Efficient Tests under a Weak Convergence Assumption

Abstract

The asymptotic validity of tests is usually established by making appropriate primitive assumptions, which imply the weak convergence of a specific function of the data, and an appeal to the continuous mapping theorem. This paper, instead, takes the weak convergence of some function of the data to a limiting random element as the starting point and studies efficiency in the class of tests that remain asymptotically valid for all models that induce the same weak limit. It is found that efficient tests in this class are simply given by efficient tests in the limiting problem-that is, with the limiting random element assumed observed-evaluated at sample analogues. Efficient tests in the limiting problem are usually straightforward to derive, even in nonstandard testing problems. What is more, their evaluation at sample analogues typically yields tests that coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model. This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests, and tests about regression coefficients under weak instruments.

A Robust Test for Weak Instruments

“…(High Level) The following limits hold as S → ∞. Assumption HL allows for a general form of W, similarly to the models in Müller (2011) andMikusheva (2010). This is our key generalization from the model in Staiger and Stock (1997), who required W to have the form ⊗ I K .…”

Conditional Linear Combination Tests for Weakly Identified Models

“…Further, under the weak convergence assumption (1) for (g g) as in (25), the test φ K T = 1{K T > χ in the sense of Müller (2011) for a family of elliptically contoured weight functions. 20 Under strong identification φ K T = 1{K T > χ 2 p 1−α } is also generally equivalent to the usual Wald tests, though we will need conditions beyond (1) to establish this.…”

Pre and post break parameter inference

“…It is in principle possible to apply limits of experiments ideas also to semiparametric models (see, for instance, Choi et al (1996) for an application to semiparametric LAN models), but the non-standard nature of the testing problem here makes this an involved enterprise beyond the scope of the paper. As a more straightforward alternative, we now apply the asymptotic efficiency concept of tests introduced by Müller (2011) to post break parameter inference in GMM models.…”

“…For an underlying GMM model, we invoke the results of Müller (2011) to obtain the corresponding asymptotic efficiency claims in the class of tests that are ''robust'' in the sense of Müller (2011).…”

“…Section 2.4 translates this result for a scalar parameter to an efficiency claim for models with a vector parameter. Section 3 considers GMM models and contains details on the application of Müller's (2011) approach to large sample efficiency. Section 4 defines the suggested test statistic in terms of partial sample GMM statistics (cf.…”