ARCH models in Python
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bashtage ENH: Ensure all ppf and cdf accept scalars
Add scalar path when missing
Add test for scalar path
Latest commit f0902e6 Nov 15, 2018


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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Module Contents


Released documentation is hosted on read the docs. Current documentation from the master branch is hosted on my github pages.

More about ARCH

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 process, e.g., FIGARCH
  • Improve docstrings where unclear or with typos
  • Provide examples, preferably in the form of IPython notebooks


Volatility Modeling

  • Mean models
    • Constant mean
    • Heterogeneous Autoregression (HAR)
    • Autoregression (AR)
    • Zero mean
    • Models with and without exogenous regressors
  • Volatility models
    • ARCH
    • GARCH
    • TARCH
    • EGARCH
    • EWMA/RiskMetrics
  • Distributions
    • Normal
    • Student's T
    • Generalized Error Distribution

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

import datetime as dt
import as web
st = dt.datetime(1990,1,1)
en = dt.datetime(2014,1,1)
data = web.get_data_yahoo('^FTSE', start=st, end=en)
returns = 100 * data['Adj Close'].pct_change().dropna()

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

Unit Root Tests

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

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


  • Bootstraps
    • IID Bootstrap
    • Stationary Bootstrap
    • Circular Block Bootstrap
    • Moving 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 datetime as dt
import pandas as pd
import as web
start = dt.datetime(1951,1,1)
end = dt.datetime(2014,1,1)
sp500 = web.get_data_yahoo('^GSPC', start=start, end=end)
start = sp500.index.min()
end = sp500.index.max()
monthly_dates = pd.date_range(start, end, freq='M')
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')

Multiple Comparison Procedures

  • 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.


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

  • Python (2.7, 3.5 - 3.7)
  • NumPy (1.13+)
  • SciPy (0.19+)
  • Pandas (0.21+)
  • statsmodels (0.8+)
  • matplotlib (2.0+), optional
  • cached-property (1.5.1+), optional

Optional Requirements

  • Numba (0.35+) will be used if available and when installed using the --no-binary option
  • jupyter and notebook are required to run the notebooks


Standard installation with a compiler requires Cython. If you do not have a compiler installed, the arch should still install. You will see a warning but this can be ignored. If you don't have a compiler, numba is strongly recommended.


Releases are available PyPI and can be installed with pip.

pip install arch

This command should work whether you have a compiler installed or not. If you want to install with the --no-binary options, use

pip install arch --install-option="--no-binary"

You can alternatively install the latest version from GitHub

pip install git+

--install-option="--no-binary" can be used to disable compilation of the extensions.


conda users can install from my channel,

conda install arch -c bashtage


Building extension using the community edition of Visual Studio is well supported for Python 3.5+. Building extensions for 64-bit Windows for use in Python 2.7 is also supported using Microsoft Visual C++ Compiler for Python 2.7. Building on other combinations of Python/Windows is more difficult and is not necessary when Numba is installed since just-in-time compiled code (Numba) runs as fast as ahead-of-time compiled extensions.


The development requirements are:

  • Cython (0.24+, if not using --no-binary)
  • py.test (For tests)
  • sphinx (to build docs)
  • guzzle_sphinx_theme (to build docs)
  • jupyter, notebook and nbsphinx (to build docs)

Installation Notes:

  1. If Cython is not installed, the package will be installed as-if --no-binary was used.
  2. Setup does not verify these requirements. Please ensure these are installed.