An interaction in a fixed effects (FE) regression is usually specified by demeaning the product term. However, this strategy does not yield a genuine within estimator. Instead, an estimator is produced that reflects unit-level differences of interacted variables whose moderators vary within units. This is desirable if the interaction of one unit-specific and one time-dependent variable is specified in FE, but it may yield problematic results if both interacted variables vary within units. Then, as algebraic transformations show, the FE interaction estimator picks up unit-specific effect heterogeneity of both variables. Accordingly, Monte Carlo experiments reveal that it is biased if one of the interacted variables is correlated with an unobserved unit-specific moderator of the other interacted variable. In light of these insights, we propose that a within interaction of two timedependent variables be estimated by first demeaning each variable and then demeaning the product term. This “double-demeaned” estimator is not subject to bias caused by unobserved effect heterogeneity. It is, however, less efficient than standard FE and only works with T>2.

中文翻译：

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固定效应回归模型中的相互作用

固定效应 (FE) 回归中的相互作用通常通过贬低乘积项来指定。但是，这种策略不会产生真正的内部估计器。相反，产生的估计量反映了相互作用变量的单位级差异，其调节因子在单位内变化。如果在有限元中指定了一个单位特定变量和一个时间相关变量的相互作用，这是可取的，但如果两个相互作用的变量在单位内发生变化，则可能会产生有问题的结果。然后，如代数变换所示，有限元相互作用估计器获取两个变量的单位特定效应异质性。因此，蒙特卡罗实验表明，如果一个交互变量与另一个交互变量的未观察到的特定于单位的调节因子相关，则它是有偏差的。鉴于这些见解，我们建议通过首先贬低每个变量然后贬低乘积项来估计两个时间相关变量的内部相互作用。这种“双重贬低”的估计量不受未观察到的效应异质性引起的偏差的影响。但是，它的效率低于标准 FE，并且仅适用于 T>2。