Autoregressive Conditional Heteroskedasticity (ARCH) and other tools for financial econometrics, written in Python (with Cython and/or Numba used to improve performance)

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• Univariate ARCH Models
• Unit Root Tests
• Cointegration Testing and Analysis
• Bootstrapping
• Multiple Comparison Tests
• Long-run Covariance Estimation

arch is Python 3 only. Version 4.8 is the final version that supported Python 2.7.

Documentation for the latest release is hosted on GitHub Pages, and documentation from the main branch is at bashtage.github.io/arch/devel.

More information about ARCH and related models is available in the notes and research available at Kevin Sheppard's site.

Contributions are welcome. There are opportunities at many levels to contribute:

• Implement new volatility processes, distributions or tests
• Improve docstrings where unclear or with typos
• Provide examples, preferably in the form of IPython notebooks

• Mean models
  • Constant mean
  • Heterogeneous Autoregression (HAR)
  • Autoregression (AR)
  • Moving Average (MA)
  • Zero mean
  • ARCH-in-mean
  • Models with and without exogenous regressors
• Volatility models
  • ARCH
  • GARCH
  • TARCH
  • EGARCH
  • APARCH
  • NGARCH
  • FIGARCH and FIAPARCH
  • HARCH
  • MIDAS Hyperbolic
  • EWMA/RiskMetrics and RiskMetrics 2006
• Distributions
  • Normal
  • Student's T
  • Skewed Student's T
  • Generalized Error Distribution
• Diagnostics
  • Residual diagnostics: excess kurtosis, Hill estimator and VaR ratio
  • VaR backtests: Kupiec, Christoffersen and the Basel traffic light

See the univariate volatility example notebook for a more complete overview.

import arch.data.sp500
data = arch.data.sp500.load()
returns = 100 * data['Adj Close'].pct_change().dropna()

from arch import arch_model
am = arch_model(returns)
res = am.fit()

• Augmented Dickey-Fuller
• Dickey-Fuller GLS
• Phillips-Perron
• KPSS
• Zivot-Andrews
• Variance Ratio tests

See the unit root testing example notebook for examples of testing series for unit roots.

• Tests
  • Engle-Granger Test
  • Phillips-Ouliaris Test
• Cointegration Vector Estimation
  • Canonical Cointegrating Regression
  • Dynamic OLS
  • Fully Modified OLS

See the cointegration testing example notebook for examples of testing series for cointegration.

• Bootstraps
  • IID Bootstrap
  • Stationary Bootstrap
  • Circular Block Bootstrap
  • Moving Block Bootstrap
  • Local Block Bootstrap
• Methods
  • Confidence interval construction
  • Covariance estimation
  • Apply method to estimate model across bootstraps
  • Generic Bootstrap iterator

See the bootstrap example notebook for examples of bootstrapping the Sharpe ratio and a Probit model from statsmodels.

# Import data
import pandas as pd
import numpy as np
import arch.data.sp500
sp500 = arch.data.sp500.load()
start = sp500.index.min()
end = sp500.index.max()
monthly_dates = pd.date_range(start, end, freq='ME')
monthly = sp500.reindex(monthly_dates, method='ffill')
returns = 100 * monthly['Adj Close'].pct_change().dropna()

# Function to compute parameters
def sharpe_ratio(x):
    mu, sigma = 12 * x.mean(), np.sqrt(12 * x.var())
    return np.array([mu, sigma, mu / sigma])

# Bootstrap confidence intervals
from arch.bootstrap import IIDBootstrap
bs = IIDBootstrap(returns)
ci = bs.conf_int(sharpe_ratio, 1000, method='percentile')

• Test of Superior Predictive Ability (SPA), also known as the Reality Check or Bootstrap Data Snooper
• Stepwise (StepM)
• Model Confidence Set (MCS)

See the multiple comparison example notebook for examples of the multiple comparison procedures.

Kernel-based estimators of long-run covariance including the Bartlett kernel which is known as Newey-West in econometrics. Automatic bandwidth selection, using the methods of Newey and West (1994) or Andrews (1991), is available for all of the covariance estimators. VAR-HAC and pre-whitened recolored estimators are also available.

from arch.covariance.kernel import Bartlett
from arch.data import nasdaq
data = nasdaq.load()
returns = data[["Adj Close"]].pct_change().dropna()

cov_est = Bartlett(returns ** 2)
# Get the long-run covariance
cov_est.cov.long_run

These requirements reflect the testing environment. It is possible that arch will work with older versions.

• Python (3.10-3.15)
• NumPy (1.22.3+)
• SciPy (1.8+)
• pandas (1.4+)
• statsmodels (0.13.1+)
• matplotlib (3+), optional

arch 9.0 is the last release that supports Python 3.10 and NumPy 1.x. Later releases will raise the other minimum versions accordingly.

• Numba (0.55+) will be used if available and when arch is installed without building the binary modules. To make sure that they are not built, install from the source distribution with the no-binary build option:

python -m pip install arch --no-binary arch -Csetup-args="-Dno-binary=true"

• jupyter and notebook are required to run the notebooks

Wheels are available for Linux, macOS and Windows, so installing with pip does not need a compiler. Building from source needs a C compiler; pip installs the build requirements (Cython, meson-python and NumPy) automatically.

Releases are available PyPI and can be installed with pip.

pip install arch

You can alternatively install the latest version from GitHub

pip install git+https://github.com/bashtage/arch.git

Adding -Csetup-args="-Dno-binary=true" to the pip install command disables compilation of the extensions.

conda users can install from conda-forge,

conda install arch-py -c conda-forge

Note: The conda-forge name is arch-py.

Building the extensions from source needs the Visual Studio Build Tools (the free community edition is enough).

The development requirements are dependency groups in pyproject.toml, which pip 25.1 or later installs. From a clone of the repository,

python -m pip install --group dev
python -m pip install -e . --no-build-isolation

installs the tools to build (Cython 3.1+, meson-python and ninja), test (pytest 9+) and lint arch, and then arch itself. Adding --group doc to the first command also installs what the documentation needs (sphinx, sphinx-immaterial and nbsphinx). doc/source/contributing.rst lists the commands that check the code.

1. Building without the binary modules (-Csetup-args="-Dno-binary=true") still needs a C compiler and Cython, since the build checks for them. The recursions then run in numba if it is installed, and in pure Python otherwise, which is much slower.