What is the model constant when the dummy variable equals 1...

What is the model constant when the dummy variable equals 1 in the following equations, where $x_1$ is a continuous variable and $x_2$ is a dummy variable with a value of 0 or 1 ?
a. $\hat{y}=4+8 x_1+3 x_2$
b. $\hat{y}=7+6 x_1+5 x_2$
c. $\hat{y}=4+8 x_1+3 x_2+4 x_1 x_2$

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Key Concepts

Dummy Variables

Dummy variables are binary indicators used in regression analysis to represent categorical data. They take values of 0 or 1 and allow the model to estimate different intercepts or slopes for different categories. When interpreting the model, the coefficient on a dummy variable shows the change in the intercept when moving from the reference group (dummy = 0) to the indicated group (dummy = 1).

Interaction Terms

Interaction terms in regression models are used to examine whether the effect of one independent variable depends on the level of another. When a continuous variable interacts with a dummy variable, it allows both the slope and sometimes the intercept to differ between the groups represented by the dummy variable. This interaction adjusts the baseline model when the dummy variable equals 1.

Regression Intercept

The regression intercept, often called the constant term, represents the expected value of the dependent variable when all independent variables are equal to zero. In models with dummy variables, the intercept typically gives the baseline level (e.g., when the dummy variable equals 0) unless the interpretation is shifted via interactions or reparameterization.

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