intro-linear-regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Intro Linear Regression\n",
    "\n",
    "A technique that measures the relationship between two variables. If we have an independent variable  X, and a dependent outcome variable  Y, linear regression allows us to determine which linear model  Y=α+βX best explains the data.\n",
    "\n",
    "Considering TSLA and SPY: We want to know how TSLA varies as a function of how SPY varies, so we will take the daily returns of each and regress them against each other.\n",
    "\n",
    "Python's statsmodels library has a built-in linear fit function. Note that this will give a line of best fit; whether or not the relationship it shows is significant is for us to determine. The output will also have some statistics about the model, such as R-squared and the F value, which may help you quantify how good the fit actually is."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import yfinance as yf\n",
    "import requests\n",
    "import datetime\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from pandas_datareader import data as pdr\n",
    "from matplotlib import style\n",
    "from math import pi\n",
    "import scipy.stats as stats\n",
    "\n",
    "from statsmodels import regression\n",
    "import statsmodels.api as sm\n",
    "import math\n",
    "\n",
    "yf.pdr_override()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First define a function that performs linear regression and plots the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linreg(X,Y):\n",
    "    # Running the linear regression\n",
    "    X = sm.add_constant(X)\n",
    "    model = regression.linear_model.OLS(Y, X).fit()\n",
    "    a = model.params[0]\n",
    "    b = model.params[1]\n",
    "    X = X[:, 1]\n",
    "\n",
    "    # Return summary of the regression and plot results\n",
    "    X2 = np.linspace(X.min(), X.max(), 100)\n",
    "    Y_hat = X2 * b + a\n",
    "    plt.scatter(X, Y, alpha=0.3) # Plot the raw data\n",
    "    plt.plot(X2, Y_hat, 'r', alpha=0.9);  # Add the regression line, colored in red\n",
    "    plt.xlabel('X Value')\n",
    "    plt.ylabel('Y Value')\n",
    "    return model.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, getting pricing data on TSLA and SPY and performing a regression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[*********************100%***********************]  1 of 1 completed\n",
      "[*********************100%***********************]  1 of 1 completed\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.205</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.202</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   64.27</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 Jun 2020</td> <th>  Prob (F-statistic):</th> <td>4.23e-14</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>01:25:42</td>     <th>  Log-Likelihood:    </th> <td>  549.18</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   251</td>      <th>  AIC:               </th> <td>  -1094.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   249</td>      <th>  BIC:               </th> <td>  -1087.</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>    0.0009</td> <td>    0.002</td> <td>    0.532</td> <td> 0.595</td> <td>   -0.002</td> <td>    0.004</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    1.9542</td> <td>    0.244</td> <td>    8.017</td> <td> 0.000</td> <td>    1.474</td> <td>    2.434</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>64.854</td> <th>  Durbin-Watson:     </th> <td>   1.998</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td> 376.770</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.865</td> <th>  Prob(JB):          </th> <td>1.53e-82</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 8.748</td> <th>  Cond. No.          </th> <td>    142.</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<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:                      y   R-squared:                       0.205\n",
       "Model:                            OLS   Adj. R-squared:                  0.202\n",
       "Method:                 Least Squares   F-statistic:                     64.27\n",
       "Date:                Mon, 08 Jun 2020   Prob (F-statistic):           4.23e-14\n",
       "Time:                        01:25:42   Log-Likelihood:                 549.18\n",
       "No. Observations:                 251   AIC:                            -1094.\n",
       "Df Residuals:                     249   BIC:                            -1087.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.0009      0.002      0.532      0.595      -0.002       0.004\n",
       "x1             1.9542      0.244      8.017      0.000       1.474       2.434\n",
       "==============================================================================\n",
       "Omnibus:                       64.854   Durbin-Watson:                   1.998\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              376.770\n",
       "Skew:                           0.865   Prob(JB):                     1.53e-82\n",
       "Kurtosis:                       8.748   Cond. No.                         142.\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "start_date = '2014-01-01'\n",
    "end_date = '2015-01-01'\n",
    "asset = pdr.get_data_yahoo('TSLA', start=start_date, end=end_date, interval = \"1d\")['Adj Close']\n",
    "benchmark = pdr.get_data_yahoo('SPY', start=start_date, end=end_date, interval = \"1d\")['Adj Close']\n",
    "\n",
    "# We have to take the percent changes to get to returns\n",
    "# Get rid of the first (0th) element because it is NAN\n",
    "r_a = asset.pct_change()[1:]\n",
    "r_b = benchmark.pct_change()[1:]\n",
    "\n",
    "linreg(r_b.values, r_a.values)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each point on the above graph represents a day, with the x-coordinate being the return of SPY, and the y-coordinate being the return of TSLA. As we can see, the line of best fit tells us that for every 1% increased return we see from the SPY, we should see an extra 1.92% from TSLA. This is expressed by the parameter  β\n",
    " , which is 1.9271 as estimated. Of course, for decresed return we will also see about double the loss in TSLA, so we haven't gained anything, we are just more volatile."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear Regression vs. Correlation\n",
    "\n",
    "- Linear regression gives us a specific linear model, but is limited to cases of linear dependence.\n",
    "- Correlation is general to linear and non-linear dependencies, but doesn't give us an actual model.\n",
    "- Both are measures of covariance.\n",
    "- Linear regression can give us relationship between Y and many independent variables by making X multidimensional."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Knowing Parameters vs. Estimates\n",
    "\n",
    "It is very important to keep in mind that all  α\n",
    "  and  β\n",
    "  parameters estimated by linear regression are just that - estimates. You can never know the underlying true parameters unless you know the physical process producing the data. The parameters you estimate today may not be the same analysis done including tomorrow's data, and the underlying true parameters may be moving. As such it is very important when doing actual analysis to pay attention to the standard error of the parameter estimates. More material on the standard error will be presented in a later lecture. One way to get a sense of how stable your parameter estimates are is to estimate them using a rolling window of data and see how much variance there is in the estimates.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### An Example Case\n",
    "\n",
    "What happens if we regress two purely random variables?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.001</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>  -0.009</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>  0.1094</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 Jun 2020</td> <th>  Prob (F-statistic):</th>  <td> 0.742</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>01:25:43</td>     <th>  Log-Likelihood:    </th> <td> -24.995</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   100</td>      <th>  AIC:               </th> <td>   53.99</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    98</td>      <th>  BIC:               </th> <td>   59.20</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>    0.5418</td> <td>    0.059</td> <td>    9.202</td> <td> 0.000</td> <td>    0.425</td> <td>    0.659</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>   -0.0346</td> <td>    0.105</td> <td>   -0.331</td> <td> 0.742</td> <td>   -0.242</td> <td>    0.173</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>70.239</td> <th>  Durbin-Watson:     </th> <td>   1.732</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>   7.918</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.153</td> <th>  Prob(JB):          </th> <td>  0.0191</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 1.656</td> <th>  Cond. No.          </th> <td>    4.15</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<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:                      y   R-squared:                       0.001\n",
       "Model:                            OLS   Adj. R-squared:                 -0.009\n",
       "Method:                 Least Squares   F-statistic:                    0.1094\n",
       "Date:                Mon, 08 Jun 2020   Prob (F-statistic):              0.742\n",
       "Time:                        01:25:43   Log-Likelihood:                -24.995\n",
       "No. Observations:                 100   AIC:                             53.99\n",
       "Df Residuals:                      98   BIC:                             59.20\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.5418      0.059      9.202      0.000       0.425       0.659\n",
       "x1            -0.0346      0.105     -0.331      0.742      -0.242       0.173\n",
       "==============================================================================\n",
       "Omnibus:                       70.239   Durbin-Watson:                   1.732\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):                7.918\n",
       "Skew:                          -0.153   Prob(JB):                       0.0191\n",
       "Kurtosis:                       1.656   Cond. No.                         4.15\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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I19Bsyc99wHspZAGl18svF/7v6oLu7sL/6wfDRURyI0jX0OdKH5vZn1NI+0yvD30IfvCD9du6u1/9r6enECR6eio/V+l3yn+v+P/UknNseonpFWPDYB87z9rMGUODlV+vwu+/6j0UvESkTZpZYmIDhVnC6fWhD8G73w1LS7C4WPh/7d/iIiwvv7K99Oe1x6dOFVoVlfYvfZ2lJQBW3OlZWuU1Vuhac3fcYaW7g85WLualQaE8aAQNVOXBpXxbrX1qvUel31fwEkmkIGMEP6GYOgp0AtuAdI8PXHFFNO/jDisr3PfjZ1g4tcCmLrClZWx5iVOzc2xghV8+f0v1ILK8DAsLhf9Lg07pvrWCUem/U6fqB7i19/IQl3OqF3RKtwfZJ+wAp+AlORCkRfDekp+XgeeLs4alHjPo6uL4SifDW89gqeSCYu4cm1mAvTtjLGAVKyvVg0qQoNPoPtUC3Pw8zMzUf52lpfCD19p4Uq1AVK01FCTAlQezWkErSKBU8GpI3hfiqxoIzGxL8cfydNFNxe6NE+EVK1tSN5uzsxP6+wv/0qJa8KrUqqq0T70uvvIWWaWW1fLyK8Gr/HdLyxFF8CoPJkFaRc0EuvJ9q/1+tQBX2q3ZEeoNE6taW4hvY18Xw4O9zC4sc+/h53N1t8FaLYJHKHQJVapaOHB+KCVKkaC1CM3mjEBag1e1llLQLsJagapaYFvbZ+01FxZgerpyYCt/ndXV8P4enZ2NB6ig3Yc1Atyzz05zPh30bejDu7rZ3N3N7AocmXyakT1n1x7ziil4tVvVQODu50VZkCQJcoFvpBah2ZxSUWdn4V9fX9wlCW5lpXKAqBR06gWv8gDXyOssLLzS8qoWBNd+rhO89iyt0GG2rsbrwKo7dHfW/nt0dUFXF8td3cy5sdTRSUdvLxv6e+nZ0Be8269WgDv/fHjXu1r+6GoeRpCdzOwMCvcJOH3GuvsDYRUqTkEv8I3eOEKzOSUT1oJXb2/cJQludbVy11zx5x8dfo7FuQUGOxxbWsJWlpmfnaOPVUa3b6qdWbi4yPTMHEefm6LfV+hhheWFJVbm5tmxqZsN5usD3Oxs/QzF8uD1rnfFHwjM7D8Cv0chZfTHwKXAD4F/F2rJYhL0An9idpHhwfVfhoHeLo7PLERXWBGpr6OjULvuqTwmd8GZO05X/kq7bi/fcxYEqLx9//EJ5hZX1o0BTs8v8WRPJ/v2jjRX5rWWVzEFPWxBOrh+D/hF4Gl3fzvwJiCzd7EIul571DeOEJFwrHXd9vd0cnxmgf6ezoYGikO5x8Naq2twsPAvZEG6hubdfd7MMLNed/8nM3tt6CWLSdAMHw0Ai2RHK123qcsKrCBIi2DczIYo3FbyXjP7DvB0uMWKT9A7A7VaixCRbIj6bmJhMG8gl9nMfgXYDNzt7rHc2250dNTHxsZCfY/yrKGRTX1MvDyf28kmIlJbGiakmdkj7j5a6blaE8ruAm4Dvu3uMwDu/g/hFDFZSpuJeZ5skoaTW+Kn8yT9WYG1uoa+Avwq8KSZ3Wlmv2Zm6en0apPSLKIOMzb2dbOxr4uD45kdLwdeCYBziysMD/Yyt7jCvYefZ2JqLu6iSUJMTM1x+0NP8aff/SkPHX2RDjOdJylVNRC4+3fc/RpgJ/C/gQ8Cz5jZX5vZ5VEVMG6hZASkQF4DoASzVlH42fMzDA/00tFhPDZ+kqUV13mSQkHuR3AK+AbwDTN7A/A1CkGhzpS7bMhCRkAz0jJPQt0S8VirKCyvOpv6uop3LlzmqRdneOOOMxJ3nkhtdbOGzOwsM/uomX2fQubQPcDFoZcsIbKQEdCMNMyTUPdVfNZayht7u5lfKsyC7evqZHp+OXHnidRXa7D4w8A1wGspdA19wt1/UG3/rMrrOkFpmCcRZBa4WgzhWKso7BrewI+PnQTAWaWrwxJ3nkh9tbqG3gr8KXCfu4e45GDypT0joBlpCID1uq/ynPEVtrWKwsa+Lt6wfRNP/Ms0x2cW+aULtnLZa8/U3zdlaq0++qEoCyLJk/QAWG/8ptGFASW40orC3NIKbzl/q1pbKdbMPYtFEqFe91VaBrzTKukVBQmu3oSy33H3p6IrjjQqz33g9bqv8prxJdKoWi2Cvwb+zsy+BvyZu0ezHqoEpj7w2rXSNAx4iyRBrTGCb5rZd4H/AoyZ2deB1ZLnPx9B+WKX5Bq3+sBrS8OAt0gS1BsjWARmgV5gIyWBIAgz2wd8kcLks//p7p+ust+vA98CftHdw11RrgFJr3GrD7w+9WOnV5IrYVlTa4xgH/B54ABwcXGGcWBm1gncDFwOjAMPm9kBdz9ctt9GCje/eajBsocu6TVu9YFLViW9EpY1tWYWfxL4DXe/sdEgUHQJcMTdjxaXrL4DuKrCfn8CfAaYb+I9QpX0dYbyOutZsk9rXUWr1qJzv+zuh1p47XOAYyWPx4vbTjOzi4Ed7v63tV7IzK43szEzG5ucnGyhSI1J+jILujmOZFXSK2FZE9s8AjProND1dG29fd19P7AfCjemCbdkr0hD1on6wCWL1O0ZrSC3qmzWs8COksfbi9vWbAT2At8zs6eAS4EDZlbxDjpxUI1bJB7q9oxWmC2Ch4HdZnYehQBwNfD+tSfd/SQwvPbYzL4H/KckZQ2BatySfmnMvlHqb7RCCwTuvmxmN1BYtroTuMXdD5nZTcCYux8I671FpCDN2TeqhEUn1DECd78LuKts26eq7HtZmGWpJI01JYlOFs6PpKdASzKEOUaQaLqpidSSlfND2TcSRG4DgfKUpZasnB9JT4GWZMhtIFBNSWrJyvmh7BsJIreBQDUlqSUr54dSoCWI3N6YJg2TxZqRhQHOJMjS+aHsG6knty2CLNaUsjLAmQRhnB8TU3Pc/fgEtz30NHc/PqHPRRIjty0CyF5NSamC7dXO8yPN+fySfbltEWRRVgY4sygrWUiSTQoEGZKVAc4sUpCWJFMgyBClCiaXgrQkWa7HCLJGC3UlV70sJGV7SZwUCDImawPgWVErSGsgWeKmQCASkWpBWtleEjeNEYjETAPJEjcFApGYaSBZ4qauIZGA1gZ0n5ycZWpukc393Zy/bbDlgd0sLWch6aRAIBLA2oDuyqrzzIlTdHTAy3PL9HV38sLLCy0N7CrbKz2ymt2lQCASwNqA7j+/MM2G3k76u7uYW1zh+PQiu88abHlgV9leyZfl7C6NEYgEsDagOz2/TF9XJwB93R1MLyxpYDcnsrxMiAKBSABrA7ob+7qYX14BYH5plY293RrYzYksZ3cpEIgEsLZ8x7bBPk4trPDSqQVOLS4zvLFHy3jkRJazuxQIRAJYG9AdGerj3C0bGOztYseWfs4+vT3dfcRSX5bX8tJgcRVZzQ6Q5mlAN9+ynN2lQFBBlrMDRKR5Wa0MqGuogixnB4iIlFMgqCDL2QEiIuUUCCrIcnaAiEg5jRFUoLVfpN2UfCBJphZBBWvZAf09nRyfWaC/p1MDxdK0teSDucUVhgd7mVtc4d7DzzMxNRd30UQAtQiqymp2QBRU+10vbzee0eefPqG2CMxsn5k9YWZHzOzGCs9/3MwOm9ljZnafme0MszwSPtV+Xy1PyQf6/NMptBaBmXUCNwOXA+PAw2Z2wN0Pl+z2KDDq7qfM7CPAnwH/PqwyrVGNJTx5q/0G8co6Rd2nt2U1+SBNn7+uA68Is0VwCXDE3Y+6+yJwB3BV6Q7ufr+7nyo+fBDYHmJ5ANVYgpiYmuPuxye47aGnufvxiYb+Nnmq/QaV5aUJyqXl89d1YL0wA8E5wLGSx+PFbdVcB3y30hNmdr2ZjZnZ2OTkZEuF0mSx2lr9giQx9baVwNYOeUo+SOLnX4muA+slImvIzD4AjAKfrfS8u+9391F3H922bVtL75WWGktcWv2CJK32m5Sa38hQP/v2jvD+t+xk396RTAYBSN7nX42uA+uFGQieBXaUPN5e3LaOmb0T+CRwpbsvhFgeID01lri0+gVJWu1XNb9oJe3zr0bXgfXCTB99GNhtZudRCABXA+8v3cHM3gR8Bdjn7i+EWJbTNFmstnYMbCYp9fbE7CLDg73rtg30dnF8JvQ6R2q0e9A0SZ9/NboOrBdai8Ddl4EbgHuAnwJ3uvshM7vJzK4s7vZZYBD4ppn92MwOhFWeNWmpscQlLU37oFTzqy0pXWdR03VgPXP3uMvQkNHRUR8bG4u7GJmWpbS60iXFS2t+ef7Sl7r78QnmFlfWtQCn55fo7+lk396RGEsm7WZmj7j7aKXnNLNYXiUNTfugsnwzkXZQ15mAAoHkQJYCW7vlabKbVKdAIJJjeR40zVIXaKsSMY9ApF3injyWNnkdNM3rIHk1ahFIZuhe083JY9dZmtZEikLuAkF5c3BkUx8TL8+reZgB+nJLUBokXy9XXUPlzcGJqXn2P/Akz03NqXmYAVo2QILS/JL1chUIypcbmJyZZ2igi+PTi1p+IAP05ZagsjZxslW56hoqbw5Ozy+zua+b6YWl09vy3DxMuzxnwMQpjdk3ml+yXq4CQXnO9Ma+Lk7OLbG575Uao2qQ6aUvd/TSPECfx0HyanIVCMprjNsG+3j6+BwXbBtk1T1zNcg01tRapS93tDRAnw25GiMoz5keGerj+redx9lD/a/KoU57PrrypCUKGqDPhly1CKByjfGisn3S3Nxdo5qaRCHuJSry2OoNQ65aBEFl4WYmqqlJFOLMvlGrt30UCCrIwkVUqZQShTiXqMhChS0pctc1FETczd12UCqlRCWuAXrNDm4ftQgqyMJkk7wuJib5oVZv+6hFUEFW8tGVSilZplZv+ygQVKGLqEiyZaXCVknU2VAKBCI5lJW0yyxW2OJIX1cgEMmZLMyTaUZagl8cc4AUCDImLSd7I6I6piz+7SrJ0mTDoJ9ZmoJfHNlQyhrKkCxOsInqmLL4t6smC/NkoLHPLE1zDuLIhlIgyJA0nexBRXVMWfzbVZOVtMtGPrM0Bb840tcVCDIkTSd7UFEdUxb/dtVkYZ4MNPaZpSn4xTEHSGMEGZKFGdHlojqmLP7tqslK2mUjn1na5hxEnQ2lFkGGZKWmVyqqY8ri366WkaF+9u0d4f1v2cm+vSOpCwLQ2Gemmfa1mbvHXYaGjI6O+tjYWNzFSKwsZr4oa6g5WTueSvJwjO1iZo+4+2jF5xQIGqMTT9KgNF2ytCtEteDoJeWaUSsQqGuoAXlKMZR0y1MWVJKl5ZoRaiAws31m9oSZHTGzGys832tm3yg+/5CZ7QqzPK3Sl0vSIk9ZUEmWlmtGaIHAzDqBm4H3AHuAa8xsT9lu1wEvuftrgC8AnwmrPO2gL5ekRZrSJbMsLdeMMFsElwBH3P2ouy8CdwBXle1zFfC14s/fAt5hZhZimVqiL5ekRd6yoJIqLdeMMAPBOcCxksfjxW0V93H3ZeAk8KrEXjO73szGzGxscnIypOLWpy+XpIXSJZMhLdeMVEwoc/f9wH4oZA3FVY6sTMSRfMjiEs1pk5ZrRpiB4FlgR8nj7cVtlfYZN7MuYDPwYohlapm+XCLSiDRcM8LsGnoY2G1m55lZD3A1cKBsnwPAbxV/fh/w9562iQ0iIikXWovA3ZfN7AbgHqATuMXdD5nZTcCYux8A/gr4upkdAU5QCBYiIhKhUMcI3P0u4K6ybZ8q+Xke+I0wyyAiIrVpZrGISM4pEIiI5JwCgYhIzikQiIjknAKBiEjOpe5+BGY2CTzd5K8PA8fbWJyky9vxQv6OOW/HC/k75nYd705331bpidQFglaY2Vi1GzNkUd6OF/J3zHk7XsjfMUdxvOoaEhHJOQUCEZGcy1sg2B93ASKWt+OF/B1z3o4X8nfMoR9vrsYIRETk1fLWIhARkTIKBCIiOZfJQGBm+8zsCTM7YmY3Vni+18y+UXz+ITPbFX0p2yfA8X7czA6b2WNmdp+Z7YyjnO1S73hL9vt1M3MzS32qYZBjNrPfLH7Oh8zstqjL2E4Bzulzzex+M3u0eF5fEUc528XMbjGzF8zs8SrPm5n9RfHv8ZiZXdzWArh7pv5RuPfBz4HzgR7gILCnbJ/fAb5c/Plq4Btxlzvk4307sKH480eyfrzF/TYCDwAPAqNxlzuCz50i+8MAAATDSURBVHg38ChwRvHxmXGXO+Tj3Q98pPjzHuCpuMvd4jG/DbgYeLzK81cA3wUMuBR4qJ3vn8UWwSXAEXc/6u6LwB3AVWX7XAV8rfjzt4B3mJlFWMZ2qnu87n6/u58qPnyQwm1D0yrI5wvwJ8BngPkoCxeSIMf8YeBmd38JwN1fiLiM7RTkeB3YVPx5M/BchOVrO3d/gMLNuaq5CvgbL3gQGDKzkXa9fxYDwTnAsZLH48VtFfdx92XgJLA1ktK1X5DjLXUdhZpFWtU93mKzeYe7/22UBQtRkM/4QuBCM/u+mT1oZvsiK137BTnePwY+YGbjFG5+9dFoihabRr/nDQn1DmWSLGb2AWAU+JW4yxIWM+sAPg9cG3NRotZFoXvoMgotvgfM7PXuPhVrqcJzDXCru3/OzN5K4Za3e919Ne6CpVEWWwTPAjtKHm8vbqu4j5l1UWhavhhJ6dovyPFiZu8EPglc6e4LEZUtDPWOdyOwF/iemT1FoT/1QMoHjIN8xuPAAXdfcvcngZ9RCAxpFOR4rwPuBHD3HwJ9FBZny6pA3/NmZTEQPAzsNrPzzKyHwmDwgbJ9DgC/Vfz5fcDfe3FEJoXqHq+ZvQn4CoUgkOa+Y6hzvO5+0t2H3X2Xu++iMCZypbuPxVPctghyTn+bQmsAMxum0FV0NMpCtlGQ430GeAeAmb2OQiCYjLSU0ToAfLCYPXQpcNLdJ9r14pnrGnL3ZTO7AbiHQvbBLe5+yMxuAsbc/QDwVxSakkcoDNBcHV+JWxPweD8LDALfLI6JP+PuV8ZW6BYEPN5MCXjM9wDvMrPDwArwCXdPZSs34PH+AfBVM/t9CgPH16a4MoeZ3U4hkA8Xxz3+COgGcPcvUxgHuQI4ApwCfrut75/iv52IiLRBFruGRESkAQoEIiI5p0AgIpJzCgQiIjmnQCAiknMKBJJrZrbDzJ40sy3Fx2cUH+8q2+9+M3t32baPmdmX6rz+TLvLLNJuCgSSa+5+DPgS8Onipk8D+939qbJdb+fV802uLm4XSTUFAhH4AnCpmX0M+LfAn1fY51vArxZnulJsMZwN/F8zGyze5+FHZvYTM3vVaqhmdpmZ/Z+Sx//DzK4t/vxmM/sHM3vEzO5p56qSIkEoEEjuufsS8AkKAeFjxcfl+5wA/hF4T3HT1cCdxdms88CvufvFFO798Lmgy5qbWTfw34H3ufubgVuA/9riIYk0JHNLTIg06T3ABIUF6+6tss9a99B3iv9fV9xuwH8zs7cBqxSWBz4L+JcA7/vatfcsxo7OYjlEIqNAILlnZm8ELqewUun/M7M7qizo9R3gC8X7HWxw90eK2/8DsA14s7svFVc97Sv73WXWt8DXnjfgkLu/tT1HI9I4dQ1JrhW7cL5EoUvoGQoL9FUaI8DdZ4D7KXTflA4SbwZeKAaBtwOV7gn9NLDHCvfLHqK4cibwBLCtuKY+ZtZtZr/QhkMTCUyBQPLuwxRWY13rDvpL4HVmVu3mPbcDF7E+EPwvYNTMfgJ8EPin8l8qZifdCTxe/P/R4vZFCkuhf8bMDgI/Bn6p1YMSaYRWHxURyTm1CEREck6BQEQk5xQIRERyToFARCTnFAhERHJOgUBEJOcUCEREcu7/A+Jn8KERUtUOAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.random.rand(100)\n",
    "Y = np.random.rand(100)\n",
    "linreg(X, Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above shows a fairly uniform cloud of points. It is important to note that even with 100 samples, the line has a visible slope due to random chance. This is why it is crucial that you use statistical tests and not visualizations to verify your results.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Further Example\n",
    "\n",
    "Now let's make Y dependent on X plus some random noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.695</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.692</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   223.1</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 Jun 2020</td> <th>  Prob (F-statistic):</th> <td>5.34e-27</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>01:25:43</td>     <th>  Log-Likelihood:    </th> <td>  22.245</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   100</td>      <th>  AIC:               </th> <td>  -40.49</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    98</td>      <th>  BIC:               </th> <td>  -35.28</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>    0.0240</td> <td>    0.037</td> <td>    0.654</td> <td> 0.515</td> <td>   -0.049</td> <td>    0.097</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.9734</td> <td>    0.065</td> <td>   14.936</td> <td> 0.000</td> <td>    0.844</td> <td>    1.103</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 3.749</td> <th>  Durbin-Watson:     </th> <td>   1.763</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.153</td> <th>  Jarque-Bera (JB):  </th> <td>   2.115</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.059</td> <th>  Prob(JB):          </th> <td>   0.347</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.298</td> <th>  Cond. No.          </th> <td>    4.15</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<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:                      y   R-squared:                       0.695\n",
       "Model:                            OLS   Adj. R-squared:                  0.692\n",
       "Method:                 Least Squares   F-statistic:                     223.1\n",
       "Date:                Mon, 08 Jun 2020   Prob (F-statistic):           5.34e-27\n",
       "Time:                        01:25:43   Log-Likelihood:                 22.245\n",
       "No. Observations:                 100   AIC:                            -40.49\n",
       "Df Residuals:                      98   BIC:                            -35.28\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.0240      0.037      0.654      0.515      -0.049       0.097\n",
       "x1             0.9734      0.065     14.936      0.000       0.844       1.103\n",
       "==============================================================================\n",
       "Omnibus:                        3.749   Durbin-Watson:                   1.763\n",
       "Prob(Omnibus):                  0.153   Jarque-Bera (JB):                2.115\n",
       "Skew:                           0.059   Prob(JB):                        0.347\n",
       "Kurtosis:                       2.298   Cond. No.                         4.15\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate ys correlated with xs by adding normally-destributed errors\n",
    "Y = X + 0.2*np.random.randn(100)\n",
    "\n",
    "linreg(X,Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a situation like this one, the line of best fit does indeed model the dependent variable Y quite well (with a high  R-squared\n",
    "  value).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evaluating and Reporting Results\n",
    "\n",
    "The regression model relies on several assumptions:\n",
    "\n",
    "- The independent variable is not random.\n",
    "- The variance of the error term is constant across observations. This is important for evaluating the goodness of the fit.\n",
    "- The errors are not autocorrelated. The Durbin-Watson statistic detects this; if it is close to 2, there is no autocorrelation.\n",
    "- The errors are normally distributed. If this does not hold, we cannot use some of the statistics, such as the F-test.\n",
    "\n",
    "If we confirm that the necessary assumptions of the regression model are satisfied, we can safely use the statistics reported to analyze the fit. For example, the  R-squared\n",
    "  value tells us the fraction of the total variation of  Y\n",
    "  that is explained by the model.\n",
    "\n",
    "When making a prediction based on the model, it's useful to report not only a single value but a confidence interval. The linear regression reports 95% confidence intervals for the regression parameters, and we can visualize what this means using the seaborn library, which plots the regression line and highlights the 95% (by default) confidence interval for the regression line."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
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 },
 "nbformat": 4,
 "nbformat_minor": 2
}