def test_innovations_mle_statespace_nonconsecutive():
# Test innovations output against state-space output.
endog = lake.copy()
endog = endog - endog.mean()
start_params = [0, 0, np.var(endog)]
p, mleres = innovations_mle(endog,
order=([0, 1], 0, [0, 1]),
demean=False,
start_params=start_params)
mod = sarimax.SARIMAX(endog, order=([0, 1], 0, [0, 1]))
# Test that the maximized log-likelihood found via applications of the
# innovations algorithm matches the log-likelihood found by the Kalman
# filter at the same parameters
res = mod.filter(p.params)
assert_allclose(-mleres.minimize_results.fun, res.llf)
# Test MLE fitting
# To avoid small numerical differences with MLE fitting, start at the
# parameters found from innovations_mle
res2 = mod.fit(start_params=p.params, disp=0)
# Test that the state space approach confirms the MLE values found by
# innovations_mle
assert_allclose(p.params, res2.params)
# Test that starting parameter estimation succeeds and isn't terrible
# (i.e. leads to the same MLE)
p2, _ = innovations_mle(endog, order=([0, 1], 0, [0, 1]), demean=False)
# (does not need to be high-precision test since it's okay if different
# starting parameters give slightly different MLE)
assert_allclose(p.params, p2.params, atol=1e-5)
def test_innovations_mle_integrated():
endog = np.r_[0, np.cumsum(lake.copy())]
start_params = [0, np.var(lake.copy())]
with assert_warns(UserWarning):
p, mleres = innovations_mle(endog, order=(1, 1, 0),
demean=False, start_params=start_params)
mod = sarimax.SARIMAX(endog, order=(1, 1, 0),
simple_differencing=True)
# Test that the maximized log-likelihood found via applications of the
# innovations algorithm matches the log-likelihood found by the Kalman
# filter at the same parameters
res = mod.filter(p.params)
assert_allclose(-mleres.minimize_results.fun, res.llf)
# Test MLE fitting
# To avoid small numerical differences with MLE fitting, start at the
# parameters found from innovations_mle
res2 = mod.fit(start_params=p.params, disp=0)
# Test that the state space approach confirms the MLE values found by
# innovations_mle
# Note: atol is required only due to precision issues on Windows
assert_allclose(p.params, res2.params, atol=1e-6)
# Test that the result is equivalent to order=(1, 0, 0) on the differenced
# data
p2, _ = innovations_mle(lake.copy(), order=(1, 0, 0), demean=False,
start_params=start_params)
# (doesn't need to be high-precision test since it's okay if different
# starting parameters give slightly different MLE)
assert_allclose(p.params, p2.params, atol=1e-5)
def test_innovations_mle_misc():
endog = np.arange(20)**2 * 1.0
# Check that when Hannan-Rissanen estimates non-stationary starting
# parameters, innovations_mle sets it to zero
hr, _ = hannan_rissanen(endog, ar_order=1, demean=False)
assert_(hr.ar_params[0] > 1)
_, res = innovations_mle(endog, order=(1, 0, 0))
assert_allclose(res.start_params[0], 0)
# Check that when Hannan-Rissanen estimates non-invertible starting
# parameters, innovations_mle sets it to zero
hr, _ = hannan_rissanen(endog, ma_order=1, demean=False)
assert_(hr.ma_params[0] > 1)
_, res = innovations_mle(endog, order=(0, 0, 1))
assert_allclose(res.start_params[0], 0)
def test_brockwell_davis_example_525():
# Difference and demean the series
endog = lake.copy()
# Use HR method to get initial coefficients for MLE
initial, _ = hannan_rissanen(endog, ar_order=1, ma_order=1, demean=True)
# Fit MLE via innovations algorithm
p, _ = innovations_mle(endog, order=(1, 0, 1), demean=True,
start_params=initial.params)
assert_allclose(p.params, [0.7446, 0.3213, 0.4750], atol=1e-4)
# Fit MLE via innovations algorithm, with default starting parameters
p, _ = innovations_mle(endog, order=(1, 0, 1), demean=True)
assert_allclose(p.params, [0.7446, 0.3213, 0.4750], atol=1e-4)
def test_innovations_mle():
# Test for basic use of Yule-Walker estimation
endog = dta['infl'].iloc[:100]
# ARMA(1, 1), no trend (since trend would imply GLS estimation)
desired_p, _ = innovations_mle(
endog, order=(1, 0, 1), demean=False)
mod = ARIMA(endog, order=(1, 0, 1), trend='n')
res = mod.fit(method='innovations_mle')
# 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 (since trend would imply GLS estimation)
desired_p, _ = innovations_mle(
endog, order=(1, 0, 0), seasonal_order=(1, 0, 0, 4), demean=False)
mod = ARIMA(endog, order=(1, 0, 0), seasonal_order=(1, 0, 0, 4), trend='n')
res = mod.fit(method='innovations_mle')
# Note: atol is required only due to precision issues on Windows
assert_allclose(res.params, desired_p.params, atol=1e-5)
def test_brockwell_davis_example_524():
# Difference and demean the series
endog = dowj.diff().iloc[1:]
# Use Burg method to get initial coefficients for MLE
initial, _ = burg(endog, ar_order=1, demean=True)
# Fit MLE via innovations algorithm
p, _ = innovations_mle(endog, order=(1, 0, 0), demean=True,
start_params=initial.params)
assert_allclose(p.ar_params, 0.4471, atol=1e-4)
def test_brockwell_davis_example_541():
# Difference and demean the series
endog = oshorts.copy()
# Use innovations MA method to get initial coefficients for MLE
initial, _ = innovations(endog, ma_order=1, demean=True)
# Fit MLE via innovations algorithm
p, _ = innovations_mle(endog, order=(0, 0, 1), demean=True,
start_params=initial[1].params)
assert_allclose(p.ma_params, -0.818, atol=1e-3)
def test_brockwell_davis_example_524_variance():
# See `test_brockwell_davis_example_524` for the main test
# TODO: the test for sigma2 fails, but the value reported by BD (0.02117)
# is suspicious. For example, the Burg results have an AR coefficient of
# 0.4371 and sigma2 = 0.1423. It seems unlikely that the small difference
# in AR coefficient would result in an order of magniture reduction in
# sigma2 (see test_burg::test_brockwell_davis_example_513). Should run
# this in the ITSM program to check its output.
endog = dowj.diff().iloc[1:]
# Use Burg method to get initial coefficients for MLE
initial, _ = burg(endog, ar_order=1, demean=True)
# Fit MLE via innovations algorithm
p, _ = innovations_mle(endog, order=(1, 0, 0), demean=True,
start_params=initial.params)
assert_allclose(p.sigma2, 0.02117, atol=1e-4)
def fit(self,
start_params=None,
transformed=True,
includes_fixed=False,
method=None,
method_kwargs=None,
gls=None,
gls_kwargs=None,
cov_type=None,
cov_kwds=None,
return_params=False,
low_memory=False):
"""
Fit (estimate) the parameters of the model.
Parameters
----------
start_params : array_like, optional
Initial guess of the solution for the loglikelihood maximization.
If None, the default is given by Model.start_params.
transformed : bool, optional
Whether or not `start_params` is already transformed. Default is
True.
includes_fixed : bool, optional
If parameters were previously fixed with the `fix_params` method,
this argument describes whether or not `start_params` also includes
the fixed parameters, in addition to the free parameters. Default
is False.
method : str, optional
The method used for estimating the parameters of the model. Valid
options include 'statespace', 'innovations_mle', 'hannan_rissanen',
'burg', 'innovations', and 'yule_walker'. Not all options are
available for every specification (for example 'yule_walker' can
only be used with AR(p) models).
method_kwargs : dict, optional
Arguments to pass to the fit function for the parameter estimator
described by the `method` argument.
gls : bool, optional
Whether or not to use generalized least squares (GLS) to estimate
regression effects. The default is False if `method='statespace'`
and is True otherwise.
gls_kwargs : dict, optional
Arguments to pass to the GLS estimation fit method. Only applicable
if GLS estimation is used (see `gls` argument for details).
cov_type : str, optional
The `cov_type` keyword governs the method for calculating the
covariance matrix of parameter estimates. Can be one of:
- 'opg' for the outer product of gradient estimator
- 'oim' for the observed information matrix estimator, calculated
using the method of Harvey (1989)
- 'approx' for the observed information matrix estimator,
calculated using a numerical approximation of the Hessian matrix.
- 'robust' for an approximate (quasi-maximum likelihood) covariance
matrix that may be valid even in the presence of some
misspecifications. Intermediate calculations use the 'oim'
method.
- 'robust_approx' is the same as 'robust' except that the
intermediate calculations use the 'approx' method.
- 'none' for no covariance matrix calculation.
Default is 'opg' unless memory conservation is used to avoid
computing the loglikelihood values for each observation, in which
case the default is 'oim'.
cov_kwds : dict or None, optional
A dictionary of arguments affecting covariance matrix computation.
**opg, oim, approx, robust, robust_approx**
- 'approx_complex_step' : bool, optional - If True, numerical
approximations are computed using complex-step methods. If False,
numerical approximations are computed using finite difference
methods. Default is True.
- 'approx_centered' : bool, optional - If True, numerical
approximations computed using finite difference methods use a
centered approximation. Default is False.
return_params : bool, optional
Whether or not to return only the array of maximizing parameters.
Default is False.
low_memory : bool, optional
If set to True, techniques are applied to substantially reduce
memory usage. If used, some features of the results object will
not be available (including smoothed results and in-sample
prediction), although out-of-sample forecasting is possible.
Default is False.
Returns
-------
ARIMAResults
Examples
--------
>>> mod = sm.tsa.arima.ARIMA(endog, order=(1, 0, 0))
>>> res = mod.fit()
>>> print(res.summary())
"""
# Determine which method to use
# 1. If method is specified, make sure it is valid
if method is not None:
self._spec_arima.validate_estimator(method)
# 2. Otherwise, use state space
# TODO: may want to consider using innovations (MLE) if possible here,
# (since in some cases it may be faster than state space), but it is
# less tested.
else:
method = 'statespace'
# Can only use fixed parameters with method='statespace'
if self._has_fixed_params and method != 'statespace':
raise ValueError('When parameters have been fixed, only the method'
' "statespace" can be used; got "%s".' % method)
# Handle kwargs related to the fit method
if method_kwargs is None:
method_kwargs = {}
required_kwargs = []
if method == 'statespace':
required_kwargs = [
'enforce_stationarity', 'enforce_invertibility',
'concentrate_scale'
]
elif method == 'innovations_mle':
required_kwargs = ['enforce_invertibility']
for name in required_kwargs:
if name in method_kwargs:
raise ValueError('Cannot override model level value for "%s"'
' when method="%s".' % (name, method))
method_kwargs[name] = getattr(self, name)
# Handle kwargs related to GLS estimation
if gls_kwargs is None:
gls_kwargs = {}
# Handle starting parameters
# TODO: maybe should have standard way of computing starting
# parameters in this class?
if start_params is not None:
if method not in ['statespace', 'innovations_mle']:
raise ValueError('Estimation method "%s" does not use starting'
' parameters, but `start_params` argument was'
' given.' % method)
method_kwargs['start_params'] = start_params
method_kwargs['transformed'] = transformed
method_kwargs['includes_fixed'] = includes_fixed
# Perform estimation, depending on whether we have exog or not
p = None
fit_details = None
has_exog = self._spec_arima.exog is not None
if has_exog or method == 'statespace':
# Use GLS if it was explicitly requested (`gls = True`) or if it
# was left at the default (`gls = None`) and the ARMA estimator is
# anything but statespace.
# Note: both GLS and statespace are able to handle models with
# integration, so we don't need to difference endog or exog here.
if has_exog and (gls or (gls is None and method != 'statespace')):
p, fit_details = estimate_gls(
self.endog,
exog=self.exog,
order=self.order,
seasonal_order=self.seasonal_order,
include_constant=False,
arma_estimator=method,
arma_estimator_kwargs=method_kwargs,
**gls_kwargs)
elif method != 'statespace':
raise ValueError('If `exog` is given and GLS is disabled'
' (`gls=False`), then the only valid'
" method is 'statespace'. Got '%s'." % method)
else:
method_kwargs.setdefault('disp', 0)
res = super(ARIMA, self).fit(return_params=return_params,
low_memory=low_memory,
cov_type=cov_type,
cov_kwds=cov_kwds,
**method_kwargs)
if not return_params:
res.fit_details = res.mlefit
else:
# Handle differencing if we have an integrated model
# (these methods do not support handling integration internally,
# so we need to manually do the differencing)
endog = self.endog
order = self._spec_arima.order
seasonal_order = self._spec_arima.seasonal_order
if self._spec_arima.is_integrated:
warnings.warn('Provided `endog` series has been differenced'
' to eliminate integration prior to parameter'
' estimation by method "%s".' % method)
endog = diff(
endog,
k_diff=self._spec_arima.diff,
k_seasonal_diff=self._spec_arima.seasonal_diff,
seasonal_periods=self._spec_arima.seasonal_periods)
if order[1] > 0:
order = (order[0], 0, order[2])
if seasonal_order[1] > 0:
seasonal_order = (seasonal_order[0], 0, seasonal_order[2],
seasonal_order[3])
# Now, estimate parameters
if method == 'yule_walker':
p, fit_details = yule_walker(endog,
ar_order=order[0],
demean=False,
**method_kwargs)
elif method == 'burg':
p, fit_details = burg(endog,
ar_order=order[0],
demean=False,
**method_kwargs)
elif method == 'hannan_rissanen':
p, fit_details = hannan_rissanen(endog,
ar_order=order[0],
ma_order=order[2],
demean=False,
**method_kwargs)
elif method == 'innovations':
p, fit_details = innovations(endog,
ma_order=order[2],
demean=False,
**method_kwargs)
# innovations computes estimates through the given order, so
# we want to take the estimate associated with the given order
p = p[-1]
elif method == 'innovations_mle':
p, fit_details = innovations_mle(endog,
order=order,
seasonal_order=seasonal_order,
demean=False,
**method_kwargs)
# In all cases except method='statespace', we now need to extract the
# parameters and, optionally, create a new results object
if p is not None:
# Need to check that fitted parameters satisfy given restrictions
if (self.enforce_stationarity
and self._spec_arima.max_reduced_ar_order > 0
and not p.is_stationary):
raise ValueError('Non-stationary autoregressive parameters'
' found with `enforce_stationarity=True`.'
' Consider setting it to False or using a'
' different estimation method, such as'
' method="statespace".')
if (self.enforce_invertibility
and self._spec_arima.max_reduced_ma_order > 0
and not p.is_invertible):
raise ValueError('Non-invertible moving average parameters'
' found with `enforce_invertibility=True`.'
' Consider setting it to False or using a'
' different estimation method, such as'
' method="statespace".')
# Build the requested results
if return_params:
res = p.params
else:
# Handle memory conservation option
if low_memory:
conserve_memory = self.ssm.conserve_memory
self.ssm.set_conserve_memory(MEMORY_CONSERVE)
# Perform filtering / smoothing
if (self.ssm.memory_no_predicted or self.ssm.memory_no_gain
or self.ssm.memory_no_smoothing):
func = self.filter
else:
func = self.smooth
res = func(p.params,
transformed=True,
includes_fixed=True,
cov_type=cov_type,
cov_kwds=cov_kwds)
# Save any details from the fit method
res.fit_details = fit_details
# Reset memory conservation
if low_memory:
self.ssm.set_conserve_memory(conserve_memory)
return res
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
