In simple regression with intercept and slope, errors are uncorrelated with mean zero and variance σ²>0. I:a 95% joint confidence region for (β₀,β₁) is bounded by an ellipse. II:Cov(β̂₀,β̂₁) does not involve σ². Which hold?
The official answer is **I only, A**, using the usual normal-theory joint confidence ellipse (or an asymptotic Wald ellipse) for a full-rank design. The covariance is −σ²x̄/S_xx and generally involves σ², so II is false. The printed assumptions specify only second moments, which by themselves do not guarantee the exact finite-sample F-based 95% coverage of that standard ellipse; normality is an additional implicit assumption in the intended construction.
More Regression Inference and Quadratic Forms questions
GATE 2024 ST Q65– In a regression with 25 observations,intercept and two predictors, the overall test uses…
GATE 2022 ST Q52– Fit Y=β₀+β₁X₁+β₂X₂+β₃X₃+ε with independent N(0,σ²) errors to 52 observations. Regression…
GATE 2021 ST Q65– Fit Y i=β₀+β₁x (1i)+⋯+β₂₂x (22i)+ε i,i=1,…,123,with iid N(0,σ²) errors. Regression sum of…
GATE 2026 ST Q48– In Y i=β₀+β₁x i1+β₂x i2+β₃x i3+ε i,i=1,…,31, the errors are iid N(0,1). The 5% overall…
GATE 2026 ST Q60– Y~N₃(0,I₃), A is the 3×3 matrix with every entry 1/3, and B=I₃−A. Which hold?
GATE 2025 ST Q46– For y i=α+βx i+ε i,i=1,…,24, iid N(0,σ²) errors, the centered sums are S xx=22.82,S…