Lars Peter Hansen Prize Lecture: Uncertainty Outside and Inside Economic Models

432 

The Nobel Prizes

reveals the outcome of Bayes’ rule conditioning on date t information where 

Mt is a date t likelihood ratio between the two original models. Since all convex 

combinations are considered, we thus allow all priors including point priors. 

Here I have considered mixtures of the two models, but the basic logic extends 

to a setting with more general a priori averages across models.

Expected utility minimization over a family of martingales provides a trac-

table way to account for this form of ambiguity aversion, as in max-min utility. 

Alternatively the minimization can be subject to penalization as in variational 

preferences. Provided that we can apply the Min-Max Theorem, we may again 

produce a (constrained or penalized) worst-case martingale distortion. The am-

biguity averse decision maker behaves as if he or she is optimizing using the 

worst-case martingale as the actual probability specification. This same martin-

gale shows up in first-order conditions for optimization and hence in equilib-

rium pricing relationships. With this as if approach I can construct a distorted 

probability starting from a concern about model misspecification. The focus on 

a worst-case distortion is the outcome of a concern for robustness to model 

misspecification.

Of course there is no “free lunch” for such an analysis. We must limit the 

family of martingales to obtain interesting outcomes. The idea of conducting a 

sensitivity analysis would seem to have broad appeal, but of course the “devil is 

in the details.” Research from control theory as reflected in Basar and Bernhard 

(1995) and Petersen et al. (2000), Hansen and Sargent (2001) and Hansen et 

al. (2006) and others has used discrepancies based on discounted versions of 

relative entropy measured by E[M

t 

log

 

M

t

⎪F

0

]. For a given date t this measure 

is the expected log-likelihood ratio under the M probability model and lends 

itself to tractable formulas for implementation.

39

 Another insightful formula-

tion is given by Chen and Epstein (2002), which targets misspecification of 

transition densities in continuous time. Either of these approaches requires 

additional parameters that restrict the search over alternative models. The sta-

tistical discrepancy measures described in Section 5 provide one way to guide 

this choice.

40

As Hansen and Sargent (2007) emphasize, it is possible to combine this mul-

tiple models approach with a multiple priors approach. This allows simultane-

ously for multiple benchmark models and potential misspecification. In addi-

tion there is ambiguity in how to weight the alternative models.

39 

See Strzalecki (2011) for an axiomatic analysis of associated preferences.

40 

See Anderson et al. (2003) for an example of this approach.

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Uncertainty Outside and Inside Economic Models 433

6.3  What Might We achieve?

For the purposes of this essay, the important outcome of this discussion is the 

ability to use ambiguity aversion or a concern about model misspecification as 

a way to generate what looks like distorted beliefs. In an application, Chamber-

lain (2000) studied individual portfolio problems from the vantage point of an 

econometrician (who could be placed inside a model) using max-min utility 

and featuring calculations of the endogenously determined worst-case models 

under plausible classes of priors. These worst-case models give candidates for 

the distorted beliefs mentioned in the previous section. A worst-case martingale 

belief distortion is part of the equilibrium calculation in the macroeconomic 

model of Ilut and Schneider (2014). These authors study simultaneously pro-

duction and pricing using a recursive max-min formulation of the type advo-

cated by Epstein and Schneider (2003) and introduce ambiguity shocks as an 

exogenous source of fluctuations.

Ambiguity aversion with unknown models provides an alternative to as-

suming large values of risk aversion parameters. This is evident from the control 

theoretic link between what is called risk sensitivity and robustness, noted in a 

variety of contexts including Jacobson (1973), Whittle (1981) and James (1992). 

Hansen and Sargent (1995) and Hansen et al. (2006) suggest a recursive for-

mulation of risk sensitivity and link it to recursive utility as developed in the 

economics literature. While the control theory literature features the equivalent 

interpretations for decision rules, Hansen et al. (1999), Anderson et al. (2003), 

Maenhout (2004) and Hansen (2011) consider its impact on security market 

prices. This link formally relies on the use of relative entropy as a measure of dis-

crepancy for martingales, but more generally I expect that ambiguity aversion 

often will have similar empirical implications to (possibly extreme) risk aversion 

for models of asset pricing. Formal axiomatic analyses can isolate behaviorally 

distinct implications. For this reason I will not overextend my claims of the ob-

servational similarity between risk and ambiguity. Axiomatic distinctions, how-

ever, are not necessarily present in actual empirical evidence.

The discussion so far produces an ambiguity component to prices in asset 

markets in addition to the familiar risk prices. There is no endogenous rationale 

for market compensations fluctuating over time. While exogenously specified 

stochastic volatility commonly used in asset pricing models also delivers fluc-

tuations, this is a rather superficial success that leaves open the question of what 

the underlying source is for the implied fluctuations. The calculations in Han-

sen (2007a) and Hansen and Sargent (2010) suggest an alternative mechanism. 

Investors concerned with the misspecification of multiple models view these 

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