ASSIGNMENT-2-Akash Singh.ipynb

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   "source": [
    "import pyforest\n",
    "import statsmodels.api as sm\n",
    "import statsmodels.stats.api as sms\n",
    "import scipy.stats as st"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "1c55Km_3bhVg"
   },
   "source": [
    "## Q1.Use GQ test to check for the presence of heteroskedasticity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "sk255gHAXpBU"
   },
   "source": [
    "## LOAD DATASET\n",
    "### The following data on consumption-expenditure and income of 20 families (in ‘000 rupees):"
   ]
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       "    "
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Income</th>\n",
       "      <th>Con-exp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>22.3</td>\n",
       "      <td>19.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>32.3</td>\n",
       "      <td>31.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>36.6</td>\n",
       "      <td>31.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12.1</td>\n",
       "      <td>12.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>42.3</td>\n",
       "      <td>40.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6.2</td>\n",
       "      <td>6.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>44.7</td>\n",
       "      <td>38.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>26.1</td>\n",
       "      <td>25.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>10.3</td>\n",
       "      <td>10.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>40.2</td>\n",
       "      <td>38.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8.1</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>34.5</td>\n",
       "      <td>33.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>38.0</td>\n",
       "      <td>33.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>14.1</td>\n",
       "      <td>13.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>16.4</td>\n",
       "      <td>14.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>24.1</td>\n",
       "      <td>21.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>30.1</td>\n",
       "      <td>29.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>28.3</td>\n",
       "      <td>25.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>18.2</td>\n",
       "      <td>17.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>20.1</td>\n",
       "      <td>19.8</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Income  Con-exp\n",
       "0     22.3     19.9\n",
       "1     32.3     31.2\n",
       "2     36.6     31.8\n",
       "3     12.1     12.1\n",
       "4     42.3     40.7\n",
       "5      6.2      6.1\n",
       "6     44.7     38.6\n",
       "7     26.1     25.5\n",
       "8     10.3     10.3\n",
       "9     40.2     38.8\n",
       "10     8.1      8.0\n",
       "11    34.5     33.1\n",
       "12    38.0     33.5\n",
       "13    14.1     13.1\n",
       "14    16.4     14.8\n",
       "15    24.1     21.6\n",
       "16    30.1     29.3\n",
       "17    28.3     25.0\n",
       "18    18.2     17.9\n",
       "19    20.1     19.8"
      ]
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   "source": [
    "d= pd.read_csv(\"C:/Users/Akash/Desktop/Mayukh sir practical/assignment2/a2d1.csv\")\n",
    "d"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### SORT THE DATA-SET BASED ON INCOME "
   ]
  },
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Income</th>\n",
       "      <th>Con-exp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6.2</td>\n",
       "      <td>6.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8.1</td>\n",
       "      <td>8.0</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>10.3</td>\n",
       "      <td>10.3</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12.1</td>\n",
       "      <td>12.1</td>\n",
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       "      <td>14.1</td>\n",
       "      <td>13.1</td>\n",
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       "      <td>16.4</td>\n",
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       "      <td>18.2</td>\n",
       "      <td>17.9</td>\n",
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       "      <th>19</th>\n",
       "      <td>20.1</td>\n",
       "      <td>19.8</td>\n",
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       "      <td>22.3</td>\n",
       "      <td>19.9</td>\n",
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       "      <td>24.1</td>\n",
       "      <td>21.6</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>26.1</td>\n",
       "      <td>25.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>28.3</td>\n",
       "      <td>25.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>30.1</td>\n",
       "      <td>29.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>32.3</td>\n",
       "      <td>31.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>34.5</td>\n",
       "      <td>33.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>36.6</td>\n",
       "      <td>31.8</td>\n",
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       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>38.0</td>\n",
       "      <td>33.5</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>40.2</td>\n",
       "      <td>38.8</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>42.3</td>\n",
       "      <td>40.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>44.7</td>\n",
       "      <td>38.6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "    Income  Con-exp\n",
       "5      6.2      6.1\n",
       "10     8.1      8.0\n",
       "8     10.3     10.3\n",
       "3     12.1     12.1\n",
       "13    14.1     13.1\n",
       "14    16.4     14.8\n",
       "18    18.2     17.9\n",
       "19    20.1     19.8\n",
       "0     22.3     19.9\n",
       "15    24.1     21.6\n",
       "7     26.1     25.5\n",
       "17    28.3     25.0\n",
       "16    30.1     29.3\n",
       "1     32.3     31.2\n",
       "11    34.5     33.1\n",
       "2     36.6     31.8\n",
       "12    38.0     33.5\n",
       "9     40.2     38.8\n",
       "4     42.3     40.7\n",
       "6     44.7     38.6"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1=data.sort_values(\"Income\")\n",
    "d1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "GNlwAVIaYCvW"
   },
   "source": [
    "### HERE, INDEPENDENT COVARIATE IS INCOME(x) AND DEPENDENT VARIABLE IS CONSUMPTION-EXPENDITURE(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5      6.2\n",
      "10     8.1\n",
      "8     10.3\n",
      "3     12.1\n",
      "13    14.1\n",
      "14    16.4\n",
      "18    18.2\n",
      "19    20.1\n",
      "0     22.3\n",
      "15    24.1\n",
      "7     26.1\n",
      "17    28.3\n",
      "16    30.1\n",
      "1     32.3\n",
      "11    34.5\n",
      "2     36.6\n",
      "12    38.0\n",
      "9     40.2\n",
      "4     42.3\n",
      "6     44.7\n",
      "Name: Income, dtype: float64\n",
      "5      6.1\n",
      "10     8.0\n",
      "8     10.3\n",
      "3     12.1\n",
      "13    13.1\n",
      "14    14.8\n",
      "18    17.9\n",
      "19    19.8\n",
      "0     19.9\n",
      "15    21.6\n",
      "7     25.5\n",
      "17    25.0\n",
      "16    29.3\n",
      "1     31.2\n",
      "11    33.1\n",
      "2     31.8\n",
      "12    33.5\n",
      "9     38.8\n",
      "4     40.7\n",
      "6     38.6\n",
      "Name: Con-exp, dtype: float64\n"
     ]
    }
   ],
   "source": [
    "x=data1[\"Income\"]\n",
    "print(x)\n",
    "y=data1[\"Con-exp\"]\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "QFCrI9dlYsc5"
   },
   "source": [
    "### PERFORMING SIMPLE LINEAR REGREESION EQUATION USING STATS MODEL\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "colab": {
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    "id": "xWGDp8aLQtVO",
    "outputId": "bce543d3-f72d-45e8-c6a8-2e32c8437971"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                Con-exp   R-squared:                       0.986\n",
      "Model:                            OLS   Adj. R-squared:                  0.985\n",
      "Method:                 Least Squares   F-statistic:                     1263.\n",
      "Date:                Mon, 30 May 2022   Prob (F-statistic):           4.00e-18\n",
      "Time:                        23:58:56   Log-Likelihood:                -32.785\n",
      "No. Observations:                  20   AIC:                             69.57\n",
      "Df Residuals:                      18   BIC:                             71.56\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          0.8471      0.703      1.204      0.244      -0.631       2.325\n",
      "Income         0.8993      0.025     35.534      0.000       0.846       0.952\n",
      "==============================================================================\n",
      "Omnibus:                        1.874   Durbin-Watson:                   2.060\n",
      "Prob(Omnibus):                  0.392   Jarque-Bera (JB):                1.113\n",
      "Skew:                          -0.236   Prob(JB):                        0.573\n",
      "Kurtosis:                       1.945   Cond. No.                         66.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x_const=sm.add_constant(x)\n",
    "model=sm.OLS(y,x_const)\n",
    "result=model.fit()\n",
    "print(result.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "j8rdKO0uRMo0",
    "outputId": "53590a60-49dd-4046-acbf-777b6f2de7ce"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5      6.422865\n",
       "10     8.131582\n",
       "8     10.110096\n",
       "3     11.728880\n",
       "13    13.527530\n",
       "14    15.595977\n",
       "18    17.214761\n",
       "19    18.923478\n",
       "0     20.901992\n",
       "15    22.520777\n",
       "7     24.319426\n",
       "17    26.297940\n",
       "16    27.916725\n",
       "1     29.895239\n",
       "11    31.873753\n",
       "2     33.762335\n",
       "12    35.021390\n",
       "9     36.999904\n",
       "4     38.888486\n",
       "6     41.046865\n",
       "dtype: float64"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred= result.predict(x_const)\n",
    "y_pred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "-zjzutkjRfzs",
    "outputId": "f18becb9-ee94-4eb3-88bd-30565ef1d620"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5    -0.322865\n",
      "10   -0.131582\n",
      "8     0.189904\n",
      "3     0.371120\n",
      "13   -0.427530\n",
      "14   -0.795977\n",
      "18    0.685239\n",
      "19    0.876522\n",
      "0    -1.001992\n",
      "15   -0.920777\n",
      "7     1.180574\n",
      "17   -1.297940\n",
      "16    1.383275\n",
      "1     1.304761\n",
      "11    1.226247\n",
      "2    -1.962335\n",
      "12   -1.521390\n",
      "9     1.800096\n",
      "4     1.811514\n",
      "6    -2.446865\n",
      "dtype: float64\n"
     ]
    }
   ],
   "source": [
    "res= y-y_pred\n",
    "print(res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rDXv_NGTZ1q_"
   },
   "source": [
    "### RESIDUAL PLOT ON INDEPENDENT VARIABLE VS RESIUAL"
   ]
  },
  {
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JfyzpYUlfsv1hSf8t6X0Z1fYO27dr7V3D85I+0kRtlQOSPijpbNVPlaRPKo/x61fbPZmM305Jj9reprWHrC9FxJdtf1fNj12/2v4+k7HrJYf/5gb5TKqxy2L6IQDgxjXeWgEAjIcgB4DCEeQAUDiCHAAKR5ADQOEIcgAoHEEOAIX7fxm4H/HsggsqAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x,res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "nCmOxQiwaFWW"
   },
   "source": [
    "#### OBSERED FANNEL SHAPED DIAGRAM i.e HETEROSKEDASTICITY IS PRESENT. NEXT, WE PERFORMED GQ-TEST TO ENSURE PRESENCE OF HETEROSKEDASTICITY"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "NLvIx4ooSDWX",
    "outputId": "6649b1b4-336d-4668-e165-4402deda9d21"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5      6.2\n",
      "10     8.1\n",
      "8     10.3\n",
      "3     12.1\n",
      "13    14.1\n",
      "14    16.4\n",
      "18    18.2\n",
      "19    20.1\n",
      "Name: Income, dtype: float64\n"
     ]
    }
   ],
   "source": [
    "c=4\n",
    "smallest = data1[:8]\n",
    "#print(smallest)\n",
    "print(smallest[\"Income\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "r34yogUlSfxO",
    "outputId": "0081d742-8384-4141-8e5d-f8988b7f0a8b"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                Con-exp   R-squared:                       0.988\n",
      "Model:                            OLS   Adj. R-squared:                  0.986\n",
      "Method:                 Least Squares   F-statistic:                     481.2\n",
      "Date:                Mon, 30 May 2022   Prob (F-statistic):           5.86e-07\n",
      "Time:                        23:59:01   Log-Likelihood:                -5.5874\n",
      "No. Observations:                   8   AIC:                             15.17\n",
      "Df Residuals:                       6   BIC:                             15.33\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          0.2350      0.605      0.389      0.711      -1.244       1.714\n",
      "Income         0.9500      0.043     21.937      0.000       0.844       1.056\n",
      "==============================================================================\n",
      "Omnibus:                        3.523   Durbin-Watson:                   1.610\n",
      "Prob(Omnibus):                  0.172   Jarque-Bera (JB):                1.472\n",
      "Skew:                          -1.043   Prob(JB):                        0.479\n",
      "Kurtosis:                       2.746   Cond. No.                         42.7\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Akash\\AppData\\Local\\Programs\\Python\\Python310\\lib\\site-packages\\scipy\\stats\\_stats_py.py:1477: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n",
      "  warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n"
     ]
    }
   ],
   "source": [
    "x1_const=sm.add_constant(smallest[\"Income\"])\n",
    "model1= sm.OLS(smallest[\"Con-exp\"],x1_const)\n",
    "result1=model1.fit()\n",
    "print(result1.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "id": "EMK2E8EnTmKf"
   },
   "outputs": [],
   "source": [
    "y1_pred= result1.predict(x1_const)\n",
    "y1=smallest[\"Con-exp\"]\n",
    "res1=y1-y1_pred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Cs0EzYqZT3dv",
    "outputId": "0187ea17-ec90-4b8a-ed3b-77dba85b345e"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.8934714887358706\n"
     ]
    }
   ],
   "source": [
    "Res1 = res1**2\n",
    "ssr1 = Res1.sum()\n",
    "print(ssr1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 300
    },
    "id": "luiI5QPFUVfE",
    "outputId": "cf46d9e0-a212-4900-af50-e4d05908a00a"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Income</th>\n",
       "      <th>Con-exp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>30.1</td>\n",
       "      <td>29.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>32.3</td>\n",
       "      <td>31.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>34.5</td>\n",
       "      <td>33.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>36.6</td>\n",
       "      <td>31.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>38.0</td>\n",
       "      <td>33.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>40.2</td>\n",
       "      <td>38.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>42.3</td>\n",
       "      <td>40.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>44.7</td>\n",
       "      <td>38.6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Income  Con-exp\n",
       "16    30.1     29.3\n",
       "1     32.3     31.2\n",
       "11    34.5     33.1\n",
       "2     36.6     31.8\n",
       "12    38.0     33.5\n",
       "9     40.2     38.8\n",
       "4     42.3     40.7\n",
       "6     44.7     38.6"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "largest = data1[-8:]\n",
    "largest\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "YkRMFGRFU_vx",
    "outputId": "3854c82a-55c4-4afc-9240-06a95dae4d28"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                Con-exp   R-squared:                       0.833\n",
      "Model:                            OLS   Adj. R-squared:                  0.806\n",
      "Method:                 Least Squares   F-statistic:                     30.00\n",
      "Date:                Mon, 30 May 2022   Prob (F-statistic):            0.00155\n",
      "Time:                        23:59:05   Log-Likelihood:                -15.076\n",
      "No. Observations:                   8   AIC:                             34.15\n",
      "Df Residuals:                       6   BIC:                             34.31\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          6.0938      5.250      1.161      0.290      -6.751      18.939\n",
      "Income         0.7641      0.140      5.477      0.002       0.423       1.106\n",
      "==============================================================================\n",
      "Omnibus:                        1.258   Durbin-Watson:                   1.852\n",
      "Prob(Omnibus):                  0.533   Jarque-Bera (JB):                0.631\n",
      "Skew:                          -0.009   Prob(JB):                        0.729\n",
      "Kurtosis:                       1.624   Cond. No.                         304.\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Akash\\AppData\\Local\\Programs\\Python\\Python310\\lib\\site-packages\\scipy\\stats\\_stats_py.py:1477: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n",
      "  warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n"
     ]
    }
   ],
   "source": [
    "x2_const = sm.add_constant(largest[\"Income\"])\n",
    "mod = sm.OLS(largest[\"Con-exp\"],x2_const)\n",
    "result2 = mod.fit()\n",
    "print(result2.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "2hyrdOW6VMKn",
    "outputId": "e988a102-1f42-4f80-c1a1-6c2dadb6b22c"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20.299525511488298"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred2 = result2.predict(x2_const)\n",
    "res2 = (largest[\"Con-exp\"]-y_pred2)\n",
    "Res2 = res2**2\n",
    "ssr2 = Res2.sum()\n",
    "ssr2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "jb7Vv9L3VdyT",
    "outputId": "bb1d618a-392c-44b7-da5c-d1a430cb063a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10.720798085552783\n"
     ]
    }
   ],
   "source": [
    "F_obs=ssr2/ssr1\n",
    "print(F_obs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "b1chi6R6Vjms",
    "outputId": "ccd55100-cf48-4bcf-83ff-606a3ee1d193"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.283865713822639\n"
     ]
    }
   ],
   "source": [
    "F_critical=st.f.ppf(0.95,6,6)\n",
    "print(F_critical)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "M8Lv2NDaVqpP",
    "outputId": "7057598e-30d1-4c16-be5e-79527b4199d7"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "we reject H0 i.e heteroskedasticity is present\n"
     ]
    }
   ],
   "source": [
    "if(F_obs>F_critical):\n",
    "  print(\"we reject H0 i.e heteroskedasticity is present\")\n",
    "else:\n",
    "  print(\"we accept H0 i.e homoskedasticity is present\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rUvwtrqLbpSY"
   },
   "source": [
    "## Q2.If heteroskedasticity is present, use any simple model for the error-variance to obtain better estimates of the parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "glCcZB4ybw9M"
   },
   "source": [
    "### PERFORMING GLS TO OBTAIN BETTER ESTIMATOR OF THE PARAMETERS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "vSD3TmjLcqeU",
    "outputId": "2f1e3312-3dac-49af-852a-28a72975ee9d"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            GLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                Con-exp   R-squared:                       0.986\n",
      "Model:                            GLS   Adj. R-squared:                  0.985\n",
      "Method:                 Least Squares   F-statistic:                     1263.\n",
      "Date:                Mon, 30 May 2022   Prob (F-statistic):           4.00e-18\n",
      "Time:                        23:59:55   Log-Likelihood:                -32.785\n",
      "No. Observations:                  20   AIC:                             69.57\n",
      "Df Residuals:                      18   BIC:                             71.56\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          0.8471      0.703      1.204      0.244      -0.631       2.325\n",
      "Income         0.8993      0.025     35.534      0.000       0.846       0.952\n",
      "==============================================================================\n",
      "Omnibus:                        1.874   Durbin-Watson:                   2.060\n",
      "Prob(Omnibus):                  0.392   Jarque-Bera (JB):                1.113\n",
      "Skew:                          -0.236   Prob(JB):                        0.573\n",
      "Kurtosis:                       1.945   Cond. No.                         66.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x_const = sm.add_constant(x)\n",
    "model3 = sm.GLS(y,x_const)\n",
    "result3 = model3.fit()\n",
    "print(result3.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rNDOfnUZdy78"
   },
   "source": [
    "## B)Q1. Use Glejser test to check for the presence of heteroskedasticity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "sdzvhb-3d8cE"
   },
   "source": [
    "### LOAD DATA-SET\n",
    "### The following data are on speed (sp) of a car and distance (dis) it covers to come to a standstill."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 206
    },
    "id": "N4jcjS16eAiU",
    "outputId": "b4862105-844d-433e-8b51-bcf5e4d5e5aa"
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
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    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>dis</th>\n",
       "      <th>sp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>8</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   dis  sp\n",
       "0    4   4\n",
       "1    2   5\n",
       "2    4   5\n",
       "3    8   5\n",
       "4    8   5"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2= pd.read_csv(\"C:/Users/Akash/Desktop/Mayukh sir practical/assignment2/a2d2.csv\")\n",
    "data2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "RpBfBSo-xKdV"
   },
   "source": [
    "### HERE SPEED OF CAR IS INDEPENDENT VARIABLE(x) and DISTANCE IS DEPENDENT VARIABLE(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "id": "p6Qzq6HTeKnQ"
   },
   "outputs": [],
   "source": [
    "x = data2[\"sp\"]\n",
    "y = data2[\"dis\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "mrjSkHfVx6T7"
   },
   "source": [
    "### PERFORMING OLS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "j7ZgesvvhOVB",
    "outputId": "55c62cd4-aeeb-4cd9-ba52-e1f35bb6e175"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                    dis   R-squared:                       0.876\n",
      "Model:                            OLS   Adj. R-squared:                  0.874\n",
      "Method:                 Least Squares   F-statistic:                     408.6\n",
      "Date:                Tue, 31 May 2022   Prob (F-statistic):           6.13e-28\n",
      "Time:                        00:02:36   Log-Likelihood:                -232.96\n",
      "No. Observations:                  60   AIC:                             469.9\n",
      "Df Residuals:                      58   BIC:                             474.1\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const        -20.4174      3.345     -6.105      0.000     -27.112     -13.722\n",
      "sp             3.1515      0.156     20.213      0.000       2.839       3.464\n",
      "==============================================================================\n",
      "Omnibus:                        4.671   Durbin-Watson:                   1.821\n",
      "Prob(Omnibus):                  0.097   Jarque-Bera (JB):                3.693\n",
      "Skew:                           0.499   Prob(JB):                        0.158\n",
      "Kurtosis:                       3.693   Cond. No.                         46.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x_const = sm.add_constant(x)\n",
    "mod = sm.OLS(y,x_const)\n",
    "result = mod.fit()\n",
    "print(result.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "L0C9lwZLhTZQ",
    "outputId": "90b4306b-a5b5-49b8-fdfa-c91ea667bdba"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercepts:  -20.41742046782962\n",
      "Slope:  3.151528220726599\n"
     ]
    }
   ],
   "source": [
    "parameters = result.params\n",
    "intercept = parameters.const\n",
    "slope = parameters.sp\n",
    "print(\"Intercepts: \", intercept)\n",
    "print(\"Slope: \",slope)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "JbdXQH-1h_fh",
    "outputId": "eb2165d5-79f8-4588-f63c-c9476622b6dd"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0   -7.811308\n",
       "1   -4.659779\n",
       "2   -4.659779\n",
       "3   -4.659779\n",
       "4   -4.659779\n",
       "dtype: float64"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred=result.predict(x_const)\n",
    "y_pred.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "tY0vvpMtiXm3",
    "outputId": "1968386b-2cdd-44a8-e2d5-8d2ee6c14d62"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    11.811308\n",
       "1     6.659779\n",
       "2     8.659779\n",
       "3    12.659779\n",
       "4    12.659779\n",
       "dtype: float64"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res2= y- y_pred\n",
    "res2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 422
    },
    "id": "x_c3TB9SiqNT",
    "outputId": "027ec19a-5f00-4734-a5a6-58815e206053"
   },
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       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt'); }\n",
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    {
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      "text/plain": [
       "Text(0.5, 1.0, 'X vs Residual plot')"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1440x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(20,6))\n",
    "\n",
    "plt.subplot(1,2,1)\n",
    "plt.scatter(y_pred,res2)\n",
    "plt.xlabel(\"y\")\n",
    "plt.ylabel(\"Residual\")\n",
    "plt.title(\"Y vs Residual plot\")\n",
    "\n",
    "plt.subplot(1,2,2)\n",
    "plt.scatter(x,res2)\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"Residual\")\n",
    "plt.title(\"X vs Residual plot\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "id": "Q-DFwsnAj0LD"
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt\\nimport numpy as np'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RES=np.absolute(res2)\n",
    "#1 |ei| = b0+b1.sqrt(1/xi)+Vi\n",
    "#2 |ei| = b0+b1.sqrt(xi)+Vi\n",
    "#3 |ei| = b0+b1.(1/xi)+Vi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "rdsZmGw5ke0u",
    "outputId": "ae8115a8-ccc7-4326-ed81-dff92e0993c3"
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt\\nimport numpy as np'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                    dis   R-squared:                       0.876\n",
      "Model:                            OLS   Adj. R-squared:                  0.874\n",
      "Method:                 Least Squares   F-statistic:                     408.6\n",
      "Date:                Tue, 31 May 2022   Prob (F-statistic):           6.13e-28\n",
      "Time:                        00:03:40   Log-Likelihood:                -232.96\n",
      "No. Observations:                  60   AIC:                             469.9\n",
      "Df Residuals:                      58   BIC:                             474.1\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const        -20.4174      3.345     -6.105      0.000     -27.112     -13.722\n",
      "sp             3.1515      0.156     20.213      0.000       2.839       3.464\n",
      "==============================================================================\n",
      "Omnibus:                        4.671   Durbin-Watson:                   1.821\n",
      "Prob(Omnibus):                  0.097   Jarque-Bera (JB):                3.693\n",
      "Skew:                           0.499   Prob(JB):                        0.158\n",
      "Kurtosis:                       3.693   Cond. No.                         46.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x11_const = sm.add_constant(np.sqrt(1/x))\n",
    "model11 = sm.OLS(RES,x11_const)\n",
    "result11 = mod.fit()\n",
    "print(result11.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "N7P9kfnplsrS",
    "outputId": "48db29f6-bb0f-47f7-90fb-f518448f3a57"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.047\n",
      "Model:                            OLS   Adj. R-squared:                  0.031\n",
      "Method:                 Least Squares   F-statistic:                     2.880\n",
      "Date:                Tue, 31 May 2022   Prob (F-statistic):             0.0951\n",
      "Time:                        00:03:42   Log-Likelihood:                -202.75\n",
      "No. Observations:                  60   AIC:                             409.5\n",
      "Df Residuals:                      58   BIC:                             413.7\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const         11.5330      1.649      6.996      0.000       8.233      14.833\n",
      "sp           -30.9150     18.217     -1.697      0.095     -67.380       5.550\n",
      "==============================================================================\n",
      "Omnibus:                       18.002   Durbin-Watson:                   1.930\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               22.022\n",
      "Skew:                           1.266   Prob(JB):                     1.65e-05\n",
      "Kurtosis:                       4.549   Cond. No.                         19.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x22_const=sm.add_constant(1/x)\n",
    "model22= sm.OLS(RES,x22_const)\n",
    "result22=model22.fit()\n",
    "print(result22.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "jXKrn_xAmW0f",
    "outputId": "4c387e46-d492-4f73-edb1-b004835b20dd"
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt\\nimport numpy as np'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.165\n",
      "Model:                            OLS   Adj. R-squared:                  0.150\n",
      "Method:                 Least Squares   F-statistic:                     11.43\n",
      "Date:                Tue, 31 May 2022   Prob (F-statistic):            0.00130\n",
      "Time:                        00:03:42   Log-Likelihood:                -198.80\n",
      "No. Observations:                  60   AIC:                             401.6\n",
      "Df Residuals:                      58   BIC:                             405.8\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const         -1.3810      3.256     -0.424      0.673      -7.898       5.136\n",
      "sp             2.5236      0.746      3.381      0.001       1.030       4.018\n",
      "==============================================================================\n",
      "Omnibus:                        8.280   Durbin-Watson:                   2.080\n",
      "Prob(Omnibus):                  0.016   Jarque-Bera (JB):                7.555\n",
      "Skew:                           0.786   Prob(JB):                       0.0229\n",
      "Kurtosis:                       3.743   Cond. No.                         17.1\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x33_const=sm.add_constant(np.sqrt(x))\n",
    "model33= sm.OLS(RES,x33_const)\n",
    "result33=model33.fit()\n",
    "print(result33.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "31obVZqgntoa",
    "outputId": "83334aca-f8ae-4eb1-b8c7-d3bdef6134ef"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.151528220726599\n"
     ]
    }
   ],
   "source": [
    "slope11=result11.params.sp\n",
    "print(slope11)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "p3_6CczPuyLz",
    "outputId": "3d4eb7c6-d87b-41d4-bc85-0600cb271083"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "20.202103979016663\n"
     ]
    }
   ],
   "source": [
    "T_obs= slope11/ 0.156 \n",
    "print(T_obs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "F5ZYbXcTwLBS",
    "outputId": "253c711e-b6a1-495f-a71d-a378d0fefff9"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.671552762153672"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T_critical=st.t.ppf(0.95,58)\n",
    "T_critical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "NIu-RLGu1jKi",
    "outputId": "4e72dd28-4c7c-4c71-960e-05b3f5e45666"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "WE REJECT H0 i.e HETEROSKEDASTICITY IS PRESENT\n"
     ]
    }
   ],
   "source": [
    "if(T_obs>T_critical):\n",
    "  print(\"WE REJECT H0 i.e HETEROSKEDASTICITY IS PRESENT\")\n",
    "else:\n",
    "  print(\"WE ACCEPECT H0 i.e HETEROSKEDASTICITY IS NOT PRESENT \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uC8FJc7p39f1"
   },
   "source": [
    "## REMEDY"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Cecxtz6i4CE5"
   },
   "source": [
    "### TRANSFORMING Y --> LOG Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "id": "GTZVj7Ld2Pgt"
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "        if (window._pyforest_update_imports_cell) { window._pyforest_update_imports_cell('import pandas as pd\\nimport matplotlib.pyplot as plt\\nimport numpy as np'); }\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_log = np.log(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "myTUQpaC4PNN"
   },
   "source": [
    "### PERFORMING GLS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "gonkeiXX4OfT",
    "outputId": "b038ce0f-ad65-41d4-b2cf-6efec659e7c0"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            GLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                    dis   R-squared:                       0.865\n",
      "Model:                            GLS   Adj. R-squared:                  0.862\n",
      "Method:                 Least Squares   F-statistic:                     370.5\n",
      "Date:                Tue, 31 May 2022   Prob (F-statistic):           7.27e-27\n",
      "Time:                        00:03:48   Log-Likelihood:                -24.427\n",
      "No. Observations:                  60   AIC:                             52.85\n",
      "Df Residuals:                      58   BIC:                             57.04\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          1.4977      0.103     14.471      0.000       1.291       1.705\n",
      "sp             0.0929      0.005     19.249      0.000       0.083       0.103\n",
      "==============================================================================\n",
      "Omnibus:                        8.532   Durbin-Watson:                   1.323\n",
      "Prob(Omnibus):                  0.014   Jarque-Bera (JB):                8.095\n",
      "Skew:                          -0.724   Prob(JB):                       0.0175\n",
      "Kurtosis:                       4.067   Cond. No.                         46.6\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "x44_const = sm.add_constant(x)\n",
    "model5 = sm.GLS(y_log,x44_const)\n",
    "result5 = model5.fit()\n",
    "print(result5.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "kwafJSlO3Y6b",
    "outputId": "7e4af608-a62b-4a69-895f-6c95cf9e3c18"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0   -0.482895\n",
       "1   -1.268910\n",
       "2   -0.575763\n",
       "3    0.117384\n",
       "4    0.117384\n",
       "dtype: float64"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_pred_gls = result5.predict(x44_const)\n",
    "\n",
    "res_gls = (y_log - y_pred_gls)\n",
    "res_gls.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ijQKoPik4ypN"
   },
   "source": [
    "### Confirming that our new gls model's error terms are homoskedastic using BP Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Y5skKsmz40Jw",
    "outputId": "cffb3843-0abb-483f-c9b5-ed76fa0482c9"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The p value of our test is 0.06\n",
      "We accept the null hypothesis\n",
      "Our errors are Homoscedastic\n"
     ]
    }
   ],
   "source": [
    "names = ['Lagrange multiplier statistic', 'p-value',\n",
    "        'f-value', 'f p-value']\n",
    "lmstat, pvalue , fvalue , fpvalue = sms.het_breuschpagan(res_gls, x44_const)\n",
    "print(\"The p value of our test is {:.2f}\".format(pvalue))\n",
    "\n",
    "\n",
    "if pvalue<=0.05:\n",
    "    print(\"Our data is heteroscedastic\")\n",
    "else:\n",
    "    print(\"We accept the null hypothesis\\nOur errors are Homoscedastic\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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