Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 19H / b

Solution

For this model, the Gauss-Markov theorem states that

β

β

: if

β = \sum_{i = 1}^{n} a_{i} Y_{i}

is unbiased for every

β

, then

var (β) \geq var (β)

Indeed,

E β = β \sum_{i} a_{i} x_{i},

so unbiasedness is equivalent to

\sum_{i} a_{i} x_{i} = 1

. Write

a_{i} = \frac{x _{i}}{S _{xx}} + c_{i} .

The constraint becomes

\sum_{i} c_{i} x_{i} = 0

. Since the errors are independent with common variance

σ^{2}

var (β) = σ^{2} i \sum a_{i}^{2} = σ^{2} (\frac{1}{S _{xx}} + \frac{2}{S _{xx}} i \sum c_{i} x_{i} + i \sum c_{i}^{2}) = \frac{σ ^{2}}{S _{xx}} + σ^{2} i \sum c_{i}^{2} \geq \frac{σ ^{2}}{S _{xx}} = var (β) .

Equality holds exactly when every

c_{i} = 0

, which gives

a_{i} = x_{i} / S_{xx}

and

β = β

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