  A fundamental problem with nonlinear models is that maximum likelihood
estimates are not guaranteed to exist. Though nonexistence is a well known
problem in the binary choice literature, 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 (pseudo-)maximum likelihood estimators used to estimate a wide
class of generalized linear models (GLMs). We show that some, but not all, of
these GLM estimators 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 models with high-dimensional parameters, such
as panel data models with multiple levels of fixed effects.
