Abstract – On the Asymptotic Size Distortion of Tests When Instruments Locally Violate the Exogeneity Assumption
In the linear instrumental variables model with possibly weak instruments we
derive the asymptotic size of testing procedures when instruments locally violate
the exogeneity assumption. We study the tests by Anderson and Rubin (1949),
Moreira (2003), and Kleibergen (2005) and their generalized empirical likelihood
versions. These tests have asymptotic size equal to nominal size when the instru-
ments are exogenous but are size distorted otherwise. While in just-identified
models all the tests that we consider are equally size distorted asymptotically,
the Anderson and Rubin (1949) type tests are less size distorted than the tests of
Moreira (2003) and Kleibergen (2005) in over-identi…ed situations. On the other
hand, we also show that there are parameter sequences under which the former
test asymptotically overrejects more frequently.
Given that strict exogeneity of instruments is often a questionable assumption,
our findings should be important to applied researchers who are concerned about
the degree of size distortion of their inference procedure. We suggest robustness
of asymptotic size under local model violations as a new alternative measure to
choose among competing testing procedures.
We also investigate the subsampling and hybrid tests introduced in Andrews
and Guggenberger (2010b) and show that they do not offer any improvement in
terms of size-distortion reduction over the Anderson and Rubin (1949) type tests.
Abstract – Distortions of Asymptotic Confidence Size in Locally Misspecified Moment Inequality Models

This paper studies the behavior under local misspeci cation of several con dence sets
(CSs) commonly used in the literature on inference in moment inequality models. We
suggest the degree of asymptotic con dence size distortion as an alternative criterium to
power to choose among competing inference methods, and apply this criterium to compare
across critical values and test statistics employed in the construction of CSs. We
nd two important results under weak assumptions. First, we show that CSs based on
subsampling and generalized moment selection (GMS, Andrews and Soares (2010)) su er
from the same degree of asymptotic con dence size distortion, despite the fact that the
latter can lead to CSs with strictly smaller expected volume under correct model speci –
cation. Second, we show that CSs based on the quasi-likelihood ratio test statistic have
asymptotic con dence size that can be an arbitrary fraction of the asymptotic con dence
size of CSs obtained by using the modi ed method of moments. Our results are supported
by Monte Carlo simulations.