This paper extends previous work on the identification of search models in which observed
worker productivity is imperfectly observed. In particular, it establishes that these models
remain identified even when employment histories are left-censored (i.e. we do not get to follow
workers from their initial job out of unemployment), as well as when workers set different reservation
wages from one another. We further show that allowing for heterogeneity in reservation
can affect the empirical estimates we obtain, specifically estimates of the rate at which workers
receive job offers.