We propose ℓ 2 -relaxation, which is a novel convex optimization problem, to tackle a forecast combination with many forecasts or a minimum variance portfolio with many assets. ℓ 2 -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.

Business decision makers are increasingly using predictive social media analytic tools in forecasting exercises but ignoring potential model uncertainty. Using data on the universe of Twitter messages, we calculate the sentiment regarding each film to understand whether these opinions affect box office opening and DVD retail sales. Our results contrasting eleven different econometric strategies including penalization methods indicate that accounting for model uncertainty can lead to large gains in forecast accuracy. While penalization methods do not outperform model averaging on forecast accuracy, evidence indicates they perform equivalently at the variable selection stage. Finally, incorporating social media data greatly improves forecast accuracy.