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Part CCSIR NET December 2023central-vs-noncentral

Central vs noncentral

Consider Y=Xβ+εY = X\beta + \varepsilon with X a fixed n × (p + 1) matrix of rank p + 1 and ε\varepsilon ~ Nn(0,σ2I)N_{n}(0, \sigma^{2}I). If β\betâ is the OLS estimator, which of the following statements are true?

  1. A.(1/σ2)YTXβ(1/\sigma^{2}) Y^{T}X\betâ has a central χp+12\chi^{2}_{p+1} distribution
  2. B.(1/σ2)(Y−Xβ(1/\sigma^{2})(Y - X\betâ)T(Y−Xβ)^{T}(Y - X\betâ) has a central χn−p−12\chi^{2}_{n-p-1} distribution✓ Correct
  3. C.XβX\betâ and (Y−Xβ(Y - X\betâ)T(Y−Xβ)^{T}(Y - X\betâ) are independently distributed✓ Correct
  4. D.(1/σ2)∑(Yi−(1/\sigma^{2}) \sum(Y_{i} - Ȳ)2)^{2} has a central χn−12\chi^{2}_{n-1} distribution

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 invariants don't determine.

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50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

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

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