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alkaline-ml / statsmodels python
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Version:
0.11.1 ▾
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# TODO: Determine which tests are valid for GLSAR, and under what conditions
# TODO: Fix issue with constant and GLS
# TODO: GLS: add options Iterative GLS, for iterative fgls if sigma is None
# TODO: GLS: default if sigma is none should be two-step GLS
# TODO: Check nesting when performing model based tests, lr, wald, lm
"""
This module implements standard regression models:
Generalized Least Squares (GLS)
Ordinary Least Squares (OLS)
Weighted Least Squares (WLS)
Generalized Least Squares with autoregressive error terms GLSAR(p)
Models are specified with an endogenous response variable and an
exogenous design matrix and are fit using their `fit` method.
Subclasses that have more complicated covariance matrices
should write over the 'whiten' method as the fit method
prewhitens the response by calling 'whiten'.
General reference for regression models:
D. C. Montgomery and E.A. Peck. "Introduction to Linear Regression
Analysis." 2nd. Ed., Wiley, 1992.
Econometrics references for regression models:
R. Davidson and J.G. MacKinnon. "Econometric Theory and Methods," Oxford,
2004.
W. Green. "Econometric Analysis," 5th ed., Pearson, 2003.
"""
from statsmodels.compat.python import lrange, lzip
from statsmodels.compat.pandas import Appender
import numpy as np
from scipy.linalg import toeplitz
from scipy import stats
from scipy import optimize
from statsmodels.tools.tools import pinv_extended
from statsmodels.tools.decorators import (cache_readonly,
cache_writable)
import statsmodels.base.model as base
import statsmodels.base.wrapper as wrap
from statsmodels.emplike.elregress import _ELRegOpts
import warnings
from statsmodels.tools.sm_exceptions import InvalidTestWarning
# need import in module instead of lazily to copy `__doc__`
from statsmodels.regression._prediction import PredictionResults
from . import _prediction as pred
__docformat__ = 'restructuredtext en'
__all__ = ['GLS', 'WLS', 'OLS', 'GLSAR', 'PredictionResults',
'RegressionResultsWrapper']
_fit_regularized_doc =\
r"""
Return a regularized fit to a linear regression model.
Parameters
----------
method : str
Either 'elastic_net' or 'sqrt_lasso'.
alpha : scalar or array_like
The penalty weight. If a scalar, the same penalty weight
applies to all variables in the model. If a vector, it
must have the same length as `params`, and contains a
penalty weight for each coefficient.
L1_wt : scalar
The fraction of the penalty given to the L1 penalty term.
Must be between 0 and 1 (inclusive). If 0, the fit is a
ridge fit, if 1 it is a lasso fit.
start_params : array_like
Starting values for ``params``.
profile_scale : bool
If True the penalized fit is computed using the profile
(concentrated) log-likelihood for the Gaussian model.
Otherwise the fit uses the residual sum of squares.
refit : bool
If True, the model is refit using only the variables that
have non-zero coefficients in the regularized fit. The
refitted model is not regularized.
**kwargs
Additional keyword arguments that contain information used when
constructing a model using the formula interface.
Returns
-------
statsmodels.base.elastic_net.RegularizedResults
The regularized results.
Notes
-----
The elastic net uses a combination of L1 and L2 penalties.
The implementation closely follows the glmnet package in R.
The function that is minimized is:
.. math::
0.5*RSS/n + alpha*((1-L1\_wt)*|params|_2^2/2 + L1\_wt*|params|_1)
where RSS is the usual regression sum of squares, n is the
sample size, and :math:`|*|_1` and :math:`|*|_2` are the L1 and L2
norms.
For WLS and GLS, the RSS is calculated using the whitened endog and
exog data.
Post-estimation results are based on the same data used to
select variables, hence may be subject to overfitting biases.
The elastic_net method uses the following keyword arguments:
maxiter : int
Maximum number of iterations
cnvrg_tol : float
Convergence threshold for line searches
zero_tol : float
Coefficients below this threshold are treated as zero.
The square root lasso approach is a variation of the Lasso
that is largely self-tuning (the optimal tuning parameter
does not depend on the standard deviation of the regression
errors). If the errors are Gaussian, the tuning parameter
can be taken to be
alpha = 1.1 * np.sqrt(n) * norm.ppf(1 - 0.05 / (2 * p))
where n is the sample size and p is the number of predictors.
The square root lasso uses the following keyword arguments:
zero_tol : float
Coefficients below this threshold are treated as zero.
The cvxopt module is required to estimate model using the square root
lasso.
References
----------
.. [*] Friedman, Hastie, Tibshirani (2008). Regularization paths for
generalized linear models via coordinate descent. Journal of
Statistical Software 33(1), 1-22 Feb 2010.
.. [*] A Belloni, V Chernozhukov, L Wang (2011). Square-root Lasso:
pivotal recovery of sparse signals via conic programming.
Biometrika 98(4), 791-806. https://arxiv.org/pdf/1009.5689.pdf
"""
def _get_sigma(sigma, nobs):
"""
Returns sigma (matrix, nobs by nobs) for GLS and the inverse of its
Cholesky decomposition. Handles dimensions and checks integrity.
If sigma is None, returns None, None. Otherwise returns sigma,
cholsigmainv.
"""
if sigma is None:
return None, None
sigma = np.asarray(sigma).squeeze()
if sigma.ndim == 0:
sigma = np.repeat(sigma, nobs)
if sigma.ndim == 1:
if sigma.shape != (nobs,):
raise ValueError("Sigma must be a scalar, 1d of length %s or a 2d "
"array of shape %s x %s" % (nobs, nobs, nobs))
cholsigmainv = 1/np.sqrt(sigma)
else:
if sigma.shape != (nobs, nobs):
raise ValueError("Sigma must be a scalar, 1d of length %s or a 2d "
"array of shape %s x %s" % (nobs, nobs, nobs))
cholsigmainv = np.linalg.cholesky(np.linalg.inv(sigma)).T
return sigma, cholsigmainv
class RegressionModel(base.LikelihoodModel):
"""
Base class for linear regression models. Should not be directly called.
Intended for subclassing.
"""
def __init__(self, endog, exog, **kwargs):
super(RegressionModel, self).__init__(endog, exog, **kwargs)
self._data_attr.extend(['pinv_wexog', 'wendog', 'wexog', 'weights'])
def initialize(self):
"""Initialize model components."""
self.wexog = self.whiten(self.exog)
self.wendog = self.whiten(self.endog)
# overwrite nobs from class Model:
self.nobs = float(self.wexog.shape[0])
self._df_model = None
self._df_resid = None
self.rank = None
@property
def df_model(self):
"""
The model degree of freedom.
The dof is defined as the rank of the regressor matrix minus 1 if a
constant is included.
"""
if self._df_model is None:
if self.rank is None:
self.rank = np.linalg.matrix_rank(self.exog)
self._df_model = float(self.rank - self.k_constant)
return self._df_model
@df_model.setter
def df_model(self, value):
self._df_model = value
@property
def df_resid(self):
"""
The residual degree of freedom.
The dof is defined as the number of observations minus the rank of
the regressor matrix.
"""
if self._df_resid is None:
if self.rank is None:
self.rank = np.linalg.matrix_rank(self.exog)
self._df_resid = self.nobs - self.rank
return self._df_resid
@df_resid.setter
def df_resid(self, value):
self._df_resid = value
def whiten(self, x):
"""
Whiten method that must be overwritten by individual models.
Parameters
----------
x : array_like
Data to be whitened.
"""
raise NotImplementedError("Subclasses must implement.")
def fit(self, method="pinv", cov_type='nonrobust', cov_kwds=None,
use_t=None, **kwargs):
"""
Full fit of the model.
The results include an estimate of covariance matrix, (whitened)
residuals and an estimate of scale.
Parameters
----------
method : str, optional
Can be "pinv", "qr". "pinv" uses the Moore-Penrose pseudoinverse
to solve the least squares problem. "qr" uses the QR
factorization.
cov_type : str, optional
See `regression.linear_model.RegressionResults` for a description
of the available covariance estimators.
cov_kwds : list or None, optional
See `linear_model.RegressionResults.get_robustcov_results` for a
description required keywords for alternative covariance
estimators.
use_t : bool, optional
Flag indicating to use the Student's t distribution when computing
p-values. Default behavior depends on cov_type. See
`linear_model.RegressionResults.get_robustcov_results` for
implementation details.
**kwargs
Additional keyword arguments that contain information used when
constructing a model using the formula interface.
Returns
-------
RegressionResults
The model estimation results.
See Also
--------
RegressionResults
The results container.
RegressionResults.get_robustcov_results
A method to change the covariance estimator used when fitting the
model.
Notes
-----
The fit method uses the pseudoinverse of the design/exogenous variables
to solve the least squares minimization.
"""
if method == "pinv":
if not (hasattr(self, 'pinv_wexog') and
hasattr(self, 'normalized_cov_params') and
hasattr(self, 'rank')):
self.pinv_wexog, singular_values = pinv_extended(self.wexog)
self.normalized_cov_params = np.dot(
self.pinv_wexog, np.transpose(self.pinv_wexog))
# Cache these singular values for use later.
self.wexog_singular_values = singular_values
self.rank = np.linalg.matrix_rank(np.diag(singular_values))
beta = np.dot(self.pinv_wexog, self.wendog)
elif method == "qr":
if not (hasattr(self, 'exog_Q') and
hasattr(self, 'exog_R') and
hasattr(self, 'normalized_cov_params') and
hasattr(self, 'rank')):
Q, R = np.linalg.qr(self.wexog)
self.exog_Q, self.exog_R = Q, R
self.normalized_cov_params = np.linalg.inv(np.dot(R.T, R))
# Cache singular values from R.
self.wexog_singular_values = np.linalg.svd(R, 0, 0)
self.rank = np.linalg.matrix_rank(R)
else:
Q, R = self.exog_Q, self.exog_R
# used in ANOVA
self.effects = effects = np.dot(Q.T, self.wendog)
beta = np.linalg.solve(R, effects)
else:
raise ValueError('method has to be "pinv" or "qr"')
if self._df_model is None:
self._df_model = float(self.rank - self.k_constant)
if self._df_resid is None:
self.df_resid = self.nobs - self.rank
if isinstance(self, OLS):
lfit = OLSResults(
self, beta,
normalized_cov_params=self.normalized_cov_params,
cov_type=cov_type, cov_kwds=cov_kwds, use_t=use_t)
else:
lfit = RegressionResults(
self, beta,
normalized_cov_params=self.normalized_cov_params,
cov_type=cov_type, cov_kwds=cov_kwds, use_t=use_t,
**kwargs)
return RegressionResultsWrapper(lfit)
def predict(self, params, exog=None):
"""
Return linear predicted values from a design matrix.
Parameters
----------
params : array_like
Parameters of a linear model.
exog : array_like, optional
Design / exogenous data. Model exog is used if None.
Returns
-------
array_like
An array of fitted values.
Notes
-----
If the model has not yet been fit, params is not optional.
"""
# JP: this does not look correct for GLMAR
# SS: it needs its own predict method
if exog is None:
exog = self.exog
return np.dot(exog, params)
def get_distribution(self, params, scale, exog=None, dist_class=None):
"""
Construct a random number generator for the predictive distribution.
Parameters
----------
params : array_like
The model parameters (regression coefficients).
scale : scalar
The variance parameter.
exog : array_like
The predictor variable matrix.
dist_class : class
A random number generator class. Must take 'loc' and 'scale'
as arguments and return a random number generator implementing
an ``rvs`` method for simulating random values. Defaults to normal.
Returns
-------
gen
Frozen random number generator object with mean and variance
determined by the fitted linear model. Use the ``rvs`` method
to generate random values.
Notes
-----
Due to the behavior of ``scipy.stats.distributions objects``,
the returned random number generator must be called with
``gen.rvs(n)`` where ``n`` is the number of observations in
the data set used to fit the model. If any other value is
used for ``n``, misleading results will be produced.
"""
fit = self.predict(params, exog)
if dist_class is None:
from scipy.stats.distributions import norm
dist_class = norm
gen = dist_class(loc=fit, scale=np.sqrt(scale))
return gen
class GLS(RegressionModel):
__doc__ = r"""
Generalized Least Squares
%(params)s
sigma : scalar or array
The array or scalar `sigma` is the weighting matrix of the covariance.
The default is None for no scaling. If `sigma` is a scalar, it is
assumed that `sigma` is an n x n diagonal matrix with the given
scalar, `sigma` as the value of each diagonal element. If `sigma`
is an n-length vector, then `sigma` is assumed to be a diagonal
matrix with the given `sigma` on the diagonal. This should be the
same as WLS.
%(extra_params)s
Attributes
----------
pinv_wexog : ndarray
`pinv_wexog` is the p x n Moore-Penrose pseudoinverse of `wexog`.
cholsimgainv : ndarray
The transpose of the Cholesky decomposition of the pseudoinverse.
df_model : float
p - 1, where p is the number of regressors including the intercept.
of freedom.
df_resid : float
Number of observations n less the number of parameters p.
llf : float
The value of the likelihood function of the fitted model.
nobs : float
The number of observations n.
normalized_cov_params : ndarray
p x p array :math:`(X^{T}\Sigma^{-1}X)^{-1}`
results : RegressionResults instance
A property that returns the RegressionResults class if fit.
sigma : ndarray
`sigma` is the n x n covariance structure of the error terms.
wexog : ndarray
Design matrix whitened by `cholsigmainv`
wendog : ndarray
Response variable whitened by `cholsigmainv`
See Also
--------
WLS : Fit a linear model using Weighted Least Squares.
OLS : Fit a linear model using Ordinary Least Squares.
Notes
-----
If sigma is a function of the data making one of the regressors
a constant, then the current postestimation statistics will not be correct.
Examples
--------
>>> import statsmodels.api as sm
>>> data = sm.datasets.longley.load(as_pandas=False)
>>> data.exog = sm.add_constant(data.exog)
>>> ols_resid = sm.OLS(data.endog, data.exog).fit().resid
>>> res_fit = sm.OLS(ols_resid[1:], ols_resid[:-1]).fit()
>>> rho = res_fit.params
`rho` is a consistent estimator of the correlation of the residuals from
an OLS fit of the longley data. It is assumed that this is the true rho
of the AR process data.
>>> from scipy.linalg import toeplitz
>>> order = toeplitz(np.arange(16))
>>> sigma = rho**order
`sigma` is an n x n matrix of the autocorrelation structure of the
data.
>>> gls_model = sm.GLS(data.endog, data.exog, sigma=sigma)
>>> gls_results = gls_model.fit()
>>> print(gls_results.summary())
""" % {'params': base._model_params_doc,
'extra_params': base._missing_param_doc + base._extra_param_doc}
def __init__(self, endog, exog, sigma=None, missing='none', hasconst=None,
**kwargs):
# TODO: add options igls, for iterative fgls if sigma is None
# TODO: default if sigma is none should be two-step GLS
sigma, cholsigmainv = _get_sigma(sigma, len(endog))
super(GLS, self).__init__(endog, exog, missing=missing,
hasconst=hasconst, sigma=sigma,
cholsigmainv=cholsigmainv, **kwargs)
# store attribute names for data arrays
self._data_attr.extend(['sigma', 'cholsigmainv'])
def whiten(self, x):
"""
GLS whiten method.
Parameters
----------
x : array_like
Data to be whitened.
Returns
-------
ndarray
The value np.dot(cholsigmainv,X).
See Also
--------
GLS : Fit a linear model using Generalized Least Squares.
"""
x = np.asarray(x)
if self.sigma is None or self.sigma.shape == ():
return x
elif self.sigma.ndim == 1:
if x.ndim == 1:
return x * self.cholsigmainv
else:
return x * self.cholsigmainv[:, None]
else:
return np.dot(self.cholsigmainv, x)
def loglike(self, params):
r"""
Compute the value of the Gaussian log-likelihood function at params.
Given the whitened design matrix, the log-likelihood is evaluated
at the parameter vector `params` for the dependent variable `endog`.
Parameters
----------
params : array_like
The model parameters.
Returns
-------
float
The value of the log-likelihood function for a GLS Model.
Notes
-----
The log-likelihood function for the normal distribution is
.. math:: -\frac{n}{2}\log\left(\left(Y-\hat{Y}\right)^{\prime}
\left(Y-\hat{Y}\right)\right)
-\frac{n}{2}\left(1+\log\left(\frac{2\pi}{n}\right)\right)
-\frac{1}{2}\log\left(\left|\Sigma\right|\right)
Y and Y-hat are whitened.
"""
# TODO: combine this with OLS/WLS loglike and add _det_sigma argument
nobs2 = self.nobs / 2.0
SSR = np.sum((self.wendog - np.dot(self.wexog, params))**2, axis=0)
llf = -np.log(SSR) * nobs2 # concentrated likelihood
llf -= (1+np.log(np.pi/nobs2))*nobs2 # with likelihood constant
if np.any(self.sigma):
# FIXME: robust-enough check? unneeded if _det_sigma gets defined
if self.sigma.ndim == 2:
det = np.linalg.slogdet(self.sigma)
llf -= .5*det[1]
else:
llf -= 0.5*np.sum(np.log(self.sigma))
# with error covariance matrix
return llf
def hessian_factor(self, params, scale=None, observed=True):
"""
Compute weights for calculating Hessian.
Parameters
----------
params : ndarray
The parameter at which Hessian is evaluated.
scale : None or float
If scale is None, then the default scale will be calculated.
Default scale is defined by `self.scaletype` and set in fit.
If scale is not None, then it is used as a fixed scale.
observed : bool
If True, then the observed Hessian is returned. If false then the
expected information matrix is returned.
Returns
-------
ndarray
A 1d weight vector used in the calculation of the Hessian.
The hessian is obtained by `(exog.T * hessian_factor).dot(exog)`.
"""
if self.sigma is None or self.sigma.shape == ():
return np.ones(self.exog.shape[0])
elif self.sigma.ndim == 1:
return self.cholsigmainv
else:
return np.diag(self.cholsigmainv)
@Appender(_fit_regularized_doc)
def fit_regularized(self, method="elastic_net", alpha=0.,
L1_wt=1., start_params=None, profile_scale=False,
refit=False, **kwargs):
# Need to adjust since RSS/n term in elastic net uses nominal
# n in denominator
if self.sigma is not None:
alpha = alpha * np.sum(1 / np.diag(self.sigma)) / len(self.endog)
rslt = OLS(self.wendog, self.wexog).fit_regularized(
method=method, alpha=alpha,
L1_wt=L1_wt,
start_params=start_params,
profile_scale=profile_scale,
refit=refit, **kwargs)
from statsmodels.base.elastic_net import (
RegularizedResults, RegularizedResultsWrapper)
rrslt = RegularizedResults(self, rslt.params)
return RegularizedResultsWrapper(rrslt)
class WLS(RegressionModel):
__doc__ = """
Weighted Least Squares
The weights are presumed to be (proportional to) the inverse of
the variance of the observations. That is, if the variables are
to be transformed by 1/sqrt(W) you must supply weights = 1/W.
%(params)s
weights : array_like, optional
A 1d array of weights. If you supply 1/W then the variables are
pre- multiplied by 1/sqrt(W). If no weights are supplied the
default value is 1 and WLS results are the same as OLS.
%(extra_params)s
Attributes
----------
weights : ndarray
The stored weights supplied as an argument.
See Also
--------
GLS : Fit a linear model using Generalized Least Squares.
OLS : Fit a linear model using Ordinary Least Squares.
Notes
-----
If the weights are a function of the data, then the post estimation
statistics such as fvalue and mse_model might not be correct, as the
package does not yet support no-constant regression.
Examples
--------
>>> import statsmodels.api as sm
>>> Y = [1,3,4,5,2,3,4]
>>> X = range(1,8)
>>> X = sm.add_constant(X)
>>> wls_model = sm.WLS(Y,X, weights=list(range(1,8)))
>>> results = wls_model.fit()
>>> results.params
array([ 2.91666667, 0.0952381 ])
>>> results.tvalues
array([ 2.0652652 , 0.35684428])
>>> print(results.t_test([1, 0]))
<T test: effect=array([ 2.91666667]), sd=array([[ 1.41224801]]), t=array([[ 2.0652652]]), p=array([[ 0.04690139]]), df_denom=5>
>>> print(results.f_test([0, 1]))
<F test: F=array([[ 0.12733784]]), p=[[ 0.73577409]], df_denom=5, df_num=1>
""" % {'params': base._model_params_doc,
'extra_params': base._missing_param_doc + base._extra_param_doc}
def __init__(self, endog, exog, weights=1., missing='none', hasconst=None,
**kwargs):
weights = np.array(weights)
if weights.shape == ():
if (missing == 'drop' and 'missing_idx' in kwargs and
kwargs['missing_idx'] is not None):
# patsy may have truncated endog
weights = np.repeat(weights, len(kwargs['missing_idx']))
else:
weights = np.repeat(weights, len(endog))
# handle case that endog might be of len == 1
if len(weights) == 1:
weights = np.array([weights.squeeze()])
else:
weights = weights.squeeze()
super(WLS, self).__init__(endog, exog, missing=missing,
weights=weights, hasconst=hasconst, **kwargs)
nobs = self.exog.shape[0]
weights = self.weights
# Experimental normalization of weights
weights = weights / np.sum(weights) * nobs
if weights.size != nobs and weights.shape[0] != nobs:
raise ValueError('Weights must be scalar or same length as design')
def whiten(self, x):
"""
Whitener for WLS model, multiplies each column by sqrt(self.weights).
Parameters
----------
x : array_like
Data to be whitened.
Returns
-------
array_like
The whitened values sqrt(weights)*X.
"""
x = np.asarray(x)
if x.ndim == 1:
return x * np.sqrt(self.weights)
elif x.ndim == 2:
return np.sqrt(self.weights)[:, None] * x
def loglike(self, params):
r"""
Compute the value of the gaussian log-likelihood function at params.
Given the whitened design matrix, the log-likelihood is evaluated
at the parameter vector `params` for the dependent variable `Y`.
Parameters
----------
params : array_like
The parameter estimates.
Returns
-------
float
The value of the log-likelihood function for a WLS Model.
Notes
--------
.. math:: -\frac{n}{2}\log SSR
-\frac{n}{2}\left(1+\log\left(\frac{2\pi}{n}\right)\right)
-\frac{1}{2}\log\left(\left|W\right|\right)
where :math:`W` is a diagonal weight matrix matrix and
:math:`SSR=\left(Y-\hat{Y}\right)^\prime W \left(Y-\hat{Y}\right)` is
the sum of the squared weighted residuals.
"""
nobs2 = self.nobs / 2.0
SSR = np.sum((self.wendog - np.dot(self.wexog, params))**2, axis=0)
llf = -np.log(SSR) * nobs2 # concentrated likelihood
llf -= (1+np.log(np.pi/nobs2))*nobs2 # with constant
llf += 0.5 * np.sum(np.log(self.weights))
return llf
def hessian_factor(self, params, scale=None, observed=True):
"""
Compute the weights for calculating the Hessian.
Parameters
----------
params : ndarray
The parameter at which Hessian is evaluated.
scale : None or float
If scale is None, then the default scale will be calculated.
Default scale is defined by `self.scaletype` and set in fit.
If scale is not None, then it is used as a fixed scale.
observed : bool
If True, then the observed Hessian is returned. If false then the
expected information matrix is returned.
Returns
-------
ndarray
A 1d weight vector used in the calculation of the Hessian.
The hessian is obtained by `(exog.T * hessian_factor).dot(exog)`.
"""
return self.weights
@Appender(_fit_regularized_doc)
def fit_regularized(self, method="elastic_net", alpha=0.,
L1_wt=1., start_params=None, profile_scale=False,
refit=False, **kwargs):
# Docstring attached below
# Need to adjust since RSS/n in elastic net uses nominal n in
# denominator
alpha = alpha * np.sum(self.weights) / len(self.weights)
rslt = OLS(self.wendog, self.wexog).fit_regularized(
method=method, alpha=alpha,
L1_wt=L1_wt,
start_params=start_params,
profile_scale=profile_scale,
refit=refit, **kwargs)
from statsmodels.base.elastic_net import (
RegularizedResults, RegularizedResultsWrapper)
rrslt = RegularizedResults(self, rslt.params)
return RegularizedResultsWrapper(rrslt)
class OLS(WLS):
__doc__ = """
Ordinary Least Squares
%(params)s
%(extra_params)s
Attributes
----------
weights : scalar
Has an attribute weights = array(1.0) due to inheritance from WLS.
See Also
--------
WLS : Fit a linear model using Weighted Least Squares.
GLS : Fit a linear model using Generalized Least Squares.
Notes
-----
No constant is added by the model unless you are using formulas.
Examples
--------
>>> import statsmodels.api as sm
>>> Y = [1,3,4,5,2,3,4]
>>> X = range(1,8)
>>> X = sm.add_constant(X)
>>> model = sm.OLS(Y,X)
>>> results = model.fit()
>>> results.params
array([ 2.14285714, 0.25 ])
>>> results.tvalues
array([ 1.87867287, 0.98019606])
>>> print(results.t_test([1, 0]))
<T test: effect=array([ 2.14285714]), sd=array([[ 1.14062282]]), t=array([[ 1.87867287]]), p=array([[ 0.05953974]]), df_denom=5>
>>> print(results.f_test(np.identity(2)))
<F test: F=array([[ 19.46078431]]), p=[[ 0.00437251]], df_denom=5, df_num=2>
""" % {'params': base._model_params_doc,
'extra_params': base._missing_param_doc + base._extra_param_doc}
# TODO: change example to use datasets. This was the point of datasets!
def __init__(self, endog, exog=None, missing='none', hasconst=None,
**kwargs):
super(OLS, self).__init__(endog, exog, missing=missing,
hasconst=hasconst, **kwargs)
if "weights" in self._init_keys:
self._init_keys.remove("weights")
def loglike(self, params, scale=None):
"""
The likelihood function for the OLS model.
Parameters
----------
params : array_like
The coefficients with which to estimate the log-likelihood.
scale : float or None
If None, return the profile (concentrated) log likelihood
(profiled over the scale parameter), else return the
log-likelihood using the given scale value.
Returns
-------
float
The likelihood function evaluated at params.
"""
nobs2 = self.nobs / 2.0
nobs = float(self.nobs)
resid = self.endog - np.dot(self.exog, params)
if hasattr(self, 'offset'):
resid -= self.offset
ssr = np.sum(resid**2)
if scale is None:
# profile log likelihood
llf = -nobs2*np.log(2*np.pi) - nobs2*np.log(ssr / nobs) - nobs2
else:
# log-likelihood
llf = -nobs2 * np.log(2 * np.pi * scale) - ssr / (2*scale)
return llf
def whiten(self, x):
"""
OLS model whitener does nothing.
Parameters
----------
x : array_like
Data to be whitened.
Returns
-------
array_like
The input array unmodified.
See Also
--------
OLS : Fit a linear model using Ordinary Least Squares.
"""
return x
def score(self, params, scale=None):
"""
Evaluate the score function at a given point.
The score corresponds to the profile (concentrated)
log-likelihood in which the scale parameter has been profiled
out.
Parameters
----------
params : array_like
The parameter vector at which the score function is
computed.
scale : float or None
If None, return the profile (concentrated) log likelihood
(profiled over the scale parameter), else return the
log-likelihood using the given scale value.
Returns
-------
ndarray
The score vector.
"""
if not hasattr(self, "_wexog_xprod"):
self._setup_score_hess()
xtxb = np.dot(self._wexog_xprod, params)
sdr = -self._wexog_x_wendog + xtxb
if scale is None:
ssr = self._wendog_xprod - 2 * np.dot(self._wexog_x_wendog.T,
params)
ssr += np.dot(params, xtxb)
return -self.nobs * sdr / ssr
else:
return -sdr / scale
def _setup_score_hess(self):
y = self.wendog
if hasattr(self, 'offset'):
y = y - self.offset
self._wendog_xprod = np.sum(y * y)
self._wexog_xprod = np.dot(self.wexog.T, self.wexog)
self._wexog_x_wendog = np.dot(self.wexog.T, y)
def hessian(self, params, scale=None):
"""
Evaluate the Hessian function at a given point.
Parameters
----------
params : array_like
The parameter vector at which the Hessian is computed.
scale : float or None
If None, return the profile (concentrated) log likelihood
(profiled over the scale parameter), else return the
log-likelihood using the given scale value.
Returns
-------
ndarray
The Hessian matrix.
"""
if not hasattr(self, "_wexog_xprod"):
self._setup_score_hess()
xtxb = np.dot(self._wexog_xprod, params)
if scale is None:
ssr = self._wendog_xprod - 2 * np.dot(self._wexog_x_wendog.T,
params)
ssr += np.dot(params, xtxb)
ssrp = -2*self._wexog_x_wendog + 2*xtxb
hm = self._wexog_xprod / ssr - np.outer(ssrp, ssrp) / ssr**2
return -self.nobs * hm / 2
else:
return -self._wexog_xprod / scale
def hessian_factor(self, params, scale=None, observed=True):
"""
Calculate the weights for the Hessian.
Parameters
----------
params : ndarray
The parameter at which Hessian is evaluated.
scale : None or float
If scale is None, then the default scale will be calculated.
Default scale is defined by `self.scaletype` and set in fit.
If scale is not None, then it is used as a fixed scale.
observed : bool
If True, then the observed Hessian is returned. If false then the
expected information matrix is returned.
Returns
-------
ndarray
A 1d weight vector used in the calculation of the Hessian.
The hessian is obtained by `(exog.T * hessian_factor).dot(exog)`.
"""
return np.ones(self.exog.shape[0])
@Appender(_fit_regularized_doc)
def fit_regularized(self, method="elastic_net", alpha=0.,
L1_wt=1., start_params=None, profile_scale=False,
refit=False, **kwargs):
# In the future we could add support for other penalties, e.g. SCAD.
if method not in ("elastic_net", "sqrt_lasso"):
msg = "Unknown method '%s' for fit_regularized" % method
raise ValueError(msg)
# Set default parameters.
defaults = {"maxiter": 50, "cnvrg_tol": 1e-10,
"zero_tol": 1e-8}
defaults.update(kwargs)
if method == "sqrt_lasso":
from statsmodels.base.elastic_net import (
RegularizedResults, RegularizedResultsWrapper
)
params = self._sqrt_lasso(alpha, refit, defaults["zero_tol"])
results = RegularizedResults(self, params)
return RegularizedResultsWrapper(results)
from statsmodels.base.elastic_net import fit_elasticnet
if L1_wt == 0:
return self._fit_ridge(alpha)
# If a scale parameter is passed in, the non-profile
# likelihood (residual sum of squares divided by -2) is used,
# otherwise the profile likelihood is used.
if profile_scale:
loglike_kwds = {}
score_kwds = {}
hess_kwds = {}
else:
loglike_kwds = {"scale": 1}
score_kwds = {"scale": 1}
hess_kwds = {"scale": 1}
return fit_elasticnet(self, method=method,
alpha=alpha,
L1_wt=L1_wt,
start_params=start_params,
loglike_kwds=loglike_kwds,
score_kwds=score_kwds,
hess_kwds=hess_kwds,
refit=refit,
check_step=False,
**defaults)
def _sqrt_lasso(self, alpha, refit, zero_tol):
try:
import cvxopt
except ImportError:
msg = 'sqrt_lasso fitting requires the cvxopt module'
raise ValueError(msg)
n = len(self.endog)
p = self.exog.shape[1]
h0 = cvxopt.matrix(0., (2*p+1, 1))
h1 = cvxopt.matrix(0., (n+1, 1))
h1[1:, 0] = cvxopt.matrix(self.endog, (n, 1))
G0 = cvxopt.spmatrix([], [], [], (2*p+1, 2*p+1))
for i in range(1, 2*p+1):
G0[i, i] = -1
G1 = cvxopt.matrix(0., (n+1, 2*p+1))
G1[0, 0] = -1
G1[1:, 1:p+1] = self.exog
G1[1:, p+1:] = -self.exog
c = cvxopt.matrix(alpha / n, (2*p + 1, 1))
c[0] = 1 / np.sqrt(n)
from cvxopt import solvers
solvers.options["show_progress"] = False
rslt = solvers.socp(c, Gl=G0, hl=h0, Gq=[G1], hq=[h1])
x = np.asarray(rslt['x']).flat
bp = x[1:p+1]
bn = x[p+1:]
params = bp - bn
if not refit:
return params
ii = np.flatnonzero(np.abs(params) > zero_tol)
rfr = OLS(self.endog, self.exog[:, ii]).fit()
params *= 0
params[ii] = rfr.params
return params
def _fit_ridge(self, alpha):
"""
Fit a linear model using ridge regression.
Parameters
----------
alpha : scalar or array_like
The penalty weight. If a scalar, the same penalty weight
applies to all variables in the model. If a vector, it
must have the same length as `params`, and contains a
penalty weight for each coefficient.
Notes
-----
Equivalent to fit_regularized with L1_wt = 0 (but implemented
more efficiently).
"""
u, s, vt = np.linalg.svd(self.exog, 0)
v = vt.T
q = np.dot(u.T, self.endog) * s
s2 = s * s
if np.isscalar(alpha):
sd = s2 + alpha * self.nobs
params = q / sd
params = np.dot(v, params)
else:
vtav = self.nobs * np.dot(vt, alpha[:, None] * v)
d = np.diag(vtav) + s2
np.fill_diagonal(vtav, d)
r = np.linalg.solve(vtav, q)
params = np.dot(v, r)
from statsmodels.base.elastic_net import RegularizedResults
return RegularizedResults(self, params)
class GLSAR(GLS):
__doc__ = """
Generalized Least Squares with AR covariance structure
%(params)s
rho : int
The order of the autoregressive covariance.
%(extra_params)s
Notes
-----
GLSAR is considered to be experimental.
The linear autoregressive process of order p--AR(p)--is defined as:
TODO
Examples
--------
>>> import statsmodels.api as sm
>>> X = range(1,8)
>>> X = sm.add_constant(X)
>>> Y = [1,3,4,5,8,10,9]
>>> model = sm.GLSAR(Y, X, rho=2)
>>> for i in range(6):
... results = model.fit()
... print("AR coefficients: {0}".format(model.rho))
... rho, sigma = sm.regression.yule_walker(results.resid,
... order=model.order)
... model = sm.GLSAR(Y, X, rho)
...
AR coefficients: [ 0. 0.]
AR coefficients: [-0.52571491 -0.84496178]
AR coefficients: [-0.6104153 -0.86656458]
AR coefficients: [-0.60439494 -0.857867 ]
AR coefficients: [-0.6048218 -0.85846157]
AR coefficients: [-0.60479146 -0.85841922]
>>> results.params
array([-0.66661205, 1.60850853])
>>> results.tvalues
array([ -2.10304127, 21.8047269 ])
>>> print(results.t_test([1, 0]))
<T test: effect=array([-0.66661205]), sd=array([[ 0.31697526]]), t=array([[-2.10304127]]), p=array([[ 0.06309969]]), df_denom=3>
>>> print(results.f_test(np.identity(2)))
<F test: F=array([[ 1815.23061844]]), p=[[ 0.00002372]], df_denom=3, df_num=2>
Or, equivalently
>>> model2 = sm.GLSAR(Y, X, rho=2)
>>> res = model2.iterative_fit(maxiter=6)
>>> model2.rho
array([-0.60479146, -0.85841922])
""" % {'params': base._model_params_doc,
'extra_params': base._missing_param_doc + base._extra_param_doc}
# TODO: Complete docstring
def __init__(self, endog, exog=None, rho=1, missing='none', hasconst=None,
**kwargs):
# this looks strange, interpreting rho as order if it is int
if isinstance(rho, np.int):
self.order = rho
self.rho = np.zeros(self.order, np.float64)
else:
self.rho = np.squeeze(np.asarray(rho))
if len(self.rho.shape) not in [0, 1]:
raise ValueError("AR parameters must be a scalar or a vector")
if self.rho.shape == ():
self.rho.shape = (1,)
self.order = self.rho.shape[0]
if exog is None:
# JP this looks wrong, should be a regression on constant
# results for rho estimate now identical to yule-walker on y
# super(AR, self).__init__(endog, add_constant(endog))
super(GLSAR, self).__init__(endog, np.ones((endog.shape[0], 1)),
missing=missing, hasconst=None,
**kwargs)
else:
super(GLSAR, self).__init__(endog, exog, missing=missing,
**kwargs)
def iterative_fit(self, maxiter=3, rtol=1e-4, **kwargs):
"""
Perform an iterative two-stage procedure to estimate a GLS model.
The model is assumed to have AR(p) errors, AR(p) parameters and
regression coefficients are estimated iteratively.
Parameters
----------
maxiter : int, optional
The number of iterations.
rtol : float, optional
Relative tolerance between estimated coefficients to stop the
estimation. Stops if max(abs(last - current) / abs(last)) < rtol.
**kwargs
Additional keyword arguments passed to `fit`.
Returns
-------
RegressionResults
The results computed using an iterative fit.
"""
# TODO: update this after going through example.
converged = False
i = -1 # need to initialize for maxiter < 1 (skip loop)
history = {'params': [], 'rho': [self.rho]}
for i in range(maxiter - 1):
if hasattr(self, 'pinv_wexog'):
del self.pinv_wexog
self.initialize()
results = self.fit()
history['params'].append(results.params)
if i == 0:
last = results.params
else:
diff = np.max(np.abs(last - results.params) / np.abs(last))
if diff < rtol:
converged = True
break
last = results.params
self.rho, _ = yule_walker(results.resid,
order=self.order, df=None)
history['rho'].append(self.rho)
# why not another call to self.initialize
# Use kwarg to insert history
if not converged and maxiter > 0:
# maxiter <= 0 just does OLS
if hasattr(self, 'pinv_wexog'):
del self.pinv_wexog
self.initialize()
# if converged then this is a duplicate fit, because we did not
# update rho
results = self.fit(history=history, **kwargs)
results.iter = i + 1
# add last fit to history, not if duplicate fit
if not converged:
results.history['params'].append(results.params)
results.iter += 1
results.converged = converged
return results
def whiten(self, x):
"""
Whiten a series of columns according to an AR(p) covariance structure.
Whitening using this method drops the initial p observations.
Parameters
----------
x : array_like
The data to be whitened.
Returns
-------
ndarray
The whitened data.
"""
# TODO: notation for AR process
x = np.asarray(x, np.float64)
_x = x.copy()
# the following loops over the first axis, works for 1d and nd
for i in range(self.order):
_x[(i + 1):] = _x[(i + 1):] - self.rho[i] * x[0:-(i + 1)]
return _x[self.order:]
def yule_walker(x, order=1, method="unbiased", df=None, inv=False,
demean=True):
"""
Estimate AR(p) parameters from a sequence using the Yule-Walker equations.
Unbiased or maximum-likelihood estimator (mle)
Parameters
----------
x : array_like
A 1d array.
order : int, optional
The order of the autoregressive process. Default is 1.
method : str, optional
Method can be 'unbiased' or 'mle' and this determines
denominator in estimate of autocorrelation function (ACF) at
lag k. If 'mle', the denominator is n=X.shape[0], if 'unbiased'
the denominator is n-k. The default is unbiased.
df : int, optional
Specifies the degrees of freedom. If `df` is supplied, then it
is assumed the X has `df` degrees of freedom rather than `n`.
Default is None.
inv : bool
If inv is True the inverse of R is also returned. Default is
False.
demean : bool
True, the mean is subtracted from `X` before estimation.
Returns
-------
rho : ndarray
AR(p) coefficients computed using the Yule-Walker method.
sigma : float
The estimate of the residual standard deviation.
See Also
--------
burg : Burg's AR estimator.
Notes
-----
See https://en.wikipedia.org/wiki/Autoregressive_moving_average_model for
further details.
Examples
--------
>>> import statsmodels.api as sm
>>> from statsmodels.datasets.sunspots import load
>>> data = load(as_pandas=False)
>>> rho, sigma = sm.regression.yule_walker(data.endog, order=4,
... method="mle")
>>> rho
array([ 1.28310031, -0.45240924, -0.20770299, 0.04794365])
>>> sigma
16.808022730464351
"""
# TODO: define R better, look back at notes and technical notes on YW.
# First link here is useful
# http://www-stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/YuleWalkerAndMore.htm
method = str(method).lower()
if method not in ["unbiased", "mle"]:
raise ValueError("ACF estimation method must be 'unbiased' or 'MLE'")
x = np.array(x, dtype=np.float64)
if demean:
x -= x.mean()
n = df or x.shape[0]
# this handles df_resid ie., n - p
adj_needed = method == "unbiased"
if x.ndim > 1 and x.shape[1] != 1:
raise ValueError("expecting a vector to estimate AR parameters")
r = np.zeros(order+1, np.float64)
r[0] = (x ** 2).sum() / n
for k in range(1, order+1):
r[k] = (x[0:-k] * x[k:]).sum() / (n - k * adj_needed)
R = toeplitz(r[:-1])
rho = np.linalg.solve(R, r[1:])
sigmasq = r[0] - (r[1:]*rho).sum()
if inv:
return rho, np.sqrt(sigmasq), np.linalg.inv(R)
else:
return rho, np.sqrt(sigmasq)
def burg(endog, order=1, demean=True):
"""
Compute Burg's AP(p) parameter estimator.
Parameters
----------
endog : array_like
The endogenous variable.
order : int, optional
Order of the AR. Default is 1.
demean : bool, optional
Flag indicating to subtract the mean from endog before estimation.
Returns
-------
rho : ndarray
The AR(p) coefficients computed using Burg's algorithm.
sigma2 : float
The estimate of the residual variance.
See Also
--------
yule_walker : Estimate AR parameters using the Yule-Walker method.
Notes
-----
AR model estimated includes a constant that is estimated using the sample
mean (see [1]_). This value is not reported.
References
----------
.. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
Examples
--------
>>> import statsmodels.api as sm
>>> from statsmodels.datasets.sunspots import load
>>> data = load(as_pandas=True)
>>> rho, sigma2 = sm.regression.linear_model.burg(data.endog, order=4)
>>> rho
array([ 1.30934186, -0.48086633, -0.20185982, 0.05501941])
>>> sigma2
271.2467306963966
"""
# Avoid circular imports
from statsmodels.tsa.stattools import levinson_durbin_pacf, pacf_burg
endog = np.squeeze(np.asarray(endog))
if endog.ndim != 1:
raise ValueError('endog must be 1-d or squeezable to 1-d.')
order = int(order)
if order < 1:
raise ValueError('order must be an integer larger than 1')
if demean:
endog = endog - endog.mean()
pacf, sigma = pacf_burg(endog, order, demean=demean)
ar, _ = levinson_durbin_pacf(pacf)
return ar, sigma[-1]
class RegressionResults(base.LikelihoodModelResults):
r"""
This class summarizes the fit of a linear regression model.
It handles the output of contrasts, estimates of covariance, etc.
Parameters
----------
model : RegressionModel
The regression model instance.
params : ndarray
The estimated parameters.
normalized_cov_params : ndarray
The normalized covariance parameters.
scale : float
The estimated scale of the residuals.
cov_type : str
The covariance estimator used in the results.
cov_kwds : dict
Additional keywords used in the covariance specification.
use_t : bool
Flag indicating to use the Student's t in inference.
**kwargs
Additional keyword arguments used to initialize the results.
Attributes
----------
pinv_wexog
See model class docstring for implementation details.
cov_type
Parameter covariance estimator used for standard errors and t-stats.
df_model
Model degrees of freedom. The number of regressors `p`. Does not
include the constant if one is present.
df_resid
Residual degrees of freedom. `n - p - 1`, if a constant is present.
`n - p` if a constant is not included.
het_scale
adjusted squared residuals for heteroscedasticity robust standard
errors. Is only available after `HC#_se` or `cov_HC#` is called.
See HC#_se for more information.
history
Estimation history for iterative estimators.
model
A pointer to the model instance that called fit() or results.
params
The linear coefficients that minimize the least squares
criterion. This is usually called Beta for the classical
linear model.
"""
_cache = {} # needs to be a class attribute for scale setter?
def __init__(self, model, params, normalized_cov_params=None, scale=1.,
cov_type='nonrobust', cov_kwds=None, use_t=None, **kwargs):
super(RegressionResults, self).__init__(
model, params, normalized_cov_params, scale)
self._cache = {}
if hasattr(model, 'wexog_singular_values'):
self._wexog_singular_values = model.wexog_singular_values
else:
self._wexog_singular_values = None
self.df_model = model.df_model
self.df_resid = model.df_resid
if cov_type == 'nonrobust':
self.cov_type = 'nonrobust'
self.cov_kwds = {
'description': 'Standard Errors assume that the ' +
'covariance matrix of the errors is correctly ' +
'specified.'}
if use_t is None:
use_t = True # TODO: class default
self.use_t = use_t
else:
if cov_kwds is None:
cov_kwds = {}
if 'use_t' in cov_kwds:
# TODO: we want to get rid of 'use_t' in cov_kwds
use_t_2 = cov_kwds.pop('use_t')
if use_t is None:
use_t = use_t_2
# TODO: warn or not?
self.get_robustcov_results(cov_type=cov_type, use_self=True,
use_t=use_t, **cov_kwds)
for key in kwargs:
setattr(self, key, kwargs[key])
def conf_int(self, alpha=.05, cols=None):
"""
Compute the confidence interval of the fitted parameters.
Parameters
----------
alpha : float, optional
The `alpha` level for the confidence interval. The default
`alpha` = .05 returns a 95% confidence interval.
cols : array_like, optional
Columns to included in returned confidence intervals.
Returns
-------
array_like
The confidence intervals.
Notes
-----
The confidence interval is based on Student's t-distribution.
"""
# keep method for docstring for now
ci = super(RegressionResults, self).conf_int(alpha=alpha, cols=cols)
return ci
@cache_readonly
def nobs(self):
"""Number of observations n."""
return float(self.model.wexog.shape[0])
@cache_readonly
def fittedvalues(self):
"""The predicted values for the original (unwhitened) design."""
return self.model.predict(self.params, self.model.exog)
@cache_readonly
def wresid(self):
"""
The residuals of the transformed/whitened regressand and regressor(s).
"""
return self.model.wendog - self.model.predict(
self.params, self.model.wexog)
@cache_readonly
def resid(self):
"""The residuals of the model."""
return self.model.endog - self.model.predict(
self.params, self.model.exog)
# TODO: fix writable example
@cache_writable()
def scale(self):
"""
A scale factor for the covariance matrix.
The Default value is ssr/(n-p). Note that the square root of `scale`
is often called the standard error of the regression.
"""
wresid = self.wresid
return np.dot(wresid, wresid) / self.df_resid
@cache_readonly
def ssr(self):
"""Sum of squared (whitened) residuals."""
wresid = self.wresid
return np.dot(wresid, wresid)
@cache_readonly
def centered_tss(self):
"""The total (weighted) sum of squares centered about the mean."""
model = self.model
weights = getattr(model, 'weights', None)
sigma = getattr(model, 'sigma', None)
if weights is not None:
mean = np.average(model.endog, weights=weights)
return np.sum(weights * (model.endog - mean)**2)
elif sigma is not None:
# Exactly matches WLS when sigma is diagonal
iota = np.ones_like(model.endog)
iota = model.whiten(iota)
mean = model.wendog.dot(iota) / iota.dot(iota)
err = model.endog - mean
err = model.whiten(err)
return np.sum(err**2)
else:
centered_endog = model.wendog - model.wendog.mean()
return np.dot(centered_endog, centered_endog)
@cache_readonly
def uncentered_tss(self):
"""
Uncentered sum of squares.
The sum of the squared values of the (whitened) endogenous response
variable.
"""
wendog = self.model.wendog
return np.dot(wendog, wendog)
@cache_readonly
def ess(self):
"""
The explained sum of squares.
If a constant is present, the centered total sum of squares minus the
sum of squared residuals. If there is no constant, the uncentered total
sum of squares is used.
"""
if self.k_constant:
return self.centered_tss - self.ssr
else:
return self.uncentered_tss - self.ssr
@cache_readonly
def rsquared(self):
"""
R-squared of the model.
This is defined here as 1 - `ssr`/`centered_tss` if the constant is
included in the model and 1 - `ssr`/`uncentered_tss` if the constant is
omitted.
"""
if self.k_constant:
return 1 - self.ssr/self.centered_tss
else:
return 1 - self.ssr/self.uncentered_tss
@cache_readonly
def rsquared_adj(self):
"""
Adjusted R-squared.
This is defined here as 1 - (`nobs`-1)/`df_resid` * (1-`rsquared`)
if a constant is included and 1 - `nobs`/`df_resid` * (1-`rsquared`) if
no constant is included.
"""
return 1 - (np.divide(self.nobs - self.k_constant, self.df_resid)
* (1 - self.rsquared))
@cache_readonly
def mse_model(self):
"""
Mean squared error the model.
The explained sum of squares divided by the model degrees of freedom.
"""
if np.all(self.df_model == 0.0):
return np.full_like(self.ess, np.nan)
return self.ess/self.df_model
@cache_readonly
def mse_resid(self):
"""
Mean squared error of the residuals.
The sum of squared residuals divided by the residual degrees of
freedom.
"""
if np.all(self.df_resid == 0.0):
return np.full_like(self.ssr, np.nan)
return self.ssr/self.df_resid
@cache_readonly
def mse_total(self):
"""
Total mean squared error.
The uncentered total sum of squares divided by the number of
observations.
"""
if np.all(self.df_resid + self.df_model == 0.0):
return np.full_like(self.centered_tss, np.nan)
if self.k_constant:
return self.centered_tss / (self.df_resid + self.df_model)
else:
return self.uncentered_tss / (self.df_resid + self.df_model)
@cache_readonly
def fvalue(self):
"""
F-statistic of the fully specified model.
Calculated as the mean squared error of the model divided by the mean
squared error of the residuals if the nonrobust covariance is used.
Otherwise computed using a Wald-like quadratic form that tests whether
all coefficients (excluding the constant) are zero.
"""
if hasattr(self, 'cov_type') and self.cov_type != 'nonrobust':
# with heteroscedasticity or correlation robustness
k_params = self.normalized_cov_params.shape[0]
mat = np.eye(k_params)
const_idx = self.model.data.const_idx
# TODO: What if model includes implicit constant, e.g. all
# dummies but no constant regressor?
# TODO: Restats as LM test by projecting orthogonalizing
# to constant?
if self.model.data.k_constant == 1:
# if constant is implicit, return nan see #2444
if const_idx is None:
return np.nan
idx = lrange(k_params)
idx.pop(const_idx)
mat = mat[idx] # remove constant
if mat.size == 0: # see #3642
return np.nan
ft = self.f_test(mat)
# using backdoor to set another attribute that we already have
self._cache['f_pvalue'] = ft.pvalue
return ft.fvalue
else:
# for standard homoscedastic case
return self.mse_model/self.mse_resid
@cache_readonly
def f_pvalue(self):
"""The p-value of the F-statistic."""
# Special case for df_model 0
if self.df_model == 0:
return np.full_like(self.fvalue, np.nan)
return stats.f.sf(self.fvalue, self.df_model, self.df_resid)
@cache_readonly
def bse(self):
"""The standard errors of the parameter estimates."""
return np.sqrt(np.diag(self.cov_params()))
@cache_readonly
def aic(self):
r"""
Akaike's information criteria.
For a model with a constant :math:`-2llf + 2(df\_model + 1)`. For a
model without a constant :math:`-2llf + 2(df\_model)`.
"""
return -2 * self.llf + 2 * (self.df_model + self.k_constant)
@cache_readonly
def bic(self):
r"""
Bayes' information criteria.
For a model with a constant :math:`-2llf + \log(n)(df\_model+1)`.
For a model without a constant :math:`-2llf + \log(n)(df\_model)`.
"""
return (-2 * self.llf + np.log(self.nobs) * (self.df_model +
self.k_constant))
@cache_readonly
def eigenvals(self):
"""
Return eigenvalues sorted in decreasing order.
"""
if self._wexog_singular_values is not None:
eigvals = self._wexog_singular_values ** 2
else:
eigvals = np.linalg.linalg.eigvalsh(np.dot(self.model.wexog.T,
self.model.wexog))
return np.sort(eigvals)[::-1]
@cache_readonly
def condition_number(self):
"""
Return condition number of exogenous matrix.
Calculated as ratio of largest to smallest eigenvalue.
"""
eigvals = self.eigenvals
return np.sqrt(eigvals[0]/eigvals[-1])
# TODO: make these properties reset bse
def _HCCM(self, scale):
H = np.dot(self.model.pinv_wexog,
scale[:, None] * self.model.pinv_wexog.T)
return H
@cache_readonly
def cov_HC0(self):
"""
Heteroscedasticity robust covariance matrix. See HC0_se.
"""
self.het_scale = self.wresid**2
cov_HC0 = self._HCCM(self.het_scale)
return cov_HC0
@cache_readonly
def cov_HC1(self):
"""
Heteroscedasticity robust covariance matrix. See HC1_se.
"""
self.het_scale = self.nobs/(self.df_resid)*(self.wresid**2)
cov_HC1 = self._HCCM(self.het_scale)
return cov_HC1
@cache_readonly
def cov_HC2(self):
"""
Heteroscedasticity robust covariance matrix. See HC2_se.
"""
# probably could be optimized
wexog = self.model.wexog
h = np.diag(wexog @ self.normalized_cov_params @ wexog.T)
self.het_scale = self.wresid**2/(1-h)
cov_HC2 = self._HCCM(self.het_scale)
return cov_HC2
@cache_readonly
def cov_HC3(self):
"""
Heteroscedasticity robust covariance matrix. See HC3_se.
"""
wexog = self.model.wexog
h = np.diag(wexog @ self.normalized_cov_params @ wexog.T)
self.het_scale = (self.wresid / (1 - h))**2
cov_HC3 = self._HCCM(self.het_scale)
return cov_HC3
@cache_readonly
def HC0_se(self):
"""
White's (1980) heteroskedasticity robust standard errors.
Notes
-----
Defined as sqrt(diag(X.T X)^(-1)X.T diag(e_i^(2)) X(X.T X)^(-1)
where e_i = resid[i].
When HC0_se or cov_HC0 is called the RegressionResults instance will
then have another attribute `het_scale`, which is in this case is just
resid**2.
"""
return np.sqrt(np.diag(self.cov_HC0))
@cache_readonly
def HC1_se(self):
"""
MacKinnon and White's (1985) heteroskedasticity robust standard errors.
Notes
-----
Defined as sqrt(diag(n/(n-p)*HC_0).
When HC1_se or cov_HC1 is called the RegressionResults instance will
then have another attribute `het_scale`, which is in this case is
n/(n-p)*resid**2.
"""
return np.sqrt(np.diag(self.cov_HC1))
@cache_readonly
def HC2_se(self):
"""
MacKinnon and White's (1985) heteroskedasticity robust standard errors.
Notes
-----
Defined as (X.T X)^(-1)X.T diag(e_i^(2)/(1-h_ii)) X(X.T X)^(-1)
where h_ii = x_i(X.T X)^(-1)x_i.T
When HC2_se or cov_HC2 is called the RegressionResults instance will
then have another attribute `het_scale`, which is in this case is
resid^(2)/(1-h_ii).
"""
return np.sqrt(np.diag(self.cov_HC2))
@cache_readonly
def HC3_se(self):
"""
MacKinnon and White's (1985) heteroskedasticity robust standard errors.
Notes
-----
Defined as (X.T X)^(-1)X.T diag(e_i^(2)/(1-h_ii)^(2)) X(X.T X)^(-1)
where h_ii = x_i(X.T X)^(-1)x_i.T.
When HC3_se or cov_HC3 is called the RegressionResults instance will
then have another attribute `het_scale`, which is in this case is
resid^(2)/(1-h_ii)^(2).
"""
return np.sqrt(np.diag(self.cov_HC3))
@cache_readonly
def resid_pearson(self):
"""
Residuals, normalized to have unit variance.
Returns
-------
array_like
The array `wresid` normalized by the sqrt of the scale to have
unit variance.
"""
if not hasattr(self, 'resid'):
raise ValueError('Method requires residuals.')
eps = np.finfo(self.wresid.dtype).eps
if np.sqrt(self.scale) < 10 * eps * self.model.endog.mean():
# do not divide if scale is zero close to numerical precision
from warnings import warn
warn("All residuals are 0, cannot compute normed residuals.",
RuntimeWarning)
return self.wresid
else:
return self.wresid / np.sqrt(self.scale)
def _is_nested(self, restricted):
"""
Parameters
----------
restricted : Result instance
The restricted model is assumed to be nested in the current
model. The result instance of the restricted model is required to
have two attributes, residual sum of squares, `ssr`, residual
degrees of freedom, `df_resid`.
Returns
-------
nested : bool
True if nested, otherwise false
Notes
-----
A most nests another model if the regressors in the smaller
model are spanned by the regressors in the larger model and
the regressand is identical.
"""
if self.model.nobs != restricted.model.nobs:
return False
full_rank = self.model.rank
restricted_rank = restricted.model.rank
if full_rank <= restricted_rank:
return False
restricted_exog = restricted.model.wexog
full_wresid = self.wresid
scores = restricted_exog * full_wresid[:, None]
score_l2 = np.sqrt(np.mean(scores.mean(0) ** 2))
# TODO: Could be improved, and may fail depending on scale of
# regressors
return np.allclose(score_l2, 0)
def compare_lm_test(self, restricted, demean=True, use_lr=False):
"""
Use Lagrange Multiplier test to test a set of linear restrictions.
Parameters
----------
restricted : Result instance
The restricted model is assumed to be nested in the
current model. The result instance of the restricted model
is required to have two attributes, residual sum of
squares, `ssr`, residual degrees of freedom, `df_resid`.
demean : bool
Flag indicating whether the demean the scores based on the
residuals from the restricted model. If True, the covariance of
the scores are used and the LM test is identical to the large
sample version of the LR test.
use_lr : bool
A flag indicating whether to estimate the covariance of the model
scores using the unrestricted model. Setting the to True improves
the power of the test.
Returns
-------
lm_value : float
The test statistic which has a chi2 distributed.
p_value : float
The p-value of the test statistic.
df_diff : int
The degrees of freedom of the restriction, i.e. difference in df
between models.
Notes
-----
The LM test examines whether the scores from the restricted model are
0. If the null is true, and the restrictions are valid, then the
parameters of the restricted model should be close to the minimum of
the sum of squared errors, and so the scores should be close to zero,
on average.
"""
import statsmodels.stats.sandwich_covariance as sw
from numpy.linalg import inv
if not self._is_nested(restricted):
raise ValueError("Restricted model is not nested by full model.")
wresid = restricted.wresid
wexog = self.model.wexog
scores = wexog * wresid[:, None]
n = self.nobs
df_full = self.df_resid
df_restr = restricted.df_resid
df_diff = (df_restr - df_full)
s = scores.mean(axis=0)
if use_lr:
scores = wexog * self.wresid[:, None]
demean = False
if demean:
scores = scores - scores.mean(0)[None, :]
# Form matters here. If homoskedastics can be sigma^2 (X'X)^-1
# If Heteroskedastic then the form below is fine
# If HAC then need to use HAC
# If Cluster, should use cluster
cov_type = getattr(self, 'cov_type', 'nonrobust')
if cov_type == 'nonrobust':
sigma2 = np.mean(wresid**2)
xpx = np.dot(wexog.T, wexog) / n
s_inv = inv(sigma2 * xpx)
elif cov_type in ('HC0', 'HC1', 'HC2', 'HC3'):
s_inv = inv(np.dot(scores.T, scores) / n)
elif cov_type == 'HAC':
maxlags = self.cov_kwds['maxlags']
s_inv = inv(sw.S_hac_simple(scores, maxlags) / n)
elif cov_type == 'cluster':
# cluster robust standard errors
groups = self.cov_kwds['groups']
# TODO: Might need demean option in S_crosssection by group?
s_inv = inv(sw.S_crosssection(scores, groups))
else:
raise ValueError('Only nonrobust, HC, HAC and cluster are ' +
'currently connected')
lm_value = n * (s @ s_inv @ s.T)
p_value = stats.chi2.sf(lm_value, df_diff)
return lm_value, p_value, df_diff
def compare_f_test(self, restricted):
"""
Use F test to test whether restricted model is correct.
Parameters
----------
restricted : Result instance
The restricted model is assumed to be nested in the
current model. The result instance of the restricted model
is required to have two attributes, residual sum of
squares, `ssr`, residual degrees of freedom, `df_resid`.
Returns
-------
f_value : float
The test statistic which has an F distribution.
p_value : float
The p-value of the test statistic.
df_diff : int
The degrees of freedom of the restriction, i.e. difference in
df between models.
Notes
-----
See mailing list discussion October 17,
This test compares the residual sum of squares of the two
models. This is not a valid test, if there is unspecified
heteroscedasticity or correlation. This method will issue a
warning if this is detected but still return the results under
the assumption of homoscedasticity and no autocorrelation
(sphericity).
"""
has_robust1 = getattr(self, 'cov_type', 'nonrobust') != 'nonrobust'
has_robust2 = (getattr(restricted, 'cov_type', 'nonrobust') !=
'nonrobust')
if has_robust1 or has_robust2:
warnings.warn('F test for comparison is likely invalid with ' +
'robust covariance, proceeding anyway',
InvalidTestWarning)
ssr_full = self.ssr
ssr_restr = restricted.ssr
df_full = self.df_resid
df_restr = restricted.df_resid
df_diff = (df_restr - df_full)
f_value = (ssr_restr - ssr_full) / df_diff / ssr_full * df_full
p_value = stats.f.sf(f_value, df_diff, df_full)
return f_value, p_value, df_diff
def compare_lr_test(self, restricted, large_sample=False):
"""
Likelihood ratio test to test whether restricted model is correct.
Parameters
----------
restricted : Result instance
The restricted model is assumed to be nested in the current model.
The result instance of the restricted model is required to have two
attributes, residual sum of squares, `ssr`, residual degrees of
freedom, `df_resid`.
large_sample : bool
Flag indicating whether to use a heteroskedasticity robust version
of the LR test, which is a modified LM test.
Returns
-------
lr_stat : float
The likelihood ratio which is chisquare distributed with df_diff
degrees of freedom.
p_value : float
The p-value of the test statistic.
df_diff : int
The degrees of freedom of the restriction, i.e. difference in df
between models.
Notes
-----
The exact likelihood ratio is valid for homoskedastic data,
and is defined as
.. math:: D=-2\\log\\left(\\frac{\\mathcal{L}_{null}}
{\\mathcal{L}_{alternative}}\\right)
where :math:`\\mathcal{L}` is the likelihood of the
model. With :math:`D` distributed as chisquare with df equal
to difference in number of parameters or equivalently
difference in residual degrees of freedom.
The large sample version of the likelihood ratio is defined as
.. math:: D=n s^{\\prime}S^{-1}s
where :math:`s=n^{-1}\\sum_{i=1}^{n} s_{i}`
.. math:: s_{i} = x_{i,alternative} \\epsilon_{i,null}
is the average score of the model evaluated using the
residuals from null model and the regressors from the
alternative model and :math:`S` is the covariance of the
scores, :math:`s_{i}`. The covariance of the scores is
estimated using the same estimator as in the alternative
model.
This test compares the loglikelihood of the two models. This
may not be a valid test, if there is unspecified
heteroscedasticity or correlation. This method will issue a
warning if this is detected but still return the results
without taking unspecified heteroscedasticity or correlation
into account.
This test compares the loglikelihood of the two models. This
may not be a valid test, if there is unspecified
heteroscedasticity or correlation. This method will issue a
warning if this is detected but still return the results
without taking unspecified heteroscedasticity or correlation
into account.
is the average score of the model evaluated using the
residuals from null model and the regressors from the
alternative model and :math:`S` is the covariance of the
scores, :math:`s_{i}`. The covariance of the scores is
estimated using the same estimator as in the alternative
model.
"""
# TODO: put into separate function, needs tests
# See mailing list discussion October 17,
if large_sample:
return self.compare_lm_test(restricted, use_lr=True)
has_robust1 = (getattr(self, 'cov_type', 'nonrobust') != 'nonrobust')
has_robust2 = (
getattr(restricted, 'cov_type', 'nonrobust') != 'nonrobust')
if has_robust1 or has_robust2:
warnings.warn('Likelihood Ratio test is likely invalid with ' +
'robust covariance, proceeding anyway',
InvalidTestWarning)
llf_full = self.llf
llf_restr = restricted.llf
df_full = self.df_resid
df_restr = restricted.df_resid
lrdf = (df_restr - df_full)
lrstat = -2*(llf_restr - llf_full)
lr_pvalue = stats.chi2.sf(lrstat, lrdf)
return lrstat, lr_pvalue, lrdf
def get_robustcov_results(self, cov_type='HC1', use_t=None, **kwargs):
"""
Create new results instance with robust covariance as default.
Parameters
----------
cov_type : str
The type of robust sandwich estimator to use. See Notes below.
use_t : bool
If true, then the t distribution is used for inference.
If false, then the normal distribution is used.
If `use_t` is None, then an appropriate default is used, which is
`True` if the cov_type is nonrobust, and `False` in all other
cases.
**kwargs
Required or optional arguments for robust covariance calculation.
See Notes below.
Returns
-------
RegressionResults
This method creates a new results instance with the
requested robust covariance as the default covariance of
the parameters. Inferential statistics like p-values and
hypothesis tests will be based on this covariance matrix.
Notes
-----
The following covariance types and required or optional arguments are
currently available:
- 'fixed scale' and optional keyword argument 'scale' which uses
a predefined scale estimate with default equal to one.
- 'HC0', 'HC1', 'HC2', 'HC3' and no keyword arguments:
heteroscedasticity robust covariance
- 'HAC' and keywords
- `maxlag` integer (required) : number of lags to use
- `kernel` callable or str (optional) : kernel
currently available kernels are ['bartlett', 'uniform'],
default is Bartlett
- `use_correction` bool (optional) : If true, use small sample
correction
- 'cluster' and required keyword `groups`, integer group indicator
- `groups` array_like, integer (required) :
index of clusters or groups
- `use_correction` bool (optional) :
If True the sandwich covariance is calculated with a small
sample correction.
If False the sandwich covariance is calculated without
small sample correction.
- `df_correction` bool (optional)
If True (default), then the degrees of freedom for the
inferential statistics and hypothesis tests, such as
pvalues, f_pvalue, conf_int, and t_test and f_test, are
based on the number of groups minus one instead of the
total number of observations minus the number of explanatory
variables. `df_resid` of the results instance is adjusted.
If False, then `df_resid` of the results instance is not
adjusted.
- 'hac-groupsum' Driscoll and Kraay, heteroscedasticity and
autocorrelation robust standard errors in panel data
keywords
- `time` array_like (required) : index of time periods
- `maxlag` integer (required) : number of lags to use
- `kernel` callable or str (optional). The available kernels
are ['bartlett', 'uniform']. The default is Bartlett.
- `use_correction` False or string in ['hac', 'cluster'] (optional).
If False the the sandwich covariance is calculated without small
sample correction. If `use_correction = 'cluster'` (default),
then the same small sample correction as in the case of
`covtype='cluster'` is used.
- `df_correction` bool (optional) The adjustment to df_resid, see
cov_type 'cluster' above
# TODO: we need more options here
- 'hac-panel' heteroscedasticity and autocorrelation robust standard
errors in panel data.
The data needs to be sorted in this case, the time series
for each panel unit or cluster need to be stacked. The
membership to a timeseries of an individual or group can
be either specified by group indicators or by increasing
time periods.
keywords
- either `groups` or `time` : array_like (required)
`groups` : indicator for groups
`time` : index of time periods
- `maxlag` integer (required) : number of lags to use
- `kernel` callable or str (optional)
currently available kernels are ['bartlett', 'uniform'],
default is Bartlett
- `use_correction` False or string in ['hac', 'cluster'] (optional)
If False the sandwich covariance is calculated without
small sample correction.
- `df_correction` bool (optional)
adjustment to df_resid, see cov_type 'cluster' above
# TODO: we need more options here
Reminder:
`use_correction` in "hac-groupsum" and "hac-panel" is not bool,
needs to be in [False, 'hac', 'cluster']
TODO: Currently there is no check for extra or misspelled keywords,
except in the case of cov_type `HCx`
"""
import statsmodels.stats.sandwich_covariance as sw
from statsmodels.base.covtype import normalize_cov_type, descriptions
cov_type = normalize_cov_type(cov_type)
if 'kernel' in kwargs:
kwargs['weights_func'] = kwargs.pop('kernel')
if 'weights_func' in kwargs and not callable(kwargs['weights_func']):
kwargs['weights_func'] = sw.kernel_dict[kwargs['weights_func']]
# TODO: make separate function that returns a robust cov plus info
use_self = kwargs.pop('use_self', False)
if use_self:
res = self
else:
res = self.__class__(
self.model, self.params,
normalized_cov_params=self.normalized_cov_params,
scale=self.scale)
res.cov_type = cov_type
# use_t might already be defined by the class, and already set
if use_t is None:
use_t = self.use_t
res.cov_kwds = {'use_t': use_t} # store for information
res.use_t = use_t
adjust_df = False
if cov_type in ['cluster', 'hac-panel', 'hac-groupsum']:
df_correction = kwargs.get('df_correction', None)
# TODO: check also use_correction, do I need all combinations?
if df_correction is not False: # i.e. in [None, True]:
# user did not explicitely set it to False
adjust_df = True
res.cov_kwds['adjust_df'] = adjust_df
# verify and set kwargs, and calculate cov
# TODO: this should be outsourced in a function so we can reuse it in
# other models
# TODO: make it DRYer repeated code for checking kwargs
if cov_type in ['fixed scale', 'fixed_scale']:
res.cov_kwds['description'] = descriptions['fixed_scale']
res.cov_kwds['scale'] = scale = kwargs.get('scale', 1.)
res.cov_params_default = scale * res.normalized_cov_params
elif cov_type.upper() in ('HC0', 'HC1', 'HC2', 'HC3'):
if kwargs:
raise ValueError('heteroscedasticity robust covariance '
'does not use keywords')
res.cov_kwds['description'] = descriptions[cov_type.upper()]
res.cov_params_default = getattr(self, 'cov_' + cov_type.upper())
elif cov_type.lower() == 'hac':
# TODO: check if required, default in cov_hac_simple
maxlags = kwargs['maxlags']
res.cov_kwds['maxlags'] = maxlags
weights_func = kwargs.get('weights_func', sw.weights_bartlett)
res.cov_kwds['weights_func'] = weights_func
use_correction = kwargs.get('use_correction', False)
res.cov_kwds['use_correction'] = use_correction
res.cov_kwds['description'] = descriptions['HAC'].format(
maxlags=maxlags,
correction=['without', 'with'][use_correction])
res.cov_params_default = sw.cov_hac_simple(
self, nlags=maxlags, weights_func=weights_func,
use_correction=use_correction)
elif cov_type.lower() == 'cluster':
# cluster robust standard errors, one- or two-way
groups = kwargs['groups']
if not hasattr(groups, 'shape'):
groups = np.asarray(groups).T
if groups.ndim >= 2:
groups = groups.squeeze()
res.cov_kwds['groups'] = groups
use_correction = kwargs.get('use_correction', True)
res.cov_kwds['use_correction'] = use_correction
if groups.ndim == 1:
if adjust_df:
# need to find number of groups
# duplicate work
self.n_groups = n_groups = len(np.unique(groups))
res.cov_params_default = sw.cov_cluster(
self, groups, use_correction=use_correction)
elif groups.ndim == 2:
if hasattr(groups, 'values'):
groups = groups.values
if adjust_df:
# need to find number of groups
# duplicate work
n_groups0 = len(np.unique(groups[:, 0]))
n_groups1 = len(np.unique(groups[:, 1]))
self.n_groups = (n_groups0, n_groups1)
n_groups = min(n_groups0, n_groups1) # use for adjust_df
# Note: sw.cov_cluster_2groups has 3 returns
res.cov_params_default = sw.cov_cluster_2groups(
self, groups, use_correction=use_correction)[0]
else:
raise ValueError('only two groups are supported')
res.cov_kwds['description'] = descriptions['cluster']
elif cov_type.lower() == 'hac-panel':
# cluster robust standard errors
res.cov_kwds['time'] = time = kwargs.get('time', None)
res.cov_kwds['groups'] = groups = kwargs.get('groups', None)
# TODO: nlags is currently required
# nlags = kwargs.get('nlags', True)
# res.cov_kwds['nlags'] = nlags
# TODO: `nlags` or `maxlags`
res.cov_kwds['maxlags'] = maxlags = kwargs['maxlags']
use_correction = kwargs.get('use_correction', 'hac')
res.cov_kwds['use_correction'] = use_correction
weights_func = kwargs.get('weights_func', sw.weights_bartlett)
res.cov_kwds['weights_func'] = weights_func
if groups is not None:
groups = np.asarray(groups)
tt = (np.nonzero(groups[:-1] != groups[1:])[0] + 1).tolist()
nobs_ = len(groups)
elif time is not None:
time = np.asarray(time)
# TODO: clumsy time index in cov_nw_panel
tt = (np.nonzero(time[1:] < time[:-1])[0] + 1).tolist()
nobs_ = len(time)
else:
raise ValueError('either time or groups needs to be given')
groupidx = lzip([0] + tt, tt + [nobs_])
self.n_groups = n_groups = len(groupidx)
res.cov_params_default = sw.cov_nw_panel(self, maxlags, groupidx,
weights_func=weights_func,
use_correction=use_correction)
res.cov_kwds['description'] = descriptions['HAC-Panel']
elif cov_type.lower() == 'hac-groupsum':
# Driscoll-Kraay standard errors
res.cov_kwds['time'] = time = kwargs['time']
# TODO: nlags is currently required
# nlags = kwargs.get('nlags', True)
# res.cov_kwds['nlags'] = nlags
# TODO: `nlags` or `maxlags`
res.cov_kwds['maxlags'] = maxlags = kwargs['maxlags']
use_correction = kwargs.get('use_correction', 'cluster')
res.cov_kwds['use_correction'] = use_correction
weights_func = kwargs.get('weights_func', sw.weights_bartlett)
res.cov_kwds['weights_func'] = weights_func
if adjust_df:
# need to find number of groups
tt = (np.nonzero(time[1:] < time[:-1])[0] + 1)
self.n_groups = n_groups = len(tt) + 1
res.cov_params_default = sw.cov_nw_groupsum(
self, maxlags, time, weights_func=weights_func,
use_correction=use_correction)
res.cov_kwds['description'] = descriptions['HAC-Groupsum']
else:
raise ValueError('cov_type not recognized. See docstring for ' +
'available options and spelling')
if adjust_df:
# Note: df_resid is used for scale and others, add new attribute
res.df_resid_inference = n_groups - 1
return res
@Appender(pred.get_prediction.__doc__)
def get_prediction(self, exog=None, transform=True, weights=None,
row_labels=None, **kwargs):
return pred.get_prediction(
self, exog=exog, transform=transform, weights=weights,
row_labels=row_labels, **kwargs)
def summary(self, yname=None, xname=None, title=None, alpha=.05):
"""
Summarize the Regression Results.
Parameters
----------
yname : str, optional
Name of endogenous (response) variable. The Default is `y`.
xname : list[str], optional
Names for the exogenous variables. Default is `var_##` for ## in
the number of regressors. Must match the number of parameters
in the model.
title : str, optional
Title for the top table. If not None, then this replaces the
default title.
alpha : float
The significance level for the confidence intervals.
Returns
-------
Summary
Instance holding the summary tables and text, which can be printed
or converted to various output formats.
See Also
--------
statsmodels.iolib.summary.Summary : A class that holds summary results.
"""
from statsmodels.stats.stattools import (
jarque_bera, omni_normtest, durbin_watson)
jb, jbpv, skew, kurtosis = jarque_bera(self.wresid)
omni, omnipv = omni_normtest(self.wresid)
eigvals = self.eigenvals
condno = self.condition_number
# TODO: Avoid adding attributes in non-__init__
self.diagn = dict(jb=jb, jbpv=jbpv, skew=skew, kurtosis=kurtosis,
omni=omni, omnipv=omnipv, condno=condno,
mineigval=eigvals[-1])
# TODO not used yet
# diagn_left_header = ['Models stats']
# diagn_right_header = ['Residual stats']
# TODO: requiring list/iterable is a bit annoying
# need more control over formatting
# TODO: default do not work if it's not identically spelled
top_left = [('Dep. Variable:', None),
('Model:', None),
('Method:', ['Least Squares']),
('Date:', None),
('Time:', None),
('No. Observations:', None),
('Df Residuals:', None),
('Df Model:', None),
]
if hasattr(self, 'cov_type'):
top_left.append(('Covariance Type:', [self.cov_type]))
rsquared_type = '' if self.k_constant else ' (uncentered)'
top_right = [('R-squared' + rsquared_type + ':',
["%#8.3f" % self.rsquared]),
('Adj. R-squared' + rsquared_type + ':',
["%#8.3f" % self.rsquared_adj]),
('F-statistic:', ["%#8.4g" % self.fvalue]),
('Prob (F-statistic):', ["%#6.3g" % self.f_pvalue]),
('Log-Likelihood:', None),
('AIC:', ["%#8.4g" % self.aic]),
('BIC:', ["%#8.4g" % self.bic])
]
diagn_left = [('Omnibus:', ["%#6.3f" % omni]),
('Prob(Omnibus):', ["%#6.3f" % omnipv]),
('Skew:', ["%#6.3f" % skew]),
('Kurtosis:', ["%#6.3f" % kurtosis])
]
diagn_right = [('Durbin-Watson:',
["%#8.3f" % durbin_watson(self.wresid)]
),
('Jarque-Bera (JB):', ["%#8.3f" % jb]),
('Prob(JB):', ["%#8.3g" % jbpv]),
('Cond. No.', ["%#8.3g" % condno])
]
if title is None:
title = self.model.__class__.__name__ + ' ' + "Regression Results"
# create summary table instance
from statsmodels.iolib.summary import Summary
smry = Summary()
smry.add_table_2cols(self, gleft=top_left, gright=top_right,
yname=yname, xname=xname, title=title)
smry.add_table_params(self, yname=yname, xname=xname, alpha=alpha,
use_t=self.use_t)
smry.add_table_2cols(self, gleft=diagn_left, gright=diagn_right,
yname=yname, xname=xname,
title="")
# add warnings/notes, added to text format only
etext = []
if hasattr(self, 'cov_type'):
etext.append(self.cov_kwds['description'])
if self.model.exog.shape[0] < self.model.exog.shape[1]:
wstr = "The input rank is higher than the number of observations."
etext.append(wstr)
if eigvals[-1] < 1e-10:
wstr = "The smallest eigenvalue is %6.3g. This might indicate "
wstr += "that there are\n"
wstr += "strong multicollinearity problems or that the design "
wstr += "matrix is singular."
wstr = wstr % eigvals[-1]
etext.append(wstr)
elif condno > 1000: # TODO: what is recommended?
wstr = "The condition number is large, %6.3g. This might "
wstr += "indicate that there are\n"
wstr += "strong multicollinearity or other numerical "
wstr += "problems."
wstr = wstr % condno
etext.append(wstr)
if etext:
etext = ["[{0}] {1}".format(i + 1, text)
for i, text in enumerate(etext)]
etext.insert(0, "Warnings:")
smry.add_extra_txt(etext)
return smry
def summary2(self, yname=None, xname=None, title=None, alpha=.05,
float_format="%.4f"):
"""
Experimental summary function to summarize the regression results.
Parameters
----------
yname : str
The name of the dependent variable (optional).
xname : list[str], optional
Names for the exogenous variables. Default is `var_##` for ## in
the number of regressors. Must match the number of parameters
in the model.
title : str, optional
Title for the top table. If not None, then this replaces the
default title.
alpha : float
The significance level for the confidence intervals.
float_format : str
The format for floats in parameters summary.
Returns
-------
Summary
Instance holding the summary tables and text, which can be printed
or converted to various output formats.
See Also
--------
statsmodels.iolib.summary2.Summary
A class that holds summary results.
"""
# Diagnostics
from statsmodels.stats.stattools import (jarque_bera,
omni_normtest,
durbin_watson)
from collections import OrderedDict
jb, jbpv, skew, kurtosis = jarque_bera(self.wresid)
omni, omnipv = omni_normtest(self.wresid)
dw = durbin_watson(self.wresid)
eigvals = self.eigenvals
condno = self.condition_number
eigvals = np.sort(eigvals) # in increasing order
diagnostic = OrderedDict([
('Omnibus:', "%.3f" % omni),
('Prob(Omnibus):', "%.3f" % omnipv),
('Skew:', "%.3f" % skew),
('Kurtosis:', "%.3f" % kurtosis),
('Durbin-Watson:', "%.3f" % dw),
('Jarque-Bera (JB):', "%.3f" % jb),
('Prob(JB):', "%.3f" % jbpv),
('Condition No.:', "%.0f" % condno)
])
# Summary
from statsmodels.iolib import summary2
smry = summary2.Summary()
smry.add_base(results=self, alpha=alpha, float_format=float_format,
xname=xname, yname=yname, title=title)
smry.add_dict(diagnostic)
# Warnings
if eigvals[-1] < 1e-10:
warn = "The smallest eigenvalue is %6.3g. This might indicate that\
there are strong multicollinearity problems or that the design\
matrix is singular." % eigvals[-1]
smry.add_text(warn)
if condno > 1000:
warn = "* The condition number is large (%.g). This might indicate \
strong multicollinearity or other numerical problems." % condno
smry.add_text(warn)
return smry
class OLSResults(RegressionResults):
"""
Results class for for an OLS model.
Parameters
----------
model : RegressionModel
The regression model instance.
params : ndarray
The estimated parameters.
normalized_cov_params : ndarray
The normalized covariance parameters.
scale : float
The estimated scale of the residuals.
cov_type : str
The covariance estimator used in the results.
cov_kwds : dict
Additional keywords used in the covariance specification.
use_t : bool
Flag indicating to use the Student's t in inference.
**kwargs
Additional keyword arguments used to initialize the results.
See Also
--------
RegressionResults
Results store for WLS and GLW models.
Notes
-----
Most of the methods and attributes are inherited from RegressionResults.
The special methods that are only available for OLS are:
- get_influence
- outlier_test
- el_test
- conf_int_el
"""
def get_influence(self):
"""
Calculate influence and outlier measures.
Returns
-------
OLSInfluence
The instance containing methods to calculate the main influence and
outlier measures for the OLS regression.
See Also
--------
statsmodels.stats.outliers_influence.OLSInfluence
A class that exposes methods to examine observation influence.
"""
from statsmodels.stats.outliers_influence import OLSInfluence
return OLSInfluence(self)
def outlier_test(self, method='bonf', alpha=.05, labels=None,
order=False, cutoff=None):
"""
Test observations for outliers according to method.
Parameters
----------
method : str
The method to use in the outlier test. Must be one of:
- `bonferroni` : one-step correction
- `sidak` : one-step correction
- `holm-sidak` :
- `holm` :
- `simes-hochberg` :
- `hommel` :
- `fdr_bh` : Benjamini/Hochberg
- `fdr_by` : Benjamini/Yekutieli
See `statsmodels.stats.multitest.multipletests` for details.
alpha : float
The familywise error rate (FWER).
labels : None or array_like
If `labels` is not None, then it will be used as index to the
returned pandas DataFrame. See also Returns below.
order : bool
Whether or not to order the results by the absolute value of the
studentized residuals. If labels are provided they will also be
sorted.
cutoff : None or float in [0, 1]
If cutoff is not None, then the return only includes observations
with multiple testing corrected p-values strictly below the cutoff.
The returned array or dataframe can be empty if t.
Returns
-------
array_like
Returns either an ndarray or a DataFrame if labels is not None.
Will attempt to get labels from model_results if available. The
columns are the Studentized residuals, the unadjusted p-value,
and the corrected p-value according to method.
Notes
-----
The unadjusted p-value is stats.t.sf(abs(resid), df) where
df = df_resid - 1.
"""
from statsmodels.stats.outliers_influence import outlier_test
return outlier_test(self, method, alpha, labels=labels,
order=order, cutoff=cutoff)
def el_test(self, b0_vals, param_nums, return_weights=0, ret_params=0,
method='nm', stochastic_exog=1):
"""
Test single or joint hypotheses using Empirical Likelihood.
Parameters
----------
b0_vals : 1darray
The hypothesized value of the parameter to be tested.
param_nums : 1darray
The parameter number to be tested.
return_weights : bool
If true, returns the weights that optimize the likelihood
ratio at b0_vals. The default is False.
ret_params : bool
If true, returns the parameter vector that maximizes the likelihood
ratio at b0_vals. Also returns the weights. The default is False.
method : str
Can either be 'nm' for Nelder-Mead or 'powell' for Powell. The
optimization method that optimizes over nuisance parameters.
The default is 'nm'.
stochastic_exog : bool
When True, the exogenous variables are assumed to be stochastic.
When the regressors are nonstochastic, moment conditions are
placed on the exogenous variables. Confidence intervals for
stochastic regressors are at least as large as non-stochastic
regressors. The default is True.
Returns
-------
tuple
The p-value and -2 times the log-likelihood ratio for the
hypothesized values.
Examples
--------
>>> import statsmodels.api as sm
>>> data = sm.datasets.stackloss.load(as_pandas=False)
>>> endog = data.endog
>>> exog = sm.add_constant(data.exog)
>>> model = sm.OLS(endog, exog)
>>> fitted = model.fit()
>>> fitted.params
>>> array([-39.91967442, 0.7156402 , 1.29528612, -0.15212252])
>>> fitted.rsquared
>>> 0.91357690446068196
>>> # Test that the slope on the first variable is 0
>>> fitted.el_test([0], [1])
>>> (27.248146353888796, 1.7894660442330235e-07)
"""
params = np.copy(self.params)
opt_fun_inst = _ELRegOpts() # to store weights
if len(param_nums) == len(params):
llr = opt_fun_inst._opt_nuis_regress(
[],
param_nums=param_nums,
endog=self.model.endog,
exog=self.model.exog,
nobs=self.model.nobs,
nvar=self.model.exog.shape[1],
params=params,
b0_vals=b0_vals,
stochastic_exog=stochastic_exog)
pval = 1 - stats.chi2.cdf(llr, len(param_nums))
if return_weights:
return llr, pval, opt_fun_inst.new_weights
else:
return llr, pval
x0 = np.delete(params, param_nums)
args = (param_nums, self.model.endog, self.model.exog,
self.model.nobs, self.model.exog.shape[1], params,
b0_vals, stochastic_exog)
if method == 'nm':
llr = optimize.fmin(opt_fun_inst._opt_nuis_regress, x0,
maxfun=10000, maxiter=10000, full_output=1,
disp=0, args=args)[1]
if method == 'powell':
llr = optimize.fmin_powell(opt_fun_inst._opt_nuis_regress, x0,
full_output=1, disp=0,
args=args)[1]
pval = 1 - stats.chi2.cdf(llr, len(param_nums))
if ret_params:
return llr, pval, opt_fun_inst.new_weights, opt_fun_inst.new_params
elif return_weights:
return llr, pval, opt_fun_inst.new_weights
else:
return llr, pval
def conf_int_el(self, param_num, sig=.05, upper_bound=None,
lower_bound=None, method='nm', stochastic_exog=True):
"""
Compute the confidence interval using Empirical Likelihood.
Parameters
----------
param_num : float
The parameter for which the confidence interval is desired.
sig : float
The significance level. Default is 0.05.
upper_bound : float
The maximum value the upper limit can be. Default is the
99.9% confidence value under OLS assumptions.
lower_bound : float
The minimum value the lower limit can be. Default is the 99.9%
confidence value under OLS assumptions.
method : str
Can either be 'nm' for Nelder-Mead or 'powell' for Powell. The
optimization method that optimizes over nuisance parameters.
The default is 'nm'.
stochastic_exog : bool
When True, the exogenous variables are assumed to be stochastic.
When the regressors are nonstochastic, moment conditions are
placed on the exogenous variables. Confidence intervals for
stochastic regressors are at least as large as non-stochastic
regressors. The default is True.
Returns
-------
lowerl : float
The lower bound of the confidence interval.
upperl : float
The upper bound of the confidence interval.
See Also
--------
el_test : Test parameters using Empirical Likelihood.
Notes
-----
This function uses brentq to find the value of beta where
test_beta([beta], param_num)[1] is equal to the critical value.
The function returns the results of each iteration of brentq at each
value of beta.
The current function value of the last printed optimization should be
the critical value at the desired significance level. For alpha=.05,
the value is 3.841459.
To ensure optimization terminated successfully, it is suggested to do
el_test([lower_limit], [param_num]).
If the optimization does not terminate successfully, consider switching
optimization algorithms.
If optimization is still not successful, try changing the values of
start_int_params. If the current function value repeatedly jumps
from a number between 0 and the critical value and a very large number
(>50), the starting parameters of the interior minimization need
to be changed.
"""
r0 = stats.chi2.ppf(1 - sig, 1)
if upper_bound is None:
upper_bound = self.conf_int(.01)[param_num][1]
if lower_bound is None:
lower_bound = self.conf_int(.01)[param_num][0]
def f(b0):
return self.el_test(np.array([b0]), np.array([param_num]),
method=method,
stochastic_exog=stochastic_exog)[0] - r0
lowerl = optimize.brenth(f, lower_bound,
self.params[param_num])
upperl = optimize.brenth(f, self.params[param_num],
upper_bound)
# ^ Seems to be faster than brentq in most cases
return (lowerl, upperl)
class RegressionResultsWrapper(wrap.ResultsWrapper):
_attrs = {
'chisq': 'columns',
'sresid': 'rows',
'weights': 'rows',
'wresid': 'rows',
'bcov_unscaled': 'cov',
'bcov_scaled': 'cov',
'HC0_se': 'columns',
'HC1_se': 'columns',
'HC2_se': 'columns',
'HC3_se': 'columns',
'norm_resid': 'rows',
}
_wrap_attrs = wrap.union_dicts(base.LikelihoodResultsWrapper._attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(
base.LikelihoodResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(RegressionResultsWrapper,
RegressionResults)