Econometric applications: discussion
In many empirical problems, it is quite possible that the exact null distribution of the relevant test statistic S(Y) will not be easy to compute analytically, even though it is nuisance-parameter-free. In this case, S(Y) is called a pivotal statistic, i. e. the null distribution of S(Y) is uniquely determined under the null hypothesis. In such cases, we will show that the MC test easily solves the size control problem, regardless of the distributional complexities involved. The above examples on normality tests and the UL hypotheses tests, all involve pivotal statistics. The problem is more complicated in the presence of nuisance parameters. We will first discuss a property related to nuisance-parameter-dependent test statistics which will prove to be fundamental in finite sample contexts.3
In the context of a right-tailed test problem, consider a statistic S(Y) whose null distribution depends on nuisance parameters and suppose it is possible to find another statistic S(Y) such that
S(Y) < S*(Y), V0 Є ©о, (23.28)
and S*(Y) is pivotal under the null. Then S(Y) is said to be boundedly pivotal. The implications of this property are as follows. From (23.28), we obtain
P0[S(Y) > c] < P[S*(Y) > c], V0 Є ©0.
Then if we calculate c such that
P[S*(Y) > c] = a, (23.29)
we solve the level constraint for the test based on S(Y). It is clear that (23.28) and
P0[S(Y) > c] < a, V0 Є ©о.
As emphasized earlier, the size control constraint is easier to deal with in the case of S*(Y) because it is pivotal. Consequently, the maximization problem
sup Pe[S(Y) > c]
has a non-trivial solution (less than 1) in the case of boundedly pivotal statistics. If this property fails to hold, the latter optimization problem may admit only the trivial solution, so that it becomes mathematically impossible to control the significance level of the test.
It is tempting to dismiss such considerations assuming they will occur only in "textbook" cases. Yet it can be shown (we will consider this issue in the next section) that similar considerations explain the poor performance of the Wald tests and confidence intervals discussed in Sections 2.1 and 2.3 above. These are problems of empirical relevance in econometric practice. In the next session, we will show that the bootstrap will also fail for such problems! For further discussion of the basic notions of statistical testing mentioned in this section, the reader may consult Lehmann (1986, ch. 3), Gourieroux and Monfort (1995), and Dufour (1990, 1997).