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Let's say we have the following regression

$Y = \alpha + \beta X + \gamma W + u$

the way I interpret $\beta$ is "the effect of $X$ on $Y$ keeping $W$ constant".

Now let's say that I have panel data, and I want to control for unobserved heterogeneity, then I add individual fixed effects to the previous equation. The interpretation is similar: "the average effect of $X$ on $Y$ across individuals".

Alternatively, I am worried about unobserved time-variant confounders, therefore I include time-fixed effects. The interpretation is now: "the average effect of $X$ on $Y$ across time".

My question is: "how to interpret $\beta$ in a case in which I include both time and individual fixed effects?"

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The interpretation is straightforward, but it comes with a lot of baggage. The interpretation cannot be separated from the assumptions. The interpretation of $\beta$ is the effect of $X$ on $Y$ for a given person at a given time. It is not an average (not a useful one, at least); it assumes the effect is constant across all people and all times. It also assumes there is no time-by-person interaction; if there is only one data point per person per time point, this interaction is not identified anyway, so it is confounded with $X$ (that is, it is impossible to separate time-by-person effects from the effect of $X$).

I don't love the "across individuals" or "across time" interpretations because the interpretation of regression coefficients involves "holding other variables constant", i.e., $\beta$ is how much $Y$ would be expected to differ between two measurements on the same person at the same time that differed on $X$ by 1. It is "across" individuals or time only in the sense that the restrictive model has required the treatment effect to be constant across all individuals and times, an unlikely assumption (though one hard to avoid).

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