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

424 

The Nobel Prizes

bound. If the minimized value is zero, the probability distortion vanishes and 

investors eventually settle on the benchmark model as being correct.

A straightforward derivation shows that even when we change the roles of 

the benchmark model and the alternative model, the counterpart to κ(M) re-

mains the same.

31

 Why is Chernoff entropy interesting? When this common 

decay rate is small, even long histories of data are not very informative about 

model differences.

32

 Elsewhere I have explored the connection between this 

Chernoff measure and Sharpe ratios commonly used in empirical finance, see 

Anderson et al. (2003) and Hansen (2007a).

33

 The Chernoff calculations are of-

ten straightforward when both models (the benchmark and perturbed models) 

are Markovian. In general, however, it can be a challenge to use this measure 

in practice without imposing considerable a priori structure on the alternative 

models.

In what follows, I will explore discrepancy measures that are similar to 

this Chernoff measure but are arguably more tractable to implement. What I 

describe builds directly on my discussion of GMM methods and extensions. 

Armed with factorization (13), approaches that I suggested for the study of SDFs 

can be adapted to the study of belief distortions. I elaborate in the discussion 

that follows.

5.3  ignored belief distortions

Let me return to GMM estimation and model misspecification. Recall that the 

justification for GMM estimation is typically deduced under the premise that 

the underlying model is correctly specified. The possibility of permanent belief 

distortions, say distortions for which κ(M) > 0, add structure to the model mis-

specification. But this is not enough structure to identify fully the belief distor-

tion unless an econometrician uses sufficient asset payoffs and prices to reveal 

the SDF. Producing bounds with this extra structure can still proceed along the 

lines of those discussed in Section 3.2.3 with some modifications. I sketch below 

one such approach.

31 

With this symmetry and other convenient properties of κ(M), we can interpret the 

measure as a metric over (equivalence classes of ) martingales.

32 

Bayesian and max-min decision theory for model selection both equate decay rates in 

type I and type II error rates.

33 

The link is most evident when a one-period (in discrete time) or local (in continuous 

time) measure of statistical discrimination is used in conjunction with a conditional nor-

mal distribution, instead of the large t measure described here.

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Uncertainty Outside and Inside Economic Models 425

Suppose the investors in the model are allowed to have distorted beliefs, and 

part of the estimation is to deduce the magnitude of the distortions. How big 

would these distortions need to be in a statistical sense in order to satisfy the 

pricing restrictions? What follows makes some progress in addressing this ques-

tion. To elaborate, consider again the basic pricing relation with distorted beliefs 

written as unconditional expectation:

 

E

M

t

+

S

t

+

M

t

S

t

⎛

⎝⎜

⎞

⎠⎟

(Y

t

+

′)Z

t

−(Q

t

′)Z

t

⎡

⎣

⎢

⎤

⎦

⎥ = 0.

 

As with our discussion of the study of SDFs without parametric restrictions, 

we allow for a multiplicity of possible martingales and impose bounds on expec-

tations of convex functions of the ratio 

M

t

+

M

t

.

To deduce restrictions on M, for notational simplicity I drop the t subscripts 

and write the pricing relation as:

 

E(m

s

′

y

− ′

q )

= 0

E(m

−1) = 0.

 (17)

To bound properties of m solve

 

inf

m

>0

E[

φ(m)]  (18)

subject to (17) where ϕ is given by equation (5). This formulation nests many 

of the so-called F-divergence measures for probability distributions including 

the well known Kullback-Leibler divergence (θ = –1, 0). A Chernoff-type mea-

sure can be imputed by computing the bound for –1 < θ < 0 and optimizing 

after an appropriate rescaling of the objective by θ(1 + θ). As in the previous 

analysis of Section 3.2.3, there may be many solutions to the equations given in 

(17). While the minimization problem selects one of these, I am interested in 

this optimization problem to see how small the objective can be in a statistical 

sense. If the infimum of the objective is small, then statistically small changes in 

distributions suffice to satisfy the pricing restrictions. Such departures allow for 

“behavior biases” that are close statistically to the benchmark probabilities used 

in generating the data.

I have just sketched an unconditional approach to this calculation by al-

lowing conditioning information to be used through the “back door” with 

the specification of Z but representing the objective and constraints in terms 

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