  When parameters are weakly identified, bounds on the parameters may provide a
valuable source of information. Existing weak identification estimation and
inference results are unable to combine weak identification with bounds. Within
a class of minimum distance models, this paper proposes identification-robust
inference that incorporates information from bounds when parameters are weakly
identified. The inference is based on limit theory that combines weak
identification theory (Andrews and Cheng (2012)) with parameter-on-the-boundary
theory (Andrews (1999)) via a new argmax theorem. This paper characterizes weak
identification in low-dimensional factor models (due to weak factors) and
demonstrates the role of the bounds and identification-robust inference in two
example factor models. This paper also demonstrates the identification-robust
inference in an empirical application: estimating the effects of a randomized
intervention on parental investments in children, where parental investments
are modeled by a factor model.
