Forecasting VaR and ES by using deep quantile regression, GANs-based scenario generation, and heterogeneous market hypothesis

Financial Innovation

Table 12 Out-of-sample ES forecast evaluation by using statistical tests

	ES_UC	ES_C	MBA	MBR	N&Z	M&F
\(\tau = 0.01\)
FTSE100	0.9230	0.5030	0.8140	0.8270	1.0000	0.7749
N225	0.8390	0.9074	0.6860	0.7160	0.7252	0.7569
SPX500	0.9240	0.1343	0.4480	0.7270	0.5524	0.7431
DAX	0.5570	0.0589	0.3450	0.2650	0.7252	0.6284
\(\tau = 0.05\)
FTSE100	0.9980	0.9850	0.9920	0.9980	1.0000	0.7630
N225	0.9870	0.7430	0.9550	0.9770	0.0846	0.4970
SPX500	1.0000	0.9780	0.9780	0.9970	1.0000	0.8440
DAX	0.8970	0.2640	0.8370	0.8890	0.0846	0.7470

The bold result indicates that the null hypothesis is not rejected. The hypothesis test form of ES_UC is: \(H_{0} :P_{t}^{\left[ \alpha \right]} = F_{t}^{\left[ \alpha \right]} ,\forall t\); \(H_{1} :ES_{\alpha ,t}^{F} \ge ES_{\alpha ,t}^{{}}\), for all t and > for some t, \(VaR_{\alpha ,t}^{F} = VaR_{\alpha ,t}^{{}}\), for all t. The hypothesis test form of ES_C is: \(H_{0} :P_{t}^{\left[ \alpha \right]} = F_{t}^{\left[ \alpha \right]} ,\forall t\); \(H_{1} :ES_{\alpha ,t}^{F} \ge ES_{\alpha ,t}^{{}}\), for all t and > for some t, \(VaR_{\alpha ,t}^{F} \ge VaR_{\alpha ,t}^{{}}\), for all t.