368 The Basics of financial economeTrics
Consistency. An estimator θˆn is a consistent estimator for θ if it converges
in probability to θ, as given by equation (C.3), that is, plimθ=ˆn θ.
The consistency of an estimator is an important property since we
can rely on the consistent estimator to systematically give sound
results with respect to the true parameter. This means that if we
increase the sample size n, we will obtain estimates that will deviate
from the parameter θ only in rare cases.Unbiased efficiency In the previous discussions, we tried to determine where
the estimator tends to. This analysis, however, left unanswered the question
of how fast the estimator gets there. For this purpose, we introduce the
notion of unbiased efficiency.
Let us suppose that two estimators θˆ and θˆ are unbiased for some
parameter θ. Then, we say that θˆ is a more efficient estimator than θˆ if it
has a smaller variance; that is,
varvarθθ(θθˆ)(< ˆ*) (C.5)
for any value of the parameter θ. Consequently, no matter what the true
parameter value is, the standard error of θˆ is always smaller than that of θˆ*.
Since they are assumed to be both unbiased, the first should be preferred.^3
Linear Unbiased estimators A particular sort of estimators are linear unbiased
estimators. We introduce them separately from the linear estimators here
because they often display appealing statistical features.
In general, linear unbiased estimators are of the form
(^) θ=∑
=aX
ˆ ii
i
n
1
To meet the condition of zero bias, the weights ai have to add to one. Due
to their lack of bias, the MSE will only consist of the variance part. With
sample size n, their variances can be easily computed as
varθ(θσˆ)=∑i=ai X
n 22
1
(^3) If the parameter consists of more than one component, then the definition of effi-
ciency in equation (C.5) needs to be extended to an expression that uses the covari-
ance matrix of the estimators rather than only the variances.