Dear Lorenz,

Regarding the omitted variable query-

Generally, the reason for omission a variable in regression result is to address dependency among the independent variables.

Further, you may analyze data feature of Hausman model selection test

Dear Lorenz

The best way to understand this problem is to program it in a matlab procedure (for instance) and then observe what happens when you use all the created dummy variables in the regression. Brajesh´s answer is near the core of the problem, but not very close. The problem is related to including ALL those DUMMY independent variables. As the picture shows, you are not including a column of ones for the intercept. You mentioned that, for the no-fixed-effects time-fixed-effects case, "as expected", one of the dummy variables was removed from the model. Do you know exactly why? Besides, why was 1980 the removed variable? First, whenever you have a set of dummy variables, you cannot include a column of ones as an independent variable because then you will get PERFECT COLLINEARITY and thus inv(X'X) does not exist (ie, any results you may get -from the routine you are using- are completely wrong). Secondly, the routine is getting rid of one of those dummies (in this case, 1980) just to be able to provide "correct results" WITHOUT a column of ones. This implies that the routine is computing the degree of multicollinearity and it is somehow detecting "the specific variable generating that problem" (subroutine?) and thus getting rid of it. Let's consider the second case, the fixed-effects time-fixed-effects case. Now you got two sets of dummy variables and thus the problem gets worse because now you have PERFECT COLLINEARITY (because just one of those sets of dummy variables is now working as a column of ones wrt the other set of dummy variables). To tell you the true, "the specific variable generating that problem" (subroutine?) is running twice, thus getting rid of two variables. CLEARLY, I suggest you should code a matlab program and then consider those two sets of dummy variables as "the whole set of dummy variables" (excluding "union", "educ" & "exp"). Then get rid of the FIRST dummy variable in the whole set of dummy variables, and save the results. Then do it again getting rid of the SECOND dummy variable, and save the results, ... and so on until getting rid of the LAST dummy variable. Then you must compare all those saved results wrt the SSR's, the AIC's, or any other econometric criterion in order to choose the best estimated model. You should NOT exclude any variable in the set { "union", "educ", "exp"}.