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\section{Methodology}
\label{sec:methodology}
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\subsection{Realized Volatility}

We construct daily realized volatility from intraday 5-minute log returns. For asset $i$ on day $t$, let $P_{i,t,j}$ denote the closing price of the $j$-th 5-minute bar, where $j = 1, \ldots, M_t$ and $M_t$ is the number of intraday observations. The 5-minute log return is:
\begin{equation}
    r_{i,t,j} = \ln\left(\frac{P_{i,t,j}}{P_{i,t,j-1}}\right).
    \label{eq:logreturn}
\end{equation}

The daily realized variance is computed as the sum of squared intraday returns \citep{Andersen2003,BarndorffNielsen2002}:
\begin{equation}
    \RV_{i,t} = \sum_{j=1}^{M_t} r_{i,t,j}^2.
    \label{eq:rv}
\end{equation}

The annualized daily realized volatility is then:
\begin{equation}
    \sigma_{i,t}^{\RV} = \sqrt{252 \cdot \RV_{i,t}}.
    \label{eq:rv_annualized}
\end{equation}

We impose a minimum threshold of $M_t \geq 20$ intraday observations per day to ensure reliable realized volatility estimation. Days with fewer observations are excluded. Under standard assumptions, as the sampling frequency increases ($M_t \to \infty$), $\RV_{i,t}$ converges to the integrated variance:
\begin{equation}
    \RV_{i,t} \xrightarrow{p} \int_0^1 \sigma_{i,t}^2(s)\, ds.
    \label{eq:rv_convergence}
\end{equation}

The 5-minute sampling frequency represents a widely accepted compromise between capturing sufficient intraday variation and avoiding microstructure noise contamination \citep{Liu2015}.

\subsection{Vector Autoregression}

Let $\mathbf{y}_t = (\RV_{1,t}, \RV_{2,t}, \ldots, \RV_{N,t})'$ denote the $N \times 1$ vector of daily realized volatilities for $N = 10$ assets. We model the joint dynamics using a VAR($p$) process:
\begin{equation}
    \mathbf{y}_t = \boldsymbol{\mu} + \sum_{\ell=1}^{p} \boldsymbol{\Phi}_\ell \mathbf{y}_{t-\ell} + \boldsymbol{\varepsilon}_t,
    \label{eq:var}
\end{equation}
where $\boldsymbol{\mu}$ is an $N \times 1$ vector of intercepts, $\boldsymbol{\Phi}_\ell$ are $N \times N$ coefficient matrices, and $\boldsymbol{\varepsilon}_t \sim (0, \boldsymbol{\Sigma})$ is the vector of reduced-form innovations with covariance matrix $\boldsymbol{\Sigma}$. We set $p = 4$ lags throughout. The VAR($p$) admits the moving average representation:
\begin{equation}
    \mathbf{y}_t = \boldsymbol{\mu}^* + \sum_{s=0}^{\infty} \boldsymbol{\Psi}_s \boldsymbol{\varepsilon}_{t-s},
    \label{eq:vma}
\end{equation}
where the $N \times N$ coefficient matrices $\boldsymbol{\Psi}_s$ are computed recursively:
\begin{equation}
    \boldsymbol{\Psi}_s = \sum_{\ell=1}^{p} \boldsymbol{\Phi}_\ell \boldsymbol{\Psi}_{s-\ell}, \quad s = 1, 2, \ldots
    \label{eq:psi_recursion}
\end{equation}
with $\boldsymbol{\Psi}_0 = \mathbf{I}_N$ and $\boldsymbol{\Phi}_\ell = \mathbf{0}$ for $\ell > p$.

\subsection{Generalized Forecast Error Variance Decomposition}

Following \citet{DieboldYilmaz2012}, we employ the generalized variance decomposition of \citet{Koop1996} and \citet{Pesaran1998}, which is invariant to the ordering of variables. The $H$-step-ahead generalized forecast error variance decomposition is defined as:
\begin{equation}
    \theta_{ij}^g(H) = \frac{\sigma_{jj}^{-1} \sum_{h=0}^{H-1} (\mathbf{e}_i' \boldsymbol{\Psi}_h \boldsymbol{\Sigma} \mathbf{e}_j)^2}{\sum_{h=0}^{H-1} \mathbf{e}_i' \boldsymbol{\Psi}_h \boldsymbol{\Sigma} \boldsymbol{\Psi}_h' \mathbf{e}_i},
    \label{eq:gfevd}
\end{equation}
where $\sigma_{jj}$ is the $j$-th diagonal element of $\boldsymbol{\Sigma}$, $\mathbf{e}_i$ is the $i$-th column of $\mathbf{I}_N$, and $H = 10$ days. Since the rows of $\theta_{ij}^g(H)$ do not necessarily sum to unity under the generalized framework, we normalize:
\begin{equation}
    \tilde{\theta}_{ij}^g(H) = \frac{\theta_{ij}^g(H)}{\sum_{j=1}^{N} \theta_{ij}^g(H)},
    \label{eq:gfevd_normalized}
\end{equation}
such that $\sum_{j=1}^{N} \tilde{\theta}_{ij}^g(H) = 1$ and $\sum_{i,j=1}^{N} \tilde{\theta}_{ij}^g(H) = N$.

\subsection{Spillover Measures}

From the normalized GFEVD matrix $\tilde{\boldsymbol{\Theta}}^g(H) = [\tilde{\theta}_{ij}^g(H)]$, we define four spillover measures.

\paragraph{Total Spillover Index.}
\begin{equation}
    \TSI(H) = \frac{\sum_{\substack{i,j=1 \\ i \neq j}}^{N} \tilde{\theta}_{ij}^g(H)}{N} \times 100.
    \label{eq:tsi}
\end{equation}

\paragraph{Directional Spillovers TO Others.} The volatility spillover transmitted by market $j$ to all other markets:
\begin{equation}
    S_{\cdot j}^g(H) = \sum_{\substack{i=1 \\ i \neq j}}^{N} \tilde{\theta}_{ij}^g(H) \times 100.
    \label{eq:to_spillover}
\end{equation}

\paragraph{Directional Spillovers FROM Others.} The volatility spillover received by market $i$ from all other markets:
\begin{equation}
    S_{i \cdot}^g(H) = \sum_{\substack{j=1 \\ j \neq i}}^{N} \tilde{\theta}_{ij}^g(H) \times 100.
    \label{eq:from_spillover}
\end{equation}

\paragraph{Net Directional Spillovers.} The net volatility spillover for market $i$:
\begin{equation}
    S_i^{\text{net}}(H) = S_{\cdot i}^g(H) - S_{i \cdot}^g(H).
    \label{eq:net_spillover}
\end{equation}
A positive (negative) value indicates that market $i$ is a net transmitter (receiver) of volatility.

\subsection{Rolling Window Analysis}

To capture time-varying dynamics, we estimate the VAR model and compute the GFEVD using a rolling window of $w = 200$ trading days, advancing by one day:
\begin{equation}
    \TSI_t(H) = \TSI(H; \mathbf{y}_{t-w+1}, \ldots, \mathbf{y}_t).
    \label{eq:rolling_tsi}
\end{equation}

\subsection{Granger Causality Tests}

For realized volatility series $\RV_{i,t}$ and $\RV_{j,t}$, we test whether lagged values of $\RV_{j,t}$ have predictive power for $\RV_{i,t}$ beyond own lags \citep{Granger1969}. The null hypothesis is tested using an $F$-statistic:
\begin{equation}
    F = \frac{(\text{RSS}_R - \text{RSS}_U) / p}{\text{RSS}_U / (T - 2p - 1)},
    \label{eq:granger_f}
\end{equation}
where $\text{RSS}_R$ and $\text{RSS}_U$ are the residual sums of squares from the restricted and unrestricted models.

\subsection{Network Representation}

We represent the volatility connectedness structure as a weighted directed graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{W})$, where the vertex set $\mathcal{V}$ corresponds to the $N$ assets, the edge set $\mathcal{E}$ contains directed edges $(j, i)$ where the GFEVD contribution $\tilde{\theta}_{ij}^g(H)$ exceeds the median of all off-diagonal elements, and edge weight equals the magnitude of the pairwise spillover. Node size is proportional to $S_{\cdot j}^g(H)$, and nodes are colored by asset class.