Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 3 / b

Solution

Conditional on the covariates and all responses used to construct

g

, the residual

ε_{i}

has conditional second moment at most

C

; conditional independence under the null is what permits this conditioning. Consequently

E [\frac{1}{n} \sum_{i = 1}^{n} ε_{i}^{2} G_{i}^{2}] \leq C E [\frac{1}{n} \sum_{i = 1}^{n} G_{i}^{2}] ⟶ 0.

\frac{1}{n} \sum_{i} ε_{i}^{2} G_{i}^{2} p 0.

\frac{1}{n} \sum_{i} ξ_{i} ε_{i}^{2} G_{i} \leq (\frac{1}{n} \sum_{i} ε_{i}^{2} G_{i}^{2})^{1/2} (\frac{1}{n} \sum_{i} ε_{i}^{2} ξ_{i}^{2})^{1/2} .

The first factor converges to zero in probability. The weak law of large numbers and part a make the second

O_{p} (1)

, so the product converges to zero in probability.

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