372 The Basics of financial economeTrics
confidence interval depends on the outcome x. By design, as we have just
shown, the interval is such that in (1 – α) × 100% of all outcomes, the inter-
val contains the true parameter and in α × 100%, it does not cover the true
parameter value.
Note that the parameter is invariant, only the interval is random.
Hypothesis Testing
Thus far in this appendix, inference on some unknown parameter meant
that we had no knowledge of its value and therefore we had to obtain an
estimate. This could either be a single point estimate or an entire confi-
dence interval. However, sometimes, one already has some idea of the value
a parameter might have or used to have. Thus, it might not be important
to obtain a particular single value or range of values for the parameter, but
instead to gain sufficient information to conclude that the parameter more
likely either belongs to a particular part of the parameter space or not. So,
instead we need to obtain information to verify whether some assumption
concerning the parameter can be supported or has to be rejected. This brings
us to the field of hypothesis testing.
To perform hypothesis testing it is essential to express the competing
statements about the value of a parameter as hypotheses. To test for these,
we develop a test statistic for which we set up a decision rule. For a spe-
cific sample, this test statistic then either assumes a value in the acceptance
region or the rejection region, regions that we describe in this chapter. Fur-
thermore, we see the two error types one can incur when testing. We see that
the hypothesis test structure allows one to control the probability of error
through what we see to be the test size or significance level. We discover
that each observation has a certain p-value expressing its significance. As a
quality criterion of a test, we introduce the power from which the uniformly
most powerful test can be defined. Furthermore, we explain what is meant
by an unbiased test—unbiasedness provides another important quality cri-
terion—as well as whether a test is consistent.
Hypotheses
Before being able to test anything, we need to express clearly what we
intend to achieve with the help of the test. For this task, it is essential that
we unambiguously formulate the possible outcomes of the test. In the realm
of hypothesis testing, we have two competing statements to decide upon.
These statements are the hypotheses of the test.