91Ó°ÊÓ

Added-variable plots This problem uses the United Nations example in Section 3.1 to demonstrate many of the properties of added-variable plots. This problem is based on the mean function $$\mathrm{E}\left(\log (\text {Fertility}) | \log (P P g d p)=x_{1}, \text {Purban}=x_{2}\right)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$$ There is nothing special about the two-predictor regression mean function, but we are using this case for simplicity. A. Show that the estimated coefficient for \(\log (P P g d p)\) is the same as the estimated slope in the added-variable plot for \(\log (P P g d p)\) after \(P\)urban. This correctly suggests that all the estimates in a multiple linear regression model are adjusted for all the other terms in the mean function. Also, show that the residuals in the added-variable plot are identical to the residuals from the mean function with both predictors. B. Show that the \(t\) -test for the coefficient for \(\log (P P g d p)\) is not quite the same from the added-variable plot and from the regression with both terms, and explain why they are slightly different.

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 \(X_{1}\) and \(X_{2}\) have sample correlation equal to zero. This means that, if \(x_{i j}, i=1, \ldots, n\) and \(j=1,2\) are the observed values of these two terms for the \(n\) cases in the data, \(\sum_{i=1}^{n}\left(x_{i 1}-\bar{x}_{1}\right)\left(x_{i 2}-\bar{x}_{2}\right)=0\) A. Give the formula for the slope of the regression for \(Y\) on \(X_{1}\), and for \(Y\) on \(X_{2}\). Give the value of the slope of the regression for \(X_{2}\) on \(X_{1}\). B. Give formulas for the residuals for the regressions of \(Y\) on \(X_{1}\) and for \(X_{2}\) on \(X_{1}\). The plot of these two sets of residuals corresponds to the added-variable plot in Figure \(3.1 \mathrm{d}\) C. Compute the slope of the regression corresponding to the added-variable plot for the regression of \(Y\) on \(X_{2}\) after \(X_{1}\), and show that this slope is exactly the same as the slope for the simple regression of \(Y\) on \(X_{2}\) ignoring \(X_{1}\). Also find the intercept for the added-variable plot.