One of the problems with use of the condition number $K$ as an indicator of possible difficulty caused by collinearity is that it gives no indication of the cause or of possible solutions. If a major aim of the regression study is the estimation of slopes $\beta_{j}$ then a finer analysis can suggest such causes and solutions. Since
$$D[\hat{\boldsymbol{\beta}}]=\mathbf{M}^{-1} \sigma^{2}=\sigma^{2} \sum\left(1 / \dot{\lambda}{i}\right) \tilde{w}{i} \dot{w}{i}^{\prime}, \quad \operatorname{Var}\left(\hat{\beta}{j}\right)=\sigma^{2} \sum \overline{\mathbf{w}}{i j}^{2} / \dot{\lambda}{i}=\sigma^{2} c_{j j}$$
and the variance proportions
$$p_{i j}=w_{i j}^{2} /\left(\lambda_{i} c_{j j}\right)$$
are measures of the relative contributions of the $i$ th eigenpair to the variance of $\hat{\beta}{j}$, since $\sum{i} p_{i j}=1$. Study of these $p_{i j}$ provides some insight into the causes of large variance inflation.

$$\hat{\boldsymbol{\beta}}{\boldsymbol{r}}=\left(\mathbf{M}+r \mathbf{I}{\mathbf{k}}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y}$$
Note that $\hat{\boldsymbol{\beta}}{r}$ is a vector, not the rth component of $\hat{\beta}$. This should not cause confusion. We can write $\boldsymbol{\beta}$, in a form which will provide more insight. Since $\mathbf{X}^{\prime} \mathbf{Y}=\mathbf{M} \hat{\boldsymbol{\beta}}, \hat{\boldsymbol{\beta}}{r}=\mathbf{Z}{r} \hat{\boldsymbol{\beta}}$ for $\mathbf{Z}{r} \equiv\left(\mathbf{M}+r \mathbf{I}{k}\right)^{-1} \mathbf{M}$. But, writing $\mathbf{M}$ in its spectral form, we get $$\mathbf{Z}{r}=\left[\sum_{i}\left(\begin{array}{c} 1 \ j_{i}+r \end{array}\right) P_{i}\right]\left[\sum_{j} \lambda_{j} P_{j}\right]=\sum_{i}\left[\begin{array}{c} \lambda_{i} \ \lambda_{i}+r \end{array}\right] P_{i}$$
so that
$$\hat{\boldsymbol{\beta}}{r}=\left[\sum{i}\left[\frac{\lambda_{i}}{\dot{\lambda}{i}+r}\right] P{i}\right]\left[\beta+\sum F_{i} \tilde{w}{i} / \sqrt{\lambda{i}}\right]=\left[\sum_{i} \frac{\lambda_{i}}{\lambda_{i}+r} P_{i}\right] \beta+\sum_{i} \frac{\sqrt{\lambda_{i}}}{\dot{\lambda}{i}+r} F{i} \tilde{\mathbf{w}}_{i^{*}}$$

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