Verifying the existence of maximum likelihood estimates for generalized linear models

A fundamental problem with nonlinear estimation models is that estimates are not guaranteed to exist. However, while non-existence is a well-studied issue for binary choice models, it presents significant challenges for other models as well and is not as well understood in more general settings. These challenges are only magnified for models that feature many fixed effects and other high-dimensional parameters. We address the current ambiguity surrounding this topic by studying the conditions that govern the existence of estimates for a wide class of generalized linear models (GLMs). We show that some, but not all, GLMs can still deliver consistent estimates of at least some of the linear parameters when these conditions fail to hold. We also demonstrate how to verify these conditions in the presence of high-dimensional fixed effects, as are often recommended in the international trade literature and in other common panel settings