Small identifying subsample when using individual specific fixed effects

Question

What would be good questions to adress when coefficients in a panel data regression change upon adding fixed effects and the identifying subsample i rather narrow?

A standard problem for fixed effects estimation is that it cannot be used when the interest is in estimation of coefficients for time-invariant covariates. The problem I face could perhaps be said to be a less extreme version of this problem, because it is not the case that no observational units displays time-variation on the covariates of interests but instead that the subset of the sample with variation is rather small (5%).

Specifically I am considering a panel of workers full population 2008-2016. The effect of occupational skill drops dramatically when adding fixed effects. High skill occupations increase wage by 40% when not controlling for individual fixed effects but only 6% when controlling for individual fixed effects. The model is a standard mincerian wage regression

$$\log w_{it} = \mu_i + \delta \cdot highskill.occupation_{it} + \mathbf x_{it}^\top \beta + \epsilon_{it},$$

with log-hourly wage as dependent variable, $\mu_i$ unobserved time constant fixed effect and high skill occupation as a dummy variable.

I am aware that the identification changes drastically when adding fixed effects because the coefficient $\delta$ is then only identified by individual workers changing occupation. In the population under consideration approximately 4-5% of the population change occupational level each year. One of my worries is naturally that this group of individuals are in some sense special in ways not controlled for. But beyond checking how many workers actually contribute to the identification I do not know what else might be a good idea to look at, hence the question.

Notice that the term "fixed effect" is here used in the econometric sense of the word, hence the mincerian wage equation is estimated using the within estimator.

Potential strategies:

Strategy 1: Compare the estimates gotten to other credible estimates from the literature.

Strategy 2: If data permits it the high skill occupational variable could be investigated. It could for example be the case that the occupations that individual workers are changing between is only a particular subset of the set of occupational groups.

Strategy 3: Perhaps it is informative to estimate and results compare a mixed effects model.

Answer 1

Are there any confounders that the fixed effects are controlling for?, would be my number one question. Instead of fixed effects notation, consider the same equation with a time-invariant unobserved variable, $A_i$

$$ \log w_{it} = \alpha + \gamma A_i + \delta D_{it} + \textbf{x}_{it}^\prime \boldsymbol\beta + u_{it} $$

Without fixed effects, the error term is $ \varepsilon_{it} = \gamma A_i + u_{it} $. If $A_i$ also explains the probability of being in a high-skill occupation, then it will cause omitted variable bias. Your problem seems very similar to the "returns to education" problem (Mostly Harmless Econometrics by Angrist and Pischke is a very good reference on this topic), i.e., unobserved ability is causing both occupation and earnings.