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

408 

The Nobel Prizes

approach, however, understates the class of possible GMM estimators in a po-

tentially important way. Hansen (1985) shows how to construct an efficiency 

bound for the much larger (infinite dimensional) class of GMM estimators. This 

efficiency bound is a greatest lower bound on the asymptotic efficiency of the 

implied GMM estimators. Not surprisingly, it is more challenging to attain this 

bound in practice. For some related but special (linear) time series problems, 

Hansen and Singleton (1996) and West et al. (2009) discuss implementation 

strategies.

There is a more extensive literature exploring these and closely related 

questions in an iid (independent and identically distributed) data setting, in-

cluding Chamberlain (1987), who looks at an even larger set of estimators. By 

connecting to an extensive statistics literature on semiparametric efficiency, he 

shows that this larger set does not improve the statistical efficiency relative to 

the GMM efficiency bound. Robinson (1987), Newey (1990), and Newey (1993) 

suggest ways to construct estimators that attain this efficiency bound for some 

important special cases.

16

 Finally, given the rich array of moment restrictions, 

there are opportunities for more flexible parameterizations of, say, a SDF pro-

cess. Suppose the conditional moment restrictions contain a finite-dimensional 

parameter vector of interest along with an infinite-dimensional (nonparamet-

ric) component. Chamberlain (1992) constructs a corresponding efficiency 

bound and Ai and Chen (2003) extend this analysis and estimation for such 

problems. While these richer efficiency results have not been shown in the time 

series environment I consider, I suspect that they can indeed be extended.

3.2.2  m

odel

 m

iSSPecificaTion

The approaches to GMM estimation that I have described so far presume a given 

parameterization of a SDF process. For instance, the analysis of GMM efficiency 

in Hansen (1982b) and Hansen (1985) and related literature presumes that the 

model is correctly specified for one value of the unknown (to the econometri-

cian) parameter. Alternatively, we may seek to find the best choice of a param-

eter value even if the pricing restrictions are only approximatively correct. In 

our paper, Hansen and Jagannathan (1997), we suggest a modification of GMM 

estimation in which appropriately scaled pricing errors are minimized. We pro-

pose this as a way to make model comparisons in economically meaningful 

ways. Recently, Gosh et al. (2012) adopt an alternative formulation of model 

16 

Relatedly, Zhang and Gijbels (2003), Kitamura et al. (2004) and Antoine et al. (2007) 

studied methods based on restricting nonparametric estimates of conditional density 

functions to attain Chamberlain (1987)’s efficiency bound in an estimation environment 

with independent and identically distributed data generation.

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Uncertainty Outside and Inside Economic Models 409

misspecification extending the approach of Stutzer (1995) described later. This 

remains an interesting and important line of investigation that parallels the dis-

cussion of model misspecification in other areas of statistics and econometrics. 

I will return to this topic later in this essay.

3.2.3  n

onPaRameTRic

 c

haRacTeRizaTion

A complementary approach to building and testing new parametric models is 

to treat the SDF process as unobserved by the econometrician. It is still possible 

to deduce empirical characterizations of such processes implied by asset market 

data. This analysis provides insights into modeling challenges by showing what 

properties a valid SDF process must possess.

It turns out that there are potentially many valid stochastic discount factors 

over a payoff horizon l:

 

s

≡

S

t

+

S

t

 

that will satisfy either (2) or the unconditional counterpart (3). For simplicity, 

focus on (3).

17

 With this in mind, let

 

′

y

= Y

t

+

( )

′ Z

t

′

q

= Q

t

( )

′ Z

t

 

where for notational simplicity, I omit the time subscripts on the left-hand side 

of this equation. In what follows I will assume some form of a Law of Large 

Numbers so that we can estimate such entities. See Hansen and Richard (1987) 

for a discussion of such issues. Rewriting (3) with this simpler notation:

 

E[sy′ – q′] = 0. 

(4)

This equation typically implies many solutions for a positive s > 0. In our pre-

vious discussion of parametric models, we excluded many solutions by adopting 

a parametric representation in terms of observables and an unknown parameter 

vector. In practice this often led to a finding that there were no solutions, that 

is no values of s solving (4), within the parametric family assumed for s. Using 

Hansen (1982b), this finding was formalized as a test of the pricing restrictions. 

The finding alone left open the question: rejecting the parametric restrictions 

17 

For conditional counterparts to some of the results I summarize see Gallant et al. 

(1990) and Cochrane and Hansen (1992).

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