Probabilistic forecast aggregation with statistical depth

This paper considers aggregation methods for interval forecasts and forecasts of cumulative distribution functions (CDFs) when there are many forecasters, and past forecast accuracy may not be known. For aggregation, the median and trimmed means have been proposed as simple and robust alternatives to the mean, with some trimmed mean approaches enabling recalibration to widen or narrow the resulting interval or CDF forecast. For interval forecast aggregation, the median and trimming are applied to each bound separately. To try to use the available information better, we treat the bounds as a bivariate point with statistical depth used to order the points in terms of centrality. The deepest point can be viewed as the median interval forecast, and the depth of each point can be used as the basis for trimming. For CDF forecasts, the literature presents aggregation methods for which the median or trimmed mean are obtained for each point on the domain of the distribution. However, if one part of a CDF forecast is outlying, the appeal of using the rest of the CDF forecast is perhaps reduced. We use functional depth to provide a measure of centrality for each CDF forecast, and hence identify the deepest function, which can be viewed as the median forecast. We also use functional depth as the basis for trimming, and consider weighted depth to control the width of the resulting aggregated interval or CDF forecast. We provide empirical illustration using data from surveys of professional macroeconomic forecasters, and an application to growth-at-risk.