Extracting group effects from individual fixed effects I have a conceptual econometrics question I was hoping to get some help with.

I have a panel of 100 individual firms over time (10 years) at the quarterly frequency across 10 countries. One of the questions I'm interested in is about whether there are fixed differences across countries with respect to the dependent variable.

The initially obvious way to me of looking at this is to drop the firm fixed effects in my regression and include country dummies instead.

But I was wondering if it's feasible/makes sense to get at the same question by averaging the firm fixed effect parameters in some way within country groups. 

And if so, would this be unambiguously better (considering that the Hausman test points to the value of including firm fixed effects)?

Thanks for your help and let me know if anything is unclear. This is my first post
Good question.

You observe $y_{ijt}$ ($i$: firm, $j$: country, $t$: time). Let us not consider time effects as they are irrelevant to our discussion. You are considering identifying "country effects" using two approaches.

In short, the two are as different as RE vs FE in standard 2D panel models, with the main difference being in what you assume about the correlation of the explanatory variables and the firm effects within country groups. (i) Dropping firm-effects and using country dummies gives consistency if firms within the same country have no fixed effects [1]. (ii) Defining country effects by the averages of firm fixed-effects allows for firm-level fixed effects [1] correlated with the regressor levels.

More detailed discussions follow.

Your first approach (of having country dummies and no firm effects) is represented by the model $y_{ijt} = c_j + X_{ijt}\beta + v_{ijt}$, with $E(v_{ijt})=0$ and $X_{ijt}$ being uncorrelated with $v_{ijt}$. If there is no firm-level correlation of $X$ and $v$ within country groups (that is, if $X_{ijt}$ is exogenous to the error $v_{ijt}$), the random-effects regression (POLS or FGLS or whatever) of this model gives consistent estimators. However, if $v_{ijt}$ contains firm-level fixed effects (say, $\mu_{ij}$) that are correlated with the level of $X_{ijt}$, this approach leads to inconsistency.

The model corresponding to your second approach is $y_{it} = c_j + \mu_{ij} + X_{ijt}\beta + e_{ijt}$, where $E(e_{ijt} | X_{ij1}, \ldots, X_{ijT})=0$ for all $t$. What is important here is that $c_j$ and $\mu_{ij}$ are not separately identified without further restrictions even under this strict exogeneity of $X_{ijt}$ (because you can add a constant to $\mu_j$ and subtract the same constant from $\mu_{ij}$). You are proposing to define $c_j$ by the further restriction that $n_j^{-1} \sum_{i=1}^{n_j} \mu_{ij}=0$, which is fine.

What is the real difference between the two approaches? For the model $y_{it} = c_j + \mu_{ij} + X_{ijt}\beta + e_{ijt}$, the difference is that the first assumes that $X_{ijt}$ is exogenous to $\mu_{ij} + e_{ijt}$ while the second assumes that $X_{ijt}$ is exogenous to $e_{ijt}$. Under the assumption that $X_{ijt}$ is strictly exogenous to $e_{ijt}$, the difference is reduced to whether $X_{ijt}$ is correlated with $\mu_{ij}$. If, within a country, a firm with high $X_{ijt}$ level has higher $\mu_{ij}$, then only the second approach gives a consistent estimator. It is all the same as any RE vs FE considerations in standard (2D) panel data models.

[1] "Fixed effects" above mean time-invariant effects that are possibly correlated with the levels of the explanatory variables.