Anon

376 The Basics of financial economeTrics

hypothesis holds. Unfortunately, however, we do not know whether we com-
mit an error or not when we are testing. We do have some control, though,
as to the probability of error given a certain hypothesis as we explain next.

Test Size

We just learned that, depending on which hypothesis is true, we can com-
mit either a type I or a type II error. Now, we will concentrate on the cor-
responding probabilities of incurring these errors.

Test size. The test size is the probability of committing a type I error.

This probability is denoted by PI(δ) for test δ.^5

We illustrate this in Figure C.1, where we display the density function
f((tX),Θ 0 ) of the test statistic t(X) under the null hypothesis. The horizontal axis
along which t(X) assumes values is subdivided into the acceptance ΔA and the

figURe C.1 Determining the Size PI(δ) of Some Test δ via the Density Function of
the Test Statistic t(X)

ΔA ΔC t(X)

PI(δ)

f(t(X),θ 0 )

(^5) The probability PI(δ) could alternatively be written as Pd
Θ 0 () 1 to indicate that we^
erroneously reject the null hypothesis even though H 0 holds.