Using the Hald cement data (Example 10.1), find the eigenvector associated with the smallest eigenvalue of \(\mathbf{X}^{\prime} \mathbf{X}\). Interpret the elements of this vector. What can you say about the source of multicollinearity in these data?

Data From Example 10.1

Hald [1952] presents data concerning the heat evolved in calories per gram of cement (y) as a function of the amount of each of four ingredients in the mix: tri- calcium aluminate (x), tricalcium silicate (x2), tetracalcium alumino ferrite (x3), and dicalcium silicate (x4). The data are shown in Appendix Table B.21. These reflect quite serious problems with multicollinearity. The VIFs are: x1: 38.496 x2: 254.423 x3: 46.868 x4: 282.513 We will use these data to illustrate the all-possible-regressions approach to vari- able selection.