14.1.2 Optimal Combinations under Linex Loss

14.1.2 Optimal Combinations under Linex Loss

In the same way that the shape of the loss function matters for the optimal forecast, it also matters to the forecast combination problem. Consider the case with Linex loss:

$$L(e)=\exp (ae)-ae-1,\text{\hspace{1em}(14.10)}$$

where a controls aversion against large positive $(a>0)$ or large negative $(a<0)$ forecast errors. Assume that the vector of forecast errors $e=y{\iota }_m-f$ is Gaussian with zero mean and covariance matrix ${\Sigma }_e$ so that the forecasts are unbiased. Consider the linear forecast combination scheme

$$f^c={\omega }_0+{\omega }^{\prime }f,\mathrm{subject}\mathrm{to}{\omega }^{\prime }{\iota }_m=1.$$

As we shall see, ω0 can be used to capture an optimal bias. Using that the error from the combination $e^c=y-f^c=y-{\omega }^{\prime }f-{\omega }_0={\omega }^{\prime }e-{\omega }_0$ , the expected loss becomes

$$E\left[L\left(e^c({\omega }_0,\omega )\right)\right]=\exp (-a{\omega }_0+\frac{a^2}{2}{\omega }^{\prime }{\Sigma }_e\omega )+a{\omega }_0-1.\text{\hspace{1em}(14.11)}$$

Taking the derivative with respect to ${\omega }_0$ and setting this to $0,$ we have

$${\omega }_0^{\ast }=\frac{a}{2}{\omega }^{\prime }{\Sigma }_e\omega .$$

In turn, inserting this into the expected loss in (14.11), we see that the weights, $\omega ,$ should be chosen to minimize ${\omega }^{\prime }{\Sigma }_e\omega ,$ subject to ${\omega }^{\prime }{\iota }_m=1$ . This is identical to the earlier constrained optimization problem under MSE loss in (14.8) and so the combination weights are given by (14.9). Although the optimal combination weights, ${\omega }^{\ast }$ , are unchanged from the case with MSE loss, the intercept ${\omega }_0^{\ast }$ accounts for the shape of the loss function and is the only term that depends on the Linex asymmetry parameter, ${}^{a.}$ The optimal combination has a bias, $\lambda ({\iota }_m^{\prime }{\Sigma }_e^{-1}{\iota }_m{)}^{-1}/2$ , that reflects the dispersion of the combined forecast error evaluated at the optimal combination weights.

In general, the combination weights need not be identical under MSE and asymmetric loss. For example, when the distribution of forecast errors is a mixture of two Gaussian distributions, the optimal combination is a solution to $m+1$ nonlinear equations in ${\omega }_0$ and $\omega .$ This setup with a mixture of distributions can be used to capture a situation where one particular forecast is correlated with the outcome only during times when other forecasts break down and so creates a role for this particular forecast as a hedge against model breakdown. See Timmermann (2006) for further discussion of this point.

14.2 ESTIMATION OF FORECAST COMBINATION WEIGHTS

The optimal forecast combination methods described in section 14.1 refer to the population models for the forecast combinations. In practice, the combination weights need to be estimated using past data, just as we do for regular forecasting problems. All the estimation issues treated earlier in the book are therefore directly relevant to the combination problem.

Once we use estimated parameters, solutions to problems such as (14.8) no longer have any optimality properties in a “risk” sense. For any forecast combination problem, there is typically no single optimal forecast method with estimated parameters. Risk functions for different estimation methods will typically depend on the datagenerating process in such a way that we prefer one method for some processes and different methods for other data-generating processes.

Treating the forecasts as data means that all issues related to how we estimate forecasting models from data apply and are relevant. Many approaches can therefore be used to estimate the combination weights such as M-estimation or method of moment approaches using the first-order conditions for minimization of loss, plug-in methods based on least squares parameter estimates, or shrinkage methods. Many of the results in the literature on forecast combination simply apply standard estimation methods to the forecast combination problem.

The “data” used in forecast combination are not the outcome of a random draw but can, rather, be regarded as unbiased, if not necessarily precise, forecasts of the outcome. This suggests imposing special restrictions on the combination schemes. Specifically, under MSE loss, linear combination schemes might restrict the combination weights to sum to 1 and be nonnegative, so that ${\omega }_i\in [0,1]$ . Simple combination schemes such as equal weighting satisfy these constraints and do not require estimation of any parameters. Equal weighting can thus be viewed as a reasonable prior when no data have been observed and plays a similar role to putting zero weights on data in forecasting models.