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

404 

The Nobel Prizes

the macroeconomy including the linkages with financial markets that are pre-

sumed to exist. This is a lot to ask in early stages of model development. Of 

course, an eventual aim is to produce a full model of stochastic equilibrium.

The econometric tools that I developed are well suited to study a rich fam-

ily of asset pricing models, among other things. Previously, Ross (1978) and 

Harrison and Kreps (1979) produced mathematical characterizations of asset 

pricing in frictionless asset pricing markets implied by the absence of arbitrage. 

Their work provides a general way to capture how financial markets value risky 

payoffs. My own research, and that with collaborators, built on this conceptual 

approach, but with an important reframing. Our explicit consideration of sto-

chastic discounting, left implicit in the Ross (1978) and Harrison and Kreps 

(1979) framework, opened the door to new ways to conduct empirical studies 

of asset pricing models using GMM and related econometric methods. I now 

describe these methods.

3.1  a GMM approach to empirical asset Pricing

A productive starting point in empirical asset pricing is

 

E

S

t

+

S

t

⎛

⎝⎜

⎞

⎠⎟

Y

t

+

⎪F

t

⎡

⎣

⎢

⎤

⎦

⎥ = Q

t

 (1)

where S > 0 is a stochastic discount factor (SDF) process. In formula (1), Y

t+l

 is a 

vector of payoffs on assets at time t + l, and Q

t

 is a vector of corresponding asset 

prices. The event collection (sigma algebra), F

t

, captures information available 

to an investor at date t. The discount factor process is stochastic in order to 

adjust market values for risk. Each realized state is discounted differently and 

this differential discounting reflects investor compensation for risk exposure. 

Rational expectations is imposed by presuming that the conditional expecta-

tion operator is consistent with the probability law that governs the actual data 

generation. With this approach a researcher does not specify formally that prob-

ability law and instead “lets the data speak.”

Relations of type (1) are premised on investment decisions made in optimal 

ways and are fundamental ingredients in stochastic economic models. The spec-

ification of a SDF process encapsulates some economics. It is constructed from 

the intertemporal marginal rates of substitution of marginal investors. Investors 

consider the choice of consuming today or investing to support opportunities to 

consume in the future. There are a variety of investment opportunities with dif-

ferential exposure to risk. Investors’ risk aversion enters the SDF and influences 

the nature of the investment that is undertaken. While I have used the language 

of financial markets, this same formulation applies to investments in physical 

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Uncertainty Outside and Inside Economic Models 405

and human capital. In a model of a stochastic equilibrium, this type of relation 

holds when evaluated at equilibrium outcomes. Relation (1) by itself is typi-

cally not sufficient to determine fully a stochastic equilibrium, so focusing on 

this relation alone leads us to a partially specified model. Additional modeling 

ingredients are required to complete the specification. The presumption is that 

whatever those details might be, the observed time series come from a stochas-

tic equilibrium that is consistent with an equation of the form (1).

Implications of relation (1), including the role of SDFs and the impact of 

conditioning information used by investors, were explored systematically in 

Hansen and Richard (1987). But the origins of this empirically tractable for-

mulation traces back to Rubinstein (1976), Lucas (1978) and Grossman and 

Shiller (1981), and the conceptual underpinnings to Ross (1978) and Harrison 

and Kreps (1979).

9

 To implement formula (1) as it stands, we need to specify the 

information set of economic agents correctly. The Law of Iterated Expectations 

allows us to understate the information available to economic agents.

10

 For in-

stance let  F

t

⊂ F

t

 denote a smaller information set used by an external analyst. 

By averaging over the finer information set F

t

 conditioned on the coarser infor-

mation set  F

t

, I obtain

 

E

S

t

+

S

t

⎛

⎝⎜

⎞

⎠⎟

Y

t

+

( )

′ − Q

t

( )

′⎪F

t

⎡

⎣

⎢

⎤

⎦

⎥ = 0.  (2)

I now slip in conditioning information through the “back door” by constructing 

a conformable matrix Z

t

 with entries in the reduced information set (that are 

F



t

 measurable). Then 

 

E

S

t

+

S

t

⎛

⎝⎜

⎞

⎠⎟

(Y

t

+

′)Z

t

−(Q

t

′)Z

t

⎪F

t



⎡

⎣

⎢

⎤

⎦

⎥ = 0.

 

9 

The concept of a SDF was first introduced in Hansen and Richard (1987). Stochastic 

discount factors are closely connected to the “risk-neutral” probabilities used in valu-

ing derivative claims. This connection is evident by dividing the one-period SDF by its 

conditional mean and using the resulting random variable to define a new one-period 

conditional probability distribution, the risk neutral distribution.

10 

In his study of interest rates, Shiller (1972) in his PhD dissertation suggested omitted 

information as a source of an “error term” for an econometrician. In Hansen and Sargent 

(1980), we built on this insight by contrasting implications for a “Shiller error-term” as 

a disturbance term to processes that are unobserved to an econometrician and enter 

structural relations. In Hansen and Sargent (1991) we show how to allow for omitted 

information in linear or log-linear time series models using quasi-likelihood methods.

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