Consider a correctly specified regression model with $p$ terms, including the intercept. Make the usual assumptions about $varepsilon$. Prove that [sum_{i=1}^{n} operatorname{Var}left(hat{y}_{i} ight)=p sigma^{2}]

Consider a correctly specified regression model with $p$ terms, including the intercept. Make the usual assumptions about $\varepsilon$. Prove that

\[\sum_{i=1}^{n} \operatorname{Var}\left(\hat{y}_{i}\right)=p \sigma^{2}\]