Multicollinearity: Ridge vs LASSO vs Elastic Net

Setup

• Two standardized predictors x1 and x2 with corr(x1, x2) = 0.99.
• X'X = [[100, 99], [99, 100]] and X'y = [120, 120].

Tasks

1. Compute the ridge (L2) estimate w_ridge = (X'X + λI)^{-1} X'y for λ = 10. Report numeric weights to 2 decimal places and explain why L2 prefers sharing weight under high collinearity.
2. Without solving the full LASSO, argue which coefficient pattern the L1 solution is likely to produce for λ1 = 10: both similar and small, one near zero and the other large, or something else? Justify using the geometry of L1 vs L2 constraint sets under high collinearity.
3. Propose elastic-net penalties (α and λ) that would stabilize selection while controlling variance. Explain how you would tune α and λ and which validation metric you would pick if the goal is sparse interpretation with minimal loss in PR-AUC.