374 The Basics of financial economeTrics
decision Rule The object of hypothesis testing is to make a decision about
these hypotheses. So, we either have to accept the null hypothesis and,
consequently, must reject the alternative hypothesis, or we reject the null
hypothesis and decide in favor of the alternative hypothesis.
A hypothesis test is designed such that the null hypothesis is maintained
until evidence provided by the sample is so strong that we have to decide
against it. This leads us to the two common ways of using the test.
With the first application, we simply want to find out whether a situa-
tion that we deemed correct actually is true. Thus, the situation under the
null hypothesis is considered the status quo or the experience that is held on
to until the support given by the sample in favor of the alternative hypoth-
esis is too strong to sustain the null hypothesis any longer.
The alternative use of the hypothesis test is to try to promote a concept
formulated by the alternative hypothesis by finding sufficient evidence in
favor of it, rendering it the more credible of the two hypotheses. In this sec-
ond approach, the aim of the tester is to reject the null hypothesis because
the situation under the alternative hypothesis is more favorable.
In the realm of hypothesis testing, the decision is generally regarded
as the process of following certain rules. We denote our decision rule by δ.
The decision δ is designed to either assume value d 0 or value d 1. Depending
on which way we are using the test, the meaning of these two values is as
follows. In the first case, the value d 0 expresses that we hold on to H 0 while
the contrarian value d 1 expresses that we reject H 0. In the second case, we
interpret d 0 as being undecided with respect to H 0 and H 1 and that proof is
not strong enough in favor of H 1. On the other hand, by d 1 , we indicate that
we reject H 0 in favor of H 1.
In general, d 1 can be interpreted as the result we obtain from the deci-
sion rule when the sample outcome is highly unreasonable under the null
hypothesis.
So, what makes us come up with either d 0 or d 1? As discussed earlier in
this chapter, we infer by first drawing a sample of some size n, X = (X 1 , X 2 ,... ,
Xn). Our decision then should be based on this sample. That is, it would be wise
to include in our decision rule the sample X such that the decision becomes a
function of the sample, (i.e., δ(X)). Then, δ(X) is a random variable due to the
randomness of X. A reasonable step would be to link our test δ(X) to a statistic,
denoted by t(X), that itself is related or equal to an estimator θˆ for the param-
eter of interest θ. Such estimators were introduced earlier in this chapter.
From now on, we will assume that our test rule δ is synonymous with
checking whether the statistic t(X) is assuming certain values or not from
which we derive decision d 0 or d 1.
As we know from point estimates, by drawing a sample X, we select a
particular value x from the sample space X. Depending on this realization x,