2.2.1.5 Piecewise Asymmetric Loss

2.2.1.5 Piecewise Asymmetric Loss

A general class of asymmetric loss functions can be constructed by letting the loss function shift at a discrete set of points, $\{{\bar{e}}_1,\dots ,{\bar{e}}_{n-1}\}$ :

$$L(e)=\{\begin{array}{cl}{L}_1(e)& \mathrm{if}e\le {\overline{e}}_1,\\ L_2(e)& \mathrm{if}{\overline{e}}_1<e\le {\overline{e}}_2,\\ & \\ ⋮& ⋮\\ L_n(e)& \mathrm{if}e>{\overline{e}}_{n-1}.\end{array}\text{\hspace{1em}(2.15)}$$

Here ${\bar{e}}_{i-1}<{\bar{e}}_i$ for $i=2,\dots ,n-1$ . It is common to set $n=2$ , choose ${\bar{e}}_1=0$ and assume that both pieces of the loss function satisfy the usual loss properties so that the loss is piecewise asymmetric around 0 and continuous (but not necessarily

Right-skewed linex loss with $a_2=1$

Left-skewed linex loss with $a_2=-1$

Figure 2.2: MSE loss versus Linex loss for different values of the Linex parameter, $a_2$

differentiable) at 0. Lin-lin loss in (2.13) is a special case of (2.15) as is the asymmetric quadratic loss function

$$L(e)=\{\begin{array}{ll}(1-\alpha )e^2& \mathrm{if}e\le 0,\\ \alpha e^2& \mathrm{if}e>0,\end{array}\text{\hspace{1em}(2.16)}$$

considered by Artis and Marcellino (2001), Newey and Powell (1987) and Weiss (1996).

A flexible class of loss functions proposed by Elliott, Komunjer, and Timmermann (2005) sets $n=2$ and ${\bar{e}}_1=0$ in (2.15), while $L_1(e)=(1-\alpha )|e{|}^p$ and $L_2(e)=\alpha |e{|}^p$ where $\mathbf{p}$ is a positive integer, and $\alpha \in (0,1)$ . This gives the EKT loss function,

$$L\left(e\right)\equiv \left[\alpha +(1-2\alpha )1\left(e<0\right)\right]|e{|}^p,\text{\hspace{1em}(2.17)}$$

where $1(e<0)$ is an indicator function that equals 1 if $e<0$ , otherwise equals 0. Letting α deviate from 0.5 produces asymmetric loss, with larger values of α indicating greater aversion to positive forecast errors. Imposing $p=1$ and $\alpha =0.5$ MAE loss is obtained. More generally, setting $p=1,(2.17)$ reduces to lin-lin loss since the loss is linear on both sides of 0, but with different slopes. Setting $p=2$ and $\alpha =0.5$ gives the MSE loss function which is therefore also nested as a special case, as is the asymmetric quadratic loss function (2.16) for $p=2,\alpha \in (0,1)$ . Hence, the EKT family of loss functions nests the loss functions in (2.11), (2.12), (2.13), and (2.16) as special cases and generalizes many of the commonly employed loss functions.

α = 0.75

Figure 2.3: MSE loss versus EKT loss with p=3 for different values of the asymmetry parameter, α.

Figure 2.3 plots the EKT loss function for p = 3, α = 0.25 (top) and $\alpha =0.75$ (bottom). Compared with MSE loss, substantial asymmetries can be generated by this loss function.

Empirically, the EKT loss function has been used to analyze forecasts of government budget deficits produced by the IMF and OECD (Elliott, Komunjer, and Timmermann, 2005), the Federal Reserve Board’s inflation forecasts (Capistrán, 2008), as well as output and inflation forecasts from the Survey of Professional Forecasters (Elliott, Komunjer, and Timmermann, 2008).