Anon

Model Estimation 285

If we have to estimate k parameters, we can compute the first k empiri-
cal moments and equate them to the corresponding first k moments. We
obtain k equations the solutions for which yield the desired parameters:

μ

μ

1

1

1

=

=

=

=

∑

∑

X

n

X

n

i

i

n

k

i

k

i

n

 (13.21)

For example, suppose we know that the Y data in our sample given by
equation (13.1) are a random extraction from a normal distribution. A nor-
mal distribution is fully characterized by two parameters, the mean and the
variance. The first moment is equal to the mean; the second moment is equal
to the variance plus the square of the first moment. In fact:

(^) ()
μ=μ
σ= −μ =− μ+μ=μ−μ
μ=σ+μ

EX EX 2 X

1

2 2 22

21

2

2

2

(13.22)

Computing the first two empirical moments for the sample data given
by equation (13.1), we obtain

m

m

1

2

156

2 642

=

=

.

.

and solving for σμ, from equation (13.22) we obtain

μ=

σ= −=

1.56

(^22) 2.642 1.56 0.2084
Therefore, the MOM estimates that our sample data are drawn from the
following normal distribution: N(1.56, 0.2084). This is the same result
obtained above using the MLE method.
generalized Method of Moments
MOM is a parametric method insofar as it assumes that the functional form
of the distribution is known. MOM has been generalized to the generalized