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I would like to gain some intuition on the differences between models using various fixed effects specifications. This question relates to this one by me: Three way fixed effects vs combining two of the effects

Consider that I have panel data on firms $i$ across two time periods $t$. Firms can belong to, let's say, subsector $s$. These subsectors are included in a broader economic classification $e$, which I call a sector. I explore the impacts of variation in tariff rates $\tau_{st}$ on relevant economic outcomes $y_{ist}$. $\beta_j$ is my parameter of interest

The first regression model I consider is of the following form, considering time and sector fixed effects : $$y_{it} = \beta_1 \cdot \tau_{st} + \gamma' X_{it} + \eta' Z_{st} + \lambda_t + \mu_e + \epsilon_{it}$$

Two alternative specifications include sector-year fixed effects:

$$y_{it} = \beta_2 \cdot \tau_{st} + \gamma' X_{it} + \eta' Z_{st} + \delta_{te} + \mu_e + \epsilon_{it}$$ $$y_{it} = \beta_3 \cdot \tau_{st} + \gamma' X_{it} + \eta' Z_{st} + \lambda_t + \delta_{te} + \epsilon_{it}$$ $$y_{it} = \beta_4 \cdot \tau_{st} + \gamma' X_{it} + \eta' Z_{st} + \delta_{te} + \epsilon_{it}$$

$X_{it}$ are time-varying firm controls, $Z_{st}$ are time-varying subsector controls It is clear that my $\beta$'s are not all equal. Are all of these models even estimable?

I know that one-way fixed effects models identify relative impacts within a certain group. Is it clear what kinds of unobservables am I controlling for in each case?

For instance, how could I control for differential time trends in each sector $e$?

Arthur Carvalho Brito
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    Unless I misinterpret your model specifications; the estimates are adjusted for differential time trends in each sector $e$ by $\delta_{te}$, not? Since this is a sector-specific time trend. The models are estimable; conditional on the sample size being large enough as the various fixed effects will gobble up some degrees of freedom. A hierarchical (or mixed) model might be more flexible as you have the outcome for firm $i$ at time $t$ nested in subsector $s$ of sector $e$. – horseoftheyear May 29 '20 at 17:51
  • @horseoftheyear Your intepretation is correct. But is it standard practice to adjust for a general time trend $\lambda_t$ if I have used $\delta_{te}$? – Arthur Carvalho Brito May 29 '20 at 18:19
  • I'm not sure if I get what a hierarchical model is, but I don't want to have different effects for each sector. As you can see, $\beta$ is not varying wich $s$ or with $e$ - there isn't even enough variation in the data for that – Arthur Carvalho Brito May 29 '20 at 18:21
  • Yes, because they filter out different effect. The $\lambda_t$ accounts for common shocks, due to the business cycle for instance, while $\delta_{te}$ accounts for the sector-specific shocks (tourism faces other shocks than insurance etc.). – horseoftheyear May 29 '20 at 18:23
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    So concerning the hierarchical model, you don't get different effects for each sector (although you can model it that way). It just models the structure of the data a bit more naturally (imho). Given your remark about the lack of variation in the data it might actually be useful as with a fixed effect model the estimates rely exclusively on the within-variation whereas a hierarchical model the estimates borrow information. See [Gelman](http://www.stat.columbia.edu/~gelman/research/published/multi2.pdf) for instance. I'll stop now before I come across as evangelical :) – horseoftheyear May 29 '20 at 18:28

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