Do unbalanced observations contribute to identification in fixed effects models? I am quite confused on how to interpret my regression results.
I want to estimate the effect of tariff changes $\tau_{st}$ in sector $s$, period $t$ on firm $i$ employment. I have two time periods (4 years apart) and want to use time and/or firm fixed effects. This is the base model:
$$y_{ist} = \beta \tau_{st} + \lambda_t + \mu_i + \epsilon_{ist}$$
The thing is that not all $i$ firms are observed in both time periods. There are either leavers, entrants, or continuing firms.
I have two questions related to the sources of identification of $\beta$ - Ideally, I would want to use cross-sectoral variation in tariff changes to identify it. Here it goes:
1 - Do leavers and entrants contribute to the estimation if I include $\mu_i$? How so? I understand they would in case I had $\mu_s$ instead. If not, would I get the same estimates by just dropping such observations?
2 - There a few firms that, although present in both $t=1$ and $t=2$, change sectors $s$ from period to period. Do these units also help with the identification of $\beta$ if I use the model above?
If I wanted to consider cases in which a firm may react to the policy change by moving to a sector with higher tariffs on the following period, which fixed effect structure would do so?
 A: I am going to change your notation slightly and explicitly write $\tau$ depending on $i$ as $\tau_{ist}$, even if in practice, $\tau_{ist}$ is only determined by $s,t$. One of the most helpful theorems for trying to understand what variation a regression is using to identify a coefficient is the Frisch-Waugh-Lovell theorem. Specialized to your problem, a version of it states that $\beta$ is equivalent to the following
$$\hat \beta = \frac{\hat{\mathbb E}[y_{ist}(\tau_{ist} - \hat \tau_{ist})]}{\hat{\mathbb E}[(\tau - \hat\tau_{ist})^2]}$$
where $\hat\tau_{ist}$ is the predicted value of $\tau_{ist}$ from the the least squares regression $\tau_{ist} = a_t + b_i + \delta_{ist}$, and $\hat{\mathbb E}$ is my notation for the sample mean. So to answer your question 1, it would suffice to show two facts:

*

*$\tau_{ist} - \hat\tau_{ist} = 0$ for any observation $i$ that is only present in one of the two time periods

*$\hat \tau_{ist}$ does not depend on the value of $\tau_{jst}$ for any observation $j$ that only shows up in one time period

Point 1 is easy enough to see: if this was not true, then by changing the parameter $b_i$ by amount $\tau_{ist} - \hat\tau_{ist}$, we could decrease the least squares objective, which would contradict that $\hat\tau_{ist}$ is the least squares fit. Point 2 is also fairly simple. The basic idea is that by the argument I just made to prove point 1, for all $j$ that only shows up in one period, we can always choose $b_i$ in such a way that $\tau_{jst} - \hat\tau_{jst} = 0$. But this means that the least squares fit for the model $\tau_{ist} = a_t + b_i + \delta_{ist}$ can always be obtained by first fitting the model only for individuals showing up in both time periods and then setting $b_j$ accordingly for the rest and importantly, this second step does not change any of the fits $\hat\tau_{ist}$ from the first step.
Once we have shown these two points, using our expression for $\hat\beta$ that observations only showing up in one period do not affect the estimate: By point 1, their contribution to the sample mean is 0, and by point 2, they do not affect $\hat\tau_{ist}$ for any $i$ that did show up in both periods.
To your point 2, there is no reason why firms that switch should not affect your estimate for $\beta$. From the perspective of OLS, this structure that you imposed that $\tau$ is sector specific is not used for estimation at all, so individuals experiencing a change in $\tau$ because they switched sectors should be treated no differently than individuals experiencing a change in $\tau$ because their sector experienced a change in $\tau$.
Finally, to your last point, it seems that a FE model might just not work if your concern is that firms are selecting into tariff levels based on their reactions to policy changes. In particular, these FE models tend to rely on a "common trends" assumption or something similar, but selection into tariffs may arise precisely because firms with different trends select differently. I do not have enough context about your problem to offer any concrete advice with any certainty, but one direction you might consider is looking for covariates that explain why firms are switching industries.
