14.1 OPTIMAL FORECAST COMBINATIONS: THEORY

14.1 OPTIMAL FORECAST COMBINATIONS: THEORY

Results on optimal combinations of forecasts are similar to those discussed in chapter 3 for the construction of an optimal forecast, the key distinction being that the conditioning information is usually restricted to a set of forecasts of the outcome, so $z=\{f_1,\dots ,f_m\}$ if we have m forecasts to combine. The optimal forecast combination, $f^{c\ast }$ , is a function of the forecasts that solves

$$\underset{f^c}{\min }E\left[L(f^c(f_1,f_2,\dots ,f_m),y)\right].\text{\hspace{1em}(14.2)}$$

Importantly, optimality of the combined forecast as defined in (14.2) is conditional on observing the forecasts, $\{f_1,f_2,\dots ,f_m\}$ rather than the underlying information sets that were used to construct the forecasts. When $f(\cdot )$ is a linear index, the combination is linear with weights ${\omega }_1,\dots ,{\omega }_m$ . More generally, we can consider any function of the individual forecasts, although this is not often done in practice.

Additional restrictions are often placed on the search for combination schemes. A typical assumption is that the forecasts are unbiased, which reduces the space of linear combination models by constraining the weights to sum to 1. Moreover, because the underlying “data” are forecasts, they can be expected to obtain nonnegative weights that sum to unity, i.e., $0\le {\omega }_i\le 1$ , for $i=1,\dots ,m$ . Such considerations can be used to reduce the relevant parameter space for the combination weights in ways that rule out a variety of methods and may offer a more attractive risk function.

In general there is no need to constrain the information set to include only the set of observed forecasts $\{f_1,\dots ,f_m\}$ . This vector could be augmented to include other observed variables, x, to extend (14.2) as follows:

$$\underset{f^c}{\min }E\left[L(f^c(f_1,f_2,\dots ,f_m,x),y)\right].\text{\hspace{1em}(14.3)}$$

Treating the forecasts as data makes forecast combination equivalent to the forecast problem described throughout this book.

Similar to the results in chapter 4, one should realize that the solution is only “optimal” in the sense that (14.2) or (14.3) is minimized. However, when the function needs to be estimated on data, we must also consider the effect of estimation error. Methods based on optimal combinations with suboptimal estimation techniques need not return the best possible forecast combination or have any optimality properties. Different estimators yield different risk functions that depend on the underlying data-generating process and typically no single optimal method will have a risk function that uniformly dominates all other methods across all possible parameter values. Instead, different families of forecasting methods typically have different risk properties in different parts of the parameter space. This will be borne out in the next section in which we discuss estimation methods for constructing the forecast combinations.