پازل ترکیب پیش بینی: توضیح نظری ساده  The forecast combination puzzle: A simple theoretical explanation

1. Introduction When several forecasts of the same event are available, it is natural to try and find a (linear) combination of these forecasts that is the ‘best’ in some sense. If we define ‘best’ in terms of the mean squared error and the variances and covariances of the forecasts are known, then optimal weights can be derived. In practice, though, these (co)variances are not known and need to be estimated. This leads to estimated optimal weights and an estimated optimal forecast combination. Empirical evidence and extensive simulations show that the estimated optimal forecast combination typically does not perform well, and that thearithmetic mean often performs better. This empirical fact has become known as the ‘forecast combination puzzle’. The history of the puzzle is elegantly summarized by Graefe, Armstrong, Jones, and Cuzán (2014, Section 4), and Smith and Wallis (2009) made a rigorous attempt to explain it, using simulations and an empirical example. They showed that the effect of the error on the estimation of the weights can be large, thus providing an empirical explanation of the forecast puzzle. Smith and Wallis (2009) use the words ‘finite-sample’ error, which suggests that this error may vanish asymptotically. However, it is not so easy to find an asymptotic justification for ignoring the noise generated by estimating the weights. To begin with, it is not clear what ‘asymptotic’ means here. What goes to infinity? The number of forecasts? If so, then the number of weights also goes to infinity. The number of observations underlying the total (but finite) set of forecasts? That would make more sense, but it would be difficult to analyze. In this paper, we provide a theoretical explanation for the empirical and simulation results of Smith and Wallis (2009) and others. The key ingredient of our approach is the specific acknowledgement that the optimal weights should be derived by taking the estimation step into account explicitly. In other words, we view the derivation and estimation of optimal weights as a joint effort, not as two separate efforts. This approach differs from (almost) all previous research, not only the study by Bates and Granger (1969), but also later contributions, important and insightful though they may be, such as those of Elliott (2011), Hansen (2008), Hsiao and Wan (2014), and Liang, Zou, Wan, and Zhang (2011). The separation of the mathematical derivation and statistical estimation can be quite dangerous. However, even though the disadvantages of such separations have been highlighted, they are still quite common in econometrics, and specifically in the modelaveraging literature, which explicitly attempts to combine model selection and estimation, so that uncertainty in the model selection procedure is not ignored when reporting properties of the estimates; see for example Magnus and De Luca (2016).