378 The Basics of financial economeTrics
to us, no matter what we use the test for. Consequently, we attempt to avoid
this type of error by trying to reduce the associated probability or, equiva-
lently, the test size.
Fortunately, the test size is something we have control over. We can
simply reduce PI(δ) through selection of an arbitrarily large acceptance
region ΔA. In the most extreme case, we set ΔA equal to the entire state
space of δ so that, virtually, we never reject the null hypothesis. However,
by inflating ΔA, we have to reduce ΔC, which generally results in an increase
in the probability PII(d 0 ) of a type II error because now it becomes more
likely for X to fall into ΔA also when θ is in Θ 1 (i.e., under the alternative
hypothesis). Thus, we are facing a trade-off between the probability of a
type I error and a type II error. A common agreement is to limit the prob-
ability of occurrence of a type I error to some real number between zero
and one. This α is referred to as the significance level. Frequently, values of
α = 0.01 or α = 0.05 are found.
Formally, the postulate for the test is PI(δ) ≤ α. So, when the null hypoth-
esis is true, in at most α of all outcomes, we will obtain a sample value x in
ΔC. Consequently, in at most α of the test runs, the test result will errone-
ously be d 1 (i.e., we decide against the null hypothesis).
The p-value
Suppose we had drawn some sample x and computed the value t(x) of the
statistic from it. It might be of interest to find out how significant this test
result is or, in other words, at which significance level this value t(x) would
still lead to decision d 0 (i.e., no rejection of the null hypothesis), while any
value greater than t(x) would result in its rejection (i.e., d 1 ). This concept
brings us to the next definition.
p-value. Suppose we have a sample realization given by x = (x 1 ,
x 2 ,... , xn). Furthermore, let δ(X) be any test with test statistic t(X)
such that the test statistic evaluated at x, t(x), is the value of the ac-
ceptance region ΔA closest to the rejection region ΔC. The p-value
determines the probability, under the null hypothesis, that in any
trial X the test statistic t(X) assumes a value in the rejection region
ΔC; that is,pP=∈θθ 00 (( ))tX ∆=C PX((δ=))d 1We can interpret the p-value as follows. Suppose we obtained a sample
outcome x such that the test statistics assumed the corresponding value t(x).
Now, we want to know the probability, given that the null hypothesis holds,