Towards a theory of point optimal testing

Abstract

This paper puts the case for the inclusion of point optimal tests in the econometrician's repertoire. They do not suit every testing situation but the current evidence, which is reviewed here, indicates that they can have extremely useful small-sample power properties. As well as being most powerful at a nominated point in the alternative hypothesis parameter space, they may also have optimum power at a number of other points and indeed be uniformly most powerful when such a test exists. Point optimal tests can also be used to trace out the maximum attainable power envelope for a given testing problem, thus providing a benchmark against which test procedures can be evaluated. In some cases, point optimal tests can be constructed from tests of a simple null hypothesis against a simple alternative. For a wide range Of models of interest to econometricians, this paper shows how one can check whether a point optimal test can be constructed in this way. When it cannot, one may wish to consider approximately point optimal tests. As an illustration, the approach is applied to the non-nested problem of testing for AR(1) disturbances against MA(1) disturbances in the linear regression model.

Power Properties of Invariant Tests for Spatial Autocorrelation in Linear Regression

“…In the present paper, invariant tests are defined with respect to F X , as, for instance, in Berenblut and Webb (1973), because this simplifies the statement of our results. Some authors (e.g., King, 1988) define invariance of tests for autocorrelation in linear regression with respect to F + X . The distinction between invariance under F X and invariance under F + X is not substantive, because tests that are invariant under F + X but not under F X are never used in practice.…”

“…i.e., it is the most powerful invariant test against the specific alternative hypothesis ρ =ρ > 0 (see King, 1988). POI tests define the power envelope of invariant tests.…”

Pre and post break parameter inference

“…The value of σ 2 β = 22 is motivated by King's (1988) discussion of overall reasonable point-optimal tests, since it turns out that for σ 2 β = 22, the best 5% level test has approximately 50% weighted average power. The uniform weighting over ρ accords to the choice in Andrews and Ploberger (1994) and is intended to generate reasonable power for all break dates ρ ∈ [0.15, 0.85].…”

Optimally Testing General Breaking Processes in Linear Time Series Models

“…One possibility is to draw on the ideas of King (1988) and simply computeĴ(Ω * ) for some constantΩ * , while being aware thatΩ * and the true Ω * will typically differ. Note that such a discrepancy does not affect the size properties of the statistic, but it will affect its power under the alternative of a time-varying {β t }.…”