System Regression Models¶

System regression estimates multiple regressions simultaneously. There are three reasons to consider system estimation instead of equation by equation estimation

Joint inference on parameters across models
Efficiency gains in some circumstances using cross-model GLS
Imposing restrictions on parameters across models

The main model is the Seemingly Unrelated Regression (SUR) Estimator. This estimator uses a modified syntax since the class allow multiple models to be specified, each with it own dependent and exogenous variables. The more structured syntax uses a dict or preferably an OrderedDict, which ensures that the order of the equations in the results is preserved, where each entry is a complete model with a dependent and exogenous (stored as a dict with keys dependent and exog.

from collections import OrderedDict
import statsmodels.api as sm
from linearmodels.datasets import fringe
from linearmodels.system import SUR
data = sm.add_constant(fringe.load())
equations = OrderedDict()
equations['earnings'] = {'dependent': data.hrearn,
                         'exog': data[['const', 'exper', 'tenure']]}
equations['benefits'] = {'dependent': data.hrbens,
                         'exog': data[['const', 'exper', 'tenure']]}
mod = SUR(equations)
mod.fit(cov_type='unadjusted')

                        System GLS Estimation Summary
==============================================================================
Estimator:                        GLS   Overall R-squared:              0.0757
No. Equations.:                     2   Cov. Estimator:             unadjusted
No. Observations:                 616   Num. Constraints:                 None
Date:                Sat, Jun 17 2017
Time:                        23:21:18


                Equation: earnings, Dependent Variable: hrearn
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
const          4.2839     0.3407     12.573     0.0000      3.6161      4.9517
exper          0.1163     0.0186     6.2478     0.0000      0.0798      0.1528
tenure        -0.0283     0.0295    -0.9598     0.3372     -0.0862      0.0295

                Equation: benefits, Dependent Variable: hrbens
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
const          0.6390     0.0449     14.220     0.0000      0.5509      0.7270
exper          0.0014     0.0025     0.5617     0.5743     -0.0034      0.0062
tenure         0.0316     0.0039     8.1254     0.0000      0.0240      0.0393

==============================================================================
SystemResults, id: 0x282ca8f7b70

In addition to SUR, the system module also contain an estimator for the Three-stage Least Squares (IV3SLS) Estimator. 3SLS is a generalization of SUR which allows variables to be either exogenous or endogenous, and when there are endogenous variables, for instruments to be used. SUR is a special case of 3SLS where there are no endogenous variables or instruments. 3SLS allows systems of IV equations to be jointly estimated.