ENH collect tools for linear restrictions, constraints, contrasts

[josef-pkt]

so far they are mainly in t_test and f_test plus the restricted estimation in GEE.

extra: naming convention

r_matrix, q_matrix is not really good (although I always remember it), used also in TheilGLS RLS #1667 uses constrand param

[josef-pkt]

Two background docs (that I never worked my way through) http://statsmodels.sourceforge.net/devel/contrasts.html http://patsy.readthedocs.org/en/latest/categorical-coding.html

[josef-pkt]

same or similar to Kerby's reparameterization, Stata has function makecns — Constrained estimation with technical details in the (P) manual

Note: Stata calculates the LS estimate in the transformed model, but reports the results in the original parameterization.

implication: cov_params has reduced rank (rank(exog) - k_constraints)

see #1131 for numerical reasons why we might want to temporarily internally transform for estimation.

Similar: Ridge Regression works with standardized and orthogonalized variables internally, in the description in articles. Also PCR can report results in the original parameterization, instead of in factor space.

[josef-pkt]

additionally on imposing linear restrictions R params = q through transformation In Stata, if q != 0 this is an affine, and not a linear transformation params = T dot params_reduced + a where T is the linear transformation matrix and a is a constant. Which means we cannot just run OLS with the reparameterized exog.

[josef-pkt]

this issue is a bit outdated

see #4361 for the contrast/restriction part and links to other issues

imposing linear or affine restrictions has been implemented in fit_constrained for model with an offset (GLM and count models)

[josef-pkt]

one item where I don't know where it should go:

base._constraints has transform_params_constraint (I found it now because it has zero unit test coverage.) to go from unconstrained to constrained estimate. This was intended somewhere for one-step methods or as starting parameter for restricted estimation, e.g. likelihood ratio test for imposing restrictions, dropping variables.

[josef-pkt]

bump for transform_params_constraint I did not remember whether or where I had parked this. Also, a reference would be helpful, I don't remember where I saw this, most likely it's in several text books. It would be useful to have also the corresponding cov_params for the constrained params

There is a version for multivariate linear model. #8722 e.g. for MANOVA type multivariate constraints L B M = C Berndt, Ernst R., and N. Eugene Savin. “Conflict among Criteria for Testing Hypotheses in the Multivariate Linear Regression Model.” Econometrica 45, no. 5 (1977): 1263–77. https://doi.org/10.2307/1914072.

Kubáček, Lubomír. “Multivariate Regression Model with Constraints.” Mathematica Slovaca 57, no. 3 (June 1, 2007): 271–96. https://doi.org/10.2478/s12175-007-0022-7.