On the uncertainty of a combined forecast: The critical role of correlation

Abstract

The purpose of this paper is to show that the effect of the zero-correlation assumption in combining forecasts can be huge, and that ignoring (positive) correlation can lead to confidence bands around the forecast combination that are much too narrow. In the typical case where three or more forecasts are combined, the estimated variance increases without bound when correlation increases. Intuitively, this is because similar forecasts provide little information if we know that they are highly correlated. Although we concentrate on forecast combinations and confidence bands, our theory applies to any statistic where the observations are linearly combined. We apply our theoretical results to explain why forecasts by central banks (in our case, the Bank of Japan and the European Central Bank) are so frequently misleadingly precise. In most cases ignoring correlation is harmful, and an estimated historical correlation or an imposed fixed correlation larger than 0.7 is required to produce credible confidence bands.