Intuition on why Fixed-effects models depend on there being variation within the unit of analysis.

[Jun Park]

I am new to econometrics and tried to find the answer on google, but couldn't find any. I've learned that Fixed-effects models depend on there being variation within each higher-level unit of analysis. For example, if there is no variation within a company's observations, it can't be used in the model.
1. This is probably only for the main predictor variable right? All other control variables wouldn't matter? For example, if company size is the main predictor and I use size fixed effects, if there is no variation in size then STATA will automatically drop those variables?
2. Is this also relevant for continuous variables?
3. In a fixed-effects logistic regression model, you cannot use observations that have no variation on y. Why is it the y variable here not x?

These might be very simple questions for most of you.. but would appreciate if anyone can help!

[Carlo Lazzaro]

Jun:
1) the -fe- estimator wipes out time-invariant variables and works poor when there's little withi-panel variation for time-varying variables;
2) yes:
3) this holds for cross-sectional logistic regression, too. It is also important to highlight that -xtlogit,fe- refers to conditional fixed effects (that differ from the -fe- estimator used in -xtreg-) due to the incidental parameter bias (see
http://www.econ.brown.edu/Faculty/To...meters1948.pdf).

[Richard Williams]

I discuss this a bit in

https://www3.nd.edu/~rwilliam/xsoc73...xedEffects.pdf

[Clyde Schechter]

Though not really in answer to O.P.'s question I would like to point out an important, but often overlooked, fact about fixed effects regressions that is relevant here.

When you have panel data, there are two different dimensions along which an explanatory variable x can have an effect on an outcome variable y. The variable y may be affected by differences in the values of x among the different panels; this is called the between-panel effect. It might also be affected by changes in the values of x over time within each panel; this is called the within-panel effect. Importantly: the two effects can be different, even in opposite directions.

Fixed effect models only estimate the within-panel effect. If your research goals include learning about between-panel effects, you cannot accomplish that with a fixed-effects model. (Regardless of what a Hausman or overidentification or other test may say about fixed versus random effects--if you need to estimate between panel effects, you must not use a fixed-effects model.) This relates to the original question in #1 in that if an explanatory variable does not vary within panels, then there is no within-panel effect to estimate, so the fixed-effects model will reject that variable.