[math] \require{textmacros} \def \bbeta {\bf \beta} \def\fat#1{\mbox{\boldmath$#1$}} \def\reminder#1{\marginpar{\rule[0pt]{1mm}{11pt}}\textbf{#1}} \def\SSigma{\bf \Sigma} \def\ttheta{\bf \theta} \def\aalpha{\bf \alpha} \def\ddelta{\bf \delta} \def\eeta{\bf \eta} \def\llambda{\bf \lambda} \def\ggamma{\bf \gamma} \def\nnu{\bf \nu} \def\vvarepsilon{\bf \varepsilon} \def\mmu{\bf \mu} \def\nnu{\bf \nu} \def\ttau{\bf \tau} \def\SSigma{\bf \Sigma} \def\TTheta{\bf \Theta} \def\XXi{\bf \Xi} \def\PPi{\bf \Pi} \def\GGamma{\bf \Gamma} \def\DDelta{\bf \Delta} \def\ssigma{\bf \sigma} \def\UUpsilon{\bf \Upsilon} \def\PPsi{\bf \Psi} \def\PPhi{\bf \Phi} \def\LLambda{\bf \Lambda} \def\OOmega{\bf \Omega} [/math]

Consider the standard linear regression model [math]Y_i = \mathbf{X}_{i,\ast} \bbeta + \varepsilon_i[/math] for [math]i=1, \ldots, n[/math] and with the [math]\varepsilon_i[/math] i.i.d. normally distributed with zero mean and a common but unknown variance. Information on the response, design matrix and relevant summary statistics are:

[[math]] \begin{eqnarray*} \mathbf{X}^{\top} = \left( \begin{array}{rrr} 2 & 1 & -2 \end{array} \right), \, \mathbf{Y}^{\top} = \left( \begin{array}{rrr} -1 & -1 & 1 \end{array} \right), \, \mathbf{X}^{\top} \mathbf{X} = \left( \begin{array}{r} 9 \end{array} \right), \mbox{ and } \, \mathbf{X}^{\top} \mathbf{Y} = \left( \begin{array}{r} -5 \end{array} \right), \end{eqnarray*} [[/math]]

from which the sample size and dimension of the covariate space are immediate.

• Evaluate the ridge regression estimator [math]\hat{\bbeta}(\lambda)[/math] with [math]\lambda=1[/math].
• Evaluate the variance of the ridge regression estimator, i.e.[math]\widehat{\mbox{Var}}[\hat{\bbeta}(\lambda)][/math], for [math]\lambda = 1[/math]. In this the error variance [math]\sigma^2[/math] is estimated by [math]n^{-1} \| \mathbf{Y} - \mathbf{X} \hat{\bbeta}(\lambda) \|_2^2[/math].
• Recall that the ridge regression estimator [math]\hat{\bbeta}(\lambda)[/math] is normally distributed. Consider the interval
  [[math]] \begin{eqnarray*} \mathcal{C} & = & \big(\hat{\bbeta}(\lambda) - 2 \{ \widehat{\mbox{Var}}[\hat{\bbeta}(\lambda)] \}^{1/2}, \, \hat{\bbeta}(\lambda) + 2 \{ \widehat{\mbox{Var}}[\hat{\bbeta}(\lambda)] \}^{1/2} \big). \end{eqnarray*} [[/math]]
  Is this a genuine (approximate) [math]95\%[/math] confidence interval for [math]\bbeta[/math]? If so, motivate. If not, what is the interpretation of this interval?
• Suppose the design matrix is augmented with an extra column identical to the first one. Is the estimate of the error variance unaffected, or not? Motivate.