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1% ============================================================================2\section{Empirical Results}3\label{sec:results}4% ============================================================================56\subsection{Full-Sample Connectedness}7\label{sec:full_sample}89\Cref{tab:connectedness_full} reports the full-sample connectedness table for log realized volatility over the 2{,}514 trading days from March 2016 to March 2026, with a VAR(4) and $H = 10$. Entry $(i, j)$ is the percentage of the ten-day forecast-error variance of instrument $i$ attributable to shocks in instrument $j$; the last column and the last two rows give the FROM, TO and NET measures of \eqref{eq:directional}--\eqref{eq:net_spillover}, each followed by its 90\% bootstrap interval.1011\begin{table}[htbp]12\centering13\caption{Full-sample connectedness table, log realized volatility, VAR(4), $H = 10$.}14\label{tab:connectedness_full}15\adjustbox{max width=\textwidth, max totalheight=0.80\textheight}{%16\begin{threeparttable}17\input{tables/tab_connectedness_full}18\begin{tablenotes}[flushleft]19\footnotesize20\item \textit{Notes:} Generalized forecast-error variance decomposition, 2016-03-11 to 2026-03-11, $T = 2{,}514$ trading days. Entry $(i,j)$ is the share (\%) of the 10-day-ahead forecast-error variance of the log realized volatility of $i$ due to shocks in $j$; diagonal entries (own shares) in bold. FROM and TO are the off-diagonal row and column sums; NET $=$ TO $-$ FROM in percentage points; the bottom-right entry of the TO row is the total spillover index. Figures in brackets are 90\% percentile intervals from a residual bootstrap with 500 replications. Source: \path{results/rev_table_connectedness_full.csv} and \path{rev_table_connectedness_full_ci.csv}.21\end{tablenotes}22\end{threeparttable}}23\end{table}2425The total spillover index is 54.8\%, with a 90\% interval of 53.1--56.9\%: slightly more than half of the forecast-error variance of the system is attributable to cross-asset shocks. Four features of the table organize the discussion.2627First, equity index futures are the dominant transmitters. ES and NQ transmit 104\% and 92\% of forecast-error variance to the other nine series and receive 68\% and 66\%, for net spillovers of $+36.0$ (interval 29.9--41.5) and $+25.6$ (19.0--31.3) percentage points. Shocks to ES explain 12\% of the variance of crude oil, 11\% of gold, 8--9\% of each exchange rate and 21\% of the realized volatility of the VIX; conversely, no other series explains more than 12\% of ES. The two contracts are also each other's largest source of variance (28--30\%), as expected for two indices of the same market.2829Second, the realized volatility of the VIX is the largest net receiver, at $-20.6$ percentage points (interval $-25.2$ to $-14.0$). Shocks to ES and NQ explain 21.2\% and 18.5\% of its forecast-error variance, while its own shocks explain only 40\%; it transmits 40\% in total, of which 12\% and 11\% to ES and NQ. This is the pattern one expects for a price that is a deterministic function of S\&P~500 option prices and whose intraday variability therefore inherits that of the underlying. It is not a statement about the information content of the VIX level for future volatility, which is a separate question \citep{BekaertHoerova2014}. Crude oil ($-14.2$, interval $-21.6$ to $-6.7$) and sterling ($-13.2$, interval $-18.0$ to $-7.4$) are the other clear net receivers; gold ($-7.2$) and USD/JPY ($-5.2$) are mild receivers whose intervals touch or include zero, and EUR/USD is neutral.3031Third, bitcoin and ether form a distinct block. Each explains 27--28\% of the other's variance, and their own shares, at 67\% and 65\%, are the largest in the table. Their exposure to the eight traditional instruments is small but not negligible in the full sample: 6.3\% of bitcoin's and 7.1\% of ether's forecast-error variance, of which ES and NQ account for about half. Their net spillovers, $+1.7$ (interval $-3.3$ to $6.4$) for bitcoin and $-2.4$ ($-7.9$ to $2.1$) for ether, are not distinguishable from zero. The earlier version of this paper reported own shares above 77\% and cross-asset shares below 0.5\% for the cryptocurrencies; those figures were produced by a panel in which the traditional series were forward-filled on weekends, which mechanically breaks the contemporaneous correlation between crypto innovations and the (artificially zero) innovations of the other series.3233Fourth, the exchange rates and commodities form a middle tier. EUR/USD receives 16.5\% of its variance from GBP/USD and 14.4\% from USD/JPY, GBP/USD receives 19.3\% from EUR/USD, and gold receives about 10\% each from the two European currencies, which is consistent with a common dollar factor in all these prices. Crude oil has the largest own share among traditional assets (54\%), receives most of its cross-asset variance from ES and NQ (22\% together) and transmits little, which is at odds with a view of oil as an originator of macroeconomic volatility over this decade.3435\Cref{fig:connectedness_heatmap} displays the matrix graphically. The block structure is visible: a dense equity--FX--commodity core, a two-node crypto block linked to the core mainly through the equity futures, and the VIX attached to the core almost exclusively through ES and NQ.3637\begin{figure}[htbp]38\centering39\includegraphics[width=0.85\textwidth]{fig2_connectedness_heatmap.png}40\caption{Full-sample variance-decomposition shares.}41\label{fig:connectedness_heatmap}42\begin{minipage}{0.95\textwidth}43\vspace{4pt}44\footnotesize\textit{Notes:} Shares (\%) of the 10-day forecast-error variance of the row instrument attributable to shocks in the column instrument, from \Cref{tab:connectedness_full}. Darker cells denote larger shares.45\end{minipage}46\end{figure}4748\subsection{Robustness of the Full-Sample Results}49\label{sec:robustness}5051\Cref{tab:robustness} re-estimates the full-sample decomposition under alternative choices. The total index lies between 53.7\% and 55.5\% for $p \in \{1, 2, 3, 4, 6\}$ and between 54.4\% and 55.3\% for $H \in \{5, 10, 20\}$; it falls to 50.0\% when realized volatility enters in levels rather than logarithms, because the covariance matrix is then dominated by a few extreme days on which comovement is weaker than on ordinary days. Including weekend crypto variance in the Monday observation leaves every measure within one percentage point of the baseline. Excluding the two cryptocurrencies raises the index to 59.0\%, which quantifies the dilution they induce, and excluding the VIX lowers it to 52.9\%.5253\begin{table}[htbp]54\centering55\caption{Robustness of full-sample connectedness to specification and sample choices.}56\label{tab:robustness}57\adjustbox{max width=\textwidth}{%58\begin{threeparttable}59\input{tables/tab_robustness}60\begin{tablenotes}[flushleft]61\footnotesize62\item \textit{Notes:} TSI in percent; ES, NQ, USDJPY, CL, BTC and VIX columns are net directional spillovers in percentage points; the last column is bitcoin's own variance share (\%). Baseline: log realized volatility, trading-day panel, VAR(4), $H = 10$. ``Weekend-inclusive crypto RV'' averages the daily variance of BTC and ETH over the calendar days since the previous trading day. Source: \path{results/rev_table_robustness.csv}.63\end{tablenotes}64\end{threeparttable}}65\end{table}6667The ranking of the main transmitters and receivers is stable: ES is the largest net transmitter in every row of the table (between $+26$ and $+42$ percentage points), NQ the second, and the realized volatility of the VIX the largest net receiver (between $-17$ and $-27$), followed by crude oil (between $-6$ and $-17$). The net position of bitcoin, by contrast, is not robust in sign: it is $+7.4$ with one lag, $+4.4$ at $H = 20$, $-0.3$ at $H = 5$, $-1.3$ when ether is excluded, $+3.0$ in 2016--2019 and $-8.7$ in 2020--2026. Bitcoin's own share falls from 78\% in 2016--2019 to 48\% in 2020--2026, the clearest sign in the table that its integration with the system increased. The sub-sample split also reverses the sign of USD/JPY, a net transmitter before 2020 ($+9.9$) and a net receiver after ($-11.3$), and raises the total index from 51.5\% to 60.9\%.6869\subsection{Time-Varying Connectedness}70\label{sec:rolling_results}7172\Cref{fig:rolling_spillover} plots the rolling total spillover index for windows of 200, 250 and 500 trading days. The three series tell the same story at different levels of smoothing. Connectedness was moderate and declining from 2016 to late 2017, reaching its minimum of 50.5\% (200-day window) in October 2017, when the cryptocurrency boom decoupled bitcoin and ether from everything else. It then fluctuated between 51\% and 61\% until February 2020 and rose sharply during the COVID-19 crash: the 200-day index moved from 60.4\% on 20 February 2020 to 69.1\% on 31 March 2020 and remained above 65\% until the end of June. The 2021 decline, the renewed increase through the 2022 tightening cycle (the 200-day index exceeds 70\% in September--October 2022), the trough of early 2024 and the climb through 2025 are all visible in the 500-day window, which ends the sample at its own maximum of 66.5\% in March 2026. The 200-day index reaches the highest value of the entire sample, 73.4\%, on 24 September 2025; the calendar-year estimates of \Cref{sec:yearly} confirm that 2025 is the most connected year of the decade, at 71.5\% (interval 67.9--75.2).7374\begin{figure}[htbp]75\centering76\includegraphics[width=\textwidth]{fig3_rolling_spillover.png}77\caption{Rolling total spillover index.}78\label{fig:rolling_spillover}79\begin{minipage}{0.95\textwidth}80\vspace{4pt}81\footnotesize\textit{Notes:} Total spillover index (\%) from a VAR(4) on log realized volatility with $H = 10$, estimated over rolling windows of 200, 250 and 500 trading days and plotted at the last day of the window. The dashed line is the full-sample estimate (54.8\%). Vertical lines mark the COVID-19 crash (16 March 2020), the first Federal Reserve rate increase of the 2022 cycle (16 March 2022), the FTX collapse (11 November 2022), the launch of spot bitcoin ETFs (11 January 2024) and the volatility spike of 5 August 2024. Source: \path{results/rev_rolling_w200.csv}, \path{rev_rolling_w250.csv}, \path{rev_rolling_w500.csv}.82\end{minipage}83\end{figure}8485Two differences from the May 2026 version deserve mention. First, the level of the rolling index is lower (mean 61.9\% for 200 trading days against 67.1\% previously) and the 500-day mean of 58.3\% is closer to the full-sample value of 54.8\%; the gap between rolling and full-sample estimates is the small-sample upward bias of a heavily parameterized VAR, which shrinks as the window lengthens (\Cref{tab:rolling_summary} in the appendix). Second, the earlier version reported a maximum of 81.6\% on 6 February 2018, interpreted as the signature of the ``Volmageddon'' episode, with the index jumping from 62\% to 82\% in two days. On the trading-day panel the index moves from 54.4\% on 2 February to 56.4\% on 6 February 2018. The earlier spike was produced by the interaction of a single extreme observation of VIX realized volatility with a panel in which most other series were forward-filled on surrounding weekend days; it is not a feature of the data.8687\subsection{Regime-Specific Connectedness}88\label{sec:regimes}8990\Cref{tab:regimes} summarizes the decomposition in four sub-samples, with bootstrap intervals; the full ten-by-ten tables are in \Cref{sec:appendix_regimes}, and \Cref{fig:net_spillovers_regimes} plots the net spillovers.9192\begin{table}[htbp]93\centering94\caption{Total and net directional spillovers by regime, with 90\% bootstrap intervals.}95\label{tab:regimes}96\begin{threeparttable}97\small98\input{tables/tab_regimes}99\begin{tablenotes}[flushleft]100\footnotesize101\item \textit{Notes:} Log realized volatility, $H = 10$; $p = 4$ except for the 85-day COVID window, where $p = 1$ (the VAR(4) estimates for that window are TSI 71.4\%, NET USDJPY $+105.3$, NET BTC $-4.6$). Brackets: 90\% percentile intervals from 500 residual-bootstrap replications. The spot-ETF era begins on 11 January 2024, the first trading day of U.S. spot bitcoin ETFs. Source: \path{results/rev_table_regimes.csv}.102\end{tablenotes}103\end{threeparttable}104\end{table}105106\begin{figure}[htbp]107\centering108\includegraphics[width=\textwidth]{fig4_net_spillovers_regimes.png}109\caption{Net directional spillovers by regime.}110\label{fig:net_spillovers_regimes}111\begin{minipage}{0.95\textwidth}112\vspace{4pt}113\footnotesize\textit{Notes:} Net directional spillovers (percentage points) from \Cref{tab:regimes}; whiskers are 90\% bootstrap intervals. Positive values denote net transmitters.114\end{minipage}115\end{figure}116117In the pre-COVID period (292 days) the equity futures dominate even more than in the full sample, with net spillovers of $+41$ and $+47$ percentage points and intervals that exclude zero by a wide margin; the European currencies and crude oil are significant net receivers; and the VIX is neutral. Bitcoin's point estimate is $+12.7$, but its interval ($-14.8$ to $+24.0$) spans zero.118119The COVID crash window (85 trading days) is the sub-sample in which inference matters most. The total index rises to 71.7\% (interval 65.9--78.1), the highest of the four regimes. USD/JPY becomes the dominant transmitter, at $+81.8$ percentage points with an interval of $+7.3$ to $+135.3$: the sign is established, the magnitude is not. The economic reading is that the yen's safe-haven appreciation and the unwinding of carry trades in March 2020 \citep{Ranaldo2010,Habib2012}, together with the dollar funding squeeze, made exchange-rate volatility the common factor of that episode. ES and NQ remain net transmitters in point estimate ($+19$ and $+33$) but with intervals that include zero, which corrects the earlier version's statement that equities ``flipped'' to receivers; that statement rested on a VAR(4) with 41 coefficients per equation estimated on 122 calendar days. Bitcoin is a net receiver in point estimate ($-14.4$), consistent with \citet{ConlonMcGee2020}, but its interval is $-45.9$ to $+32.9$. The only other significant result of the window is that the realized volatility of the VIX is a strong net receiver ($-46.6$, interval $-61.4$ to $-11.2$).120121In the 2022 tightening cycle (254 days) the hierarchy is flatter. ES remains a significant transmitter ($+25.4$) but NQ is neutral; USD/JPY ($-18.6$) and crude oil ($-21.6$) are significant receivers; the European currencies and gold have positive but insignificant point estimates. Bitcoin's net spillover is $-3.4$ with an interval of $-16.7$ to $+12.4$. The earlier version reported $+7.9$ for bitcoin in this regime and described a transition to net transmitter; on the trading-day panel with log volatility the point estimate is negative and, more importantly, the interval shows that the data cannot distinguish either sign from zero in a one-year window.122123The spot-ETF era (544 days) is the regime with the sharpest crypto results. The equity futures are back at the top ($+29.4$ and $+28.7$, both significant), crude oil remains a significant receiver ($-19.9$), and bitcoin and ether are both significant net receivers, at $-12.4$ (interval $-20.4$ to $-2.1$) and $-19.5$ ($-27.7$ to $-9.2$). As cryptocurrencies became accessible through regulated equity-market vehicles, their realized volatility became more responsive to shocks in the traditional system without the reverse becoming true.124125\subsection{Crypto Integration by Calendar Year}126\label{sec:yearly}127128\Cref{tab:yearly} and \Cref{fig:yearly_integration} follow the integration of the cryptocurrencies year by year. The share of bitcoin's forecast-error variance explained by the eight traditional instruments (that is, FROM excluding ether) was 13.8\% in 2016, 8.7\% in 2017 and 6.8\% in 2019, with an intermediate 22.8\% in 2018. From 2020 onwards it never falls below 20\%: 34.4\% in 2020, 20--26\% in 2021--2024, and 46.2\% in 2025. The pairwise shares with ES rise in both directions, from under 1\% in 2017 and 2019 to 10.0\% (BTC from ES) and 7.7\% (ES from BTC) in 2025, so that integration is genuinely two-sided at the pairwise level. The net spillover of bitcoin, however, is negative in every year from 2020 to 2025 (between $-1.6$ and $-14.6$), while it was positive in 2016, 2018 and 2019; none of the yearly intervals excludes zero for bitcoin, but the interval for ether excludes zero in 2025 ($-20.2$, interval $-31.4$ to $-4.5$). The total index by year rises from 51--55\% in 2016--2019 to 61--63\% in 2020--2024 and 71.5\% in 2025.129130\begin{table}[htbp]131\centering132\caption{Connectedness by calendar year, log realized volatility, VAR(2), $H = 10$.}133\label{tab:yearly}134\adjustbox{max width=\textwidth}{%135\begin{threeparttable}136\input{tables/tab_yearly}137\begin{tablenotes}[flushleft]138\footnotesize139\item \textit{Notes:} 2016 covers 11 March--31 December. TSI in percent and net spillovers in percentage points, with 90\% intervals from 300 bootstrap replications in brackets. ``FROM BTC excl.\ ETH'' is the share of bitcoin's forecast-error variance due to the eight non-crypto instruments. BTC $\leftarrow$ ES is the share of bitcoin's variance due to ES shocks and ES $\leftarrow$ BTC the share of ES variance due to bitcoin shocks. Source: \path{results/rev_table_yearly.csv}.140\end{tablenotes}141\end{threeparttable}}142\end{table}143144\begin{figure}[htbp]145\centering146\includegraphics[width=\textwidth]{fig8_yearly_crypto_integration.png}147\caption{Bitcoin's integration with traditional assets by calendar year.}148\label{fig:yearly_integration}149\begin{minipage}{0.95\textwidth}150\vspace{4pt}151\footnotesize\textit{Notes:} Panel (a): net directional spillovers of bitcoin and ES by year with 90\% bootstrap intervals. Panel (b): share of bitcoin's 10-day forecast-error variance explained by the eight traditional instruments, and pairwise shares between bitcoin and ES. From \Cref{tab:yearly}.152\end{minipage}153\end{figure}154155\Cref{fig:net_directional_btc_es} shows the same evolution in rolling windows. Bitcoin's 500-day net spillover drifts from slightly negative in 2018 to a maximum near $+15$ percentage points at the end of 2019, falls through 2020--2021, and remains between $-5$ and $-20$ from 2022 to the end of the sample, finishing at $-19.1$ in March 2026; it is positive in only 27\% of the 500-day windows and 34\% of the 200-day windows. ES is a net transmitter in every window, between $+20$ and $+57$ percentage points.156157\begin{figure}[htbp]158\centering159\includegraphics[width=\textwidth]{fig6_net_directional_btc_es.png}160\caption{Rolling net directional spillovers of bitcoin and ES.}161\label{fig:net_directional_btc_es}162\begin{minipage}{0.95\textwidth}163\vspace{4pt}164\footnotesize\textit{Notes:} Net directional spillovers (percentage points) for bitcoin (purple) and ES (blue) from VAR(4) estimates on log realized volatility over rolling windows of 200 (thin lines) and 500 (thick lines) trading days. Vertical lines as in \Cref{fig:rolling_spillover}. Source: \path{results/rev_rolling_w200.csv}, \path{rev_rolling_w500.csv}.165\end{minipage}166\end{figure}167168\subsection{Network Representation}169\label{sec:network}170171\Cref{fig:network_graph} draws the net pairwise directional spillovers of \eqref{eq:net_pairwise} that exceed the 60th percentile of all pairwise values in absolute terms (1.0 percentage point). The graph has a hub-and-spoke structure: ES and NQ send net volatility to every other traditional instrument and to the VIX, the thickest edge being ES $\to$ VIX; EUR/USD sends net volatility to GBP/USD, USD/JPY and gold; and gold sends a small net flow to crude oil. Bitcoin and ether are isolated at this threshold because their net pairwise spillovers with every traditional instrument---and with each other---are smaller than one percentage point in the full sample, even though their gross exposures to equities are not zero. The network therefore summarizes the asymmetric transmission among traditional assets rather than the gross linkages of the crypto block, which \Cref{sec:yearly} shows to be growing.172173\begin{figure}[htbp]174\centering175\includegraphics[width=0.8\textwidth]{fig7_network_graph.png}176\caption{Network of net pairwise directional spillovers, full sample.}177\label{fig:network_graph}178\begin{minipage}{0.95\textwidth}179\vspace{4pt}180\footnotesize\textit{Notes:} Nodes are the ten instruments, coloured by asset class, with size proportional to TO spillover. A directed edge from $i$ to $j$ is drawn when $\mathrm{NET}_{i \to j}$ exceeds 1.0 percentage point (the 60th percentile of absolute net pairwise spillovers); edge width is proportional to its value. From the decomposition of \Cref{tab:connectedness_full}.181\end{minipage}182\end{figure}183184\subsection{Granger Causality}185\label{sec:granger_results}186187\Cref{tab:granger_causality} reports Granger-causality tests on daily log realized volatility with four lags over the common panel. Equity futures Granger-cause the realized volatility of the VIX ($F = 19.9$ for ES and $14.4$ for NQ, $p < 0.001$ under both conventional and HAC inference), and the reverse direction is also significant though weaker ($F = 5.3$ and $5.8$), confirming the asymmetric two-way link of \Cref{sec:full_sample}. ES and NQ do not Granger-cause each other beyond their own lags, which is expected for two indices whose volatilities move almost simultaneously. None of ES, NQ or the VIX Granger-causes bitcoin's realized volatility ($p$ between 0.26 and 0.66). In the other direction, bitcoin Granger-causes the realized volatility of the VIX ($F = 5.1$, $p < 0.001$) and, marginally, of ES ($F = 2.3$, $p = 0.054$; HAC $p = 0.024$), and there is two-way predictability between bitcoin and gold. These statistics must be read with two cautions. The sums of the coefficients on the causing variable are between $-0.07$ and $+0.03$, so that the predictive content, while statistically detectable in a sample of 2{,}500 days, is economically small relative to own-lag persistence. And sixteen tests are reported, so that one or two rejections at the 5\% level would be expected under the global null. The session-ordered regressions of \Cref{sec:sessions} provide a sharper test of the direction of information flow between bitcoin and U.S. equities.188189\begin{table}[htbp]190\centering191\caption{Granger-causality tests on daily log realized volatility, four lags.}192\label{tab:granger_causality}193\begin{threeparttable}194\small195\input{tables/tab_granger}196\begin{tablenotes}[flushleft]197\footnotesize198\item \textit{Notes:} Common trading-day panel, 2016-03-11 to 2026-03-11. $F$ is the conventional statistic of \eqref{eq:granger}; Wald (HAC) uses a Newey--West covariance matrix with ten lags and is distributed $\chi^2(4)$ under the null. \sym{***}, \sym{**}, \sym{*}: significant at 1\%, 5\%, 10\%. Source: \path{results/rev_table_granger.csv}.199\end{tablenotes}200\end{threeparttable}201\end{table}202203\subsection{Intraday Periodicity and Trading Sessions}204\label{sec:sessions}205206\Cref{fig:hourly_volatility} plots the mean absolute 5-minute return by hour of the day in UTC for the VIX, ES, NQ and bitcoin over the common sample, after conversion of the Eastern-time stamps. The equity futures display the familiar pattern of the U.S. cash session \citep{Wood1985,Harris1986}: a trough around 04:00 UTC during the late Asian hours, a rise from 07:00--08:00 UTC when European markets open, a sharp peak in the hour of the U.S. cash open (13:30--14:30 UTC depending on daylight-saving time), a second peak in the hour before the cash close, and a decline after 20:00 UTC. The peak-to-trough ratio is 3.9 for ES and 4.5 for NQ. The VIX, which is disseminated from 07:15--08:15 UTC, is relatively quiet during European hours and three times as volatile in the hour of the U.S. cash open as at its 10:00 UTC minimum; because its ``returns'' are changes in an implied-volatility index, its hourly profile also reflects the option market's reaction to the cash open. Bitcoin's profile is nearly flat: the mean absolute return ranges from 0.240\% at 07:00 UTC to 0.287\% at 18:00 UTC, a peak-to-trough ratio of 1.2, with a mild hump during U.S. hours consistent with \citet{Eross2019}.207208\begin{figure}[htbp]209\centering210\includegraphics[width=\textwidth]{fig5_hourly_volatility.png}211\caption{Intraday periodicity of absolute 5-minute returns by hour of the day (UTC).}212\label{fig:hourly_volatility}213\begin{minipage}{0.95\textwidth}214\vspace{4pt}215\footnotesize\textit{Notes:} Mean absolute 5-minute log return (\%) by UTC hour, 11 March 2016 to 11 March 2026, after conversion of the vendor's U.S. Eastern time stamps with daylight-saving rules. Shaded areas mark the Asian (00:00--08:00), European (08:00--14:30) and U.S. (14:30--21:00) sessions. The VIX is disseminated only from 03:15 Eastern time. Source: \path{results/rev_intraday_hourly_absret_utc.csv}.216\end{minipage}217\end{figure}218219\Cref{tab:sessions} translates the profiles into shares of daily variance. The U.S. session generates 53\% of the daily variance of ES and 54\% of NQ, the European session 31--32\%, and the Asian session 14--16\%. For the VIX the U.S. session accounts for 63\% of the variance realized during its dissemination hours. For bitcoin the three sessions contribute 37\%, 25\% and 38\%: the smaller European share reflects the shorter session (6.5 hours against 8 for the Asian session) rather than lower volatility per hour, and the U.S. session is the most volatile per unit of time. Bitcoin's weekend volatility is lower than its weekday volatility but far from negligible: Saturday and Sunday account for 22.1\% of the variance of an average week (28.6\% would correspond to a uniform distribution), and mean annualized realized volatility is 96--99\% on weekend days against 104--106\% on weekdays, in line with the day-of-week effects of \citet{Baur2019}. The earlier version of this paper, which read the Eastern-time stamps as UTC, attributed the equity-futures peak to the European session and the VIX peak to 09:00--10:00 UTC; both statements described the U.S. cash open shifted by four to five hours.220221\begin{table}[htbp]222\centering223\caption{Volatility by trading session, 2016-03-11 to 2026-03-11.}224\label{tab:sessions}225\begin{threeparttable}226\small227\input{tables/tab_sessions}228\begin{tablenotes}[flushleft]229\footnotesize230\item \textit{Notes:} Weekdays only. Mean $|r|$ is the average absolute 5-minute return in the session; annualized volatility is the mean over days of $\sqrt{252 \cdot \RV^{s}_{i,t}}$, where $\RV^{s}_{i,t}$ is the session realized variance of \eqref{eq:session_rv}; share of day is the mean over days of the session's share of the variance realized in the three sessions. The VIX has no Asian-session bars except a few minutes before 08:00 UTC in winter. Source: \path{results/rev_intraday_sessions.csv}.231\end{tablenotes}232\end{threeparttable}233\end{table}234235\Cref{tab:sessions_predictive} reports the session-ordered predictive regressions of \eqref{eq:session_predictive}. Bitcoin's Asian- and European-session variance does not help predict the U.S.-session variance of ES or NQ on the same day once four lags of the latter are included: the HAC Wald statistics have $p$-values between 0.16 and 0.48 and the incremental $R^2$ is below 0.0012. Bitcoin's earlier-session variance does predict the U.S.-session variance of the VIX ($p < 0.001$), but the incremental $R^2$ is 0.007 and the same-day coefficient is 0.06. In the reverse direction the evidence is stronger and uniform: the previous day's U.S.-session variance of ES, NQ and the VIX predicts bitcoin's Asian-session variance with coefficients of 0.08--0.10 and $p$-values below 0.002, and ES's own Asian-session (Globex overnight) variance predicts bitcoin's U.S.-session variance. For comparison, ES's overnight variance predicts its own U.S.-session variance with a coefficient of 0.38 and an incremental $R^2$ of 0.046, an order of magnitude larger than any cross-asset effect.236237\begin{table}[htbp]238\centering239\caption{Session-ordered predictive regressions for log session realized variance.}240\label{tab:sessions_predictive}241\adjustbox{max width=\textwidth}{%242\begin{threeparttable}243\input{tables/tab_sessions_predictive}244\begin{tablenotes}[flushleft]245\footnotesize246\item \textit{Notes:} Each row is a regression of the log realized variance of the target session on four lags of itself and on the log realized variance of the predictor session on the same day and three previous days, \eqref{eq:session_predictive}. ``Coef.\ (same day)'' is $\hat{\gamma}_0$ with its HAC standard error; Wald (HAC) tests the joint nullity of the four predictor coefficients and is $\chi^2(4)$ under the null (Newey--West, ten lags); $\Delta R^2$ is the increase in $R^2$ relative to the own-lags regression. In ``(t$-$1)'' rows the predictor is the U.S. session of the previous trading day. \sym{***}, \sym{**}, \sym{*}: significant at 1\%, 5\%, 10\%. Source: \path{results/rev_intraday_session_predictive.csv}.247\end{tablenotes}248\end{threeparttable}}249\end{table}250251\Cref{tab:weekend_btc} addresses the specific hypothesis that bitcoin's weekend trading provides a continuous information channel for Monday equity volatility. Regressing the log U.S.-session variance of ES on Mondays on Friday's ES and bitcoin variances, the addition of bitcoin's Saturday--Sunday variance has a coefficient of $-0.04$ (standard error 0.05) and raises $R^2$ by less than 0.001; adding bitcoin's Monday Asian-session variance leaves both crypto coefficients insignificant at the 5\% level. Over 501 Mondays, weekend bitcoin volatility carries no detectable incremental information about the volatility of the U.S. equity market when it reopens.252253\begin{table}[htbp]254\centering255\caption{Does weekend bitcoin volatility predict Monday U.S.-session volatility of ES?}256\label{tab:weekend_btc}257\adjustbox{max width=\textwidth}{%258\begin{threeparttable}259\input{tables/tab_weekend_btc}260\begin{tablenotes}[flushleft]261\footnotesize262\item \textit{Notes:} Dependent variable: log U.S.-session realized variance of ES on Mondays, 2016--2026. Regressors: log Friday U.S.-session variance of ES, log Friday daily variance of bitcoin (coefficient not shown), log Saturday-plus-Sunday variance of bitcoin and log Monday Asian-session variance of bitcoin. HAC standard errors (four lags) in parentheses. \sym{***}, \sym{**}, \sym{*}: significant at 1\%, 5\%, 10\%. Source: \path{results/rev_intraday_weekend_btc_monday_es.csv}.263\end{tablenotes}264\end{threeparttable}}265\end{table}266