Although the FE specification requires that any included variable has to vary across time so you could not include just an interaction of any dummies (some $d_i*g_i$ interaction would be time invariant and would not be possible to include it in FE model - although there are other panel models that could handle such interaction term as well) Where $\alpha_i$ would be panel fixed effects $T_t*A_t$ would be interaction of a treatment period dummy $T_t$ which is 1 during treatment period and 0 outside treatment period and $A_i$ which is dummy for assignment in either treatment (1) or control group (0), and together the interaction term tells us if in a given $t$ and individual was recieving treatment $t$. A DiD can be specified as (see Mostly Harmless Econometrics by Angrist and Pischke): In fact the widely used differences-in-differences (DiD) estimator relies on it. Generally speaking you can include interaction terms in panel data. When you multiply two dummies you are creating what is called an interaction term. Depends on what the dummies are and what is the specification of the model you are using.
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