Two return predictors are correlated at 0.95 and both carry real signal. Explain what lasso does with them, what ridge does, and what elastic net changes. State the elastic net penalty and which part fixes the problem.

Two return predictors are correlated at 0.95 and both carry real signal. Explain what lasso does with them, what ridge does, and what elastic net changes. State the elastic net penalty and which part fixes the problem.

Approach: Compare the behaviour of the L1 and L2 penalties on two nearly identical columns, then write the mixed penalty and identify the term that ties their coefficients together.

Lasso keeps one of the pair almost arbitrarily and zeroes the other, ridge splits the weight roughly evenly, and elastic net with penalty lambda*(alpha*||b||_1 + (1 - alpha)*||b||_2^2) keeps both with similar coefficients because the strictly convex L2 part creates the grouping effect. With two columns correlated at 0.95 the L1 objective is nearly flat along the line b_1 + b_2 = c, so tiny sampling noise decides which coefficient survives and the selection flips between bootstrap resamples. That instability matters on a desk because the research note then claims one factor works and the other does not, when the data supports no such statement. The L2 term is strictly convex along that flat direction with a unique minimum at equal weights, so it stabilises the solution and bounds the coefficient difference by a quantity proportional to sqrt(1 - correlation). Elastic net still zeroes genuinely useless features because the L1 corner is retained. The cost is two hyperparameters, and alpha must be selected on the same purged validation scheme as lambda or the extra configurations inflate the selection bias.

Follow-up: How would you compute the bound on |b_1 - b_2| in terms of the sample correlation and the L2 weight, and what does it become at correlation exactly one?

Key concepts: elastic net, grouping effect, correlated predictors, L2 penalty.