Consider the simple linear regression model:
Y = β₀ + βâ‚X + εᵢ, i = 1, 2, ..., n
where εᵢ are iid with mean 0 and variance σ². The least squares estimator, β̂₀ and β̂â‚, is obtained by solving the following normal equations simultaneously for β₀ and βâ‚:
ΣY = nβ₀ + ΣXᵢ
ΣXáµ¢Y = ΣXᵢβ₀ + ΣX²ᵢβâ‚, i = 1, 2, ..., n
(a) Show that:
ΣXᵢ - nX̄ = ΣY - nȲ
ΣXᵢ² - nX̄² = ΣX²ᵢ - nX̄²
X̄ = ΣXᵢ / n
Ȳ = ΣY / n
(b) Let YÌ‚ = β̂₀ + β̂â‚X and ȳ = Y - YÌ„. Show that:
Σȳ = 0
ΣXᵢȳ = 0
(c) Show that:
ΣXᵢεᵢ = 0
(d) Show that:
ΣYεᵢ = 0
(e) Show that the regression line YÌ‚ = β̂₀ + β̂â‚X always goes through the point (XÌ„, YÌ„).
