Anon

284 The Basics of financial economeTrics

the variables z are independent from the errors ε and are correlated with x.
The variables z are referred to as instruments.
Instruments lead to a consistent estimator of the regression parameters.
In fact, it can be demonstrated that the following (k + 1)-vector is a consis-
tent IV estimator of the regression parameters:

bZ=()''XZyb=()ββ 0 ,...,'k (13.20)

In general, the IV estimator is not efficient. Efficiency of the IV estimators
improves if the instruments are highly correlated with the regressors. In prac-
tice, it might be very difficult to find instruments for a given regression model.

Method of Moments

The method of moments (MOM) estimation approach is the oldest estima-
tion method for estimating the parameters of a population. The intuition
behind the MOM is simple: MOM estimates the parameters of a probability
distribution by equating its moments with the empirical moments computed
for the sample. The MOM assumes that (1) we know the form of the dis-
tribution of the population from which the sample has been extracted and
(2) moments can be expressed in terms of the parameters of the distribution.
In general, this latter condition is satisfied for all usual distributions.^6
Given a random variable X with a given distribution PP= ()θ, where θ
is a k-vector of parameters, the jth moment of P is defined as

μμjj= ()θ =EX()j

Suppose now that we have a sample of n variables independently extracted
from the same distribution P. The jth empirical moment of P, defined as

m

X

j n

i

j

i

n

= =

∑

1

is a function of the data. It is known that empirical moments are consistent
estimators of moments.

(^6) However, theoretically it might be difficult to define a distribution in terms of
parameters. For example, a distribution could be defined as the solution of a differ-
ential equation. This could make it very difficult to establish a relationship between
moments and parameters.