286 The Basics of financial economeTrics
method of moments (GMM).^7 GMM does not assume complete knowledge
of the distribution but seeks to optimize a number of parameters of the
distribution.
To understand how the GMM works, let’s first go back to the previous
example and compute both the third empirical moment and the third theo-
retical moment in function of μσ,. The theoretical moment is obtained by
expanding the basic definition of moment as follows:
μμμμμμ33 3(^3233)
= ()=−()()+
=−()+−()+
EX EX
EX XXX
EX EX E
()− +
=−()+− ()+
μμ μμμμ2332(^33) (^3) ()XE−μμ^23 + μ
Given that EXEX()()−μ=−0, μ =σ
(^22)
, the theoretical third moment is
μσ 3 =+ 3 23 μμ
Let’s compare theoretical and empirical moments:
μ= =
μ= =
μ= =
m
m
m
1.5600 1.5600
2.6420 2.6420
4.7717 4.7718
11
22
33The theoretical and empirical values of the first two moments are obviously
identical because of equation (13.21). Note also that the theoretical and
empirical values of the third moment also almost coincide. If we compute
the differences,
(^) g
m
m
m
11
22
33
=
μ−
μ−
μ−
(13.23)
we obtain g≈ 0.
In order to illustrate the GMM, let’s use the sample of data shown in
Table 13.2. Notice that the values for X are the same as in Table 13.1 but the
(^7) The GMM framework was proposed in 1982 by Lars Hansen, the 2013 corecipient
of the Nobel Prize in Economic Sciences. See Lars P. Hansen, “Large Sample Prop-
erties of Generalized Methods of Moments Estimators,” Econometrica 50 (1982):
1029–1054.