MultinomialResults.f_test(r_matrix, cov_p=None, scale=1.0, invcov=None)

Compute the F-test for a joint linear hypothesis.

This is a special case of wald_test that always uses the F distribution.


r_matrix : array-like, str, or tuple

  • array : An r x k array where r is the number of restrictions to test and k is the number of regressors. It is assumed that the linear combination is equal to zero.
  • str : The full hypotheses to test can be given as a string. See the examples.
  • tuple : A tuple of arrays in the form (R, q), q can be either a scalar or a length k row vector.

cov_p : array-like, optional

An alternative estimate for the parameter covariance matrix. If None is given, self.normalized_cov_params is used.

scale : float, optional

Default is 1.0 for no scaling.

invcov : array-like, optional

A q x q array to specify an inverse covariance matrix based on a restrictions matrix.


res : ContrastResults instance

The results for the test are attributes of this results instance.


The matrix r_matrix is assumed to be non-singular. More precisely,

r_matrix (pX pX.T) r_matrix.T

is assumed invertible. Here, pX is the generalized inverse of the design matrix of the model. There can be problems in non-OLS models where the rank of the covariance of the noise is not full.


>>> import numpy as np
>>> import statsmodels.api as sm
>>> data = sm.datasets.longley.load()
>>> data.exog = sm.add_constant(data.exog)
>>> results = sm.OLS(data.endog, data.exog).fit()
>>> A = np.identity(len(results.params))
>>> A = A[1:,:]

This tests that each coefficient is jointly statistically significantly different from zero.

>>> print(results.f_test(A))
<F test: F=array([[ 330.28533923]]), p=4.984030528700946e-10, df_denom=9, df_num=6>

Compare this to

>>> results.fvalue
>>> results.f_pvalue
>>> B = np.array(([0,0,1,-1,0,0,0],[0,0,0,0,0,1,-1]))

This tests that the coefficient on the 2nd and 3rd regressors are equal and jointly that the coefficient on the 5th and 6th regressors are equal.

>>> print(results.f_test(B))
<F test: F=array([[ 9.74046187]]), p=0.005605288531708235, df_denom=9, df_num=2>

Alternatively, you can specify the hypothesis tests using a string

>>> from statsmodels.datasets import longley
>>> from statsmodels.formula.api import ols
>>> dta = longley.load_pandas().data
>>> formula = 'TOTEMP ~ GNPDEFL + GNP + UNEMP + ARMED + POP + YEAR'
>>> results = ols(formula, dta).fit()
>>> hypotheses = '(GNPDEFL = GNP), (UNEMP = 2), (YEAR/1829 = 1)'
>>> f_test = results.f_test(hypotheses)
>>> print(f_test)
<F test: F=array([[ 144.17976065]]), p=6.322026217355609e-08, df_denom=9, df_num=3>