How can I choose type of regression for panel data? I would like to do a research thesis where I investigate whether exists a linear relationship among a ratio commonly used by firms and a set of dependent variables. I know this is very general but I'm looking to have a clear understanding of how the statistical analysis should be run so that my results make sense. From my current idea I have a single dependent variable - the ratio - , 6 independent numeric variables which depend on the country of origin of the company and are constant over time, plus some financial variables which should serve as control variables. I want to randomly collect data from a specific database for a 10-year period so I will probably end up with an unbalanced panel.
I have zero experience with econometrics but after some research it seems that there are 3 main methods to deal with panel data for my purposes: 1) Pooled OLS - which to my understanding is basically the standard linear regression applied to panel data –  ; 2) Fixed Effects ; 3) Random Effects . I noticed that some papers related to my idea test two or three models and compare them while others just choose one - without really explaining why - and assess the results. My first concern is which option is better. I think that using more than a model can lead to more "robust" results, but if I don't get significant coefficients in one of them comparisons might be more difficult. On the other hand, I fear choosing only one model might lead to criticism such as "you used this model which assumes x so your results are weak".
Also, I see that many papers say something along the lines of "we control for year and industry fixed effects". Can this be done regardless of 1) 2) 3) regression method or only by using a specific model? I'm trying to understand when different models should be used conceptually rather than their correctness from a statistic POV, since it seems they all have some advantages and drawbacks. For example, the textbook "Econometric Analysis of Panel Data" by Baltagi says "FE estimator cannot estimate the effect of any time-invariant variable like sex, race, religion, schooling or union participation." But if I have 6 time-invariant independent variables, does this mean I should never use FE, even if the Hausman test tells me otherwise? Any help would be appreciated, thank you in advance and sorry for the length of the thread.
 A: This is a big question with no right answer. A lot depends on your audience and field. The fact that you are citing Baltagi suggests you might be in the econ area. If so, you are going to have a hard time convincing colleagues/readers that random effects are appropriate unless you want to do something that absolutely cannot be done in a fixed effects model. Estimating the effect of time-invariant or firm-invariant predictors is an example of something that is a no-go in fixed effects models. But many in the econ audience probably would argue that you can't trust those effects anyway because of endogeneity (correlations between level-2 covariates and the random intercepts).
Note that you can get around this problem and estimate firm effects if you have an even higher level of nesting. Let's say firms are nested within countries. By including the country-specific means of the firm variables in the random intercepts model, these firm-level predictors become within-firms effects. This is a known way to get around lower-level endogeneity in random effects models, which was elaborated upon by Mundlak (1978).
One thing you should be careful about is including firm and year fixed effects in the same model. Recent methodological work shows that models like this ("two-way fixed effects") have some problems. See work by Kropko & Kubinec (2020) and Imai & Kim (2021). Note that within a random effects model where firm is the level-2 cluster variable, you could easily include time fixed effects (0/1 indicator variables for each year of data). But in random effects models, time is often treated continuously and included as a random slope, especially if the trend in the outcome is systematic and varies across firms. This is known as a growth curve model.
References
Mundlak, Y. 1978. On the pooling of time series and cross section data. Econometrica 46: 69–85
