2.1.2 Common Properties of Loss Functions

2.1.2 Common Properties of Loss Functions

Reasonable loss functions are grounded in economic decision problems. Under the utility-maximizing approach, loss functions inherit well-known properties from the utility function. Rather than deriving loss functions from first principles, however, it is common practice to instead use loss functions with a “reasonable shape.” For the loss function to be “reasonable,” a set of minimal properties should hold. Other properties such as symmetry or homogeneity may suggest broad families of loss functions with certain desirable characteristics. We cover both types of properties below.

Trade-offs between different forecast errors when $f\ne y$ are quantified by the loss function. To capture the notion that bigger errors imply bigger losses, often it is imposed that the loss is nondecreasing as the forecast moves further away from the outcome. Mathematically, this means that $L\left(f_2,y\right)\ge L\left(f_1,y\right)$ for either $f_2>f_1>y$ or $f_2<f_1<y$ for all real $y.$ Nearly all loss functions used in practice have this feature.

For loss functions that depend only on the forecast error, $e=y-f$ , and thus take the form $L(f,y)=L(e)$ , Granger (1999) summarized these requirements:

$$L(0)=0(\mathrm{minimallossof}0);\text{\hspace{1em}(2.3a)}$$

$L\left(e\right)\ge 0$ for all e;

(2.3b)

L (e) is nonincreasing in e for $e<0$ and nondecreasing in e for $e>0$

$$L\left(e_1\right)\le L\left(e_2\right)\mathrm{if}{e}_2<e_1<0,L\left(e_1\right)\le L\left(e_2\right)\mathrm{if}{e}_2>e_1>0.\text{\hspace{1em}(2.3c)}$$

As in the case with more general loss, $L(f,y)$ , condition (2.3a) simply normalizes the loss associated with the perfect forecast $(y=f)$ to be 0. The second condition states that imperfect forecasts $(y\ne f)$ generate larger loss than perfect ones. Most common loss functions depend only on e; see section 2.2 for examples.

Other properties of loss functions such as homogeneity, symmetry, differentiability, and boundedness can be used to define broad classes of loss functions. We next review these.

Homogeneity can be used to define classes of loss functions that lead to the same decisions. Homogeneous loss functions factor in such a way that

$$L(af,ay)=h(a)L(f,y),\text{\hspace{1em}(2.4)}$$

for some positive function $h(a)$ , where the degree of homogeneity does not matter. For loss functions that depend only on the forecast error, homogeneity amounts to $L(ae)=h(a)L(e)$ for some positive function $h(a)$ . Homogeneity is a useful property when solving for optimal forecasts since the optimal forecast will be invariant to different values of $h(a)$

Symmetry of the loss function refers to symmetry of the forecast around y. It is the property that, for all $f_{:}$ ,

$$L(y-f,y)=L(y+f,y).\text{\hspace{1em}(2.5)}$$

For loss functions that depend only on the forecast error, symmetry reduces to $L(-e)=L(e)$ , so that over- and underpredictions of the same magnitude lead to identical loss.

Most empirical work in economic forecasting assumes symmetric loss. This choice reflects the difficulties in putting numbers on the relative cost of over- and underpredictions. Construction of a loss function requires a deeper understanding of the forecaster’s objectives and this may be difficult to accomplish. Still, the implicit choice of MSE loss by the majority of studies in the forecasting literature seems difficult to justify on economic grounds. As noted by Granger and Newbold (1986, page 125), “an assumption of symmetry about the conditional mean. . . is likely to be an easy one to accept. . . an assumption of symmetry for the cost function is much less acceptable.”

Differentiability of the loss function with respect to the forecast is again a regularity condition that is useful and helps simplify numerically the search for optimal forecasts. However, this condition may not be desirable and is certainly not required for a loss function to be well defined. In general, a finite numbers of points where the loss function fails to be differentiable will not cause undue problems at the estimation stage. However, when the loss function is extremely irregular, different methods are required for understanding the statistical properties of the loss function (see the maximum utility estimator in chapter 12).

Finally, loss functions may be bounded or unbounded. As a practical matter, there is often no obvious reason to let the weight the loss function places on very large forecast errors increase without bound. For example, the squared error loss function examined below assigns very different losses to forecasts of, say, US inflation that result in errors of 100% versus 500% even though it is not obvious that the associated losses should really be very different since both forecasts would lead to very similar actions. Unbounded loss functions can create technical problems for the analysis of forecasts as the expected loss may not exist, so most results in decision theory are derived under the assumption of bounded loss. In practice, forecasts are usually bounded and extremely large forecasts typically get trimmed as they are deemed implausible.