364 The Basics of financial economeTrics
due to the uncertainty associated with each individual draw. For the param-
eter θ and the estimator θˆ, we define the sample error as ()θ−ˆ θ. Now, a most
often preferred estimator should yield an expected sample error of zero.
This expected value is defined as
Eθ(θ−ˆ θ) (C.1)
and referred to as bias.^1 If the expression in equation (C.1) is different from
zero, the estimator is said to be a biased estimator while it is an unbiased
estimator if the expected value in equation (C.1) is zero.
The subscript θ in equation (C.1) indicates that the expected value is com-
puted based on the distribution with parameter θ whose value is unknown.
Technically, however, the computation of the expected value is feasible for a
general θ.
Mean Squared error Bias as a quality criterion tells us about the expected
deviation of the estimator from the parameter. However, the bias fails to
inform us about the variability or spread of the estimator. For a reliable
inference for the parameter value, we should prefer an estimator with rather
small variability or, in other words, high precision.
Assume that we repeatedly, say m times, draw samples of given size
n. Using estimator θˆ for each of these samples, we compute the respec-
tive estimate θˆt of parameter θ, where t = 1, 2,... , m. From these m esti-
mates, we then obtain an empirical distribution of the estimates including
an empirical spread given by the sample distribution of the estimates. We
know that with increasing sample length n, the empirical distribution will
eventually look like the normal distribution for most estimators. However,
regardless of any empirical distribution of estimates, an estimator has a
theoretical sampling distribution for each sample size n. So, the random
estimator is, as a random variable, distributed by the law of the sampling
distribution. The empirical and the sampling distribution will look more
alike the larger is n. The sampling distribution provides us with a theoreti-
cal measure of spread of the estimator which is called the standard error
(SE). This is a measure that can often be found listed together with the
observed estimate.
To completely eliminate the variance, one could simply take a constant
θ=ˆ c as the estimator for some parameter. However, this not reasonable
since it is insensitive to sample information and thus remains unchanged
(^1) We assume in this chapter that the estimators and the elements of the sample have
finite variance, and in particular the expression in equation (C.1) is well-defined.