(First off, bonus points if you can write my question title more clearly)
My project right now is to research citizenship status and income.
At first my regression was like this:
$$ INCOME = \beta0 + \beta1STATUS $$
Now, I am going to do one with control variables like this: $$ INCOME = \beta0 + \beta1STATUS + \beta2EDUCATION + \beta3ZIPCODE$$ When I'm looking at the coefficient and standard error for immigration status, what does the regression of earnings on immigration status and all my control variables tell me compared to a regression of just earnings on immigration status? Generally in regressions, how does adding more variables change standard errors and coefficients?