Heterogeneity of time invariant characteristic in event study model

Question

I am trying to estimate the impact that the onset of a medical disease has on a number of outcomes, call them $O$. To do this, I am using an event study model with individual and time fixed effects... \begin{equation} O_{i,t} = \alpha + \sum_{\substack{k=S \\ k\neq -1}}^{F}{\mu_k} + Z_{i,t}\delta + \sigma_i + \epsilon_{i,t} \end{equation}

I have theoretical reason to believe that characteristics of the individual at the onset of the individual may moderate the effect of disease onset, and so I would like to explore heterogeneity in the treatment effect by this at-diagnosis characteristic. For instance, an individual's marital status at diagnosis may moderate his response following diagnosis.

I apologize if this is a trivial question but how can I incorporate this time invariant characteristic into a model with the individual effect? Should I simply run subsample analysis by group? And if so, what is a good strategy for dealing with continuous (but fixed) heterogeneity -- for instance, heterogeneity of income at diagnosis? Should I drop the individual fixed effect?

I appreciate all of your assistance and thoughts in advance.

Answer 1

If you have a time-constant covariate $X$, you could include it but drop the individual fixed effects $\sigma_i$. (There could be perfect multicollinearity if you kept them.) Since you mentioned that $X$ might have a moderating effect w.r.t. $Z$, you would include interaction terms between $X$ and $Z$. So you would have something like $$ O_{i,t} = \alpha + \beta_t + \gamma x_i + \delta Z_{i,t} + \theta (x_i\cdot Z_{i,t}) + \epsilon_{i,t} $$ with $\beta_t$ being time fixed effects. You could set $\beta_1=0$ for identification. (One more linear restriction might be needed.) From your verbal description of the problem, I am not sure what $\sum_{\substack{k=S \\ k\neq -1}}^{F}{\mu_k}$ is meant to represent, so I did not include it in the equation above.

_{(I might be completely wrong. It has been a long while since I did anything with panel data models.)}