# Source code for statsmodels.stats.oaxaca

```# TODO Non-Linear Regressions can be used
# TODO Further decomposition of the two_fold parameters i.e.
# the delta method for further two_fold detail
"""

This class implements Oaxaca-Blinder Decomposition. It returns
a OaxacaResults Class:

OaxacaBlinder:
Two-Fold (two_fold)
Three-Fold (three_fold)

OaxacaResults:
Table Summary (summary)

Oaxaca-Blinder is a statistical method that is used to explain
the differences between two mean values. The idea is to show
from two mean values what can be explained by the data and
what cannot by using OLS regression frameworks.

"The original use by Oaxaca's was to explain the wage
differential between two different groups of workers,
but the method has since been applied to numerous other
topics." (Wikipedia)

The model is designed to accept two endogenous response variables
and two exogenous explanitory variables. They are then fit using
the specific type of decomposition that you want.

The method was famously used in Card and Krueger's paper
"School Quality and Black-White Relative Earnings: A Direct Assessment" (1992)

General reference for Oaxaca-Blinder:

B. Jann "The Blinder-Oaxaca decomposition for linear
regression models," The Stata Journal, 2008.

Econometrics references for regression models:

E. M. Kitagawa  "Components of a Difference Between Two Rates"
Journal of the American Statistical Association, 1955.

A. S. Blinder "Wage Discrimination: Reduced Form and Structural
Estimates," The Journal of Human Resources, 1973.
"""
from textwrap import dedent

import numpy as np

from statsmodels.regression.linear_model import OLS

[docs]class OaxacaBlinder(object):
"""
Class to perform Oaxaca-Blinder Decomposition.

Parameters
----------
endog : array_like
The endogenous variable or the dependent variable that you are trying
to explain.
exog : array_like
The exogenous variable(s) or the independent variable(s) that you are
using to explain the endogenous variable.
bifurcate : {int, str}
The column of the exogenous variable(s) on which to split. This would
generally be the group that you wish to explain the two means for.
Int of the column for a NumPy array or int/string for the name of
the column in Pandas.
hasconst : bool, optional
Indicates whether the two exogenous variables include a user-supplied
constant. If True, a constant is assumed. If False, a constant is added
at the start. If nothing is supplied, then True is assumed.
swap : bool, optional
Imitates the STATA Oaxaca command by allowing users to choose to swap
groups. Unlike STATA, this is assumed to be True instead of False
cov_type : str, optional
See regression.linear_model.RegressionResults for a description of the
available covariance estimators
cov_kwds : dict, optional
See linear_model.RegressionResults.get_robustcov_results for a
description required keywords for alternative covariance estimators

Notes
-----
Please check if your data includes at constant. This will still run, but
will return incorrect values if set incorrectly.

You can access the models by using their code as an attribute, e.g.,
_t_model for the total model, _f_model for the first model, _s_model for
the second model.

Examples
--------
>>> import numpy as np
>>> import statsmodels.api as sm

'3' is the column of which we want to explain or which indicates
the two groups. In this case, it is if you rent.

>>> model = sm.OaxacaBlinder(df.endog, df.exog, 3, hasconst = False)
>>> model.two_fold().summary()
Oaxaca-Blinder Two-fold Effects
Unexplained Effect: 27.94091
Explained Effect: 130.80954
Gap: 158.75044

>>> model.three_fold().summary()
Oaxaca-Blinder Three-fold Effects
Endowments Effect: 321.74824
Coefficient Effect: 75.45371
Interaction Effect: -238.45151
Gap: 158.75044
"""

def __init__(
self,
endog,
exog,
bifurcate,
hasconst=True,
swap=True,
cov_type="nonrobust",
cov_kwds=None,
):
if str(type(exog)).find("pandas") != -1:
bifurcate = exog.columns.get_loc(bifurcate)
endog, exog = np.array(endog), np.array(exog)

self.two_fold_type = None
self.bifurcate = bifurcate
self.cov_type = cov_type
self.cov_kwds = cov_kwds
self.neumark = np.delete(exog, bifurcate, axis=1)
self.exog = exog
self.hasconst = hasconst
bi_col = exog[:, bifurcate]
endog = np.column_stack((bi_col, endog))
bi = np.unique(bi_col)
self.bi_col = bi_col

# split the data along the bifurcate axis, the issue is you need to
# delete it after you fit the model for the total model.
exog_f = exog[np.where(exog[:, bifurcate] == bi)]
exog_s = exog[np.where(exog[:, bifurcate] == bi)]
endog_f = endog[np.where(endog[:, 0] == bi)]
endog_s = endog[np.where(endog[:, 0] == bi)]
exog_f = np.delete(exog_f, bifurcate, axis=1)
exog_s = np.delete(exog_s, bifurcate, axis=1)
endog_f = endog_f[:, 1]
endog_s = endog_s[:, 1]
self.endog = endog[:, 1]

self.len_f, self.len_s = len(endog_f), len(endog_s)
self.gap = endog_f.mean() - endog_s.mean()

if swap and self.gap < 0:
endog_f, endog_s = endog_s, endog_f
exog_f, exog_s = exog_s, exog_f
self.gap = endog_f.mean() - endog_s.mean()
bi, bi = bi, bi

self.bi = bi

if hasconst is False:

self.exog_f_mean = np.mean(exog_f, axis=0)
self.exog_s_mean = np.mean(exog_s, axis=0)

self._f_model = OLS(endog_f, exog_f).fit(
cov_type=cov_type, cov_kwds=cov_kwds
)
self._s_model = OLS(endog_s, exog_s).fit(
cov_type=cov_type, cov_kwds=cov_kwds
)

[docs]    def variance(self, decomp_type, n=5000, conf=0.99):
"""
A helper function to calculate the variance/std. Used to keep
the decomposition functions cleaner
"""
if self.submitted_n is not None:
n = self.submitted_n
if self.submitted_conf is not None:
conf = self.submitted_conf
if self.submitted_weight is not None:
submitted_weight = [
self.submitted_weight,
1 - self.submitted_weight,
]
bi = self.bi
bifurcate = self.bifurcate
endow_eff_list = []
coef_eff_list = []
int_eff_list = []
exp_eff_list = []
unexp_eff_list = []
for _ in range(0, n):
endog = np.column_stack((self.bi_col, self.endog))
exog = self.exog
amount = len(endog)

samples = np.random.randint(0, high=amount, size=amount)
endog = endog[samples]
exog = exog[samples]
neumark = np.delete(exog, bifurcate, axis=1)

exog_f = exog[np.where(exog[:, bifurcate] == bi)]
exog_s = exog[np.where(exog[:, bifurcate] == bi)]
endog_f = endog[np.where(endog[:, 0] == bi)]
endog_s = endog[np.where(endog[:, 0] == bi)]
exog_f = np.delete(exog_f, bifurcate, axis=1)
exog_s = np.delete(exog_s, bifurcate, axis=1)
endog_f = endog_f[:, 1]
endog_s = endog_s[:, 1]
endog = endog[:, 1]

two_fold_type = self.two_fold_type

if self.hasconst is False:

_f_model = OLS(endog_f, exog_f).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
_s_model = OLS(endog_s, exog_s).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
exog_f_mean = np.mean(exog_f, axis=0)
exog_s_mean = np.mean(exog_s, axis=0)

if decomp_type == 3:
endow_eff = (exog_f_mean - exog_s_mean) @ _s_model.params
coef_eff = exog_s_mean @ (_f_model.params - _s_model.params)
int_eff = (exog_f_mean - exog_s_mean) @ (
_f_model.params - _s_model.params
)

endow_eff_list.append(endow_eff)
coef_eff_list.append(coef_eff)
int_eff_list.append(int_eff)

elif decomp_type == 2:
len_f = len(exog_f)
len_s = len(exog_s)

if two_fold_type == "cotton":
t_params = (len_f / (len_f + len_s) * _f_model.params) + (
len_s / (len_f + len_s) * _s_model.params
)

elif two_fold_type == "reimers":
t_params = 0.5 * (_f_model.params + _s_model.params)

elif two_fold_type == "self_submitted":
t_params = (
submitted_weight * _f_model.params
+ submitted_weight * _s_model.params
)

elif two_fold_type == "nuemark":
_t_model = OLS(endog, neumark).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
t_params = _t_model.params

else:
_t_model = OLS(endog, exog).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
t_params = np.delete(_t_model.params, bifurcate)

unexplained = (exog_f_mean @ (_f_model.params - t_params)) + (
exog_s_mean @ (t_params - _s_model.params)
)

explained = (exog_f_mean - exog_s_mean) @ t_params

unexp_eff_list.append(unexplained)
exp_eff_list.append(explained)

high, low = int(n * conf), int(n * (1 - conf))
if decomp_type == 3:
return [
np.std(np.sort(endow_eff_list)[low:high]),
np.std(np.sort(coef_eff_list)[low:high]),
np.std(np.sort(int_eff_list)[low:high]),
]
elif decomp_type == 2:
return [
np.std(np.sort(unexp_eff_list)[low:high]),
np.std(np.sort(exp_eff_list)[low:high]),
]

[docs]    def three_fold(self, std=False, n=None, conf=None):
"""
Calculates the three-fold Oaxaca Blinder Decompositions

Parameters
----------
std: boolean, optional
If true, bootstrapped standard errors will be calculated.
n: int, optional
A amount of iterations to calculate the bootstrapped
standard errors. This defaults to 5000.
conf: float, optional
This is the confidence required for the standard error
calculation. Defaults to .99, but could be anything less
than or equal to one. One is heavy discouraged, due to the
extreme outliers inflating the variance.

Returns
-------
OaxacaResults
A results container for the three-fold decomposition.
"""
self.submitted_n = n
self.submitted_conf = conf
self.submitted_weight = None
std_val = None
self.endow_eff = (
self.exog_f_mean - self.exog_s_mean
) @ self._s_model.params
self.coef_eff = self.exog_s_mean @ (
self._f_model.params - self._s_model.params
)
self.int_eff = (self.exog_f_mean - self.exog_s_mean) @ (
self._f_model.params - self._s_model.params
)

if std is True:
std_val = self.variance(3)

return OaxacaResults(
(self.endow_eff, self.coef_eff, self.int_eff, self.gap),
3,
std_val=std_val,
)

[docs]    def two_fold(
self,
std=False,
two_fold_type="pooled",
submitted_weight=None,
n=None,
conf=None,
):
"""
Calculates the two-fold or pooled Oaxaca Blinder Decompositions

Methods
-------
std: boolean, optional
If true, bootstrapped standard errors will be calculated.

two_fold_type: string, optional
This method allows for the specific calculation of the
non-discriminatory model. There are four different types
available at this time. pooled, cotton, reimers, self_submitted.
Pooled is assumed and if a non-viable parameter is given,
pooled will be ran.

pooled - This type assumes that the pooled model's parameters
(a normal regression) is the non-discriminatory model.
This includes the indicator variable. This is generally
the best idea. If you have economic justification for
using others, then use others.

nuemark - This is similar to the pooled type, but the regression
is not done including the indicator variable.

cotton - This type uses the adjusted in Cotton (1988), which
accounts for the undervaluation of one group causing the
overevalution of another. It uses the sample size weights for
a linear combination of the two model parameters

reimers - This type uses a linear combination of the two
models with both parameters being 50% of the
non-discriminatory model.

self_submitted - This allows the user to submit their
own weights. Please be sure to put the weight of the larger mean
group only. This should be submitted in the
submitted_weights variable.

submitted_weight: int/float, required only for self_submitted,
This is the submitted weight for the larger mean. If the
weight for the larger mean is p, then the weight for the
other mean is 1-p. Only submit the first value.

n: int, optional
A amount of iterations to calculate the bootstrapped
standard errors. This defaults to 5000.
conf: float, optional
This is the confidence required for the standard error
calculation. Defaults to .99, but could be anything less
than or equal to one. One is heavy discouraged, due to the
extreme outliers inflating the variance.

Returns
-------
OaxacaResults
A results container for the two-fold decomposition.
"""
self.submitted_n = n
self.submitted_conf = conf
std_val = None
self.two_fold_type = two_fold_type
self.submitted_weight = submitted_weight

if two_fold_type == "cotton":
self.t_params = (
self.len_f / (self.len_f + self.len_s) * self._f_model.params
) + (self.len_s / (self.len_f + self.len_s) * self._s_model.params)

elif two_fold_type == "reimers":
self.t_params = 0.5 * (self._f_model.params + self._s_model.params)

elif two_fold_type == "self_submitted":
if submitted_weight is None:
submitted_weight = [submitted_weight, 1 - submitted_weight]
self.t_params = (
submitted_weight * self._f_model.params
+ submitted_weight * self._s_model.params
)

elif two_fold_type == "nuemark":
self._t_model = OLS(self.endog, self.neumark).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
self.t_params = self._t_model.params

else:
self._t_model = OLS(self.endog, self.exog).fit(
cov_type=self.cov_type, cov_kwds=self.cov_kwds
)
self.t_params = np.delete(self._t_model.params, self.bifurcate)

self.unexplained = (
self.exog_f_mean @ (self._f_model.params - self.t_params)
) + (self.exog_s_mean @ (self.t_params - self._s_model.params))
self.explained = (self.exog_f_mean - self.exog_s_mean) @ self.t_params

if std is True:
std_val = self.variance(2)

return OaxacaResults(
(self.unexplained, self.explained, self.gap), 2, std_val=std_val
)

[docs]class OaxacaResults:
"""
This class summarizes the fit of the OaxacaBlinder model.

Use .summary() to get a table of the fitted values or
use .params to receive a list of the values
use .std to receive a list of the standard errors

If a two-fold model was fitted, this will return
unexplained effect, explained effect, and the
mean gap. The list will always be of the following order
and type. If standard error was asked for, then standard error
calculations will also be included for each variable after each
calculated effect.

unexplained : float
This is the effect that cannot be explained by the data at hand.
This does not mean it cannot be explained with more.
explained : float
This is the effect that can be explained using the data.
gap : float
This is the gap in the mean differences of the two groups.

If a three-fold model was fitted, this will
return characteristic effect, coefficient effect
interaction effect, and the mean gap. The list will
be of the following order and type. If standard error was asked
for, then standard error calculations will also be included for
each variable after each calculated effect.

endowment effect : float
This is the effect due to the group differences in
predictors
coefficient effect : float
This is the effect due to differences of the coefficients
of the two groups
interaction effect : float
This is the effect due to differences in both effects
existing at the same time between the two groups.
gap : float
This is the gap in the mean differences of the two groups.

Attributes
----------
params
A list of all values for the fitted models.
std
A list of standard error calculations.
"""

def __init__(self, results, model_type, std_val=None):
self.params = results
self.std = std_val
self.model_type = model_type

[docs]    def summary(self):
"""
Print a summary table with the Oaxaca-Blinder effects
"""
if self.model_type == 2:
if self.std is None:
print(
dedent(
f"""\
Oaxaca-Blinder Two-fold Effects
Unexplained Effect: {self.params:.5f}
Explained Effect: {self.params:.5f}
Gap: {self.params:.5f}"""
)
)
else:
print(
dedent(
"""\
Oaxaca-Blinder Two-fold Effects
Unexplained Effect: {:.5f}
Unexplained Standard Error: {:.5f}
Explained Effect: {:.5f}
Explained Standard Error: {:.5f}
Gap: {:.5f}""".format(
self.params,
self.std,
self.params,
self.std,
self.params,
)
)
)
if self.model_type == 3:
if self.std is None:
print(
dedent(
f"""\
Oaxaca-Blinder Three-fold Effects
Endowment Effect: {self.params:.5f}
Coefficient Effect: {self.params:.5f}
Interaction Effect: {self.params:.5f}
Gap: {self.params:.5f}"""
)
)
else:
print(
dedent(
f"""\
Oaxaca-Blinder Three-fold Effects
Endowment Effect: {self.params:.5f}
Endowment Standard Error: {self.std:.5f}
Coefficient Effect: {self.params:.5f}
Coefficient Standard Error: {self.std:.5f}
Interaction Effect: {self.params:.5f}
Interaction Standard Error: {self.std:.5f}
Gap: {self.params:.5f}"""
)
)
```