Dummy's, significant and not significant I have in my model two dummy's, for a categorical variable with 3 categories (native, EU-immigrant and non-EU-immigrant). The reference category is 'native'. These are merely control variables. 
Now I find a significant coefficient for one dummy, and not for the other. See last 2 rows here. Note these are log odds.
    L1PRED  S.E.
Response    R   

Fixed Part      
cons    -1.775  0.080
female  0.431   0.042
age 0.016   0.001
FA  1.053   0.052
TV  1.778   0.112
second  -0.277  0.052
third   -0.269  0.070
fourth  -0.298  0.093
fifth   -0.163  0.102
EU_imm  -0.011  0.092
nEU_imm 0.953   0.439

I was wondering what the right thing to do here is? Because any control variable that has no effect is usually left out of the model. But I cannot do that here, I think? 
Would it be most common to recode 'native' and 'EU-immigrants' into one category, since there is no significant effect on the outcome?
 A: It is not one of your control variables that is not significant but rather one level of that variable that is not significantly different from your baseline, as you point out. 
In this situation, I would not change anything and report coefficients for both levels. Recoding is an option, but unless there is a good reason to do it, I would not. 
You state that "any control variable that has no effect is usually left out of the model", this is not standard practice in every (any?) field and if there is a good theoretical reason to think that the variable might have an impact on the outcome I would keep it in the model for this reason, whether it is significant or not. For more on variable selection, this question is helpful.
A: Adding to the good comments and answers so far, here are some reasons to include control variables even if not significant:
1) If you expected a large effect and you get a small one, that is important to know
2) Adding the control variable may affect the relationship between the other independent variables and the dependent variable
And here is a reason not to combine "native" and "EU immigrant": it loses information. As is you have evidence that non-EU immigrants are different from EU immigrants and natives. Part of that is lost if you combine the levels. 
