Regularization choices for modeling contribution per order (p=50)

Context: You are building a linear model for contribution per order (continuous outcome) with about p = 50 covariates that include:

• Highly correlated marketing dummy variables (e.g., overlapping campaigns, channels)
• Weather variables
• Daypart indicators

Assume predictors are standardized and that a binary treatment indicator D (e.g., exposed vs. not exposed to a marketing action) is of substantive interest for inference.

Tasks

1. Lasso vs. Ridge

• Explain the bias–variance trade‑offs of L1 (Lasso) and L2 (Ridge).
• Contrast their variable selection behavior under correlated groups of predictors.
• Discuss how each affects uncertainty quantification for treatment effects, including best practices to avoid bias in the estimated treatment coefficient.

2. Elastic Net and tuning for valid inference

• Describe when Elastic Net strictly dominates using Lasso or Ridge alone in this setting.
• Explain how you would tune α and λ via cross‑validation, and how to keep inference valid after model selection (e.g., post‑selection refitting, stability selection).

3. Interactions and heterogeneity

• Discuss how regularization interacts with collinearity when you include treatment×covariate interactions (D×X), and the risk of shrinking true heterogeneous effects to zero.