Implied Probabilities in GMM Estimators

THECONVENTIONAL WAY to estimate a distribution function is to assume it belongs to a
class parameterized by a finite-dimensional vector and then estimate the unknown
parameter vector. In many cases, e.g., regression models, part of the assumption is of the
form: a given function of the data and of the parameter vector has a zero mean. In this
note, we consider estimating distribution functions using only assumptions of this type
(moment restrictions). We do not assume that the distribution function belongs to a
finite-dimensional parametric class. The motivation for this exercise is that moment
restrictions are often implied by theory (a good example is asset pricing models), but
distributional assumptions typically are not. We will write the moment restriction on the
true distribution function F, as jG(x, 8,)dF0(x) = 0, where x E Bn,G: Bnx BP-+ Bq
is a given function, and 8, E BPis an unknown parameter vector.
The obvious nonparametric estimator of the distribution function is the sample
distribution function. This does not use the information in the moment restrictions. A
nonparametric estimator that does use the information in the moment restrictions is
given by what Manski (1988) calk the second form of the analogy principle. This
principle is to select an estimator 0 of 8, by minimizing the distance (relatiye to some
metric)-between the sample distribution function-and a distribution function F satisfying
jG(x,O)dF(x) = 0. The distribution function F is therefore the member of the class
{Fl(38)lG(x, O)dF(x) = 0) that is closest to the samplePistribution function.*The first
form of the analogy principle is to choose the estimator 8 which makes jG(x, 8) dFs(x)
as close as possible to zero, relative to some metric on Bq,where FSdenotes the sample
distribution function. GMM (Hansen (1982)) is the leading example of this form of the
analogy principle. It is not immediately obvious that this yields a distribution function
estimator. To explain how it does and to describe the estimator are the purposes of this
note.