  Bartik regressions use locations' differential exposure to nationwide
sector-level shocks as an instrument to estimate the effect of a location-level
treatment on an outcome. In the canonical Bartik design, locations'
differential exposure to industry-level employment shocks are used as an
instrument to measure the effect of their employment evolution on their wage
evolution. Some recent papers studying Bartik designs have assumed that the
sector-level shocks are exogenous and all have the same expectation. This
second assumption may sometimes be implausible. For instance, there could be
industries whose employment is more likely to grow than that of other
industries. We replace that second assumption by parallel trends assumptions.
Under our assumptions, Bartik regressions identify weighted sums of
location-specific effects, with weights that may be negative. Accordingly, such
regressions may be misleading in the presence of heterogeneous effects, an
issue that was not present under the assumptions maintained in previous papers.
Estimating the weights attached to Bartik regressions is a way to assess their
robustness to heterogeneous effects. We also propose an alternative estimator
that is robust to location-specific effects. Finally, we revisit two
applications. In both cases, Bartik regressions have fairly large negative
weights attached to them. Our alternative estimator is substantially different
from the Bartik regression coefficient in one application.
