ch6_Exercicio_4.ipynb

{
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  {
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   "outputs": [],
   "source": [
    "'''\n",
    "Introdução à Econometria - Uma abordagem moderna (Tradução da 6 edição norte-americana)\n",
    "Autor: WOOLDRIDGE, J. M.\n",
    "Editora: CENGAGE LEARNING\n",
    "\n",
    "Cap. 6: Análise de regressão múltipla: problemas adicionais\n",
    "Exercício 4, item (ii), reprodução da equação estimada.\n",
    "\n",
    "O objetivo deste arquivo é apenas ilustrar a diferença entre usar \":\" e \"*\" em modelos com termos de interação.\n",
    "Por exemplo, o modelo abaixo inclui um termo de interação entre x2 e x3:\n",
    "y = B0 + B1*x1 + B2*(x2*x3) + u\n",
    "\n",
    "\n",
    "Arquivo com os dados: wage2.xls\n",
    "\n",
    "Arquivo com dados em:\n",
    "http://students.cengage.com.br/dashboard/private/livroView.jsf;jsessionid=95E9AD889A4A4B7ABBD2A5251F1E14BE?id=104577\n",
    "\n",
    "Em caso de dúvidas ou problemas, solicitamos, por gentileza, entrar em contato pelo e-mail:\n",
    "python.economia@gmail.com\n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_excel('wage2.xls',\n",
    "                   header=None,\n",
    "                   usecols=[4, 5, 6, 14, 15, 16],\n",
    "                   names=['educ', 'exper', 'tenure', 'meduc', 'feduc', 'lwage' ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>educ</th>\n",
       "      <th>exper</th>\n",
       "      <th>tenure</th>\n",
       "      <th>meduc</th>\n",
       "      <th>feduc</th>\n",
       "      <th>lwage</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>12</td>\n",
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       "      <td>8</td>\n",
       "      <td>8</td>\n",
       "      <td>6.645091</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>18</td>\n",
       "      <td>11</td>\n",
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       "      <td>14</td>\n",
       "      <td>14</td>\n",
       "      <td>6.694562</td>\n",
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       "      <td>14</td>\n",
       "      <td>6.715384</td>\n",
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       "      <th>3</th>\n",
       "      <td>12</td>\n",
       "      <td>13</td>\n",
       "      <td>7</td>\n",
       "      <td>12</td>\n",
       "      <td>12</td>\n",
       "      <td>6.476973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>11</td>\n",
       "      <td>14</td>\n",
       "      <td>5</td>\n",
       "      <td>6</td>\n",
       "      <td>11</td>\n",
       "      <td>6.331502</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "   educ  exper  tenure meduc feduc     lwage\n",
       "0    12     11       2     8     8  6.645091\n",
       "1    18     11      16    14    14  6.694562\n",
       "2    14     11       9    14    14  6.715384\n",
       "3    12     13       7    12    12  6.476973\n",
       "4    11     14       5     6    11  6.331502"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head()"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
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    {
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     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 935 entries, 0 to 934\n",
      "Data columns (total 6 columns):\n",
      " #   Column  Non-Null Count  Dtype  \n",
      "---  ------  --------------  -----  \n",
      " 0   educ    935 non-null    int64  \n",
      " 1   exper   935 non-null    int64  \n",
      " 2   tenure  935 non-null    int64  \n",
      " 3   meduc   935 non-null    object \n",
      " 4   feduc   935 non-null    object \n",
      " 5   lwage   935 non-null    float64\n",
      "dtypes: float64(1), int64(3), object(2)\n",
      "memory usage: 44.0+ KB\n"
     ]
    }
   ],
   "source": [
    "df.info()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8, 14, 12, 6, 13, 16, 10, 11, 18, 17, 15, '.', 9, 7, 2, 5, 4, 3, 1,\n",
       "       0], dtype=object)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['meduc'].unique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8, 14, 12, 11, '.', 5, 10, 16, 18, 7, 9, 15, 6, 13, 17, 4, 2, 3, 0],\n",
       "      dtype=object)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "df['feduc'].unique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['meduc'].replace({'.': np.nan}, inplace=True)\n",
    "df['feduc'].replace({'.': np.nan}, inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 935 entries, 0 to 934\n",
      "Data columns (total 6 columns):\n",
      " #   Column  Non-Null Count  Dtype  \n",
      "---  ------  --------------  -----  \n",
      " 0   educ    935 non-null    int64  \n",
      " 1   exper   935 non-null    int64  \n",
      " 2   tenure  935 non-null    int64  \n",
      " 3   meduc   857 non-null    float64\n",
      " 4   feduc   741 non-null    float64\n",
      " 5   lwage   935 non-null    float64\n",
      "dtypes: float64(3), int64(3)\n",
      "memory usage: 44.0 KB\n"
     ]
    }
   ],
   "source": [
    "df.info()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['pareduc'] = df['meduc'] + df['feduc']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>educ</th>\n",
       "      <th>exper</th>\n",
       "      <th>tenure</th>\n",
       "      <th>meduc</th>\n",
       "      <th>feduc</th>\n",
       "      <th>lwage</th>\n",
       "      <th>pareduc</th>\n",
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       "      <th>0</th>\n",
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       "      <td>8.0</td>\n",
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       "      <td>16.0</td>\n",
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       "      <td>18</td>\n",
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       "      <td>6.694562</td>\n",
       "      <td>28.0</td>\n",
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       "      <th>2</th>\n",
       "      <td>14</td>\n",
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       "      <td>14.0</td>\n",
       "      <td>14.0</td>\n",
       "      <td>6.715384</td>\n",
       "      <td>28.0</td>\n",
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       "      <th>3</th>\n",
       "      <td>12</td>\n",
       "      <td>13</td>\n",
       "      <td>7</td>\n",
       "      <td>12.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>6.476973</td>\n",
       "      <td>24.0</td>\n",
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       "      <th>4</th>\n",
       "      <td>11</td>\n",
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       "      <td>6.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>6.331502</td>\n",
       "      <td>17.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "   educ  exper  tenure  meduc  feduc     lwage  pareduc\n",
       "0    12     11       2    8.0    8.0  6.645091     16.0\n",
       "1    18     11      16   14.0   14.0  6.694562     28.0\n",
       "2    14     11       9   14.0   14.0  6.715384     28.0\n",
       "3    12     13       7   12.0   12.0  6.476973     24.0\n",
       "4    11     14       5    6.0   11.0  6.331502     17.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>lwage</td>      <th>  R-squared:         </th> <td>   0.169</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.164</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   36.44</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 29 Jun 2022</td> <th>  Prob (F-statistic):</th> <td>9.44e-28</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>21:40:08</td>     <th>  Log-Likelihood:    </th> <td> -329.77</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   722</td>      <th>  AIC:               </th> <td>   669.5</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   717</td>      <th>  BIC:               </th> <td>   692.4</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     4</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "        <td></td>          <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>    <td>    5.6465</td> <td>    0.130</td> <td>   43.582</td> <td> 0.000</td> <td>    5.392</td> <td>    5.901</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>educ</th>         <td>    0.0468</td> <td>    0.010</td> <td>    4.462</td> <td> 0.000</td> <td>    0.026</td> <td>    0.067</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>educ:pareduc</th> <td>    0.0008</td> <td>    0.000</td> <td>    3.677</td> <td> 0.000</td> <td>    0.000</td> <td>    0.001</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>exper</th>        <td>    0.0189</td> <td>    0.004</td> <td>    4.786</td> <td> 0.000</td> <td>    0.011</td> <td>    0.027</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>tenure</th>       <td>    0.0102</td> <td>    0.003</td> <td>    3.413</td> <td> 0.001</td> <td>    0.004</td> <td>    0.016</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>21.840</td> <th>  Durbin-Watson:     </th> <td>   1.837</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>  34.056</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.251</td> <th>  Prob(JB):          </th> <td>4.03e-08</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.939</td> <th>  Cond. No.          </th> <td>2.85e+03</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 2.85e+03. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                  lwage   R-squared:                       0.169\n",
       "Model:                            OLS   Adj. R-squared:                  0.164\n",
       "Method:                 Least Squares   F-statistic:                     36.44\n",
       "Date:                Wed, 29 Jun 2022   Prob (F-statistic):           9.44e-28\n",
       "Time:                        21:40:08   Log-Likelihood:                -329.77\n",
       "No. Observations:                 722   AIC:                             669.5\n",
       "Df Residuals:                     717   BIC:                             692.4\n",
       "Df Model:                           4                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "================================================================================\n",
       "                   coef    std err          t      P>|t|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------\n",
       "Intercept        5.6465      0.130     43.582      0.000       5.392       5.901\n",
       "educ             0.0468      0.010      4.462      0.000       0.026       0.067\n",
       "educ:pareduc     0.0008      0.000      3.677      0.000       0.000       0.001\n",
       "exper            0.0189      0.004      4.786      0.000       0.011       0.027\n",
       "tenure           0.0102      0.003      3.413      0.001       0.004       0.016\n",
       "==============================================================================\n",
       "Omnibus:                       21.840   Durbin-Watson:                   1.837\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               34.056\n",
       "Skew:                          -0.251   Prob(JB):                     4.03e-08\n",
       "Kurtosis:                       3.939   Cond. No.                     2.85e+03\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 2.85e+03. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Reprodução da equação estimada no item (ii)\n",
    "modelo = smf.ols(formula='lwage ~ educ + educ:pareduc + exper + tenure', data=df)\n",
    "reg = modelo.fit()\n",
    "reg.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nota: Veja o que ocorre se utilizamos * nos termos de interação\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept       6.132853\n",
       "educ:pareduc    0.001462\n",
       "exper           0.013902\n",
       "tenure          0.011225\n",
       "dtype: float64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Para ilustração, aqui retiramos a variável educ\n",
    "modelo = smf.ols(formula='lwage ~ educ:pareduc + exper + tenure', data=df)  # uso de : nos termos de interação\n",
    "reg = modelo.fit()\n",
    "reg.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept       4.937661\n",
       "educ            0.097113\n",
       "pareduc         0.033214\n",
       "educ:pareduc   -0.001568\n",
       "exper           0.019557\n",
       "tenure          0.010308\n",
       "dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "modelo = smf.ols(formula='lwage ~ educ*pareduc + exper + tenure', data=df)  # uso de * nos termos de interação\n",
    "reg = modelo.fit()\n",
    "reg.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''Note que quando utilizamos * no termo de interação as variáveis educ e pareduc foram incluídas individualmente mesmo não\n",
    "aparecendo na formula passada para smf.ols()'''"
   ]
  }
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