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

420 

The Nobel Prizes

beliefs inside the model for two reasons. First, investor beliefs may differ from 

those implied by the model even if other components of the model are correctly 

specified. For instance, when historical evidence is weak, there is scope for be-

liefs that are different from those revealed by infinite histories of data. Second, 

if some of the model ingredients are not correct but only approximations, then 

the use of model-based beliefs based on an appeal to rational expectations is less 

compelling. Instead there is a rationale for the actors inside the model to adjust 

their beliefs in face of potential misspecification.

For reasons of tractability and pedagogical simplicity, throughout this and 

the next section I use a baseline probability model to represent conditional ex-

pectations, but not necessarily the beliefs of the people inside the model. Pre-

suming that economic actors use the baseline model with full confidence would 

give rise to a rational expectations formulation, but I will explore departures 

from this approach. I present a tractable way to analyze how varying beliefs 

will alter this baseline probability model. Also, I will continue my focus on the 

channel by which SDFs affect asset values. A SDF and the associated risk prices, 

however, are only well-defined relative to a baseline model. Alterations in beliefs 

affect SDFs in ways that can imitate risk aversion. They also can provide an ad-

ditional source of fluctuations in asset values.

My aim in this section is to study whether statistically small changes in be-

liefs can imitate what appears to be a large amount of risk aversion. While I fea-

ture the role of statistical discipline, explicit considerations of both learning and 

market discipline also come into play when there are heterogeneous consumers. 

For many environments there may well be an intriguing interplay between these 

model ingredients, but I find it revealing to narrow my focus. As is evident from 

recent work by Blume and Easley (2006), Kogan et al. (2011) and Borovička 

(2013), distorted beliefs can sometimes survive in the long run. Presumably 

when statistical evidence for discriminating among models is weak, the impact 

of market selection, whereby there is a competitive advantage of confidently 

model specification a model. A model builder may impose these restrictions prior to 

looking at the data. The expectations become “rational” once the model is fit to data, 

assuming that the model is correctly specified. I used GMM and related methods to ex-

amine only a portion of the implications of a fully specified, fully solved model. In such 

applications, an empirical economist is not able to use a model solution to deduce the be-

liefs of economic actors. Instead these methods presume that the beliefs of the economic 

actors are consistent with historical data as revealed by the Law of Large Numbers. This 

approach presumes that part of the model is correctly specified, and the data are used as 

part of the implementation of the rational expectations restrictions.

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Uncertainty Outside and Inside Economic Models 421

knowing the correct model, will at the very least be sluggish. In both this and the 

next section, I am revisiting a theme considered by Hansen (2007a).

5.1  Martingale Models of belief Perturbations

Consider again the asset pricing formula but now under an altered or perturbed 

belief relative to a baseline probability model:

 

E

S

t

+

S

t

⎛

⎝⎜

⎞

⎠⎟

Y

t

+

⎪F

t

⎡

⎣

⎢

⎤

⎦

⎥ = Q

t

 (12)

where the Ẽ is used to denote the perturbed expectation operator and S̃ is the 

SDF derived under the altered expectations. Mathematically, it is most conve-

nient to represent beliefs in an intertemporal environment using a strictly posi-

tive (with probability one) stochastic process M with a unit expectation for all t 

≥ 0. Specifically, construct the altered conditional expectations via the formula:

 

E B

τ

⎪F

t

⎡⎣

⎤⎦ = E

M

τ

M

t

⎛

⎝⎜

⎞

⎠⎟

B

τ

⎪F

t

⎡

⎣

⎢

⎤

⎦

⎥

 

for any bounded random variable B

τ

 in the date τ ≥ t information set F

τ

. The 

martingale restriction imposed on M is necessary for the conditional expecta-

tions for different calendar dates to be consistent.

29

Using a positive martingale M to represent perturbed expectations we re-

write (12) as:

 

E

M

t

+

S

t

+

M

t

S

t

⎛

⎝⎜

⎞

⎠⎟

Y

t

+

⎪F

t

⎡

⎣

⎢

⎤

⎦

⎥ = Q

t

 

which matches our original pricing formula (1) provided that

 

S = MS̃. (13)

29 

The date zero expectation of random variable B

t

 that is in the F

t

 information set may 

be computed in multiple ways

 

E B

t

⎪F

0

⎡⎣

⎤⎦ = E

M

τ

M

0

⎛

⎝⎜

⎞

⎠⎟

B

t

⎪F

0

⎡

⎣

⎢

⎤

⎦

⎥ = E

M

t

M

0

⎛

⎝⎜

⎞

⎠⎟

B

t

⎪F

0

⎡

⎣

⎢

⎤

⎦

⎥

 

 

for any τ ≥ t. For this equality to hold for all bounded random variables B

t

 in the date t 

information set, E(M

τ

 ⎪F

t

) = M

t

. This verifies that M is a martingale relative to {F

t

 : t ≥ 0}.

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