GARCH(1,1) with GED Innovations

A fast C++ implementation of GARCH(1,1) model with Generalized Error Distribution (GED) innovations, with Python bindings via pybind11.

Features

• Fast C++ Implementation: Core algorithms implemented in C++ for optimal performance
• GED Innovations: Support for Generalized Error Distribution for heavy-tailed financial returns
• Easy Python Interface: Simple and intuitive Python API similar to popular packages like arch
• Comprehensive Output: Get fitted volatilities, residuals, forecasts, and model statistics
• Parameter Validation: Built-in parameter constraints and stationarity checks

Installation

From PyPI (recommended)

pip install garch11_ged

From Source

git clone https://github.com/yourusername/garch11_ged.git
cd garch11_ged
pip install .

Development Installation

git clone https://github.com/yourusername/garch11_ged.git
cd garch11_ged
pip install -e ".[dev]"

Quick Start

import numpy as np
import garch11_ged

# Generate sample return data
np.random.seed(42)
returns = np.random.normal(0, 0.01, 1000)

# Fit GARCH(1,1) model with GED innovations
model = garch11_ged.GARCH11_GED()
model.fit(returns)

# Print model summary
print(model.summary())

# Get parameter estimates
params = model.parameters
print(f"Parameters: {params}")

# Get fitted volatilities
volatilities = model.volatilities

# Forecast volatility
forecast = model.predict_volatility(horizon=10)
print(f"10-step forecast: {forecast}")

API Reference

GARCH11_GED Class

Constructor

model = garch11_ged.GARCH11_GED()
model = garch11_ged.GARCH11_GED(mu, omega, alpha, beta, nu)

Methods

• fit(returns, max_iter=1000, tolerance=1e-6): Fit the model to data
• predict_volatility(horizon=1): Forecast volatility
• get_parameters(): Get parameter estimates [mu, omega, alpha, beta, nu]
• get_volatilities(): Get fitted conditional volatilities
• get_residuals(): Get fitted residuals
• get_standardized_residuals(): Get standardized residuals
• get_log_likelihood(): Get log-likelihood value
• get_aic(): Get AIC (Akaike Information Criterion)
• get_bic(): Get BIC (Bayesian Information Criterion)
• summary(): Get detailed model summary

Properties

• parameters: Model parameters
• volatilities: Fitted volatilities
• residuals: Fitted residuals
• standardized_residuals: Standardized residuals
• log_likelihood: Log-likelihood value
• aic: AIC value
• bic: BIC value
• fitted: Whether model is fitted

Convenience Functions

# Fit model in one line
model = garch11_ged.fit_garch(returns)

Model Specification

The GARCH(1,1) model with GED innovations is specified as:

Mean Equation:

r_t = μ + ε_t

Variance Equation:

σ²_t = ω + α·ε²_{t-1} + β·σ²_{t-1}

Innovation Distribution:

ε_t | F_{t-1} ~ GED(0, σ_t, ν)

Where:

• μ is the mean return
• ω > 0 is the constant term
• α ≥ 0 is the ARCH parameter
• β ≥ 0 is the GARCH parameter
• α + β < 1 for stationarity
• ν > 0 is the GED shape parameter

Performance

This implementation is significantly faster than pure Python implementations:

import time
import numpy as np

# Generate large dataset
returns = np.random.normal(0, 0.01, 10000)

# Time the fitting
start_time = time.time()
model = garch11_ged.GARCH11_GED()
model.fit(returns)
end_time = time.time()

print(f"Fitting time: {end_time - start_time:.3f} seconds")

Requirements

• Python ≥ 3.7
• NumPy ≥ 1.17.0
• C++14 compatible compiler

License

This project is licensed under the MIT License - see the LICENSE file for details.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

Citation

If you use this package in your research, please cite:

@software{garch11_ged,
  title={GARCH(1,1) with GED Innovations},
  author={Your Name},
  year={2024},
  url={https://github.com/yourusername/garch11_ged}
}

Acknowledgments

This implementation is based on the GARCH modeling framework from the rugarch R package by Alexios Ghalanos.