Anon

370 The Basics of financial economeTrics

just a single number. This may increase the likelihood that the true param-
eter is one of the many values included in the interval.
If, in one extreme case, we select as an interval the entire parameter
space, the true parameter will definitely lie inside of it. Instead, if we choose
our interval to consist of one value only, the probability of this interval
containing the true value approaches zero and we end up with the same
situation as with the point estimator. So, there is a trade-off between a high
probability of the interval Iθ containing the true parameter value, achieved
through increasing the interval’s width, and the precision gained by a very
narrow interval.
As in our discussion of point estimates in the previous section, we
should use the information provided by the sample. Hence, it should be rea-
sonable that the interval bounds depend on the sample in some way. Then
technically each interval bound is a function that maps the sample space,
denoted by X, into the parameter space since the sample is some outcome in
the sample space and the interval bound transforms the sample into a value
in the parameter space representing the minimum or maximum parameter
value suggested by the interval. Because the interval depends on the sample
X = (X 1 , X 2 ,... , Xn), and since the sample is random, the interval [l(X),
u(X)] is also random. We can derive the probability of the interval lying
beyond the true parameter (i.e., either completely below or above) from the
sample distribution. These two possible errors occur exactly if either u(x) <
θ or θ < l(x).
Our objective is then to construct an interval so as to minimize the
probability of these errors occurring. Suppose we want this probability
of error to be equal to α. For example, we may select α = 0.05 such that
in 5% of all outcomes, the true parameter will not be covered by the
interval. Let

pPl=θ((<lX)) and pPu=<((uX))θ

Then, it must be that

Pl([θ∉()Xu,(Xp)])=+l pu=α

Now let’s provide two important definitions: a confidence level and con-
fidence interval.

definition of a Confidence Level For some parameter θ, let the probability of the
interval not containing the true parameter value be given by the probability