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Least Squares: Residuals & Orthogonality
Move a candidate line; only the OLS line makes the residuals orthogonal to the regressors and the SSR minimal.
Day 20: Projections, Orthogonality & the Geometry of Least Squarescandidate lineOLS linepositive residualnegative residual
SSR (candidate)
43.49
OLS minimum 24.20
R² (candidate)
0.629
Σ e
-12.53
0 at OLS
Σ x·e
-92.95
0 at OLS
Move the candidate line. Vertical segments are residuals. Only at the OLS line are the residuals orthogonal to the regressors: Σe = 0 and Σx·e = 0, and the SSR is minimal (right panel: SSR as a function of the slope, with the intercept re-optimised).