The Centers for Medicare & Medicaid Services (CMS) and the National Quality Forum have endorsed the 30-day mortality rate as an important indicator of hospital quality. Concerns have been raised, however, as to whether post-discharge mortality rates are reasonable measures of hospital quality as they consider the frequency of an event that occurs after a patient is discharged and no longer under the watch and care of hospital staff. Estimating the causal effect of length-of-stay (LOS) on post-discharge mortality from retrospective data introduces a number of econometric challenges. We describe three potential sources of (endogeneity and censoring) biases and propose an approach that provides conservative estimates of the true treatment effect. Using a large dataset comprised of all hospital encounters of every Medicare Fee-for-Service patient with acute myocardial infarction from 2000 to 2011, we find evidence that an increase in LOS is associated with a decrease in 30-day mortality rates. An additional day in the hospital could decrease 30-day mortality rates by over 6%. Moreover, we find that, from a social planner's perspective, the gains achieved in reducing mortality rates likely exceed the cost of keeping the patients in the hospital for an additional day.