R² Bounds for Predictive Models: What Univariate Properties Tell us About Multivariate Predictability

A long-standing puzzle in macroeconomic forecasting has been that a wide variety of multivariate models have struggled to out-predict univariate models consistently. We seek an explanation for this puzzle in terms of population properties. We derive bounds for the predictive R² of the true, but unknown, multivariate model from univariate ARMA parameters alone. These bounds can be quite tight, implying little forecasting gain even if we knew the true multivariate model. We illustrate using CPI inflation data. Supplementary materials for this article are available online.