Pooled OLS vs FE and Two way FE

[Saif Sheikh]

So I ran couple of different models to see how does position influence overall rating of a player. And the results I got are not what I expected. So I am curious if I did something wrong. In the attached screenshot, you could notice that coefficients on forward and center are positive and significant, but in FE model it becomes insignificant and negative. So I am really curious what happened there.
Another interesting thing in my results is the dramatic bias penalties has without controlling for time. And when I control for time and entitiy differences in Two-way FE coefficient on center becomes positive. So I am just curious to know what's going on here.

[Clyde Schechter]

There is nothing surprising. There is no reason to expect these results to resemble each other at all. You have fit three different models to the data and, predictably, gotten three different results. Occasionally, more or less by coincidence, they resemble each other, but not here.

The difference between the OLS and FE model is that the FE model is estimating within panel (I guess your panel here is player?) So in the OLS analysis, the coefficient of "Center" estimates the difference in rating between players who are centers and players who are not centers. By contrast, in the FE model, thee coefficient of "Center" estimates the change in rating for any given player if he/she changes from a non-Center position to a Center position. There is no reason these to be the same, or even close, or even in the same direction.

For a very clear example where within-panel and between-panel changes run in opposite directions, run this code and study the results:

clear
set obs 5
gen panel_id = _n
expand 2

set seed 1234
by panel_id , sort: gen y = 4*panel_id - _n + 3 + rnormal(0, 0.5)
by panel_id: gen x = panel_id + _n

xtset panel_id

xtreg y x, fe
regress y x

//    GRAPH THE DATA TO SHOW WHAT'S HAPPENING
separate y, by(panel_id)

graph twoway connect y? x || lfit y x

Now, as between the simple FE and the two-way FE model, this is a matter of having added extra variables to the model. Whenever you add new variables to a model, you can expect the results to change, unless the new variables are completely independent of the original ones. It is essentially the difference between a model with some covariates and a different model with none. When the change is large enough to flip the direction of an effect, it is sometimes referred to as Simpson's paradox, though really there is nothing paradoxical about it. If you want to learn more about Simpson's paradox, the Wikipedia page on it is rather good.