Residual Deep Learning for Realized Volatility Using a Hybrid HAR-LSTM Model with ARIMA Features and SHAP Interpretability

Classical linear models such as the Heterogeneous Autoregressive Realized Volatility (HAR-RV) model are difficult to beat at short horizons, yet cannot capture nonlinear, regime-dependent dynamics, while pure deep learning approaches typically overfit. This paper proposes a residual-learning hybrid in which a Long Short-Term Memory (LSTM) network learns nonlinear corrections to a HAR-RV baseline augmented with an ARIMA log-volatility forecast, complemented by SHAP interpretability. Using daily data for the S&P 500 and the CEE Fund (NYSE: CEE) over 2019–2024, the model targets the one-day-ahead update of a five-day squared-return volatility window; the mechanical persistence of this overlapping target is quantified through a horizon-consistent GARCH(1,1) benchmark and a non-overlapping five-day-ahead target. The hybrid attains R² = 0.741 (S&P 500) and 0.846 (CEE Fund). It is statistically indistinguishable from ARIMA, improves on HAR-RV under squared-error and HMSE loss (S&P 500) and QLIKE loss (CEE Fund), and significantly outperforms a pure LSTM in both markets on this overlapping target (Diebold–Mariano p ≤ 0.027), a ranking preserved under expanding-window, ablation and sensitivity analyses; on the non-overlapping target the hybrid and the pure LSTM are statistically indistinguishable. SHAP attribution concentrates in lags t−1 to t−4 and contrasts the intraday range (S&P 500) with the daily volatility component (CEE Fund).