  We investigate the performance and sampling variability of estimated forecast
combinations, with particular attention given to the combination of forecast
distributions. Unknown parameters in the forecast combination are optimized
according to criterion functions based on proper scoring rules, which are
chosen to reward the form of forecast accuracy that matters for the problem at
hand, and forecast performance is measured using the out-of-sample expectation
of said scoring rule. Our results provide novel insights into the behavior of
estimated forecast combinations. Firstly, we show that, asymptotically, the
sampling variability in the performance of standard forecast combinations is
determined solely by estimation of the constituent models, with estimation of
the combination weights contributing no sampling variability whatsoever, at
first order. Secondly, we show that, if computationally feasible, forecast
combinations produced in a single step -- in which the constituent model and
combination function parameters are estimated jointly -- have superior
predictive accuracy and lower sampling variability than standard forecast
combinations -- where constituent model and combination function parameters are
estimated in two steps. These theoretical insights are demonstrated
numerically, both in simulation settings and in an extensive empirical
illustration using a time series of S&P500 returns.
