GARCH-LSTM Hybrid Model

• GARCH-LSTM is a hybrid model that combines GARCH econometric techniques with LSTM networks to forecast financial volatility accurately while preserving stylized facts.
• Its architecture embeds a modular GARCH kernel within the LSTM output gate, capturing volatility clustering, asymmetry, and long memory effects.
• Empirical evaluations demonstrate that GARCH-LSTM achieves 3% lower MAE and 10% lower MSE compared to standalone models, enhancing risk quantile forecasting.

The GARCH-LSTM model is a hybrid methodology integrating Generalized Autoregressive Conditional Heteroskedasticity (GARCH) modeling paradigms with Long Short-Term Memory (LSTM) neural networks for volatility forecasting in financial time series. This synthesis is grounded in a theoretical equivalence between GARCH models and neural network architectures and designed to embed volatility stylized facts directly into neural frameworks, thereby enhancing both the predictive accuracy and interpretability of volatility forecasts (Zhao et al., 2024).

1. Theoretical Equivalence Between GARCH and Neural Network Models

The foundation of GARCH-LSTM modeling is the mathematical equivalence between classical GARCH processes and recursive neural networks (RNNs). Specifically, the GARCH(1,1) volatility update,

$$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2,$$

is algebraically and structurally analogous to a scalar RNN cell lacking activation function and output transformation. Stylized facts traditionally associated with GARCH models are thus directly encoded in neural network counterparts:

• Volatility Clustering: Encoded via GARCH(1,1) recursion as a linear RNN.
• Leverage Effect (asymmetry): GJR-GARCH as RNN with an additional indicator input for innovation sign.
• Long Memory: FI-GARCH as a convolutional neural network (CNN) with fractional integration implemented by a sliding window of weights.

Both stochastic and NN models can use identical likelihood-based loss functions for estimation, e.g., negative log-likelihood for Gaussian or Student-t innovation processes: $$-\log \mathcal{L}(\epsilon_t; \Theta) = \frac{1}{2} \log \hat{\sigma}_t^2(\Theta) + \frac{\epsilon_t^2}{2\hat{\sigma}_t^2(\Theta)}$$ or the heavier-tailed Student-t version.

2. Architecture and Integration Scheme of GARCH-LSTM

The GARCH-LSTM architecture is constructed by modularly embedding the GARCH "kernel"—the NN equivalent of a GARCH (or GARCH-family) recursion—within the LSTM’s output gate. The essential recursion is: $$\begin{aligned} f_t & = \sigma_g(W_f * \epsilon_{t-1} + U_f * \sigma_{t-1}^2 + b_f) \ i_t & = \sigma_g(W_i * \epsilon_{t-1} + U_i * \sigma_{t-1}^2 + b_i) \ o_t & = \mathcal{K}_{garch}(\epsilon_{t-1}, \sigma_{t-1}^2; \Theta) \ \tilde{c}_t & = \sigma_c(W_c * \epsilon_{t-1} + U_c * \sigma_{t-1}^2 + b_c) \ c_t & = f_t \odot c_{t-1} + i_t \odot \tilde{c}_t \ \sigma_t^2 & = o_t \odot (1 + w * \tanh(c_t)) \end{aligned}$$ where $\mathcal{K}_{garch}$ is parameterized for GARCH, GJR-GARCH, or FI-GARCH forms, and $w$ serves as a mixing parameter between pure GARCH and LSTM-modulated output.

This structure allows:

• GARCH mechanisms to dominate (for $w=0$),
• or LSTM-modulated volatility forecasts (for $w>0$), capturing nonlinear temporal dependencies without discarding traditional stylized facts.

3. Stylized Facts Preservation and Enhancement

By embedding GARCH kernels inside LSTM frameworks, the model ensures direct and interpretable coding of stylized facts:

• Volatility Clustering: Natural consequence of the GARCH recursion inherited in output gate.
• Asymmetric Responses: Achievable through modular kernel swaps (e.g., GJR-GARCH).
• Long Memory: FI-GARCH convolutional kernels can be plugged in, enabling non-exponential decay and hyperbolic persistence.

Moreover, the LSTM’s memory and gate structure enable augmentation beyond classical GARCH reach—amplifying the nonlinearity and depth of learned patterns.

4. Empirical Evaluation and Comparative Results

Empirical evaluation on synthetic and real financial datasets demonstrates that:

• NN counterparts of GARCH models attain near-identical parameter recovery fidelity as classic GARCH estimators.
• Out-of-sample volatility forecast accuracy (in MAE and MSE) is statistically indistinguishable between vanilla GARCH and NN GARCH for standard tasks.
• Incorporation of maximum likelihood-based losses (“N-loss”, “T-loss”) significantly improves volatility forecast performance relative to classic MSE objectives.

Most notably, the GARCH-LSTM hybrid achieves:

• 3% lower MAE and 10% lower MSE than the best alternative methods, across major real-world datasets and forecast horizons.
• Superior stability and sample efficiency.
• Robust violation rates in VaR estimation tasks (approx. 5%, closely matching theoretical expectations), with improved stability relative to standalone GARCH and NN models.

Component	Classic GARCH	NN Counterpart	In GARCH-LSTM
Model Structure	Linear recursion	Scalar RNN/CNN	GARCH kernel in output gate, LSTM cell modulates output
Parameters	$\omega, \alpha, \beta$	Same ($\omega, \alpha, \beta$)	Tuned by backpropagation
Stylized Facts	By design	By equivalence	By kernel selection
Forecast Output	$\sigma_t^2$	$\sigma_t^2$	LSTM-modulated $\sigma_t^2$

5. Interpretability, Generalization, and Practical Implications

A key feature of GARCH-LSTM is interpretability: since the NN is constructed from the GARCH kernel, all outputs and memory remain tied to well-understood econometric mechanisms. Selecting the kernel (GARCH, GJR-GARCH, FI-GARCH) tunes the stylized facts, while LSTM memory enables adaptation to market regimes, nonlinear effects, and structural breaks.

This hybrid approach, validated in direct empirical comparison, offers:

• Enhanced robustness in volatility and risk quantile forecasting.
• Stability across forecast horizons and market environments.
• An interpretable and technically sound alternative to both pure econometric and deep neural models.

6. Limitations and Further Directions

The current GARCH-LSTM implementation requires careful architectural selection (LSTM depth, mixing weights, kernel specification). While parameter recovery and stylized fact preservation are empirically robust, further improvement in loss function design (e.g., direct risk quantile targeting) and extension to multivariate or realized-volatility settings is plausible, as suggested by related research (Zhao et al., 2024).

Continued exploration of kernel modularity and gating (for additional stylized facts), as well as generalization to more complex asset classes, is warranted for practitioners seeking state-of-the-art volatility forecast accuracy and risk management reliability in evolving financial markets.

7. Summary and Conceptual Synthesis

GARCH-LSTM represents an interpretable, modular, and empirically validated approach to volatility forecasting. It leverages the rigor of GARCH-type models for stylized facts and the expressivity of LSTM networks for nonlinear and regime-dependent effects. The theoretical equivalence and modular architecture ensure that practitioners can construct models with domain-grounded interpretability and enhanced predictive performance, providing a technically robust framework for uncertainty quantification in financial economics (Zhao et al., 2024).