Part BCSIR NET December 2024projection-shrinks-variance-and-creates-covariance

Projection shrinks variance and creates covariance

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

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

Projection shrinks variance and creates covariance

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