Chapter 20 Generalized Method of Moments Estimators and ...

EXT 10-16 Web Extension 10
Comparing Equations 20.A.3 and 20.A.4 with Equations 20.A.1 and 20.A.2
shows that plim(b ~ 0) and plim(b ~ 1) appear in Equations 20.A.3 and 20.A.4 just
where b 0 and b 1 do in Equations 20.A.1 and 20.A.2. If Equations 20.A.3 and
20.A.4 have a unique solution for plim and plim , then plim and
plim (b ~ (b ~ 0) (b ~ 1) (b ~ 0) = b 0
1) = b 1 .
There is one case in which the solution of Equations 20.A.3 and 20.A.4 for
plim(b ~ 0) and plim(b ~ 1) is not unique. If X takes on only one value, and is therefore
perfectly collinear with the intercept term, Equation 20.A.4 is equivalent to Equation
20.A.3: Equation 20.A.4 reduces to Equation 20.A.3 if we divide both sides
of Equation 20.A.4 by the constant value of X. In this case, we really have only
one equation in two unknowns. And, in this special case of perfect multicollinearity,
the two equations do not yield a unique solution for plim and plim .
Otherwise, the solution is unique, so plim(b ~ (b ~
and plim (b ~ 0) (b ~ 1)
0) = b 0 1) = b 1 .
Thus, barring perfect multicollinearity, when the sample moments converge
in probability to their population expectations, b ~ and b ~ 0 1 are consistent estimators.
Barring perfect collinearity–like problems, method of moments estimators
are generally consistent when the sample moments converge to their population
values. Because it is the Law of Large Numbers that ensures that sample means
converge in probability to their population analogs, the Law of Large Numbers is
key to the consistency of method of moments estimators. As noted earlier, infinite
variances in a DGP endanger the applicability of the Law of Large Numbers.
Endnotes
1
1. n gX ie i will converge in probability to zero if its expectation is zero and its variance
goes to zero as n grows. By assumption, E(X i e i ) = 0, so the first criterion is met. The
second criterion will also be met if we assume var(X (equal to E(X 2 is a finite,
nonzero constant, t 4 , so that varA 1 i e 2 i e i )
i ))
n gX ie iB = t 4 >n.