  The classical tests in the instrumental variable model can behave arbitrarily
if the data is contaminated. For instance, one outlying observation can be
enough to change the outcome of a test. We develop a framework to construct
testing procedures that are robust to weak instruments, outliers and
heavy-tailed errors in the instrumental variable model. The framework is
constructed upon M-estimators. By deriving the influence functions of the
classical weak instrument robust tests, such as the Anderson-Rubin test, K-test
and the conditional likelihood ratio (CLR) test, we prove their unbounded
sensitivity to infinitesimal contamination. Therefore, we construct
contamination resistant/robust alternatives. In particular, we show how to
construct a robust CLR statistic based on Mallows type M-estimators and show
that its asymptotic distribution is the same as that of the (classical) CLR
statistic. The theoretical results are corroborated by a simulation study.
Finally, we revisit three empirical studies affected by outliers and
demonstrate how the new robust tests can be used in practice.
