{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Discrete Choice Models Overview" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:31.090213Z", "iopub.status.busy": "2021-02-02T06:54:31.089374Z", "iopub.status.idle": "2021-02-02T06:54:32.126310Z", "shell.execute_reply": "2021-02-02T06:54:32.125071Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import statsmodels.api as sm" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Data\n", "\n", "Load data from Spector and Mazzeo (1980). Examples follow Greene's Econometric Analysis Ch. 21 (5th Edition)." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.131534Z", "iopub.status.busy": "2021-02-02T06:54:32.130185Z", "iopub.status.idle": "2021-02-02T06:54:32.146721Z", "shell.execute_reply": "2021-02-02T06:54:32.147790Z" } }, "outputs": [], "source": [ "spector_data = sm.datasets.spector.load(as_pandas=False)\n", "spector_data.exog = sm.add_constant(spector_data.exog, prepend=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Inspect the data:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.155574Z", "iopub.status.busy": "2021-02-02T06:54:32.154150Z", "iopub.status.idle": "2021-02-02T06:54:32.163599Z", "shell.execute_reply": "2021-02-02T06:54:32.164532Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 2.66 20. 0. 1. ]\n", " [ 2.89 22. 0. 1. ]\n", " [ 3.28 24. 0. 1. ]\n", " [ 2.92 12. 0. 1. ]\n", " [ 4. 21. 0. 1. ]]\n", "[0. 0. 0. 0. 1.]\n" ] } ], "source": [ "print(spector_data.exog[:5,:])\n", "print(spector_data.endog[:5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear Probability Model (OLS)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.168600Z", "iopub.status.busy": "2021-02-02T06:54:32.167361Z", "iopub.status.idle": "2021-02-02T06:54:32.186859Z", "shell.execute_reply": "2021-02-02T06:54:32.187999Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Parameters: [0.46385168 0.01049512 0.37855479]\n" ] } ], "source": [ "lpm_mod = sm.OLS(spector_data.endog, spector_data.exog)\n", "lpm_res = lpm_mod.fit()\n", "print('Parameters: ', lpm_res.params[:-1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Logit Model" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "execution": { "iopub.execute_input": "2021-02-02T06:54:32.192566Z", "iopub.status.busy": "2021-02-02T06:54:32.191048Z", "iopub.status.idle": "2021-02-02T06:54:32.203632Z", "shell.execute_reply": "2021-02-02T06:54:32.204604Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Parameters: [ 2.82611259 0.09515766 2.37868766 -13.02134686]\n" ] } ], "source": [ "logit_mod = sm.Logit(spector_data.endog, spector_data.exog)\n", "logit_res = logit_mod.fit(disp=0)\n", "print('Parameters: ', logit_res.params)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Marginal Effects" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.209389Z", "iopub.status.busy": "2021-02-02T06:54:32.207851Z", "iopub.status.idle": "2021-02-02T06:54:32.220744Z", "shell.execute_reply": "2021-02-02T06:54:32.221843Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " Logit Marginal Effects \n", "=====================================\n", "Dep. Variable: y\n", "Method: dydx\n", "At: overall\n", "==============================================================================\n", " dy/dx std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 0.3626 0.109 3.313 0.001 0.148 0.577\n", "x2 0.0122 0.018 0.686 0.493 -0.023 0.047\n", "x3 0.3052 0.092 3.304 0.001 0.124 0.486\n", "==============================================================================\n" ] } ], "source": [ "margeff = logit_res.get_margeff()\n", "print(margeff.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As in all the discrete data models presented below, we can print a nice summary of results:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.226605Z", "iopub.status.busy": "2021-02-02T06:54:32.225066Z", "iopub.status.idle": "2021-02-02T06:54:32.247704Z", "shell.execute_reply": "2021-02-02T06:54:32.248771Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " Logit Regression Results \n", "==============================================================================\n", "Dep. Variable: y No. Observations: 32\n", "Model: Logit Df Residuals: 28\n", "Method: MLE Df Model: 3\n", "Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.3740\n", "Time: 06:54:32 Log-Likelihood: -12.890\n", "converged: True LL-Null: -20.592\n", "Covariance Type: nonrobust LLR p-value: 0.001502\n", "==============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 2.8261 1.263 2.238 0.025 0.351 5.301\n", "x2 0.0952 0.142 0.672 0.501 -0.182 0.373\n", "x3 2.3787 1.065 2.234 0.025 0.292 4.465\n", "const -13.0213 4.931 -2.641 0.008 -22.687 -3.356\n", "==============================================================================\n" ] } ], "source": [ "print(logit_res.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Probit Model " ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.252984Z", "iopub.status.busy": "2021-02-02T06:54:32.251663Z", "iopub.status.idle": "2021-02-02T06:54:32.269567Z", "shell.execute_reply": "2021-02-02T06:54:32.270682Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully.\n", " Current function value: 0.400588\n", " Iterations 6\n", "Parameters: [ 1.62581004 0.05172895 1.42633234 -7.45231965]\n", "Marginal effects: \n", " Probit Marginal Effects \n", "=====================================\n", "Dep. Variable: y\n", "Method: dydx\n", "At: overall\n", "==============================================================================\n", " dy/dx std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 0.3608 0.113 3.182 0.001 0.139 0.583\n", "x2 0.0115 0.018 0.624 0.533 -0.025 0.048\n", "x3 0.3165 0.090 3.508 0.000 0.140 0.493\n", "==============================================================================\n" ] } ], "source": [ "probit_mod = sm.Probit(spector_data.endog, spector_data.exog)\n", "probit_res = probit_mod.fit()\n", "probit_margeff = probit_res.get_margeff()\n", "print('Parameters: ', probit_res.params)\n", "print('Marginal effects: ')\n", "print(probit_margeff.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Multinomial Logit" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Load data from the American National Election Studies:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.275515Z", "iopub.status.busy": "2021-02-02T06:54:32.274002Z", "iopub.status.idle": "2021-02-02T06:54:32.304154Z", "shell.execute_reply": "2021-02-02T06:54:32.305074Z" } }, "outputs": [], "source": [ "anes_data = sm.datasets.anes96.load(as_pandas=False)\n", "anes_exog = anes_data.exog\n", "anes_exog = sm.add_constant(anes_exog, prepend=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Inspect the data:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.309848Z", "iopub.status.busy": "2021-02-02T06:54:32.308511Z", "iopub.status.idle": "2021-02-02T06:54:32.317469Z", "shell.execute_reply": "2021-02-02T06:54:32.318516Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[-2.30258509 7. 36. 3. 1. ]\n", " [ 5.24755025 3. 20. 4. 1. ]\n", " [ 3.43720782 2. 24. 6. 1. ]\n", " [ 4.4200447 3. 28. 6. 1. ]\n", " [ 6.46162441 5. 68. 6. 1. ]]\n", "[6. 1. 1. 1. 0.]\n" ] } ], "source": [ "print(anes_data.exog[:5,:])\n", "print(anes_data.endog[:5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Fit MNL model:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.322993Z", "iopub.status.busy": "2021-02-02T06:54:32.321682Z", "iopub.status.idle": "2021-02-02T06:54:32.354319Z", "shell.execute_reply": "2021-02-02T06:54:32.355415Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully.\n", " Current function value: 1.548647\n", " Iterations 7\n", "[[-1.15359746e-02 -8.87506530e-02 -1.05966699e-01 -9.15567017e-02\n", " -9.32846040e-02 -1.40880692e-01]\n", " [ 2.97714352e-01 3.91668642e-01 5.73450508e-01 1.27877179e+00\n", " 1.34696165e+00 2.07008014e+00]\n", " [-2.49449954e-02 -2.28978371e-02 -1.48512069e-02 -8.68134503e-03\n", " -1.79040689e-02 -9.43264870e-03]\n", " [ 8.24914421e-02 1.81042758e-01 -7.15241904e-03 1.99827955e-01\n", " 2.16938850e-01 3.21925702e-01]\n", " [ 5.19655317e-03 4.78739761e-02 5.75751595e-02 8.44983753e-02\n", " 8.09584122e-02 1.08894083e-01]\n", " [-3.73401677e-01 -2.25091318e+00 -3.66558353e+00 -7.61384309e+00\n", " -7.06047825e+00 -1.21057509e+01]]\n" ] } ], "source": [ "mlogit_mod = sm.MNLogit(anes_data.endog, anes_exog)\n", "mlogit_res = mlogit_mod.fit()\n", "print(mlogit_res.params)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Poisson\n", "\n", "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. " ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.360019Z", "iopub.status.busy": "2021-02-02T06:54:32.358443Z", "iopub.status.idle": "2021-02-02T06:54:32.396228Z", "shell.execute_reply": "2021-02-02T06:54:32.397201Z" } }, "outputs": [], "source": [ "rand_data = sm.datasets.randhie.load(as_pandas=False)\n", "rand_exog = rand_data.exog.view(float).reshape(len(rand_data.exog), -1)\n", "rand_exog = sm.add_constant(rand_exog, prepend=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Fit Poisson model: " ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.401477Z", "iopub.status.busy": "2021-02-02T06:54:32.400166Z", "iopub.status.idle": "2021-02-02T06:54:32.559139Z", "shell.execute_reply": "2021-02-02T06:54:32.560147Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully.\n", " Current function value: 3.091609\n", " Iterations 6\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " Poisson Regression Results \n", "==============================================================================\n", "Dep. Variable: y No. Observations: 20190\n", "Model: Poisson Df Residuals: 20180\n", "Method: MLE Df Model: 9\n", "Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.06343\n", "Time: 06:54:32 Log-Likelihood: -62420.\n", "converged: True LL-Null: -66647.\n", "Covariance Type: nonrobust LLR p-value: 0.000\n", "==============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0525 0.003 -18.216 0.000 -0.058 -0.047\n", "x2 -0.2471 0.011 -23.272 0.000 -0.268 -0.226\n", "x3 0.0353 0.002 19.302 0.000 0.032 0.039\n", "x4 -0.0346 0.002 -21.439 0.000 -0.038 -0.031\n", "x5 0.2717 0.012 22.200 0.000 0.248 0.296\n", "x6 0.0339 0.001 60.098 0.000 0.033 0.035\n", "x7 -0.0126 0.009 -1.366 0.172 -0.031 0.005\n", "x8 0.0541 0.015 3.531 0.000 0.024 0.084\n", "x9 0.2061 0.026 7.843 0.000 0.155 0.258\n", "const 0.7004 0.011 62.741 0.000 0.678 0.722\n", "==============================================================================\n" ] } ], "source": [ "poisson_mod = sm.Poisson(rand_data.endog, rand_exog)\n", "poisson_res = poisson_mod.fit(method=\"newton\")\n", "print(poisson_res.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Negative Binomial\n", "\n", "The negative binomial model gives slightly different results. " ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:32.564614Z", "iopub.status.busy": "2021-02-02T06:54:32.563266Z", "iopub.status.idle": "2021-02-02T06:54:33.278087Z", "shell.execute_reply": "2021-02-02T06:54:33.277697Z" } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/travis/build/statsmodels/statsmodels/statsmodels/base/model.py:568: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals\n", " ConvergenceWarning)\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " NegativeBinomial Regression Results \n", "==============================================================================\n", "Dep. Variable: y No. Observations: 20190\n", "Model: NegativeBinomial Df Residuals: 20180\n", "Method: MLE Df Model: 9\n", "Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.01845\n", "Time: 06:54:33 Log-Likelihood: -43384.\n", "converged: False LL-Null: -44199.\n", "Covariance Type: nonrobust LLR p-value: 0.000\n", "==============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0579 0.006 -9.515 0.000 -0.070 -0.046\n", "x2 -0.2678 0.023 -11.802 0.000 -0.312 -0.223\n", "x3 0.0412 0.004 9.938 0.000 0.033 0.049\n", "x4 -0.0381 0.003 -11.216 0.000 -0.045 -0.031\n", "x5 0.2691 0.030 8.985 0.000 0.210 0.328\n", "x6 0.0382 0.001 26.080 0.000 0.035 0.041\n", "x7 -0.0441 0.020 -2.201 0.028 -0.083 -0.005\n", "x8 0.0173 0.036 0.478 0.632 -0.054 0.088\n", "x9 0.1782 0.074 2.399 0.016 0.033 0.324\n", "const 0.6635 0.025 26.786 0.000 0.615 0.712\n", "alpha 1.2930 0.019 69.477 0.000 1.256 1.329\n", "==============================================================================\n" ] } ], "source": [ "mod_nbin = sm.NegativeBinomial(rand_data.endog, rand_exog)\n", "res_nbin = mod_nbin.fit(disp=False)\n", "print(res_nbin.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Alternative solvers\n", "\n", "The default method for fitting discrete data MLE models is Newton-Raphson. You can use other solvers by using the ``method`` argument: " ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:33.282888Z", "iopub.status.busy": "2021-02-02T06:54:33.281913Z", "iopub.status.idle": "2021-02-02T06:54:33.787905Z", "shell.execute_reply": "2021-02-02T06:54:33.787396Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully.\n", " Current function value: 1.548647\n", " Iterations: 111\n", " Function evaluations: 117\n", " Gradient evaluations: 117\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " MNLogit Regression Results \n", "==============================================================================\n", "Dep. Variable: y No. Observations: 944\n", "Model: MNLogit Df Residuals: 908\n", "Method: MLE Df Model: 30\n", "Date: Tue, 02 Feb 2021 Pseudo R-squ.: 0.1648\n", "Time: 06:54:33 Log-Likelihood: -1461.9\n", "converged: True LL-Null: -1750.3\n", "Covariance Type: nonrobust LLR p-value: 1.822e-102\n", "==============================================================================\n", " y=1 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0115 0.034 -0.337 0.736 -0.079 0.056\n", "x2 0.2977 0.094 3.180 0.001 0.114 0.481\n", "x3 -0.0249 0.007 -3.823 0.000 -0.038 -0.012\n", "x4 0.0825 0.074 1.121 0.262 -0.062 0.227\n", "x5 0.0052 0.018 0.295 0.768 -0.029 0.040\n", "const -0.3734 0.630 -0.593 0.553 -1.608 0.861\n", "------------------------------------------------------------------------------\n", " y=2 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0888 0.039 -2.266 0.023 -0.166 -0.012\n", "x2 0.3917 0.108 3.619 0.000 0.180 0.604\n", "x3 -0.0229 0.008 -2.893 0.004 -0.038 -0.007\n", "x4 0.1810 0.085 2.123 0.034 0.014 0.348\n", "x5 0.0479 0.022 2.149 0.032 0.004 0.092\n", "const -2.2509 0.763 -2.949 0.003 -3.747 -0.755\n", "------------------------------------------------------------------------------\n", " y=3 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.1060 0.057 -1.858 0.063 -0.218 0.006\n", "x2 0.5734 0.159 3.617 0.000 0.263 0.884\n", "x3 -0.0149 0.011 -1.311 0.190 -0.037 0.007\n", "x4 -0.0072 0.126 -0.057 0.955 -0.255 0.240\n", "x5 0.0576 0.034 1.713 0.087 -0.008 0.123\n", "const -3.6656 1.157 -3.169 0.002 -5.932 -1.399\n", "------------------------------------------------------------------------------\n", " y=4 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0916 0.044 -2.091 0.037 -0.177 -0.006\n", "x2 1.2788 0.129 9.921 0.000 1.026 1.531\n", "x3 -0.0087 0.008 -1.031 0.302 -0.025 0.008\n", "x4 0.1998 0.094 2.123 0.034 0.015 0.384\n", "x5 0.0845 0.026 3.226 0.001 0.033 0.136\n", "const -7.6139 0.958 -7.951 0.000 -9.491 -5.737\n", "------------------------------------------------------------------------------\n", " y=5 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.0933 0.039 -2.371 0.018 -0.170 -0.016\n", "x2 1.3470 0.117 11.494 0.000 1.117 1.577\n", "x3 -0.0179 0.008 -2.352 0.019 -0.033 -0.003\n", "x4 0.2169 0.085 2.552 0.011 0.050 0.384\n", "x5 0.0810 0.023 3.524 0.000 0.036 0.126\n", "const -7.0605 0.844 -8.362 0.000 -8.715 -5.406\n", "------------------------------------------------------------------------------\n", " y=6 coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "x1 -0.1409 0.042 -3.343 0.001 -0.223 -0.058\n", "x2 2.0701 0.143 14.435 0.000 1.789 2.351\n", "x3 -0.0094 0.008 -1.160 0.246 -0.025 0.007\n", "x4 0.3219 0.091 3.534 0.000 0.143 0.500\n", "x5 0.1089 0.025 4.304 0.000 0.059 0.158\n", "const -12.1058 1.060 -11.421 0.000 -14.183 -10.028\n", "==============================================================================\n" ] } ], "source": [ "mlogit_res = mlogit_mod.fit(method='bfgs', maxiter=250)\n", "print(mlogit_res.summary())" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }