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I know that if the coefficient of determination $R^2$ is 0, then the estimated regression line is horizontal. However, in the case of multiple regression, how would you show that $\hat \beta_0=\bar y$ and $\hat \beta_1 = ...=\hat \beta_p=0$, assuming that $\hat y=Z\hat \beta$ where $Z$ is a $n\times(p+1)$ matrix

Christoph Hanck
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Hints:

When all the other regressors except the intercept are zero, you effectively only regress on a constant. What are the fitted values of that regression? What are then the residuals $e$?

How does the $R^2$, defined as $$ R^2=1-\frac{e'e}{(y-\bar{y})'(y-\bar{y})}, $$ then look like?

Christoph Hanck
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