Using formulas to specify models¶
Formulas can be used to specify models using mostly standard formulaic syntax. Since system estimation is more complicated than the specification of a single model, there are two methods available to specify a system:
Dictionary of formulas
Single formula separated using {}
These examples use data on fringe benefits from F. Vella (1993), “A Simple Estimator for Simultaneous Models with Censored Endogenous Regressors” which appears in Wooldridge (2002). The model consists of two equations, one for hourly wage and the other for hourly benefits. The initial model uses the same regressors in both equations.
[1]:
import numpy as np
import pandas as pd
from linearmodels.datasets import fringe
data = fringe.load()
Dictionary¶
The dictionary syntax is virtually identical to standard formulaic syntax where each equation is specified in a key-value pair where the key is the equation label and the value is the formula. It is recommended to use an OrderedDict which will preserve equation order in results. Keys must be strings.
[2]:
from collections import OrderedDict
formula = OrderedDict()
formula["benefits"] = (
"hrbens ~ educ + exper + expersq + union + south + nrtheast + nrthcen + male"
)
formula["earnings"] = "hrearn ~ educ + exper + expersq + nrtheast + married + male"
[3]:
from linearmodels.system import SUR
mod = SUR.from_formula(formula, data)
print(mod.fit(cov_type="unadjusted"))
System GLS Estimation Summary
===================================================================================
Estimator: GLS Overall R-squared: 0.6951
No. Equations.: 2 McElroy's R-squared: 0.2197
No. Observations: 616 Judge's (OLS) R-squared: 0.1873
Date: Sat, Mar 22 2025 Berndt's R-squared: 0.3775
Time: 18:45:14 Dhrymes's R-squared: 0.6950
Cov. Estimator: unadjusted
Num. Constraints: None
Equation: benefits, Dependent Variable: hrbens
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.0346 0.0050 6.8605 0.0000 0.0247 0.0445
exper 0.0350 0.0063 5.5974 0.0000 0.0228 0.0473
expersq -0.0006 0.0001 -4.4326 0.0000 -0.0009 -0.0003
union 0.3682 0.0474 7.7659 0.0000 0.2753 0.4612
south -0.1775 0.0586 -3.0289 0.0025 -0.2923 -0.0626
nrtheast -0.1224 0.0714 -1.7132 0.0867 -0.2624 0.0176
nrthcen -0.1433 0.0621 -2.3074 0.0210 -0.2650 -0.0216
male 0.2518 0.0490 5.1435 0.0000 0.1559 0.3478
Equation: earnings, Dependent Variable: hrearn
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.3547 0.0357 9.9280 0.0000 0.2847 0.4247
exper -0.0750 0.0492 -1.5260 0.1270 -0.1714 0.0213
expersq 0.0038 0.0011 3.4600 0.0005 0.0016 0.0059
nrtheast -0.7127 0.4473 -1.5934 0.1111 -1.5894 0.1640
married 0.4472 0.3920 1.1410 0.2539 -0.3210 1.2155
male 1.8899 0.3904 4.8408 0.0000 1.1247 2.6551
==============================================================================
Covariance Estimator:
Homoskedastic (Unadjusted) Covariance (Debiased: False, GLS: True)
Curly Braces¶
The same formula can be expressed in a single string by surrounding each equation with braces {}.
[4]:
braces_formula = """
{hrbens ~ educ + exper + expersq + union + south + nrtheast + nrthcen + male}
{hrearn ~ educ + exper + expersq + nrtheast + married + male}
"""
braces_mod = SUR.from_formula(braces_formula, data)
braces_res = braces_mod.fit(cov_type="unadjusted")
print(braces_res)
System GLS Estimation Summary
===================================================================================
Estimator: GLS Overall R-squared: 0.6951
No. Equations.: 2 McElroy's R-squared: 0.2197
No. Observations: 616 Judge's (OLS) R-squared: 0.1873
Date: Sat, Mar 22 2025 Berndt's R-squared: 0.3775
Time: 18:45:14 Dhrymes's R-squared: 0.6950
Cov. Estimator: unadjusted
Num. Constraints: None
Equation: hrbens, Dependent Variable: hrbens
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.0346 0.0050 6.8605 0.0000 0.0247 0.0445
exper 0.0350 0.0063 5.5974 0.0000 0.0228 0.0473
expersq -0.0006 0.0001 -4.4326 0.0000 -0.0009 -0.0003
union 0.3682 0.0474 7.7659 0.0000 0.2753 0.4612
south -0.1775 0.0586 -3.0289 0.0025 -0.2923 -0.0626
nrtheast -0.1224 0.0714 -1.7132 0.0867 -0.2624 0.0176
nrthcen -0.1433 0.0621 -2.3074 0.0210 -0.2650 -0.0216
male 0.2518 0.0490 5.1435 0.0000 0.1559 0.3478
Equation: hrearn, Dependent Variable: hrearn
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.3547 0.0357 9.9280 0.0000 0.2847 0.4247
exper -0.0750 0.0492 -1.5260 0.1270 -0.1714 0.0213
expersq 0.0038 0.0011 3.4600 0.0005 0.0016 0.0059
nrtheast -0.7127 0.4473 -1.5934 0.1111 -1.5894 0.1640
married 0.4472 0.3920 1.1410 0.2539 -0.3210 1.2155
male 1.8899 0.3904 4.8408 0.0000 1.1247 2.6551
==============================================================================
Covariance Estimator:
Homoskedastic (Unadjusted) Covariance (Debiased: False, GLS: True)
Labeled Formulas¶
When using the curly brace formula specification, the equation names are determined by the dependent variable names. When names are repeated as is the case in some datasets (e.g. a SUR on GDP of multiple countries) then the equation labels will be modified until they are unique. This can produce meaningless equation labels, and so it is possible to pass an equation label using the syntax
{label : dep ~ exog}
[5]:
labeled_formula = """
{benefits: hrbens ~ educ + exper + expersq + union + south + nrtheast + nrthcen + male}
{earnings: hrearn ~ educ + exper + expersq + nrtheast + married + male}
"""
labels_mod = SUR.from_formula(labeled_formula, data)
labeled_res = labels_mod.fit(cov_type="unadjusted")
print("Unlabeled")
print(braces_res.equation_labels)
print("Labeled")
print(labeled_res.equation_labels)
Unlabeled
['hrbens', 'hrearn']
Labeled
['benefits', 'earnings']
Other Options¶
Estimation Weights¶
SUR supports weights which are assumed to be proportional to the inverse variance of the data so that
Weights can be passed using a DataFrame where each column.
Here the results are printed to ensure that the estimates are different from those in the standard GLS model.
[6]:
random_weights = np.random.chisquare(5, size=(616, 2))
random_weights = pd.DataFrame(random_weights, columns=["benefits", "earnings"])
weighted_mod = SUR.from_formula(formula, data, weights=random_weights)
print(weighted_mod.fit())
System GLS Estimation Summary
===================================================================================
Estimator: GLS Overall R-squared: 0.6722
No. Equations.: 2 McElroy's R-squared: 0.2114
No. Observations: 616 Judge's (OLS) R-squared: 0.1833
Date: Sat, Mar 22 2025 Berndt's R-squared: 0.4844
Time: 18:45:14 Dhrymes's R-squared: 0.6721
Cov. Estimator: robust
Num. Constraints: None
Equation: benefits, Dependent Variable: hrbens
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.0309 0.0044 6.9868 0.0000 0.0222 0.0396
exper 0.0376 0.0056 6.7220 0.0000 0.0267 0.0486
expersq -0.0007 0.0001 -5.3023 0.0000 -0.0009 -0.0004
union 0.3427 0.0457 7.4991 0.0000 0.2532 0.4323
south -0.1514 0.0537 -2.8169 0.0048 -0.2567 -0.0460
nrtheast -0.0577 0.0640 -0.9020 0.3671 -0.1832 0.0677
nrthcen -0.0404 0.0561 -0.7191 0.4721 -0.1504 0.0696
male 0.2014 0.0454 4.4377 0.0000 0.1125 0.2904
Equation: earnings, Dependent Variable: hrearn
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
educ 0.3393 0.1006 3.3722 0.0007 0.1421 0.5365
exper -0.0398 0.2173 -0.1833 0.8546 -0.4656 0.3860
expersq 0.0025 0.0057 0.4427 0.6580 -0.0087 0.0137
nrtheast -0.5360 0.5539 -0.9676 0.3332 -1.6216 0.5497
married 0.4078 0.3678 1.1090 0.2674 -0.3130 1.1286
male 2.1240 0.2551 8.3266 0.0000 1.6240 2.6239
==============================================================================
Covariance Estimator:
Heteroskedastic (Robust) Covariance (Debiased: False, GLS: True)
Prespecified Residual Covariance¶
Like a standard SUR, it is possible to pass a prespecified residual covariance for use in the GLS step. This is done using the keyword argument sigma in the from_formula method, and is otherwise identical to passing one to the standard SUR.