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
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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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I just realized that Z should be $nx(p+1)$. I think in this case $e'e=(y-\bar y)'(y-\bar y)$ ? – carolineklj Nov 26 '20 at 16:37
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also would all the fitted values $\hat y$ just be $\bar y$ ? – carolineklj Nov 26 '20 at 17:15
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Indeed, see, e.g., https://stats.stackexchange.com/questions/201919/utility-of-the-frisch-waugh-theorem. The desired result then follows directly, right? – Christoph Hanck Nov 27 '20 at 05:17
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Yes, thank you! – carolineklj Nov 27 '20 at 17:20
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welcome - if your question has been answered, please consider clicking on the check mark to indicate it has been dealt with – Christoph Hanck Nov 27 '20 at 17:41