Explaining changes caused by dummy variables quantitatively

Question

Consider the following linear model \begin{equation} Y_{j}=\beta_{1}+\beta_{2}E_{j}+\beta_{3}B_{j}+\epsilon_{j} \tag{*} \end{equation} where $Y_{j}$ represents the natural logarithm of the annual salary of an employee currently, $E_{j}$ the number of finished years at an educational institution, and $B_{j}$ is the natural logarithm of the initial salary of the employee.

I have regressed $(*)$ on a large data set, and compared the coefficients $\beta_{2}, \beta_{3}$, to another model that includes two additional dummy variables, $D_{1j}$ (gender) and $D_{2j}$ (minority). Although they have not changed drastically, there still is some change. Now, I have no trouble giving a qualitative explanation as to why this is the case. What I am looking for is this:

Is there some calculation, perhaps a test, that will facilitate a qualitative explanation of why the coefficients of $(*)$ may change when one adds two dummy variables?

Answer 1

I'm not quite sure I follow your question. I see two possibilities:

1. You can test to see if the relationship (i.e., if $\beta_2$ and $\beta_3$ change) between salary and education & initial salary differ based on gender and minority status, by including and testing interaction terms. I discuss interactions further here: Interaction in generalized linear model. The model you would fit is:
   \begin{align} Y_{j} = &\beta_{1} + \beta_{2}E_{j} + \beta_{3}B_{j} + \beta_4D_{1j} + \beta_5D_{2j} + \\ &\beta_6E_{j}D_{1j} + \beta_7E_{j}D_{2j} + \beta_8B_{j}D_{1j} + \beta_9B_{j}D_{2j} + \varepsilon_{j} \end{align} The tests of the interaction terms are the tests of $\beta_6$, $\beta_7$, $\beta_8$, and $\beta_9$. As far as which test to use, I gather your dummies are single (1 df) variables, so if you want to test them individually, you can use the $t$-tests of the coefficients that come with typical regression output. (By contrast, if minority status were multiple dummies to compare various minorities each with the ethnic / racial majority, then you would need to drop all of them and test the nested model with an $F$-test.) If you wanted to test if a dummy (say, gender), had an impact on the effect of education or initial salary together, you would need to drop both interaction terms and use the $F$-test for the nested model. I explain the use of the $F$-test for multiple dummies in an interaction here: Testing for moderation with continuous vs. categorical moderators.

2. If you are wondering why the effects already in your model (i.e., $\beta_2$ and $\beta_3$) could have changed when you added the dummies, the answer is endogeneity. That is, your variables were correlated with your dummies. To learn more about this, it may help you to read my answer here: Is the difference between 'controlling for' and 'ignoring' other variables in multiple regression?