$ℓ2$-Relaxation: With Applications to Forecast Combination and Portfolio Analysis

We propose $ℓ2$-relaxation, which is a novel convex optimization problem, to tackle forecast combination with many forecasts or minimum variance portfolio with many assets. $ℓ2$-relaxation minimizes the squared Euclidean norm of the weight vector subject to a set of relaxed linear inequalities to balance the bias and variance. It delivers optimality with approximately equal within-group weights when latent block equicorrelation patterns dominate the high dimensional sample variance-covariance matrix of the individual forecast errors or the assets. Its wide applicability is highlighted in three real data examples in microeconomics, macroeconomics, and finance.