The Power of Tests for Detecting $p$-Hacking
$p$-Hacking can undermine the validity of empirical studies. A flourishing empirical literature investigates the prevalence of $p$-hacking based on the empirical distribution of reported $p$-values across studies. Interpreting results in this literature requires a careful understanding of the power of methods used to detect different types of $p$-hacking. We theoretically study the implications of likely forms of $p$-hacking on the distribution of reported $p$-values and the power of existing methods for detecting it. Power can be quite low, depending crucially on the particular $p$-hacking strategy and the distribution of actual effects tested by the studies. Publication bias can enhance the power for testing the joint null hypothesis of no $p$-hacking and no publication bias. We relate the power of the tests to the costs of $p$-hacking and show that power tends to be larger when $p$-hacking is very costly. Monte Carlo simulations support our theoretical results.