Backward Elimination.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fa220d07",
   "metadata": {},
   "source": [
    "### Importing the libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "e14895ba",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1152d3d6",
   "metadata": {},
   "source": [
    "### Import datasets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "e661dd94",
   "metadata": {},
   "outputs": [],
   "source": [
    "data = pd.read_csv(\"50_Startups.csv\")\n",
    "y = data.iloc[:,:-1]\n",
    "w = data.iloc[:,-1:]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fb9980d",
   "metadata": {},
   "source": [
    "### Encoding the categorical data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "c76ddeae",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import OneHotEncoder\n",
    "from sklearn.compose import ColumnTransformer\n",
    "ct = ColumnTransformer(transformers=[(\"encoder\",OneHotEncoder(),[3])],remainder=\"passthrough\")\n",
    "y = np.array(ct.fit_transform(y)).astype(\"int32\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "871bfa81",
   "metadata": {},
   "source": [
    "### Adding the one's column for backward elimination"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "13e832be",
   "metadata": {},
   "outputs": [],
   "source": [
    "# y = np.append(arr=np.ones((50,1)).astype(\"int32\"),values=y,axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c78a59a4",
   "metadata": {},
   "source": [
    "### Backward elimination.."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "ea677459",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>Profit</td>      <th>  R-squared:         </th> <td>   0.951</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.945</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   169.9</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 04 Jul 2021</td> <th>  Prob (F-statistic):</th> <td>1.34e-27</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:32:53</td>     <th>  Log-Likelihood:    </th> <td> -525.38</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1063.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    44</td>      <th>  BIC:               </th> <td>   1074.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     5</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>const</th> <td> 5.008e+04</td> <td> 6952.617</td> <td>    7.204</td> <td> 0.000</td> <td> 3.61e+04</td> <td> 6.41e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>   42.0063</td> <td> 3256.058</td> <td>    0.013</td> <td> 0.990</td> <td>-6520.148</td> <td> 6604.161</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>  240.7605</td> <td> 3338.877</td> <td>    0.072</td> <td> 0.943</td> <td>-6488.304</td> <td> 6969.825</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>    <td>    0.8060</td> <td>    0.046</td> <td>   17.368</td> <td> 0.000</td> <td>    0.712</td> <td>    0.900</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x4</th>    <td>   -0.0270</td> <td>    0.052</td> <td>   -0.517</td> <td> 0.608</td> <td>   -0.132</td> <td>    0.078</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x5</th>    <td>    0.0270</td> <td>    0.017</td> <td>    1.574</td> <td> 0.123</td> <td>   -0.008</td> <td>    0.062</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.783</td> <th>  Durbin-Watson:     </th> <td>   1.283</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  21.267</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.948</td> <th>  Prob(JB):          </th> <td>2.41e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.572</td> <th>  Cond. No.          </th> <td>1.47e+06</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, 1.47e+06. 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:                 Profit   R-squared:                       0.951\n",
       "Model:                            OLS   Adj. R-squared:                  0.945\n",
       "Method:                 Least Squares   F-statistic:                     169.9\n",
       "Date:                Sun, 04 Jul 2021   Prob (F-statistic):           1.34e-27\n",
       "Time:                        23:32:53   Log-Likelihood:                -525.38\n",
       "No. Observations:                  50   AIC:                             1063.\n",
       "Df Residuals:                      44   BIC:                             1074.\n",
       "Df Model:                           5                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       5.008e+04   6952.617      7.204      0.000    3.61e+04    6.41e+04\n",
       "x1            42.0063   3256.058      0.013      0.990   -6520.148    6604.161\n",
       "x2           240.7605   3338.877      0.072      0.943   -6488.304    6969.825\n",
       "x3             0.8060      0.046     17.368      0.000       0.712       0.900\n",
       "x4            -0.0270      0.052     -0.517      0.608      -0.132       0.078\n",
       "x5             0.0270      0.017      1.574      0.123      -0.008       0.062\n",
       "==============================================================================\n",
       "Omnibus:                       14.783   Durbin-Watson:                   1.283\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.267\n",
       "Skew:                          -0.948   Prob(JB):                     2.41e-05\n",
       "Kurtosis:                       5.572   Cond. No.                     1.47e+06\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, 1.47e+06. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm \n",
    "X_opt = y[:,0:6]\n",
    "regressor_OLS = sm.OLS(w,X_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4048b871",
   "metadata": {},
   "source": [
    "### Removing the variable with highest p-value (X1) and fitting the variables again"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "id": "0646fe0c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>Profit</td>      <th>  R-squared:         </th> <td>   0.951</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.946</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   217.2</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 04 Jul 2021</td> <th>  Prob (F-statistic):</th> <td>8.49e-29</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:46:09</td>     <th>  Log-Likelihood:    </th> <td> -525.38</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1061.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    45</td>      <th>  BIC:               </th> <td>   1070.</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>const</th> <td> 5.011e+04</td> <td> 6647.901</td> <td>    7.537</td> <td> 0.000</td> <td> 3.67e+04</td> <td> 6.35e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>  220.1847</td> <td> 2900.553</td> <td>    0.076</td> <td> 0.940</td> <td>-5621.828</td> <td> 6062.197</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>    0.8060</td> <td>    0.046</td> <td>   17.606</td> <td> 0.000</td> <td>    0.714</td> <td>    0.898</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>    <td>   -0.0270</td> <td>    0.052</td> <td>   -0.523</td> <td> 0.604</td> <td>   -0.131</td> <td>    0.077</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x4</th>    <td>    0.0270</td> <td>    0.017</td> <td>    1.592</td> <td> 0.118</td> <td>   -0.007</td> <td>    0.061</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.759</td> <th>  Durbin-Watson:     </th> <td>   1.282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  21.173</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.948</td> <th>  Prob(JB):          </th> <td>2.53e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.563</td> <th>  Cond. No.          </th> <td>1.40e+06</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, 1.4e+06. 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:                 Profit   R-squared:                       0.951\n",
       "Model:                            OLS   Adj. R-squared:                  0.946\n",
       "Method:                 Least Squares   F-statistic:                     217.2\n",
       "Date:                Sun, 04 Jul 2021   Prob (F-statistic):           8.49e-29\n",
       "Time:                        23:46:09   Log-Likelihood:                -525.38\n",
       "No. Observations:                  50   AIC:                             1061.\n",
       "Df Residuals:                      45   BIC:                             1070.\n",
       "Df Model:                           4                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       5.011e+04   6647.901      7.537      0.000    3.67e+04    6.35e+04\n",
       "x1           220.1847   2900.553      0.076      0.940   -5621.828    6062.197\n",
       "x2             0.8060      0.046     17.606      0.000       0.714       0.898\n",
       "x3            -0.0270      0.052     -0.523      0.604      -0.131       0.077\n",
       "x4             0.0270      0.017      1.592      0.118      -0.007       0.061\n",
       "==============================================================================\n",
       "Omnibus:                       14.759   Durbin-Watson:                   1.282\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.173\n",
       "Skew:                          -0.948   Prob(JB):                     2.53e-05\n",
       "Kurtosis:                       5.563   Cond. No.                     1.40e+06\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, 1.4e+06. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm \n",
    "X_opt = y[:,[0,2,3,4,5]]\n",
    "regressor_OLS = sm.OLS(w,X_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "id": "973d7ef7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>Profit</td>      <th>  R-squared:         </th> <td>   0.948</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.945</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   279.4</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 04 Jul 2021</td> <th>  Prob (F-statistic):</th> <td>1.59e-29</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:48:40</td>     <th>  Log-Likelihood:    </th> <td> -526.75</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1062.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    46</td>      <th>  BIC:               </th> <td>   1069.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     3</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>const</th> <td> 5.459e+04</td> <td> 6122.005</td> <td>    8.917</td> <td> 0.000</td> <td> 4.23e+04</td> <td> 6.69e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td> 1111.2045</td> <td> 2893.065</td> <td>    0.384</td> <td> 0.703</td> <td>-4712.234</td> <td> 6934.642</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>    0.8609</td> <td>    0.031</td> <td>   28.127</td> <td> 0.000</td> <td>    0.799</td> <td>    0.922</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>    <td>   -0.0527</td> <td>    0.050</td> <td>   -1.057</td> <td> 0.296</td> <td>   -0.153</td> <td>    0.048</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>14.252</td> <th>  Durbin-Watson:     </th> <td>   1.197</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  19.190</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.953</td> <th>  Prob(JB):          </th> <td>6.81e-05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.362</td> <th>  Cond. No.          </th> <td>6.72e+05</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, 6.72e+05. 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:                 Profit   R-squared:                       0.948\n",
       "Model:                            OLS   Adj. R-squared:                  0.945\n",
       "Method:                 Least Squares   F-statistic:                     279.4\n",
       "Date:                Sun, 04 Jul 2021   Prob (F-statistic):           1.59e-29\n",
       "Time:                        23:48:40   Log-Likelihood:                -526.75\n",
       "No. Observations:                  50   AIC:                             1062.\n",
       "Df Residuals:                      46   BIC:                             1069.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       5.459e+04   6122.005      8.917      0.000    4.23e+04    6.69e+04\n",
       "x1          1111.2045   2893.065      0.384      0.703   -4712.234    6934.642\n",
       "x2             0.8609      0.031     28.127      0.000       0.799       0.922\n",
       "x3            -0.0527      0.050     -1.057      0.296      -0.153       0.048\n",
       "==============================================================================\n",
       "Omnibus:                       14.252   Durbin-Watson:                   1.197\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               19.190\n",
       "Skew:                          -0.953   Prob(JB):                     6.81e-05\n",
       "Kurtosis:                       5.362   Cond. No.                     6.72e+05\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, 6.72e+05. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm \n",
    "X_opt = y[:,[0,2,3,4]]\n",
    "regressor_OLS = sm.OLS(w,X_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "0358d8bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>Profit</td>      <th>  R-squared:         </th> <td>   0.947</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.944</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   417.5</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 04 Jul 2021</td> <th>  Prob (F-statistic):</th> <td>1.19e-30</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:50:58</td>     <th>  Log-Likelihood:    </th> <td> -527.35</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1061.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    47</td>      <th>  BIC:               </th> <td>   1066.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</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>const</th> <td> 4.876e+04</td> <td> 2653.077</td> <td>   18.377</td> <td> 0.000</td> <td> 4.34e+04</td> <td> 5.41e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td> 1158.9995</td> <td> 2896.303</td> <td>    0.400</td> <td> 0.691</td> <td>-4667.610</td> <td> 6985.609</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>    <td>    0.8530</td> <td>    0.030</td> <td>   28.691</td> <td> 0.000</td> <td>    0.793</td> <td>    0.913</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>13.425</td> <th>  Durbin-Watson:     </th> <td>   1.122</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.001</td> <th>  Jarque-Bera (JB):  </th> <td>  17.626</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.907</td> <th>  Prob(JB):          </th> <td>0.000149</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.273</td> <th>  Cond. No.          </th> <td>2.02e+05</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, 2.02e+05. 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:                 Profit   R-squared:                       0.947\n",
       "Model:                            OLS   Adj. R-squared:                  0.944\n",
       "Method:                 Least Squares   F-statistic:                     417.5\n",
       "Date:                Sun, 04 Jul 2021   Prob (F-statistic):           1.19e-30\n",
       "Time:                        23:50:58   Log-Likelihood:                -527.35\n",
       "No. Observations:                  50   AIC:                             1061.\n",
       "Df Residuals:                      47   BIC:                             1066.\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       4.876e+04   2653.077     18.377      0.000    4.34e+04    5.41e+04\n",
       "x1          1158.9995   2896.303      0.400      0.691   -4667.610    6985.609\n",
       "x2             0.8530      0.030     28.691      0.000       0.793       0.913\n",
       "==============================================================================\n",
       "Omnibus:                       13.425   Durbin-Watson:                   1.122\n",
       "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               17.626\n",
       "Skew:                          -0.907   Prob(JB):                     0.000149\n",
       "Kurtosis:                       5.273   Cond. No.                     2.02e+05\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, 2.02e+05. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm \n",
    "X_opt = y[:,[0,2,3]]\n",
    "regressor_OLS = sm.OLS(w,X_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "703b0337",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>         <td>Profit</td>      <th>  R-squared:         </th> <td>   0.014</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>  -0.007</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>  0.6575</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 04 Jul 2021</td> <th>  Prob (F-statistic):</th>  <td> 0.421</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:51:26</td>     <th>  Log-Likelihood:    </th> <td> -600.31</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    50</td>      <th>  AIC:               </th> <td>   1205.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    48</td>      <th>  BIC:               </th> <td>   1208.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</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>const</th> <td> 1.088e+05</td> <td> 6936.739</td> <td>   15.689</td> <td> 0.000</td> <td> 9.49e+04</td> <td> 1.23e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td> 9943.2135</td> <td> 1.23e+04</td> <td>    0.811</td> <td> 0.421</td> <td>-1.47e+04</td> <td> 3.46e+04</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.077</td> <th>  Durbin-Watson:     </th> <td>   0.058</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.962</td> <th>  Jarque-Bera (JB):  </th> <td>   0.123</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.080</td> <th>  Prob(JB):          </th> <td>   0.940</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.817</td> <th>  Cond. No.          </th> <td>    2.42</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                 Profit   R-squared:                       0.014\n",
       "Model:                            OLS   Adj. R-squared:                 -0.007\n",
       "Method:                 Least Squares   F-statistic:                    0.6575\n",
       "Date:                Sun, 04 Jul 2021   Prob (F-statistic):              0.421\n",
       "Time:                        23:51:26   Log-Likelihood:                -600.31\n",
       "No. Observations:                  50   AIC:                             1205.\n",
       "Df Residuals:                      48   BIC:                             1208.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const       1.088e+05   6936.739     15.689      0.000    9.49e+04    1.23e+05\n",
       "x1          9943.2135   1.23e+04      0.811      0.421   -1.47e+04    3.46e+04\n",
       "==============================================================================\n",
       "Omnibus:                        0.077   Durbin-Watson:                   0.058\n",
       "Prob(Omnibus):                  0.962   Jarque-Bera (JB):                0.123\n",
       "Skew:                           0.080   Prob(JB):                        0.940\n",
       "Kurtosis:                       2.817   Cond. No.                         2.42\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm \n",
    "X_opt = y[:,[0,2]]\n",
    "regressor_OLS = sm.OLS(w,X_opt).fit()\n",
    "regressor_OLS.summary()"
   ]
  }
 ],
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