Lecture/STA130_F23_C8.ipynb

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    "# Multiple Linear Regression (also called Multi-Linear or ~~Multivariate~~ regression)\n",
    "- Previously, we did **Simple linear regression**\n",
    "- This time...\n",
    "\n",
    "$$ AmazonPrice = \\beta_0 + \\beta_1 ListPrice + \\beta_H Height + \\beta_W Width + \\beta_T \\underbrace{I(HardorPaper == H)}_{\\text{indicator function}}$$ \n",
    "\n",
    "The indicator function is a $1$ if the book type is hardback.\n",
    "\n",
    "$$ y = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + \\beta_3 x_3 + \\beta_4 \\underbrace{I(x_4 == H)}_{\\text{indicator function}}$$ \n",
    "\n",
    "- The $x's$ are called **covariates** (in statistics) or **features** (in machine learning)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "bcb37372",
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   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Title</th>\n",
       "      <th>Author</th>\n",
       "      <th>List Price</th>\n",
       "      <th>Amazon Price</th>\n",
       "      <th>Hard_or_Paper</th>\n",
       "      <th>NumPages</th>\n",
       "      <th>Publisher</th>\n",
       "      <th>Pub year</th>\n",
       "      <th>ISBN-10</th>\n",
       "      <th>Height</th>\n",
       "      <th>Width</th>\n",
       "      <th>Thick</th>\n",
       "      <th>Weight_oz</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1,001 Facts that Will Scare the S#*t Out of Yo...</td>\n",
       "      <td>Cary McNeal</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>304.0</td>\n",
       "      <td>Adams Media</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>1605506249</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.8</td>\n",
       "      <td>11.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>21: Bringing Down the House - Movie Tie-In: Th...</td>\n",
       "      <td>Ben Mezrich</td>\n",
       "      <td>15.00</td>\n",
       "      <td>10.20</td>\n",
       "      <td>P</td>\n",
       "      <td>273.0</td>\n",
       "      <td>Free Press</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>1416564195</td>\n",
       "      <td>8.4</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.7</td>\n",
       "      <td>7.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>100 Best-Loved Poems (Dover Thrift Editions)</td>\n",
       "      <td>Smith</td>\n",
       "      <td>1.50</td>\n",
       "      <td>1.50</td>\n",
       "      <td>P</td>\n",
       "      <td>96.0</td>\n",
       "      <td>Dover Publications</td>\n",
       "      <td>1995.0</td>\n",
       "      <td>486285537</td>\n",
       "      <td>8.3</td>\n",
       "      <td>5.2</td>\n",
       "      <td>0.3</td>\n",
       "      <td>4.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1421: The Year China Discovered America</td>\n",
       "      <td>Gavin Menzies</td>\n",
       "      <td>15.99</td>\n",
       "      <td>10.87</td>\n",
       "      <td>P</td>\n",
       "      <td>672.0</td>\n",
       "      <td>Harper Perennial</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>61564893</td>\n",
       "      <td>8.8</td>\n",
       "      <td>6.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>28.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1493: Uncovering the New World Columbus Created</td>\n",
       "      <td>Charles C. Mann</td>\n",
       "      <td>30.50</td>\n",
       "      <td>16.77</td>\n",
       "      <td>P</td>\n",
       "      <td>720.0</td>\n",
       "      <td>Knopf</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>307265722</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>1.4</td>\n",
       "      <td>22.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>320</th>\n",
       "      <td>Where the Sidewalk Ends</td>\n",
       "      <td>Shel Silverstein</td>\n",
       "      <td>18.99</td>\n",
       "      <td>12.24</td>\n",
       "      <td>H</td>\n",
       "      <td>192.0</td>\n",
       "      <td>HarperCollins</td>\n",
       "      <td>2004.0</td>\n",
       "      <td>60572345</td>\n",
       "      <td>9.3</td>\n",
       "      <td>6.6</td>\n",
       "      <td>1.1</td>\n",
       "      <td>24.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>321</th>\n",
       "      <td>White Privilege</td>\n",
       "      <td>Paula S. Rothenberg</td>\n",
       "      <td>27.55</td>\n",
       "      <td>27.55</td>\n",
       "      <td>P</td>\n",
       "      <td>160.0</td>\n",
       "      <td>Worth Publishers</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1429233443</td>\n",
       "      <td>9.1</td>\n",
       "      <td>6.1</td>\n",
       "      <td>0.7</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>322</th>\n",
       "      <td>Why I wore lipstick</td>\n",
       "      <td>Geralyn Lucas</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>224.0</td>\n",
       "      <td>St Martin's Griffin</td>\n",
       "      <td>2005.0</td>\n",
       "      <td>031233446X</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>6.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>323</th>\n",
       "      <td>Worlds Together, Worlds Apart: A History of th...</td>\n",
       "      <td>Robert Tignor</td>\n",
       "      <td>97.50</td>\n",
       "      <td>97.50</td>\n",
       "      <td>P</td>\n",
       "      <td>480.0</td>\n",
       "      <td>W. W. Norton &amp; Company</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>393934942</td>\n",
       "      <td>10.7</td>\n",
       "      <td>8.9</td>\n",
       "      <td>0.9</td>\n",
       "      <td>14.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>324</th>\n",
       "      <td>Wuthering Heights</td>\n",
       "      <td>Emily Bronte</td>\n",
       "      <td>16.99</td>\n",
       "      <td>4.95</td>\n",
       "      <td>P</td>\n",
       "      <td>344.0</td>\n",
       "      <td>CreatSpace</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1463533411</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>16.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>325 rows × 13 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                 Title               Author  \\\n",
       "0    1,001 Facts that Will Scare the S#*t Out of Yo...          Cary McNeal   \n",
       "1    21: Bringing Down the House - Movie Tie-In: Th...          Ben Mezrich   \n",
       "2         100 Best-Loved Poems (Dover Thrift Editions)                Smith   \n",
       "3              1421: The Year China Discovered America        Gavin Menzies   \n",
       "4      1493: Uncovering the New World Columbus Created      Charles C. Mann   \n",
       "..                                                 ...                  ...   \n",
       "320                            Where the Sidewalk Ends     Shel Silverstein   \n",
       "321                                    White Privilege  Paula S. Rothenberg   \n",
       "322                                Why I wore lipstick        Geralyn Lucas   \n",
       "323  Worlds Together, Worlds Apart: A History of th...        Robert Tignor   \n",
       "324                                  Wuthering Heights         Emily Bronte   \n",
       "\n",
       "     List Price  Amazon Price Hard_or_Paper  NumPages               Publisher  \\\n",
       "0         12.95          5.18             P     304.0             Adams Media   \n",
       "1         15.00         10.20             P     273.0              Free Press   \n",
       "2          1.50          1.50             P      96.0      Dover Publications   \n",
       "3         15.99         10.87             P     672.0        Harper Perennial   \n",
       "4         30.50         16.77             P     720.0                   Knopf   \n",
       "..          ...           ...           ...       ...                     ...   \n",
       "320       18.99         12.24             H     192.0           HarperCollins   \n",
       "321       27.55         27.55             P     160.0        Worth Publishers   \n",
       "322       12.95          5.18             P     224.0     St Martin's Griffin   \n",
       "323       97.50         97.50             P     480.0  W. W. Norton & Company   \n",
       "324       16.99          4.95             P     344.0              CreatSpace   \n",
       "\n",
       "     Pub year     ISBN-10  Height  Width  Thick  Weight_oz  \n",
       "0      2010.0  1605506249     7.8    5.5    0.8       11.2  \n",
       "1      2008.0  1416564195     8.4    5.5    0.7        7.2  \n",
       "2      1995.0   486285537     8.3    5.2    0.3        4.0  \n",
       "3      2008.0    61564893     8.8    6.0    1.6       28.8  \n",
       "4      2011.0   307265722     8.0    5.2    1.4       22.4  \n",
       "..        ...         ...     ...    ...    ...        ...  \n",
       "320    2004.0    60572345     9.3    6.6    1.1       24.0  \n",
       "321    2011.0  1429233443     9.1    6.1    0.7        8.0  \n",
       "322    2005.0  031233446X     8.0    5.4    0.7        6.4  \n",
       "323    2010.0   393934942    10.7    8.9    0.9       14.4  \n",
       "324    2011.0  1463533411     7.8    5.1    1.0       16.0  \n",
       "\n",
       "[325 rows x 13 columns]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "amazonbooks = pd.read_csv(\"amazonbooks.csv\", encoding=\"ISO-8859-1\")\n",
    "amazonbooks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "6636778c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.formula.api as smf\n",
    "fit_model = smf.ols(formula='Q(\"Amazon Price\") ~ Q(\"List Price\") + Height + Width + C(Hard_or_Paper)', \n",
    "                    data=amazonbooks).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8d40bd75",
   "metadata": {
    "scrolled": true
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    {
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       " 't_test_pairwise',\n",
       " 'tvalues',\n",
       " 'uncentered_tss',\n",
       " 'use_t',\n",
       " 'wald_test',\n",
       " 'wald_test_terms',\n",
       " 'wresid']"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(fit_model)"
   ]
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   "cell_type": "code",
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       " 'whiten']"
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   "source": [
    "dir(fit_model.model)"
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  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "b827b2be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Q(\"List Price\") + Height + Width + C(Hard_or_Paper)'"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model.model.formula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "584e7bbc",
   "metadata": {},
   "source": [
    "$$ AmazonPrice = \\beta_0 + \\beta_1 ListPrice + \\require{enclose} \\enclose{horizontalstrike}{\\beta_H Height + \\beta_W Width} + \\beta_P \\underbrace{I(HardorPaper == P)}_{\\text{indicator function}}$$ \n",
    "\n",
    "The p-values in the table below are for the null hypothesis that the coefficient if 0. If a p-value is not quite small it does not provide evidence against this null hypthesis. Since the `Height` and `Width` variables do not have small p-values, we do not have evidence against the assumption that the coefficients of these variables in the linear model form are $0$.  Therefore, we remove these variables from the model, because if the coefficients were zero (an assumption we have not provided evidence against), then $0$ times those variables is just $0$... and they disappear from the model..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "b522a8ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.911</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.909</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   799.5</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>3.32e-163</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:23:50</td>      <th>  Log-Likelihood:    </th> <td> -873.34</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   319</td>       <th>  AIC:               </th> <td>   1757.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   314</td>       <th>  BIC:               </th> <td>   1776.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     4</td>       <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>     <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "            <td></td>               <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>             <td>   -2.7254</td> <td>    2.371</td> <td>   -1.150</td> <td> 0.251</td> <td>   -7.390</td> <td>    1.939</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[T.P]</th> <td>    2.0305</td> <td>    0.498</td> <td>    4.081</td> <td> 0.000</td> <td>    1.051</td> <td>    3.009</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>       <td>    0.8463</td> <td>    0.018</td> <td>   46.533</td> <td> 0.000</td> <td>    0.811</td> <td>    0.882</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Height</th>                <td>   -0.3326</td> <td>    0.289</td> <td>   -1.151</td> <td> 0.251</td> <td>   -0.901</td> <td>    0.236</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Width</th>                 <td>    0.2316</td> <td>    0.306</td> <td>    0.756</td> <td> 0.450</td> <td>   -0.371</td> <td>    0.834</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>100.776</td> <th>  Durbin-Watson:     </th> <td>   1.968</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1617.753</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.818</td>  <th>  Prob(JB):          </th> <td>    0.00</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>13.910</td>  <th>  Cond. No.          </th> <td>    282.</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:      Q(\"Amazon Price\")   R-squared:                       0.911\n",
       "Model:                            OLS   Adj. R-squared:                  0.909\n",
       "Method:                 Least Squares   F-statistic:                     799.5\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          3.32e-163\n",
       "Time:                        14:23:50   Log-Likelihood:                -873.34\n",
       "No. Observations:                 319   AIC:                             1757.\n",
       "Df Residuals:                     314   BIC:                             1776.\n",
       "Df Model:                           4                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "Intercept                -2.7254      2.371     -1.150      0.251      -7.390       1.939\n",
       "C(Hard_or_Paper)[T.P]     2.0305      0.498      4.081      0.000       1.051       3.009\n",
       "Q(\"List Price\")           0.8463      0.018     46.533      0.000       0.811       0.882\n",
       "Height                   -0.3326      0.289     -1.151      0.251      -0.901       0.236\n",
       "Width                     0.2316      0.306      0.756      0.450      -0.371       0.834\n",
       "==============================================================================\n",
       "Omnibus:                      100.776   Durbin-Watson:                   1.968\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1617.753\n",
       "Skew:                           0.818   Prob(JB):                         0.00\n",
       "Kurtosis:                      13.910   Cond. No.                         282.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "8342cade",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.903</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.903</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   3002.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>2.82e-165</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:24:27</td>      <th>  Log-Likelihood:    </th> <td> -897.98</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   324</td>       <th>  AIC:               </th> <td>   1800.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   322</td>       <th>  BIC:               </th> <td>   1808.</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>Intercept</th>       <td>   -2.4070</td> <td>    0.354</td> <td>   -6.791</td> <td> 0.000</td> <td>   -3.104</td> <td>   -1.710</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th> <td>    0.8298</td> <td>    0.015</td> <td>   54.789</td> <td> 0.000</td> <td>    0.800</td> <td>    0.860</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>114.617</td> <th>  Durbin-Watson:     </th> <td>   1.978</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1830.463</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.992</td>  <th>  Prob(JB):          </th> <td>    0.00</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>14.474</td>  <th>  Cond. No.          </th> <td>    38.5</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:      Q(\"Amazon Price\")   R-squared:                       0.903\n",
       "Model:                            OLS   Adj. R-squared:                  0.903\n",
       "Method:                 Least Squares   F-statistic:                     3002.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          2.82e-165\n",
       "Time:                        14:24:27   Log-Likelihood:                -897.98\n",
       "No. Observations:                 324   AIC:                             1800.\n",
       "Df Residuals:                     322   BIC:                             1808.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "===================================================================================\n",
       "                      coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------\n",
       "Intercept          -2.4070      0.354     -6.791      0.000      -3.104      -1.710\n",
       "Q(\"List Price\")     0.8298      0.015     54.789      0.000       0.800       0.860\n",
       "==============================================================================\n",
       "Omnibus:                      114.617   Durbin-Watson:                   1.978\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1830.463\n",
       "Skew:                           0.992   Prob(JB):                         0.00\n",
       "Kurtosis:                      14.474   Cond. No.                         38.5\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# this was last time...\n",
    "smf.ols(formula='Q(\"Amazon Price\") ~ Q(\"List Price\")', data=amazonbooks).fit().summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "a1b9aa04",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.921</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.920</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   719.6</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>1.67e-167</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:35:10</td>      <th>  Log-Likelihood:    </th> <td> -813.63</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   314</td>       <th>  AIC:               </th> <td>   1639.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   308</td>       <th>  BIC:               </th> <td>   1662.</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>Intercept</th>             <td>   -0.8980</td> <td>    0.812</td> <td>   -1.106</td> <td> 0.270</td> <td>   -2.496</td> <td>    0.700</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[T.P]</th> <td>    1.2895</td> <td>    0.542</td> <td>    2.380</td> <td> 0.018</td> <td>    0.223</td> <td>    2.356</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>       <td>    0.8552</td> <td>    0.016</td> <td>   54.369</td> <td> 0.000</td> <td>    0.824</td> <td>    0.886</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                 <td>   -1.8302</td> <td>    1.133</td> <td>   -1.615</td> <td> 0.107</td> <td>   -4.060</td> <td>    0.400</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Weight_oz</th>             <td>   -0.0664</td> <td>    0.044</td> <td>   -1.519</td> <td> 0.130</td> <td>   -0.153</td> <td>    0.020</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>NumPages</th>              <td>   -0.0012</td> <td>    0.002</td> <td>   -0.491</td> <td> 0.624</td> <td>   -0.006</td> <td>    0.004</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>121.423</td> <th>  Durbin-Watson:     </th> <td>   2.118</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1675.391</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 1.177</td>  <th>  Prob(JB):          </th> <td>    0.00</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>14.068</td>  <th>  Cond. No.          </th> <td>2.67e+03</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 2.67e+03. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:      Q(\"Amazon Price\")   R-squared:                       0.921\n",
       "Model:                            OLS   Adj. R-squared:                  0.920\n",
       "Method:                 Least Squares   F-statistic:                     719.6\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          1.67e-167\n",
       "Time:                        14:35:10   Log-Likelihood:                -813.63\n",
       "No. Observations:                 314   AIC:                             1639.\n",
       "Df Residuals:                     308   BIC:                             1662.\n",
       "Df Model:                           5                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "Intercept                -0.8980      0.812     -1.106      0.270      -2.496       0.700\n",
       "C(Hard_or_Paper)[T.P]     1.2895      0.542      2.380      0.018       0.223       2.356\n",
       "Q(\"List Price\")           0.8552      0.016     54.369      0.000       0.824       0.886\n",
       "Thick                    -1.8302      1.133     -1.615      0.107      -4.060       0.400\n",
       "Weight_oz                -0.0664      0.044     -1.519      0.130      -0.153       0.020\n",
       "NumPages                 -0.0012      0.002     -0.491      0.624      -0.006       0.004\n",
       "==============================================================================\n",
       "Omnibus:                      121.423   Durbin-Watson:                   2.118\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1675.391\n",
       "Skew:                           1.177   Prob(JB):                         0.00\n",
       "Kurtosis:                      14.068   Cond. No.                     2.67e+03\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 2.67e+03. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model2 = smf.ols(formula='Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + Weight_oz + NumPages + C(Hard_or_Paper)', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model2.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "bc3c89ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Q(\"List Price\") + Height + Width + C(Hard_or_Paper)'"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model.model.formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "d3864e3d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + Weight_oz + NumPages + C(Hard_or_Paper)'"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model2.model.formula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e61f660f",
   "metadata": {},
   "source": [
    "![](https://www.jcpcarchives.org/userfiles/values-of-p-Inference.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "29237bd9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.921</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.920</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   902.3</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>2.37e-169</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:38:55</td>      <th>  Log-Likelihood:    </th> <td> -816.27</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   315</td>       <th>  AIC:               </th> <td>   1643.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   310</td>       <th>  BIC:               </th> <td>   1661.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     4</td>       <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>     <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "            <td></td>               <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>             <td>   -0.6885</td> <td>    0.736</td> <td>   -0.935</td> <td> 0.350</td> <td>   -2.137</td> <td>    0.760</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[T.P]</th> <td>    1.1079</td> <td>    0.450</td> <td>    2.463</td> <td> 0.014</td> <td>    0.223</td> <td>    1.993</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>       <td>    0.8540</td> <td>    0.016</td> <td>   54.641</td> <td> 0.000</td> <td>    0.823</td> <td>    0.885</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                 <td>   -2.3027</td> <td>    0.737</td> <td>   -3.124</td> <td> 0.002</td> <td>   -3.753</td> <td>   -0.853</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Weight_oz</th>             <td>   -0.0672</td> <td>    0.043</td> <td>   -1.580</td> <td> 0.115</td> <td>   -0.151</td> <td>    0.016</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>122.742</td> <th>  Durbin-Watson:     </th> <td>   2.114</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1760.075</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 1.178</td>  <th>  Prob(JB):          </th> <td>    0.00</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>14.338</td>  <th>  Cond. No.          </th> <td>    132.</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:      Q(\"Amazon Price\")   R-squared:                       0.921\n",
       "Model:                            OLS   Adj. R-squared:                  0.920\n",
       "Method:                 Least Squares   F-statistic:                     902.3\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          2.37e-169\n",
       "Time:                        14:38:55   Log-Likelihood:                -816.27\n",
       "No. Observations:                 315   AIC:                             1643.\n",
       "Df Residuals:                     310   BIC:                             1661.\n",
       "Df Model:                           4                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "Intercept                -0.6885      0.736     -0.935      0.350      -2.137       0.760\n",
       "C(Hard_or_Paper)[T.P]     1.1079      0.450      2.463      0.014       0.223       1.993\n",
       "Q(\"List Price\")           0.8540      0.016     54.641      0.000       0.823       0.885\n",
       "Thick                    -2.3027      0.737     -3.124      0.002      -3.753      -0.853\n",
       "Weight_oz                -0.0672      0.043     -1.580      0.115      -0.151       0.016\n",
       "==============================================================================\n",
       "Omnibus:                      122.742   Durbin-Watson:                   2.114\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1760.075\n",
       "Skew:                           1.178   Prob(JB):                         0.00\n",
       "Kurtosis:                      14.338   Cond. No.                         132.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model3 = smf.ols(formula='Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + Weight_oz + C(Hard_or_Paper)', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model3.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "215736e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   1280.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>1.93e-177</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:39:29</td>      <th>  Log-Likelihood:    </th> <td> -855.77</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th> <td>   1720.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   319</td>       <th>  BIC:               </th> <td>   1735.</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>Intercept</th>             <td>   -1.0255</td> <td>    0.766</td> <td>   -1.339</td> <td> 0.181</td> <td>   -2.532</td> <td>    0.481</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[T.P]</th> <td>    1.1440</td> <td>    0.464</td> <td>    2.464</td> <td> 0.014</td> <td>    0.231</td> <td>    2.057</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>       <td>    0.8663</td> <td>    0.014</td> <td>   61.780</td> <td> 0.000</td> <td>    0.839</td> <td>    0.894</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                 <td>   -3.1085</td> <td>    0.561</td> <td>   -5.541</td> <td> 0.000</td> <td>   -4.212</td> <td>   -2.005</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>101.699</td> <th>  Durbin-Watson:     </th> <td>   2.083</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1295.817</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.897</td>  <th>  Prob(JB):          </th> <td>4.14e-282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>12.647</td>  <th>  Cond. No.          </th> <td>    115.</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:      Q(\"Amazon Price\")   R-squared:                       0.923\n",
       "Model:                            OLS   Adj. R-squared:                  0.923\n",
       "Method:                 Least Squares   F-statistic:                     1280.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          1.93e-177\n",
       "Time:                        14:39:29   Log-Likelihood:                -855.77\n",
       "No. Observations:                 323   AIC:                             1720.\n",
       "Df Residuals:                     319   BIC:                             1735.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "Intercept                -1.0255      0.766     -1.339      0.181      -2.532       0.481\n",
       "C(Hard_or_Paper)[T.P]     1.1440      0.464      2.464      0.014       0.231       2.057\n",
       "Q(\"List Price\")           0.8663      0.014     61.780      0.000       0.839       0.894\n",
       "Thick                    -3.1085      0.561     -5.541      0.000      -4.212      -2.005\n",
       "==============================================================================\n",
       "Omnibus:                      101.699   Durbin-Watson:                   2.083\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1295.817\n",
       "Skew:                           0.897   Prob(JB):                    4.14e-282\n",
       "Kurtosis:                      12.647   Cond. No.                         115.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model4 = smf.ols(formula='Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + C(Hard_or_Paper)', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model4.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "b8ddba14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + C(Hard_or_Paper)'"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model4.model.formula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7260d89f",
   "metadata": {},
   "source": [
    "$$AmazonPrice =  \\enclose{horizontalstrike}{\\beta_0}0 + \\beta_L ListPrice + \\beta_T Thick + \\beta_P I(Hard\\_or\\_Paper==P) + \\beta_P I(Hard\\_or\\_Paper==H)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05af0023",
   "metadata": {},
   "source": [
    "Using the `-1` notation in the formula didn't actually quite do what I wanted!\n",
    "I wanted to have not intercept and just an indicator of paperback, like this (as opposed to the above):\n",
    "\n",
    "$$AmazonPrice =  \\enclose{horizontalstrike}{\\beta_0}0 + \\beta_L ListPrice + \\beta_T Thick + \\beta_P I(Hard\\_or\\_Paper==P)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "c1489075",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   1280.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>1.93e-177</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:48:23</td>      <th>  Log-Likelihood:    </th> <td> -855.77</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th> <td>   1720.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   319</td>       <th>  BIC:               </th> <td>   1735.</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>C(Hard_or_Paper)[H]</th> <td>   -1.0255</td> <td>    0.766</td> <td>   -1.339</td> <td> 0.181</td> <td>   -2.532</td> <td>    0.481</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[P]</th> <td>    0.1185</td> <td>    0.547</td> <td>    0.217</td> <td> 0.829</td> <td>   -0.958</td> <td>    1.195</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>     <td>    0.8663</td> <td>    0.014</td> <td>   61.780</td> <td> 0.000</td> <td>    0.839</td> <td>    0.894</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>               <td>   -3.1085</td> <td>    0.561</td> <td>   -5.541</td> <td> 0.000</td> <td>   -4.212</td> <td>   -2.005</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>101.699</td> <th>  Durbin-Watson:     </th> <td>   2.083</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1295.817</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.897</td>  <th>  Prob(JB):          </th> <td>4.14e-282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>12.647</td>  <th>  Cond. No.          </th> <td>    123.</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:      Q(\"Amazon Price\")   R-squared:                       0.923\n",
       "Model:                            OLS   Adj. R-squared:                  0.923\n",
       "Method:                 Least Squares   F-statistic:                     1280.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          1.93e-177\n",
       "Time:                        14:48:23   Log-Likelihood:                -855.77\n",
       "No. Observations:                 323   AIC:                             1720.\n",
       "Df Residuals:                     319   BIC:                             1735.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=======================================================================================\n",
       "                          coef    std err          t      P>|t|      [0.025      0.975]\n",
       "---------------------------------------------------------------------------------------\n",
       "C(Hard_or_Paper)[H]    -1.0255      0.766     -1.339      0.181      -2.532       0.481\n",
       "C(Hard_or_Paper)[P]     0.1185      0.547      0.217      0.829      -0.958       1.195\n",
       "Q(\"List Price\")         0.8663      0.014     61.780      0.000       0.839       0.894\n",
       "Thick                  -3.1085      0.561     -5.541      0.000      -4.212      -2.005\n",
       "==============================================================================\n",
       "Omnibus:                      101.699   Durbin-Watson:                   2.083\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1295.817\n",
       "Skew:                           0.897   Prob(JB):                    4.14e-282\n",
       "Kurtosis:                      12.647   Cond. No.                         123.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model5 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + C(Hard_or_Paper)', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model5.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "07098a10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Q(\"List Price\") + Thick + C(Hard_or_Paper)'"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model4.model.formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "77447934",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   1280.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>1.93e-177</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:50:22</td>      <th>  Log-Likelihood:    </th> <td> -855.77</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th> <td>   1720.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   319</td>       <th>  BIC:               </th> <td>   1735.</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>Intercept</th>             <td>   -1.0255</td> <td>    0.766</td> <td>   -1.339</td> <td> 0.181</td> <td>   -2.532</td> <td>    0.481</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>C(Hard_or_Paper)[T.P]</th> <td>    1.1440</td> <td>    0.464</td> <td>    2.464</td> <td> 0.014</td> <td>    0.231</td> <td>    2.057</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>       <td>    0.8663</td> <td>    0.014</td> <td>   61.780</td> <td> 0.000</td> <td>    0.839</td> <td>    0.894</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                 <td>   -3.1085</td> <td>    0.561</td> <td>   -5.541</td> <td> 0.000</td> <td>   -4.212</td> <td>   -2.005</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>101.699</td> <th>  Durbin-Watson:     </th> <td>   2.083</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1295.817</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.897</td>  <th>  Prob(JB):          </th> <td>4.14e-282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>12.647</td>  <th>  Cond. No.          </th> <td>    115.</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:      Q(\"Amazon Price\")   R-squared:                       0.923\n",
       "Model:                            OLS   Adj. R-squared:                  0.923\n",
       "Method:                 Least Squares   F-statistic:                     1280.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          1.93e-177\n",
       "Time:                        14:50:22   Log-Likelihood:                -855.77\n",
       "No. Observations:                 323   AIC:                             1720.\n",
       "Df Residuals:                     319   BIC:                             1735.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "Intercept                -1.0255      0.766     -1.339      0.181      -2.532       0.481\n",
       "C(Hard_or_Paper)[T.P]     1.1440      0.464      2.464      0.014       0.231       2.057\n",
       "Q(\"List Price\")           0.8663      0.014     61.780      0.000       0.839       0.894\n",
       "Thick                    -3.1085      0.561     -5.541      0.000      -4.212      -2.005\n",
       "==============================================================================\n",
       "Omnibus:                      101.699   Durbin-Watson:                   2.083\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1295.817\n",
       "Skew:                           0.897   Prob(JB):                    4.14e-282\n",
       "Kurtosis:                      12.647   Cond. No.                         115.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model4.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "7ea442ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.923</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   1280.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>1.93e-177</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>14:57:02</td>      <th>  Log-Likelihood:    </th> <td> -855.77</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th> <td>   1720.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   319</td>       <th>  BIC:               </th> <td>   1735.</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>Hard_or_Paper == \"P\"[False]</th> <td>   -1.0255</td> <td>    0.766</td> <td>   -1.339</td> <td> 0.181</td> <td>   -2.532</td> <td>    0.481</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Hard_or_Paper == \"P\"[True]</th>  <td>    0.1185</td> <td>    0.547</td> <td>    0.217</td> <td> 0.829</td> <td>   -0.958</td> <td>    1.195</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\")</th>             <td>    0.8663</td> <td>    0.014</td> <td>   61.780</td> <td> 0.000</td> <td>    0.839</td> <td>    0.894</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                       <td>   -3.1085</td> <td>    0.561</td> <td>   -5.541</td> <td> 0.000</td> <td>   -4.212</td> <td>   -2.005</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>101.699</td> <th>  Durbin-Watson:     </th> <td>   2.083</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1295.817</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.897</td>  <th>  Prob(JB):          </th> <td>4.14e-282</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>12.647</td>  <th>  Cond. No.          </th> <td>    123.</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:      Q(\"Amazon Price\")   R-squared:                       0.923\n",
       "Model:                            OLS   Adj. R-squared:                  0.923\n",
       "Method:                 Least Squares   F-statistic:                     1280.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):          1.93e-177\n",
       "Time:                        14:57:02   Log-Likelihood:                -855.77\n",
       "No. Observations:                 323   AIC:                             1720.\n",
       "Df Residuals:                     319   BIC:                             1735.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "===============================================================================================\n",
       "                                  coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------------\n",
       "Hard_or_Paper == \"P\"[False]    -1.0255      0.766     -1.339      0.181      -2.532       0.481\n",
       "Hard_or_Paper == \"P\"[True]      0.1185      0.547      0.217      0.829      -0.958       1.195\n",
       "Q(\"List Price\")                 0.8663      0.014     61.780      0.000       0.839       0.894\n",
       "Thick                          -3.1085      0.561     -5.541      0.000      -4.212      -2.005\n",
       "==============================================================================\n",
       "Omnibus:                      101.699   Durbin-Watson:                   2.083\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1295.817\n",
       "Skew:                           0.897   Prob(JB):                    4.14e-282\n",
       "Kurtosis:                      12.647   Cond. No.                         123.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model6 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + (Hard_or_Paper==\"P\")', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model6.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "id": "1ef9c2ed",
   "metadata": {},
   "outputs": [],
   "source": [
    "amazonbooks['Paperback'] = (amazonbooks['Hard_or_Paper']=='P').astype(int)\n",
    "# edit -- I like using hardback better than paperback...\n",
    "# - there's more paperback, so having an indicator of hardback is more like a boost for the smaller book type\n",
    "# - Hardback has a better significance than paperback\n",
    "amazonbooks['Hardback'] = (amazonbooks['Hard_or_Paper']=='H').astype(int)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "id": "be48402e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared (uncentered):</th>      <td>   0.963</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared (uncentered):</th> <td>   0.963</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th>          <td>   2805.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th>          <td>2.37e-229</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:49:31</td>      <th>  Log-Likelihood:    </th>          <td> -855.79</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th>          <td>   1718.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   320</td>       <th>  BIC:               </th>          <td>   1729.</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>Q(\"List Price\")</th> <td>    0.8673</td> <td>    0.013</td> <td>   66.095</td> <td> 0.000</td> <td>    0.842</td> <td>    0.893</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>           <td>   -3.0112</td> <td>    0.336</td> <td>   -8.968</td> <td> 0.000</td> <td>   -3.672</td> <td>   -2.351</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Hardback</th>        <td>   -1.1581</td> <td>    0.459</td> <td>   -2.523</td> <td> 0.012</td> <td>   -2.061</td> <td>   -0.255</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>99.007</td> <th>  Durbin-Watson:     </th> <td>   2.079</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>1262.070</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.860</td> <th>  Prob(JB):          </th> <td>8.81e-275</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>12.530</td> <th>  Cond. No.          </th> <td>    59.1</td> \n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] R² is computed without centering (uncentered) since the model does not contain a constant.<br/>[2] 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:      Q(\"Amazon Price\")   R-squared (uncentered):                   0.963\n",
       "Model:                            OLS   Adj. R-squared (uncentered):              0.963\n",
       "Method:                 Least Squares   F-statistic:                              2805.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):                   2.37e-229\n",
       "Time:                        15:49:31   Log-Likelihood:                         -855.79\n",
       "No. Observations:                 323   AIC:                                      1718.\n",
       "Df Residuals:                     320   BIC:                                      1729.\n",
       "Df Model:                           3                                                  \n",
       "Covariance Type:            nonrobust                                                  \n",
       "===================================================================================\n",
       "                      coef    std err          t      P>|t|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------\n",
       "Q(\"List Price\")     0.8673      0.013     66.095      0.000       0.842       0.893\n",
       "Thick              -3.0112      0.336     -8.968      0.000      -3.672      -2.351\n",
       "Hardback           -1.1581      0.459     -2.523      0.012      -2.061      -0.255\n",
       "==============================================================================\n",
       "Omnibus:                       99.007   Durbin-Watson:                   2.079\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1262.070\n",
       "Skew:                           0.860   Prob(JB):                    8.81e-275\n",
       "Kurtosis:                      12.530   Cond. No.                         59.1\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] R² is computed without centering (uncentered) since the model does not contain a constant.\n",
       "[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#fit_model7 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + Paperback', \n",
    "fit_model7 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + Hardback', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model7.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "id": "74f9f77d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + Hardback'"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model7.model.formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "5c15453a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.232</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.220</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   18.62</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>3.62e-16</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:49:35</td>      <th>  Log-Likelihood:    </th> <td> -1208.2</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   314</td>       <th>  AIC:               </th> <td>   2428.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   308</td>       <th>  BIC:               </th> <td>   2451.</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>Intercept</th> <td>  -33.6768</td> <td>    8.112</td> <td>   -4.152</td> <td> 0.000</td> <td>  -49.638</td> <td>  -17.716</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>     <td>   -6.3227</td> <td>    2.771</td> <td>   -2.282</td> <td> 0.023</td> <td>  -11.775</td> <td>   -0.870</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Weight_oz</th> <td>    0.4550</td> <td>    0.166</td> <td>    2.741</td> <td> 0.006</td> <td>    0.128</td> <td>    0.782</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Height</th>    <td>    3.0343</td> <td>    0.936</td> <td>    3.241</td> <td> 0.001</td> <td>    1.192</td> <td>    4.877</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Width</th>     <td>    3.5401</td> <td>    0.952</td> <td>    3.718</td> <td> 0.000</td> <td>    1.667</td> <td>    5.414</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Paperback</th> <td>    2.9002</td> <td>    1.614</td> <td>    1.797</td> <td> 0.073</td> <td>   -0.276</td> <td>    6.076</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>376.037</td> <th>  Durbin-Watson:     </th> <td>   2.113</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>24357.299</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 5.355</td>  <th>  Prob(JB):          </th> <td>    0.00</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>44.797</td>  <th>  Cond. No.          </th> <td>    217.</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:      Q(\"Amazon Price\")   R-squared:                       0.232\n",
       "Model:                            OLS   Adj. R-squared:                  0.220\n",
       "Method:                 Least Squares   F-statistic:                     18.62\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):           3.62e-16\n",
       "Time:                        15:49:35   Log-Likelihood:                -1208.2\n",
       "No. Observations:                 314   AIC:                             2428.\n",
       "Df Residuals:                     308   BIC:                             2451.\n",
       "Df Model:                           5                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    -33.6768      8.112     -4.152      0.000     -49.638     -17.716\n",
       "Thick         -6.3227      2.771     -2.282      0.023     -11.775      -0.870\n",
       "Weight_oz      0.4550      0.166      2.741      0.006       0.128       0.782\n",
       "Height         3.0343      0.936      3.241      0.001       1.192       4.877\n",
       "Width          3.5401      0.952      3.718      0.000       1.667       5.414\n",
       "Paperback      2.9002      1.614      1.797      0.073      -0.276       6.076\n",
       "==============================================================================\n",
       "Omnibus:                      376.037   Durbin-Watson:                   2.113\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):            24357.299\n",
       "Skew:                           5.355   Prob(JB):                         0.00\n",
       "Kurtosis:                      44.797   Cond. No.                         217.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model8 = smf.ols(formula='Q(\"Amazon Price\") ~ Thick + Weight_oz + Height+ Width + Paperback', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model8.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "16964cbe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared:         </th> <td>   0.224</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared:    </th> <td>   0.214</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th> <td>   22.31</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th> <td>3.36e-16</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:07:46</td>      <th>  Log-Likelihood:    </th> <td> -1209.9</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   314</td>       <th>  AIC:               </th> <td>   2430.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   309</td>       <th>  BIC:               </th> <td>   2448.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     4</td>       <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>     <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  -28.6618</td> <td>    7.644</td> <td>   -3.750</td> <td> 0.000</td> <td>  -43.702</td> <td>  -13.622</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>     <td>   -7.3698</td> <td>    2.719</td> <td>   -2.711</td> <td> 0.007</td> <td>  -12.719</td> <td>   -2.020</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Weight_oz</th> <td>    0.4505</td> <td>    0.167</td> <td>    2.704</td> <td> 0.007</td> <td>    0.123</td> <td>    0.778</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Height</th>    <td>    3.0165</td> <td>    0.940</td> <td>    3.210</td> <td> 0.001</td> <td>    1.168</td> <td>    4.865</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Width</th>     <td>    3.2323</td> <td>    0.940</td> <td>    3.439</td> <td> 0.001</td> <td>    1.383</td> <td>    5.082</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>381.525</td> <th>  Durbin-Watson:     </th> <td>   2.076</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>25810.105</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 5.476</td>  <th>  Prob(JB):          </th> <td>    0.00</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>46.044</td>  <th>  Cond. No.          </th> <td>    204.</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:      Q(\"Amazon Price\")   R-squared:                       0.224\n",
       "Model:                            OLS   Adj. R-squared:                  0.214\n",
       "Method:                 Least Squares   F-statistic:                     22.31\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):           3.36e-16\n",
       "Time:                        15:07:46   Log-Likelihood:                -1209.9\n",
       "No. Observations:                 314   AIC:                             2430.\n",
       "Df Residuals:                     309   BIC:                             2448.\n",
       "Df Model:                           4                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    -28.6618      7.644     -3.750      0.000     -43.702     -13.622\n",
       "Thick         -7.3698      2.719     -2.711      0.007     -12.719      -2.020\n",
       "Weight_oz      0.4505      0.167      2.704      0.007       0.123       0.778\n",
       "Height         3.0165      0.940      3.210      0.001       1.168       4.865\n",
       "Width          3.2323      0.940      3.439      0.001       1.383       5.082\n",
       "==============================================================================\n",
       "Omnibus:                      381.525   Durbin-Watson:                   2.076\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):            25810.105\n",
       "Skew:                           5.476   Prob(JB):                         0.00\n",
       "Kurtosis:                      46.044   Cond. No.                         204.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model9 = smf.ols(formula='Q(\"Amazon Price\") ~ Thick + Weight_oz + Height + Width', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model9.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "4f37a538",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared (uncentered):</th>      <td>   0.592</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared (uncentered):</th> <td>   0.588</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th>          <td>   150.5</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th>          <td>2.91e-60</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:06:55</td>      <th>  Log-Likelihood:    </th>          <td> -1217.1</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   314</td>       <th>  AIC:               </th>          <td>   2440.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   311</td>       <th>  BIC:               </th>          <td>   2451.</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>Thick</th>     <td>  -11.7762</td> <td>    2.314</td> <td>   -5.089</td> <td> 0.000</td> <td>  -16.329</td> <td>   -7.223</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Weight_oz</th> <td>    0.7890</td> <td>    0.141</td> <td>    5.578</td> <td> 0.000</td> <td>    0.511</td> <td>    1.067</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Width</th>     <td>    2.4681</td> <td>    0.312</td> <td>    7.899</td> <td> 0.000</td> <td>    1.853</td> <td>    3.083</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>396.530</td> <th>  Durbin-Watson:     </th> <td>   2.073</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>28642.710</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 5.846</td>  <th>  Prob(JB):          </th> <td>    0.00</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>48.305</td>  <th>  Cond. No.          </th> <td>    52.8</td> \n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] R² is computed without centering (uncentered) since the model does not contain a constant.<br/>[2] 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:      Q(\"Amazon Price\")   R-squared (uncentered):                   0.592\n",
       "Model:                            OLS   Adj. R-squared (uncentered):              0.588\n",
       "Method:                 Least Squares   F-statistic:                              150.5\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):                    2.91e-60\n",
       "Time:                        15:06:55   Log-Likelihood:                         -1217.1\n",
       "No. Observations:                 314   AIC:                                      2440.\n",
       "Df Residuals:                     311   BIC:                                      2451.\n",
       "Df Model:                           3                                                  \n",
       "Covariance Type:            nonrobust                                                  \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Thick        -11.7762      2.314     -5.089      0.000     -16.329      -7.223\n",
       "Weight_oz      0.7890      0.141      5.578      0.000       0.511       1.067\n",
       "Width          2.4681      0.312      7.899      0.000       1.853       3.083\n",
       "==============================================================================\n",
       "Omnibus:                      396.530   Durbin-Watson:                   2.073\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):            28642.710\n",
       "Skew:                           5.846   Prob(JB):                         0.00\n",
       "Kurtosis:                      48.305   Cond. No.                         52.8\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] R² is computed without centering (uncentered) since the model does not contain a constant.\n",
       "[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model10 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Thick + Weight_oz + Width', \n",
    "                     data=amazonbooks).fit()\n",
    "fit_model10.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "id": "1f2f52ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") + Thick + Hardback'"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model7.model.formula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "696617e5",
   "metadata": {},
   "source": [
    "$$\\large \\hat y = \\beta_L ListPrice + \\beta_T Thick + \\beta_P Paperback$$\n",
    "\n",
    "$$\\large \\hat y = 0.86 ListPrice - 3.69 Thick + 0.70 Paperback$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "f33c3330",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.884999999999998"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the original paperback based model\n",
    "0.86 * 12.95 - 3.69 * 0.8 + 0.70 *1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "id": "3db2fcb5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Q(\"List Price\")    0.867349\n",
       "Thick             -3.011230\n",
       "Hardback          -1.158125\n",
       "dtype: float64"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model7.params"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25ce2012",
   "metadata": {},
   "source": [
    "$$\\large \\hat y = \\beta_L ListPrice + \\beta_T Thick + \\beta_P Hardback$$\n",
    "\n",
    "$$\\large \\hat y = 0.867 ListPrice - 3.91 Thick -1.158 Hardback$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "7da22c06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.8901407366203"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model7.predict()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "id": "3d51552c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.81965"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# model 7 changed to hardback\n",
    "0.867 * 12.95 - 3.01 * 0.8 -1.15 *0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "id": "b25bedb0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.823188806135462"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model7.predict()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "57ec555f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ Thick + Weight_oz + Height + Width'"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model9.model.formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "8c6e4661",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Q(\"Amazon Price\") ~ -1 + Thick + Weight_oz + Width'"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model10.model.formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "a04ffa0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept   -28.661822\n",
       "Thick        -7.369776\n",
       "Weight_oz     0.450477\n",
       "Height        3.016454\n",
       "Width         3.232274\n",
       "dtype: float64"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model9.params"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f75c0a25",
   "metadata": {},
   "source": [
    "$$\\large \\hat y = Intercept + \\beta_T Thick + \\beta_W Weight\\_oz + \\beta_H Height + \\beta_W Width $$\n",
    "\n",
    "$$\\large \\hat y = -28.66 - 7.36 Thick + 0.45 Weight\\_oz + 3.01 Height + 3.23 Width $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "9341f5d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.734999999999996"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# y-hat for the first row observation\n",
    "-28.66 - 7.36*0.8 + 0.45 * 11.2 + 3.01 * 7.8 + 3.23 *5.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "bd5f9911",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.79354246101503"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model9.predict()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "95ab7321",
   "metadata": {},
   "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>Title</th>\n",
       "      <th>Author</th>\n",
       "      <th>List Price</th>\n",
       "      <th>Amazon Price</th>\n",
       "      <th>Hard_or_Paper</th>\n",
       "      <th>NumPages</th>\n",
       "      <th>Publisher</th>\n",
       "      <th>Pub year</th>\n",
       "      <th>ISBN-10</th>\n",
       "      <th>Height</th>\n",
       "      <th>Width</th>\n",
       "      <th>Thick</th>\n",
       "      <th>Weight_oz</th>\n",
       "      <th>Paperback</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1,001 Facts that Will Scare the S#*t Out of Yo...</td>\n",
       "      <td>Cary McNeal</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>304.0</td>\n",
       "      <td>Adams Media</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>1605506249</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.8</td>\n",
       "      <td>11.2</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>21: Bringing Down the House - Movie Tie-In: Th...</td>\n",
       "      <td>Ben Mezrich</td>\n",
       "      <td>15.00</td>\n",
       "      <td>10.20</td>\n",
       "      <td>P</td>\n",
       "      <td>273.0</td>\n",
       "      <td>Free Press</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>1416564195</td>\n",
       "      <td>8.4</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.7</td>\n",
       "      <td>7.2</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>100 Best-Loved Poems (Dover Thrift Editions)</td>\n",
       "      <td>Smith</td>\n",
       "      <td>1.50</td>\n",
       "      <td>1.50</td>\n",
       "      <td>P</td>\n",
       "      <td>96.0</td>\n",
       "      <td>Dover Publications</td>\n",
       "      <td>1995.0</td>\n",
       "      <td>486285537</td>\n",
       "      <td>8.3</td>\n",
       "      <td>5.2</td>\n",
       "      <td>0.3</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1421: The Year China Discovered America</td>\n",
       "      <td>Gavin Menzies</td>\n",
       "      <td>15.99</td>\n",
       "      <td>10.87</td>\n",
       "      <td>P</td>\n",
       "      <td>672.0</td>\n",
       "      <td>Harper Perennial</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>61564893</td>\n",
       "      <td>8.8</td>\n",
       "      <td>6.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>28.8</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1493: Uncovering the New World Columbus Created</td>\n",
       "      <td>Charles C. Mann</td>\n",
       "      <td>30.50</td>\n",
       "      <td>16.77</td>\n",
       "      <td>P</td>\n",
       "      <td>720.0</td>\n",
       "      <td>Knopf</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>307265722</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>1.4</td>\n",
       "      <td>22.4</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>320</th>\n",
       "      <td>Where the Sidewalk Ends</td>\n",
       "      <td>Shel Silverstein</td>\n",
       "      <td>18.99</td>\n",
       "      <td>12.24</td>\n",
       "      <td>H</td>\n",
       "      <td>192.0</td>\n",
       "      <td>HarperCollins</td>\n",
       "      <td>2004.0</td>\n",
       "      <td>60572345</td>\n",
       "      <td>9.3</td>\n",
       "      <td>6.6</td>\n",
       "      <td>1.1</td>\n",
       "      <td>24.0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>321</th>\n",
       "      <td>White Privilege</td>\n",
       "      <td>Paula S. Rothenberg</td>\n",
       "      <td>27.55</td>\n",
       "      <td>27.55</td>\n",
       "      <td>P</td>\n",
       "      <td>160.0</td>\n",
       "      <td>Worth Publishers</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1429233443</td>\n",
       "      <td>9.1</td>\n",
       "      <td>6.1</td>\n",
       "      <td>0.7</td>\n",
       "      <td>8.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>322</th>\n",
       "      <td>Why I wore lipstick</td>\n",
       "      <td>Geralyn Lucas</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>224.0</td>\n",
       "      <td>St Martin's Griffin</td>\n",
       "      <td>2005.0</td>\n",
       "      <td>031233446X</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>6.4</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>323</th>\n",
       "      <td>Worlds Together, Worlds Apart: A History of th...</td>\n",
       "      <td>Robert Tignor</td>\n",
       "      <td>97.50</td>\n",
       "      <td>97.50</td>\n",
       "      <td>P</td>\n",
       "      <td>480.0</td>\n",
       "      <td>W. W. Norton &amp; Company</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>393934942</td>\n",
       "      <td>10.7</td>\n",
       "      <td>8.9</td>\n",
       "      <td>0.9</td>\n",
       "      <td>14.4</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>324</th>\n",
       "      <td>Wuthering Heights</td>\n",
       "      <td>Emily Bronte</td>\n",
       "      <td>16.99</td>\n",
       "      <td>4.95</td>\n",
       "      <td>P</td>\n",
       "      <td>344.0</td>\n",
       "      <td>CreatSpace</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1463533411</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>16.0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>325 rows × 14 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                 Title               Author  \\\n",
       "0    1,001 Facts that Will Scare the S#*t Out of Yo...          Cary McNeal   \n",
       "1    21: Bringing Down the House - Movie Tie-In: Th...          Ben Mezrich   \n",
       "2         100 Best-Loved Poems (Dover Thrift Editions)                Smith   \n",
       "3              1421: The Year China Discovered America        Gavin Menzies   \n",
       "4      1493: Uncovering the New World Columbus Created      Charles C. Mann   \n",
       "..                                                 ...                  ...   \n",
       "320                            Where the Sidewalk Ends     Shel Silverstein   \n",
       "321                                    White Privilege  Paula S. Rothenberg   \n",
       "322                                Why I wore lipstick        Geralyn Lucas   \n",
       "323  Worlds Together, Worlds Apart: A History of th...        Robert Tignor   \n",
       "324                                  Wuthering Heights         Emily Bronte   \n",
       "\n",
       "     List Price  Amazon Price Hard_or_Paper  NumPages               Publisher  \\\n",
       "0         12.95          5.18             P     304.0             Adams Media   \n",
       "1         15.00         10.20             P     273.0              Free Press   \n",
       "2          1.50          1.50             P      96.0      Dover Publications   \n",
       "3         15.99         10.87             P     672.0        Harper Perennial   \n",
       "4         30.50         16.77             P     720.0                   Knopf   \n",
       "..          ...           ...           ...       ...                     ...   \n",
       "320       18.99         12.24             H     192.0           HarperCollins   \n",
       "321       27.55         27.55             P     160.0        Worth Publishers   \n",
       "322       12.95          5.18             P     224.0     St Martin's Griffin   \n",
       "323       97.50         97.50             P     480.0  W. W. Norton & Company   \n",
       "324       16.99          4.95             P     344.0              CreatSpace   \n",
       "\n",
       "     Pub year     ISBN-10  Height  Width  Thick  Weight_oz  Paperback  \n",
       "0      2010.0  1605506249     7.8    5.5    0.8       11.2          1  \n",
       "1      2008.0  1416564195     8.4    5.5    0.7        7.2          1  \n",
       "2      1995.0   486285537     8.3    5.2    0.3        4.0          1  \n",
       "3      2008.0    61564893     8.8    6.0    1.6       28.8          1  \n",
       "4      2011.0   307265722     8.0    5.2    1.4       22.4          1  \n",
       "..        ...         ...     ...    ...    ...        ...        ...  \n",
       "320    2004.0    60572345     9.3    6.6    1.1       24.0          0  \n",
       "321    2011.0  1429233443     9.1    6.1    0.7        8.0          1  \n",
       "322    2005.0  031233446X     8.0    5.4    0.7        6.4          1  \n",
       "323    2010.0   393934942    10.7    8.9    0.9       14.4          1  \n",
       "324    2011.0  1463533411     7.8    5.1    1.0       16.0          1  \n",
       "\n",
       "[325 rows x 14 columns]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "amazonbooks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "dafead48",
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.express as px\n",
    "import plotly.graph_objects as go\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "id": "037c9320",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 0.960867144102875\n"
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       "var gd = document.getElementById('5989c5d5-0a70-4330-8d41-aac178479ef9');\n",
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       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
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       "    x.observe(notebookContainer, {childList: true});\n",
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       "    x.observe(outputEl, {childList: true});\n",
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     "output_type": "display_data"
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   ],
   "source": [
    "# fit_model7\n",
    "y_7 = fit_model7.model.endog\n",
    "print(\"r =\",np.corrcoef(y_7, fit_model7.predict())[0,1])\n",
    "fig = px.scatter(x=y_7, y=fit_model7.predict(),\n",
    "                 labels={\"x\": \"y\", \"y\": \"y-hat\"},)\n",
    "fig.add_trace(go.Scatter(x=[0,140], y=[0,140], name=\"y=x\", line_shape='linear'))\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "id": "3395320c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9232656686164151"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
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  {
   "cell_type": "code",
   "execution_count": 104,
   "id": "ccda1fcc",
   "metadata": {},
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     "name": "stdout",
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     "text": [
      "r = 0.4733900350278686\n"
     ]
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       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# fit_model9\n",
    "y_9 = fit_model9.model.endog\n",
    "print(\"r =\",np.corrcoef(y_9, fit_model9.predict())[0,1])\n",
    "fig = px.scatter(x=y_9, y=fit_model9.predict(),\n",
    "                 labels={\"x\": \"y\", \"y\": \"y-hat\"},)\n",
    "fig.add_trace(go.Scatter(x=[0,140], y=[0,140], name=\"y=x\", line_shape='linear'))\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "e2294b77",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 0.4346690986911411\n"
     ]
    },
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       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# fit_model10\n",
    "y_10 = fit_model10.model.endog\n",
    "print(\"r =\",np.corrcoef(y_10, fit_model10.predict())[0,1])\n",
    "fig = px.scatter(x=y_10, y=fit_model10.predict(),\n",
    "                 labels={\"x\": \"y\", \"y\": \"y-hat\"},)\n",
    "fig.add_trace(go.Scatter(x=[0,140], y=[0,140], name=\"y=x\", line_shape='linear'))\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "8402a561",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9216, 0.22372899999999998, 0.188356)"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# y versus y-hat correlation for different model fits\n",
    "\n",
    "# model 7 # model 9 # model 10\n",
    "0.960**2, 0.473**2, 0.434**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56c173be",
   "metadata": {},
   "source": [
    "$R^2$ is called the coefficient of determination for a linear model\n",
    "- $R^2$ is the \"proportion of variation explained by the model\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "id": "2b3786d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>    <td>Q(\"Amazon Price\")</td> <th>  R-squared (uncentered):</th>      <td>   0.967</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>        <th>  Adj. R-squared (uncentered):</th> <td>   0.966</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>   <th>  F-statistic:       </th>          <td>   1306.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Mon, 08 May 2023</td>  <th>  Prob (F-statistic):</th>          <td>5.21e-229</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:53:54</td>      <th>  Log-Likelihood:    </th>          <td> -840.89</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   323</td>       <th>  AIC:               </th>          <td>   1696.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   316</td>       <th>  BIC:               </th>          <td>   1722.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     7</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>Q(\"List Price\")</th>                 <td>    0.7693</td> <td>    0.056</td> <td>   13.774</td> <td> 0.000</td> <td>    0.659</td> <td>    0.879</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick</th>                           <td>   -4.6291</td> <td>    0.686</td> <td>   -6.747</td> <td> 0.000</td> <td>   -5.979</td> <td>   -3.279</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\"):Thick</th>           <td>    0.0996</td> <td>    0.042</td> <td>    2.369</td> <td> 0.018</td> <td>    0.017</td> <td>    0.182</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Paperback</th>                       <td>   -4.2854</td> <td>    1.059</td> <td>   -4.048</td> <td> 0.000</td> <td>   -6.368</td> <td>   -2.202</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\"):Paperback</th>       <td>    0.3562</td> <td>    0.079</td> <td>    4.535</td> <td> 0.000</td> <td>    0.202</td> <td>    0.511</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Thick:Paperback</th>                 <td>    7.8834</td> <td>    1.555</td> <td>    5.070</td> <td> 0.000</td> <td>    4.824</td> <td>   10.943</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Q(\"List Price\"):Thick:Paperback</th> <td>   -0.4716</td> <td>    0.087</td> <td>   -5.426</td> <td> 0.000</td> <td>   -0.643</td> <td>   -0.301</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>100.989</td> <th>  Durbin-Watson:     </th> <td>   2.049</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td>1023.142</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.966</td>  <th>  Prob(JB):          </th> <td>6.72e-223</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td>11.502</td>  <th>  Cond. No.          </th> <td>    381.</td> \n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] R² is computed without centering (uncentered) since the model does not contain a constant.<br/>[2] 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:      Q(\"Amazon Price\")   R-squared (uncentered):                   0.967\n",
       "Model:                            OLS   Adj. R-squared (uncentered):              0.966\n",
       "Method:                 Least Squares   F-statistic:                              1306.\n",
       "Date:                Mon, 08 May 2023   Prob (F-statistic):                   5.21e-229\n",
       "Time:                        15:53:54   Log-Likelihood:                         -840.89\n",
       "No. Observations:                 323   AIC:                                      1696.\n",
       "Df Residuals:                     316   BIC:                                      1722.\n",
       "Df Model:                           7                                                  \n",
       "Covariance Type:            nonrobust                                                  \n",
       "===================================================================================================\n",
       "                                      coef    std err          t      P>|t|      [0.025      0.975]\n",
       "---------------------------------------------------------------------------------------------------\n",
       "Q(\"List Price\")                     0.7693      0.056     13.774      0.000       0.659       0.879\n",
       "Thick                              -4.6291      0.686     -6.747      0.000      -5.979      -3.279\n",
       "Q(\"List Price\"):Thick               0.0996      0.042      2.369      0.018       0.017       0.182\n",
       "Paperback                          -4.2854      1.059     -4.048      0.000      -6.368      -2.202\n",
       "Q(\"List Price\"):Paperback           0.3562      0.079      4.535      0.000       0.202       0.511\n",
       "Thick:Paperback                     7.8834      1.555      5.070      0.000       4.824      10.943\n",
       "Q(\"List Price\"):Thick:Paperback    -0.4716      0.087     -5.426      0.000      -0.643      -0.301\n",
       "==============================================================================\n",
       "Omnibus:                      100.989   Durbin-Watson:                   2.049\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1023.142\n",
       "Skew:                           0.966   Prob(JB):                    6.72e-223\n",
       "Kurtosis:                      11.502   Cond. No.                         381.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] R² is computed without centering (uncentered) since the model does not contain a constant.\n",
       "[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fit_model11 = smf.ols(formula='Q(\"Amazon Price\") ~ -1 + Q(\"List Price\") * Thick * Paperback', \n",
    "                    data=amazonbooks).fit()\n",
    "# fit_model7.summary() # model has 3 covariates only\n",
    "fit_model11.summary() # model 11 has 7 covariates"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af18449b",
   "metadata": {},
   "source": [
    "# Interactions! Automatically create new data columns to help predict!\n",
    "\n",
    "$$\\large \\hat y = \\beta_L ListPrice * \\beta_T Thick * \\beta_P Paperback$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3724f96",
   "metadata": {},
   "source": [
    "- For model 7: $R^2$ reported in the summary function was 0.963\n",
    "- For model 11: $R^2$ reported in the summary function was 0.967"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "id": "c790f8f7",
   "metadata": {},
   "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>Title</th>\n",
       "      <th>Author</th>\n",
       "      <th>List Price</th>\n",
       "      <th>Amazon Price</th>\n",
       "      <th>Hard_or_Paper</th>\n",
       "      <th>NumPages</th>\n",
       "      <th>Publisher</th>\n",
       "      <th>Pub year</th>\n",
       "      <th>ISBN-10</th>\n",
       "      <th>Height</th>\n",
       "      <th>Width</th>\n",
       "      <th>Thick</th>\n",
       "      <th>Weight_oz</th>\n",
       "      <th>Paperback</th>\n",
       "      <th>Hardback</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1,001 Facts that Will Scare the S#*t Out of Yo...</td>\n",
       "      <td>Cary McNeal</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>304.0</td>\n",
       "      <td>Adams Media</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>1605506249</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.8</td>\n",
       "      <td>11.2</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>21: Bringing Down the House - Movie Tie-In: Th...</td>\n",
       "      <td>Ben Mezrich</td>\n",
       "      <td>15.00</td>\n",
       "      <td>10.20</td>\n",
       "      <td>P</td>\n",
       "      <td>273.0</td>\n",
       "      <td>Free Press</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>1416564195</td>\n",
       "      <td>8.4</td>\n",
       "      <td>5.5</td>\n",
       "      <td>0.7</td>\n",
       "      <td>7.2</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>100 Best-Loved Poems (Dover Thrift Editions)</td>\n",
       "      <td>Smith</td>\n",
       "      <td>1.50</td>\n",
       "      <td>1.50</td>\n",
       "      <td>P</td>\n",
       "      <td>96.0</td>\n",
       "      <td>Dover Publications</td>\n",
       "      <td>1995.0</td>\n",
       "      <td>486285537</td>\n",
       "      <td>8.3</td>\n",
       "      <td>5.2</td>\n",
       "      <td>0.3</td>\n",
       "      <td>4.0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1421: The Year China Discovered America</td>\n",
       "      <td>Gavin Menzies</td>\n",
       "      <td>15.99</td>\n",
       "      <td>10.87</td>\n",
       "      <td>P</td>\n",
       "      <td>672.0</td>\n",
       "      <td>Harper Perennial</td>\n",
       "      <td>2008.0</td>\n",
       "      <td>61564893</td>\n",
       "      <td>8.8</td>\n",
       "      <td>6.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>28.8</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1493: Uncovering the New World Columbus Created</td>\n",
       "      <td>Charles C. Mann</td>\n",
       "      <td>30.50</td>\n",
       "      <td>16.77</td>\n",
       "      <td>P</td>\n",
       "      <td>720.0</td>\n",
       "      <td>Knopf</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>307265722</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>1.4</td>\n",
       "      <td>22.4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>320</th>\n",
       "      <td>Where the Sidewalk Ends</td>\n",
       "      <td>Shel Silverstein</td>\n",
       "      <td>18.99</td>\n",
       "      <td>12.24</td>\n",
       "      <td>H</td>\n",
       "      <td>192.0</td>\n",
       "      <td>HarperCollins</td>\n",
       "      <td>2004.0</td>\n",
       "      <td>60572345</td>\n",
       "      <td>9.3</td>\n",
       "      <td>6.6</td>\n",
       "      <td>1.1</td>\n",
       "      <td>24.0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>321</th>\n",
       "      <td>White Privilege</td>\n",
       "      <td>Paula S. Rothenberg</td>\n",
       "      <td>27.55</td>\n",
       "      <td>27.55</td>\n",
       "      <td>P</td>\n",
       "      <td>160.0</td>\n",
       "      <td>Worth Publishers</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1429233443</td>\n",
       "      <td>9.1</td>\n",
       "      <td>6.1</td>\n",
       "      <td>0.7</td>\n",
       "      <td>8.0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>322</th>\n",
       "      <td>Why I wore lipstick</td>\n",
       "      <td>Geralyn Lucas</td>\n",
       "      <td>12.95</td>\n",
       "      <td>5.18</td>\n",
       "      <td>P</td>\n",
       "      <td>224.0</td>\n",
       "      <td>St Martin's Griffin</td>\n",
       "      <td>2005.0</td>\n",
       "      <td>031233446X</td>\n",
       "      <td>8.0</td>\n",
       "      <td>5.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>6.4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>323</th>\n",
       "      <td>Worlds Together, Worlds Apart: A History of th...</td>\n",
       "      <td>Robert Tignor</td>\n",
       "      <td>97.50</td>\n",
       "      <td>97.50</td>\n",
       "      <td>P</td>\n",
       "      <td>480.0</td>\n",
       "      <td>W. W. Norton &amp; Company</td>\n",
       "      <td>2010.0</td>\n",
       "      <td>393934942</td>\n",
       "      <td>10.7</td>\n",
       "      <td>8.9</td>\n",
       "      <td>0.9</td>\n",
       "      <td>14.4</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>324</th>\n",
       "      <td>Wuthering Heights</td>\n",
       "      <td>Emily Bronte</td>\n",
       "      <td>16.99</td>\n",
       "      <td>4.95</td>\n",
       "      <td>P</td>\n",
       "      <td>344.0</td>\n",
       "      <td>CreatSpace</td>\n",
       "      <td>2011.0</td>\n",
       "      <td>1463533411</td>\n",
       "      <td>7.8</td>\n",
       "      <td>5.1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>16.0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>325 rows × 15 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                 Title               Author  \\\n",
       "0    1,001 Facts that Will Scare the S#*t Out of Yo...          Cary McNeal   \n",
       "1    21: Bringing Down the House - Movie Tie-In: Th...          Ben Mezrich   \n",
       "2         100 Best-Loved Poems (Dover Thrift Editions)                Smith   \n",
       "3              1421: The Year China Discovered America        Gavin Menzies   \n",
       "4      1493: Uncovering the New World Columbus Created      Charles C. Mann   \n",
       "..                                                 ...                  ...   \n",
       "320                            Where the Sidewalk Ends     Shel Silverstein   \n",
       "321                                    White Privilege  Paula S. Rothenberg   \n",
       "322                                Why I wore lipstick        Geralyn Lucas   \n",
       "323  Worlds Together, Worlds Apart: A History of th...        Robert Tignor   \n",
       "324                                  Wuthering Heights         Emily Bronte   \n",
       "\n",
       "     List Price  Amazon Price Hard_or_Paper  NumPages               Publisher  \\\n",
       "0         12.95          5.18             P     304.0             Adams Media   \n",
       "1         15.00         10.20             P     273.0              Free Press   \n",
       "2          1.50          1.50             P      96.0      Dover Publications   \n",
       "3         15.99         10.87             P     672.0        Harper Perennial   \n",
       "4         30.50         16.77             P     720.0                   Knopf   \n",
       "..          ...           ...           ...       ...                     ...   \n",
       "320       18.99         12.24             H     192.0           HarperCollins   \n",
       "321       27.55         27.55             P     160.0        Worth Publishers   \n",
       "322       12.95          5.18             P     224.0     St Martin's Griffin   \n",
       "323       97.50         97.50             P     480.0  W. W. Norton & Company   \n",
       "324       16.99          4.95             P     344.0              CreatSpace   \n",
       "\n",
       "     Pub year     ISBN-10  Height  Width  Thick  Weight_oz  Paperback  \\\n",
       "0      2010.0  1605506249     7.8    5.5    0.8       11.2          1   \n",
       "1      2008.0  1416564195     8.4    5.5    0.7        7.2          1   \n",
       "2      1995.0   486285537     8.3    5.2    0.3        4.0          1   \n",
       "3      2008.0    61564893     8.8    6.0    1.6       28.8          1   \n",
       "4      2011.0   307265722     8.0    5.2    1.4       22.4          1   \n",
       "..        ...         ...     ...    ...    ...        ...        ...   \n",
       "320    2004.0    60572345     9.3    6.6    1.1       24.0          0   \n",
       "321    2011.0  1429233443     9.1    6.1    0.7        8.0          1   \n",
       "322    2005.0  031233446X     8.0    5.4    0.7        6.4          1   \n",
       "323    2010.0   393934942    10.7    8.9    0.9       14.4          1   \n",
       "324    2011.0  1463533411     7.8    5.1    1.0       16.0          1   \n",
       "\n",
       "     Hardback  \n",
       "0           0  \n",
       "1           0  \n",
       "2           0  \n",
       "3           0  \n",
       "4           0  \n",
       "..        ...  \n",
       "320         1  \n",
       "321         0  \n",
       "322         0  \n",
       "323         0  \n",
       "324         0  \n",
       "\n",
       "[325 rows x 15 columns]"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "amazonbooks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e67e3ed0",
   "metadata": {},
   "outputs": [],
   "source": [
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