In an hypothesis testing problem involving nuisance
parameters for which boundedly complete sufficient
statistics exist under the null hypothesis, the
class of all similar regions for the problem is
characterized by the conditional distribution of the
data given these sufficient statistics. If there
exists a one-to-one transformation
y → (t, u) of
the data, y, to the sufficient
statistic, t, and a second vector
of statistics, u, that is
independent of t under the null
hypothesis, then the statistic u
itself characterizes the class of similar regions.
This paper applies this idea to five testing
problems of interest in econometrics. In each case
we obtain the density of the relevant statistic
under the null hypothesis, when it is free of
nuisance parameters, and under the alternative.
Using the density under the alternative, we discuss
the power properties of the class of similar tests
for each problem. Other applications are also
suggested.