Factor-Adjusted Ridge Prediction Using Large-Dimensional Mixed-Effects Models

Idiosyncratic information from a large number of predictors is often omitted in sparse forecasting. This leaves room for improvement if the population model is dense, that is, when the large-dimensional idiosyncratic components have a non-trivial aggregate predictive power but each one is negligible. We propose a factor-adjusted ridge (FaR) estimator with either a closed-form ridge factor or a ridge factor selected by a bias-corrected factor-adjusted k-fold cross-validation procedure to exploit systematically the idiosyncratic information using a large-dimensional mixed-effects model. Our methods allow for the presence of autoregressors, approximate factors, and many predictors, possibly more than the sample size in the estimation. Using spectral analysis for large dimensional random matrices, we show that our ridge forecasts are asymptotically optimal in terms of limiting predictive loss for true means among the class of rotation-equivariant Tikhonov estimators with general quadratic penalties. Our asymptotic theory allows non-parametric distributions for martingale difference errors and idiosyncratic random coefficients, and adapts to the cross-sectional and temporal dependence structures of the large-dimensional predictors. A simulation study shows the efficiency of our FaR estimators under dense models and their robustness against sparsity. We apply our methods to forecast the growth rate of US industrial production using the FRED-MD database. Our FaR forecasts consistently outperform the principal-components-based forecasts and the factor-adjusted LASSO forecasts