Question

3. Prove that the sample covariance between the fitted values and the residuals ûi is always zero in the simple linear regression model with an intercept. Show all of the steps in your derivation.

Answer 1

The sample covariance between the fitted values and the residuals is

$cov(\hat{y}_i,\hat{u}_i)=(1/(n-1))\sum_{i=1}^{n}(\hat{y}_i-\bar{y})(\hat{u}_i-\bar{u}); we\:know\:\bar{u} = 0,\hat{u}_i = y_i-\hat{y}_i.\\ (1/n-1)\sum_{i=1}^{n}(\hat{y}_i-\bar{y})(y_i-\hat{y}_i) = \sum_{i=1}^{n}(\hat{y}_iy_i -\bar{y}y_i+\bar{y}\hat{y}_i-\hat{y}_i^2) [\because ignoring\: (1/n-1)\: term]= \sum_{i=1}^{n}\hat{y}_iy_i -\bar{y}\sum_{i=1}^{n}(y_i-\hat{y}_i) -\sum_{i=1}^{n}\hat{y}_i^2 =\\ \sum_{i=1}^{n}\hat{y}_i(y_i-\hat{y}_i)-\bar{y}\sum_{i=1}^{n}\hat{u}_i = \sum_{i=1}^{n}(\hat{\beta}x_i)\hat{u}_i (\because \sum_{i=1}^{n}\hat{u}_i=n\bar{u}=0;\hat{y}_i=\hat{\beta}x_i) =\\ \hat{\beta}\sum_{i=1}^{n}x_i\hat{u_i} = 0 (\because \sum_{i=1}^{n}x_i\hat{u}_i=0)$