<![CDATA[This study develops a hybrid ARX–GARCH–LSTM approach to estimate the volatility of PGAS stock during the 2010–2025 period. Stock volatility exhibits characteristics such as heteroskedasticity, volatility clustering, and nonlinear patterns, requiring an approach capable of capturing volatility dynamics more accurately. The proposed approach integrates the ARX model to capture the influence of external factors, the GARCH model to model time-varying volatility, and Long Short-Term Memory (LSTM) to learn nonlinear patterns from the residual/errors of the GARCH model. The modeling process begins by transforming stock prices into log returns, followed by ARX estimation to purify returns from the influence of exogenous variables. The ARX residuals are then modeled using GARCH(1,1), and the residual/errors generated by the GARCH model are subsequently used as input for the LSTM model to construct the hybrid ARX–GARCH–LSTM model. The results show that the hybrid ARX–GARCH–LSTM model outperforms the GARCH and baseline LSTM models with an RMSE value of 0.004250, an MAE value of 0.003077, and an R² value of 0.827293. Compared to the GARCH model, the hybrid approach reduces RMSE by 42.43% and MAE by 48.53%, while increasing the R² value by 72.83%. These findings indicate that the integration of statistical models and deep learning methods can improve the accuracy of stock volatility estimation and potentially support investment decision-making and financial risk management.]]>
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