Bayesian Factor-adjusted Sparse Regression.

Many sparse regression methods are based on the assumption that covariates are weakly correlated, which unfortunately do not hold in many economic and financial datasets. To address this challenge, we model the strongly-correlated covariates by a factor structure: strong correlations among covariates are explained by common factors and the remaining variations are interpreted as idiosyncratic components. We then propose a factor-adjusted sparse regression model with both common factors and idiosyncratic components as decorrelated covariates and develop a semi-Bayesian method. Parameter estimation rate-optimality and model selection consistency are established by non-asymptotic analyses. We show on simulated data that the semi-Bayesian method outperforms its Lasso analogue, manifests insensitivity to the overestimates of the number of common factors, pays a negligible price when covariates are not correlated, scales up well with increasing sample size, dimensionality and sparsity, and converges fast to the equilibrium of the posterior distribution. Numerical results on a real dataset of U.S. bond risk premia and macroeconomic indicators also lend strong supports to the proposed method.