This paper extends the familiar notion of fixed effects to nonlinear structures with infinite-dimensional unobservables, like preferences. The main result is that a generalized version of differencing identifies local average responses (LARs) in nonseparable structures. In contrast to existing results, this does not require either substantial restrictions on functional form or independence between the persistent unobservables and the explanatory variables of interest, and it requires only two time periods. On the other hand, the results are confined to the subpopulation of “stayers” (Chamberlain, 1982), i.e., the population for which the explanatory variables do not change over time. We extend the basic framework to include time trends and dynamics in the explanatory variables, and we show how distributional effects as well as average partial effects are identified. Our approach also allows endogeneity in the transitory unobservables. Furthermore, we show that this new identification principle can be applied to well-known objects like the slope coefficient in the semiparametric panel data binary choice model with fixed effects. Finally, we suggest estimators for the local average response and average partial effect, and we analyze their large- and finite-sample behavior.