4. Analyzing prices using the AR model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Importing the relevant packages\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.graphics.tsaplots as sgt\n",
    "from statsmodels.tsa.arima_model import ARMA\n",
    "from scipy.stats.distributions import chi2\n",
    "import statsmodels.tsa.stattools as sts \n",
    "import seaborn as sns\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Importing the Data and Pre-processing "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "raw_csv_data = pd.read_csv(\"Index2018.csv\") \n",
    "df_comp=raw_csv_data.copy()\n",
    "df_comp.date = pd.to_datetime(df_comp.date, dayfirst = True)\n",
    "df_comp.set_index(\"date\", inplace=True)\n",
    "df_comp=df_comp.asfreq('b')\n",
    "df_comp=df_comp.fillna(method='ffill')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_comp['market_value']=df_comp.ftse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "del df_comp['spx']\n",
    "del df_comp['dax']\n",
    "del df_comp['ftse']\n",
    "del df_comp['nikkei']\n",
    "size = int(len(df_comp)*0.8)\n",
    "df, df_test = df_comp.iloc[:size], df_comp.iloc[size:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The ACF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sgt.plot_acf(df.market_value, zero = False, lags = 40)\n",
    "plt.title(\"ACF for Prices\", size = 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The PACF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sgt.plot_pacf(df.market_value, lags = 40, alpha = 0.05, zero = False, method = ('ols'))\n",
    "plt.title(\"PACF for Prices\", size = 20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The AR(1) Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar = ARMA(df.market_value, order=(1,0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "results_ar = model_ar.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>            <td>ARMA(1, 0)</td>    <th>  Log Likelihood     </th> <td>-27622.598</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.258</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55251.196</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>10:03:35</td>     <th>  BIC                </th>  <td>55270.760</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55258.052</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th></th>                 <td>- 04-05-2013</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>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>              <td> 5057.0038</td> <td>  486.461</td> <td>   10.395</td> <td> 0.000</td> <td> 4103.558</td> <td> 6010.450</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.market_value</th> <td>    0.9985</td> <td>    0.001</td> <td> 1298.863</td> <td> 0.000</td> <td>    0.997</td> <td>    1.000</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<caption>Roots</caption>\n",
       "<tr>\n",
       "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.1</th> <td>           1.0015</td> <td>          +0.0000j</td> <td>           1.0015</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                              ARMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:           market_value   No. Observations:                 5021\n",
       "Model:                     ARMA(1, 0)   Log Likelihood              -27622.598\n",
       "Method:                       css-mle   S.D. of innovations             59.258\n",
       "Date:                Fri, 27 Sep 2019   AIC                          55251.196\n",
       "Time:                        10:03:35   BIC                          55270.760\n",
       "Sample:                    01-07-1994   HQIC                         55258.052\n",
       "                         - 04-05-2013                                         \n",
       "======================================================================================\n",
       "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------------\n",
       "const               5057.0038    486.461     10.395      0.000    4103.558    6010.450\n",
       "ar.L1.market_value     0.9985      0.001   1298.863      0.000       0.997       1.000\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.0015           +0.0000j            1.0015            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results_ar.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Higher-Lag AR Models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>            <td>ARMA(2, 0)</td>    <th>  Log Likelihood     </th> <td>-27621.866</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.249</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55251.731</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>10:03:37</td>     <th>  BIC                </th>  <td>55277.817</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55260.872</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th></th>                 <td>- 04-05-2013</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>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>              <td> 5036.7744</td> <td>  495.444</td> <td>   10.166</td> <td> 0.000</td> <td> 4065.721</td> <td> 6007.828</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.market_value</th> <td>    0.9814</td> <td>    0.014</td> <td>   69.595</td> <td> 0.000</td> <td>    0.954</td> <td>    1.009</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L2.market_value</th> <td>    0.0171</td> <td>    0.014</td> <td>    1.211</td> <td> 0.226</td> <td>   -0.011</td> <td>    0.045</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<caption>Roots</caption>\n",
       "<tr>\n",
       "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.1</th> <td>           1.0015</td> <td>          +0.0000j</td> <td>           1.0015</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.2</th> <td>         -58.4573</td> <td>          +0.0000j</td> <td>          58.4573</td> <td>           0.5000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                              ARMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:           market_value   No. Observations:                 5021\n",
       "Model:                     ARMA(2, 0)   Log Likelihood              -27621.866\n",
       "Method:                       css-mle   S.D. of innovations             59.249\n",
       "Date:                Fri, 27 Sep 2019   AIC                          55251.731\n",
       "Time:                        10:03:37   BIC                          55277.817\n",
       "Sample:                    01-07-1994   HQIC                         55260.872\n",
       "                         - 04-05-2013                                         \n",
       "======================================================================================\n",
       "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------------\n",
       "const               5036.7744    495.444     10.166      0.000    4065.721    6007.828\n",
       "ar.L1.market_value     0.9814      0.014     69.595      0.000       0.954       1.009\n",
       "ar.L2.market_value     0.0171      0.014      1.211      0.226      -0.011       0.045\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.0015           +0.0000j            1.0015            0.0000\n",
       "AR.2          -58.4573           +0.0000j           58.4573            0.5000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_2 = ARMA(df.market_value, order=(2,0))\n",
    "results_ar_2 = model_ar_2.fit()\n",
    "results_ar_2.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>            <td>ARMA(3, 0)</td>    <th>  Log Likelihood     </th> <td>-27615.871</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.178</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55241.742</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>10:03:37</td>     <th>  BIC                </th>  <td>55274.349</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55253.168</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th></th>                 <td>- 04-05-2013</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>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>              <td> 5089.6412</td> <td>  520.376</td> <td>    9.781</td> <td> 0.000</td> <td> 4069.723</td> <td> 6109.560</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.market_value</th> <td>    0.9806</td> <td>    0.014</td> <td>   69.619</td> <td> 0.000</td> <td>    0.953</td> <td>    1.008</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L2.market_value</th> <td>   -0.0309</td> <td>    0.020</td> <td>   -1.563</td> <td> 0.118</td> <td>   -0.070</td> <td>    0.008</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L3.market_value</th> <td>    0.0489</td> <td>    0.014</td> <td>    3.466</td> <td> 0.001</td> <td>    0.021</td> <td>    0.077</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<caption>Roots</caption>\n",
       "<tr>\n",
       "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.1</th> <td>           1.0013</td> <td>          -0.0000j</td> <td>           1.0013</td> <td>          -0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.2</th> <td>          -0.1850</td> <td>          -4.5161j</td> <td>           4.5199</td> <td>          -0.2565</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.3</th> <td>          -0.1850</td> <td>          +4.5161j</td> <td>           4.5199</td> <td>           0.2565</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                              ARMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:           market_value   No. Observations:                 5021\n",
       "Model:                     ARMA(3, 0)   Log Likelihood              -27615.871\n",
       "Method:                       css-mle   S.D. of innovations             59.178\n",
       "Date:                Fri, 27 Sep 2019   AIC                          55241.742\n",
       "Time:                        10:03:37   BIC                          55274.349\n",
       "Sample:                    01-07-1994   HQIC                         55253.168\n",
       "                         - 04-05-2013                                         \n",
       "======================================================================================\n",
       "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------------\n",
       "const               5089.6412    520.376      9.781      0.000    4069.723    6109.560\n",
       "ar.L1.market_value     0.9806      0.014     69.619      0.000       0.953       1.008\n",
       "ar.L2.market_value    -0.0309      0.020     -1.563      0.118      -0.070       0.008\n",
       "ar.L3.market_value     0.0489      0.014      3.466      0.001       0.021       0.077\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.0013           -0.0000j            1.0013           -0.0000\n",
       "AR.2           -0.1850           -4.5161j            4.5199           -0.2565\n",
       "AR.3           -0.1850           +4.5161j            4.5199            0.2565\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_3 = ARMA(df.market_value, order=(3,0))\n",
    "results_ar_3 = model_ar_3.fit()\n",
    "results_ar_3.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>            <td>ARMA(4, 0)</td>    <th>  Log Likelihood     </th> <td>-27602.863</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.024</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55217.726</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>10:03:38</td>     <th>  BIC                </th>  <td>55256.854</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55231.437</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th></th>                 <td>- 04-05-2013</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>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>              <td> 5493.1989</td> <td>  959.292</td> <td>    5.726</td> <td> 0.000</td> <td> 3613.022</td> <td> 7373.376</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.market_value</th> <td>    0.9771</td> <td>    0.014</td> <td>   69.567</td> <td> 0.000</td> <td>    0.950</td> <td>    1.005</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L2.market_value</th> <td>   -0.0286</td> <td>    0.020</td> <td>   -1.455</td> <td> 0.146</td> <td>   -0.067</td> <td>    0.010</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L3.market_value</th> <td>   -0.0224</td> <td>    0.020</td> <td>   -1.135</td> <td> 0.256</td> <td>   -0.061</td> <td>    0.016</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L4.market_value</th> <td>    0.0729</td> <td>    0.014</td> <td>    5.171</td> <td> 0.000</td> <td>    0.045</td> <td>    0.100</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<caption>Roots</caption>\n",
       "<tr>\n",
       "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.1</th> <td>           1.0009</td> <td>          -0.0000j</td> <td>           1.0009</td> <td>          -0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.2</th> <td>           0.9543</td> <td>          -2.0876j</td> <td>           2.2954</td> <td>          -0.1818</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.3</th> <td>           0.9543</td> <td>          +2.0876j</td> <td>           2.2954</td> <td>           0.1818</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.4</th> <td>          -2.6026</td> <td>          -0.0000j</td> <td>           2.6026</td> <td>          -0.5000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                              ARMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:           market_value   No. Observations:                 5021\n",
       "Model:                     ARMA(4, 0)   Log Likelihood              -27602.863\n",
       "Method:                       css-mle   S.D. of innovations             59.024\n",
       "Date:                Fri, 27 Sep 2019   AIC                          55217.726\n",
       "Time:                        10:03:38   BIC                          55256.854\n",
       "Sample:                    01-07-1994   HQIC                         55231.437\n",
       "                         - 04-05-2013                                         \n",
       "======================================================================================\n",
       "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------------\n",
       "const               5493.1989    959.292      5.726      0.000    3613.022    7373.376\n",
       "ar.L1.market_value     0.9771      0.014     69.567      0.000       0.950       1.005\n",
       "ar.L2.market_value    -0.0286      0.020     -1.455      0.146      -0.067       0.010\n",
       "ar.L3.market_value    -0.0224      0.020     -1.135      0.256      -0.061       0.016\n",
       "ar.L4.market_value     0.0729      0.014      5.171      0.000       0.045       0.100\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.0009           -0.0000j            1.0009           -0.0000\n",
       "AR.2            0.9543           -2.0876j            2.2954           -0.1818\n",
       "AR.3            0.9543           +2.0876j            2.2954            0.1818\n",
       "AR.4           -2.6026           -0.0000j            2.6026           -0.5000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_4 = ARMA(df.market_value, order=[4,0])\n",
    "results_ar_4 = model_ar_4.fit()\n",
    "results_ar_4.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LLR Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def LLR_test(mod_1, mod_2, DF=1):\n",
    "    L1 = mod_1.fit().llf\n",
    "    L2 = mod_2.fit().llf\n",
    "    LR = (2*(L2-L1))\n",
    "    p = chi2.sf(LR, DF).round(3)\n",
    "    return p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparing Higher-Lag AR Models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LLR_test(model_ar_2, model_ar_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "LLR_test(model_ar_3, model_ar_4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_4 = ARMA(df.market_value, order=[4,0])\n",
    "results_ar_4 = model_ar_4.fit()\n",
    "print(results_ar_4.summary()) \n",
    "print (\"LLR test: \" + str(LLR_test(model_ar_3, model_ar_4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_5 = ARMA(df.market_value, order=(5,0))\n",
    "results_ar_5 = model_ar_5.fit()\n",
    "print(results_ar_5.summary())\n",
    "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_4, model_ar_5)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_6 = ARMA(df.market_value, order=(6,0))\n",
    "results_ar_6 = model_ar_6.fit()\n",
    "print(results_ar_6.summary())\n",
    "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_5, model_ar_6)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_7 = ARMA(df.market_value, order=(7,0))\n",
    "results_ar_7 = model_ar_7.fit()\n",
    "print(results_ar_7.summary())\n",
    "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_6, model_ar_7)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_8 = ARMA(df.market_value, order=(8,0))\n",
    "results_ar_8 = model_ar_8.fit()\n",
    "print(results_ar_8.summary())\n",
    "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_7, model_ar_8)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"LLR test: \" + str(LLR_test(model_ar, model_ar_7, DF = 6)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analysing the Residuals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['res_price'] = results_ar_7.resid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.res_price.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.res_price.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sts.adfuller(df.res_price)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sgt.plot_acf(df.res_price, zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for Prices\",size=24)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.res_price[1:].plot(figsize=(20,5))\n",
    "plt.title(\"Residuals of Prices\",size=24)\n",
    "plt.show()"
   ]
  }
 ],
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