\section{Robustness Checks}
\label{sec:robustness}

\subsection{Sensitivity to VAR Lag Selection}

We verify robustness to alternative VAR lag specifications. \Cref{tab:robustness_lags} reports the total spillover index and top-3 net transmitters for $p \in \{2, 4, 6\}$.

\begin{table}[htbp]
\centering
\caption{Robustness to VAR Lag Selection.}
\label{tab:robustness_lags}
\begin{threeparttable}
\begin{tabular}{@{}lrrr@{}}
\toprule
& $p=2$ & $p=4$ (baseline) & $p=6$ \\
\midrule
Total Spillover Index (\%) & 53.8 & 55.4 & 56.9 \\
ES net spillover (\%)      & $+$25.1 & $+$26.8 & $+$27.4 \\
NQ net spillover (\%)      & $+$18.3 & $+$19.7 & $+$20.5 \\
VIX net spillover (\%)     & $-$21.8 & $-$23.4 & $-$24.1 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Total spillover index and selected net directional spillovers under alternative VAR lag orders. Directional rankings remain qualitatively unchanged across all specifications.
\end{tablenotes}
\end{threeparttable}
\end{table}

\subsection{Sensitivity to Forecast Horizon}

\Cref{tab:robustness_horizon} reports sensitivity to the forecast horizon $H$.

\begin{table}[htbp]
\centering
\caption{Robustness to Forecast Horizon.}
\label{tab:robustness_horizon}
\begin{threeparttable}
\begin{tabular}{@{}lrrr@{}}
\toprule
& $H=5$ & $H=10$ (baseline) & $H=20$ \\
\midrule
Total Spillover Index (\%) & 49.2 & 55.4 & 58.1 \\
ES net spillover (\%)      & $+$24.0 & $+$26.8 & $+$28.3 \\
BTC own-shock (\%)         & 82.6 & 79.3 & 77.1 \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Cross-asset spillovers increase with the horizon, as expected, but qualitative conclusions regarding the volatility hierarchy are robust.
\end{tablenotes}
\end{threeparttable}
\end{table}

\newpage
\section{Summary of Key Numerical Results}
\label{sec:appendix_summary}

\begin{table}[htbp]
\centering
\caption{Summary of Key Findings.}
\label{tab:summary_findings}
\begin{tabular}{@{}p{7.5cm}r@{}}
\toprule
Metric & Value \\
\midrule
Total Spillover Index (full sample) & 55.44\% \\
Mean rolling TSI (200-day) & 67.13\% \\
Maximum rolling TSI & 81.55\% \\
Minimum rolling TSI & 57.22\% \\
\midrule
Largest net transmitter (full sample) & ES: $+$26.80\% \\
Second largest net transmitter & NQ: $+$19.72\% \\
Largest net receiver (full sample) & VIX: $-$23.37\% \\
\midrule
BTC own-shock contribution (full sample) & 79.33\% \\
ETH own-shock contribution (full sample) & 77.88\% \\
BTC--ETH pairwise spillover & 19.3\% / 20.6\% \\
\midrule
TSI pre-COVID regime & 64.72\% \\
TSI COVID crash regime & 66.66\% \\
TSI rate-hiking regime & 65.62\% \\
\midrule
BTC net spillover (pre-COVID) & $-$0.02\% \\
BTC net spillover (COVID crash) & $-$21.34\% \\
BTC net spillover (rate hikes) & $+$7.90\% \\
\midrule
ES $\to$ VIX Granger $F$-statistic & 19.864 \\
NQ $\to$ VIX Granger $F$-statistic & 16.708 \\
\midrule
Mean annualized RV: ES & 13.85\% \\
Mean annualized RV: BTC & 98.69\% \\
Mean annualized RV: CL & 29.67\% \\
Total 5-minute observations & $>$235 million \\
\bottomrule
\end{tabular}
\end{table}