Anon

380 The Basics of financial economeTrics

Quality Criteria of a Test

So far, we have learned how to construct a test for a given problem. In gen-
eral, we formulate the two competing hypotheses and look for an appropri-
ate test statistic to base our decision rule on and we are then done. However,
in general, there is no unique test for any given pair of hypotheses. That is,
we may find tests that are more suitable than others for our endeavor. How
can we define what we mean by “suitable”? To answer this question, we will
discuss the following quality criteria.

power of a Test Previously, we were introduced to the size of a test that
may be equal to α. As we know, this value controls the probability of com-
mitting a type I error. So far, however, we may have several tests meeting a
required test size α. The criterion selecting the most suitable ones among
them involves the type II error. Recall that the type II error describes the
failure of rejection of the null hypothesis when it actually is wrong. So, for
parameter values θ ∈ Θ 1 , our test should produce decision d 1 with as high
a probability as possible in order to yield as small as possible a probability
of a type II error, PII(d 0 ). In the following definition, we present a criterion
that accounts for this ability of a test.

Power of a test. The power of a test is the probability of rejecting the

null hypothesis when it is actually wrong (i.e., when the alternative

hypothesis holds). Formally, this is written as PXθ 1 ((δ=))d 1.^7

For illustrational purposes, we focus on Figure C.4 where we depict the
parameter-dependent probability P(δ(X) = d 1 ) of some arbitrary test δ, over
the parameter space Θ. The solid part of the figure, computed over the set
Θ 1 , represents the power of the test. As we can see, here, the power increases
for parameter values further away from Θ 0 (i.e., increasing θ). If the power
were rising more steeply, the test would become more powerful. This brings
us to the next concept.

Uniformly Most powerful Test In the following illustration, let us only con-
sider tests of size α. That is, none of these tests incurs a type I error with
greater probability than α. For each of these tests, we determine the respec-
tive power function (i.e., the probability of rejecting the null hypothesis,

(^7) The index θ 1 of the probability measure P indicates that the alternative hypothesis
holds (i.e., the true parameter is a value in Θ 1 ).