spb/wp46_uqo

1\section{Tail risk and multivariate extensions}\label{sec:tail}23\subsection{Value-at-risk and expected shortfall}4\input{sections/results_tail_body}56\subsection{Covariance forecasts (Tier C)}7Table~\ref{tab:tierc} scores one-step covariance forecasts for two ETF baskets---an equity basket (SPY, QQQ, IWM, DIA) and a8cross-asset basket (SPY, TLT, GLD, USO, EEM)---against the realized covariance built from five-minute returns plus the9overnight outer product, over the test period (2{,}722 origins per basket). Every forecast of every model is symmetric10positive definite (the smallest eigenvalue over all origins is strictly positive, Appendix~\ref{app:extra}), so the matrix11losses are well defined. With the multivariate QLIKE of \citet{laurent2013loss} and the CCC model as the reference, dynamic12correlations matter where correlations move: on the equity basket DCC and asymmetric DCC reduce the loss by13\TierCEquityEtfDccOneOneRatio{} (ratio to CCC; Diebold--Mariano $p<0.001$) and scalar BEKK by14\TierCEquityEtfScalarBekkOneOneRatio{}, while the EWMA covariance is within sampling error of CCC and the 252-day rolling15covariance is 14\% worse; on the cross-asset basket, whose pairwise correlations are lower and less persistent, DCC and16asymmetric DCC improve on CCC by only 3\% (still $p<0.001$ given 2{,}722 origins), BEKK by 2\%, and the rolling covariance is1715\% worse. The asymmetric extension adds nothing over DCC on either basket (the asymmetric parameter $g$ is small and the18two loss series are indistinguishable), and the 90\% model confidence set keeps all six models on the equity basket but drops19the rolling covariance and CCC on the cross-asset basket. The Frobenius loss and the QLIKE of the equally weighted20portfolio variance rank the models in the same order. Multivariate models are not competitors of the univariate ones: they21answer a different question, and their cost---one DCC re-estimation takes about half a second for four assets, asymmetric DCC22two seconds---is modest.2324\begin{table}[tp]\centering\footnotesize25\caption{Tier C: one-step covariance forecasts, test period 2016--2026. Ratios are relative to CCC-GARCH(1,1); stars mark26Diebold--Mariano significance versus CCC (*** 1\%, ** 5\%, * 10\%); MCS $p$ from the 90\% model confidence set over the six27models (multivariate QLIKE); refit time is the median wall time of one re-estimation.}\label{tab:tierc}28\adjustbox{max width=\textwidth}{\input{tables/tab_tier_c}}29\end{table}30