Is it possible to have the interaction of a time-constant continuous variable with a time-varying binary treatment in Fixed effect models? The outcome is a continious variable and I am doing a panel data study.
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$\begingroup$ Please edit your question to say more about the situation that you have in mind, in particular the nature of the outcome variable. Is it a continuous outcome in a time series, a time-to-event survival-type model, or something else? You probably can include an interaction in those situations but the interpretation might pose problems depending on the nature of the data. Please provide that information by editing the question, as comments are easy to overlook and can get deleted. $\endgroup$– EdMCommented Mar 5, 2022 at 21:45
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A fixed-effects model for continuous outcome $Y$ with panel data on individuals $i$ over times $t$ can be written:
$$Y_{it}=\alpha_i + \beta' X_t + u_{it} ,$$
where $\alpha_i$ is an individual-specific (unobserved) "fixed effect," $\beta$ is a vector of regression coefficients corresponding to a vector of observed time-varying covariates $X_t$, and $u_{it}$ are error terms.
Provided that the time-constant continuous predictor and the time-varying binary treatment are observed covariates $X_t$, then there is no problem with including their interaction, which is just their product, as another predictor. That's standard practice in regression models. There might be concern whether the effect of the interaction term at time $t$ only applies to time $t$, but that's a concern about the continuous and binary-treatment predictors in a panel model as well.
The fixed-effects panel model described above would not, however, be appropriate if the effect of the binary treatment depended on an interaction with an unobserved "time-constant continuous variable" that's encompassed in the individual-specific "fixed effects."
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$\begingroup$ thanks for the answer. Not sure if I can follow the last paragraph. $\endgroup$ Commented Mar 6, 2022 at 16:33
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$\begingroup$ Is there a way to check the last point? $\endgroup$ Commented Mar 6, 2022 at 16:41
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$\begingroup$ The last paragraph is an attempt to consider heterogeneity among individuals' responses to treatment due to unobserved individual-specific factors. In a mixed-effects model (in non-econometrics terminology; see this page and its links), that might be handled by random slopes for the effect of treatment. The later chapters of Hanck et al might provide some guidance in the context of econometrics. $\endgroup$– EdMCommented Mar 6, 2022 at 16:56
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$\begingroup$ Thanks so much. I fitted three models, between, within and mixed. As the coefficients for within and between models were not the same, I decided to report the within model. $\endgroup$ Commented Mar 6, 2022 at 17:31
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$\begingroup$ @TaherehDehdarirad you should carefully consider why the "within" and "between" models disagreed, as that might indicate some critical part of the situation that the "within" model isn't capturing. $\endgroup$– EdMCommented Mar 6, 2022 at 17:37