Suppose we have a regression in which we want to fit the mean function (3.1). Following the outline in Section 3.1, suppose that the two terms X1 and X2 have sample correlation equal to zero. This means that, if xij , i = 1,…,n, and j = 1, 2 are the observed values of these two terms for the n cases in the data,
1. Give the formula for the slope of the regression for Y on X1, and for Y on X2. Give the value of the slope of the regression for X2 on X1.
2. Give formulas for the residuals for the regressions of Y on X1 and for X2 on X1. The plot of these two sets of residuals corresponds to the added-variable plot in Figure 3.1d.
3. Compute the slope of the regression corresponding to the added-variable plot for the regression of Y on X2 after X1, and show that this slope is exactly the same as the slope for the simple regression of Y on X2 ignoring X1. Also find the intercept for the added-variable plot.