Interactions with Panel Data: Implications of Different Specification Strategies

An interaction in a fixed effects (FE) regression is usually specified by demeaning the product term. How­ever, algebraic transformations reveal that this strategy does not yield a within-unit estimator. Instead, the standard FE interaction estimator reflects unit-level differences of both interacted variables. This property allows interactions of a time-constant variable and a time-varying variable in a Fixed Effects regression, but may yield unwanted results if both variables vary within units: when, for example, the interaction between income and having children on life satisfaction is modeled, the standard FE interaction estimator not only measures how within-household changes in income influence the effect of having children. It also measures how this effect differs between households with different (time-averaged) income levels. Therefore, it includes the moderating influence of time-constant unobservables correlated with income (e.g., traditionalism, conservatism) on the effect of having children.

We introduce and discuss different demeaning strategies as alternatives to the standard FE interaction estimator for panel data. Each of these alternatives portrays different causal mechanisms, has different substantive implications, and will lead to different results. This is revealed on the basis of an SHP research example dealing with the impact that children have on the marital wage premium.

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