Part CCSIR NET December 2023central-vs-noncentral

Central vs noncentral

Consider $Y = X β + ε$ with X a fixed n × (p + 1) matrix of rank p + 1 and $ε$ ~ $N_{n} (0, σ^{2} I)$ . If $β$ ̂ is the OLS estimator, which of the following statements are true?

A. $(1/ σ^{2}) Y^{T} X β$ ̂ has a central $χ_{p + 1}^{2}$ distribution
B. $(1/ σ^{2}) (Y - X β$ ̂ $)^{T} (Y - X β$ ̂) has a central $χ_{n - p - 1}^{2}$ distribution
C. $X β$ ̂ and $(Y - X β$ ̂ $)^{T} (Y - X β$ ̂) are independently distributed
D. $(1/ σ^{2}) \sum (Y_{i} -$ Ȳ $)^{2}$ has a central $χ_{n - 1}^{2}$ distribution

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The trap it tests

Invariants don't determine

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

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Related counterexample: OLS is the BLUE in every linear model

Central vs noncentral

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