class statsmodels.regression.linear_model.GLSAR(endog, exog=None, rho=1, missing='none', **kwargs)[source]

A regression model with an AR(p) covariance structure.

  • endog (array-like) – 1-d endogenous response variable. The dependent variable.
  • exog (array-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
  • rho (int) – Order of the autoregressive covariance
  • missing (str) – 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 (None 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.


>>> 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 =
...     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])


GLSAR is considered to be experimental. The linear autoregressive process of order p–AR(p)–is defined as: TODO


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.
hessian(params) The Hessian matrix of the model
hessian_factor(params[, scale, observed]) Weights for calculating Hessian
information(params) Fisher information matrix of model
initialize() Initialize (possibly re-initialize) a Model instance.
iterative_fit([maxiter, rtol]) Perform an iterative two-stage procedure to estimate a GLS model.
loglike(params) Returns 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.
whiten(X) Whiten a series of columns according to an AR(p) covariance structure.


df_model The model degree of freedom, defined as the rank of the regressor matrix minus 1 if a constant is included.
df_resid The residual degree of freedom, defined as the number of observations minus the rank of the regressor matrix.
endog_names Names of endogenous variables
exog_names Names of exogenous variables