01- Usage of the ARCH package#
This setup code is required to run in an IPython notebook
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_style("darkgrid")
plt.rc("figure", figsize=(16, 6))
plt.rc("savefig", dpi=90)
plt.rc("font", family="sans-serif")
plt.rc("font", size=14)
Data#
These examples make use of S&P 500 data from Yahoo! that is available from arch.data.sp500.
import datetime as dt
import sys
import arch.data.sp500
import numpy as np
import pandas as pd
from arch import arch_model
data = arch.data.sp500.load()
market = data["Adj Close"]
returns = 100 * market.pct_change().dropna()
returns
Date
1999-01-05 1.358200
1999-01-06 2.214041
1999-01-07 -0.205133
1999-01-08 0.422136
1999-01-11 -0.879151
...
2018-12-24 -2.711225
2018-12-26 4.959374
2018-12-27 0.856268
2018-12-28 -0.124158
2018-12-31 0.849248
Name: Adj Close, Length: 5030, dtype: float64
returns.plot()
<Axes: xlabel='Date'>
arch_model?
Signature:
arch_model(
y: 'ArrayLike | None',
x: 'ArrayLike2D | None' = None,
mean: "Literal['Constant', 'Zero', 'LS', 'AR', 'ARX', 'HAR', 'HARX', 'constant', 'zero']" = 'Constant',
lags: 'None | int | list[int] | Int32Array | Int64Array' = 0,
vol: "Literal['GARCH', 'ARCH', 'EGARCH', 'FIGARCH', 'APARCH', 'HARCH', 'FIGARCH']" = 'GARCH',
p: 'int | list[int]' = 1,
o: 'int' = 0,
q: 'int' = 1,
power: 'float' = 2.0,
dist: "Literal['normal', 'gaussian', 't', 'studentst', 'skewstudent', 'skewt', 'ged', 'generalized error']" = 'normal',
hold_back: 'int | None' = None,
rescale: 'bool | None' = None,
) -> 'HARX'
Docstring:
Initialization of common ARCH model specifications
Parameters
----------
y : {ndarray, Series, None}
The dependent variable
x : {np.array, DataFrame}, optional
Exogenous regressors. Ignored if model does not permit exogenous
regressors.
mean : str, optional
Name of the mean model. Currently supported options are: 'Constant',
'Zero', 'LS', 'AR', 'ARX', 'HAR' and 'HARX'
lags : int or list (int), optional
Either a scalar integer value indicating lag length or a list of
integers specifying lag locations.
vol : str, optional
Name of the volatility model. Currently supported options are:
'GARCH' (default), 'ARCH', 'EGARCH', 'FIGARCH', 'APARCH' and 'HARCH'
p : int, optional
Lag order of the symmetric innovation
o : int, optional
Lag order of the asymmetric innovation
q : int, optional
Lag order of lagged volatility or equivalent
power : float, optional
Power to use with GARCH and related models
dist : int, optional
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
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
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 : ARCHModel
Configured ARCH model
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.
File: ~/Documents/Projects/STI_FX_Intervention/.venv/lib/python3.9/site-packages/arch/univariate/mean.py
Type: function
Does anything beat GARCH(1,1)#
am = arch_model(returns, vol="Garch", p=1, o=0, q=1, dist="Normal")
res = am.fit(update_freq=5)
Iteration: 5, Func. Count: 35, Neg. LLF: 6970.277818445006
Iteration: 10, Func. Count: 63, Neg. LLF: 6936.718477482898
Optimization terminated successfully (Exit mode 0)
Current function value: 6936.718476989025
Iterations: 11
Function evaluations: 68
Gradient evaluations: 11
print(res.summary())
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Adj Close R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -6936.72
Distribution: Normal AIC: 13881.4
Method: Maximum Likelihood BIC: 13907.5
No. Observations: 5030
Date: Tue, Apr 18 2023 Df Residuals: 5029
Time: 14:30:29 Df Model: 1
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0564 1.149e-02 4.906 9.302e-07 [3.384e-02,7.887e-02]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
omega 0.0175 4.683e-03 3.738 1.854e-04 [8.328e-03,2.669e-02]
alpha[1] 0.1022 1.301e-02 7.852 4.105e-15 [7.665e-02, 0.128]
beta[1] 0.8852 1.380e-02 64.125 0.000 [ 0.858, 0.912]
============================================================================
Covariance estimator: robust
type(res)
arch.univariate.base.ARCHModelResult
am.fit?
Signature:
am.fit(
update_freq: 'int' = 1,
disp: "bool | Literal['off', 'final']" = 'final',
starting_values: 'ArrayLike1D | None' = None,
cov_type: "Literal['robust', 'classic']" = 'robust',
show_warning: 'bool' = True,
first_obs: 'int | DateLike | None' = None,
last_obs: 'int | DateLike | None' = None,
tol: 'float | None' = None,
options: 'dict[str, Any] | None' = None,
backcast: 'None | float | Float64Array' = None,
) -> 'ARCHModelResult'
Docstring:
Estimate model parameters
Parameters
----------
update_freq : int, optional
Frequency of iteration updates. Output is generated every
`update_freq` iterations. Set to 0 to disable iterative output.
disp : {bool, "off", "final"}
Either 'final' to print optimization result or 'off' to display
nothing. If using a boolean, False is "off" and True is "final"
starting_values : ndarray, optional
Array of starting values to use. If not provided, starting values
are constructed by the model components.
cov_type : str, optional
Estimation method of parameter covariance. Supported options are
'robust', which does not assume the Information Matrix Equality
holds and 'classic' which does. In the ARCH literature, 'robust'
corresponds to Bollerslev-Wooldridge covariance estimator.
show_warning : bool, optional
Flag indicating whether convergence warnings should be shown.
first_obs : {int, str, datetime, Timestamp}
First observation to use when estimating model
last_obs : {int, str, datetime, Timestamp}
Last observation to use when estimating model
tol : float, optional
Tolerance for termination.
options : dict, optional
Options to pass to `scipy.optimize.minimize`. Valid entries
include 'ftol', 'eps', 'disp', and 'maxiter'.
backcast : {float, ndarray}, optional
Value to use as backcast. Should be measure :math:`\sigma^2_0`
since model-specific non-linear transformations are applied to
value before computing the variance recursions.
Returns
-------
results : ARCHModelResult
Object containing model results
Notes
-----
A ConvergenceWarning is raised if SciPy's optimizer indicates
difficulty finding the optimum.
Parameters are optimized using SLSQP.
File: ~/Documents/Projects/STI_FX_Intervention/.venv/lib/python3.9/site-packages/arch/univariate/base.py
Type: method
Exercise:#
Change the code above to fit a zeromean model
Basic Forecasting#
Forecasts can be generated for standard GARCH(p,q) processes using any of the three forecast generation methods:
Analytical
Simulation-based
Bootstrap-based
Be default forecasts will only be produced for the final observation in the sample so that they are out-of-sample.
Forecasts start with specifying the model and estimating parameters.
forecasts = res.forecast(reindex=False)
Forecasts are contained in an ARCHModelForecast object which has 4 attributes:
mean- The forecast meansresidual_variance- The forecast residual variances, that is \(E_t[\epsilon_{t+h}^2]\)variance- The forecast variance of the process, \(E_t[r_{t+h}^2]\). The variance will differ from the residual variance whenever the model has mean dynamics, e.g., in an AR process.simulations- An object that contains detailed information about the simulations used to generate forecasts. Only used if the forecastmethodis set to'simulation'or'bootstrap'. If using'analytical'(the default), this isNone.
The three main outputs are all returned in DataFrames with columns of the form h.# where # is the number of steps ahead. That is, h.1 corresponds to one-step ahead forecasts while h.10 corresponds to 10-steps ahead.
The default forecast only produces 1-step ahead forecasts.
print(forecasts.mean.iloc[-3:])
print(forecasts.variance.iloc[-3:])
h.1
Date
2018-12-31 0.056353
h.1
Date
2018-12-31 3.59647
Longer horizon forecasts can be computed by passing the parameter horizon.
forecasts = res.forecast(horizon=5, reindex=False)
print(forecasts.mean.iloc[-3:])
print(forecasts.variance.iloc[-3:])
h.1 h.2 h.3 h.4 h.5
Date
2018-12-31 0.056353 0.056353 0.056353 0.056353 0.056353
h.1 h.2 h.3 h.4 h.5
Date
2018-12-31 3.59647 3.568502 3.540887 3.513621 3.4867
If you fail to set reindex you will see a warning.
forecasts = res.forecast(horizon=5, reindex=True)
forecasts.mean
| h.1 | h.2 | h.3 | h.4 | h.5 | |
|---|---|---|---|---|---|
| Date | |||||
| 1999-01-05 | NaN | NaN | NaN | NaN | NaN |
| 1999-01-06 | NaN | NaN | NaN | NaN | NaN |
| 1999-01-07 | NaN | NaN | NaN | NaN | NaN |
| 1999-01-08 | NaN | NaN | NaN | NaN | NaN |
| 1999-01-11 | NaN | NaN | NaN | NaN | NaN |
| ... | ... | ... | ... | ... | ... |
| 2018-12-24 | NaN | NaN | NaN | NaN | NaN |
| 2018-12-26 | NaN | NaN | NaN | NaN | NaN |
| 2018-12-27 | NaN | NaN | NaN | NaN | NaN |
| 2018-12-28 | NaN | NaN | NaN | NaN | NaN |
| 2018-12-31 | 0.056353 | 0.056353 | 0.056353 | 0.056353 | 0.056353 |
5030 rows × 5 columns
When not specified, or if reindex is True, then values that are not computed are nan-filled.
print(forecasts.residual_variance.iloc[-3:])
h.1 h.2 h.3 h.4 h.5
Date
2018-12-27 NaN NaN NaN NaN NaN
2018-12-28 NaN NaN NaN NaN NaN
2018-12-31 3.59647 3.568502 3.540887 3.513621 3.4867
Alternative Forecast Generation Schemes#
Fixed Window Forecasting#
Fixed-windows forecasting uses data up to a specified date to generate all forecasts after that date. This can be implemented by passing the entire data in when initializing the model and then using last_obs when calling fit. forecast() will, by default, produce forecasts after this final date.
Note last_obs follow Python sequence rules so that the actual date in last_obs is not in the sample.
res = am.fit(last_obs="2011-1-1", update_freq=5)
forecasts = res.forecast(horizon=5, reindex=False)
print(forecasts.variance.dropna().head())
Iteration: 5, Func. Count: 34, Neg. LLF: 4578.713304790317
Iteration: 10, Func. Count: 63, Neg. LLF: 4555.338449449297
Optimization terminated successfully (Exit mode 0)
Current function value: 4555.285110045399
Iterations: 14
Function evaluations: 83
Gradient evaluations: 14
h.1 h.2 h.3 h.4 h.5
Date
2010-12-31 0.381757 0.390905 0.399988 0.409008 0.417964
2011-01-03 0.451724 0.460381 0.468976 0.477512 0.485987
2011-01-04 0.428416 0.437236 0.445994 0.454691 0.463326
2011-01-05 0.420554 0.429429 0.438242 0.446993 0.455683
2011-01-06 0.402483 0.411486 0.420425 0.429301 0.438115
Rolling Window Forecasting#
Rolling window forecasts use a fixed sample length and then produce one-step from the final observation. These can be implemented using first_obs and last_obs.
index = returns.index
start_loc = 0
end_loc = np.where(index >= "2010-1-1")[0].min()
forecasts = {}
for i in tqdm(range(20)):
sys.stdout.write(".")
sys.stdout.flush()
res = am.fit(first_obs=i, last_obs=i + end_loc, disp="off")
temp = res.forecast(horizon=3, reindex=False).variance
fcast = temp.iloc[0]
forecasts[fcast.name] = fcast
print()
print(pd.DataFrame(forecasts).T)
....................
h.1 h.2 h.3
2009-12-31 0.615314 0.621743 0.628133
2010-01-04 0.751747 0.757343 0.762905
2010-01-05 0.710453 0.716315 0.722142
2010-01-06 0.666244 0.672346 0.678411
2010-01-07 0.634424 0.640706 0.646949
2010-01-08 0.600109 0.606595 0.613040
2010-01-11 0.565514 0.572212 0.578869
2010-01-12 0.599560 0.606051 0.612501
2010-01-13 0.608309 0.614748 0.621148
2010-01-14 0.575065 0.581756 0.588406
2010-01-15 0.629890 0.636245 0.642561
2010-01-19 0.695074 0.701042 0.706974
2010-01-20 0.737154 0.742908 0.748627
2010-01-21 0.954167 0.958725 0.963255
2010-01-22 1.253453 1.256401 1.259332
2010-01-25 1.178691 1.182043 1.185374
2010-01-26 1.112205 1.115886 1.119545
2010-01-27 1.051295 1.055327 1.059335
2010-01-28 1.085678 1.089512 1.093324
2010-01-29 1.085786 1.089593 1.093378
Recursive Forecast Generation#
Recursive is similar to rolling except that the initial observation does not change. This can be easily implemented by dropping the first_obs input.
import numpy as np
import pandas as pd
index = returns.index
start_loc = 0
end_loc = np.where(index >= "2010-1-1")[0].min()
forecasts = {}
for i in range(20):
sys.stdout.write(".")
sys.stdout.flush()
res = am.fit(last_obs=i + end_loc, disp="off")
temp = res.forecast(horizon=3, reindex=False).variance
fcast = temp.iloc[0]
forecasts[fcast.name] = fcast
print()
print(pd.DataFrame(forecasts).T)
....................
h.1 h.2 h.3
2009-12-31 0.615314 0.621743 0.628133
2010-01-04 0.751723 0.757321 0.762885
2010-01-05 0.709956 0.715791 0.721591
2010-01-06 0.666057 0.672146 0.678197
2010-01-07 0.634503 0.640776 0.647011
2010-01-08 0.600417 0.606893 0.613329
2010-01-11 0.565684 0.572369 0.579014
2010-01-12 0.599963 0.606438 0.612874
2010-01-13 0.608558 0.614982 0.621366
2010-01-14 0.575020 0.581639 0.588217
2010-01-15 0.629696 0.635989 0.642244
2010-01-19 0.694735 0.700656 0.706541
2010-01-20 0.736509 0.742193 0.747842
2010-01-21 0.952751 0.957245 0.961713
2010-01-22 1.251145 1.254049 1.256936
2010-01-25 1.176864 1.180162 1.183441
2010-01-26 1.110848 1.114497 1.118124
2010-01-27 1.050102 1.054077 1.058028
2010-01-28 1.084669 1.088454 1.092216
2010-01-29 1.085003 1.088783 1.092541