Learning volatility-adaptive forgetting via GARCH-modulated LSTM for stock price forecasting

Financial time series exhibit multi-scale, high-noise, and non-stationary volatility dynamics. Decomposition--integration frameworks built on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) have advanced this field, yet existing CEEMDAN hybrids remain unstable: high-frequency mode mixing contaminates downstream predictors, and the heterogeneous volatility of decomposed components is not matched by any volatility-adaptive mechanism. To address these gaps, this paper proposes CEEMDAN-SE-SSA-VMGL. CEEMDAN adaptively decomposes the original series, and sample-entropy-driven K-means clustering groups the intrinsic mode functions into high-, mid-, and low-frequency and residual components. Singular spectrum analysis (SSA) is then applied to the high-frequency group, whose singular-value decomposition of the trajectory matrix and principal-component reconstruction decouple intertwined pseudo-oscillatory modes and alleviate mode mixing. We further design a plug-and-play robust LSTM variant, the volatility-modulated GARCH-LSTM (VMGL), which embeds a differentiable GARCH(1,1) layer inside the LSTM cell as a risk gate; the estimated conditional volatility multiplicatively scales the forget-gate activation, letting memory decay adapt to the instantaneous volatility while remaining fully end-to-end trainable. Component predictions are linearly combined. On five US and Chinese financial series (CSI 300, CSI 500, AAPL, GOOGL, S&P 500), the proposed model is compared with eleven benchmark architectures, lowering the forecasting error by 9% to 24% relative to the best benchmark and by 60% to 70% relative to a standard LSTM. Diebold--Mariano and model-confidence-set (MCS) tests confirm the statistical significance and stability of the gains, spectral analyses verify the SSA denoising efficacy, and backtesting demonstrates the framework's economic utility.

This paper proposes a novel hybrid model, termed GARCH-FIS, for recursive rolling multi-step forecasting of financial time series. It integrates a Fuzzy Inference System (FIS) with a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to jointly address nonlinear dynamics and time-varying volatility. The core innovation is a dynamic parameter adaptation mechanism for the FIS, specifically activated within the multi-step forecasting cycle. In this process, the conditional volatility estimated by a rolling window GARCH model is continuously translated into a price volatility measure. At each forecasting step, this measure, alongside the updated mean of the sliding window data -- which now incorporates the most recent predicted price -- jointly determines the parameters of the FIS membership functions for the next prediction. Consequently, the granularity of the fuzzy inference adapts as the forecast horizon extends: membership functions are automatically widened during high-volatility market regimes to bolster robustness and narrowed during stable periods to enhance precision. This constitutes a fundamental advancement over a static one-step-ahead prediction setup. Furthermore, the model's fuzzy rule base is automatically constructed from data using the Wang-Mendel method, promoting interpretability and adaptability. Empirical evaluation, focused exclusively on multi-step forecasting performance across ten diverse financial assets, demonstrates that the proposed GARCH-FIS model significantly outperforms benchmark models -- including Support Vector Regression(SVR), Long Short-Term Memory networks(LSTM), and an ARIMA-GARCH econometric model -- in terms of predictive accuracy and stability, while effectively mitigating error accumulation in extended recursive forecasts.

Hybrid statistical-neural pipelines often assume that a successful statistical first stage leaves a cleaner and more learnable residual target. We examine that assumption in short-horizon volatility forecasting through a signal-forecast-system diagnostic framework. Across five liquid U.S. assets, a volatility-aligned HAR-style model outperforms AR, MA, and ARIMA. Within expanding training windows, the pre-standardization fitted residual process used to construct residual-LSTM sequences has about 82% lower variance than the corresponding target and near-zero lag-1 autocorrelation; independently, rolling pseudo-out-of-sample HAR errors show about 74% variance reduction and similarly weak lag-1 dependence. Residual-only LSTM augmentation nevertheless raises mean squared error from 0.3049 to 0.3594 on average, with deterioration on every asset. Pure LSTM records the lowest selected pseudo-out-of-sample MSE, 0.2649, while the residual hybrid requires substantially more end-to-end runtime without improving accuracy. We describe this pattern as forecaster-preconditioner asymmetry: first-stage forecasting success and statistical residual simplification need not translate into useful downstream neural preconditioning.