GATE 2021 ST – Question 18
In Y_i=β₀+β₁x_i+ε_i,i=1,…,n,n≥3,the errors are iid with mean zero and variance σ². Let β̂₀,β̂₁ be OLS estimators,x̄=Σx_i/n,S₁=Σ(x_i−x̄)²,S₂=Σy_i(x_i−x̄). For real c,find Var(β̂₀+c).
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Correct answer: (B) σ²(1/n+x̄²/S₁)
Explanation
OLS gives β̂₀=Ȳ−x̄β̂₁,with Var Ȳ=σ²/n,Var β̂₁=σ²/S₁ and zero covariance. Thus Var β̂₀=σ²(1/n+x̄²/S₁). Adding a fixed c does not alter variance. **B**,assuming S₁>0 as required to fit a unique slope.