8. The ARIMA and SARIMAX model.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Importing the relevant packages"
   ]
  },
  {
   "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",
    "import statsmodels.tsa.stattools as sts\n",
    "from statsmodels.tsa.arima_model import ARIMA\n",
    "from scipy.stats.distributions import chi2 \n",
    "from math import sqrt\n",
    "import seaborn as sns\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " "
   ]
  },
  {
   "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": [
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The LLR Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def LLR_test(mod_1, mod_2, DF = 1):\n",
    "    L1 = mod_1.llf\n",
    "    L2 = mod_2.llf\n",
    "    LR = (2*(L2-L1))    \n",
    "    p = chi2.sf(LR, DF).round(3)\n",
    "    return p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating Returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "df['returns'] = df.market_value.pct_change(1)*100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ARIMA(1,1,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARIMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>  <td>D.market_value</td>  <th>  No. Observations:  </th>    <td>5020</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>          <td>ARIMA(1, 1, 1)</td>  <th>  Log Likelihood     </th> <td>-27603.666</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.134</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 29 Nov 2019</td> <th>  AIC                </th>  <td>55215.333</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>11:04:13</td>     <th>  BIC                </th>  <td>55241.418</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-10-1994</td>    <th>  HQIC               </th>  <td>55224.473</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>    0.5655</td> <td>    0.682</td> <td>    0.829</td> <td> 0.407</td> <td>   -0.772</td> <td>    1.903</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.D.market_value</th> <td>    0.7475</td> <td>    0.070</td> <td>   10.652</td> <td> 0.000</td> <td>    0.610</td> <td>    0.885</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1.D.market_value</th> <td>   -0.7936</td> <td>    0.064</td> <td>  -12.378</td> <td> 0.000</td> <td>   -0.919</td> <td>   -0.668</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>           1.3378</td> <td>          +0.0000j</td> <td>           1.3378</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>MA.1</th> <td>           1.2601</td> <td>          +0.0000j</td> <td>           1.2601</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                             ARIMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:         D.market_value   No. Observations:                 5020\n",
       "Model:                 ARIMA(1, 1, 1)   Log Likelihood              -27603.666\n",
       "Method:                       css-mle   S.D. of innovations             59.134\n",
       "Date:                Fri, 29 Nov 2019   AIC                          55215.333\n",
       "Time:                        11:04:13   BIC                          55241.418\n",
       "Sample:                    01-10-1994   HQIC                         55224.473\n",
       "                         - 04-05-2013                                         \n",
       "========================================================================================\n",
       "                           coef    std err          z      P>|z|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------------\n",
       "const                    0.5655      0.682      0.829      0.407      -0.772       1.903\n",
       "ar.L1.D.market_value     0.7475      0.070     10.652      0.000       0.610       0.885\n",
       "ma.L1.D.market_value    -0.7936      0.064    -12.378      0.000      -0.919      -0.668\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.3378           +0.0000j            1.3378            0.0000\n",
       "MA.1            1.2601           +0.0000j            1.2601            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_1_i_1_ma_1 = ARIMA(df.market_value, order=(1,1,1))\n",
    "results_ar_1_i_1_ma_1 = model_ar_1_i_1_ma_1.fit()\n",
    "results_ar_1_i_1_ma_1.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Residuals of the ARIMA(1,1,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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": [
    "df['res_ar_1_i_1_ma_1'] = results_ar_1_i_1_ma_1.resid\n",
    "sgt.plot_acf(df.res_ar_1_i_1_ma_1, zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for ARIMA(1,1,1)\",size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "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": [
    "df['res_ar_1_i_1_ma_1'] = results_ar_1_i_1_ma_1.resid.iloc[:]\n",
    "sgt.plot_acf(df.res_ar_1_i_1_ma_1[1:], zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for ARIMA(1,1,1)\",size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Higher-Lag ARIMA Models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_1_i_1_ma_2 = ARIMA(df.market_value, order=(1,1,2))\n",
    "results_ar_1_i_1_ma_2 = model_ar_1_i_1_ma_2.fit()\n",
    "model_ar_1_i_1_ma_3 = ARIMA(df.market_value, order=(1,1,3))\n",
    "results_ar_1_i_1_ma_3 = model_ar_1_i_1_ma_3.fit()\n",
    "model_ar_2_i_1_ma_1 = ARIMA(df.market_value, order=(2,1,1))\n",
    "results_ar_2_i_1_ma_1 = model_ar_2_i_1_ma_1.fit()\n",
    "model_ar_3_i_1_ma_1 = ARIMA(df.market_value, order=(3,1,1))\n",
    "results_ar_3_i_1_ma_1 = model_ar_3_i_1_ma_1.fit()\n",
    "model_ar_3_i_1_ma_2 = ARIMA(df.market_value, order=(3,1,2))\n",
    "results_ar_3_i_1_ma_2 = model_ar_3_i_1_ma_2.fit(start_ar_lags=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ARIMA(1,1,1):  \t LL =  -27603.666412765113 \t AIC =  55215.332825530226\n",
      "ARIMA(1,1,2):  \t LL =  -27600.081863116407 \t AIC =  55210.163726232815\n",
      "ARIMA(1,1,3):  \t LL =  -27590.42403257934 \t AIC =  55192.84806515868\n",
      "ARIMA(2,1,1):  \t LL =  -27599.80748784726 \t AIC =  55209.61497569452\n",
      "ARIMA(3,1,1):  \t LL =  -27592.05537170055 \t AIC =  55196.1107434011\n",
      "ARIMA(3,1,2):  \t LL =  -27590.660808977023 \t AIC =  55195.32161795405\n"
     ]
    }
   ],
   "source": [
    "print(\"ARIMA(1,1,1):  \\t LL = \", results_ar_1_i_1_ma_1.llf, \"\\t AIC = \", results_ar_1_i_1_ma_1.aic)\n",
    "print(\"ARIMA(1,1,2):  \\t LL = \", results_ar_1_i_1_ma_2.llf, \"\\t AIC = \", results_ar_1_i_1_ma_2.aic)\n",
    "print(\"ARIMA(1,1,3):  \\t LL = \", results_ar_1_i_1_ma_3.llf, \"\\t AIC = \", results_ar_1_i_1_ma_3.aic)\n",
    "print(\"ARIMA(2,1,1):  \\t LL = \", results_ar_2_i_1_ma_1.llf, \"\\t AIC = \", results_ar_2_i_1_ma_1.aic)\n",
    "print(\"ARIMA(3,1,1):  \\t LL = \", results_ar_3_i_1_ma_1.llf, \"\\t AIC = \", results_ar_3_i_1_ma_1.aic)\n",
    "print(\"ARIMA(3,1,2):  \\t LL = \", results_ar_3_i_1_ma_2.llf, \"\\t AIC = \", results_ar_3_i_1_ma_2.aic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "LLR test p-value = 0.0\n"
     ]
    }
   ],
   "source": [
    "print(\"\\nLLR test p-value = \" + str(LLR_test(results_ar_1_i_1_ma_2, results_ar_1_i_1_ma_3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "LLR test p-value = 0.0\n"
     ]
    }
   ],
   "source": [
    "print(\"\\nLLR test p-value = \" + str(LLR_test(results_ar_1_i_1_ma_1, results_ar_1_i_1_ma_3, DF = 2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "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": [
    "df['res_ar_1_i_1_ma_3'] = results_ar_1_i_1_ma_3.resid\n",
    "sgt.plot_acf(df.res_ar_1_i_1_ma_3[1:], zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for ARIMA(1,1,3)\", size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "model_ar_5_i_1_ma_1 = ARIMA(df.market_value, order=(5,1,1))\n",
    "results_ar_5_i_1_ma_1 = model_ar_5_i_1_ma_1.fit(start_ar_lags=11)\n",
    "model_ar_6_i_1_ma_3 = ARIMA(df.market_value, order=(6,1,3))\n",
    "results_ar_6_i_1_ma_3 = model_ar_6_i_1_ma_3.fit(start_ar_lags=11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARIMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>  <td>D.market_value</td>  <th>  No. Observations:  </th>    <td>5020</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>          <td>ARIMA(5, 1, 1)</td>  <th>  Log Likelihood     </th> <td>-27586.512</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>58.932</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 29 Nov 2019</td> <th>  AIC                </th>  <td>55189.024</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>11:04:28</td>     <th>  BIC                </th>  <td>55241.193</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-10-1994</td>    <th>  HQIC               </th>  <td>55207.305</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>    0.5658</td> <td>    0.690</td> <td>    0.820</td> <td> 0.413</td> <td>   -0.787</td> <td>    1.919</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.D.market_value</th> <td>    0.4011</td> <td>    0.159</td> <td>    2.530</td> <td> 0.011</td> <td>    0.090</td> <td>    0.712</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L2.D.market_value</th> <td>   -0.0445</td> <td>    0.016</td> <td>   -2.856</td> <td> 0.004</td> <td>   -0.075</td> <td>   -0.014</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L3.D.market_value</th> <td>   -0.0543</td> <td>    0.017</td> <td>   -3.163</td> <td> 0.002</td> <td>   -0.088</td> <td>   -0.021</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L4.D.market_value</th> <td>    0.0586</td> <td>    0.019</td> <td>    3.069</td> <td> 0.002</td> <td>    0.021</td> <td>    0.096</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L5.D.market_value</th> <td>   -0.0581</td> <td>    0.014</td> <td>   -4.118</td> <td> 0.000</td> <td>   -0.086</td> <td>   -0.030</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1.D.market_value</th> <td>   -0.4214</td> <td>    0.158</td> <td>   -2.659</td> <td> 0.008</td> <td>   -0.732</td> <td>   -0.111</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>          -1.7609</td> <td>          -0.0000j</td> <td>           1.7609</td> <td>          -0.5000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.2</th> <td>           1.4633</td> <td>          -0.8605j</td> <td>           1.6975</td> <td>          -0.0846</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.3</th> <td>           1.4633</td> <td>          +0.8605j</td> <td>           1.6975</td> <td>           0.0846</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.4</th> <td>          -0.0790</td> <td>          -1.8393j</td> <td>           1.8410</td> <td>          -0.2568</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>AR.5</th> <td>          -0.0790</td> <td>          +1.8393j</td> <td>           1.8410</td> <td>           0.2568</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>MA.1</th> <td>           2.3733</td> <td>          +0.0000j</td> <td>           2.3733</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                             ARIMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:         D.market_value   No. Observations:                 5020\n",
       "Model:                 ARIMA(5, 1, 1)   Log Likelihood              -27586.512\n",
       "Method:                       css-mle   S.D. of innovations             58.932\n",
       "Date:                Fri, 29 Nov 2019   AIC                          55189.024\n",
       "Time:                        11:04:28   BIC                          55241.193\n",
       "Sample:                    01-10-1994   HQIC                         55207.305\n",
       "                         - 04-05-2013                                         \n",
       "========================================================================================\n",
       "                           coef    std err          z      P>|z|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------------\n",
       "const                    0.5658      0.690      0.820      0.413      -0.787       1.919\n",
       "ar.L1.D.market_value     0.4011      0.159      2.530      0.011       0.090       0.712\n",
       "ar.L2.D.market_value    -0.0445      0.016     -2.856      0.004      -0.075      -0.014\n",
       "ar.L3.D.market_value    -0.0543      0.017     -3.163      0.002      -0.088      -0.021\n",
       "ar.L4.D.market_value     0.0586      0.019      3.069      0.002       0.021       0.096\n",
       "ar.L5.D.market_value    -0.0581      0.014     -4.118      0.000      -0.086      -0.030\n",
       "ma.L1.D.market_value    -0.4214      0.158     -2.659      0.008      -0.732      -0.111\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1           -1.7609           -0.0000j            1.7609           -0.5000\n",
       "AR.2            1.4633           -0.8605j            1.6975           -0.0846\n",
       "AR.3            1.4633           +0.8605j            1.6975            0.0846\n",
       "AR.4           -0.0790           -1.8393j            1.8410           -0.2568\n",
       "AR.5           -0.0790           +1.8393j            1.8410            0.2568\n",
       "MA.1            2.3733           +0.0000j            2.3733            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results_ar_5_i_1_ma_1.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ARIMA(1,1,3):  \t LL =  -27590.42403257934 \t AIC =  55192.84806515868\n",
      "ARIMA(5,1,1):  \t LL =  -27586.5118883202 \t AIC =  55189.0237766404\n",
      "ARIMA(6,1,3):  \t LL =  -27583.565320401154 \t AIC =  55189.13064080231\n"
     ]
    }
   ],
   "source": [
    "print(\"ARIMA(1,1,3):  \\t LL = \", results_ar_1_i_1_ma_3.llf, \"\\t AIC = \", results_ar_1_i_1_ma_3.aic)\n",
    "print(\"ARIMA(5,1,1):  \\t LL = \", results_ar_5_i_1_ma_1.llf, \"\\t AIC = \", results_ar_5_i_1_ma_1.aic)\n",
    "print(\"ARIMA(6,1,3):  \\t LL = \", results_ar_6_i_1_ma_3.llf, \"\\t AIC = \", results_ar_6_i_1_ma_3.aic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "LLR test p-value = 0.018\n"
     ]
    }
   ],
   "source": [
    "print(\"\\nLLR test p-value = \" + str(LLR_test(results_ar_1_i_1_ma_3, results_ar_6_i_1_ma_3, DF = 5)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "LLR test p-value = 0.117\n"
     ]
    }
   ],
   "source": [
    "print(\"\\nLLR test p-value = \" + str(LLR_test(results_ar_5_i_1_ma_1, results_ar_6_i_1_ma_3, DF = 3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "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": [
    "df['res_ar_5_i_1_ma_1'] = results_ar_5_i_1_ma_1.resid\n",
    "sgt.plot_acf(df.res_ar_5_i_1_ma_1[1:], zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for ARIMA(5,1,1)\", size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Models with Higher Levels of Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['delta_prices']=df.market_value.diff(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>   <td>delta_prices</td>   <th>  No. Observations:  </th>    <td>5020</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>            <td>ARMA(1, 1)</td>    <th>  Log Likelihood     </th> <td>-27603.666</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.134</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 29 Nov 2019</td> <th>  AIC                </th>  <td>55215.333</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>11:04:30</td>     <th>  BIC                </th>  <td>55241.418</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-10-1994</td>    <th>  HQIC               </th>  <td>55224.473</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>    0.5655</td> <td>    0.682</td> <td>    0.829</td> <td> 0.407</td> <td>   -0.772</td> <td>    1.903</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.delta_prices</th> <td>    0.7475</td> <td>    0.070</td> <td>   10.652</td> <td> 0.000</td> <td>    0.610</td> <td>    0.885</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1.delta_prices</th> <td>   -0.7936</td> <td>    0.064</td> <td>  -12.378</td> <td> 0.000</td> <td>   -0.919</td> <td>   -0.668</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>           1.3378</td> <td>          +0.0000j</td> <td>           1.3378</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>MA.1</th> <td>           1.2601</td> <td>          +0.0000j</td> <td>           1.2601</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:           delta_prices   No. Observations:                 5020\n",
       "Model:                     ARMA(1, 1)   Log Likelihood              -27603.666\n",
       "Method:                       css-mle   S.D. of innovations             59.134\n",
       "Date:                Fri, 29 Nov 2019   AIC                          55215.333\n",
       "Time:                        11:04:30   BIC                          55241.418\n",
       "Sample:                    01-10-1994   HQIC                         55224.473\n",
       "                         - 04-05-2013                                         \n",
       "======================================================================================\n",
       "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
       "--------------------------------------------------------------------------------------\n",
       "const                  0.5655      0.682      0.829      0.407      -0.772       1.903\n",
       "ar.L1.delta_prices     0.7475      0.070     10.652      0.000       0.610       0.885\n",
       "ma.L1.delta_prices    -0.7936      0.064    -12.378      0.000      -0.919      -0.668\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.3378           +0.0000j            1.3378            0.0000\n",
       "MA.1            1.2601           +0.0000j            1.2601            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_delta_ar_1_i_1_ma_1 = ARIMA(df.delta_prices[1:], order=(1,0,1))\n",
    "results_delta_ar_1_i_1_ma_1 = model_delta_ar_1_i_1_ma_1.fit()\n",
    "results_delta_ar_1_i_1_ma_1.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-32.24409349570748,\n",
       " 0.0,\n",
       " 5,\n",
       " 5014,\n",
       " {'1%': -3.4316548765428174,\n",
       "  '5%': -2.8621166146845334,\n",
       "  '10%': -2.5670769326348926},\n",
       " 54845.96104221891)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sts.adfuller(df.delta_prices[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARIMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>  <td>D2.market_value</td> <th>  No. Observations:  </th>    <td>5019</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>          <td>ARIMA(1, 2, 1)</td>  <th>  Log Likelihood     </th> <td>-27614.160</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.273</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 29 Nov 2019</td> <th>  AIC                </th>  <td>55236.319</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>11:04:34</td>     <th>  BIC                </th>  <td>55262.403</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-11-1994</td>    <th>  HQIC               </th>  <td>55245.459</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>   -0.0001</td> <td>    0.001</td> <td>   -0.202</td> <td> 0.840</td> <td>   -0.001</td> <td>    0.001</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.D2.market_value</th> <td>   -0.0178</td> <td>    0.014</td> <td>   -1.262</td> <td> 0.207</td> <td>   -0.045</td> <td>    0.010</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1.D2.market_value</th> <td>   -1.0000</td> <td>    0.001</td> <td>-1712.465</td> <td> 0.000</td> <td>   -1.001</td> <td>   -0.999</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>         -56.1249</td> <td>          +0.0000j</td> <td>          56.1249</td> <td>           0.5000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>MA.1</th> <td>           1.0000</td> <td>          +0.0000j</td> <td>           1.0000</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                             ARIMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:        D2.market_value   No. Observations:                 5019\n",
       "Model:                 ARIMA(1, 2, 1)   Log Likelihood              -27614.160\n",
       "Method:                       css-mle   S.D. of innovations             59.273\n",
       "Date:                Fri, 29 Nov 2019   AIC                          55236.319\n",
       "Time:                        11:04:34   BIC                          55262.403\n",
       "Sample:                    01-11-1994   HQIC                         55245.459\n",
       "                         - 04-05-2013                                         \n",
       "=========================================================================================\n",
       "                            coef    std err          z      P>|z|      [0.025      0.975]\n",
       "-----------------------------------------------------------------------------------------\n",
       "const                    -0.0001      0.001     -0.202      0.840      -0.001       0.001\n",
       "ar.L1.D2.market_value    -0.0178      0.014     -1.262      0.207      -0.045       0.010\n",
       "ma.L1.D2.market_value    -1.0000      0.001  -1712.465      0.000      -1.001      -0.999\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1          -56.1249           +0.0000j           56.1249            0.5000\n",
       "MA.1            1.0000           +0.0000j            1.0000            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_1_i_2_ma_1 = ARIMA(df.market_value, order=(1,2,1))\n",
    "results_ar_1_i_2_ma_1 = model_ar_1_i_2_ma_1.fit(start_ar_lags=10)\n",
    "results_ar_1_i_2_ma_1.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "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": [
    "df['res_ar_1_i_2_ma_1'] = results_ar_1_i_2_ma_1.resid.iloc[:]\n",
    "sgt.plot_acf(df.res_ar_1_i_2_ma_1[2:], zero = False, lags = 40)\n",
    "plt.title(\"ACF Of Residuals for ARIMA(1,2,1)\",size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ARIMAX "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>ARIMA Model Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>  <td>D.market_value</td>  <th>  No. Observations:  </th>    <td>5020</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>          <td>ARIMA(1, 1, 1)</td>  <th>  Log Likelihood     </th> <td>-27603.556</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.132</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 29 Nov 2019</td> <th>  AIC                </th>  <td>55217.112</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>11:04:36</td>     <th>  BIC                </th>  <td>55249.718</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>           <td>01-10-1994</td>    <th>  HQIC               </th>  <td>55228.538</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>   -0.5885</td> <td>    2.655</td> <td>   -0.222</td> <td> 0.825</td> <td>   -5.793</td> <td>    4.616</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>spx</th>                  <td>    0.0011</td> <td>    0.002</td> <td>    0.451</td> <td> 0.652</td> <td>   -0.004</td> <td>    0.006</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1.D.market_value</th> <td>    0.7456</td> <td>    0.070</td> <td>   10.577</td> <td> 0.000</td> <td>    0.607</td> <td>    0.884</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1.D.market_value</th> <td>   -0.7916</td> <td>    0.065</td> <td>  -12.272</td> <td> 0.000</td> <td>   -0.918</td> <td>   -0.665</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>           1.3412</td> <td>          +0.0000j</td> <td>           1.3412</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>MA.1</th> <td>           1.2632</td> <td>          +0.0000j</td> <td>           1.2632</td> <td>           0.0000</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                             ARIMA Model Results                              \n",
       "==============================================================================\n",
       "Dep. Variable:         D.market_value   No. Observations:                 5020\n",
       "Model:                 ARIMA(1, 1, 1)   Log Likelihood              -27603.556\n",
       "Method:                       css-mle   S.D. of innovations             59.132\n",
       "Date:                Fri, 29 Nov 2019   AIC                          55217.112\n",
       "Time:                        11:04:36   BIC                          55249.718\n",
       "Sample:                    01-10-1994   HQIC                         55228.538\n",
       "                         - 04-05-2013                                         \n",
       "========================================================================================\n",
       "                           coef    std err          z      P>|z|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------------\n",
       "const                   -0.5885      2.655     -0.222      0.825      -5.793       4.616\n",
       "spx                      0.0011      0.002      0.451      0.652      -0.004       0.006\n",
       "ar.L1.D.market_value     0.7456      0.070     10.577      0.000       0.607       0.884\n",
       "ma.L1.D.market_value    -0.7916      0.065    -12.272      0.000      -0.918      -0.665\n",
       "                                    Roots                                    \n",
       "=============================================================================\n",
       "                  Real          Imaginary           Modulus         Frequency\n",
       "-----------------------------------------------------------------------------\n",
       "AR.1            1.3412           +0.0000j            1.3412            0.0000\n",
       "MA.1            1.2632           +0.0000j            1.2632            0.0000\n",
       "-----------------------------------------------------------------------------\n",
       "\"\"\""
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ar_1_i_1_ma_1_Xspx = ARIMA(df.market_value, exog = df.spx, order=(1,1,1))\n",
    "results_ar_1_i_1_ma_1_Xspx = model_ar_1_i_1_ma_1_Xspx.fit()\n",
    "results_ar_1_i_1_ma_1_Xspx.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## SARIMAX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "from statsmodels.tsa.statespace.sarimax import SARIMAX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Statespace 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>SARIMAX(1, 0, 1)x(2, 0, 1, 5)</td> <th>  Log Likelihood     </th> <td>-26712.642</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>                  <td>Fri, 29 Nov 2019</td>        <th>  AIC                </th>  <td>53439.284</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                      <td>11:04:55</td>            <th>  BIC                </th>  <td>53484.933</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Sample:</th>                   <td>01-07-1994</td>           <th>  HQIC               </th>  <td>53455.280</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th></th>                         <td>- 04-05-2013</td>          <th>                     </th>      <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>              <td>opg</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>spx</th>      <td>    2.7041</td> <td>    0.035</td> <td>   76.450</td> <td> 0.000</td> <td>    2.635</td> <td>    2.773</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.L1</th>    <td>    0.9998</td> <td>    0.000</td> <td> 2270.454</td> <td> 0.000</td> <td>    0.999</td> <td>    1.001</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.L1</th>    <td>   -0.3116</td> <td>    0.010</td> <td>  -32.698</td> <td> 0.000</td> <td>   -0.330</td> <td>   -0.293</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.S.L5</th>  <td>    0.9651</td> <td>    0.011</td> <td>   86.972</td> <td> 0.000</td> <td>    0.943</td> <td>    0.987</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ar.S.L10</th> <td>    0.0346</td> <td>    0.010</td> <td>    3.335</td> <td> 0.001</td> <td>    0.014</td> <td>    0.055</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ma.S.L5</th>  <td>   -0.9988</td> <td>    0.006</td> <td> -170.936</td> <td> 0.000</td> <td>   -1.010</td> <td>   -0.987</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>sigma2</th>   <td> 2444.6455</td> <td>   28.648</td> <td>   85.334</td> <td> 0.000</td> <td> 2388.496</td> <td> 2500.795</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Ljung-Box (Q):</th>          <td>95.55</td> <th>  Jarque-Bera (JB):  </th> <td>4202.36</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Q):</th>                <td>0.00</td>  <th>  Prob(JB):          </th>  <td>0.00</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Heteroskedasticity (H):</th> <td>1.23</td>  <th>  Skew:              </th>  <td>-0.52</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(H) (two-sided):</th>    <td>0.00</td>  <th>  Kurtosis:          </th>  <td>7.36</td>  \n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Covariance matrix calculated using the outer product of gradients (complex-step)."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                                 Statespace Model Results                                \n",
       "=========================================================================================\n",
       "Dep. Variable:                      market_value   No. Observations:                 5021\n",
       "Model:             SARIMAX(1, 0, 1)x(2, 0, 1, 5)   Log Likelihood              -26712.642\n",
       "Date:                           Fri, 29 Nov 2019   AIC                          53439.284\n",
       "Time:                                   11:04:55   BIC                          53484.933\n",
       "Sample:                               01-07-1994   HQIC                         53455.280\n",
       "                                    - 04-05-2013                                         \n",
       "Covariance Type:                             opg                                         \n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "spx            2.7041      0.035     76.450      0.000       2.635       2.773\n",
       "ar.L1          0.9998      0.000   2270.454      0.000       0.999       1.001\n",
       "ma.L1         -0.3116      0.010    -32.698      0.000      -0.330      -0.293\n",
       "ar.S.L5        0.9651      0.011     86.972      0.000       0.943       0.987\n",
       "ar.S.L10       0.0346      0.010      3.335      0.001       0.014       0.055\n",
       "ma.S.L5       -0.9988      0.006   -170.936      0.000      -1.010      -0.987\n",
       "sigma2      2444.6455     28.648     85.334      0.000    2388.496    2500.795\n",
       "===================================================================================\n",
       "Ljung-Box (Q):                       95.55   Jarque-Bera (JB):              4202.36\n",
       "Prob(Q):                              0.00   Prob(JB):                         0.00\n",
       "Heteroskedasticity (H):               1.23   Skew:                            -0.52\n",
       "Prob(H) (two-sided):                  0.00   Kurtosis:                         7.36\n",
       "===================================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n",
       "\"\"\""
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_sarimax = SARIMAX(df.market_value, exog = df.spx, order=(1,0,1), seasonal_order = (2,0,1,5))\n",
    "results_sarimax = model_sarimax.fit()\n",
    "results_sarimax.summary()"
   ]
  }
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