\text { (Total } 9 \text { points) Suppose we estimate a linear regression equation } Y_{i}=\beta_{0}+ \beta_{1} X_{i}+u_{i} \text { by OLS. } \text { Show that } \frac{1}{n} \sum_{i=1}^{n} \hat{u}_{i}=0 \text {, where the } \hat{u}_{i} \text { 's are the regression } ) (3 points) Suppose we regress X; on û; by OLS, including a constant term in the regression. Show that the estimated coefficient on û; is equal to zero. (3 points) Suppose we regress Y; on the predicted value Y; by OLS,including a constant term in the regression. Show that the estimated coefficient on Y; is equal to one.

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