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1% ============================================================================2\section{Data}3\label{sec:data}4% ============================================================================56\subsection{Sources and Instruments}78The raw data are 5-minute open--high--low--close--volume bars supplied by FirstRate Data and stored in four DuckDB databases: continuous front-month futures (ES, NQ, GC, CL), spot exchange rates (EUR/USD, USD/JPY, GBP/USD), spot cryptocurrency prices in U.S. dollars (BTC, ETH) and the CBOE Volatility Index. Together the ten series contain about 235 million bars between January 2008 and March 2026. \Cref{tab:asset_universe} lists the instruments. They were chosen to be the most liquid representative of each asset class, to overlap over a decade, and to differ in trading hours: the futures trade almost continuously from Sunday evening to Friday afternoon with a daily one-hour maintenance break, the exchange rates trade continuously from Sunday evening to Friday evening, the cryptocurrencies trade every hour of every day, and the VIX is disseminated by CBOE only between 03:15 and 16:15 Eastern time (09:30--16:15 before April 2016) on U.S. trading days.910\begin{table}[htbp]11\centering12\caption{Asset universe.}13\label{tab:asset_universe}14\adjustbox{max width=\textwidth}{%15\begin{threeparttable}16\begin{tabular}{@{}lllll@{}}17\toprule18Asset class & Symbol & Instrument & Venue & Trading hours \\19\midrule20\multirow{2}{*}{Equity} & ES & E-mini S\&P 500 futures & CME Globex & Sun 18:00--Fri 17:00 ET, break 17:00--18:00 \\21 & NQ & E-mini Nasdaq 100 futures & CME Globex & Sun 18:00--Fri 17:00 ET, break 17:00--18:00 \\[3pt]22\multirow{3}{*}{FX} & EURUSD & Euro / U.S.\ dollar & Interbank spot & Sun 17:00--Fri 17:00 ET \\23 & USDJPY & U.S.\ dollar / Japanese yen & Interbank spot & Sun 17:00--Fri 17:00 ET \\24 & GBPUSD & British pound / U.S.\ dollar & Interbank spot & Sun 17:00--Fri 17:00 ET \\[3pt]25\multirow{2}{*}{Commodity} & GC & Gold futures & COMEX & Sun 18:00--Fri 17:00 ET, break 17:00--18:00 \\26 & CL & WTI crude oil futures & NYMEX & Sun 18:00--Fri 17:00 ET, break 17:00--18:00 \\[3pt]27\multirow{2}{*}{Crypto} & BTC & Bitcoin, spot, USD & Composite of exchanges & 24 hours, 7 days \\28 & ETH & Ether, spot, USD & Composite of exchanges & 24 hours, 7 days \\[3pt]29Volatility & VIX & CBOE Volatility Index & CBOE & 03:15--16:15 ET on U.S. trading days \\30\bottomrule31\end{tabular}32\begin{tablenotes}[flushleft]33\footnotesize34\item \textit{Notes:} 5-minute bars from FirstRate Data; futures are continuous front-month series. ET denotes U.S. Eastern time. The VIX is an index, not a traded instrument; its ``realized volatility'' is the realized volatility of 5-minute changes in the index.35\end{tablenotes}36\end{threeparttable}}37\end{table}3839\subsection{Time Stamps and Alignment}40\label{sec:alignment}4142All FirstRate time stamps are expressed in U.S. Eastern time, which we verified from the location of the daily Globex maintenance break (17:00--18:00) in the futures series and of the opening bar of the VIX series (03:15 or 09:30). A calendar day is therefore an Eastern-time day, and the daily realized volatility of a futures contract on day $t$ aggregates the bars between 00:00 and 24:00 Eastern of that day, that is, the second part of the overnight Globex session that opened the previous evening, the cash session, and the first hours of the following Globex session. This definition is common to all exchange-traded series and to the cryptocurrencies, whose trading is uninterrupted. The earlier version of this paper treated the time stamps as UTC; the intraday results of \Cref{sec:sessions} are based on an explicit conversion from Eastern time to UTC.4344A second alignment issue arises in the construction of the panel used for the VAR. Bitcoin and ether have a realized volatility on every calendar day, the futures and exchange rates have one on Sunday evenings (computed from a few hours of thin trading) but not on Saturdays, and the VIX has one only on CBOE trading days. The panel used in this version comprises the trading days on which all ten instruments have an own realized volatility computed from their own bars, which in practice is the CBOE calendar: 2{,}514 days from 11 March 2016, the first day with ether data, to 11 March 2026. No value is forward-filled, and Saturday and Sunday crypto volatility does not enter the baseline. The May 2026 version instead built a panel on all 3{,}651 calendar days of the same period, filling missing values forward for up to five days; 1{,}044 of its rows were weekends and 1{,}137 contained at least one forward-filled value, so that on more than 30\% of the observations eight of the ten series had, by construction, zero innovation. \Cref{sec:appendix_audit} quantifies the consequences. As a robustness check we also use a weekend-inclusive crypto measure, in which the realized variance of bitcoin and ether on a trading day is the average daily variance over the calendar days elapsed since the previous trading day, so that Monday carries Saturday and Sunday information.4546\subsection{Descriptive Statistics of 5-Minute Returns}4748\Cref{tab:descriptive_stats} summarizes the 5-minute log returns over each instrument's full history. Returns more than twenty sample standard deviations from the mean were removed before computing realized volatility, as a guard against data errors; the symmetric bounds in the minimum and maximum columns are the trace of this filter, which removes a very small number of observations but may also remove genuine extreme moves on crash days, a limitation discussed in \Cref{sec:limitations}. Three patterns stand out. Cryptocurrencies and the VIX have 5-minute standard deviations an order of magnitude larger than equity futures and exchange rates (0.60--0.69\% against 0.03--0.08\%), with crude oil the most volatile traditional asset (0.15\%). All distributions are heavy-tailed, with excess kurtosis between 20 and 63. Skewness is small and negative for equity futures and crude oil, small and positive for the exchange rates and the cryptocurrencies, and clearly positive for the VIX ($+1.00$), whose 5-minute changes are dominated by upward jumps.4950\begin{table}[htbp]51\centering52\caption{Descriptive statistics of 5-minute log returns.}53\label{tab:descriptive_stats}54\adjustbox{max width=\textwidth}{%55\begin{threeparttable}56\begin{tabular}{@{}llr@{\quad}l@{\quad}lrrrr@{\quad}r@{}}57\toprule58Asset & Class & $N$ (M) & Start & End & Std (\%) & Skew & Kurt & Min (\%) & Max (\%) \\59\midrule60ES & Equity & 1.28 & 2008-01 & 2026-03 & 0.073 & $-0.05$ & 35.3 & $-1.50$ & 1.50 \\61NQ & Equity & 1.28 & 2008-01 & 2026-03 & 0.084 & $-0.03$ & 30.1 & $-1.70$ & 1.70 \\62EURUSD & FX & 1.21 & 2010-01 & 2026-03 & 0.032 & $+0.12$ & 20.4 & $-0.63$ & 0.64 \\63USDJPY & FX & 1.20 & 2010-01 & 2026-03 & 0.034 & $-0.12$ & 27.7 & $-0.70$ & 0.70 \\64GBPUSD & FX & 1.20 & 2010-01 & 2026-03 & 0.033 & $+0.09$ & 21.8 & $-0.69$ & 0.69 \\65GC & Commodity & 1.29 & 2008-01 & 2026-03 & 0.067 & $-0.20$ & 25.3 & $-1.36$ & 1.36 \\66CL & Commodity & 1.29 & 2008-01 & 2026-03 & 0.150 & $-0.14$ & 38.2 & $-3.28$ & 3.35 \\67BTC & Crypto & 1.32 & 2013-04 & 2026-03 & 0.648 & $+0.02$ & 55.5 & $-14.90$ & 14.84 \\68ETH & Crypto & 1.01 & 2016-03 & 2026-03 & 0.597 & $+0.03$ & 63.0 & $-12.66$ & 12.73 \\69VIX & Volatility & 0.55 & 2008-01 & 2026-03 & 0.688 & $+1.00$ & 52.6 & $-14.58$ & 14.65 \\70\bottomrule71\end{tabular}72\begin{tablenotes}[flushleft]73\footnotesize74\item \textit{Notes:} 5-minute log returns $r_{i,t,j}$ over each instrument's full history after removal of returns exceeding twenty sample standard deviations in absolute value. $N$ is in millions of observations; Std is the standard deviation; Kurt is excess kurtosis. Std, Min and Max are in percent. Source: \path{results/table1_descriptive_stats.csv}.75\end{tablenotes}76\end{threeparttable}}77\end{table}7879\subsection{Daily Realized Volatility}8081\Cref{fig:rv_timeseries} plots the annualized daily realized volatility of the ten instruments over their full histories, and \Cref{tab:rv_descriptives} summarizes the series over the common 2016--2026 trading-day panel. Mean annualized realized volatility ranges from 7\% for EUR/USD to 34\% for crude oil among traditional assets, and exceeds 100\% for bitcoin, ether and the VIX. The medians are far below the means for the cryptocurrencies (53\% against 103\% for bitcoin), a sign of the extreme right tail: the largest bitcoin values, above 1{,}000\% annualized, occur in April--May and December 2017 and reflect 5-minute prices that oscillated by 5--10\% between consecutive bars on a composite feed of exchanges with large and shifting premia; we verified on an independent one-minute feed from the same vendor that these values are a property of the source data and not of our processing. Skewness in levels is between 1.9 and 5.9; after the log transformation it falls to between 0.2 and 1.4 and excess kurtosis to between 0.1 and 4.0. Log realized volatility is strongly persistent, with first-order autocorrelations between 0.55 (VIX) and 0.91 (BTC) and autocorrelations at lag 22 between 0.12 and 0.66. Augmented Dickey--Fuller tests reject a unit root at the 1\% level for all series except ether (5\%), while KPSS tests reject level stationarity for all series except EUR/USD; the combination is the familiar signature of long memory in realized volatility \citep{Andersen2003,Corsi2009}, which a finite-order VAR approximates rather than reproduces.8283\begin{figure}[htbp]84\centering85\includegraphics[width=\textwidth]{fig1_rv_timeseries.png}86\caption{Daily annualized realized volatility, 2008--2026.}87\label{fig:rv_timeseries}88\begin{minipage}{0.95\textwidth}89\vspace{4pt}90\footnotesize\textit{Notes:} Annualized realized volatility $\sigma_{i,t} = \sqrt{252 \sum_j r_{i,t,j}^2}$ from 5-minute returns, Eastern-time calendar days with at least 20 bars. The $y$-axis scale differs across panels. The March 2020 spike is visible in every traditional asset; the 2017 spikes in bitcoin and ether reflect the quality of the composite spot feed in that year (see text).91\end{minipage}92\end{figure}9394\begin{table}[htbp]95\centering96\caption{Daily realized volatility over the common trading-day panel, 2016-03-11 to 2026-03-11.}97\label{tab:rv_descriptives}98\adjustbox{max width=\textwidth}{%99\begin{threeparttable}100\input{tables/tab_rv_descriptives}101\begin{tablenotes}[flushleft]102\footnotesize103\item \textit{Notes:} $T = 2{,}514$ trading days. Mean, median and maximum of annualized realized volatility in percent. Skewness in levels and skewness, excess kurtosis and first-order autocorrelation $\rho_1$ of log realized volatility. ADF is the augmented Dickey--Fuller statistic with a constant and AIC-selected lags (\sym{***}, \sym{**}: rejection of a unit root at 1\% and 5\%; all series reject at 1\% except ETH, $p = 0.04$). KPSS is the level-stationarity statistic with automatic bandwidth; \sym{*} marks rejection of stationarity at the 5\% level. Source: \path{results/rev_table_rv_descriptives.csv}.104\end{tablenotes}105\end{threeparttable}}106\end{table}107