GATE 2020 ST – Question 34
For Yi = α + βxi + εi, i = 1,…,7, the xi are fixed and the errors are independent with mean zero and a common finite variance. OLS estimates are α̂ and β̂.
Data summaries: Σxi = 0; Σxi² = 28; Σxiyi = 28; Σyi = 21; Σyi² = 91.
Find Corr(α̂,β̂).
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Correct answer: 0 to 0
Explanation
For an intercept-and-slope fit with equal independent error variances, Cov(α̂,β̂)=−σ²x̄/Sxx. Here x̄=0, so covariance is zero. Both estimates have positive variance under the usual nondegenerate error model; their correlation is therefore0. The response summaries are not needed for this design-based covariance.