def test_studentst(self):
dist = StudentsT()
v = 4.0
ll1 = dist.loglikelihoood(np.array([v]), self.resids, self.sigma2)
# Direct calculation of PDF, then log
constant = np.exp(gammaln(0.5 * (v + 1)) - gammaln(0.5 * v))
pdf = constant / np.sqrt(np.pi * (v - 2) * self.sigma2)
pdf *= (1 + self.resids ** 2.0 / (self.sigma2 * (v - 2))) ** (
-(v + 1) / 2)
ll2 = np.log(pdf).sum()
assert_almost_equal(ll1, ll2)
assert_equal(dist.num_params, 1)
bounds = dist.bounds(self.resids)
assert_equal(len(bounds), 1)
A, b = dist.constraints()
assert_equal(A.shape, (2, 1))
k = stats.kurtosis(self.resids, fisher=False)
sv = max((4.0 * k - 6.0) / (k - 3.0) if k > 3.75 else 12.0, 4.0)
assert_array_equal(dist.starting_values(self.resids), np.array([sv]))
assert_raises(ValueError, dist.simulate, np.array([1.5]))
def test_warnings_nonstationary(self):
garch = GARCH()
parameters = np.array([0.1, 0.2, 0.8, 4.0])
studt = StudentsT()
warnings.simplefilter('always', UserWarning)
with pytest.warns(InitialValueWarning):
garch.simulate(parameters, 1000, studt.simulate([4.0]))
def test_warnings(self):
garch = GARCH()
parameters = np.array([0.1, 0.2, 0.8, 4.0])
studt = StudentsT()
with warnings.catch_warnings(record=True) as w:
garch.simulate(parameters, 1000, studt.simulate([4.0]))
assert_equal(len(w), 1)
harch = HARCH(lags=[1, 5, 22])
parameters = np.array([0.1, 0.2, 0.4, 0.5])
with warnings.catch_warnings(record=True) as w:
harch.simulate(parameters, 1000, studt.simulate([4.0]))
assert_equal(len(w), 1)
def test_warnings(self):
garch = GARCH()
parameters = np.array([0.1, 0.2, 0.8, 4.0])
studt = StudentsT()
with warnings.catch_warnings(record=True) as w:
garch.simulate(parameters, 1000, studt.simulate([4.0]))
assert_equal(len(w), 1)
harch = HARCH(lags=[1, 5, 22])
parameters = np.array([0.1, 0.2, 0.4, 0.5])
with warnings.catch_warnings(record=True) as w:
harch.simulate(parameters, 1000, studt.simulate([4.0]))
assert_equal(len(w), 1)
def test_studentst(self):
dist = StudentsT()
v = 4.0
ll1 = dist.loglikelihood(np.array([v]), self.resids, self.sigma2)
# Direct calculation of PDF, then log
constant = np.exp(gammaln(0.5 * (v + 1)) - gammaln(0.5 * v))
pdf = constant / np.sqrt(np.pi * (v - 2) * self.sigma2)
pdf *= (1 + self.resids**2.0 / (self.sigma2 * (v - 2)))**(-(v + 1) / 2)
ll2 = np.log(pdf).sum()
assert_almost_equal(ll1, ll2)
assert_equal(dist.num_params, 1)
bounds = dist.bounds(self.resids)
assert_equal(len(bounds), 1)
a, b = dist.constraints()
assert_equal(a.shape, (2, 1))
k = stats.kurtosis(self.resids, fisher=False)
sv = max((4.0 * k - 6.0) / (k - 3.0) if k > 3.75 else 12.0, 4.0)
assert_array_equal(dist.starting_values(self.resids), np.array([sv]))
with pytest.raises(ValueError):
dist.simulate(np.array([1.5]))
def test_egarch(self):
cm = ConstantMean(self.y)
cm.volatility = EGARCH()
res = cm.fit(update_freq=0, disp=DISPLAY)
summ = res.summary()
assert "Df Model: 1" in str(summ)
cm.distribution = StudentsT()
cm.fit(update_freq=0, disp=DISPLAY)
def test_invalid_params():
pits = np.arange(1, 100.0) / 100.0
dist = Normal()
with pytest.raises(ValueError):
dist.ppf(pits, [1.0])
dist = StudentsT()
with pytest.raises(ValueError):
dist.ppf(pits, [1.0])
def test_ar_plot(self):
ar = ARX(self.y, lags=1, volatility=GARCH(), distribution=StudentsT())
res = ar.fit(disp=DISPLAY, update_freq=5, cov_type='mle')
res.plot()
res.plot(annualize='D')
res.plot(annualize='W')
res.plot(annualize='M')
with pytest.raises(ValueError):
res.plot(annualize='unknown')
res.plot(scale=360)
res.hedgehog_plot(start=500)
res.hedgehog_plot(start=500, type='mean')
res.hedgehog_plot(type='volatility')
res.hedgehog_plot(start=500, method='simulation', simulations=100)
def test_ar_plot(self):
ar = ARX(self.y, lags=1, volatility=GARCH(), distribution=StudentsT())
res = ar.fit(disp=DISPLAY, update_freq=5, cov_type="mle")
res.plot()
res.plot(annualize="D")
res.plot(annualize="W")
res.plot(annualize="M")
with pytest.raises(ValueError):
res.plot(annualize="unknown")
res.plot(scale=360)
res.hedgehog_plot(start=500)
res.hedgehog_plot(start=500, plot_type="mean")
res.hedgehog_plot(plot_type="volatility")
res.hedgehog_plot(start=500, method="simulation", simulations=100)
res.hedgehog_plot(plot_type="volatility", method="bootstrap")
def arch_model(
y: Optional[ArrayLike],
x: Optional[ArrayLike] = None,
mean: str = "Constant",
lags: Optional[Union[int, List[int], NDArray]] = 0,
vol: str = "Garch",
p: Union[int, List[int]] = 1,
o: int = 0,
q: int = 1,
power: float = 2.0,
dist: str = "Normal",
hold_back: Optional[int] = None,
rescale: Optional[bool] = None,
) -> HARX:
"""
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', 'FIARCH' 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.
"""
known_mean = ("zero", "constant", "harx", "har", "ar", "arx", "ls")
known_vol = ("arch", "figarch", "garch", "harch", "constant", "egarch")
known_dist = (
"normal",
"gaussian",
"studentst",
"t",
"skewstudent",
"skewt",
"ged",
"generalized error",
)
mean = mean.lower()
vol = vol.lower()
dist = dist.lower()
if mean not in known_mean:
raise ValueError("Unknown model type in mean")
if vol.lower() not in known_vol:
raise ValueError("Unknown model type in vol")
if dist.lower() not in known_dist:
raise ValueError("Unknown model type in dist")
if mean == "harx":
am = HARX(y, x, lags, hold_back=hold_back, rescale=rescale)
elif mean == "har":
am = HARX(y, None, lags, hold_back=hold_back, rescale=rescale)
elif mean == "arx":
am = ARX(y, x, lags, hold_back=hold_back, rescale=rescale)
elif mean == "ar":
am = ARX(y, None, lags, hold_back=hold_back, rescale=rescale)
elif mean == "ls":
am = LS(y, x, hold_back=hold_back, rescale=rescale)
elif mean == "constant":
am = ConstantMean(y, hold_back=hold_back, rescale=rescale)
else: # mean == "zero"
am = ZeroMean(y, hold_back=hold_back, rescale=rescale)
if vol in ("arch", "garch", "figarch",
"egarch") and not isinstance(p, int):
raise TypeError(
"p must be a scalar int for all volatility processes except HARCH."
)
if vol == "constant":
v: VolatilityProcess = ConstantVariance()
elif vol == "arch":
assert isinstance(p, int)
v = ARCH(p=p)
elif vol == "figarch":
assert isinstance(p, int)
v = FIGARCH(p=p, q=q)
elif vol == "garch":
assert isinstance(p, int)
v = GARCH(p=p, o=o, q=q, power=power)
elif vol == "egarch":
assert isinstance(p, int)
v = EGARCH(p=p, o=o, q=q)
else: # vol == 'harch'
v = HARCH(lags=p)
if dist in ("skewstudent", "skewt"):
d: Distribution = SkewStudent()
elif dist in ("studentst", "t"):
d = StudentsT()
elif dist in ("ged", "generalized error"):
d = GeneralizedError()
else: # ('gaussian', 'normal')
d = Normal()
am.volatility = v
am.distribution = d
return am
def test_warnings_nonstationary_harch(self):
studt = StudentsT()
harch = HARCH(lags=[1, 5, 22])
parameters = np.array([0.1, 0.2, 0.4, 0.5])
with pytest.warns(InitialValueWarning):
harch.simulate(parameters, 1000, studt.simulate([4.0]))
def __init__(self, random_state=None):
DistributionMixin.__init__(self)
T.__init__(self, random_state)
from arch.univariate.distribution import (
GeneralizedError,
Normal,
SkewStudent,
StudentsT,
)
DISTRIBUTIONS = [
(SkewStudent(), [6, -0.1]),
(SkewStudent(), [6, -0.5]),
(SkewStudent(), [6, 0.1]),
(SkewStudent(), [6, 0.5]),
(GeneralizedError(), [1.5]),
(GeneralizedError(), [2.1]),
(StudentsT(), [6]),
(StudentsT(), [7]),
(Normal(), None),
]
@pytest.mark.parametrize("dist, params", DISTRIBUTIONS)
def test_moment(dist, params):
"""
Ensures that Distribtion.moment and .partial_moment agree
with numeric integrals for order n=0,...,5 and z=+/-1,...,+/-5
Parameters
----------
dist : distribution.Distribution
The distribution whose moments are being checked
def test_ar(self):
ar = ARX(self.y, lags=3)
params = np.array([1.0, 0.4, 0.3, 0.2, 1.0])
data = ar.simulate(params, self.T)
assert_equal(self.y, ar.y)
bounds = ar.bounds()
for b in bounds:
assert_equal(b[0], -np.inf)
assert_equal(b[1], np.inf)
assert_equal(len(bounds), 4)
assert_equal(ar.num_params, 4)
assert_true(ar.constant)
a, b = ar.constraints()
assert_equal(a, np.empty((0, 4)))
assert_equal(b, np.empty(0))
res = ar.fit()
nobs = 1000 - 3
rhs = np.ones((nobs, 4))
y = self.y
lhs = y[3:1000]
for i in range(3, 1000):
rhs[i - 3, 1] = y[i - 1]
rhs[i - 3, 2] = y[i - 2]
rhs[i - 3, 3] = y[i - 3]
params = np.linalg.pinv(rhs).dot(lhs)
assert_almost_equal(params, res.params[:-1])
forecasts = res.forecast(horizon=5)
direct = pd.DataFrame(index=np.arange(y.shape[0]),
columns=['h.' + str(i + 1) for i in range(5)],
dtype=np.float64)
params = res.params.iloc[:-1]
for i in range(2, y.shape[0]):
fcast = np.zeros(y.shape[0] + 5)
fcast[:y.shape[0]] = y.copy()
for h in range(1, 6):
reg = np.array([
1.0, fcast[i + h - 1], fcast[i + h - 2], fcast[i + h - 3]
])
fcast[i + h] = reg.dot(params)
direct.iloc[i, :] = fcast[i + 1:i + 6]
assert_frame_equal(direct, forecasts)
assert_equal(ar.first_obs, 3)
assert_equal(ar.last_obs, 1000)
assert_equal(ar.hold_back, None)
assert_equal(ar.lags, np.array([[0, 1, 2], [1, 2, 3]]))
assert_equal(ar.nobs, 997)
assert_equal(ar.name, 'AR')
assert_equal(ar.use_rotated, False)
ar.__repr__()
ar._repr_html_()
ar = ARX(self.y_df, lags=5)
ar.__repr__()
ar = ARX(self.y_series, lags=5)
ar.__repr__()
res = ar.fit()
assert_true(isinstance(res.resid, pd.Series))
assert_true(isinstance(res.conditional_volatility, pd.Series))
# Smoke tests
summ = ar.fit().summary()
ar = ARX(self.y, lags=1, volatility=GARCH(), distribution=StudentsT())
res = ar.fit(update_freq=5, cov_type='mle')
res.param_cov
res.plot()
res.plot(annualize='D')
res.plot(annualize='W')
res.plot(annualize='M')
assert_raises(ValueError, res.plot, annualize='unknown')
res.plot(scale=360)
res.hedgehog_plot()
def test_ar(self):
ar = ARX(self.y, lags=3)
params = np.array([1.0, 0.4, 0.3, 0.2, 1.0])
data = ar.simulate(params, self.T)
assert len(data) == self.T
assert_equal(self.y, ar.y)
bounds = ar.bounds()
for b in bounds:
assert_equal(b[0], -np.inf)
assert_equal(b[1], np.inf)
assert_equal(len(bounds), 4)
assert_equal(ar.num_params, 4)
assert ar.constant
a, b = ar.constraints()
assert_equal(a, np.empty((0, 4)))
assert_equal(b, np.empty(0))
res = ar.fit(disp=DISPLAY)
nobs = 1000 - 3
rhs = np.ones((nobs, 4))
y = self.y
lhs = y[3:1000]
for i in range(3, 1000):
rhs[i - 3, 1] = y[i - 1]
rhs[i - 3, 2] = y[i - 2]
rhs[i - 3, 3] = y[i - 3]
params = np.linalg.pinv(rhs).dot(lhs)
assert_almost_equal(params, res.params[:-1])
forecasts = res.forecast(horizon=5)
direct = pd.DataFrame(index=np.arange(y.shape[0]),
columns=['h.' + str(i + 1) for i in range(5)],
dtype=np.float64)
params = res.params.iloc[:-1]
for i in range(2, y.shape[0]):
fcast = np.zeros(y.shape[0] + 5)
fcast[:y.shape[0]] = y.copy()
for h in range(1, 6):
reg = np.array([1.0, fcast[i + h - 1],
fcast[i + h - 2], fcast[i + h - 3]])
fcast[i + h] = reg.dot(params)
direct.iloc[i, :] = fcast[i + 1:i + 6]
assert isinstance(forecasts, ARCHModelForecast)
# TODO
# assert_frame_equal(direct, forecasts)
assert_equal(ar.hold_back, None)
assert_equal(ar.lags, np.array([[0, 1, 2], [1, 2, 3]]))
assert_equal(ar.name, 'AR')
assert_equal(ar.use_rotated, False)
ar.__repr__()
ar._repr_html_()
ar = ARX(self.y_df, lags=5)
ar.__repr__()
ar = ARX(self.y_series, lags=5)
ar.__repr__()
res = ar.fit(disp=DISPLAY)
assert isinstance(res.resid, pd.Series)
assert isinstance(res.conditional_volatility, pd.Series)
# Smoke bootstrap
summ = ar.fit(disp=DISPLAY).summary()
assert 'Constant Variance' in str(summ)
ar = ARX(self.y, lags=1, volatility=GARCH(), distribution=StudentsT())
res = ar.fit(disp=DISPLAY, update_freq=5, cov_type='mle')
res.param_cov
sims = res.forecast(horizon=5, method='simulation')
assert isinstance(sims.simulations.residual_variances, np.ndarray)
assert isinstance(sims.simulations.residuals, np.ndarray)
assert isinstance(sims.simulations.values, np.ndarray)
assert isinstance(sims.simulations.variances, np.ndarray)
def arch_model(y,
x=None,
mean='Constant',
lags=0,
vol='Garch',
p=1,
o=0,
q=1,
power=2.0,
dist='Normal',
hold_back=None):
"""
Convenience function to simplify initialization of ARCH models
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', 'ARX' 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), "EGARCH', 'ARCH' 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.
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.
"""
known_mean = ('zero', 'constant', 'harx', 'har', 'ar', 'arx', 'ls')
known_vol = ('arch', 'garch', 'harch', 'constant', 'egarch')
known_dist = ('normal', 'gaussian', 'studentst', 't', 'skewstudent',
'skewt', 'ged', 'generalized error')
mean = mean.lower()
vol = vol.lower()
dist = dist.lower()
if mean not in known_mean:
raise ValueError('Unknown model type in mean')
if vol.lower() not in known_vol:
raise ValueError('Unknown model type in vol')
if dist.lower() not in known_dist:
raise ValueError('Unknown model type in dist')
if mean == 'zero':
am = ZeroMean(y, hold_back=hold_back)
elif mean == 'constant':
am = ConstantMean(y, hold_back=hold_back)
elif mean == 'harx':
am = HARX(y, x, lags, hold_back=hold_back)
elif mean == 'har':
am = HARX(y, None, lags, hold_back=hold_back)
elif mean == 'arx':
am = ARX(y, x, lags, hold_back=hold_back)
elif mean == 'ar':
am = ARX(y, None, lags, hold_back=hold_back)
else:
am = LS(y, x, hold_back=hold_back)
if vol == 'constant':
v = ConstantVariance()
elif vol == 'arch':
v = ARCH(p=p)
elif vol == 'garch':
v = GARCH(p=p, o=o, q=q, power=power)
elif vol == 'egarch':
v = EGARCH(p=p, o=o, q=q)
else: # vol == 'harch'
v = HARCH(lags=p)
if dist in ('skewstudent', 'skewt'):
d = SkewStudent()
elif dist in ('studentst', 't'):
d = StudentsT()
elif dist in ('ged', 'generalized error'):
d = GeneralizedError()
else: # ('gaussian', 'normal')
d = Normal()
am.volatility = v
am.distribution = d
return am
def test_egarch(self):
cm = ConstantMean(self.y)
cm.volatility = EGARCH()
cm.fit(update_freq=0, disp=DISPLAY)
cm.distribution = StudentsT()
cm.fit(update_freq=0, disp=DISPLAY)
import pytest
from arch.univariate.distribution import (SkewStudent, Normal, StudentsT,
GeneralizedError)
from numpy import exp, inf, log, nan, pi
from numpy.testing import assert_equal, assert_almost_equal
from scipy.integrate import quad
from scipy.special import gammaln
DISTRIBUTIONS = [(SkewStudent(), [6, -0.1]), (SkewStudent(), [6, -0.5]),
(SkewStudent(), [6, 0.1]), (SkewStudent(), [6, 0.5]),
(GeneralizedError(), [1.5]), (GeneralizedError(), [2.1]),
(StudentsT(), [6]), (StudentsT(), [7]), (Normal(), None)]
@pytest.mark.parametrize('dist, params', DISTRIBUTIONS)
def test_moment(dist, params):
"""
Ensures that Distribtion.moment and .partial_moment agree
with numeric integrals for order n=0,...,5 and z=+/-1,...,+/-5
Parameters
----------
dist : distribution.Distribution
The distribution whose moments are being checked
params : List
List of parameters
"""
assert_equal(dist.moment(-1, params), nan)
assert_equal(dist.partial_moment(-1, 0., params), nan)