We address the problem of selecting the common factors that are relevant for forecasting macroeconomic variables. In economic forecasting using diffusion indexes, the factors are ordered, according to their importance, in terms of relative variability, and are the same for each variable to predict, that is, the process of selecting the factors is not supervised by the predictand. We propose a simple and operational supervised method, based on selecting the factors on the basis of their significance in the regression of the predictand on the predictors. Given a potentially large number of predictors, we consider linear transformations obtained by principal components analysis. The orthogonality of the components implies that the standard t -statistics for the inclusion of a particular component are independent, and thus applying a selection procedure that takes into account the multiplicity of the hypotheses tests is both correct and computationally feasible. We focus on three main multiple testing procedures: Holm's sequential method, controlling the familywise error rate, the Benjamini–Hochberg method, controlling the false discovery rate, and a procedure for incorporating prior information on the ordering of the components, based on weighting the p-values according to the eigenvalues associated to the components. We compare the empirical performances of these methods with the classical diffusion index (DI) approach proposed by Stock and Watson, conducting a pseudo-real-time forecasting exercise, assessing the predictions of eight macroeconomic variables using factors extracted from an U.S. dataset consisting of 121 quarterly time series. The overall conclusion is that nature is tricky, but essentially benign: the information that is relevant for prediction is effectively condensed by the first few factors. However, variable selection, leading to exclude some of the low-order principal components, can lead to a sizable improvement in forecasting in specific cases. Only in one instance, real personal income, we were able to detect a significant contribution from high-order components.
