288 The Basics of financial economeTrics
Now, given that a normal distribution has only two parameters, there
is no way we can fit exactly the three first moments obtaining g = 0. How-
ever, we can try to estimate the parameters of the normal distribution with
some optimization criterion that involves the first three (or eventually more)
moments. This is the essence of the GMM: optimizing using more moments
than parameters. How can we define an optimization criterion? It is rea-
sonable to base an optimization criterion on some linear combination of
the products of the differences between theoretical and empirical moments.
Therefore, if we write the vector of these differences as in equation (13.22),
an optimization criterion could minimize the expression
QY()μσ,, =gW' g (13.24)
where W is a positive definite symmetric weighting matrix. An expression of
the form in equation (13.24) is called a quadratic form.
The choice of the matrix W is a critical point. Each choice of W appor-
tions the weights to each of the three moments. To illustrate, let’s assume
that we give the same weight to each moment and therefore we choose W
equal to the identity matrix. Hence, QY()μσ,, =gg'. If we minimize this
expression, for example using the MATLAB function fminsearch, we obtain
the following estimates for the model’s parameters:
μ
σ
σ=
=
=
1 7354
2.
1.2930
1.6718
If we compute the theoretical moments, we obtain the following com-
parison:
m
m
m1.7354 1.8600
1.7354 4.5220
13.9298 13.9608
11
22
33μ= =
μ= =
μ= =and therefore the vector g becomes:
g–0. 3411
–1. 9543
–9. 2691
=
and the objective function Q assumes the value Q = 0.0425, while its initial
value was Q = 2.5528.
The above illustration includes the key elements of the GMM. The GMM
is based on identifying a number of independent conditions that involve both