This project introduces the GNAR-HARX model, which combines Generalised Network Autoregressive (GNAR) structure with Heterogeneous Autoregressive (HAR) dynamics and exogenous predictors such as implied volatility. The model is designed for forecasting realised volatility by capturing both temporal persistence and cross-sectional spillovers in financial markets. We apply it to daily realised variance data for ten international stock indices, generating one-step-ahead forecasts in a rolling window over an out-of-sample period of approximately 16 years (2005-2020). Forecast accuracy is evaluated using the Quasi-Likelihood (QLIKE) loss and mean squared error (MSE), and we compare global, standard, and local variants across different network structures and exogenous specifications. The best model found by QLIKE is a local GNAR-HAR without exogenous variables, while the lowest MSE is achieved by a standard GNAR-HARX with implied volatility. Fully connected networks consistently outperform dynamically estimated graphical lasso networks. Overall, local and standard GNAR-HAR(X) models deliver the strongest forecasts, though at the cost of more parameters than the parsimonious global variant, which nevertheless remains competitive. Across all cases, GNAR-HAR(X) models outperform univariate HAR(X) benchmarks, which often require more parameters than the GNAR-based specifications. While the top model found by QLIKE does not use exogenous variables, implied volatility and overnight returns emerge as the most useful predictors when included.

翻译：本研究提出了GNAR-HARX模型，该模型将广义网络自回归（GNAR）结构与异质自回归（HAR）动态特性及隐含波动率等外生预测因子相结合。该模型旨在通过捕捉金融市场中的时间持续性与横截面溢出效应来预测已实现波动率。我们将其应用于十个国际股票指数的日度已实现方差数据，在约16年（2005-2020年）的样本外期间通过滚动窗口生成一步超前预测。预测精度采用拟似然（QLIKE）损失和均方误差（MSE）进行评估，并比较了不同网络结构和外生变量设定下的全局、标准及局部模型变体。QLIKE准则下的最优模型为不含外生变量的局部GNAR-HAR模型，而最低MSE则由包含隐含波动率的标准GNAR-HARX模型实现。全连接网络在各类设定中均持续优于动态估计的图套索网络。总体而言，局部与标准GNAR-HAR(X)模型提供了最强的预测能力，但其代价是比简约的全局变体需要更多参数——尽管后者仍保持竞争力。在所有案例中，GNAR-HAR(X)模型均优于单变量HAR(X)基准模型，后者通常比基于GNAR的设定需要更多参数。虽然QLIKE准则选出的最优模型未使用外生变量，但隐含波动率与隔夜收益率在纳入模型时被证明是最有效的预测因子。