Suppose I am interested in whether a students age affects their performance, and I run the following regression:
$performance(i) = \beta_1 age(i) + \beta_2 Female(i)+\beta_3 classsize(i)+\varepsilon(i), $
where $performance(i)$ is student i's performance, $Female(i)$ is a dummy variable which is one if the student is female and $classsize(i)$ is the number of students in the same class.
I am interested (only) in $\beta_1$. If I collect the data from all students of a given school, the problem of correlated errors across a given class arises. That is $classize(i)=classsize(j)$ for all student $i,j$ who are in the same class. Therefore, if I was interested in the effect of class size on performance, I would need to cluster the standard errors on class level (i.e. a categorical variables which groups the classes). If I am not interested in $\beta_3$ at all, (but I want to include it in the regression because I worry about omitted variable bias), would I still need to cluster the data on class level? What if I include a categorial variable in the regression ("class fixed effect")?
More generally: Is standard-error-clustering related to the variable(s) of interest OR to the general setup of the data?
Edit: Suppose the data is collected for 100 schools that were randomly chosen, and for each of those schools, every student enters the survey.