Specification:

The empirical unconditional distribution of out-of-sample events may not be symmetric around zero - especially because of long term increases or decreases in one or more series k. We must however still be able to put losses with respect to the long and short positions in the US Dollar against series k and portfolio P on equal footing when assigning percentile levels to large movements on both extremes. To impose symmetry in the percentile assignments we first construct the event sets:  

$$X^{\pm_{(k,P)}}(\chi) = \left\{t\vert\pm x_t^{_{(k,P)}} \gt \chi,t\in[2,1001]\right\}.$$ (40)

Using sets  we can implicitly define exceedence levels $\chi^{\pm_{(k,P)}}(\varphi)$ 

$${{\char93 }(X^{\pm_{(k,P)}}(\chi^{\pm_{(k,P)}}(\varphi))-{1\... ...+ {1\over2}\left({\varphi\over100}\right), \quad\varphi \gt 50,$$ (41)

where the exceedence percentile is $\varphi$. The right hand side of expression  imposes a mapping so that the loss making data (relative to any portfolio and position thereof) will always cover a $50\%$ range of percentile values. (Cases of x_t^(k,P) = 0 may be ignored or assigned arbitrarily to either of sets $X^{\pm_{(k,P)}}(0)$, without substantially affecting the results.)

Next, using expressions  and we can directly construct

$$\ell^{\pm_{(k,P)}}(\varphi) = {\sum_{t\in X^{\pm_{(k,P)}}(\v... ...L}_{t-1}^{_{(k,P)}}. \over{\char93 }(X^{\pm_{(k,P)}}(\varphi))}$$ (42)

which is the mean log-likelihood of all loss exceedences over volatility percentile $\varphi$ for a long (+) or short (-) position in the US Dollar with respect to the series k or portfolio P.

Finally, as specified by expressions  and , we construct the univariate average mean log-likelihood

$$\ell_u(\varphi) = {1\over20}\left(\sum_{k=1}^{10}\ell^{+_{(k)}}(\varphi) + \sum_{k=1}^{10}\ell^{-_{(k)}}(\varphi)\right)$$ (43)

and the multivariate average mean log-likelihood

$$\ell_m(\varphi) = {1\over2}\left(\ell^{+_{(P)}}(\varphi) + \ell^{-_{(P)}}(\varphi)\right)$$ (44)

plotted in figures 5u and 5m respectively for all candidate models. The construction of expression  guarantees that what is true naturally for the $50\%$ confidence level (based on the forecast distribution) is also true for the $50\%$ percentile level (of the empirical out-of-sample distribution) - that this level marks the division of profitable and loss making events in the out-of-sample. As a consequence of this and symmetrization (consideration of both long and short, positions) the value of $\ell_u(\varphi = 50)$ is based on all 10000 out-of-sample events x_t^(k) $(t\in[2,1001])$ and the value $\ell_m(\varphi = 50)$ is based on all 1000 out-of-sample events x_t^(P) $(t\in[2,1001])$ - as was the case for $\ell_u(c = 50)$ and $\ell_m(c = 50)$ specified in the last section.