I want to check my understanding of random effects with regards to the reference group in a regression. Let's say I want to predict earnings for a population on the basis of individual characteristics such as race, sex, and education and I build the following initial model.

$Earnings_i = \beta_0 + \beta_1 * black_i + \beta_2 * asian_i + \beta_3 * hispanic_i + \beta_4 * educ_i + \varepsilon_i$

The race variables are binary and the reference group for this regression are survey respondents who are white. For simplicity, we assume that black, hispanic, asian and white individuals comprise the entirety of racial/ethnic variation across the sample. Now, let's say I want to make this a multilevel model and add random effects grouped by $State_j$. So now I have

$Earnings_{ij} = \beta_{0j} + \beta_{1j} * black_{ij} + \beta_{2j} * asian_{ij} + \beta_{3j} * hispanic_{ij} + \beta_{4j} * educ_{ij} + \varepsilon_{ij}$


$\beta_{0j} = \gamma_{0j} + b_{0j}$

$\beta_{1j} = \gamma_{1j} + b_{1j}$

$\beta_{2j} = \gamma_{2j} + b_{2j}$

$\beta_{3j} = \gamma_{3j} + b_{3j}$

$\beta_{4j} = \gamma_{4j} + b_{4j}$

such that $\gamma_{ij}$ is the fixed effect and $b_{ij}$ is the random effect.

My understanding is that with random effects on all variables and the slope, I'm accounting for different effects of $black$, $hispanic$, $asian$, and education across states. Is it fair to say that the different effect of $white$ across different states is being captured in the random effect $b_{0j}$, especially since I have accounted for all other races contained within the sample population?


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