This paper considers hypothesis testing problems in semiparametric models which may be non- regular for certain values of a potentially infinite dimensional nuisance parameter. I establish that, under mild regularity conditions, tests based on the efficient score function provide locally uniform size control and enjoy minimax optimality properties. This approach is applicable to situations with (i) identification failures, (ii) boundary problems and (iii) distortions induced by the use of regularised estimators. Full details are worked out for two examples: a single index model where the link function may be relatively flat and a linear simultaneous equations model that is (weakly) identified by non-Gaussian errors. In practice the tests are easy to implement and rely on standard χ2 critical values. I illustrate the approach by using the linear simultaneous equations model to examine the labour supply decisions of men in the US. I find a small but positive effect of wage increases on hours worked for hourly paid workers, but no effect for salaried workers.
Awarded the SNDE Young Scholars Award for the best paper presented at the 2021 SNDE Workshop for Young Researchers