Suppose that all the parameters in a linear model are orthogonal (Section 2.2.4).

a. When the model contains an intercept term, show that orthogonality implies that each column in X after the first (for the intercept) has mean 0; i.e., each explanatory variable is centered. Thus, based on the previous exercise, explain why each pair of explanatory variables is uncorrelated.

b. When the explanatory variables for the model are all centered, explain why the intercept estimate does not change as the variables are added to the linear predictor. Show that that estimate equals ȳ in each case.