# -*- coding: utf-8 -*-
"""
Dynamic factor model
Author: Chad Fulton
License: Simplified-BSD
"""
from collections import OrderedDict
import numpy as np
from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper
from .tools import (
is_invertible, prepare_exog,
constrain_stationary_univariate, unconstrain_stationary_univariate,
constrain_stationary_multivariate, unconstrain_stationary_multivariate
)
from statsmodels.multivariate.pca import PCA
from statsmodels.regression.linear_model import OLS
from statsmodels.tsa.vector_ar.var_model import VAR
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tools.tools import Bunch
from statsmodels.tools.data import _is_using_pandas
from statsmodels.tsa.tsatools import lagmat
from statsmodels.tools.decorators import cache_readonly
import statsmodels.base.wrapper as wrap
from statsmodels.compat.pandas import Appender
[docs]class DynamicFactor(MLEModel):
r"""
Dynamic factor model
Parameters
----------
endog : array_like
The observed time-series process :math:`y`
exog : array_like, optional
Array of exogenous regressors for the observation equation, shaped
nobs x k_exog.
k_factors : int
The number of unobserved factors.
factor_order : int
The order of the vector autoregression followed by the factors.
error_cov_type : {'scalar', 'diagonal', 'unstructured'}, optional
The structure of the covariance matrix of the observation error term,
where "unstructured" puts no restrictions on the matrix, "diagonal"
requires it to be any diagonal matrix (uncorrelated errors), and
"scalar" requires it to be a scalar times the identity matrix. Default
is "diagonal".
error_order : int, optional
The order of the vector autoregression followed by the observation
error component. Default is None, corresponding to white noise errors.
error_var : bool, optional
Whether or not to model the errors jointly via a vector autoregression,
rather than as individual autoregressions. Has no effect unless
`error_order` is set. Default is False.
enforce_stationarity : bool, optional
Whether or not to transform the AR parameters to enforce stationarity
in the autoregressive component of the model. Default is True.
**kwargs
Keyword arguments may be used to provide default values for state space
matrices or for Kalman filtering options. See `Representation`, and
`KalmanFilter` for more details.
Attributes
----------
exog : array_like, optional
Array of exogenous regressors for the observation equation, shaped
nobs x k_exog.
k_factors : int
The number of unobserved factors.
factor_order : int
The order of the vector autoregression followed by the factors.
error_cov_type : {'diagonal', 'unstructured'}
The structure of the covariance matrix of the error term, where
"unstructured" puts no restrictions on the matrix and "diagonal"
requires it to be a diagonal matrix (uncorrelated errors).
error_order : int
The order of the vector autoregression followed by the observation
error component.
error_var : bool
Whether or not to model the errors jointly via a vector autoregression,
rather than as individual autoregressions. Has no effect unless
`error_order` is set.
enforce_stationarity : bool, optional
Whether or not to transform the AR parameters to enforce stationarity
in the autoregressive component of the model. Default is True.
Notes
-----
The dynamic factor model considered here is in the so-called static form,
and is specified:
.. math::
y_t & = \Lambda f_t + B x_t + u_t \\
f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \\
u_t & = C_1 u_{t-1} + \dots + C_1 f_{t-q} + \varepsilon_t
where there are `k_endog` observed series and `k_factors` unobserved
factors. Thus :math:`y_t` is a `k_endog` x 1 vector and :math:`f_t` is a
`k_factors` x 1 vector.
:math:`x_t` are optional exogenous vectors, shaped `k_exog` x 1.
:math:`\eta_t` and :math:`\varepsilon_t` are white noise error terms. In
order to identify the factors, :math:`Var(\eta_t) = I`. Denote
:math:`Var(\varepsilon_t) \equiv \Sigma`.
Options related to the unobserved factors:
- `k_factors`: this is the dimension of the vector :math:`f_t`, above.
To exclude factors completely, set `k_factors = 0`.
- `factor_order`: this is the number of lags to include in the factor
evolution equation, and corresponds to :math:`p`, above. To have static
factors, set `factor_order = 0`.
Options related to the observation error term :math:`u_t`:
- `error_order`: the number of lags to include in the error evolution
equation; corresponds to :math:`q`, above. To have white noise errors,
set `error_order = 0` (this is the default).
- `error_cov_type`: this controls the form of the covariance matrix
:math:`\Sigma`. If it is "dscalar", then :math:`\Sigma = \sigma^2 I`. If
it is "diagonal", then
:math:`\Sigma = \text{diag}(\sigma_1^2, \dots, \sigma_n^2)`. If it is
"unstructured", then :math:`\Sigma` is any valid variance / covariance
matrix (i.e. symmetric and positive definite).
- `error_var`: this controls whether or not the errors evolve jointly
according to a VAR(q), or individually according to separate AR(q)
processes. In terms of the formulation above, if `error_var = False`,
then the matrices :math:C_i` are diagonal, otherwise they are general
VAR matrices.
References
----------
.. [*] Lütkepohl, Helmut. 2007.
New Introduction to Multiple Time Series Analysis.
Berlin: Springer.
"""
def __init__(self, endog, k_factors, factor_order, exog=None,
error_order=0, error_var=False, error_cov_type='diagonal',
enforce_stationarity=True, **kwargs):
# Model properties
self.enforce_stationarity = enforce_stationarity
# Factor-related properties
self.k_factors = k_factors
self.factor_order = factor_order
# Error-related properties
self.error_order = error_order
self.error_var = error_var and error_order > 0
self.error_cov_type = error_cov_type
# Exogenous data
(self.k_exog, exog) = prepare_exog(exog)
# Note: at some point in the future might add state regression, as in
# SARIMAX.
self.mle_regression = self.k_exog > 0
# We need to have an array or pandas at this point
if not _is_using_pandas(endog, None):
endog = np.asanyarray(endog, order='C')
# Save some useful model orders, internally used
k_endog = endog.shape[1] if endog.ndim > 1 else 1
self._factor_order = max(1, self.factor_order) * self.k_factors
self._error_order = self.error_order * k_endog
# Calculate the number of states
k_states = self._factor_order
k_posdef = self.k_factors
if self.error_order > 0:
k_states += self._error_order
k_posdef += k_endog
# We can still estimate the model with no dynamic state (e.g. SUR), we
# just need to have one state that does nothing.
self._unused_state = False
if k_states == 0:
k_states = 1
k_posdef = 1
self._unused_state = True
# Test for non-multivariate endog
if k_endog < 2:
raise ValueError('The dynamic factors model is only valid for'
' multivariate time series.')
# Test for too many factors
if self.k_factors >= k_endog:
raise ValueError('Number of factors must be less than the number'
' of endogenous variables.')
# Test for invalid error_cov_type
if self.error_cov_type not in ['scalar', 'diagonal', 'unstructured']:
raise ValueError('Invalid error covariance matrix type'
' specification.')
# By default, initialize as stationary
kwargs.setdefault('initialization', 'stationary')
# Initialize the state space model
super(DynamicFactor, self).__init__(
endog, exog=exog, k_states=k_states, k_posdef=k_posdef, **kwargs
)
# Set as time-varying model if we have exog
if self.k_exog > 0:
self.ssm._time_invariant = False
# Initialize the components
self.parameters = OrderedDict()
self._initialize_loadings()
self._initialize_exog()
self._initialize_error_cov()
self._initialize_factor_transition()
self._initialize_error_transition()
self.k_params = sum(self.parameters.values())
# Cache parameter vector slices
def _slice(key, offset):
length = self.parameters[key]
param_slice = np.s_[offset:offset + length]
offset += length
return param_slice, offset
offset = 0
self._params_loadings, offset = _slice('factor_loadings', offset)
self._params_exog, offset = _slice('exog', offset)
self._params_error_cov, offset = _slice('error_cov', offset)
self._params_factor_transition, offset = (
_slice('factor_transition', offset))
self._params_error_transition, offset = (
_slice('error_transition', offset))
# Update _init_keys attached by super
self._init_keys += ['k_factors', 'factor_order', 'error_order',
'error_var', 'error_cov_type',
'enforce_stationarity'] + list(kwargs.keys())
def _initialize_loadings(self):
# Initialize the parameters
self.parameters['factor_loadings'] = self.k_endog * self.k_factors
# Setup fixed components of state space matrices
if self.error_order > 0:
start = self._factor_order
end = self._factor_order + self.k_endog
self.ssm['design', :, start:end] = np.eye(self.k_endog)
# Setup indices of state space matrices
self._idx_loadings = np.s_['design', :, :self.k_factors]
def _initialize_exog(self):
# Initialize the parameters
self.parameters['exog'] = self.k_exog * self.k_endog
# If we have exog effects, then the obs intercept needs to be
# time-varying
if self.k_exog > 0:
self.ssm['obs_intercept'] = np.zeros((self.k_endog, self.nobs))
# Setup indices of state space matrices
self._idx_exog = np.s_['obs_intercept', :self.k_endog, :]
def _initialize_error_cov(self):
if self.error_cov_type == 'scalar':
self._initialize_error_cov_diagonal(scalar=True)
elif self.error_cov_type == 'diagonal':
self._initialize_error_cov_diagonal(scalar=False)
elif self.error_cov_type == 'unstructured':
self._initialize_error_cov_unstructured()
def _initialize_error_cov_diagonal(self, scalar=False):
# Initialize the parameters
self.parameters['error_cov'] = 1 if scalar else self.k_endog
# Setup fixed components of state space matrices
# Setup indices of state space matrices
k_endog = self.k_endog
k_factors = self.k_factors
idx = np.diag_indices(k_endog)
if self.error_order > 0:
matrix = 'state_cov'
idx = (idx[0] + k_factors, idx[1] + k_factors)
else:
matrix = 'obs_cov'
self._idx_error_cov = (matrix,) + idx
def _initialize_error_cov_unstructured(self):
# Initialize the parameters
k_endog = self.k_endog
self.parameters['error_cov'] = int(k_endog * (k_endog + 1) / 2)
# Setup fixed components of state space matrices
# Setup indices of state space matrices
self._idx_lower_error_cov = np.tril_indices(self.k_endog)
if self.error_order > 0:
start = self.k_factors
end = self.k_factors + self.k_endog
self._idx_error_cov = (
np.s_['state_cov', start:end, start:end])
else:
self._idx_error_cov = np.s_['obs_cov', :, :]
def _initialize_factor_transition(self):
order = self.factor_order * self.k_factors
k_factors = self.k_factors
# Initialize the parameters
self.parameters['factor_transition'] = (
self.factor_order * self.k_factors**2)
# Setup fixed components of state space matrices
# VAR(p) for factor transition
if self.k_factors > 0:
if self.factor_order > 0:
self.ssm['transition', k_factors:order, :order - k_factors] = (
np.eye(order - k_factors))
self.ssm['selection', :k_factors, :k_factors] = np.eye(k_factors)
# Identification requires constraining the state covariance to an
# identity matrix
self.ssm['state_cov', :k_factors, :k_factors] = np.eye(k_factors)
# Setup indices of state space matrices
self._idx_factor_transition = np.s_['transition', :k_factors, :order]
def _initialize_error_transition(self):
# Initialize the appropriate situation
if self.error_order == 0:
self._initialize_error_transition_white_noise()
else:
# Generic setup fixed components of state space matrices
# VAR(q) for error transition
# (in the individual AR case, we still have the VAR(q) companion
# matrix structure, but force the coefficient matrices to be
# diagonal)
k_endog = self.k_endog
k_factors = self.k_factors
_factor_order = self._factor_order
_error_order = self._error_order
_slice = np.s_['selection',
_factor_order:_factor_order + k_endog,
k_factors:k_factors + k_endog]
self.ssm[_slice] = np.eye(k_endog)
_slice = np.s_[
'transition',
_factor_order + k_endog:_factor_order + _error_order,
_factor_order:_factor_order + _error_order - k_endog]
self.ssm[_slice] = np.eye(_error_order - k_endog)
# Now specialized setups
if self.error_var:
self._initialize_error_transition_var()
else:
self._initialize_error_transition_individual()
def _initialize_error_transition_white_noise(self):
# Initialize the parameters
self.parameters['error_transition'] = 0
# No fixed components of state space matrices
# Setup indices of state space matrices (just an empty slice)
self._idx_error_transition = np.s_['transition', 0:0, 0:0]
def _initialize_error_transition_var(self):
k_endog = self.k_endog
_factor_order = self._factor_order
_error_order = self._error_order
# Initialize the parameters
self.parameters['error_transition'] = _error_order * k_endog
# Fixed components already setup above
# Setup indices of state space matrices
# Here we want to set all of the elements of the coefficient matrices,
# the same as in a VAR specification
self._idx_error_transition = np.s_[
'transition',
_factor_order:_factor_order + k_endog,
_factor_order:_factor_order + _error_order]
def _initialize_error_transition_individual(self):
k_endog = self.k_endog
_error_order = self._error_order
# Initialize the parameters
self.parameters['error_transition'] = _error_order
# Fixed components already setup above
# Setup indices of state space matrices
# Here we want to set only the diagonal elements of the coefficient
# matrices, and we want to set them in order by equation, not by
# matrix (i.e. set the first element of the first matrix's diagonal,
# then set the first element of the second matrix's diagonal, then...)
# The basic setup is a tiled list of diagonal indices, one for each
# coefficient matrix
idx = np.tile(np.diag_indices(k_endog), self.error_order)
# Now we need to shift the rows down to the correct location
row_shift = self._factor_order
# And we need to shift the columns in an increasing way
col_inc = self._factor_order + np.repeat(
[i * k_endog for i in range(self.error_order)], k_endog)
idx[0] += row_shift
idx[1] += col_inc
# Make a copy (without the row shift) so that we can easily get the
# diagonal parameters back out of a generic coefficients matrix array
idx_diag = idx.copy()
idx_diag[0] -= row_shift
idx_diag[1] -= self._factor_order
idx_diag = idx_diag[:, np.lexsort((idx_diag[1], idx_diag[0]))]
self._idx_error_diag = (idx_diag[0], idx_diag[1])
# Finally, we want to fill the entries in in the correct order, which
# is to say we want to fill in lexicographically, first by row then by
# column
idx = idx[:, np.lexsort((idx[1], idx[0]))]
self._idx_error_transition = np.s_['transition', idx[0], idx[1]]
[docs] def clone(self, endog, exog=None, **kwargs):
return self._clone_from_init_kwds(endog, exog=exog, **kwargs)
@property
def _res_classes(self):
return {'fit': (DynamicFactorResults, DynamicFactorResultsWrapper)}
@property
def start_params(self):
params = np.zeros(self.k_params, dtype=np.float64)
endog = self.endog.copy()
mask = ~np.any(np.isnan(endog), axis=1)
endog = endog[mask]
# 1. Factor loadings (estimated via PCA)
if self.k_factors > 0:
# Use principal components + OLS as starting values
res_pca = PCA(endog, ncomp=self.k_factors)
mod_ols = OLS(endog, res_pca.factors)
res_ols = mod_ols.fit()
# Using OLS params for the loadings tends to gives higher starting
# log-likelihood.
params[self._params_loadings] = res_ols.params.T.ravel()
# params[self._params_loadings] = res_pca.loadings.ravel()
# However, using res_ols.resid tends to causes non-invertible
# starting VAR coefficients for error VARs
# endog = res_ols.resid
endog = endog - np.dot(res_pca.factors, res_pca.loadings.T)
# 2. Exog (OLS on residuals)
if self.k_exog > 0:
mod_ols = OLS(endog, exog=self.exog)
res_ols = mod_ols.fit()
# In the form: beta.x1.y1, beta.x2.y1, beta.x1.y2, ...
params[self._params_exog] = res_ols.params.T.ravel()
endog = res_ols.resid
# 3. Factors (VAR on res_pca.factors)
stationary = True
if self.k_factors > 1 and self.factor_order > 0:
# 3a. VAR transition (OLS on factors estimated via PCA)
mod_factors = VAR(res_pca.factors)
res_factors = mod_factors.fit(maxlags=self.factor_order, ic=None,
trend='nc')
# Save the parameters
params[self._params_factor_transition] = (
res_factors.params.T.ravel())
# Test for stationarity
coefficient_matrices = (
params[self._params_factor_transition].reshape(
self.k_factors * self.factor_order, self.k_factors
).T
).reshape(self.k_factors, self.k_factors, self.factor_order).T
stationary = is_invertible([1] + list(-coefficient_matrices))
elif self.k_factors > 0 and self.factor_order > 0:
# 3b. AR transition
Y = res_pca.factors[self.factor_order:]
X = lagmat(res_pca.factors, self.factor_order, trim='both')
params_ar = np.linalg.pinv(X).dot(Y)
stationary = is_invertible(np.r_[1, -params_ar.squeeze()])
params[self._params_factor_transition] = params_ar[:, 0]
# Check for stationarity
if not stationary and self.enforce_stationarity:
raise ValueError('Non-stationary starting autoregressive'
' parameters found with `enforce_stationarity`'
' set to True.')
# 4. Errors
if self.error_order == 0:
if self.error_cov_type == 'scalar':
params[self._params_error_cov] = endog.var(axis=0).mean()
elif self.error_cov_type == 'diagonal':
params[self._params_error_cov] = endog.var(axis=0)
elif self.error_cov_type == 'unstructured':
cov_factor = np.diag(endog.std(axis=0))
params[self._params_error_cov] = (
cov_factor[self._idx_lower_error_cov].ravel())
elif self.error_var:
mod_errors = VAR(endog)
res_errors = mod_errors.fit(maxlags=self.error_order, ic=None,
trend='nc')
# Test for stationarity
coefficient_matrices = (
np.array(res_errors.params.T).ravel().reshape(
self.k_endog * self.error_order, self.k_endog
).T
).reshape(self.k_endog, self.k_endog, self.error_order).T
stationary = is_invertible([1] + list(-coefficient_matrices))
if not stationary and self.enforce_stationarity:
raise ValueError('Non-stationary starting error autoregressive'
' parameters found with'
' `enforce_stationarity` set to True.')
# Get the error autoregressive parameters
params[self._params_error_transition] = (
np.array(res_errors.params.T).ravel())
# Get the error covariance parameters
if self.error_cov_type == 'scalar':
params[self._params_error_cov] = (
res_errors.sigma_u.diagonal().mean())
elif self.error_cov_type == 'diagonal':
params[self._params_error_cov] = res_errors.sigma_u.diagonal()
elif self.error_cov_type == 'unstructured':
try:
cov_factor = np.linalg.cholesky(res_errors.sigma_u)
except np.linalg.LinAlgError:
cov_factor = np.eye(res_errors.sigma_u.shape[0]) * (
res_errors.sigma_u.diagonal().mean()**0.5)
cov_factor = np.eye(res_errors.sigma_u.shape[0]) * (
res_errors.sigma_u.diagonal().mean()**0.5)
params[self._params_error_cov] = (
cov_factor[self._idx_lower_error_cov].ravel())
else:
error_ar_params = []
error_cov_params = []
for i in range(self.k_endog):
mod_error = ARIMA(endog[:, i], order=(self.error_order, 0, 0),
trend='n', enforce_stationarity=True)
res_error = mod_error.fit(method='burg')
error_ar_params += res_error.params[:self.error_order].tolist()
error_cov_params += res_error.params[-1:].tolist()
params[self._params_error_transition] = np.r_[error_ar_params]
params[self._params_error_cov] = np.r_[error_cov_params]
return params
@property
def param_names(self):
param_names = []
endog_names = self.endog_names
# 1. Factor loadings
param_names += [
'loading.f%d.%s' % (j+1, endog_names[i])
for i in range(self.k_endog)
for j in range(self.k_factors)
]
# 2. Exog
# Recall these are in the form: beta.x1.y1, beta.x2.y1, beta.x1.y2, ...
param_names += [
'beta.%s.%s' % (self.exog_names[j], endog_names[i])
for i in range(self.k_endog)
for j in range(self.k_exog)
]
# 3. Error covariances
if self.error_cov_type == 'scalar':
param_names += ['sigma2']
elif self.error_cov_type == 'diagonal':
param_names += [
'sigma2.%s' % endog_names[i]
for i in range(self.k_endog)
]
elif self.error_cov_type == 'unstructured':
param_names += [
'cov.chol[%d,%d]' % (i + 1, j + 1)
for i in range(self.k_endog)
for j in range(i+1)
]
# 4. Factor transition VAR
param_names += [
'L%d.f%d.f%d' % (i+1, k+1, j+1)
for j in range(self.k_factors)
for i in range(self.factor_order)
for k in range(self.k_factors)
]
# 5. Error transition VAR
if self.error_var:
param_names += [
'L%d.e(%s).e(%s)' % (i+1, endog_names[k], endog_names[j])
for j in range(self.k_endog)
for i in range(self.error_order)
for k in range(self.k_endog)
]
else:
param_names += [
'L%d.e(%s).e(%s)' % (i+1, endog_names[j], endog_names[j])
for j in range(self.k_endog)
for i in range(self.error_order)
]
return param_names
@property
def state_names(self):
names = []
endog_names = self.endog_names
# Factors and lags
names += [
(('f%d' % (j + 1)) if i == 0 else ('f%d.L%d' % (j + 1, i)))
for i in range(max(1, self.factor_order))
for j in range(self.k_factors)]
if self.error_order > 0:
names += [
(('e(%s)' % endog_names[j]) if i == 0
else ('e(%s).L%d' % (endog_names[j], i)))
for i in range(self.error_order)
for j in range(self.k_endog)]
if self._unused_state:
names += ['dummy']
return names
def transform_params(self, unconstrained):
"""
Transform unconstrained parameters used by the optimizer to constrained
parameters used in likelihood evaluation
Parameters
----------
unconstrained : array_like
Array of unconstrained parameters used by the optimizer, to be
transformed.
Returns
-------
constrained : array_like
Array of constrained parameters which may be used in likelihood
evaluation.
Notes
-----
Constrains the factor transition to be stationary and variances to be
positive.
"""
unconstrained = np.array(unconstrained, ndmin=1)
dtype = unconstrained.dtype
constrained = np.zeros(unconstrained.shape, dtype=dtype)
# 1. Factor loadings
# The factor loadings do not need to be adjusted
constrained[self._params_loadings] = (
unconstrained[self._params_loadings])
# 2. Exog
# The regression coefficients do not need to be adjusted
constrained[self._params_exog] = (
unconstrained[self._params_exog])
# 3. Error covariances
# If we have variances, force them to be positive
if self.error_cov_type in ['scalar', 'diagonal']:
constrained[self._params_error_cov] = (
unconstrained[self._params_error_cov]**2)
# Otherwise, nothing needs to be done
elif self.error_cov_type == 'unstructured':
constrained[self._params_error_cov] = (
unconstrained[self._params_error_cov])
# 4. Factor transition VAR
# VAR transition: optionally force to be stationary
if self.enforce_stationarity and self.factor_order > 0:
# Transform the parameters
unconstrained_matrices = (
unconstrained[self._params_factor_transition].reshape(
self.k_factors, self._factor_order))
# This is always an identity matrix, but because the transform
# done prior to update (where the ssm representation matrices
# change), it may be complex
cov = self.ssm['state_cov', :self.k_factors, :self.k_factors].real
coefficient_matrices, variance = (
constrain_stationary_multivariate(unconstrained_matrices, cov))
constrained[self._params_factor_transition] = (
coefficient_matrices.ravel())
else:
constrained[self._params_factor_transition] = (
unconstrained[self._params_factor_transition])
# 5. Error transition VAR
# VAR transition: optionally force to be stationary
if self.enforce_stationarity and self.error_order > 0:
# Joint VAR specification
if self.error_var:
unconstrained_matrices = (
unconstrained[self._params_error_transition].reshape(
self.k_endog, self._error_order))
start = self.k_factors
end = self.k_factors + self.k_endog
cov = self.ssm['state_cov', start:end, start:end].real
coefficient_matrices, variance = (
constrain_stationary_multivariate(
unconstrained_matrices, cov))
constrained[self._params_error_transition] = (
coefficient_matrices.ravel())
# Separate AR specifications
else:
coefficients = (
unconstrained[self._params_error_transition].copy())
for i in range(self.k_endog):
start = i * self.error_order
end = (i + 1) * self.error_order
coefficients[start:end] = constrain_stationary_univariate(
coefficients[start:end])
constrained[self._params_error_transition] = coefficients
else:
constrained[self._params_error_transition] = (
unconstrained[self._params_error_transition])
return constrained
def untransform_params(self, constrained):
"""
Transform constrained parameters used in likelihood evaluation
to unconstrained parameters used by the optimizer.
Parameters
----------
constrained : array_like
Array of constrained parameters used in likelihood evaluation, to
be transformed.
Returns
-------
unconstrained : array_like
Array of unconstrained parameters used by the optimizer.
"""
constrained = np.array(constrained, ndmin=1)
dtype = constrained.dtype
unconstrained = np.zeros(constrained.shape, dtype=dtype)
# 1. Factor loadings
# The factor loadings do not need to be adjusted
unconstrained[self._params_loadings] = (
constrained[self._params_loadings])
# 2. Exog
# The regression coefficients do not need to be adjusted
unconstrained[self._params_exog] = (
constrained[self._params_exog])
# 3. Error covariances
# If we have variances, force them to be positive
if self.error_cov_type in ['scalar', 'diagonal']:
unconstrained[self._params_error_cov] = (
constrained[self._params_error_cov]**0.5)
# Otherwise, nothing needs to be done
elif self.error_cov_type == 'unstructured':
unconstrained[self._params_error_cov] = (
constrained[self._params_error_cov])
# 3. Factor transition VAR
# VAR transition: optionally force to be stationary
if self.enforce_stationarity and self.factor_order > 0:
# Transform the parameters
constrained_matrices = (
constrained[self._params_factor_transition].reshape(
self.k_factors, self._factor_order))
cov = self.ssm['state_cov', :self.k_factors, :self.k_factors].real
coefficient_matrices, variance = (
unconstrain_stationary_multivariate(
constrained_matrices, cov))
unconstrained[self._params_factor_transition] = (
coefficient_matrices.ravel())
else:
unconstrained[self._params_factor_transition] = (
constrained[self._params_factor_transition])
# 5. Error transition VAR
# VAR transition: optionally force to be stationary
if self.enforce_stationarity and self.error_order > 0:
# Joint VAR specification
if self.error_var:
constrained_matrices = (
constrained[self._params_error_transition].reshape(
self.k_endog, self._error_order))
start = self.k_factors
end = self.k_factors + self.k_endog
cov = self.ssm['state_cov', start:end, start:end].real
coefficient_matrices, variance = (
unconstrain_stationary_multivariate(
constrained_matrices, cov))
unconstrained[self._params_error_transition] = (
coefficient_matrices.ravel())
# Separate AR specifications
else:
coefficients = (
constrained[self._params_error_transition].copy())
for i in range(self.k_endog):
start = i * self.error_order
end = (i + 1) * self.error_order
coefficients[start:end] = (
unconstrain_stationary_univariate(
coefficients[start:end]))
unconstrained[self._params_error_transition] = coefficients
else:
unconstrained[self._params_error_transition] = (
constrained[self._params_error_transition])
return unconstrained
def _validate_can_fix_params(self, param_names):
super(DynamicFactor, self)._validate_can_fix_params(param_names)
ix = np.cumsum(list(self.parameters.values()))[:-1]
(_, _, _, factor_transition_names, error_transition_names) = [
arr.tolist() for arr in np.array_split(self.param_names, ix)]
if self.enforce_stationarity and self.factor_order > 0:
if self.k_factors > 1 or self.factor_order > 1:
fix_all = param_names.issuperset(factor_transition_names)
fix_any = (
len(param_names.intersection(factor_transition_names)) > 0)
if fix_any and not fix_all:
raise ValueError(
'Cannot fix individual factor transition parameters'
' when `enforce_stationarity=True`. In this case,'
' must either fix all factor transition parameters or'
' none.')
if self.enforce_stationarity and self.error_order > 0:
if self.error_var or self.error_order > 1:
fix_all = param_names.issuperset(error_transition_names)
fix_any = (
len(param_names.intersection(error_transition_names)) > 0)
if fix_any and not fix_all:
raise ValueError(
'Cannot fix individual error transition parameters'
' when `enforce_stationarity=True`. In this case,'
' must either fix all error transition parameters or'
' none.')
def update(self, params, transformed=True, includes_fixed=False,
complex_step=False):
"""
Update the parameters of the model
Updates the representation matrices to fill in the new parameter
values.
Parameters
----------
params : array_like
Array of new parameters.
transformed : bool, optional
Whether or not `params` is already transformed. If set to False,
`transform_params` is called. Default is True..
Returns
-------
params : array_like
Array of parameters.
Notes
-----
Let `n = k_endog`, `m = k_factors`, and `p = factor_order`. Then the
`params` vector has length
:math:`[n \times m] + [n] + [m^2 \times p]`.
It is expanded in the following way:
- The first :math:`n \times m` parameters fill out the factor loading
matrix, starting from the [0,0] entry and then proceeding along rows.
These parameters are not modified in `transform_params`.
- The next :math:`n` parameters provide variances for the error_cov
errors in the observation equation. They fill in the diagonal of the
observation covariance matrix, and are constrained to be positive by
`transofrm_params`.
- The next :math:`m^2 \times p` parameters are used to create the `p`
coefficient matrices for the vector autoregression describing the
factor transition. They are transformed in `transform_params` to
enforce stationarity of the VAR(p). They are placed so as to make
the transition matrix a companion matrix for the VAR. In particular,
we assume that the first :math:`m^2` parameters fill the first
coefficient matrix (starting at [0,0] and filling along rows), the
second :math:`m^2` parameters fill the second matrix, etc.
"""
params = self.handle_params(params, transformed=transformed,
includes_fixed=includes_fixed)
# 1. Factor loadings
# Update the design / factor loading matrix
self.ssm[self._idx_loadings] = (
params[self._params_loadings].reshape(self.k_endog, self.k_factors)
)
# 2. Exog
if self.k_exog > 0:
exog_params = params[self._params_exog].reshape(
self.k_endog, self.k_exog).T
self.ssm[self._idx_exog] = np.dot(self.exog, exog_params).T
# 3. Error covariances
if self.error_cov_type in ['scalar', 'diagonal']:
self.ssm[self._idx_error_cov] = (
params[self._params_error_cov])
elif self.error_cov_type == 'unstructured':
error_cov_lower = np.zeros((self.k_endog, self.k_endog),
dtype=params.dtype)
error_cov_lower[self._idx_lower_error_cov] = (
params[self._params_error_cov])
self.ssm[self._idx_error_cov] = (
np.dot(error_cov_lower, error_cov_lower.T))
# 4. Factor transition VAR
self.ssm[self._idx_factor_transition] = (
params[self._params_factor_transition].reshape(
self.k_factors, self.factor_order * self.k_factors))
# 5. Error transition VAR
if self.error_var:
self.ssm[self._idx_error_transition] = (
params[self._params_error_transition].reshape(
self.k_endog, self._error_order))
else:
self.ssm[self._idx_error_transition] = (
params[self._params_error_transition])
[docs]class DynamicFactorResults(MLEResults):
"""
Class to hold results from fitting an DynamicFactor model.
Parameters
----------
model : DynamicFactor instance
The fitted model instance
Attributes
----------
specification : dictionary
Dictionary including all attributes from the DynamicFactor model
instance.
coefficient_matrices_var : array
Array containing autoregressive lag polynomial coefficient matrices,
ordered from lowest degree to highest.
See Also
--------
statsmodels.tsa.statespace.kalman_filter.FilterResults
statsmodels.tsa.statespace.mlemodel.MLEResults
"""
def __init__(self, model, params, filter_results, cov_type=None,
**kwargs):
super(DynamicFactorResults, self).__init__(model, params,
filter_results, cov_type,
**kwargs)
self.df_resid = np.inf # attribute required for wald tests
self.specification = Bunch(**{
# Model properties
'k_endog': self.model.k_endog,
'enforce_stationarity': self.model.enforce_stationarity,
# Factor-related properties
'k_factors': self.model.k_factors,
'factor_order': self.model.factor_order,
# Error-related properties
'error_order': self.model.error_order,
'error_var': self.model.error_var,
'error_cov_type': self.model.error_cov_type,
# Other properties
'k_exog': self.model.k_exog
})
# Polynomials / coefficient matrices
self.coefficient_matrices_var = None
if self.model.factor_order > 0:
ar_params = (
np.array(self.params[self.model._params_factor_transition]))
k_factors = self.model.k_factors
factor_order = self.model.factor_order
self.coefficient_matrices_var = (
ar_params.reshape(k_factors * factor_order, k_factors).T
).reshape(k_factors, k_factors, factor_order).T
self.coefficient_matrices_error = None
if self.model.error_order > 0:
ar_params = (
np.array(self.params[self.model._params_error_transition]))
k_endog = self.model.k_endog
error_order = self.model.error_order
if self.model.error_var:
self.coefficient_matrices_error = (
ar_params.reshape(k_endog * error_order, k_endog).T
).reshape(k_endog, k_endog, error_order).T
else:
mat = np.zeros((k_endog, k_endog * error_order))
mat[self.model._idx_error_diag] = ar_params
self.coefficient_matrices_error = (
mat.T.reshape(error_order, k_endog, k_endog))
@property
def factors(self):
"""
Estimates of unobserved factors
Returns
-------
out : Bunch
Has the following attributes shown in Notes.
Notes
-----
The output is a bunch of the following format:
- `filtered`: a time series array with the filtered estimate of
the component
- `filtered_cov`: a time series array with the filtered estimate of
the variance/covariance of the component
- `smoothed`: a time series array with the smoothed estimate of
the component
- `smoothed_cov`: a time series array with the smoothed estimate of
the variance/covariance of the component
- `offset`: an integer giving the offset in the state vector where
this component begins
"""
# If present, level is always the first component of the state vector
out = None
spec = self.specification
if spec.k_factors > 0:
offset = 0
end = spec.k_factors
res = self.filter_results
out = Bunch(
filtered=res.filtered_state[offset:end],
filtered_cov=res.filtered_state_cov[offset:end, offset:end],
smoothed=None, smoothed_cov=None,
offset=offset)
if self.smoothed_state is not None:
out.smoothed = self.smoothed_state[offset:end]
if self.smoothed_state_cov is not None:
out.smoothed_cov = (
self.smoothed_state_cov[offset:end, offset:end])
return out
@cache_readonly
def coefficients_of_determination(self):
"""
Coefficients of determination (:math:`R^2`) from regressions of
individual estimated factors on endogenous variables.
Returns
-------
coefficients_of_determination : array
A `k_endog` x `k_factors` array, where
`coefficients_of_determination[i, j]` represents the :math:`R^2`
value from a regression of factor `j` and a constant on endogenous
variable `i`.
Notes
-----
Although it can be difficult to interpret the estimated factor loadings
and factors, it is often helpful to use the coefficients of
determination from univariate regressions to assess the importance of
each factor in explaining the variation in each endogenous variable.
In models with many variables and factors, this can sometimes lend
interpretation to the factors (for example sometimes one factor will
load primarily on real variables and another on nominal variables).
See Also
--------
plot_coefficients_of_determination
"""
from statsmodels.tools import add_constant
spec = self.specification
coefficients = np.zeros((spec.k_endog, spec.k_factors))
which = 'filtered' if self.smoothed_state is None else 'smoothed'
for i in range(spec.k_factors):
exog = add_constant(self.factors[which][i])
for j in range(spec.k_endog):
endog = self.filter_results.endog[j]
coefficients[j, i] = OLS(endog, exog).fit().rsquared
return coefficients
def plot_coefficients_of_determination(self, endog_labels=None,
fig=None, figsize=None):
"""
Plot the coefficients of determination
Parameters
----------
endog_labels : bool, optional
Whether or not to label the endogenous variables along the x-axis
of the plots. Default is to include labels if there are 5 or fewer
endogenous variables.
fig : Matplotlib Figure instance, optional
If given, subplots are created in this figure instead of in a new
figure. Note that the grid will be created in the provided
figure using `fig.add_subplot()`.
figsize : tuple, optional
If a figure is created, this argument allows specifying a size.
The tuple is (width, height).
Notes
-----
Produces a `k_factors` x 1 plot grid. The `i`th plot shows a bar plot
of the coefficients of determination associated with factor `i`. The
endogenous variables are arranged along the x-axis according to their
position in the `endog` array.
See Also
--------
coefficients_of_determination
"""
from statsmodels.graphics.utils import _import_mpl, create_mpl_fig
_import_mpl()
fig = create_mpl_fig(fig, figsize)
spec = self.specification
# Should we label endogenous variables?
if endog_labels is None:
endog_labels = spec.k_endog <= 5
# Plot the coefficients of determination
coefficients_of_determination = self.coefficients_of_determination
plot_idx = 1
locations = np.arange(spec.k_endog)
for coeffs in coefficients_of_determination.T:
# Create the new axis
ax = fig.add_subplot(spec.k_factors, 1, plot_idx)
ax.set_ylim((0, 1))
ax.set(title='Factor %i' % plot_idx, ylabel=r'$R^2$')
bars = ax.bar(locations, coeffs)
if endog_labels:
width = bars[0].get_width()
ax.xaxis.set_ticks(locations + width / 2)
ax.xaxis.set_ticklabels(self.model.endog_names)
else:
ax.set(xlabel='Endogenous variables')
ax.xaxis.set_ticks([])
plot_idx += 1
return fig
@Appender(MLEResults.summary.__doc__)
def summary(self, alpha=.05, start=None, separate_params=True):
from statsmodels.iolib.summary import summary_params
spec = self.specification
# Create the model name
model_name = []
if spec.k_factors > 0:
if spec.factor_order > 0:
model_type = ('DynamicFactor(factors=%d, order=%d)' %
(spec.k_factors, spec.factor_order))
else:
model_type = 'StaticFactor(factors=%d)' % spec.k_factors
model_name.append(model_type)
if spec.k_exog > 0:
model_name.append('%d regressors' % spec.k_exog)
else:
model_name.append('SUR(%d regressors)' % spec.k_exog)
if spec.error_order > 0:
error_type = 'VAR' if spec.error_var else 'AR'
model_name.append('%s(%d) errors' % (error_type, spec.error_order))
summary = super(DynamicFactorResults, self).summary(
alpha=alpha, start=start, model_name=model_name,
display_params=not separate_params
)
if separate_params:
indices = np.arange(len(self.params))
def make_table(self, mask, title, strip_end=True):
res = (self, self.params[mask], self.bse[mask],
self.zvalues[mask], self.pvalues[mask],
self.conf_int(alpha)[mask])
param_names = [
'.'.join(name.split('.')[:-1]) if strip_end else name
for name in
np.array(self.data.param_names)[mask].tolist()
]
return summary_params(res, yname=None, xname=param_names,
alpha=alpha, use_t=False, title=title)
k_endog = self.model.k_endog
k_exog = self.model.k_exog
k_factors = self.model.k_factors
factor_order = self.model.factor_order
_factor_order = self.model._factor_order
_error_order = self.model._error_order
# Add parameter tables for each endogenous variable
loading_indices = indices[self.model._params_loadings]
loading_masks = []
exog_indices = indices[self.model._params_exog]
exog_masks = []
for i in range(k_endog):
# 1. Factor loadings
# Recall these are in the form:
# 'loading.f1.y1', 'loading.f2.y1', 'loading.f1.y2', ...
loading_mask = (
loading_indices[i * k_factors:(i + 1) * k_factors])
loading_masks.append(loading_mask)
# 2. Exog
# Recall these are in the form:
# beta.x1.y1, beta.x2.y1, beta.x1.y2, ...
exog_mask = exog_indices[i * k_exog:(i + 1) * k_exog]
exog_masks.append(exog_mask)
# Create the table
mask = np.concatenate([loading_mask, exog_mask])
title = "Results for equation %s" % self.model.endog_names[i]
table = make_table(self, mask, title)
summary.tables.append(table)
# Add parameter tables for each factor
factor_indices = indices[self.model._params_factor_transition]
factor_masks = []
if factor_order > 0:
for i in range(k_factors):
start = i * _factor_order
factor_mask = factor_indices[start: start + _factor_order]
factor_masks.append(factor_mask)
# Create the table
title = "Results for factor equation f%d" % (i+1)
table = make_table(self, factor_mask, title)
summary.tables.append(table)
# Add parameter tables for error transitions
error_masks = []
if spec.error_order > 0:
error_indices = indices[self.model._params_error_transition]
for i in range(k_endog):
if spec.error_var:
start = i * _error_order
end = (i + 1) * _error_order
else:
start = i * spec.error_order
end = (i + 1) * spec.error_order
error_mask = error_indices[start:end]
error_masks.append(error_mask)
# Create the table
title = ("Results for error equation e(%s)" %
self.model.endog_names[i])
table = make_table(self, error_mask, title)
summary.tables.append(table)
# Error covariance terms
error_cov_mask = indices[self.model._params_error_cov]
table = make_table(self, error_cov_mask,
"Error covariance matrix", strip_end=False)
summary.tables.append(table)
# Add a table for all other parameters
masks = []
for m in (loading_masks, exog_masks, factor_masks,
error_masks, [error_cov_mask]):
m = np.array(m).flatten()
if len(m) > 0:
masks.append(m)
masks = np.concatenate(masks)
inverse_mask = np.array(list(set(indices).difference(set(masks))))
if len(inverse_mask) > 0:
table = make_table(self, inverse_mask, "Other parameters",
strip_end=False)
summary.tables.append(table)
return summary
class DynamicFactorResultsWrapper(MLEResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(DynamicFactorResultsWrapper, # noqa:E305
DynamicFactorResults)