Residual Deep Learning for Realized Volatility Using a Hybrid HAR-LSTM Model with ARIMA Features and SHAP Interpretability
- 1Department of Finance, Banking and Economic Analysis, Faculty of Economics and Business Administration, University of Craiova, 200585 Craiova, Romania
- 2“Eugeniu Carada” Doctoral School of Economic Sciences, Faculty of Economics and Business Administration, University of Craiova, 200585 Craiova, Romania
Abstract
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).
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BibTeX
@article{spulbar2026residual,
title = {{Residual Deep Learning for Realized Volatility Using a Hybrid HAR-LSTM Model with ARIMA Features and SHAP Interpretability}},
author = {Cristi Spulbar and Cezar Cătălin Ene},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203661},
url = {https://www.mdpi.com/2227-7390/14/20/3661},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Residual Deep Learning for Realized Volatility Using a Hybrid HAR-LSTM Model with ARIMA Features and SHAP Interpretability
AU - Cristi Spulbar
AU - Cezar Cătălin Ene
PY - 2026
DA - 2026/10/09
JO - Mathematics
DO - 10.3390/math14203661
UR - https://www.mdpi.com/2227-7390/14/20/3661
AB - 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).
SN - 2227-7390
ER -