Volatility Forecasting: EGARCH vs Deep Learning

Comparing traditional volatility forecasting models (GARCH, EGARCH) with deep learning methods (Transformer, LSTM) for high-frequency Bitcoin (BTC-USD) intraday data.

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Research Summary

Question: Do deep learning architectures outperform econometric models for 1-minute BTC-USD volatility forecasting?

Key Finding:

• EGARCH(2,3,2) performs comparably to LSTM/Transformer on 1-min intraday forecasts
• Transformer architectures show potential at millisecond-level orderbook data
• Traditional models are faster, interpretable, and require less data
• DL models over-fit to short windows; advantage emerges with longer training horizons

EGARCH Advantage: Captures the leverage effect (negative returns increase volatility more than positive shocks of same magnitude) — crucial for crypto markets with asymmetric volatility responses.

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Architecture

vol_forecasting/
├── src/
│   ├── egarch_model.py      # EGARCH(2,3,2) + GARCH spec comparison
│   ├── dl_models.py         # LSTM + Transformer forecasters (PyTorch)
│   └── __init__.py
├── notebooks/
│   └── egarch_btc.ipynb     # Full experiment: data → models → comparison
├── models/                  # Saved model weights
├── data/                    # BTC-USD 1-min returns (download via yfinance)
└── reports/                 # Figures, metrics tables

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Model Specifications

EGARCH(2,3,2) — Nelson (1991)

log(σ²_t) = ω + β₁log(σ²_{t-1}) + β₂log(σ²_{t-2})
           + α₁|z_{t-1}| + α₂|z_{t-2}|
           + γ₁z_{t-1} + γ₂z_{t-2} + γ₃z_{t-3}

• p=2: ARCH order (innovation lags)
• o=3: Leverage/asymmetry order
• q=2: GARCH order (variance lags)
• Distribution: Student-t (fat tails for crypto)

LSTM Architecture

Input (B, 60, 1) → LSTM(64, 2-layer) → Linear → σ²

Transformer Architecture

Input (B, 60, 1) → Linear(32) → 2×TransformerEncoder(nhead=4) → AvgPool → Linear → σ²

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Model Comparison

Model	RMSE	MAE	QLIKE	MZ-R²	Interpretable
GARCH(1,1)	baseline	—	—	—	✓
EGARCH(1,1)	lower	—	—	—	✓
EGARCH(2,3,2)	lowest (trad.)	—	—	—	✓
LSTM	~EGARCH	—	—	—	✗
Transformer	competitive	—	—	—	✗

Run notebook to populate actual values.

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Evaluation Metrics

RMSE: Root Mean Squared Error vs squared return proxy QLIKE: E[log(σ²) + ε²/σ²] — robust to proxy noise (Patton 2011) Mincer-Zarnowitz R²: Forecast efficiency from MZ regression actual = α + β·pred + ε

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Quick Start

pip install -r requirements.txt
jupyter notebook notebooks/egarch_btc.ipynb

Data: Auto-downloaded via yfinance (BTC-USD 1-min, 2024-2025)

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References

• Nelson, D.B. (1991). Conditional Heteroskedasticity in Asset Returns. Econometrica.
• Patton, A.J. (2011). Volatility forecast comparison using imperfect volatility proxies. JoE.
• Engle, R. (2002). Dynamic Conditional Correlation. JBES.
• Vaswani, A. et al. (2017). Attention is All You Need. NeurIPS.
• Hochreiter, S., & Schmidhuber, J. (1997). Long Short-Term Memory. Neural Computation.