L2-Relaxation: With Applications to Forecast Combination and Portfolio Analysis

TL;DR

This paper introduces L2-relaxation, a convex optimization approach for forecast combination and portfolio selection that effectively handles large-scale problems with latent group structures, balancing bias and variance.

Contribution

The paper proposes a novel L2-relaxation method that minimizes the squared Euclidean norm of weights under relaxed constraints, offering a new solution for high-dimensional forecast and portfolio problems.

Findings

L2-relaxation achieves roughly equal within-group weights in latent group structures.

The method demonstrates optimality when the number of units grows faster than the time dimension.

Finite sample performance is excellent in Monte Carlo simulations and real data applications.

Abstract

This paper tackles forecast combination with many forecasts or minimum variance portfolio selection with many assets. A novel convex problem called L2-relaxation is proposed. In contrast to standard formulations, L2-relaxation minimizes the squared Euclidean norm of the weight vector subject to a set of relaxed linear inequality constraints. The magnitude of relaxation, controlled by a tuning parameter, balances the bias and variance. When the variance-covariance (VC) matrix of the individual forecast errors or financial assets exhibits latent group structures -- a block equicorrelation matrix plus a VC for idiosyncratic noises, the solution to L2-relaxation delivers roughly equal within-group weights. Optimality of the new method is established under the asymptotic framework when the number of the cross-sectional units $N$ potentially grows much faster than the time dimension $T$. Excellent finite sample performance of our method is demonstrated in Monte Carlo simulations. Its wide applicability is highlighted in three real data examples concerning empirical applications of microeconomics, macroeconomics, and finance.