness parameter κ() is held constant independent for j = 0, 1, 2, ... , n − 1, but we exclude the random param- of , then robustness vanishes in the continuous time limit as eter θ from the construction. The resulting family {Gj : j = → 0. By contrast, we let κ()= κd as in Problem 3.1 so that 0, 1, 2, ... , n − 1} captures the reduced information structure for penalization used in making robust prior/posterior adjustments a decision maker that does not know the parameter realization. scales with . Take small limits, and find For simplicity, let µ (θ)= θ. 2 2 2 Suppose that the date 0 prior for θ is normal with mean dbt 1 |ς| + (1 − t)qt |ς| = qt + . m0 and variance q0. Only data on Y are used to make infer- dt 2κ |ς|2 2 d ences about θ. The posterior conditioned on Gt is normal with mean mt and variance qt . The Bayesian recursive updating for The first term on the right-hand side is the recursive robust- mt+ is ness adjustment that remains present in a continuous time limit because of how we scale the robustness penalty as a function mt+ − mt = of . q t log Y − log Y − m + |ς|2 , [6] |ς|2 + q t+ t t 2 D. Risk Aversion and Smooth Ambiguity. We next explore “smooth t ambiguity adjustments” from ref. 4 over different time intervals, including ones that are arbitrarily small. For X that is Ft measurable, define a certainty equivalent operator §The calculations build in part on the prior work in ref. 15, which first discussed produc- tive ways for robustness concerns to persist in a continuous time limiting specification. − 1 h −α() i α() Our analysis here addresses these issues more generally and forges connections to a A,t (X ) ≡ E X |Gt , broader collection of previous contributions.

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A 2 log − = = a eedon depend may t | | | 2 2 2 t |ς ≥ log V (1 + = (1 + A α | en mohabgiyadjustments ambiguity smooth Define . t 0} 2 |ς h ν t A β + t log = | mut oipsn ifiie ambiguity “infinite” imposing to amounts ,t 2 (θ ea bevbesohsi rcs and process stochastic observable an be − otne opa oeo icutn in discounting of role a play to continues − q (V )dt Y [R t t t )q )q t t ,t Y ) and + t t saatdto adapted is Y t (A V (1 + 2 2 t t ny h eiinmkrscon- maker’s decision The only. t θ ς + α (γ n let and , t n atrsaeso othe to aversion captures and {F h · − )] , q dW − t t t } (γ + )m A 1)|ς A t etefitaingener- filtration the be 1 , α() t t V = θ. {G + = | − {G T 2 Y a B t + α 1)| given, 1 } t t etesm for same the be t , . h } etefiltration the be α() ς ysetting by clstevari- the scales n aifisa satisfies and n 1 | 2 hc nthe in which , Y hl we while , + 2 1 q , |ς t 2 | = α() 2 . γ [9] = . where vlto ftecniuto au rcs.Ntc that Notice process. value continuation the of evolution Eq. of sign equal of write logarithms, property taking homogeneity and the recursion Using this substitution. of elasticity ral to adapted process sumption where and where operators: adjustment ambiguity adjustment. lognormal familiar the captures evolution This value tinuation aversion. misspecification prior and/or for equation preference HJB an maker’s derive the will decision We the processes. consumption of over construction recursive give (Ω, aifistecmol sdadcnein recursion convenient and that used Suppose commonly the discounting. satisfies subjective and consumption diate satisfies Lemma, efis td h oa ikadjustment, risk local the study first we where edrv huitcly h ii ftergthn ieo the of side right-hand the of limit the (heuristically) derive We and risk define we 3–5, sections in analyses the by Motivated con- the of logarithm the use to convenient most it find We Let interme- includes that specification preference a consider We V {V F V t , ,t w ∗ = t w v {F > δ {W = lim } R ↓0 t ,t B ,t exp = oicroaers vrin mohabgiyaversion, ambiguity smooth aversion, risk incorporate to ,t ∗ is t ,t ∗ ν [1 lim = }, = 0 t ↓0 is t 1 G = 0 (v : (θ (w stesbetv icutrate, discount subjective the is − − P). 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APPLIED ECONOMIC MATHEMATICS SCIENCES With these intermediate calculations, we construct the local More generally, E (exp [−αh νt (θ)]|Gt ) is the moment- counterpart to Eq. 10 by dividing both sides of the equation by generating function for νt when viewed as a function of θ and taking small limits, resulting in conditioned on Gt . In general, a moment-generating function is not guaranteed to be finite for all αh . Even when the moment- generating function is finite for small αh , there may only be a δ Ct 1−ρ 0 = − 1 compact interval of αh for which the function remains finite. 1 − ρ Vt The existence of a finite upper bound for αh has a recognizable connection to breakdown points in dynamic control theory. 1 γ 2 − log E (exp [−αh νt (θ)] |Gt )− |ςt | . [11] Remark 5.3: Recursive Eq. 11 also gives the continuous time αh 2 counterpart to a discrete time recursion in ref. 5 that captures two forms of robustness. The first term on the right-hand side comes from discounting, As we discussed previously, smooth ambiguity adjustment Eq. intermediate consumption, and intertemporal substitution. The 12 has a dual interpretation as the outcome of prior/posterior second term captures uncertainty to the drift induced by the uncertainty adjustment. For the likelihood adjustment, there is unknown parameter vector θ. We motivate this either as a robust a well-known link between risk sensitivity and robustness dating prior/posterior adjustment or as an aversion to “smooth ambigu- back to ref. 3.# Capture the likelihood uncertainty by represent- ity” in the unknown drift. The third term captures the local risk h adjustment coming from exposure to the underlying Brownian ing dWt = ht dt + dWt under a change of probability measure, h motion. for which dWt is a Brownian increment and ht is a local drift Using this limiting recursion for an HJB equation requires that distortion. This adjustment alters the implied value function νt we use a value function and its derivatives to deduce formulas for by ςt · ht . Minimizing the local evolution of the value function by κf 2 νt (θ) and ςt as functions of the relevant Markov-state vector. the choice of ht subject to a penalization, 2 |ht | , illustrates a We conclude with four connections to related literature in link between robustness and risk sensitivity that is familiar in the control theory and economics. control literature. Remark 5.1: Recursion Eq. 11 provides a continuous time ana- Remark 5.4: To construct the filtration {Gt : t ≥ 0} in prac- log to discrete time recursions in refs. 6 and 7. Moreover, it shows tice and use it for solving a control problem, we must produce how to incorporate ambiguity aversion into the continuous time a recursive solution for a filtering or estimation problem. While specifications of refs. 17 and 18. we posed the analysis as one for which θ is an unknown param- Remark 5.2: The ambiguity adjustment eter, in fact, this parameter could be an actual process designed to capture time variation in some underlying parameters. The G 1 sigma algebra t could condition on the entire process or just − log E (exp [−αh νt (θ)] |Gt ) [12] the process up to time t. Examples of recursive constructions αh include Zakai equations, Kalman filtering, or Wohham filtering depending on the application. for the drift is a smooth counterpart to a continuous time ambiguity adjustment with period-by-period constraints when ACKNOWLEDGMENTS. We thank Michael Barnett, William Brock, Peter Hansen, Massimo Marinacci, and Grace Tsiang for helpful comments. L.P.H. the random set Ht = {νt (θ): θ ∈ Θ} is a compact subset of a acknowledges Alfred P. Sloan Foundation Grant G-2018-11113 for the Euclidean space with probability 1. As αh becomes arbitrarily support. large, the smooth ambiguity adjustment converges to the mini- mum over Ht . The result is a continuation value recursion of the form considered in ref. 19. #A continuous time analysis is, for instance, in ref. 9.

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