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

406 

The Nobel Prizes

Under an asset pricing interpretation, (Y

t+l

)′Z

t

 is a synthetic payoff vector 

with a corresponding price vector (Q

t

)′Z

t

. Finally, we may form the uncondi-

tional expectation by averaging over the coarser conditioning information set 

F



t

:

 

E

S

t

+

S

t

⎛

⎝⎜

⎞

⎠⎟

Y

t

+

( )

′ − Q

t

( )

′ |F

t



⎡

⎣

⎢

⎤

⎦

⎥ = 0.  (3)

This becomes an estimation problem once we parameterize the SDF in terms 

of observables and unknown parameters to be estimated.

Hansen and Singleton (1982) is an initial example of this approach.

11

 In 

that work we consider the case in which the SDF process can be constructed 

from observables along with some unknown parameters. Economics comes 

into play in justifying the construction of the SDF process and sometimes in 

the construction of returns to investment. From an econometric perspective, 

time series versions of Laws of Large Numbers and Central Limit Theorems give 

us approximate ways to estimate parameters and test restrictions as in Hansen 

(1982b).

In Hansen (1982b), I also studied statistical efficiency for a class of GMM 

estimators given a particular choice of Z in a manner that extends an approach 

due to Sargan (1958, 1959).

12

 When (3) has more equations than unknown pa-

rameters, multiple GMM estimators are the outcome of using (at least implic-

itly) alternative linear combinations of these equations equal to the number of 

parameters. Since there are many possible ways to embark on this construction, 

there is a family of GMM estimators. This family of estimators has an attainable 

efficiency bound derived and reported in Hansen (1982b).

13

 When the number 

11 

An earlier application of GMM inference is found in my work Hansen and Hodrick 

(1980). In that paper we studied the empirical relationship between the logarithm of a 

future spot exchange and the logarithm of the current forward rate and other possible 

predictors. We applied ordinary least squares in our work, but with corrected standard 

errors. Others were tempted to (and in fact did) apply generalized least squares (GLS) to 

“correct for” serial correlation, but applied in this setting GLS is statistically inconsistent. 

The counterpart to the moment conditions studied here are the least squares orthogo-

nality conditions. The contract interval played the role of l in this least squares analysis 

and was typically larger than one. In subsequent work, Hansen and Hodrick (1983), we 

used a SDF formulation to motivate further empirical characterizations, which led us 

to confront over-identification. See also Bilson (1981) and Fama (1984) who featured a 

cross-currency analysis.

12 

See Arellano (2002) for a nice discussion relating GMM estimation to the earlier work 

of Sargan.

13 

See Hansen (2007b) for a pedagogical discussion of GMM estimation including discus-

sions of large sample statistical efficiency and tests.

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Uncertainty Outside and Inside Economic Models 407

of equations exceeds the number of free parameters, there is also a direct way to 

test equations not used formally in estimation. While nesting estimators into a 

general GMM framework has great pedagogical value, I was particularly inter-

ested in applying a GMM approach to problems requiring new estimators as in 

many of the applications to financial economics and elsewhere.

14

Notice that the model, as written down in equation (3), is only partially 

specified. Typically we cannot invert this relation, or even its conditional coun-

terpart, to deduce a full time series evolution for economic aggregates and fi-

nancial variables.

15

 Other relations would have to be included in order to obtain 

a full solution to the problem.

3.2  Further econometric Challenges

I now digress temporarily and discuss some econometric extensions that I and 

others contributed to.

3.2.1  S

emiPaRameTRic

 e

fficiency

Since the model is only partially specified, the estimation challenge leads di-

rectly to what is formally called a semiparametric problem. Implicitly the re-

mainder of the model can be posed in a nonparametric manner. This gives 

rise to a problem with a finite-dimensional parameter vector of interest and an 

infinite-dimensional “nuisance” parameter vector representing the remainder 

of the model. This opens the door to the study of semiparametric efficiency of 

a large class of estimators as will be evident from the discussion that follows. In 

typical GMM problems, the actual introduction of the nuisance parameters can 

be sidestepped.

Relation (2) conditions on the information set of economic agents. We have 

great flexibility in choosing the matrix process Z. The entries of Z

t

 should be 

in the  F

t

  information set, but this still leaves many options when building a 

Z process. This flexibility gives rise to an infinite class of estimators. In Han-

sen (1982b), I studied statistical efficiency given a particular choice of Z. This 

14 

Other econometricians have subsequently found value in unifying the treatment of 

GMM estimators into a broader type of extremum estimators. This, however, misses 

some of the special features of statistical efficiency within a GMM framework and does 

not address the issue of how to construct meaningful estimators from economic models.

15 

For those reluctant to work with partially specified models, Lucas (1978) showed how 

to close a special case of this model by considering an endowment economy. But from 

an empirical standpoint, it is often not necessary to take the endowment nature of the 

economy literally. The consumption from the endowment economy may be conceived of 

as the equilibrium outcome of a model with production and preserves the same pricing 

relations.

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