The linear regression model is \(y=\beta_{1}+\beta_{2} x+e\). Let \(\bar{y}\) be the sample mean of the \(y\)-values and \(\bar{x}\) the average of the \(x\)-values. Create variables \(\tilde{y}=y-\bar{y}\) and \(\tilde{x}=x-\bar{x}\). Let \(\tilde{y}=\alpha \tilde{x}+e\).

a. Show, algebraically, that the least squares estimator of \(\alpha\) is identical to the least square estimator of \(\beta_{2}\).

b. Show, algebraically, that the least squares residuals from \(\tilde{y}=\alpha \tilde{x}+e\) are the same as the least squares residuals from the original linear model \(y=\beta_{1}+\beta_{2} x+e\).