Part BCSIR NET December 2024fitting-shrinks-residuals-and-correlates-them

Fitting shrinks residuals and correlates them

Consider a multiple linear regression model $Y_{i} = β_{0} + β_{1} x_{i 1} + \dots + β_{p} x_{i} p + ε_{i}, 1 \leq i \leq n, n > (p + 1)$ , where the errors $ε_{i}^{'} s$ are uncorrelated with zero mean and finite variance $σ^{2} > 0$ . Here, $Y_{i}$ is the i-th response. Let Ŷ $_{i}$ be the i-th predicted response by the least squares estimation method, and let $ε$ ̂ $_{i} = Y_{i} -$ Ŷ $_{i}, 1 \leq i \leq n$ . Then, which of the following statements is true?

A. $V a r (ε$ ̂ $_{i}) \leq V a r (ε_{i}), 1 \leq i \leq n$ ✓
B. $C o v (ε$ ̂ $_{i}, ε$ ̂ $_{k}) = C o v (ε_{i}, ε_{k})$ , for all i ≠ k = 1, 2, …, n
C.Var(Ŷ $_{i}) = V a r (Y_{i}), 1 \leq i \leq n$
D.E(Ŷ $_{i}) < E (Y_{i}), 1 \leq i \leq n$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: OLS is the BLUE in every linear model

Fitting shrinks residuals and correlates them

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