A scoring rule for factor and autoregressive models under misspecification

Factor models (FM) are now widely used for forecasting with large set of time series. Another class of models, which can be easily estimated and used in a large dimensional setting, is multivariate autoregressive models (MAR), where independent autoregressive processes are assumed for the series in the panel. We compare the forecasting abilities of FM and MAR models when assuming both models are misspecified and the data generating process is a vector autoregressive model. We establish which conditions need to be satisfied for a FM to overperform MAR in terms of mean square forecasting error. The condition indicates in presence of misspecification that FM is not always overperforming MAR and that the FM predictive performance depends crucially on the parameter values of the data generating process. Building on the theoretical relationship between FM and MAR predictive performances, we provide a scoring rule which can be evaluated on the data to either select the model, or combine the models in forecasting exercises. Some numerical illustrations are provided both on simulated data and on wel-known large economic datasets. The empirical results show that the frequency of the true positive signals is larger when FM and MAR forecasting performances differ substantially and it decreases as the horizon increases.