Lasso Regression Sparsity and KKT Optimality Conditions

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Convex Optimization / Regularization.

Your Task
The Lasso estimator minimizes: $J(\beta) = \frac{1}{2n} \|y - X\beta\|_2^2 + \lambda \|\beta\|_1$ where $\|\beta\|_1 = \sum_{j=1}^p |\beta_j|$.
(1) State the subdifferential $\partial \|\beta_j\|$ and derive the Karush-Kuhn-Tucker (KKT) optimality conditions.
(2) In the orthogonal design case ($X^T X = n I_p$), prove that Lasso yields the soft-thresholding operator: $\hat{\beta}_j^{\text{lasso}} = \text{sign}(\hat{\beta}_j^{\text{OLS}}) \max(0, |\hat{\beta}_j^{\text{OLS}}| - \lambda)$

Constraints

• The design matrix $X \in \mathbb{R}^{n \times p}$ has full column rank $p < n$.
• Disturbance terms satisfy standard zero-mean conditional homoskedasticity unless stated otherwise.