For panel data models including time-invariant variables, this paper proposes a new Hausman pretest estimator of the internal instruments of Hausman-Taylor estimator. It assumes Mundlak and Krishnakumar linear specification for the endogeneity of random individual effects. Furthermore, the paper evaluates the biases of currently used estimators: repeated between, ordinary least squares, two-stage restricted between, Oaxaca-Geisler estimator, fixed effect vector decomposition, and generalized least squares. Some of these may lead to erroneous conclusions regarding the statistical significance of the estimated parameter values of time-invariant variables, especially when time-invariant variables are correlated with the individual effects.