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I am estimating two regressions. The first is a regression with only dummy variables (i.e. $y = B_0 + B_1D_1 + B_2D_2$), and the second is with two independent variables (i.e. $y = B_0 + B_1X_1 + B_2X_2 + B_3D_1 + B_4D_2$).

The $D_1$ and $D_2$ are dummy variables, whereas the $X_1$ and $X_2$ are continuous. Would there be a difference in the meaning/interpretation of the B1 in the first regression and the B3 in the second regression (both the coefficient of the first dummy variable)?

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It's comparable between the meanings/interpretations of the $B_1$ in the first regression and the $B_3$ in the second regression.

You can interpret them like below:

$B_1$ in the first regression: the difference in y of $D_1$ compared with reference category

$B_3$ in the second regression: the difference in y of $D_1$ compared with reference category adjusted for the covariates ($X_1$ and $X_2$) in the model

If $X_1$ and $X_2$ are confounding variables or effect modifiers with statistical significance, There may be a difference in the results of $B_1$(in the 1st regression) and $B_3$(in the 2nd regression).

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