This paper describes how to reparametrize low-dimensional factor models with one or two factors to fit weak identification theory developed for generalized method of moments models. Some identification-robust tests, here called “plug-in” tests, require a reparametrization to distinguish weakly identified parameters from strongly identified parameters. The reparametrizations in this paper make plug-in tests available for subvector hypotheses in low-dimensional factor models with one or two factors. Simulations show that the plug-in tests are less conservative than identification-robust tests that use the original parametrization. An empirical application to a factor model of parental investments in children is included.

We propose a simple test for moment inequalities that has exact size in normal models with known variance and has uniformly asymptotically exact size under asymptotic normality. The test compares the quasi-likelihood ratio statistic to a chi-squared critical value, where the degree of freedom is the rank of the inequalities that are active in finite samples. The test requires no simulation and thus is computationally fast and especially suitable for constructing confidence sets for parameters by test inversion. It uses no tuning parameter for moment selection and yet still adapts to the slackness of the moment inequalities. Furthermore, we show how the test can be easily adapted to inference on subvectors in the common empirical setting of conditional moment inequalities with nuisance parameters entering linearly. User-friendly Matlab code to implement the test is provided.

Certain extremum estimators have asymptotic distributions that are non‐Gaussian, yet characterizable as the distribution of the arg max of a Gaussian process. This paper presents high‐level sufficient conditions under which such asymptotic distributions admit a continuous distribution function. The plausibility of the sufficient conditions is demonstrated by verifying them in three examples, namely, maximum score estimation, empirical risk minimization, and threshold regression estimation. In turn, the continuity result buttresses several recently proposed inference procedures whose validity seems to require a result of the kind established herein. A notable feature of the high‐level assumptions is that one of them is designed to enable us to employ the Cameron–Martin theorem. In a leading special case, the assumption in question is demonstrably weak and appears to be close to minimal.