Section 12a

12.1 Introduction, 12.2 Why Select?

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Problem 12a.1

Show that the square of the total bias ∥μ−E[Xβ^]∥2\left\| \mathbf{\mu }-\mathbb {E}\left[\mathbf{X} \widehat{\mathbf{\beta }}\right]\right\|^{2} is equal to the first term in (12.8).

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Problem 12a.2

Consider the regression model

Yi=α+γ1xi+γ2zi+εi(i=1,…,n)(12.13) Y_{i} = \alpha +\gamma _{1} x_{i}+\gamma _{2} z_{i}+\varepsilon _{i} \quad (i = 1, \ldots , n) \tag {12.13}

where ∑ixi=∑izi=0\sum_{i} x_{i} = \sum_{i} z_{i} = 0 and ∑ixi2=∑izi2=1\sum_{i} x_{i}^{2} = \sum_{i} z_{i}^{2} = 1. Suppose that we want to predict the response YY corresponding to a vector (x,z)(x, z) using a predictor of the form

Y^=Yˉ+γ~1x \widehat{Y} = \bar{Y}+\tilde{\gamma }_{1} x

where γ~1\tilde{\gamma }_{1} is the least squares estimate obtained by fitting the model

Yi=α+γ1xi+εi(i=1,…,n) Y_{i} = \alpha +\gamma _{1} x_{i}+\varepsilon _{i} \quad (i = 1, \ldots , n)
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(a)

Assuming that the model (12.13) is correct, show that the expected model error E[ME]=E[(Y^−α−γ1x−γ2z)2]\mathbb {E}\left[\mathrm{ME}\right] = \mathbb {E}\left[\left(\widehat{Y}-\alpha -\gamma_{1} x-\gamma_{2} z\right)^{2}\right] of the predictor Y^\widehat{Y} is given by

E[ME]=σ2(n−1+x2)+γ22(rx−z)2, \mathbb {E}\left[\mathrm{ME}\right] = \sigma ^{2}\left(n^{-1}+x^{2}\right)+\gamma _{2}^{2}(r x-z)^{2},

where r=∑ixizir = \sum_{i} x_{i} z_{i}.

(b)

Under what circumstances will Y^\widehat{Y} be a better predictor than the predictor based on both xx and zz?