Abstract We propose ℓ2-relaxation, which is a novel convex optimization problem, to tackle a forecast combination with many forecasts or a 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.

This paper provides a novel mechanism for identifying and estimating latent group structures in panel data using penalized techniques. We consider both linear and nonlinear models where the regression coefficients are heterogeneous across groups but homogeneous within a group and the group membership is unknown. Two approaches are considered—penalized profile likelihood (PPL) estimation for the general nonlinear models without endogenous regressors, and penalized GMM (PGMM) estimation for linear models with endogeneity. In both cases, we develop a new variant of Lasso called classifier‐Lasso (C‐Lasso) that serves to shrink individual coefficients to the unknown group‐specific coefficients. C‐Lasso achieves simultaneous classification and consistent estimation in a single step and the classification exhibits the desirable property of uniform consistency. For PPL estimation, C‐Lasso also achieves the oracle property so that group‐specific parameter estimators are asymptotically equivalent to infeasible estimators that use individual group identity information. For PGMM estimation, the oracle property of C‐Lasso is preserved in some special cases. Simulations demonstrate good finite‐sample performance of the approach in both classification and estimation. Empirical applications to both linear and nonlinear models are presented.