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

Projection shrinks variance and creates covariance

Consider a multiple linear regression model Yi=β0+β1xi1+⋯+βpxip+εi,1≤i≤n,n>(p+1)Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1), where errors εi′s\varepsilon_{i}'s are uncorrelated with zero mean and finite variance σ2>0\sigma^{2} > 0. Here, YiY_{i} is the i-th response. Let Ŷi_{i} be the i-th predicted response by the least squares estimation method, and let ε\varepsilon̂i=Yi−_{i} = Y_{i} - Ŷi,1≤i≤n_{i}, 1 \le i \le n. Then, which of the following statements is true?

  1. A.Var(Ŷi)≤Var(Yi),1≤i≤n_{i}) \le Var(Y_{i}), 1 \le i \le n✓ Correct
  2. B.Cov(Ŷi_{i}, Ŷk)=Cov(Yi,Yk),1≤i<k≤n_{k}) = Cov(Y_{i}, Y_{k}), 1 \le i < k \le n
  3. C.Var(εVar(\varepsilon̂i)=Var(εi),1≤i≤n_{i}) = Var(\varepsilon_{i}), 1 \le i \le n
  4. D.E(εE(\varepsilon̂i)<E(εi),1≤i≤n_{i}) < E(\varepsilon_{i}), 1 \le i \le 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, and why this one tests dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: OLS is the BLUE in every linear model

More on this topic

The chapter behind this: Linear models, Gauss–Markov and ANOVA — free to read

From Linear Models and Multivariate › Gauss–Markov, regression, ANOVA basics

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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