% ============================================================================
\section{Data}
\label{sec:data}
% ============================================================================

\subsection{Data Sources and Description}

We employ 5-minute OHLCV (Open, High, Low, Close, Volume) data from four DuckDB databases covering four major asset classes. The databases collectively contain over 235 million intraday observations spanning January 2008 to March 2026. \Cref{tab:asset_universe} details the 10 representative instruments selected for the analysis.

\begin{table}[htbp]
\centering
\caption{Asset Universe.}
\label{tab:asset_universe}
\begin{threeparttable}
\begin{tabular}{@{}llll@{}}
\toprule
Asset Class & Symbol & Description & Source \\
\midrule
\multirow{2}{*}{Equity}     & ES      & E-Mini S\&P 500 Futures    & Futures \\
                             & NQ      & E-Mini Nasdaq 100 Futures  & Futures \\[3pt]
\multirow{3}{*}{FX}         & EURUSD  & Euro / U.S.\ Dollar        & FX \\
                             & USDJPY  & U.S.\ Dollar / Japanese Yen & FX \\
                             & GBPUSD  & British Pound / U.S.\ Dollar & FX \\[3pt]
\multirow{2}{*}{Commodity}  & GC      & Gold Futures (COMEX)       & Futures \\
                             & CL      & WTI Crude Oil Futures      & Futures \\[3pt]
\multirow{2}{*}{Crypto}     & BTC     & Bitcoin (spot)             & Crypto \\
                             & ETH     & Ethereum (spot)            & Crypto \\[3pt]
Volatility                   & VIX     & CBOE Volatility Index      & Index \\
\bottomrule
\end{tabular}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} All data are 5-minute OHLCV bars stored in DuckDB databases. Source column indicates the database of origin.
\end{tablenotes}
\end{threeparttable}
\end{table}

The selection of assets is guided by: (i)~liquidity and representativeness within each asset class; (ii)~sufficient overlap in sample periods; (iii)~diversity in trading hours; and (iv)~economic significance for volatility transmission analysis.

\subsection{Descriptive Statistics of Intraday Returns}

\Cref{tab:descriptive_stats} presents descriptive statistics for 5-minute log returns across all 10 assets.

\begin{table}[htbp]
\centering
\caption{Descriptive Statistics of 5-Minute Log Returns.}
\label{tab:descriptive_stats}
\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}llr@{\quad}l@{\quad}lrrrr@{\quad}r@{}}
\toprule
Asset & Class & $N$ (M) & Start & End & Std (\%) & Skew & Kurt & Min (\%) & Max (\%) \\
\midrule
ES     & Equity     & 1.28 & 2008-01 & 2026-03 & 0.073 & $-0.05$ & 35.3 & $-1.50$ & 1.50 \\
NQ     & Equity     & 1.28 & 2008-01 & 2026-03 & 0.084 & $-0.03$ & 30.1 & $-1.70$ & 1.70 \\
EURUSD & FX         & 1.21 & 2010-01 & 2026-03 & 0.032 & $+0.12$ & 20.4 & $-0.63$ & 0.64 \\
USDJPY & FX         & 1.20 & 2010-01 & 2026-03 & 0.034 & $-0.12$ & 27.7 & $-0.70$ & 0.70 \\
GBPUSD & FX         & 1.20 & 2010-01 & 2026-03 & 0.033 & $+0.09$ & 21.8 & $-0.69$ & 0.69 \\
GC     & Commodity  & 1.29 & 2008-01 & 2026-03 & 0.067 & $-0.20$ & 25.3 & $-1.36$ & 1.36 \\
CL     & Commodity  & 1.29 & 2008-01 & 2026-03 & 0.150 & $-0.14$ & 38.2 & $-3.28$ & 3.35 \\
BTC    & Crypto     & 1.32 & 2013-04 & 2026-03 & 0.648 & $+0.02$ & 55.5 & $-14.90$ & 14.84 \\
ETH    & Crypto     & 1.01 & 2016-03 & 2026-03 & 0.597 & $+0.03$ & 63.0 & $-12.66$ & 12.73 \\
VIX    & Volatility & 0.55 & 2008-01 & 2026-03 & 0.688 & $+1.00$ & 52.6 & $-14.58$ & 14.65 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes:} Descriptive statistics for 5-minute log returns $r_{i,t,j} = \ln(P_{i,t,j}/P_{i,t,j-1})$. $N$~is the total number of observations in millions. Std~is the standard deviation. Skew~is the skewness. Kurt~is the excess kurtosis. All values in percentage points except $N$.
\end{tablenotes}
\end{threeparttable}
\end{table}

Several patterns emerge. First, there is a clear hierarchy in return volatility: cryptocurrencies (BTC: 0.648\%, ETH: 0.597\%) and VIX (0.688\%) exhibit an order of magnitude higher 5-minute return volatility compared to equity futures (ES: 0.073\%, NQ: 0.084\%) and FX pairs (EURUSD: 0.032\%). Crude oil (CL: 0.150\%) is the most volatile among traditional assets. Second, all return distributions exhibit substantial excess kurtosis, ranging from 20.4 (EURUSD) to 63.0 (ETH), indicating heavy tails far in excess of the normal distribution. Third, skewness patterns differ systematically: equity futures and crude oil exhibit slight negative skewness (leverage effects), VIX displays pronounced positive skewness ($+1.00$), and cryptocurrency returns are approximately symmetric.

\subsection{Realized Volatility Construction}

We compute daily realized volatility for each asset using Equations~\eqref{eq:rv} and~\eqref{eq:rv_annualized}, requiring a minimum of 20 intraday observations per day. \Cref{fig:rv_timeseries} displays the time series of annualized daily realized volatility for all 10 assets.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig1_rv_timeseries.png}
\caption{Daily Realized Volatility Time Series (Annualized, 5-Minute Frequency).}
\label{fig:rv_timeseries}
\begin{minipage}{0.95\textwidth}
\vspace{4pt}
\footnotesize\textit{Notes:} Each panel displays the annualized daily realized volatility $\sigma_{i,t}^{\RV} = \sqrt{252 \cdot \sum_{j} r_{i,t,j}^2}$ for one asset. Note the different $y$-axis scales across panels. The COVID-19 crash (March 2020) generates the most prominent volatility spike across all traditional assets.
\end{minipage}
\end{figure}

The mean annualized realized volatilities are: ES~(13.85\%), NQ~(16.40\%), EURUSD~(6.96\%), USDJPY~(7.71\%), GBPUSD~(7.16\%), GC~(14.00\%), CL~(29.67\%), BTC~(98.69\%), ETH~(110.58\%), and VIX~(105.96\%).