Forecasting Using Supervised Factors and Idiosyncratic Elements

We extend the Three-Pass Regression Filter (3PRF) in two key dimensions: (1) accommodating weak factors and, (2) allowing for a correlation between the target variable and the predictors, even after adjusting for common factors, driven by correlations in the idiosyncratic components of the covariates and the prediction target. Our theoretical contribution is to establish the consistency of 3PRF under these flexible assumptions, showing that relevant factors can be consistently estimated even when they are weak, albeit at slower rates. Stronger relevant factors improve 3PRF convergence to the infeasible best forecast, while weaker relevant factors dampen it. Conversely, stronger irrelevant factors hinder the rate of convergence, whereas weaker irrelevant factors enhance it. We compare 3PRF with Principal Component Regression (PCR), highlighting scenarios where 3PRF performs better. Methodologically, we extend 3PRF by integrating a LASSO step to develop the 3PRF LASSO estimator, which effectively captures the target's dependency on the predictors' idiosyncratic components. We derive the rate at which the average prediction error from this step converges to zero, accounting for generated regressor effects. Simulation results confirm that 3PRF performs well under these broad assumptions, with the LASSO step delivering a substantial gain. In an empirical application using the FRED-QD dataset, 3PRF LASSO delivers reliable forecasts of key macroeconomic variables across multiple horizons.