  We study the incidental parameter problem for the ``three-way'' Poisson
{Pseudo-Maximum Likelihood} (``PPML'') estimator recently recommended for
identifying the effects of trade policies and in other panel data gravity
settings. Despite the number and variety of fixed effects involved, we confirm
PPML is consistent for fixed $T$ and we show it is in fact the only estimator
among a wide range of PML gravity estimators that is generally consistent in
this context when $T$ is fixed. At the same time, asymptotic confidence
intervals in fixed-$T$ panels are not correctly centered at the true point
estimates, and cluster-robust variance estimates used to construct standard
errors are generally biased as well. We characterize each of these biases
analytically and show both numerically and empirically that they are salient
even for real-data settings with a large number of countries. We also offer
practical remedies that can be used to obtain more reliable inferences of the
effects of trade policies and other time-varying gravity variables, which we
make available via an accompanying Stata package called ppml_fe_bias.
