def test_statespace():
# Test for basic use of Yule-Walker estimation
endog = dta['infl'].iloc[:100]
# ARMA(1, 1), no trend
desired_p, _ = statespace(endog, order=(1, 0, 1), include_constant=False)
mod = ARIMA(endog, order=(1, 0, 1), trend='n')
res = mod.fit(method='statespace')
# Note: atol is required only due to precision issues on Windows
assert_allclose(res.params, desired_p.params, atol=1e-5)
# ARMA(1, 2), with trend
desired_p, _ = statespace(endog, order=(1, 0, 2), include_constant=True)
mod = ARIMA(endog, order=(1, 0, 2), trend='c')
res = mod.fit(method='statespace')
# Note: atol is required only due to precision issues on Windows
assert_allclose(res.params, desired_p.params, atol=1e-5)
# SARMA(1, 0)x(1, 0)4, no trend
desired_p, _spec = statespace(endog,
order=(1, 0, 0),
seasonal_order=(1, 0, 0, 4),
include_constant=False)
mod = ARIMA(endog, order=(1, 0, 0), seasonal_order=(1, 0, 0, 4), trend='n')
res = mod.fit(method='statespace')
# Note: atol is required only due to precision issues on Windows
assert_allclose(res.params, desired_p.params, atol=1e-5)
def test_start_params():
endog = lake.copy()
# Test for valid use of starting parameters
p, _ = statespace(endog, order=(1, 0, 0), start_params=[0, 0, 1.])
p, _ = statespace(endog,
order=(1, 0, 0),
start_params=[0, 1., 1.],
enforce_stationarity=False)
p, _ = statespace(endog,
order=(0, 0, 1),
start_params=[0, 1., 1.],
enforce_invertibility=False)
# Test for invalid use of starting parameters
assert_raises(ValueError,
statespace,
endog,
order=(1, 0, 0),
start_params=[0, 1., 1.])
assert_raises(ValueError,
statespace,
endog,
order=(0, 0, 1),
start_params=[0, 1., 1.])
def test_basic():
endog = lake.copy()
exog = np.arange(1, len(endog) + 1) * 1.0
# Test default options (include_constant=True, concentrate_scale=False)
p, res = statespace(endog,
exog=exog,
order=(1, 0, 0),
include_constant=True,
concentrate_scale=False)
mod_ss = sarimax.SARIMAX(endog, exog=add_constant(exog), order=(1, 0, 0))
res_ss = mod_ss.filter(p.params)
assert_allclose(res.statespace_results.llf, res_ss.llf)
# Test include_constant=False
p, res = statespace(endog,
exog=exog,
order=(1, 0, 0),
include_constant=False,
concentrate_scale=False)
mod_ss = sarimax.SARIMAX(endog, exog=exog, order=(1, 0, 0))
res_ss = mod_ss.filter(p.params)
assert_allclose(res.statespace_results.llf, res_ss.llf)
# Test concentrate_scale=True
p, res = statespace(endog,
exog=exog,
order=(1, 0, 0),
include_constant=True,
concentrate_scale=True)
mod_ss = sarimax.SARIMAX(endog,
exog=add_constant(exog),
order=(1, 0, 0),
concentrate_scale=True)
res_ss = mod_ss.filter(p.params)
assert_allclose(res.statespace_results.llf, res_ss.llf)
def gls(endog,
exog=None,
order=(0, 0, 0),
seasonal_order=(0, 0, 0, 0),
include_constant=None,
n_iter=None,
max_iter=50,
tolerance=1e-8,
arma_estimator='innovations_mle',
arma_estimator_kwargs=None):
"""
Estimate ARMAX parameters by GLS.
Parameters
----------
endog : array_like
Input time series array.
exog : array_like, optional
Array of exogenous regressors. If not included, then `include_constant`
must be True, and then `exog` will only include the constant column.
order : tuple, optional
The (p,d,q) order of the ARIMA model. Default is (0, 0, 0).
seasonal_order : tuple, optional
The (P,D,Q,s) order of the seasonal ARIMA model.
Default is (0, 0, 0, 0).
include_constant : bool, optional
Whether to add a constant term in `exog` if it's not already there.
The estimate of the constant will then appear as one of the `exog`
parameters. If `exog` is None, then the constant will represent the
mean of the process. Default is True if the specified model does not
include integration and False otherwise.
n_iter : int, optional
Optionally iterate feasible GSL a specific number of times. Default is
to iterate to convergence. If set, this argument overrides the
`max_iter` and `tolerance` arguments.
max_iter : int, optional
Maximum number of feasible GLS iterations. Default is 50. If `n_iter`
is set, it overrides this argument.
tolerance : float, optional
Tolerance for determining convergence of feasible GSL iterations. If
`iter` is set, this argument has no effect.
Default is 1e-8.
arma_estimator : str, optional
The estimator used for estimating the ARMA model. This option should
not generally be used, unless the default method is failing or is
otherwise unsuitable. Not all values will be valid, depending on the
specified model orders (`order` and `seasonal_order`). Possible values
are:
* 'innovations_mle' - can be used with any specification
* 'statespace' - can be used with any specification
* 'hannan_rissanen' - can be used with any ARMA non-seasonal model
* 'yule_walker' - only non-seasonal consecutive
autoregressive (AR) models
* 'burg' - only non-seasonal, consecutive autoregressive (AR) models
* 'innovations' - only non-seasonal, consecutive moving
average (MA) models.
The default is 'innovations_mle'.
arma_estimator_kwargs : dict, optional
Arguments to pass to the ARMA estimator.
Returns
-------
parameters : SARIMAXParams object
Contains the parameter estimates from the final iteration.
other_results : Bunch
Includes eight components: `spec`, `params`, `converged`,
`differences`, `iterations`, `arma_estimator`, 'arma_estimator_kwargs',
and `arma_results`.
Notes
-----
The primary reference is [1]_, section 6.6. In particular, the
implementation follows the iterative procedure described in section 6.6.2.
Construction of the transformed variables used to compute the GLS estimator
described in section 6.6.1 is done via an application of the innovations
algorithm (rather than explicit construction of the transformation matrix).
Note that if the specified model includes integration, both the `endog` and
`exog` series will be differenced prior to estimation and a warning will
be issued to alert the user.
References
----------
.. [1] Brockwell, Peter J., and Richard A. Davis. 2016.
Introduction to Time Series and Forecasting. Springer.
"""
# Handle n_iter
if n_iter is not None:
max_iter = n_iter
tolerance = np.inf
# Default for include_constant is True if there is no integration and
# False otherwise
integrated = order[1] > 0 or seasonal_order[1] > 0
if include_constant is None:
include_constant = not integrated
elif include_constant and integrated:
raise ValueError('Cannot include a constant in an integrated model.')
# Handle including the constant (need to do it now so that the constant
# parameter can be included in the specification as part of `exog`.)
if include_constant:
exog = np.ones_like(endog) if exog is None else add_constant(exog)
# Create the SARIMAX specification
spec = SARIMAXSpecification(endog,
exog=exog,
order=order,
seasonal_order=seasonal_order)
endog = spec.endog
exog = spec.exog
# Handle integration
if spec.is_integrated:
# TODO: this is the approach suggested by BD (see Remark 1 in
# section 6.6.2 and Example 6.6.3), but maybe there are some cases
# where we don't want to force this behavior on the user?
warnings.warn('Provided `endog` and `exog` series have been'
' differenced to eliminate integration prior to GLS'
' parameter estimation.')
endog = diff(endog,
k_diff=spec.diff,
k_seasonal_diff=spec.seasonal_diff,
seasonal_periods=spec.seasonal_periods)
exog = diff(exog,
k_diff=spec.diff,
k_seasonal_diff=spec.seasonal_diff,
seasonal_periods=spec.seasonal_periods)
augmented = np.c_[endog, exog]
# Validate arma_estimator
spec.validate_estimator(arma_estimator)
if arma_estimator_kwargs is None:
arma_estimator_kwargs = {}
# Step 1: OLS
mod_ols = OLS(endog, exog)
res_ols = mod_ols.fit()
exog_params = res_ols.params
resid = res_ols.resid
# 0th iteration parameters
p = SARIMAXParams(spec=spec)
p.exog_params = exog_params
if spec.max_ar_order > 0:
p.ar_params = np.zeros(spec.k_ar_params)
if spec.max_seasonal_ar_order > 0:
p.seasonal_ar_params = np.zeros(spec.k_seasonal_ar_params)
if spec.max_ma_order > 0:
p.ma_params = np.zeros(spec.k_ma_params)
if spec.max_seasonal_ma_order > 0:
p.seasonal_ma_params = np.zeros(spec.k_seasonal_ma_params)
p.sigma2 = res_ols.scale
ar_params = p.ar_params
seasonal_ar_params = p.seasonal_ar_params
ma_params = p.ma_params
seasonal_ma_params = p.seasonal_ma_params
sigma2 = p.sigma2
# Step 2 - 4: iterate feasible GLS to convergence
arma_results = [None]
differences = [None]
parameters = [p]
converged = False if n_iter is None else None
i = 0
for i in range(1, max_iter + 1):
prev = exog_params
# Step 2: ARMA
# TODO: allow estimator-specific kwargs?
if arma_estimator == 'yule_walker':
p_arma, res_arma = yule_walker(resid,
ar_order=spec.ar_order,
demean=False,
**arma_estimator_kwargs)
elif arma_estimator == 'burg':
p_arma, res_arma = burg(resid,
ar_order=spec.ar_order,
demean=False,
**arma_estimator_kwargs)
elif arma_estimator == 'innovations':
out, res_arma = innovations(resid,
ma_order=spec.ma_order,
demean=False,
**arma_estimator_kwargs)
p_arma = out[-1]
elif arma_estimator == 'hannan_rissanen':
p_arma, res_arma = hannan_rissanen(resid,
ar_order=spec.ar_order,
ma_order=spec.ma_order,
demean=False,
**arma_estimator_kwargs)
else:
# For later iterations, use a "warm start" for parameter estimates
# (speeds up estimation and convergence)
start_params = (None if i == 1 else np.r_[ar_params, ma_params,
seasonal_ar_params,
seasonal_ma_params,
sigma2])
# Note: in each case, we do not pass in the order of integration
# since we have already differenced the series
tmp_order = (spec.order[0], 0, spec.order[2])
tmp_seasonal_order = (spec.seasonal_order[0], 0,
spec.seasonal_order[2],
spec.seasonal_order[3])
if arma_estimator == 'innovations_mle':
p_arma, res_arma = innovations_mle(
resid,
order=tmp_order,
seasonal_order=tmp_seasonal_order,
demean=False,
start_params=start_params,
**arma_estimator_kwargs)
else:
p_arma, res_arma = statespace(
resid,
order=tmp_order,
seasonal_order=tmp_seasonal_order,
include_constant=False,
start_params=start_params,
**arma_estimator_kwargs)
ar_params = p_arma.ar_params
seasonal_ar_params = p_arma.seasonal_ar_params
ma_params = p_arma.ma_params
seasonal_ma_params = p_arma.seasonal_ma_params
sigma2 = p_arma.sigma2
arma_results.append(res_arma)
# Step 3: GLS
# Compute transformed variables that satisfy OLS assumptions
# Note: In section 6.1.1 of Brockwell and Davis (2016), these
# transformations are developed as computed by left multiplcation
# by a matrix T. However, explicitly constructing T and then
# performing the left-multiplications does not scale well when nobs is
# large. Instead, we can retrieve the transformed variables as the
# residuals of the innovations algorithm (the `normalize=True`
# argument applies a Prais-Winsten-type normalization to the first few
# observations to ensure homoskedasticity). Brockwell and Davis
# mention that they also take this approach in practice.
tmp, _ = arma_innovations.arma_innovations(augmented,
ar_params=ar_params,
ma_params=ma_params,
normalize=True)
u = tmp[:, 0]
x = tmp[:, 1:]
# OLS on transformed variables
mod_gls = OLS(u, x)
res_gls = mod_gls.fit()
exog_params = res_gls.params
resid = endog - np.dot(exog, exog_params)
# Construct the parameter vector for the iteration
p = SARIMAXParams(spec=spec)
p.exog_params = exog_params
if spec.max_ar_order > 0:
p.ar_params = ar_params
if spec.max_seasonal_ar_order > 0:
p.seasonal_ar_params = seasonal_ar_params
if spec.max_ma_order > 0:
p.ma_params = ma_params
if spec.max_seasonal_ma_order > 0:
p.seasonal_ma_params = seasonal_ma_params
p.sigma2 = sigma2
parameters.append(p)
# Check for convergence
difference = np.abs(exog_params - prev)
differences.append(difference)
if n_iter is None and np.all(difference < tolerance):
converged = True
break
else:
if n_iter is None:
warnings.warn('Feasible GLS failed to converge in %d iterations.'
' Consider increasing the maximum number of'
' iterations using the `max_iter` argument or'
' reducing the required tolerance using the'
' `tolerance` argument.' % max_iter)
# Construct final results
p = parameters[-1]
other_results = Bunch({
'spec': spec,
'params': parameters,
'converged': converged,
'differences': differences,
'iterations': i,
'arma_estimator': arma_estimator,
'arma_estimator_kwargs': arma_estimator_kwargs,
'arma_results': arma_results,
})
return p, other_results