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Slope through the origin

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A hedge ratio $b$ is fitted by least squares for the model $y = b x$ , with no intercept, to the points $(1, 3)$ , $(2, 5)$ , $(3, 8)$ and $(4, 9)$ . What is $b$ ?

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Hint

Minimise the sum of squared errors directly; the intercept does not exist, so nothing is centred.

Solution

The sum of squared errors is $\sum (y - b x)^{2}$ , and setting its derivative in $b$ to zero gives $b = \sum x y / \sum x^{2}$ .
Here $\sum x y = 3 + 10 + 24 + 36 = 73$ .
And $\sum x^{2} = 1 + 4 + 9 + 16 = 30$ , so $b = \frac{73}{30}$ .

Answer: $\frac{73}{30}$

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