class statsmodels.regression.linear_model.GLS(endog, exog, sigma=None, missing='none', hasconst=None, **kwargs)[source]

Generalized least squares model with a general covariance structure.


1-d endogenous response variable. The dependent variable.


A nobs x k array where nobs is the number of observations and k is the number of regressors. An intercept is not included by default and should be added by the user. See

sigmascalar 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.


Available options are ‘none’, ‘drop’, and ‘raise’. If ‘none’, no nan checking is done. If ‘drop’, any observations with nans are dropped. If ‘raise’, an error is raised. Default is ‘none.’

hasconstNone or bool

Indicates whether the RHS includes a user-supplied constant. If True, a constant is not checked for and k_constant is set to 1 and all result statistics are calculated as if a constant is present. If False, a constant is not checked for and k_constant is set to 0.


If sigma is a function of the data making one of the regressors a constant, then the current postestimation statistics will not be correct.


>>> import numpy as np
>>> 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 =
>>> print(gls_results.summary())

pinv_wexog is the p x n Moore-Penrose pseudoinverse of wexog.


The transpose of the Cholesky decomposition of the pseudoinverse.


The model degree of freedom, defined as the rank of the regressor matrix minus 1 if a constant is included.


The residual degree of freedom, defined as the number of observations minus the rank of the regressor matrix.


The value of the likelihood function of the fitted model.


The number of observations n.


p x p array \((X^{T}\Sigma^{-1}X)^{-1}\)

resultsRegressionResults instance

A property that returns the RegressionResults class if fit.


sigma is the n x n covariance structure of the error terms.


Design matrix whitened by cholsigmainv


Response variable whitened by cholsigmainv


fit([method, cov_type, cov_kwds, use_t])

Full fit of the model.

fit_regularized([method, alpha, L1_wt, …])

Return a regularized fit to a linear regression model.

from_formula(formula, data[, subset, drop_cols])

Create a Model from a formula and dataframe.

get_distribution(params, scale[, exog, …])

Returns a random number generator for the predictive distribution.


The Hessian matrix of the model

hessian_factor(params[, scale, observed])

Weights for calculating Hessian


Fisher information matrix of model


Initialize (possibly re-initialize) a Model instance.


Returns the value of the Gaussian log-likelihood function at params.

predict(params[, exog])

Return linear predicted values from a design matrix.


Score vector of model.


GLS whiten method.