Given a linear regression model obtained by ordinary least squares, prove that the sample covariance between the fitted values and the residuals is zero.
2026-03-30 00:18:30.1774829910
Covariance between fitted values and residuals
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I assume that you meant their dot product is zero. If so, let $\hat{y} = Hy$ where $H = X(X'X)^{-1}X'$ (from OLS). Also, note that $H$ is idempotent, i.e. $H^2 = H.$ Then, $$\hat y' . e = y'H(I - H)y = y'(H - H^2)y = y'(H - H)y = 0.$$