Section 9d

9.6 Effect of Random Explanatory Variables (9.6.1-9.6.6), 9.7 Collinearity: 9.7.1 Effect on the Variances of the Estimated Coefficients, 9.7.2 Variance Inflation Factors, 9.7.3 Variances and Eigenvalues, 9.7.4 Perturbation Theory, 9.7.5 Collinearity and Prediction

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Problem 9d.1

Prove that in a regression with two explanatory variables $x$ and $z$ ,

\operatorname {Var}\left[\widehat{\beta }_{0}\right] = \frac{\sigma ^{2}}{n}\left[1+\frac{1}{1-r^{2}}\left(\frac{1}{\mathrm{CV}_{x}^{2}}-\frac{2 r}{\mathrm{CV}_{x} \mathrm{CV}_{z}}+\frac{1}{\mathrm{CV}_{z}^{2}}\right)\right]

Problem 9d.2

Prove that $R_{j}^{\widehat{y}} = 0$ if and only in $\mathrm{x}^{*(j)}$ is ortogonal to the columns of $\mathbf{X}^{*(j)}$ .

Problem 9d.3

Show that $\mathbf{R}_{x x}$ is positive-definite provided that $\mathbf{X}$ has full rank and contains a column of 1's.

Problem 9d.4

Show that the eigenvalues $\lambda_{j}$ of $\mathbf{R}_{x x}$ satisfy

1 \leq \lambda _{j} \leq p-1

Hence prove that $\mathrm{VIF}_{j} \leq \kappa^{2}$ , where $\kappa$ is the condition number of $\mathbf{R}_{x x}$ .

Problem 9d.5

Prove that

\frac{\partial }{\partial x_{i}} \sum _{i = 1}^{n}\left(x_{i}-\bar{x}\right)\left(Y_{i}-\bar{Y}\right) = Y_{i}-\bar{Y}

and

\frac{\partial }{\partial x_{i}} \sum _{i = 1}^{n}\left(x_{i}-\bar{x}\right)^{2} = 2\left(x_{i}-\bar{x}\right) .