# statsmodels.regression.linear_model.GLS¶

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

Generalized Least Squares

Parameters
endogarray_like

A 1-d endogenous response variable. The dependent variable.

exogarray_like

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 statsmodels.tools.add_constant.

sigma

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.

missingstr

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’.

hasconst

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.

**kwargs

Extra arguments that are used to set model properties when using the formula interface.

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
>>> 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_wexogndarray

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

cholsimgainvndarray

The transpose of the Cholesky decomposition of the pseudoinverse.

df_modelfloat

The model degree of freedom.

df_residfloat

The residual degree of freedom.

llffloat

The value of the likelihood function of the fitted model.

nobsfloat

The number of observations n.

normalized_cov_paramsndarray

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

resultsRegressionResults instance

A property that returns the RegressionResults class if fit.

sigmandarray

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

wexogndarray

Design matrix whitened by cholsigmainv

wendogndarray

Response variable whitened by cholsigmainv

Methods

 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, …]) Construct a random number generator for the predictive distribution. hessian(params) The Hessian matrix of the model. hessian_factor(params[, scale, observed]) Compute weights for calculating Hessian. information(params) Fisher information matrix of model. Initialize model components. loglike(params) Compute the value of the Gaussian log-likelihood function at params. predict(params[, exog]) Return linear predicted values from a design matrix. score(params) Score vector of model. GLS whiten method.

Methods

 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, …]) Construct a random number generator for the predictive distribution. hessian(params) The Hessian matrix of the model. hessian_factor(params[, scale, observed]) Compute weights for calculating Hessian. information(params) Fisher information matrix of model. Initialize model components. loglike(params) Compute the value of the Gaussian log-likelihood function at params. predict(params[, exog]) Return linear predicted values from a design matrix. score(params) Score vector of model. GLS whiten method.

Properties

 df_model The model degree of freedom. df_resid The residual degree of freedom. endog_names Names of endogenous variables. exog_names Names of exogenous variables.