spb/wp46_uqo

1One-day 1\% and 5\% VaR and expected-shortfall forecasts are backtested on every instrument for every specification2(Table~\ref{tab:tailrisk}, Figure~\ref{fig:var}). Here the innovation law is the whole story, and H2 is supported in both3halves. With Gaussian innovations, GARCH(1,1) breaches its 1\% VaR on 1.84\% of days at the median instrument and the4unconditional-coverage test rejects it on 90\% of the instruments; the EWMA filter is worse (2.18\%, 95\% rejected). The5Student-$t$ law brings the median breach rate to 1.30\% and the rejection share to 36\%; the skewed Student-$t$ to 1.21\%6and 10\%, the best calibration of any parametric specification at the 1\% level, with the lowest FZ0 joint loss among the7daily models. The same ordering holds inside every architecture---the asymmetric, fractional and component variants with8Student-$t$ innovations all sit between 1.25\% and 1.40\% with rejection shares of 30--50\%---so that the choice of9architecture moves the 1\% calibration far less than the choice of law. At the 5\% level the ranking flips, as it must10for a symmetric $t$ fitted to skewed returns: the Gaussian GARCH(1,1) breaches 5.12\% of days and is rejected on 21\% of11instruments while the Student-$t$ version over-covers the tail (5.87\%, rejected on 52\%); only the skewed law is12acceptable at both levels (5.33\%, 14\%). Realized-measure models improve the joint VaR--ES score: HEAVY-$t$ and Realized13GARCH-$t$ have the lowest FZ0 at 1\% (1.29 and 1.30 against 1.37 for GARCH(1,1)-$t$) with rejection shares of 22--25\%.14The dynamic-quantile test, which also checks that breaches are unpredictable from past breaches and the VaR level, rejects15more often (40--70\% of instruments for most specifications): the surviving clustering of exceedances is a property of the16weekly re-estimation and the 1{,}000-day window rather than of any particular model. The distribution-free models (HAR,17historical variance) obtain their quantiles from the empirical standardized residuals of the window; they are well18calibrated unconditionally but fail the dynamic test on two thirds of the instruments.1920\begin{table}[tp]\centering\scriptsize21\caption{One-day VaR and ES backtests, test period, representative specifications and the four laws of GARCH(1,1).22hit \% = median across instruments of the realized exceedance rate; UC / CC / DQ rej.\ \% = share of instruments on which23the Kupiec, Christoffersen and Engle--Manganelli tests reject at 5\%; FZ0 ratio = mean across instruments of the FZ0 joint24VaR--ES loss relative to GARCH(1,1)-$t$ (origins with a degenerate ES are excluded and counted in Appendix~\ref{app:extra}).}\label{tab:tailrisk}25\adjustbox{max width=\textwidth}{\input{tables/tab_tailrisk}}26\end{table}2728\begin{figure}[tp]\centering29\includegraphics[width=\textwidth]{figures/fig_var.pdf}30\caption{Realized VaR exceedance rates across the 42 instruments by specification, at 1\% (left) and 5\% (right): marker =31median, bar = interquartile range; solid line = nominal level, grey band = $\pm20\%$ of the nominal level; the right-hand column32of each panel is the share of instruments on which the Kupiec unconditional-coverage test rejects at 5\%.}\label{fig:var}33\end{figure}34