14.1.1 Optimal Combinations under MSE Loss

14.1.1 Optimal Combinations under MSE Loss

The majority of results in the forecast combination literature assume MSE loss. To establish intuition for the gains from forecast combination under MSE loss, consider two individual forecasts, $f_1,f_2$ , with associated forecast errors $e_1=y-f_1$ , $e_2=y-f_2$ . Assuming that both forecasts are unbiased, we have $E\left[e_i\right]=0.$ Denote the variances of the forecast errors by ${\sigma }_i^2,i=1,2$ and their covariance by ${\sigma }_{12}$

Since the individual forecasts are unbiased, the combined forecasts will also be unbiased if the combination weights add up to 1. Consider, therefore, a combination of the two forecasts that uses weights $(\omega ,1-\omega )$ :

$$f^c=\omega f_1+\left(1-\omega \right)f_2.\text{\hspace{1em}(14.4)}$$

The associated forecast error is a weighted average of the individual forecast errors:

$$e(\omega )=y-\omega f_1-\left(1-\omega \right)f_2=\omega e_1+(1-\omega )e_2.\text{\hspace{1em}(14.5)}$$

Using that $E\left[e(\omega )\right]=0$ , the MSE loss becomes

$$\mathrm{MSE}(e(\omega ))={\omega }^2{\sigma }_1^2+(1-\omega {)}^2{\sigma }_2^2+2\omega (1-\omega ){\sigma }_{12}.\text{\hspace{1em}(14.6)}$$

The MSE loss in (14.6) is thus a weighted average of the elements of the variance– covariance matrix. Solving for the optimal combination weights, we have

$${\omega }^{\ast }=\frac{{\sigma }_2^2-{\sigma }_{12}}{{\sigma }_1^2+{\sigma }_2^2-2{\sigma }_{12}}.\text{\hspace{1em}(14.7)}$$

It follows from (14.7) that if the forecast errors have equal variance, ${\sigma }_1^2={\sigma }_2^2$ , so neither forecast dominates the other, it is optimal to assign equal weights to the forecasts, regardless of their correlation.

Intuitively, greater weight is generally assigned to the more precise model, i.e., the one with the smallest ${\sigma }_i^2$ . If the forecasts errors are weakly correlated $({\sigma }_{12}\approx 0)$ , the best combination uses a weighted average that weights the individual forecasts in proportion with the inverse of their MSE values:

$${\omega }^{\ast }\approx \frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}=\frac{{\sigma }_1^{-2}}{{\sigma }_1^{-2}+{\sigma }_2^{-2}}.$$

Note that the combination weight can be negative in (14.7) if ${\sigma }_{12}>{\sigma }_1^2$ or ${\sigma }_{12}>{\sigma }_2^2$ Indeed, if forecasts are strongly correlated and variances are sufficiently different, neither weight will fall between 0 and 1, e.g., ${\omega }^{\ast }>1,1-{\omega }^{\ast }<0$ . A negative weight on a forecast does not mean that it has no value to the forecaster. It means that the forecast can be used to offset the prediction errors of other models.

Next, consider the optimal forecast combination weights for the more general case with m forecasts. Suppose the joint distribution for the vector of forecast errors $e={\iota }_{m}y-f$ (where ${\iota }_m$ is an $(m\times 1)$ vector of 1s) has zero mean (unbiased forecasts) and variance covariance ${\Sigma }_e=E[e{e}^{\prime }]$ . Minimizing the MSE subject to the weights adding up to 1 amounts to solving

$${\mathbf{\omega }}^{\ast }=\arg \underset{\mathbf{\omega }}{\min }\left\{{\mathbf{\omega }}^{\prime }{\mathbf{\Sigma }}_e\mathbf{\omega }\right\},\mathrm{subject}\mathrm{to}{\mathbf{\omega }}^{\prime }{\mathbf{\iota }}_m=1.\text{\hspace{1em}(14.8)}$$

The resulting optimal combination weights are given by

$${\omega }^{\ast }=({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}{\Sigma }_e^{-1}{\iota }_m;\text{\hspace{1em}(14.9)}$$

see Bates and Granger (1969). The associated expected loss is ${\omega }^{\ast \prime }{\Sigma }_e{\omega }^{\ast }=$ $({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}$ . Elliott and Timmermann (2005) show that the optimality of these weights holds for much broader classes of loss functions, including asymmetric loss, provided that the errors are drawn from elliptically symmetric distributions.

Equal weights play a special role in the forecast combination literature. Equation (14.9) allows us to analyze when such weights are optimal in population. One interesting case arises when the individual forecast errors have identical variance, ${\sigma }^2$ , and identical pairwise correlations, $\rho .$ Then

$$\begin{array}{r}{\Sigma }_e^{-1}{\iota }_m=\frac{{\iota }_m}{{\sigma }^2(1+(m-1)\rho )},\\ ({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}=\frac{{\sigma }^2(1+(m-1)\rho )}{m},\end{array}$$

and so equal weights are optimal:

$${\omega }^{\ast }=\frac{1}{m}{\iota }_m.$$

Such a situation could hold to a close approximation when all forecasting models are based on similar data and hence produce forecasts with roughly the same accuracy. More generally, the optimal combination weights are equivalent to simple averaging over the forecasts when the unit vector lies in the eigenspace of ${\Sigma }_e$ . To see this, note that when ${\Sigma }_e{\iota }_m=\lambda {\iota }_m$ for scalar $\lambda$ , then ${\Sigma }_e^{-1}{\iota }_m={\lambda }^{-1}{\iota }_m$ , and

$$\begin{array}{l}{\omega }^{\ast }=({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}{\Sigma }_e^{-1}{\iota }_m\\ =({\lambda }^{-1}{\iota }_m^{\prime }{\iota }_m{)}^{-1}{\lambda }^{-1}{\iota }_m\\ =m^{-1}{\iota }_m.\end{array}$$

That the unit vector lies in the eigenspace of ${\Sigma }_e$ is both a sufficient and a necessary condition for equal weights to be optimal. To see this, note that for the optimal weights ${\omega }^{\ast }$ to be an eigenvector, we require that ${\Sigma }_e{\omega }^{\ast }=\lambda {\omega }^{\ast }$ for a scalar λ. However, ${\Sigma }_e{\omega }^{\ast }=({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}{\bar{\Sigma }}_e{\Sigma }_e^{-1}{\iota }_m=({\iota }_m^{\prime }{\hat{\Sigma }}_{\epsilon }^{-1}{\iota }_m{)}^{-1}{\iota }_m$ and $\lambda {\omega }^{\ast }=\lambda ({\iota }_m{\Sigma }_e^{-1}{\iota }_m{)}^{-1}{\Sigma }_e^{-1}{\iota }_m.$ Equating and dividing both sides by $\lambda ({\iota }_m{\Sigma }_e^{-1}{\iota }_m{)}^{-1}$ , we have ${\lambda }^{-1}{\iota }_m={\Sigma }_e^{-1}{\iota }_m,$ so for ${\omega }^{\ast }$ to be an eigenvector, ${\iota }_m$ must be an eigenvector. In the case with identical forecast error variances and identical pairwise correlations, the unit vector clearly lies in the eigenspace of ${\Sigma }_e$ because the sums of the rows of ${\Sigma }_e$ are identical across each row. The space of variance covariances for which equal weighting is optimal is, however, much wider than this special case. Variances and covariances need not be the same for each row of ${\Sigma }_e;$ it is required only that the sums across rows must be equal.