GATE 2024 ST – Question 28

Regression Analysis · Linear Models and Estimation · 1 mark · Multiple choice

In y_i=β₀+β₁x_i+ε_i, the errors are uncorrelated with common mean zero and variance σ²>0. Let ȳ be the response mean and β̂₁ the least-squares slope. Which describes Cov(ȳ,β̂₁)?

Negative
Positive
Zero
Does not exist

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Show answer and explanation

Correct answer: (C) Zero

Explanation

For a nonconstant fixed design, β̂₁=Σ(x_i−x̄)y_i/S_xx. Cov(ȳ,β̂₁)=σ²Σ(x_i−x̄)/(nS_xx)=0. The centered predictor weights sum to zero. **C**.

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