Using a modified version of the Frisch-Waugh-Lovell partialling out result, I show how the standard Mundlak regression for panel data can be extended to deliver ‘generalized fixed effects’ estimators, where coefficients on a subset of explanatory variables vary by cross-sectional unit. The characterization shows that only fitted values from individual specific regressions need to be controlled for—not the large set of interactions between unit-specific dummy variables and explanatory variables. In the special case of a single additive heterogeneity, the fitted values are simply the unit-specific time averages of the explanatory variables. The general results have practical applications, particularly when testing the standard fixed effects estimator against heterogeneous trend models. I derive a new, fully robust test that can be viewed as a regression-based Hausman test for choosing the level of heterogeneity. As an example, I reconsider some recently proposed difference-in-differences methods with staggered interventions and heterogeneous treatment effects when the parallel trends assumption is possibly violated.