GARCH-NN Equivalence in Volatility Forecasting

• The paper demonstrates that a vanilla RNN without nonlinearity can exactly replicate the GARCH(1,1) model, providing a formal equivalence between statistical and neural approaches.
• It introduces dynamic neural parameterization that generalizes classical GARCH models, enabling adaptive modeling of regime shifts and time-varying volatility in financial data.
• Empirical results show that hybrid GARCH-NN models outperform traditional GARCH benchmarks in forecasting accuracy and risk metric validations.

GARCH-NN equivalence describes the relationship between generalized autoregressive conditional heteroskedasticity (GARCH) models—a cornerstone of volatility modeling in econometrics—and neural network (NN) architectures, particularly in the context of volatility forecasting for financial time series. Recent research demonstrates precise conditions where specific neural network structures can be constructed to be mathematically equivalent to GARCH-type processes, while also laying the foundation for a unified modeling paradigm that leverages both the statistical properties of GARCH and the flexibility of NNs.

1. Formal Equivalence Between GARCH and Neural Networks

The crux of the GARCH-NN equivalence is that the recursive (autoregressive) structure central to GARCH models can be instantiated identically as a neural network, typically an RNN cell without nonlinearity. For the canonical GARCH(1,1) process:

$$\begin{aligned} r_t &= \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{D}(0, \sigma_t^2) \ \sigma_t^2 &= \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{aligned}$$

This update is equivalent to an RNN update:

$h_t = \omega + \alpha x_{t-1} + \beta h_{t-1}$

where the RNN "hidden state" $h_t$ is the conditional variance, $x_{t-1}$ is innovation squared, and the weights $(\omega, \alpha, \beta)$ directly correspond to GARCH parameters, provided no activation is used. Thus, a vanilla RNN with scalar hidden states, linear update, and appropriate inputs is structurally identical to GARCH(1,1) (Zhao et al., 29 Jan 2024).

Other GARCH variants map similarly:

• GJR-GARCH: RNN cell with extra input for a sign-based indicator (leverage effect).
• FI-GARCH: 1D CNN with fractional decaying coefficients as convolutional kernels to mimic long memory.

This formalizes the claim: if the NN weights, structure, and training loss (i.e., the conditional MLE-based likelihood used in GARCH estimation) are identical, the NN is a "GARCH in disguise" (Zhao et al., 29 Jan 2024).

2. Generalizing GARCH via Neural Parameterization

The equivalence is not the end, but a baseline. Neural GARCH models (Yin et al., 2022) generalize the GARCH formulation by allowing the parameters $(\omega_t, \alpha_t, \beta_t)$ to vary over time, dynamically generated by a neural network, typically an RNN (GRU or LSTM):

$$\sigma_t^2 = \omega_t + \alpha_t r_{t-1}^2 + \beta_t \sigma_{t-1}^2, \quad (\omega_t, \alpha_t, \beta_t) = NN(\mathrm{history}; \theta)$$

In this construction:

• Classical GARCH is a special case, recovered when the NN outputs constant parameters.
• Dynamic parameterization grants the ability to model nonstationarity, regime shifts, and evolving clustering in financial time series, which stationary GARCH cannot (Yin et al., 2022).

Variational inference (via a sequence VAE or VRNN) is typically used for parameter estimation when parameters are latent and time-varying (see Section 4 below).

3. Stylized Facts, Model Structure, and Embedding in NNs

The GARCH-NN equivalence underlines that stylized facts central to GARCH—volatility clustering, leverage, persistence, and long memory—can be embedded architecturally in neural nets:

• GARCH/short memory: scalar RNN
• GJR-GARCH/asymmetry: RNN with sign indicators
• FI-GARCH/long memory: shallow 1D CNN with decaying kernel

The "GARCH-NN" methodology proposed in (Zhao et al., 29 Jan 2024) constructs hybrid models, e.g., GARCH-LSTM, where the NN counterpart of a GARCH process is used as a kernel within an LSTM architecture, enforcing stylized financial priors within a larger NN framework.

Table: Model Construction Mode

Stylized Fact	GARCH Model	NN Equivalent
Clustering	GARCH(1,1)	Scalar RNN
Leverage	GJR-GARCH (indicator input)	RNN w/ sign-modulated in
Long Memory	FI-GARCH (frac diff.)	1D CNN (decay kernel)

4. Training, Inference, and Implementation

For strict GARCH-NN equivalence, it is integral to use the likelihood-based loss function exploited in GARCH MLE (e.g., log-likelihood for Gaussian or Student’s t innovations), rather than plain MSE. For example, the negative log-likelihood for Gaussian innovations is:

$$-\log \mathcal{L}(\epsilon_t) = \frac{1}{2} \log \hat{\sigma}_t^2 + \frac{\epsilon_t^2}{2\hat{\sigma}_t^2}.$$

Neural GARCH models with time-varying parameters (Yin et al., 2022) use a latent variable approach within a variational autoencoder (VAE) or VRNN framework. The variational posterior for the time-varying coefficients is inferred with the evidence lower bound (ELBO):

$$\mathrm{ELBO}(\theta, \phi) = \sum_{t=1}^{T}\mathbb{E}_{\gamma_{t} \sim q_\phi} [\log P_\theta(\mathbf{r}_t | \boldsymbol{\gamma}_t)] - KL(q_\phi(\boldsymbol{\gamma}_t|\cdot)\,\|\,P_\theta(\boldsymbol{\gamma}_t|\cdot)),$$

where the generative and inference networks are RNN-MLP stacks parameterizing the prior/posterior of the GARCH coefficients.

Choice of innovations (Gaussian vs. Student’s t) is encoded both in classical and NN architectures for robust handling of heavy tails.

5. Empirical Results and Comparative Performance

Empirical studies validate that NN architectures with embedded GARCH equivalents (e.g., GARCH-LSTM (Zhao et al., 29 Jan 2024), Neural GARCH (Yin et al., 2022)) provide forecasting accuracy at least as strong as classical GARCH—and usually significantly higher and more robust than both classic GARCH and unconstrained deep learning models (LSTM, Transformer). Notable empirical features:

• NN models with GARCH-like structure reliably match GARCH in simulation parameter recovery.
• In real financial returns data, GARCH-LSTM or Neural GARCH with dynamic coefficients outperforms baselines, notably in log-likelihood-based and risk-metric-based (VaR) validations across multiple assets, horizons, and data conditions.
• Smooth and interpretable GARCH-NN parameters maintain self-regularization (e.g., $\alpha_t + \beta_t < 1$), adaptively tracking regime changes without explicit model restarts or reparametrization (Yin et al., 2022).

6. Limitations, Interpretation, and Theoretical Boundaries

The GARCH-NN equivalence is exact when the neural network has the same architecture, fixed weights, and training loss as GARCH, and is sufficiently constrained (i.e., no extraneous nonlinearity or degrees of freedom). In practical settings:

• If the NN is unconstrained, it can perform strictly more general, nonlinear, or non-parametric modeling, but may overfit or lose model interpretability.
• Hybrid models such as GARCH-Informed Neural Networks (GINN) (Xu et al., 30 Sep 2024) introduce regularization terms in the loss that enforce closeness to GARCH while permitting the NN to capture additional structure. Strict equivalence disappears, but these hybrids often dominate both pure approaches empirically.
• A plausible implication is that GARCH’s domain priors should be embedded not solely via feature engineering but in model structure or regularization, to harness both the statistical properties of GARCH and the flexibility of NNs.

7. Applications and Extensions

Beyond volatility prediction, the GARCH-NN equivalence facilitates:

• Model interpretability and trust for practitioners through transparent linkages to econometric models.
• Modular hybrid design: arbitrary GARCH kernels can be plugged into LSTM and Transformer architectures, enabling volatility clustering, leverage, and long-memory to be captured natively.
• Efficient calibration: e.g., for option pricing, ANN calibration methods can act as universal approximators of GARCH-type price models, vastly accelerating calibration processes compared to brute-force simulation (Kim et al., 2023).
• Multivariate extensions: Neural GARCH can generalize to the diagonal BEKK structure in multivariate returns, permitting time-varying, data-driven modeling of covariance matrices.

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Summary:

GARCH-NN equivalence encompasses both formal, structural mappings (GARCH as a special RNN/CNN case) and generalizations (Neural GARCH with dynamic, history-adaptive parameters). The approach enables NNs to replicate, extend, and regularize GARCH-type volatility modeling, bridging statistical theory and machine learning through architectural and loss-based integration. Empirical evidence consistently supports the hybrid methodology as yielding superior, more robust forecasts in financial volatility estimation, while maintaining interpretability and statistical fidelity (Yin et al., 2022, Zhao et al., 29 Jan 2024, Xu et al., 30 Sep 2024).