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statsmodels.regression.linear_model.GLS

class statsmodels.regression.linear_model.GLS(endog, exog, sigma=None)[source]

Generalized least squares model with a general covariance structure.

Parameters :

endog : array-like

endog is a 1-d vector that contains the response/independent variable

exog : array-like

exog is a n x p vector where n is the number of observations and p is the number of regressors/dependent variables including the intercept if one is included in the data.

sigma : scalar or array

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.

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 numpy as np
>>> import statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> 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()

Attributes

pinv_wexog array pinv_wexog is the p x n Moore-Penrose pseudoinverse of wexog.
cholsimgainv array 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 array p x p array (X^{T}\Sigma^{-1}X)^{-1}
results RegressionResults instance A property that returns the RegressionResults class if fit.
sigma array sigma is the n x n covariance structure of the error terms.
wexog array Design matrix whitened by cholsigmainv
wendog array Response variable whitened by cholsigmainv

Methods

fit(**kwargs[, method]) Full fit of the model.
hessian(params) The Hessian matrix of the model
information(params) Fisher information matrix of model
initialize()
loglike(params) Returns the value of the gaussian loglikelihood function at params.
predict(params[, exog]) Return linear predicted values from a design matrix.
score(params) Score vector of model.
whiten(X) GLS whiten method.

Attributes

endog_names
exog_names

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