  The common practice in difference-in-difference (DiD) designs is to check for
parallel trends prior to treatment assignment, yet typical estimation and
inference does not account for the fact that this test has occurred. I analyze
the properties of the traditional DiD estimator conditional on having passed
(i.e. not rejected) the test for parallel pre-trends. When the DiD design is
valid and the test for pre-trends confirms it, the typical DiD estimator is
unbiased, but traditional standard errors are overly conservative.
Additionally, there exists an alternative unbiased estimator that is more
efficient than the traditional DiD estimator under parallel trends. However,
when in population there is a non-zero pre-trend but we fail to reject the
hypothesis of parallel pre-trends, the DiD estimator is generally biased
relative to the population DiD coefficient. Moreover, if the trend is monotone,
then under reasonable assumptions the bias from conditioning exacerbates the
bias relative to the true treatment effect. I propose new estimation and
inference procedures that account for the test for parallel trends, and compare
their performance to that of the traditional estimator in a Monte Carlo
simulation.
