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1\section{Robustness Checks}2\label{sec:robustness}34\subsection{Sensitivity to VAR Lag Selection}56We verify robustness to alternative VAR lag specifications. \Cref{tab:robustness_lags} reports the total spillover index and top-3 net transmitters for $p \in \{2, 4, 6\}$.78\begin{table}[htbp]9\centering10\caption{Robustness to VAR Lag Selection.}11\label{tab:robustness_lags}12\begin{threeparttable}13\begin{tabular}{@{}lrrr@{}}14\toprule15& $p=2$ & $p=4$ (baseline) & $p=6$ \\16\midrule17Total Spillover Index (\%) & 53.8 & 55.4 & 56.9 \\18ES net spillover (\%) & $+$25.1 & $+$26.8 & $+$27.4 \\19NQ net spillover (\%) & $+$18.3 & $+$19.7 & $+$20.5 \\20VIX net spillover (\%) & $-$21.8 & $-$23.4 & $-$24.1 \\21\bottomrule22\end{tabular}23\begin{tablenotes}[flushleft]24\footnotesize25\item \textit{Notes:} Total spillover index and selected net directional spillovers under alternative VAR lag orders. Directional rankings remain qualitatively unchanged across all specifications.26\end{tablenotes}27\end{threeparttable}28\end{table}2930\subsection{Sensitivity to Forecast Horizon}3132\Cref{tab:robustness_horizon} reports sensitivity to the forecast horizon $H$.3334\begin{table}[htbp]35\centering36\caption{Robustness to Forecast Horizon.}37\label{tab:robustness_horizon}38\begin{threeparttable}39\begin{tabular}{@{}lrrr@{}}40\toprule41& $H=5$ & $H=10$ (baseline) & $H=20$ \\42\midrule43Total Spillover Index (\%) & 49.2 & 55.4 & 58.1 \\44ES net spillover (\%) & $+$24.0 & $+$26.8 & $+$28.3 \\45BTC own-shock (\%) & 82.6 & 79.3 & 77.1 \\46\bottomrule47\end{tabular}48\begin{tablenotes}[flushleft]49\footnotesize50\item \textit{Notes:} Cross-asset spillovers increase with the horizon, as expected, but qualitative conclusions regarding the volatility hierarchy are robust.51\end{tablenotes}52\end{threeparttable}53\end{table}5455\newpage56\section{Summary of Key Numerical Results}57\label{sec:appendix_summary}5859\begin{table}[htbp]60\centering61\caption{Summary of Key Findings.}62\label{tab:summary_findings}63\begin{tabular}{@{}p{7.5cm}r@{}}64\toprule65Metric & Value \\66\midrule67Total Spillover Index (full sample) & 55.44\% \\68Mean rolling TSI (200-day) & 67.13\% \\69Maximum rolling TSI & 81.55\% \\70Minimum rolling TSI & 57.22\% \\71\midrule72Largest net transmitter (full sample) & ES: $+$26.80\% \\73Second largest net transmitter & NQ: $+$19.72\% \\74Largest net receiver (full sample) & VIX: $-$23.37\% \\75\midrule76BTC own-shock contribution (full sample) & 79.33\% \\77ETH own-shock contribution (full sample) & 77.88\% \\78BTC--ETH pairwise spillover & 19.3\% / 20.6\% \\79\midrule80TSI pre-COVID regime & 64.72\% \\81TSI COVID crash regime & 66.66\% \\82TSI rate-hiking regime & 65.62\% \\83\midrule84BTC net spillover (pre-COVID) & $-$0.02\% \\85BTC net spillover (COVID crash) & $-$21.34\% \\86BTC net spillover (rate hikes) & $+$7.90\% \\87\midrule88ES $\to$ VIX Granger $F$-statistic & 19.864 \\89NQ $\to$ VIX Granger $F$-statistic & 16.708 \\90\midrule91Mean annualized RV: ES & 13.85\% \\92Mean annualized RV: BTC & 98.69\% \\93Mean annualized RV: CL & 29.67\% \\94Total 5-minute observations & $>$235 million \\95\bottomrule96\end{tabular}97\end{table}98