Discrete Choice Models Overview¶
[1]:
import numpy as np
import statsmodels.api as sm
Data¶
Load data from Spector and Mazzeo (1980). Examples follow Greene’s Econometric Analysis Ch. 21 (5th Edition).
[2]:
spector_data = sm.datasets.spector.load(as_pandas=False)
spector_data.exog = sm.add_constant(spector_data.exog, prepend=False)
Inspect the data:
[3]:
print(spector_data.exog[:5,:])
print(spector_data.endog[:5])
[[ 2.66 20. 0. 1. ]
[ 2.89 22. 0. 1. ]
[ 3.28 24. 0. 1. ]
[ 2.92 12. 0. 1. ]
[ 4. 21. 0. 1. ]]
[0. 0. 0. 0. 1.]
Linear Probability Model (OLS)¶
[4]:
lpm_mod = sm.OLS(spector_data.endog, spector_data.exog)
lpm_res = lpm_mod.fit()
print('Parameters: ', lpm_res.params[:-1])
Parameters: [0.46385168 0.01049512 0.37855479]
Logit Model¶
[5]:
logit_mod = sm.Logit(spector_data.endog, spector_data.exog)
logit_res = logit_mod.fit(disp=0)
print('Parameters: ', logit_res.params)
Parameters: [ 2.82611259 0.09515766 2.37868766 -13.02134686]
Marginal Effects
[6]:
margeff = logit_res.get_margeff()
print(margeff.summary())
Logit Marginal Effects
=====================================
Dep. Variable: y
Method: dydx
At: overall
==============================================================================
dy/dx std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 0.3626 0.109 3.313 0.001 0.148 0.577
x2 0.0122 0.018 0.686 0.493 -0.023 0.047
x3 0.3052 0.092 3.304 0.001 0.124 0.486
==============================================================================
As in all the discrete data models presented below, we can print a nice summary of results:
[7]:
print(logit_res.summary())
Logit Regression Results
==============================================================================
Dep. Variable: y No. Observations: 32
Model: Logit Df Residuals: 28
Method: MLE Df Model: 3
Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.3740
Time: 06:54:32 Log-Likelihood: -12.890
converged: True LL-Null: -20.592
Covariance Type: nonrobust LLR p-value: 0.001502
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 2.8261 1.263 2.238 0.025 0.351 5.301
x2 0.0952 0.142 0.672 0.501 -0.182 0.373
x3 2.3787 1.065 2.234 0.025 0.292 4.465
const -13.0213 4.931 -2.641 0.008 -22.687 -3.356
==============================================================================
Probit Model¶
[8]:
probit_mod = sm.Probit(spector_data.endog, spector_data.exog)
probit_res = probit_mod.fit()
probit_margeff = probit_res.get_margeff()
print('Parameters: ', probit_res.params)
print('Marginal effects: ')
print(probit_margeff.summary())
Optimization terminated successfully.
Current function value: 0.400588
Iterations 6
Parameters: [ 1.62581004 0.05172895 1.42633234 -7.45231965]
Marginal effects:
Probit Marginal Effects
=====================================
Dep. Variable: y
Method: dydx
At: overall
==============================================================================
dy/dx std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 0.3608 0.113 3.182 0.001 0.139 0.583
x2 0.0115 0.018 0.624 0.533 -0.025 0.048
x3 0.3165 0.090 3.508 0.000 0.140 0.493
==============================================================================
Multinomial Logit¶
Load data from the American National Election Studies:
[9]:
anes_data = sm.datasets.anes96.load(as_pandas=False)
anes_exog = anes_data.exog
anes_exog = sm.add_constant(anes_exog, prepend=False)
Inspect the data:
[10]:
print(anes_data.exog[:5,:])
print(anes_data.endog[:5])
[[-2.30258509 7. 36. 3. 1. ]
[ 5.24755025 3. 20. 4. 1. ]
[ 3.43720782 2. 24. 6. 1. ]
[ 4.4200447 3. 28. 6. 1. ]
[ 6.46162441 5. 68. 6. 1. ]]
[6. 1. 1. 1. 0.]
Fit MNL model:
[11]:
mlogit_mod = sm.MNLogit(anes_data.endog, anes_exog)
mlogit_res = mlogit_mod.fit()
print(mlogit_res.params)
Optimization terminated successfully.
Current function value: 1.548647
Iterations 7
[[-1.15359746e-02 -8.87506530e-02 -1.05966699e-01 -9.15567017e-02
-9.32846040e-02 -1.40880692e-01]
[ 2.97714352e-01 3.91668642e-01 5.73450508e-01 1.27877179e+00
1.34696165e+00 2.07008014e+00]
[-2.49449954e-02 -2.28978371e-02 -1.48512069e-02 -8.68134503e-03
-1.79040689e-02 -9.43264870e-03]
[ 8.24914421e-02 1.81042758e-01 -7.15241904e-03 1.99827955e-01
2.16938850e-01 3.21925702e-01]
[ 5.19655317e-03 4.78739761e-02 5.75751595e-02 8.44983753e-02
8.09584122e-02 1.08894083e-01]
[-3.73401677e-01 -2.25091318e+00 -3.66558353e+00 -7.61384309e+00
-7.06047825e+00 -1.21057509e+01]]
Poisson¶
Load the Rand data. Note that this example is similar to Cameron and Trivedi’s Microeconometrics Table 20.5, but it is slightly different because of minor changes in the data.
[12]:
rand_data = sm.datasets.randhie.load(as_pandas=False)
rand_exog = rand_data.exog.view(float).reshape(len(rand_data.exog), -1)
rand_exog = sm.add_constant(rand_exog, prepend=False)
Fit Poisson model:
[13]:
poisson_mod = sm.Poisson(rand_data.endog, rand_exog)
poisson_res = poisson_mod.fit(method="newton")
print(poisson_res.summary())
Optimization terminated successfully.
Current function value: 3.091609
Iterations 6
Poisson Regression Results
==============================================================================
Dep. Variable: y No. Observations: 20190
Model: Poisson Df Residuals: 20180
Method: MLE Df Model: 9
Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.06343
Time: 06:54:32 Log-Likelihood: -62420.
converged: True LL-Null: -66647.
Covariance Type: nonrobust LLR p-value: 0.000
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0525 0.003 -18.216 0.000 -0.058 -0.047
x2 -0.2471 0.011 -23.272 0.000 -0.268 -0.226
x3 0.0353 0.002 19.302 0.000 0.032 0.039
x4 -0.0346 0.002 -21.439 0.000 -0.038 -0.031
x5 0.2717 0.012 22.200 0.000 0.248 0.296
x6 0.0339 0.001 60.098 0.000 0.033 0.035
x7 -0.0126 0.009 -1.366 0.172 -0.031 0.005
x8 0.0541 0.015 3.531 0.000 0.024 0.084
x9 0.2061 0.026 7.843 0.000 0.155 0.258
const 0.7004 0.011 62.741 0.000 0.678 0.722
==============================================================================
Negative Binomial¶
The negative binomial model gives slightly different results.
[14]:
mod_nbin = sm.NegativeBinomial(rand_data.endog, rand_exog)
res_nbin = mod_nbin.fit(disp=False)
print(res_nbin.summary())
/home/travis/build/statsmodels/statsmodels/statsmodels/base/model.py:568: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
ConvergenceWarning)
NegativeBinomial Regression Results
==============================================================================
Dep. Variable: y No. Observations: 20190
Model: NegativeBinomial Df Residuals: 20180
Method: MLE Df Model: 9
Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.01845
Time: 06:54:33 Log-Likelihood: -43384.
converged: False LL-Null: -44199.
Covariance Type: nonrobust LLR p-value: 0.000
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0579 0.006 -9.515 0.000 -0.070 -0.046
x2 -0.2678 0.023 -11.802 0.000 -0.312 -0.223
x3 0.0412 0.004 9.938 0.000 0.033 0.049
x4 -0.0381 0.003 -11.216 0.000 -0.045 -0.031
x5 0.2691 0.030 8.985 0.000 0.210 0.328
x6 0.0382 0.001 26.080 0.000 0.035 0.041
x7 -0.0441 0.020 -2.201 0.028 -0.083 -0.005
x8 0.0173 0.036 0.478 0.632 -0.054 0.088
x9 0.1782 0.074 2.399 0.016 0.033 0.324
const 0.6635 0.025 26.786 0.000 0.615 0.712
alpha 1.2930 0.019 69.477 0.000 1.256 1.329
==============================================================================
Alternative solvers¶
The default method for fitting discrete data MLE models is Newton-Raphson. You can use other solvers by using the method argument:
[15]:
mlogit_res = mlogit_mod.fit(method='bfgs', maxiter=250)
print(mlogit_res.summary())
Optimization terminated successfully.
Current function value: 1.548647
Iterations: 111
Function evaluations: 117
Gradient evaluations: 117
MNLogit Regression Results
==============================================================================
Dep. Variable: y No. Observations: 944
Model: MNLogit Df Residuals: 908
Method: MLE Df Model: 30
Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.1648
Time: 06:54:33 Log-Likelihood: -1461.9
converged: True LL-Null: -1750.3
Covariance Type: nonrobust LLR p-value: 1.822e-102
==============================================================================
y=1 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0115 0.034 -0.337 0.736 -0.079 0.056
x2 0.2977 0.094 3.180 0.001 0.114 0.481
x3 -0.0249 0.007 -3.823 0.000 -0.038 -0.012
x4 0.0825 0.074 1.121 0.262 -0.062 0.227
x5 0.0052 0.018 0.295 0.768 -0.029 0.040
const -0.3734 0.630 -0.593 0.553 -1.608 0.861
------------------------------------------------------------------------------
y=2 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0888 0.039 -2.266 0.023 -0.166 -0.012
x2 0.3917 0.108 3.619 0.000 0.180 0.604
x3 -0.0229 0.008 -2.893 0.004 -0.038 -0.007
x4 0.1810 0.085 2.123 0.034 0.014 0.348
x5 0.0479 0.022 2.149 0.032 0.004 0.092
const -2.2509 0.763 -2.949 0.003 -3.747 -0.755
------------------------------------------------------------------------------
y=3 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.1060 0.057 -1.858 0.063 -0.218 0.006
x2 0.5734 0.159 3.617 0.000 0.263 0.884
x3 -0.0149 0.011 -1.311 0.190 -0.037 0.007
x4 -0.0072 0.126 -0.057 0.955 -0.255 0.240
x5 0.0576 0.034 1.713 0.087 -0.008 0.123
const -3.6656 1.157 -3.169 0.002 -5.932 -1.399
------------------------------------------------------------------------------
y=4 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0916 0.044 -2.091 0.037 -0.177 -0.006
x2 1.2788 0.129 9.921 0.000 1.026 1.531
x3 -0.0087 0.008 -1.031 0.302 -0.025 0.008
x4 0.1998 0.094 2.123 0.034 0.015 0.384
x5 0.0845 0.026 3.226 0.001 0.033 0.136
const -7.6139 0.958 -7.951 0.000 -9.491 -5.737
------------------------------------------------------------------------------
y=5 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.0933 0.039 -2.371 0.018 -0.170 -0.016
x2 1.3470 0.117 11.494 0.000 1.117 1.577
x3 -0.0179 0.008 -2.352 0.019 -0.033 -0.003
x4 0.2169 0.085 2.552 0.011 0.050 0.384
x5 0.0810 0.023 3.524 0.000 0.036 0.126
const -7.0605 0.844 -8.362 0.000 -8.715 -5.406
------------------------------------------------------------------------------
y=6 coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
x1 -0.1409 0.042 -3.343 0.001 -0.223 -0.058
x2 2.0701 0.143 14.435 0.000 1.789 2.351
x3 -0.0094 0.008 -1.160 0.246 -0.025 0.007
x4 0.3219 0.091 3.534 0.000 0.143 0.500
x5 0.1089 0.025 4.304 0.000 0.059 0.158
const -12.1058 1.060 -11.421 0.000 -14.183 -10.028
==============================================================================