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$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X_1+\hat{\beta_2}X_2 , \quad\text{$X_1$, and $X_2$ are indicator variables}$$

In ANOVA, regression on one factor with 3 levels, how to show the following.

$$Cov(\hat{\beta_1},\hat{\beta_2})=Var(\hat{\beta_0})$$

Appreciate for any help.

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I'd do it by considering a model with no intercept and three indicator variables $\gamma_0 X_0+\gamma_1 X_2+\gamma_2 X_2$. The $\hat\gamma_i$ are independent (they are just the three means), and you can define $\beta_0=\gamma_0$, $ \beta_1=\gamma_1-\gamma_0$, and $\beta_2=\gamma_2-\gamma_0$.

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