Introduction to ARCH Models¶

ARCH models are a popular class of volatility models that use observed values of returns or residuals as volatility shocks. A basic GARCH model is specified as

\begin{eqnarray} r_t & = & \mu + \epsilon_t \\ \epsilon_t & = & \sigma_t e_t \\ \sigma^2_t & = & \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma^2_{t-1} \end{eqnarray}

A complete ARCH model is divided into three components:

a mean model, e.g., a constant mean or an ARX;
a volatility process, e.g., a GARCH or an EGARCH process; and
a distribution for the standardized residuals.

In most applications, the simplest method to construct this model is to use the constructor function arch_model()

import datetime as dt

import pandas_datareader.data as web

from arch import arch_model

start = dt.datetime(2000, 1, 1)
end = dt.datetime(2014, 1, 1)
sp500 = web.DataReader('^GSPC', 'yahoo', start=start, end=end)
returns = 100 * sp500['Adj Close'].pct_change().dropna()
am = arch_model(returns)

Alternatively, the same model can be manually assembled from the building blocks of an ARCH model

from arch import ConstantMean, GARCH, Normal

am = ConstantMean(returns)
am.volatility = GARCH(1, 0, 1)
am.distribution = Normal()

In either case, model parameters are estimated using

res = am.fit()

with the following output

Iteration:      1,   Func. Count:      6,   Neg. LLF: 5159.58323938
Iteration:      2,   Func. Count:     16,   Neg. LLF: 5156.09760149
Iteration:      3,   Func. Count:     24,   Neg. LLF: 5152.29989336
Iteration:      4,   Func. Count:     31,   Neg. LLF: 5146.47531817
Iteration:      5,   Func. Count:     38,   Neg. LLF: 5143.86337547
Iteration:      6,   Func. Count:     45,   Neg. LLF: 5143.02096168
Iteration:      7,   Func. Count:     52,   Neg. LLF: 5142.24105141
Iteration:      8,   Func. Count:     60,   Neg. LLF: 5142.07138907
Iteration:      9,   Func. Count:     67,   Neg. LLF: 5141.416653
Iteration:     10,   Func. Count:     73,   Neg. LLF: 5141.39212288
Iteration:     11,   Func. Count:     79,   Neg. LLF: 5141.39023885
Iteration:     12,   Func. Count:     85,   Neg. LLF: 5141.39023359
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 5141.39023359
            Iterations: 12
            Function evaluations: 85
            Gradient evaluations: 12

print(res.summary())

yields

                     Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable:              Adj Close   R-squared:                      -0.001
Mean Model:             Constant Mean   Adj. R-squared:                 -0.001
Vol Model:                      GARCH   Log-Likelihood:               -5141.39
Distribution:                  Normal   AIC:                           10290.8
Method:            Maximum Likelihood   BIC:                           10315.4
                                        No. Observations:                 3520
Date:                Fri, Dec 02 2016   Df Residuals:                     3516
Time:                        22:22:28   Df Model:                            4
                                  Mean Model
==============================================================================
                 coef    std err          t      P>|t|        95.0% Conf. Int.
------------------------------------------------------------------------------
mu             0.0531  1.487e-02      3.569  3.581e-04   [2.392e-02,8.220e-02]
                               Volatility Model
==============================================================================
                 coef    std err          t      P>|t|        95.0% Conf. Int.
------------------------------------------------------------------------------
omega          0.0156  4.932e-03      3.155  1.606e-03   [5.892e-03,2.523e-02]
alpha[1]       0.0879  1.140e-02      7.710  1.260e-14     [6.554e-02,  0.110]
beta[1]        0.9014  1.183e-02     76.163      0.000       [  0.878,  0.925]
==============================================================================

Covariance estimator: robust

Model Constructor¶

While models can be carefully specified using the individual components, most common specifications can be specified using a simple model constructor.

Initialization of common ARCH model specifications

Parameters:¶

y: ndarray | DataFrame | Series | None¶

The dependent variable

x: ndarray | DataFrame | None = None¶

Exogenous regressors. Ignored if model does not permit exogenous regressors.

mean: 'Constant' | 'Zero' | 'LS' | 'AR' | 'ARX' | 'HAR' | 'HARX' | 'constant' | 'zero' = 'Constant'¶

Name of the mean model. Currently supported options are: ‘Constant’, ‘Zero’, ‘LS’, ‘AR’, ‘ARX’, ‘HAR’ and ‘HARX’

lags: int | list[int] | ndarray[Any, dtype[int32]] | ndarray[Any, dtype[longlong]] | None = 0¶

Either a scalar integer value indicating lag length or a list of integers specifying lag locations.

Name of the volatility model. Currently supported options are: ‘GARCH’ (default), ‘ARCH’, ‘EGARCH’, ‘FIGARCH’, ‘APARCH’ and ‘HARCH’

p: int | list[int] = 1¶

Lag order of the symmetric innovation

o: int = 0¶

Lag order of the asymmetric innovation

q: int = 1¶

Lag order of lagged volatility or equivalent

power: float = 2.0¶

Power to use with GARCH and related models

Name of the error distribution. Currently supported options are:

Normal: ‘normal’, ‘gaussian’ (default)

Students’s t: ‘t’, ‘studentst’

Skewed Student’s t: ‘skewstudent’, ‘skewt’

Generalized Error Distribution: ‘ged’, ‘generalized error”

hold_back: int | None = None¶

Number of observations at the start of the sample to exclude when estimating model parameters. Used when comparing models with different lag lengths to estimate on the common sample.

rescale: bool | None = None¶

Flag indicating whether to automatically rescale data if the scale of the data is likely to produce convergence issues when estimating model parameters. If False, the model is estimated on the data without transformation. If True, than y is rescaled and the new scale is reported in the estimation results.

Returns:¶

model – Configured ARCH model

Return type:¶

arch.univariate.base.ARCHModel

Examples

>>> import datetime as dt
>>> import pandas_datareader.data as web
>>> djia = web.get_data_fred('DJIA')
>>> returns = 100 * djia['DJIA'].pct_change().dropna()

A basic GARCH(1,1) with a constant mean can be constructed using only the return data

>>> from arch.univariate import arch_model
>>> am = arch_model(returns)

Alternative mean and volatility processes can be directly specified

>>> am = arch_model(returns, mean='AR', lags=2, vol='harch', p=[1, 5, 22])

This example demonstrates the construction of a zero mean process with a TARCH volatility process and Student t error distribution

>>> am = arch_model(returns, mean='zero', p=1, o=1, q=1,
...                 power=1.0, dist='StudentsT')

Notes

Input that are not relevant for a particular specification, such as lags when mean=’zero’, are silently ignored.