3.7 P-P Plots

The evaluation of the predictive power across all possible confidence levels $ \alpha\in[0,1]$ can be carried out with the help of a transformation of the empirical distribution $ \{L^{(k)}\mid 0\le
k\le N-1\}$. If $ F$ is the true distribution function of the loss $ L$ within the holding period $ h$, then the random quantity $ F(L)$ is (approximately) uniformly distributed on $ [0,1]$. Therefore we check the values $ F_e\big[L(t)\big]$ for $ t_0\le t\le t_1$, where $ F_e$ is the empirical distribution. If the prediction quality of the model is adequate, these values should not differ significantly from a sample with size $ 250\,(t_1-t_0+1)$ from a uniform distribution on $ [0,1]$.

The P-P plot of the transformed distribution against the uniform distribution (which represents the distribution function of the transformed empirical distribution) should therefore be located as closely to the main diagonal as possible. The mean squared deviation from the uniform distribution (MSD) summed over all quantile levels can serve as an indicator of the predictive power of a quantile-based risk measure like VaR. The 8626 XFGpp.xpl quantlet creates a P-P plot and calculates the MSD indicator.