In class, our professor said that when it comes to deciding between estimating panel regression with fixed and random effects we should not just blindly follow the Hausman test, but also think about how we expect the omitted variables to behave based on economic intuition.
But I don't understand how we are supposed to this. Are there any examples of situations where just from an economic perspective fixed effects are better than random effects or the other way around?
Here is an example where just from an economic perspective fixed effects are better than random effects.
Suppose you have panel data and you want to regress earnings $y$ on some observable characteristics $X$ of an individual like education, tenure, experience, age, birthplace, etc. The regression you would estimate is
$$y_{it} = \alpha + X'_{it} \beta + \epsilon_{it}$$
where the error term $\epsilon_{it} = \alpha_i + \eta_{it}$, is a function of individual heterogeneity $\alpha_i$, which is not varying over time and some random shock $\eta_{it}$.
Pooled ordinary least squares and random effects assume that the observable characteristics and the individual heterogeneity component are uncorrelated, $Cov(\alpha_i,X_{it})=0$. As you know this does not hold when there is a correlation between your controls $X$ and the error term, which will bias your estimates - that's the standard omitted variables bias.
Does the assumption $Cov(\alpha_i,X_{it})=0$ hold in the earnings context?
In this context, your economic intuition will be useful. You may think of $\alpha_i$ as individual ability, which is unobserved by the econometrician but potentially correlated with some of the observed individual characteristics $X$, such as education or tenure. So, the $\alpha_i$ correlate with the regressors $X_{it}$, and the assumption $Cov(\alpha_i,X_{it})=0$, is violated. Then, a fixed effect approach, which effectively fits such intercepts will be more convincing.
