We study the properties of two specification tests that have been applied to a variety of estimators in the context of value-added measures (VAMs) of teacher and school quality: the Hausman test for choosing between student-level random and fixed effects, and a test for feedback (sometimes called a "falsification test"). We discuss theoretical properties of the tests to serve as background, and propose parsimonious one-degree-of-freedom versions. An extensive simulation study provides important further insight into the VAM setting. Unfortunately, although both the Hausman and feedback tests have good power for detecting the kinds of nonrandom assignment that can invalidate VAM estimates, they also reject in situations where estimated VAMs perform very well in terms of ranking teachers. Consequently, the tests must be used with caution when student tracking is used to form classrooms.