# 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
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)

                        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.