Let (X_{i}) denote a binary variable, and consider the regression (Y_{i}=beta_{0}+) (beta_{1} X_{i}+u_{i}). Let (bar{Y}_{0}) denote the

Question:

Let $X_{i}$ denote a binary variable, and consider the regression $Y_{i}=\beta_{0}+$ $\beta_{1} X_{i}+u_{i}$. Let $\bar{Y}_{0}$ denote the sample mean for observations with $X=0$, and let $\bar{Y}_{1}$ denote the sample mean for observations with $X=1$. Show that $\hat{\beta}_{0}=\bar{Y}_{0}, \hat{\beta}_{0}+\hat{\beta}_{1}=\bar{Y}_{1}$, and $\hat{\beta}_{1}=\bar{Y}_{1}-\bar{Y}_{0}$.

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