Interpreting interactions in linear fixed-effect regression models: When fixed-effect estimates are no longer within-effects

Fixed-effect regression models use within-firm variation to identify coefficient estimates, which is advantageous for mitigating certain endogeneity concerns and ruling out spurious relationships. I demonstrate that fixed-effect regression models with interaction terms (and by extension quadratic or higher-degree terms) confound within-firm and between-firm variation in identifying interaction coefficient estimates. Thus, in these specifications coefficient estimates lack a desirable property of standard fixed-effect estimates. I substantiate this concern using simulations and an empirical example. I also demonstrate how segmented regression aids assessing whether within-firm or between-firm variation identifies interaction coefficient estimates in fixed-effect models.