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\section{Empirical Results}
\label{sec:results}
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\subsection{Full-Sample Connectedness Analysis}

\Cref{tab:connectedness_full} presents the full-sample Diebold--Yilmaz connectedness table based on a VAR(4) model with generalized variance decomposition at a 10-day forecast horizon. The sample encompasses the common period across all 10 assets (2016--2026).

\begin{table}[htbp]
\centering
\caption{Full-Sample Diebold--Yilmaz Connectedness Table.}
\label{tab:connectedness_full}
\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}l*{10}{r}r@{}}
\toprule
& \multicolumn{10}{c}{FROM (shock origin)} & \\
\cmidrule(lr){2-11}
TO & ES & NQ & EURUSD & USDJPY & GBPUSD & GC & CL & BTC & ETH & VIX & FROM \\
\midrule
ES      & 28.6 & 25.5 & 8.4 & 6.6 & 7.1 & 9.1 & 8.4 & 0.2 & 0.2 & 5.9 & 71.4 \\
NQ      & 26.7 & 27.6 & 8.7 & 6.5 & 7.4 & 9.0 & 8.1 & 0.2 & 0.2 & 5.5 & 72.4 \\
EURUSD  & 11.4 & 11.2 & 25.7 & 14.3 & 17.4 & 11.0 & 8.0 & 0.0 & 0.2 & 0.7 & 74.3 \\
USDJPY  & 8.9  & 8.7  & 15.7 & 38.6 & 11.6 & 9.1 & 5.6 & 0.1 & 0.3 & 1.5 & 61.4 \\
GBPUSD  & 11.1 & 10.6 & 18.7 & 11.7 & 29.6 & 9.8 & 7.7 & 0.1 & 0.2 & 0.5 & 70.4 \\
GC      & 11.8 & 11.4 & 12.7 & 9.5  & 9.6  & 34.5 & 8.0 & 0.2 & 0.2 & 2.1 & 65.5 \\
CL      & 12.9 & 11.2 & 7.9  & 5.0  & 7.1  & 9.3 & 45.3 & 0.0 & 0.1 & 1.2 & 54.7 \\
BTC     & 0.1  & 0.1  & 0.1  & 0.2  & 0.1  & 0.3 & 0.1 & 79.3 & 19.3 & 0.4 & 20.7 \\
ETH     & 0.2  & 0.3  & 0.2  & 0.3  & 0.1  & 0.2 & 0.2 & 20.6 & 77.9 & 0.1 & 22.1 \\
VIX     & 15.0 & 13.2 & 2.2  & 2.5  & 1.4  & 4.8 & 1.4 & 0.4 & 0.4 & 58.6 & 41.4 \\
\midrule
TO      & 98.2 & 92.2 & 74.7 & 56.4 & 61.8 & 62.6 & 47.6 & 21.9 & 21.1 & 18.0 & \textbf{55.4} \\
NET     & $+$26.8 & $+$19.7 & $+$0.4 & $-$5.0 & $-$8.7 & $-$2.9 & $-$7.2 & $+$1.2 & $-$1.0 & $-$23.4 & \\
\bottomrule
\end{tabular}
\end{adjustbox}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Generalized forecast error variance decomposition from a VAR(4) model at horizon $H=10$. Entry $(i,j)$ is the percentage of the 10-day-ahead forecast error variance of asset $i$ attributable to shocks in asset $j$. FROM = total spillover received. TO = total spillover transmitted. NET = TO $-$ FROM. The lower-right entry (55.4) is the Total Spillover Index.
\end{tablenotes}
\end{threeparttable}
\end{table}

The total spillover index is 55.44\%, indicating that more than half of all forecast error variance in the system is attributable to cross-asset shocks rather than own-asset dynamics. Several key findings emerge:

\textit{Equity futures dominate volatility transmission.} ES and NQ are the largest net volatility transmitters, with net directional spillovers of $+26.80\%$ and $+19.72\%$, respectively. Together, they transmit a combined 190.4\% of directional spillovers to other markets, reflecting the central role of U.S.\ equity markets in global price discovery.

\textit{VIX is a net receiver, not a transmitter.} Despite its role as the ``fear gauge,'' VIX is the largest net receiver of volatility shocks ($-23.37\%$). Its forecast error variance is explained primarily by ES (15.0\%) and NQ (13.2\%) shocks, consistent with VIX being derived from S\&P 500 options.

\textit{Crypto markets form an isolated cluster.} BTC and ETH exhibit remarkably high own-shock contributions (79.3\% and 77.9\%) and negligible cross-asset spillovers ($<0.5\%$). The pairwise BTC--ETH spillover is substantial ($\approx$20\%), but connections to traditional assets are minimal.

\textit{FX and commodity markets show moderate interconnectedness.} EURUSD receives significant spillovers from GBPUSD (17.4\%) and USDJPY (14.3\%), reflecting the common U.S.\ dollar factor. CL is the most self-dependent among traditional assets (45.3\% own contribution).

\Cref{fig:connectedness_heatmap} provides a visual representation of the connectedness matrix.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{fig2_connectedness_heatmap.png}
\caption{Diebold--Yilmaz Connectedness Heatmap (Full Sample).}
\label{fig:connectedness_heatmap}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Color intensity represents the magnitude of pairwise forecast error variance contributions. Three distinct clusters emerge: (i)~a tightly connected equity--FX--commodity block; (ii)~an isolated crypto cluster (BTC--ETH); and (iii)~VIX as a peripheral receiver connected primarily to equities.
\end{minipage}
\end{figure}

\subsection{Time-Varying Connectedness}

\Cref{fig:rolling_spillover} presents the rolling 200-day total spillover index over the 2016--2026 period.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig3_rolling_spillover.png}
\caption{Rolling 200-Day Total Spillover Index with Event Markers.}
\label{fig:rolling_spillover}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Total spillover index from a VAR(4) model with generalized variance decomposition at $H=10$, estimated over rolling 200-day windows. Dashed horizontal line: full-period mean (67.13\%). Vertical lines mark: COVID-19 crash (March 2020), Federal Reserve rate hikes (March 2022), crypto winter (June 2022), FTX collapse (November 2022).
\end{minipage}
\end{figure}

The rolling total spillover index exhibits substantial time variation, ranging from 57.22\% to 81.55\% (mean: 67.13\%). Several episodes are noteworthy:

\begin{enumerate}[label=(\roman*)]
    \item \textbf{February 2018 (TSI = 81.55\%):} The maximum coincides with the ``Volmageddon'' event, when short-volatility ETPs imploded, triggering cascading volatility across all asset classes.
    \item \textbf{COVID-19 crash (March 2020):} The TSI initially dips during the acute phase before rising sharply, reflecting initial de-correlation followed by re-coupling through coordinated monetary policy responses.
    \item \textbf{2022 rate-hiking cycle:} The TSI rises steadily to 72--75\%, reflecting the macro-driven nature of the volatility environment.
    \item \textbf{Post-FTX collapse (November 2022):} The TSI shows a temporary decline, consistent with the crypto winter being contained within the crypto ecosystem.
    \item \textbf{2024--2025 elevation:} The TSI trends upward, potentially reflecting growing crypto integration through spot Bitcoin ETF approvals and institutional participation.
\end{enumerate}

\subsection{Regime-Specific Connectedness}

We estimate separate connectedness tables for three distinct regimes: pre-COVID (January 2019--February 2020), COVID crash (March--June 2020), and 2022 rate-hiking cycle (January--December 2022).

\subsubsection{Pre-COVID Regime (January 2019--February 2020)}

\begin{table}[htbp]
\centering
\caption{Connectedness Table --- Pre-COVID Regime (2019-01 to 2020-02).}
\label{tab:connectedness_precovid}
\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}l*{10}{r}r@{}}
\toprule
& ES & NQ & EURUSD & USDJPY & GBPUSD & GC & CL & BTC & ETH & VIX & FROM \\
\midrule
ES      & 31.3 & 29.0 & 4.1 & 9.4 & 1.6 & 8.5 & 6.5 & 0.0 & 0.7 & 8.9 & 68.7 \\
NQ      & 30.5 & 29.2 & 4.2 & 9.8 & 1.7 & 8.5 & 6.8 & 0.1 & 0.7 & 8.6 & 70.8 \\
EURUSD  & 12.8 & 13.1 & 24.1 & 12.9 & 13.6 & 12.3 & 9.5 & 0.2 & 0.3 & 1.4 & 75.9 \\
USDJPY  & 21.6 & 20.8 & 7.1 & 20.5 & 3.2 & 11.9 & 7.5 & 0.1 & 0.3 & 7.2 & 79.5 \\
GBPUSD  & 8.6 & 8.4 & 18.0 & 7.8 & 37.5 & 8.8 & 7.8 & 0.4 & 0.8 & 2.0 & 62.5 \\
GC      & 15.9 & 16.2 & 9.1 & 14.7 & 4.6 & 25.0 & 9.4 & 0.7 & 0.2 & 4.3 & 75.0 \\
CL      & 17.1 & 17.9 & 7.4 & 12.9 & 5.1 & 11.0 & 25.7 & 0.2 & 0.2 & 2.5 & 74.3 \\
BTC     & 7.1 & 5.6 & 0.2 & 1.4 & 0.3 & 2.2 & 1.7 & 73.0 & 6.8 & 1.7 & 27.0 \\
ETH     & 6.7 & 6.0 & 0.6 & 1.4 & 0.4 & 1.7 & 2.4 & 24.6 & 55.2 & 1.0 & 44.8 \\
VIX     & 24.9 & 22.9 & 1.8 & 8.0 & 0.3 & 5.5 & 3.3 & 0.9 & 1.2 & 31.4 & 68.6 \\
\midrule
TO  & 145.2 & 139.7 & 52.6 & 78.1 & 30.6 & 70.4 & 54.8 & 27.0 & 11.1 & 37.7 & \textbf{64.7} \\
NET & $+$76.4 & $+$68.9 & $-$23.3 & $-$1.4 & $-$31.8 & $-$4.6 & $-$19.6 & $-$0.0 & $-$33.7 & $-$30.9 & \\
\bottomrule
\end{tabular}
\end{adjustbox}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Same methodology as \Cref{tab:connectedness_full}. TSI = 64.72\%.
\end{tablenotes}
\end{threeparttable}
\end{table}

During the pre-COVID regime, the volatility hierarchy is dominated overwhelmingly by U.S.\ equity futures. ES and NQ exhibit net spillovers of $+76.4\%$ and $+68.9\%$---far exceeding their full-sample values.

\subsubsection{COVID Crash Regime (March--June 2020)}

\begin{table}[htbp]
\centering
\caption{Connectedness Table --- COVID Crash Regime (2020-03 to 2020-06).}
\label{tab:connectedness_covid}
\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}l*{10}{r}r@{}}
\toprule
& ES & NQ & EURUSD & USDJPY & GBPUSD & GC & CL & BTC & ETH & VIX & FROM \\
\midrule
ES      & 20.1 & 20.6 & 10.9 & 16.2 & 9.6 & 7.2 & 5.6 & 4.4 & 2.4 & 3.1 & 79.9 \\
NQ      & 19.4 & 21.1 & 10.8 & 16.7 & 9.4 & 7.3 & 5.8 & 3.9 & 2.2 & 3.3 & 78.9 \\
EURUSD  & 8.5 & 9.7 & 21.8 & 19.9 & 17.8 & 10.2 & 7.6 & 1.0 & 1.5 & 2.1 & 78.2 \\
USDJPY  & 5.6 & 7.9 & 14.4 & 33.6 & 11.1 & 7.3 & 3.9 & 5.4 & 3.0 & 8.0 & 66.4 \\
GBPUSD  & 11.1 & 11.3 & 17.0 & 18.6 & 23.9 & 7.9 & 5.4 & 2.1 & 2.0 & 0.6 & 76.1 \\
GC      & 8.1 & 9.6 & 13.0 & 14.4 & 9.2 & 23.6 & 12.4 & 3.9 & 4.3 & 1.6 & 76.4 \\
CL      & 4.6 & 5.2 & 8.7 & 4.5 & 6.5 & 17.2 & 49.0 & 1.5 & 2.2 & 0.6 & 51.0 \\
BTC     & 2.6 & 2.8 & 5.9 & 16.2 & 2.2 & 6.3 & 6.5 & 34.8 & 16.0 & 6.9 & 65.2 \\
ETH     & 3.4 & 3.2 & 4.0 & 8.2 & 1.1 & 3.1 & 4.4 & 17.3 & 51.6 & 3.8 & 48.4 \\
VIX     & 5.5 & 7.2 & 7.6 & 12.0 & 5.6 & 0.7 & 0.7 & 4.3 & 2.5 & 54.0 & 46.0 \\
\midrule
TO  & 68.7 & 77.3 & 92.2 & 126.7 & 72.3 & 67.2 & 52.1 & 43.9 & 36.2 & 30.0 & \textbf{66.7} \\
NET & $-$11.2 & $-$1.6 & $+$14.0 & $+$60.3 & $-$3.7 & $-$9.2 & $+$1.1 & $-$21.3 & $-$12.3 & $-$16.1 & \\
\bottomrule
\end{tabular}
\end{adjustbox}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Same methodology as \Cref{tab:connectedness_full}. TSI = 66.66\%.
\end{tablenotes}
\end{threeparttable}
\end{table}

The COVID crash regime reveals a dramatic restructuring of the volatility hierarchy. USDJPY emerges as the dominant volatility transmitter ($+60.29\%$ net), reflecting massive safe-haven demand for the Japanese yen. ES flips from its usual transmitter role to a net receiver ($-11.2\%$), and BTC becomes a substantial net receiver ($-21.3\%$), contradicting the narrative of Bitcoin as a safe-haven asset.

\subsubsection{Rate-Hiking Regime (January--December 2022)}

\begin{table}[htbp]
\centering
\caption{Connectedness Table --- Rate-Hiking Regime (2022-01 to 2022-12).}
\label{tab:connectedness_ratehikes}
\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}l*{10}{r}r@{}}
\toprule
& ES & NQ & EURUSD & USDJPY & GBPUSD & GC & CL & BTC & ETH & VIX & FROM \\
\midrule
ES      & 21.9 & 19.5 & 11.7 & 3.8 & 10.0 & 12.2 & 9.4 & 3.9 & 3.4 & 4.3 & 78.1 \\
NQ      & 20.7 & 21.7 & 11.5 & 3.6 & 9.4 & 11.8 & 9.9 & 4.2 & 3.6 & 3.7 & 78.4 \\
EURUSD  & 11.4 & 11.2 & 22.9 & 8.6 & 17.0 & 13.5 & 10.4 & 2.5 & 1.8 & 0.7 & 77.1 \\
USDJPY  & 6.2 & 5.8 & 13.3 & 38.9 & 11.3 & 10.3 & 6.0 & 3.0 & 2.6 & 2.6 & 61.1 \\
GBPUSD  & 9.4 & 8.6 & 18.5 & 8.7 & 31.4 & 12.0 & 7.5 & 1.9 & 1.6 & 0.5 & 68.6 \\
GC      & 12.9 & 12.0 & 14.2 & 6.7 & 11.1 & 22.1 & 11.5 & 3.9 & 3.8 & 1.9 & 77.9 \\
CL      & 11.6 & 11.6 & 12.0 & 5.0 & 8.8 & 13.7 & 30.1 & 2.9 & 2.4 & 1.8 & 69.9 \\
BTC     & 3.4 & 2.9 & 1.4 & 1.3 & 0.7 & 2.3 & 1.2 & 43.4 & 38.6 & 4.7 & 56.6 \\
ETH     & 3.5 & 3.1 & 1.3 & 1.2 & 0.8 & 2.7 & 1.5 & 38.2 & 43.9 & 3.8 & 56.1 \\
VIX     & 5.1 & 3.9 & 2.8 & 4.9 & 2.3 & 3.2 & 2.2 & 4.1 & 3.9 & 67.6 & 32.4 \\
\midrule
TO  & 84.3 & 78.6 & 86.6 & 43.9 & 71.5 & 81.7 & 59.7 & 64.5 & 61.5 & 23.9 & \textbf{65.6} \\
NET & $+$6.2 & $+$0.2 & $+$9.5 & $-$17.2 & $+$2.8 & $+$3.8 & $-$10.2 & $+$7.9 & $+$5.4 & $-$8.5 & \\
\bottomrule
\end{tabular}
\end{adjustbox}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Same methodology as \Cref{tab:connectedness_full}. TSI = 65.62\%.
\end{tablenotes}
\end{threeparttable}
\end{table}

The 2022 regime reveals that \textbf{BTC transitions to a net volatility transmitter} ($+7.9\%$), with ETH also becoming a net transmitter ($+5.4\%$). This marks a significant departure from the pre-COVID period. EURUSD emerges as the largest net transmitter ($+9.5\%$), driven by the EUR/USD depreciation toward parity. The BTC TO spillover of 64.5\% during this regime---compared to 27.0\% pre-COVID---represents a dramatic increase in crypto's influence.

\Cref{fig:net_spillovers_regimes} compares net directional spillovers across regimes.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig4_net_spillovers_regimes.png}
\caption{Net Directional Spillovers by Market Regime.}
\label{fig:net_spillovers_regimes}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Net directional spillovers (TO $-$ FROM) for each asset across three regimes. Positive values indicate net transmitters; negative values indicate net receivers. TSI for each regime shown in legend.
\end{minipage}
\end{figure}

\subsection{Volatility Hierarchy Evolution and Crypto Integration}

\Cref{fig:net_directional_btc_es} presents the rolling 200-day net directional spillovers for BTC and ES.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig6_net_directional_btc_es.png}
\caption{Rolling Net Directional Spillovers: BTC vs.\ ES.}
\label{fig:net_directional_btc_es}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Rolling 200-day net directional spillovers for Bitcoin (purple) and E-Mini S\&P 500 (blue). Positive values indicate net transmission. Shaded area distinguishes BTC net transmitter (above zero) from net receiver (below zero) episodes.
\end{minipage}
\end{figure}

The evolution reveals a gradual but unmistakable trend toward greater crypto integration. In the early period (2017--2018), BTC alternates between small positive and negative net spillovers. During the COVID crash, BTC temporarily becomes a significant net receiver. By late 2025, BTC's rolling net spillover stabilizes at approximately $+4.58\%$, while ES is near-neutral at $-0.62\%$, suggesting a more balanced volatility hierarchy.

\subsection{Network Visualization}

\Cref{fig:network_graph} presents the volatility transmission network, where directed edges represent pairwise spillovers exceeding the median threshold (5.7\%).

\begin{figure}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{fig7_network_graph.png}
\caption{Volatility Transmission Network.}
\label{fig:network_graph}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Nodes represent assets, colored by asset class (blue: equity, green: FX, orange: commodity, purple: crypto, brown: VIX). Node size is proportional to total TO spillover. Directed edges represent pairwise GFEVD contributions above the median threshold. Two distinct communities emerge: a densely connected traditional asset core and an isolated crypto cluster.
\end{minipage}
\end{figure}

The network reveals a core--periphery architecture: traditional assets form a densely connected core with bidirectional edges, while cryptocurrencies exist as a separate cluster with strong internal connections but no edges crossing to the traditional asset core. VIX occupies a peripheral position, receiving shocks from equities but transmitting little.

\subsection{Trading Session Analysis}

\Cref{fig:hourly_volatility} displays the average absolute 5-minute return by hour of day (UTC) for VIX, ES, NQ, and BTC.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig5_hourly_volatility.png}
\caption{Intraday Volatility Pattern by Asset and Trading Session.}
\label{fig:hourly_volatility}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Average absolute 5-minute return (\%) by hour of day (UTC). Shaded regions: Asian (00:00--08:00 UTC), European (08:00--14:30 UTC), U.S.\ (14:30--21:00 UTC). Bitcoin's volatility is remarkably uniform across all hours, contrasting with the session-dependent patterns of ES, NQ, and VIX.
\end{minipage}
\end{figure}

\textit{Equity futures (ES, NQ)} display a pronounced peak during the European session (09:00--14:00 UTC), reflecting the overlap between European market openings and the lead-up to the U.S.\ cash open. \textit{VIX} shows an even more concentrated pattern, with a sharp peak at 09:00--10:00 UTC. \textit{Bitcoin} exhibits a remarkably flat intraday profile (0.24\%--0.27\%), reflecting its decentralized, globally distributed market structure.

\Cref{tab:granger_causality} presents Granger causality test results.

\begin{table}[htbp]
\centering
\caption{Granger Causality Tests for Realized Volatility.}
\label{tab:granger_causality}
\begin{threeparttable}
\begin{tabular}{@{}llrrll@{}}
\toprule
Direction & & $F$-stat & $p$-value & Sig. \\
\midrule
ES $\to$ VIX   &  & 19.864 & $<$0.001 & \stmark \\
NQ $\to$ VIX   &  & 16.708 & $<$0.001 & \stmark \\
VIX $\to$ ES   &  & 4.303  & 0.002 & \stmark \\
VIX $\to$ NQ   &  & 4.181  & 0.002 & \stmark \\
ES $\to$ NQ    &  & 4.449  & 0.001 & \stmark \\
VIX $\to$ BTC  &  & 1.642  & 0.161 & \\
ES $\to$ BTC   &  & 0.581  & 0.676 & \\
NQ $\to$ BTC   &  & 0.441  & 0.779 & \\
BTC $\to$ VIX  &  & 1.796  & 0.127 & \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Granger causality tests with 4 lags applied to daily realized volatility series. \sym{***}~Significant at 1\%. The dominant direction is ES $\to$ VIX ($F=19.86$), confirming that equity futures lead VIX. All BTC pairings are insignificant.
\end{tablenotes}
\end{threeparttable}
\end{table}

The strongest causal relationship runs from ES to VIX ($F = 19.86$, $p < 0.001$), confirming that equity futures are the primary drivers of VIX movements. The reverse direction (VIX $\to$ ES) is significant but weaker ($F = 4.30$), suggesting bidirectional but asymmetric causality. All pairwise tests involving BTC are insignificant at the 5\% level, reinforcing the network analysis finding that crypto volatility operates independently of traditional market dynamics in the full sample.