I show that specification tests for correlated fixed effects developed by Hausman and Taylor extend in an analogous way to panel data sets with endogenous regressors. Given panel data, different sets of instrumental variables can be used to construct the test. For a simple class of models, the test in many cases is asymptotically more efficient if an incomplete set of instruments is used. However, in small samples one may do better using the complete set of instruments. Monte Carlo results demonstrate the likely gains for different assumptions about the degree of between and within variance in the data.