Section 3f

3.7 Introducing Further Explanatory Variables: 3.7.1 General Theory, 3.7.2 One Extra Variable

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Problem 3f.1

Prove that

YTRY−YTRGY=σ2γ^GT(Var⁡[γ^G])−1γ^G \mathbf{Y}^{T} \mathbf{R Y}-\mathbf{Y}^{T} \mathbf{R}_{G} \mathbf{Y} = \sigma ^{2} \widehat{\gamma }_{G}^{T}\left(\operatorname {Var}\left[\widehat{\mathbf{\gamma }}_{G}\right]\right)^{-1} \widehat{\gamma }_{G}
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Problem 3f.2

Prove that γ^G\widehat{\gamma }_{G} can be obtained by replacing Y\mathbf{Y} by Y−Zγ\mathbf{Y}-\mathbf{Z} \gamma in YTRY\mathbf{Y}^{T} \mathbf{R Y} and minimizing with respect to γ\gamma. Show further that the minimum value thus obtained is YTRGY\mathbf{Y}^{T} \mathbf{R}_{G} \mathbf{Y}.

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Problem 3f.3

If β^G=(β^G,j)\widehat{\mathbf{\beta }}_{G} = \left(\widehat{\beta }_{G, j}\right) and β^=(β^j)\widehat{\mathbf{\beta }} = \left(\widehat{\beta }_{j}\right), use Theorem 3.6 (iv) to prove that

Var⁡[β^G,j]≥Var⁡[β^j] \operatorname {Var}\left[\widehat{\beta }_{G, j}\right] \geq \operatorname {Var}\left[\widehat{\beta }_{j}\right]
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Problem 3f.4

Given that Y1,Y2,…,YnY_{1}, Y_{2}, \ldots , Y_{n} are independently distributed as N(θ,σ2)N\left(\theta , \sigma^{2}\right), find the least squares estimate of θ\theta.

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(a)

Use Theorem 3.6 to find the least squares estimates and the residual sum of squares for the augmented model

Yi=θ+γxi+εi(i=1,2,…,n) Y_{i} = \theta +\gamma x_{i}+\varepsilon _{i} \quad (i = 1,2, \ldots , n)

where the εi\varepsilon_{i} are independently distributed as N(0,σ2)N\left(0, \sigma^{2}\right).

(b)

Verify the formulae for the least square estimates of θ\theta and γ\gamma by differentiating the usual sum of squares.