├── README.md
├── 0. Updating The Dataset.ipynb
├── 11. Auto ARIMA.ipynb
├── 10. The GARCH model.ipynb
└── 4. Analyzing prices using the AR model.ipynb


/README.md:
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1 | # Time-series-analysis-in-Python
2 | I perform time series analysis of data from scratch. I also implement The Autoregressive (AR) Model, The Moving Average (MA) Model, The Autoregressive Moving Average (ARMA) Model, The Autoregressive Integrated Moving Average (ARIMA) Model, The ARCH Model, The GARCH model, Auto ARIMA, forecasting and exploring a business case.
3 | 
4 | Index2018.csv is the dataset used for time series analysis in this project
5 | 


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/0. Updating The Dataset.ipynb:
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  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "code",
  5 |    "execution_count": 1,
  6 |    "metadata": {},
  7 |    "outputs": [],
  8 |    "source": [
  9 |     "# Importing the necessary package \n",
 10 |     "import yfinance "
 11 |    ]
 12 |   },
 13 |   {
 14 |    "cell_type": "code",
 15 |    "execution_count": 2,
 16 |    "metadata": {},
 17 |    "outputs": [],
 18 |    "source": [
 19 |     "# Ignoring warning messages\n",
 20 |     "import warnings \n",
 21 |     "warnings.filterwarnings(\"ignore\")"
 22 |    ]
 23 |   },
 24 |   {
 25 |    "cell_type": "code",
 26 |    "execution_count": 3,
 27 |    "metadata": {},
 28 |    "outputs": [
 29 |     {
 30 |      "name": "stdout",
 31 |      "output_type": "stream",
 32 |      "text": [
 33 |       "[*********************100%***********************]  4 of 4 downloaded\n"
 34 |      ]
 35 |     }
 36 |    ],
 37 |    "source": [
 38 |     "# Using the .download() method to get our data\n",
 39 |     "\n",
 40 |     "raw_data = yfinance.download (tickers = \"^GSPC ^FTSE ^N225 ^GDAXI\", start = \"1994-01-07\", end = \"2019-09-27\", interval = \"1d\", group_by = 'ticker', auto_adjust = True, treads = True)\n",
 41 |     "\n",
 42 |     "# tickers -> The time series we are interested in - (in our case, these are the S&P, FTSE, NIKKEI and DAX)\n",
 43 |     "# start -> The starting date of our data set\n",
 44 |     "# end -> The ending date of our data set (at the time of upload, this is the current date)\n",
 45 |     "# interval -> The distance in time between two recorded observations. Since we're using daily closing prices, we set it equal to \"1d\", which indicates 1 day. \n",
 46 |     "# group_by -> The way we want to group the scraped data. Usually we want it to be \"ticker\", so that we have all the information about a time series in 1 variable.\n",
 47 |     "# auto_adjust -> Automatically adjust the closing prices for each period. \n",
 48 |     "# treads - > Whether to use threads for mass downloading. "
 49 |    ]
 50 |   },
 51 |   {
 52 |    "cell_type": "code",
 53 |    "execution_count": 4,
 54 |    "metadata": {},
 55 |    "outputs": [],
 56 |    "source": [
 57 |     "# Creating a back up copy in case we remove/alter elements of the data by mistake\n",
 58 |     "df_comp = raw_data.copy()"
 59 |    ]
 60 |   },
 61 |   {
 62 |    "cell_type": "code",
 63 |    "execution_count": 5,
 64 |    "metadata": {},
 65 |    "outputs": [],
 66 |    "source": [
 67 |     "# Adding new columns to the data set\n",
 68 |     "df_comp['spx'] = df_comp['^GSPC'].Close\n",
 69 |     "df_comp['dax'] = df_comp['^GDAXI'].Close\n",
 70 |     "df_comp['ftse'] = df_comp['^FTSE'].Close\n",
 71 |     "df_comp['nikkei'] = df_comp['^N225'].Close"
 72 |    ]
 73 |   },
 74 |   {
 75 |    "cell_type": "code",
 76 |    "execution_count": 6,
 77 |    "metadata": {
 78 |     "scrolled": true
 79 |    },
 80 |    "outputs": [],
 81 |    "source": [
 82 |     "df_comp = df_comp.iloc[1:] # Removing the first elements, since we always start 1 period before the first, due to time zone differences of closing prices\n",
 83 |     "del df_comp['^N225']  # Removing the original tickers of the data set\n",
 84 |     "del df_comp['^GSPC']\n",
 85 |     "del df_comp['^GDAXI']\n",
 86 |     "del df_comp['^FTSE']\n",
 87 |     "df_comp=df_comp.asfreq('b') # Setting the frequency of the data\n",
 88 |     "df_comp=df_comp.fillna(method='ffill') # Filling any missing values"
 89 |    ]
 90 |   },
 91 |   {
 92 |    "cell_type": "code",
 93 |    "execution_count": 7,
 94 |    "metadata": {},
 95 |    "outputs": [
 96 |     {
 97 |      "name": "stdout",
 98 |      "output_type": "stream",
 99 |      "text": [
100 |       "               spx      dax    ftse    nikkei\n",
101 |       "                                             \n",
102 |       "Date                                         \n",
103 |       "1994-01-07  469.90  2224.95  3446.0  18124.01\n",
104 |       "1994-01-10  475.27  2225.00  3440.6  18443.44\n",
105 |       "1994-01-11  474.13  2228.10  3413.8  18485.25\n",
106 |       "1994-01-12  474.17  2182.06  3372.0  18793.88\n",
107 |       "1994-01-13  472.47  2142.37  3360.0  18577.26\n",
108 |       "                spx       dax    ftse    nikkei\n",
109 |       "                                               \n",
110 |       "Date                                           \n",
111 |       "2019-09-20  2992.07  12468.01  7344.9  22079.09\n",
112 |       "2019-09-23  2991.78  12342.33  7326.1  22079.09\n",
113 |       "2019-09-24  2966.60  12307.15  7291.4  22098.84\n",
114 |       "2019-09-25  2984.87  12234.18  7290.0  22020.15\n",
115 |       "2019-09-26  2977.62  12288.54  7351.1  22048.24\n"
116 |      ]
117 |     }
118 |    ],
119 |    "source": [
120 |     "print (df_comp.head()) # Displaying the first 5 elements to make sure the data was scraped correctly\n",
121 |     "print (df_comp.tail()) # Making sure the last day we're including in the series are correct"
122 |    ]
123 |   }
124 |  ],
125 |  "metadata": {
126 |   "kernelspec": {
127 |    "display_name": "Python 3",
128 |    "language": "python",
129 |    "name": "python3"
130 |   },
131 |   "language_info": {
132 |    "codemirror_mode": {
133 |     "name": "ipython",
134 |     "version": 3
135 |    },
136 |    "file_extension": ".py",
137 |    "mimetype": "text/x-python",
138 |    "name": "python",
139 |    "nbconvert_exporter": "python",
140 |    "pygments_lexer": "ipython3",
141 |    "version": "3.7.4"
142 |   }
143 |  },
144 |  "nbformat": 4,
145 |  "nbformat_minor": 2
146 | }
147 | 


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/11. Auto ARIMA.ipynb:
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  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "### Packages"
  8 |    ]
  9 |   },
 10 |   {
 11 |    "cell_type": "code",
 12 |    "execution_count": 1,
 13 |    "metadata": {},
 14 |    "outputs": [],
 15 |    "source": [
 16 |     "import numpy as np\n",
 17 |     "import pandas as pd\n",
 18 |     "import scipy\n",
 19 |     "import statsmodels.api as sm\n",
 20 |     "import matplotlib.pyplot as plt\n",
 21 |     "import seaborn as sns\n",
 22 |     "import sklearn\n",
 23 |     "from statsmodels.tsa.arima_model import ARIMA\n",
 24 |     "from arch import arch_model\n",
 25 |     "import seaborn as sns\n",
 26 |     "import yfinance\n",
 27 |     "import warnings\n",
 28 |     "warnings.filterwarnings(\"ignore\")\n",
 29 |     "sns.set()"
 30 |    ]
 31 |   },
 32 |   {
 33 |    "cell_type": "markdown",
 34 |    "metadata": {},
 35 |    "source": [
 36 |     "### Loading the data"
 37 |    ]
 38 |   },
 39 |   {
 40 |    "cell_type": "code",
 41 |    "execution_count": 2,
 42 |    "metadata": {},
 43 |    "outputs": [
 44 |     {
 45 |      "name": "stdout",
 46 |      "output_type": "stream",
 47 |      "text": [
 48 |       "[*********************100%***********************]  4 of 4 downloaded\n"
 49 |      ]
 50 |     }
 51 |    ],
 52 |    "source": [
 53 |     "raw_data = yfinance.download (tickers = \"^GSPC ^FTSE ^N225 ^GDAXI\", start = \"1994-01-07\", end = \"2018-01-29\", \n",
 54 |     "                              interval = \"1d\", group_by = 'ticker', auto_adjust = True, treads = True)"
 55 |    ]
 56 |   },
 57 |   {
 58 |    "cell_type": "code",
 59 |    "execution_count": 3,
 60 |    "metadata": {},
 61 |    "outputs": [],
 62 |    "source": [
 63 |     "df_comp = raw_data.copy()"
 64 |    ]
 65 |   },
 66 |   {
 67 |    "cell_type": "code",
 68 |    "execution_count": 4,
 69 |    "metadata": {},
 70 |    "outputs": [],
 71 |    "source": [
 72 |     "df_comp['spx'] = df_comp['^GSPC'].Close[:]\n",
 73 |     "df_comp['dax'] = df_comp['^GDAXI'].Close[:]\n",
 74 |     "df_comp['ftse'] = df_comp['^FTSE'].Close[:]\n",
 75 |     "df_comp['nikkei'] = df_comp['^N225'].Close[:]"
 76 |    ]
 77 |   },
 78 |   {
 79 |    "cell_type": "code",
 80 |    "execution_count": 5,
 81 |    "metadata": {
 82 |     "scrolled": true
 83 |    },
 84 |    "outputs": [],
 85 |    "source": [
 86 |     "df_comp = df_comp.iloc[1:]\n",
 87 |     "del df_comp['^N225']\n",
 88 |     "del df_comp['^GSPC']\n",
 89 |     "del df_comp['^GDAXI']\n",
 90 |     "del df_comp['^FTSE']\n",
 91 |     "df_comp=df_comp.asfreq('b')\n",
 92 |     "df_comp=df_comp.fillna(method='ffill')"
 93 |    ]
 94 |   },
 95 |   {
 96 |    "cell_type": "markdown",
 97 |    "metadata": {},
 98 |    "source": [
 99 |     "### Creating Returns"
100 |    ]
101 |   },
102 |   {
103 |    "cell_type": "code",
104 |    "execution_count": 6,
105 |    "metadata": {},
106 |    "outputs": [],
107 |    "source": [
108 |     "df_comp['ret_spx'] = df_comp.spx.pct_change(1)*100\n",
109 |     "df_comp['ret_ftse'] = df_comp.ftse.pct_change(1)*100\n",
110 |     "df_comp['ret_dax'] = df_comp.dax.pct_change(1)*100\n",
111 |     "df_comp['ret_nikkei'] = df_comp.nikkei.pct_change(1)*100"
112 |    ]
113 |   },
114 |   {
115 |    "cell_type": "markdown",
116 |    "metadata": {},
117 |    "source": [
118 |     "### Splitting the Data"
119 |    ]
120 |   },
121 |   {
122 |    "cell_type": "code",
123 |    "execution_count": 7,
124 |    "metadata": {},
125 |    "outputs": [],
126 |    "source": [
127 |     "size = int(len(df_comp)*0.8)\n",
128 |     "df, df_test = df_comp.iloc[:size], df_comp.iloc[size:]"
129 |    ]
130 |   },
131 |   {
132 |    "cell_type": "markdown",
133 |    "metadata": {},
134 |    "source": [
135 |     "### Fitting a Model"
136 |    ]
137 |   },
138 |   {
139 |    "cell_type": "code",
140 |    "execution_count": 8,
141 |    "metadata": {},
142 |    "outputs": [],
143 |    "source": [
144 |     "from pmdarima.arima import auto_arima"
145 |    ]
146 |   },
147 |   {
148 |    "cell_type": "code",
149 |    "execution_count": 9,
150 |    "metadata": {},
151 |    "outputs": [],
152 |    "source": [
153 |     "model_auto = auto_arima(df.ret_ftse[1:])"
154 |    ]
155 |   },
156 |   {
157 |    "cell_type": "code",
158 |    "execution_count": 10,
159 |    "metadata": {},
160 |    "outputs": [
161 |     {
162 |      "data": {
163 |       "text/plain": [
164 |        "ARIMA(maxiter=50, method='lbfgs', order=(2, 0, 5), out_of_sample_size=0,\n",
165 |        "   scoring='mse', scoring_args=None, seasonal_order=(0, 0, 0, 1),\n",
166 |        "   start_params=None, suppress_warnings=False, trend=None,\n",
167 |        "   with_intercept=True)"
168 |       ]
169 |      },
170 |      "execution_count": 10,
171 |      "metadata": {},
172 |      "output_type": "execute_result"
173 |     }
174 |    ],
175 |    "source": [
176 |     "model_auto"
177 |    ]
178 |   },
179 |   {
180 |    "cell_type": "code",
181 |    "execution_count": 11,
182 |    "metadata": {},
183 |    "outputs": [
184 |     {
185 |      "data": {
186 |       "text/html": [
187 |        "<table class=\"simpletable\">\n",
188 |        "<caption>Statespace Model Results</caption>\n",
189 |        "<tr>\n",
190 |        "  <th>Dep. Variable:</th>           <td>y</td>        <th>  No. Observations:  </th>   <td>5020</td>   \n",
191 |        "</tr>\n",
192 |        "<tr>\n",
193 |        "  <th>Model:</th>           <td>SARIMAX(2, 0, 5)</td> <th>  Log Likelihood     </th> <td>-7885.690</td>\n",
194 |        "</tr>\n",
195 |        "<tr>\n",
196 |        "  <th>Date:</th>            <td>Fri, 03 Jan 2020</td> <th>  AIC                </th> <td>15789.380</td>\n",
197 |        "</tr>\n",
198 |        "<tr>\n",
199 |        "  <th>Time:</th>                <td>10:19:03</td>     <th>  BIC                </th> <td>15848.070</td>\n",
200 |        "</tr>\n",
201 |        "<tr>\n",
202 |        "  <th>Sample:</th>                  <td>0</td>        <th>  HQIC               </th> <td>15809.946</td>\n",
203 |        "</tr>\n",
204 |        "<tr>\n",
205 |        "  <th></th>                      <td> - 5020</td>     <th>                     </th>     <td> </td>    \n",
206 |        "</tr>\n",
207 |        "<tr>\n",
208 |        "  <th>Covariance Type:</th>        <td>opg</td>       <th>                     </th>     <td> </td>    \n",
209 |        "</tr>\n",
210 |        "</table>\n",
211 |        "<table class=\"simpletable\">\n",
212 |        "<tr>\n",
213 |        "      <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",
214 |        "</tr>\n",
215 |        "<tr>\n",
216 |        "  <th>intercept</th> <td>    0.0309</td> <td>    0.024</td> <td>    1.289</td> <td> 0.197</td> <td>   -0.016</td> <td>    0.078</td>\n",
217 |        "</tr>\n",
218 |        "<tr>\n",
219 |        "  <th>ar.L1</th>     <td>    0.1766</td> <td>    0.039</td> <td>    4.544</td> <td> 0.000</td> <td>    0.100</td> <td>    0.253</td>\n",
220 |        "</tr>\n",
221 |        "<tr>\n",
222 |        "  <th>ar.L2</th>     <td>   -0.8128</td> <td>    0.035</td> <td>  -22.984</td> <td> 0.000</td> <td>   -0.882</td> <td>   -0.743</td>\n",
223 |        "</tr>\n",
224 |        "<tr>\n",
225 |        "  <th>ma.L1</th>     <td>   -0.2005</td> <td>    0.038</td> <td>   -5.239</td> <td> 0.000</td> <td>   -0.275</td> <td>   -0.125</td>\n",
226 |        "</tr>\n",
227 |        "<tr>\n",
228 |        "  <th>ma.L2</th>     <td>    0.7654</td> <td>    0.037</td> <td>   20.436</td> <td> 0.000</td> <td>    0.692</td> <td>    0.839</td>\n",
229 |        "</tr>\n",
230 |        "<tr>\n",
231 |        "  <th>ma.L3</th>     <td>   -0.0953</td> <td>    0.012</td> <td>   -8.246</td> <td> 0.000</td> <td>   -0.118</td> <td>   -0.073</td>\n",
232 |        "</tr>\n",
233 |        "<tr>\n",
234 |        "  <th>ma.L4</th>     <td>    0.0112</td> <td>    0.009</td> <td>    1.229</td> <td> 0.219</td> <td>   -0.007</td> <td>    0.029</td>\n",
235 |        "</tr>\n",
236 |        "<tr>\n",
237 |        "  <th>ma.L5</th>     <td>   -0.1113</td> <td>    0.009</td> <td>  -12.960</td> <td> 0.000</td> <td>   -0.128</td> <td>   -0.094</td>\n",
238 |        "</tr>\n",
239 |        "<tr>\n",
240 |        "  <th>sigma2</th>    <td>    1.3550</td> <td>    0.014</td> <td>   94.014</td> <td> 0.000</td> <td>    1.327</td> <td>    1.383</td>\n",
241 |        "</tr>\n",
242 |        "</table>\n",
243 |        "<table class=\"simpletable\">\n",
244 |        "<tr>\n",
245 |        "  <th>Ljung-Box (Q):</th>          <td>69.64</td> <th>  Jarque-Bera (JB):  </th> <td>6575.67</td>\n",
246 |        "</tr>\n",
247 |        "<tr>\n",
248 |        "  <th>Prob(Q):</th>                <td>0.00</td>  <th>  Prob(JB):          </th>  <td>0.00</td>  \n",
249 |        "</tr>\n",
250 |        "<tr>\n",
251 |        "  <th>Heteroskedasticity (H):</th> <td>2.00</td>  <th>  Skew:              </th>  <td>-0.18</td> \n",
252 |        "</tr>\n",
253 |        "<tr>\n",
254 |        "  <th>Prob(H) (two-sided):</th>    <td>0.00</td>  <th>  Kurtosis:          </th>  <td>8.60</td>  \n",
255 |        "</tr>\n",
256 |        "</table><br/><br/>Warnings:<br/>[1] Covariance matrix calculated using the outer product of gradients (complex-step)."
257 |       ],
258 |       "text/plain": [
259 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
260 |        "\"\"\"\n",
261 |        "                           Statespace Model Results                           \n",
262 |        "==============================================================================\n",
263 |        "Dep. Variable:                      y   No. Observations:                 5020\n",
264 |        "Model:               SARIMAX(2, 0, 5)   Log Likelihood               -7885.690\n",
265 |        "Date:                Fri, 03 Jan 2020   AIC                          15789.380\n",
266 |        "Time:                        10:19:03   BIC                          15848.070\n",
267 |        "Sample:                             0   HQIC                         15809.946\n",
268 |        "                               - 5020                                         \n",
269 |        "Covariance Type:                  opg                                         \n",
270 |        "==============================================================================\n",
271 |        "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
272 |        "------------------------------------------------------------------------------\n",
273 |        "intercept      0.0309      0.024      1.289      0.197      -0.016       0.078\n",
274 |        "ar.L1          0.1766      0.039      4.544      0.000       0.100       0.253\n",
275 |        "ar.L2         -0.8128      0.035    -22.984      0.000      -0.882      -0.743\n",
276 |        "ma.L1         -0.2005      0.038     -5.239      0.000      -0.275      -0.125\n",
277 |        "ma.L2          0.7654      0.037     20.436      0.000       0.692       0.839\n",
278 |        "ma.L3         -0.0953      0.012     -8.246      0.000      -0.118      -0.073\n",
279 |        "ma.L4          0.0112      0.009      1.229      0.219      -0.007       0.029\n",
280 |        "ma.L5         -0.1113      0.009    -12.960      0.000      -0.128      -0.094\n",
281 |        "sigma2         1.3550      0.014     94.014      0.000       1.327       1.383\n",
282 |        "===================================================================================\n",
283 |        "Ljung-Box (Q):                       69.64   Jarque-Bera (JB):              6575.67\n",
284 |        "Prob(Q):                              0.00   Prob(JB):                         0.00\n",
285 |        "Heteroskedasticity (H):               2.00   Skew:                            -0.18\n",
286 |        "Prob(H) (two-sided):                  0.00   Kurtosis:                         8.60\n",
287 |        "===================================================================================\n",
288 |        "\n",
289 |        "Warnings:\n",
290 |        "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n",
291 |        "\"\"\""
292 |       ]
293 |      },
294 |      "execution_count": 11,
295 |      "metadata": {},
296 |      "output_type": "execute_result"
297 |     }
298 |    ],
299 |    "source": [
300 |     "model_auto.summary()"
301 |    ]
302 |   },
303 |   {
304 |    "cell_type": "markdown",
305 |    "metadata": {},
306 |    "source": [
307 |     "### Important Arguments"
308 |    ]
309 |   },
310 |   {
311 |    "cell_type": "code",
312 |    "execution_count": 14,
313 |    "metadata": {
314 |     "scrolled": true
315 |    },
316 |    "outputs": [],
317 |    "source": [
318 |     "model_auto = auto_arima(df_comp.ret_ftse[1:], exogenous = df_comp[['ret_spx', 'ret_dax', 'ret_nikkei']][1:], m = 5,\n",
319 |     "                       max_order = None, max_p = 7, max_q = 7, max_d = 2, max_P = 4, max_Q = 4, max_D = 2,\n",
320 |     "                       maxiter = 50, alpha = 0.05, n_jobs = -1, trend = 'ct', information_criterion = 'oob',\n",
321 |     "                       out_of_sample = int(len(df_comp)*0.2))\n",
322 |     "\n",
323 |     "\n",
324 |     "# !!! Important Note: In pdmarima v1.5.2, out_of_sample_size is replaced with out_of_sample, so make sure to use the latter!\n",
325 |     "\n",
326 |     "\n",
327 |     "# exogenous -> outside factors (e.g other time series)\n",
328 |     "# m -> seasonal cycle length\n",
329 |     "# max_order -> maximum amount of variables to be used in the regression (p + q)\n",
330 |     "# max_p -> maximum AR components\n",
331 |     "# max_q -> maximum MA components\n",
332 |     "# max_d -> maximum Integrations\n",
333 |     "# maxiter -> maximum iterations we're giving the model to converge the coefficients (becomes harder as the order increases)\n",
334 |     "# alpha -> level of significance, default is 5%, which we should be using most of the time\n",
335 |     "# n_jobs -> how many models to fit at a time (-1 indicates \"as many as possible\")\n",
336 |     "# trend -> \"ct\" usually\n",
337 |     "# information_criterion -> 'aic', 'aicc', 'bic', 'hqic', 'oob' \n",
338 |     "#        (Akaike Information Criterion, Corrected Akaike Information Criterion,\n",
339 |     "#        Bayesian Information Criterion, Hannan-Quinn Information Criterion, or\n",
340 |     "#        \"out of bag\"--for validation scoring--respectively)\n",
341 |     "# out_of_smaple -> validates the model selection (pass the entire dataset, and set 20% to be the out_of_sample_size)"
342 |    ]
343 |   },
344 |   {
345 |    "cell_type": "code",
346 |    "execution_count": 15,
347 |    "metadata": {},
348 |    "outputs": [
349 |     {
350 |      "data": {
351 |       "text/html": [
352 |        "<table class=\"simpletable\">\n",
353 |        "<caption>Statespace Model Results</caption>\n",
354 |        "<tr>\n",
355 |        "  <th>Dep. Variable:</th>                 <td>y</td>               <th>  No. Observations:  </th>   <td>6276</td>   \n",
356 |        "</tr>\n",
357 |        "<tr>\n",
358 |        "  <th>Model:</th>           <td>SARIMAX(2, 0, 2)x(1, 0, 1, 5)</td> <th>  Log Likelihood     </th> <td>-6340.661</td>\n",
359 |        "</tr>\n",
360 |        "<tr>\n",
361 |        "  <th>Date:</th>                  <td>Fri, 03 Jan 2020</td>        <th>  AIC                </th> <td>12705.322</td>\n",
362 |        "</tr>\n",
363 |        "<tr>\n",
364 |        "  <th>Time:</th>                      <td>10:34:50</td>            <th>  BIC                </th> <td>12786.256</td>\n",
365 |        "</tr>\n",
366 |        "<tr>\n",
367 |        "  <th>Sample:</th>                   <td>01-10-1994</td>           <th>  HQIC               </th> <td>12733.364</td>\n",
368 |        "</tr>\n",
369 |        "<tr>\n",
370 |        "  <th></th>                         <td>- 01-29-2018</td>          <th>                     </th>     <td> </td>    \n",
371 |        "</tr>\n",
372 |        "<tr>\n",
373 |        "  <th>Covariance Type:</th>              <td>opg</td>              <th>                     </th>     <td> </td>    \n",
374 |        "</tr>\n",
375 |        "</table>\n",
376 |        "<table class=\"simpletable\">\n",
377 |        "<tr>\n",
378 |        "       <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",
379 |        "</tr>\n",
380 |        "<tr>\n",
381 |        "  <th>intercept</th>  <td>   -0.0043</td> <td>    0.009</td> <td>   -0.500</td> <td> 0.617</td> <td>   -0.021</td> <td>    0.013</td>\n",
382 |        "</tr>\n",
383 |        "<tr>\n",
384 |        "  <th>drift</th>      <td> -1.45e-06</td> <td> 2.72e-06</td> <td>   -0.533</td> <td> 0.594</td> <td>-6.78e-06</td> <td> 3.88e-06</td>\n",
385 |        "</tr>\n",
386 |        "<tr>\n",
387 |        "  <th>ret_spx</th>    <td>    0.0958</td> <td>    0.006</td> <td>   16.947</td> <td> 0.000</td> <td>    0.085</td> <td>    0.107</td>\n",
388 |        "</tr>\n",
389 |        "<tr>\n",
390 |        "  <th>ret_dax</th>    <td>    0.5582</td> <td>    0.005</td> <td>  113.729</td> <td> 0.000</td> <td>    0.549</td> <td>    0.568</td>\n",
391 |        "</tr>\n",
392 |        "<tr>\n",
393 |        "  <th>ret_nikkei</th> <td>    0.0702</td> <td>    0.004</td> <td>   16.547</td> <td> 0.000</td> <td>    0.062</td> <td>    0.079</td>\n",
394 |        "</tr>\n",
395 |        "<tr>\n",
396 |        "  <th>ar.L1</th>      <td>   -0.1932</td> <td>    0.101</td> <td>   -1.918</td> <td> 0.055</td> <td>   -0.390</td> <td>    0.004</td>\n",
397 |        "</tr>\n",
398 |        "<tr>\n",
399 |        "  <th>ar.L2</th>      <td>    0.5083</td> <td>    0.059</td> <td>    8.643</td> <td> 0.000</td> <td>    0.393</td> <td>    0.624</td>\n",
400 |        "</tr>\n",
401 |        "<tr>\n",
402 |        "  <th>ma.L1</th>      <td>    0.0863</td> <td>    0.101</td> <td>    0.855</td> <td> 0.393</td> <td>   -0.112</td> <td>    0.284</td>\n",
403 |        "</tr>\n",
404 |        "<tr>\n",
405 |        "  <th>ma.L2</th>      <td>   -0.5369</td> <td>    0.056</td> <td>   -9.535</td> <td> 0.000</td> <td>   -0.647</td> <td>   -0.427</td>\n",
406 |        "</tr>\n",
407 |        "<tr>\n",
408 |        "  <th>ar.S.L5</th>    <td>    0.0705</td> <td>    0.339</td> <td>    0.208</td> <td> 0.836</td> <td>   -0.595</td> <td>    0.736</td>\n",
409 |        "</tr>\n",
410 |        "<tr>\n",
411 |        "  <th>ma.S.L5</th>    <td>   -0.0987</td> <td>    0.339</td> <td>   -0.291</td> <td> 0.771</td> <td>   -0.764</td> <td>    0.567</td>\n",
412 |        "</tr>\n",
413 |        "<tr>\n",
414 |        "  <th>sigma2</th>     <td>    0.4404</td> <td>    0.004</td> <td>   99.131</td> <td> 0.000</td> <td>    0.432</td> <td>    0.449</td>\n",
415 |        "</tr>\n",
416 |        "</table>\n",
417 |        "<table class=\"simpletable\">\n",
418 |        "<tr>\n",
419 |        "  <th>Ljung-Box (Q):</th>          <td>84.27</td> <th>  Jarque-Bera (JB):  </th> <td>15325.76</td>\n",
420 |        "</tr>\n",
421 |        "<tr>\n",
422 |        "  <th>Prob(Q):</th>                <td>0.00</td>  <th>  Prob(JB):          </th>   <td>0.00</td>  \n",
423 |        "</tr>\n",
424 |        "<tr>\n",
425 |        "  <th>Heteroskedasticity (H):</th> <td>0.54</td>  <th>  Skew:              </th>   <td>0.25</td>  \n",
426 |        "</tr>\n",
427 |        "<tr>\n",
428 |        "  <th>Prob(H) (two-sided):</th>    <td>0.00</td>  <th>  Kurtosis:          </th>   <td>10.64</td> \n",
429 |        "</tr>\n",
430 |        "</table><br/><br/>Warnings:<br/>[1] Covariance matrix calculated using the outer product of gradients (complex-step)."
431 |       ],
432 |       "text/plain": [
433 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
434 |        "\"\"\"\n",
435 |        "                                 Statespace Model Results                                \n",
436 |        "=========================================================================================\n",
437 |        "Dep. Variable:                                 y   No. Observations:                 6276\n",
438 |        "Model:             SARIMAX(2, 0, 2)x(1, 0, 1, 5)   Log Likelihood               -6340.661\n",
439 |        "Date:                           Fri, 03 Jan 2020   AIC                          12705.322\n",
440 |        "Time:                                   10:34:50   BIC                          12786.256\n",
441 |        "Sample:                               01-10-1994   HQIC                         12733.364\n",
442 |        "                                    - 01-29-2018                                         \n",
443 |        "Covariance Type:                             opg                                         \n",
444 |        "==============================================================================\n",
445 |        "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
446 |        "------------------------------------------------------------------------------\n",
447 |        "intercept     -0.0043      0.009     -0.500      0.617      -0.021       0.013\n",
448 |        "drift       -1.45e-06   2.72e-06     -0.533      0.594   -6.78e-06    3.88e-06\n",
449 |        "ret_spx        0.0958      0.006     16.947      0.000       0.085       0.107\n",
450 |        "ret_dax        0.5582      0.005    113.729      0.000       0.549       0.568\n",
451 |        "ret_nikkei     0.0702      0.004     16.547      0.000       0.062       0.079\n",
452 |        "ar.L1         -0.1932      0.101     -1.918      0.055      -0.390       0.004\n",
453 |        "ar.L2          0.5083      0.059      8.643      0.000       0.393       0.624\n",
454 |        "ma.L1          0.0863      0.101      0.855      0.393      -0.112       0.284\n",
455 |        "ma.L2         -0.5369      0.056     -9.535      0.000      -0.647      -0.427\n",
456 |        "ar.S.L5        0.0705      0.339      0.208      0.836      -0.595       0.736\n",
457 |        "ma.S.L5       -0.0987      0.339     -0.291      0.771      -0.764       0.567\n",
458 |        "sigma2         0.4404      0.004     99.131      0.000       0.432       0.449\n",
459 |        "===================================================================================\n",
460 |        "Ljung-Box (Q):                       84.27   Jarque-Bera (JB):             15325.76\n",
461 |        "Prob(Q):                              0.00   Prob(JB):                         0.00\n",
462 |        "Heteroskedasticity (H):               0.54   Skew:                             0.25\n",
463 |        "Prob(H) (two-sided):                  0.00   Kurtosis:                        10.64\n",
464 |        "===================================================================================\n",
465 |        "\n",
466 |        "Warnings:\n",
467 |        "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n",
468 |        "\"\"\""
469 |       ]
470 |      },
471 |      "execution_count": 15,
472 |      "metadata": {},
473 |      "output_type": "execute_result"
474 |     }
475 |    ],
476 |    "source": [
477 |     "model_auto.summary()"
478 |    ]
479 |   }
480 |  ],
481 |  "metadata": {
482 |   "kernelspec": {
483 |    "display_name": "Python 3",
484 |    "language": "python",
485 |    "name": "python3"
486 |   },
487 |   "language_info": {
488 |    "codemirror_mode": {
489 |     "name": "ipython",
490 |     "version": 3
491 |    },
492 |    "file_extension": ".py",
493 |    "mimetype": "text/x-python",
494 |    "name": "python",
495 |    "nbconvert_exporter": "python",
496 |    "pygments_lexer": "ipython3",
497 |    "version": "3.6.7"
498 |   }
499 |  },
500 |  "nbformat": 4,
501 |  "nbformat_minor": 2
502 | }
503 | 


--------------------------------------------------------------------------------
/10. The GARCH model.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "## Importing the relevant packages"
  8 |    ]
  9 |   },
 10 |   {
 11 |    "cell_type": "code",
 12 |    "execution_count": 1,
 13 |    "metadata": {},
 14 |    "outputs": [],
 15 |    "source": [
 16 |     "import pandas as pd\n",
 17 |     "import numpy as np\n",
 18 |     "import matplotlib.pyplot as plt\n",
 19 |     "import statsmodels.graphics.tsaplots as sgt\n",
 20 |     "import statsmodels.tsa.stattools as sts\n",
 21 |     "from statsmodels.tsa.arima_model import ARIMA\n",
 22 |     "from scipy.stats.distributions import chi2 \n",
 23 |     "from arch import arch_model\n",
 24 |     "from math import sqrt\n",
 25 |     "import seaborn as sns\n",
 26 |     "sns.set()"
 27 |    ]
 28 |   },
 29 |   {
 30 |    "cell_type": "markdown",
 31 |    "metadata": {},
 32 |    "source": [
 33 |     " "
 34 |    ]
 35 |   },
 36 |   {
 37 |    "cell_type": "markdown",
 38 |    "metadata": {},
 39 |    "source": [
 40 |     "## Importing the Data and Pre-processing "
 41 |    ]
 42 |   },
 43 |   {
 44 |    "cell_type": "code",
 45 |    "execution_count": 2,
 46 |    "metadata": {},
 47 |    "outputs": [],
 48 |    "source": [
 49 |     "raw_csv_data = pd.read_csv(\"Index2018.csv\") \n",
 50 |     "df_comp=raw_csv_data.copy()\n",
 51 |     "df_comp.date = pd.to_datetime(df_comp.date, dayfirst = True)\n",
 52 |     "df_comp.set_index(\"date\", inplace=True)\n",
 53 |     "df_comp=df_comp.asfreq('b')\n",
 54 |     "df_comp=df_comp.fillna(method='ffill')"
 55 |    ]
 56 |   },
 57 |   {
 58 |    "cell_type": "code",
 59 |    "execution_count": 3,
 60 |    "metadata": {},
 61 |    "outputs": [],
 62 |    "source": [
 63 |     "df_comp['market_value']=df_comp.ftse"
 64 |    ]
 65 |   },
 66 |   {
 67 |    "cell_type": "code",
 68 |    "execution_count": 4,
 69 |    "metadata": {},
 70 |    "outputs": [],
 71 |    "source": [
 72 |     "del df_comp['spx']\n",
 73 |     "del df_comp['dax']\n",
 74 |     "del df_comp['ftse']\n",
 75 |     "del df_comp['nikkei']\n",
 76 |     "size = int(len(df_comp)*0.8)\n",
 77 |     "df, df_test = df_comp.iloc[:size], df_comp.iloc[size:]"
 78 |    ]
 79 |   },
 80 |   {
 81 |    "cell_type": "code",
 82 |    "execution_count": 5,
 83 |    "metadata": {},
 84 |    "outputs": [],
 85 |    "source": [
 86 |     "import warnings\n",
 87 |     "warnings.filterwarnings(\"ignore\")"
 88 |    ]
 89 |   },
 90 |   {
 91 |    "cell_type": "markdown",
 92 |    "metadata": {},
 93 |    "source": [
 94 |     " "
 95 |    ]
 96 |   },
 97 |   {
 98 |    "cell_type": "markdown",
 99 |    "metadata": {},
100 |    "source": [
101 |     "## The LLR Test"
102 |    ]
103 |   },
104 |   {
105 |    "cell_type": "code",
106 |    "execution_count": 6,
107 |    "metadata": {},
108 |    "outputs": [],
109 |    "source": [
110 |     "def LLR_test(mod_1, mod_2, DF = 1):\n",
111 |     "    L1 = mod_1.fit(start_ar_lags = 11).llf\n",
112 |     "    L2 = mod_2.fit(start_ar_lags = 11).llf\n",
113 |     "    LR = (2*(L2-L1))    \n",
114 |     "    p = chi2.sf(LR, DF).round(3)\n",
115 |     "    return p"
116 |    ]
117 |   },
118 |   {
119 |    "cell_type": "markdown",
120 |    "metadata": {},
121 |    "source": [
122 |     "## Creating Returns"
123 |    ]
124 |   },
125 |   {
126 |    "cell_type": "code",
127 |    "execution_count": 7,
128 |    "metadata": {
129 |     "scrolled": true
130 |    },
131 |    "outputs": [],
132 |    "source": [
133 |     "df['returns'] = df.market_value.pct_change(1)*100"
134 |    ]
135 |   },
136 |   {
137 |    "cell_type": "markdown",
138 |    "metadata": {},
139 |    "source": [
140 |     "## The Simple GARCH Model"
141 |    ]
142 |   },
143 |   {
144 |    "cell_type": "code",
145 |    "execution_count": 8,
146 |    "metadata": {},
147 |    "outputs": [
148 |     {
149 |      "name": "stdout",
150 |      "output_type": "stream",
151 |      "text": [
152 |       "Iteration:      5,   Func. Count:     39,   Neg. LLF: 6972.734752397977\n",
153 |       "Iteration:     10,   Func. Count:     73,   Neg. LLF: 6970.088043671654\n",
154 |       "Optimization terminated successfully.    (Exit mode 0)\n",
155 |       "            Current function value: 6970.058366189876\n",
156 |       "            Iterations: 13\n",
157 |       "            Function evaluations: 91\n",
158 |       "            Gradient evaluations: 13\n"
159 |      ]
160 |     },
161 |     {
162 |      "data": {
163 |       "text/html": [
164 |        "<table class=\"simpletable\">\n",
165 |        "<caption>Constant Mean - GARCH Model Results</caption>\n",
166 |        "<tr>\n",
167 |        "  <th>Dep. Variable:</th>       <td>returns</td>      <th>  R-squared:         </th>  <td>  -0.001</td> \n",
168 |        "</tr>\n",
169 |        "<tr>\n",
170 |        "  <th>Mean Model:</th>       <td>Constant Mean</td>   <th>  Adj. R-squared:    </th>  <td>  -0.001</td> \n",
171 |        "</tr>\n",
172 |        "<tr>\n",
173 |        "  <th>Vol Model:</th>            <td>GARCH</td>       <th>  Log-Likelihood:    </th> <td>  -6970.06</td>\n",
174 |        "</tr>\n",
175 |        "<tr>\n",
176 |        "  <th>Distribution:</th>        <td>Normal</td>       <th>  AIC:               </th> <td>   13948.1</td>\n",
177 |        "</tr>\n",
178 |        "<tr>\n",
179 |        "  <th>Method:</th>        <td>Maximum Likelihood</td> <th>  BIC:               </th> <td>   13974.2</td>\n",
180 |        "</tr>\n",
181 |        "<tr>\n",
182 |        "  <th></th>                        <td></td>          <th>  No. Observations:  </th>    <td>5020</td>   \n",
183 |        "</tr>\n",
184 |        "<tr>\n",
185 |        "  <th>Date:</th>           <td>Thu, Sep 19 2019</td>  <th>  Df Residuals:      </th>    <td>5016</td>   \n",
186 |        "</tr>\n",
187 |        "<tr>\n",
188 |        "  <th>Time:</th>               <td>17:53:26</td>      <th>  Df Model:          </th>      <td>4</td>    \n",
189 |        "</tr>\n",
190 |        "</table>\n",
191 |        "<table class=\"simpletable\">\n",
192 |        "<caption>Mean Model</caption>\n",
193 |        "<tr>\n",
194 |        "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
195 |        "</tr>\n",
196 |        "<tr>\n",
197 |        "  <th>mu</th> <td>    0.0466</td> <td>1.183e-02</td> <td>    3.939</td> <td>8.187e-05</td> <td>[2.342e-02,6.981e-02]</td>\n",
198 |        "</tr>\n",
199 |        "</table>\n",
200 |        "<table class=\"simpletable\">\n",
201 |        "<caption>Volatility Model</caption>\n",
202 |        "<tr>\n",
203 |        "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
204 |        "</tr>\n",
205 |        "<tr>\n",
206 |        "  <th>omega</th>    <td>    0.0109</td> <td>3.004e-03</td> <td>    3.640</td> <td>2.724e-04</td> <td>[5.048e-03,1.682e-02]</td>\n",
207 |        "</tr>\n",
208 |        "<tr>\n",
209 |        "  <th>alpha[1]</th> <td>    0.0835</td> <td>1.071e-02</td> <td>    7.794</td> <td>6.476e-15</td>  <td>[6.249e-02,  0.104]</td> \n",
210 |        "</tr>\n",
211 |        "<tr>\n",
212 |        "  <th>beta[1]</th>  <td>    0.9089</td> <td>1.148e-02</td> <td>   79.168</td>   <td>0.000</td>     <td>[  0.886,  0.931]</td>  \n",
213 |        "</tr>\n",
214 |        "</table><br/><br/>Covariance estimator: robust"
215 |       ],
216 |       "text/plain": [
217 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
218 |        "\"\"\"\n",
219 |        "                     Constant Mean - GARCH Model Results                      \n",
220 |        "==============================================================================\n",
221 |        "Dep. Variable:                returns   R-squared:                      -0.001\n",
222 |        "Mean Model:             Constant Mean   Adj. R-squared:                 -0.001\n",
223 |        "Vol Model:                      GARCH   Log-Likelihood:               -6970.06\n",
224 |        "Distribution:                  Normal   AIC:                           13948.1\n",
225 |        "Method:            Maximum Likelihood   BIC:                           13974.2\n",
226 |        "                                        No. Observations:                 5020\n",
227 |        "Date:                Thu, Sep 19 2019   Df Residuals:                     5016\n",
228 |        "Time:                        17:53:26   Df Model:                            4\n",
229 |        "                                 Mean Model                                 \n",
230 |        "============================================================================\n",
231 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
232 |        "----------------------------------------------------------------------------\n",
233 |        "mu             0.0466  1.183e-02      3.939  8.187e-05 [2.342e-02,6.981e-02]\n",
234 |        "                              Volatility Model                              \n",
235 |        "============================================================================\n",
236 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
237 |        "----------------------------------------------------------------------------\n",
238 |        "omega          0.0109  3.004e-03      3.640  2.724e-04 [5.048e-03,1.682e-02]\n",
239 |        "alpha[1]       0.0835  1.071e-02      7.794  6.476e-15   [6.249e-02,  0.104]\n",
240 |        "beta[1]        0.9089  1.148e-02     79.168      0.000     [  0.886,  0.931]\n",
241 |        "============================================================================\n",
242 |        "\n",
243 |        "Covariance estimator: robust\n",
244 |        "\"\"\""
245 |       ]
246 |      },
247 |      "execution_count": 8,
248 |      "metadata": {},
249 |      "output_type": "execute_result"
250 |     }
251 |    ],
252 |    "source": [
253 |     "model_garch_1_1 = arch_model(df.returns[1:], mean = \"Constant\", vol = \"GARCH\", p = 1, q = 1)\n",
254 |     "results_garch_1_1 = model_garch_1_1.fit(update_freq = 5)\n",
255 |     "results_garch_1_1.summary()"
256 |    ]
257 |   },
258 |   {
259 |    "cell_type": "markdown",
260 |    "metadata": {},
261 |    "source": [
262 |     "## Higher-Lag GARCH Models"
263 |    ]
264 |   },
265 |   {
266 |    "cell_type": "code",
267 |    "execution_count": 9,
268 |    "metadata": {},
269 |    "outputs": [
270 |     {
271 |      "name": "stdout",
272 |      "output_type": "stream",
273 |      "text": [
274 |       "Iteration:      5,   Func. Count:     44,   Neg. LLF: 6978.593988486672\n",
275 |       "Iteration:     10,   Func. Count:     80,   Neg. LLF: 6970.063553697997\n",
276 |       "Optimization terminated successfully.    (Exit mode 0)\n",
277 |       "            Current function value: 6970.058366227228\n",
278 |       "            Iterations: 12\n",
279 |       "            Function evaluations: 95\n",
280 |       "            Gradient evaluations: 12\n"
281 |      ]
282 |     },
283 |     {
284 |      "data": {
285 |       "text/html": [
286 |        "<table class=\"simpletable\">\n",
287 |        "<caption>Constant Mean - GARCH Model Results</caption>\n",
288 |        "<tr>\n",
289 |        "  <th>Dep. Variable:</th>       <td>returns</td>      <th>  R-squared:         </th>  <td>  -0.001</td> \n",
290 |        "</tr>\n",
291 |        "<tr>\n",
292 |        "  <th>Mean Model:</th>       <td>Constant Mean</td>   <th>  Adj. R-squared:    </th>  <td>  -0.001</td> \n",
293 |        "</tr>\n",
294 |        "<tr>\n",
295 |        "  <th>Vol Model:</th>            <td>GARCH</td>       <th>  Log-Likelihood:    </th> <td>  -6970.06</td>\n",
296 |        "</tr>\n",
297 |        "<tr>\n",
298 |        "  <th>Distribution:</th>        <td>Normal</td>       <th>  AIC:               </th> <td>   13950.1</td>\n",
299 |        "</tr>\n",
300 |        "<tr>\n",
301 |        "  <th>Method:</th>        <td>Maximum Likelihood</td> <th>  BIC:               </th> <td>   13982.7</td>\n",
302 |        "</tr>\n",
303 |        "<tr>\n",
304 |        "  <th></th>                        <td></td>          <th>  No. Observations:  </th>    <td>5020</td>   \n",
305 |        "</tr>\n",
306 |        "<tr>\n",
307 |        "  <th>Date:</th>           <td>Thu, Sep 19 2019</td>  <th>  Df Residuals:      </th>    <td>5015</td>   \n",
308 |        "</tr>\n",
309 |        "<tr>\n",
310 |        "  <th>Time:</th>               <td>17:54:06</td>      <th>  Df Model:          </th>      <td>5</td>    \n",
311 |        "</tr>\n",
312 |        "</table>\n",
313 |        "<table class=\"simpletable\">\n",
314 |        "<caption>Mean Model</caption>\n",
315 |        "<tr>\n",
316 |        "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
317 |        "</tr>\n",
318 |        "<tr>\n",
319 |        "  <th>mu</th> <td>    0.0466</td> <td>1.184e-02</td> <td>    3.938</td> <td>8.219e-05</td> <td>[2.341e-02,6.982e-02]</td>\n",
320 |        "</tr>\n",
321 |        "</table>\n",
322 |        "<table class=\"simpletable\">\n",
323 |        "<caption>Volatility Model</caption>\n",
324 |        "<tr>\n",
325 |        "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
326 |        "</tr>\n",
327 |        "<tr>\n",
328 |        "  <th>omega</th>    <td>    0.0109</td> <td>2.908e-03</td> <td>    3.761</td> <td>1.696e-04</td> <td>[5.236e-03,1.663e-02]</td>\n",
329 |        "</tr>\n",
330 |        "<tr>\n",
331 |        "  <th>alpha[1]</th> <td>    0.0835</td> <td>1.189e-02</td> <td>    7.019</td> <td>2.231e-12</td>  <td>[6.017e-02,  0.107]</td> \n",
332 |        "</tr>\n",
333 |        "<tr>\n",
334 |        "  <th>beta[1]</th>  <td>    0.9089</td> <td>    0.188</td> <td>    4.845</td> <td>1.268e-06</td>   <td>[  0.541,  1.277]</td>  \n",
335 |        "</tr>\n",
336 |        "<tr>\n",
337 |        "  <th>beta[2]</th>    <td>0.0000</td>   <td>    0.180</td>   <td>0.000</td>   <td>    1.000</td>   <td>[ -0.352,  0.352]</td>  \n",
338 |        "</tr>\n",
339 |        "</table><br/><br/>Covariance estimator: robust"
340 |       ],
341 |       "text/plain": [
342 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
343 |        "\"\"\"\n",
344 |        "                     Constant Mean - GARCH Model Results                      \n",
345 |        "==============================================================================\n",
346 |        "Dep. Variable:                returns   R-squared:                      -0.001\n",
347 |        "Mean Model:             Constant Mean   Adj. R-squared:                 -0.001\n",
348 |        "Vol Model:                      GARCH   Log-Likelihood:               -6970.06\n",
349 |        "Distribution:                  Normal   AIC:                           13950.1\n",
350 |        "Method:            Maximum Likelihood   BIC:                           13982.7\n",
351 |        "                                        No. Observations:                 5020\n",
352 |        "Date:                Thu, Sep 19 2019   Df Residuals:                     5015\n",
353 |        "Time:                        17:54:06   Df Model:                            5\n",
354 |        "                                 Mean Model                                 \n",
355 |        "============================================================================\n",
356 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
357 |        "----------------------------------------------------------------------------\n",
358 |        "mu             0.0466  1.184e-02      3.938  8.219e-05 [2.341e-02,6.982e-02]\n",
359 |        "                              Volatility Model                              \n",
360 |        "============================================================================\n",
361 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
362 |        "----------------------------------------------------------------------------\n",
363 |        "omega          0.0109  2.908e-03      3.761  1.696e-04 [5.236e-03,1.663e-02]\n",
364 |        "alpha[1]       0.0835  1.189e-02      7.019  2.231e-12   [6.017e-02,  0.107]\n",
365 |        "beta[1]        0.9089      0.188      4.845  1.268e-06     [  0.541,  1.277]\n",
366 |        "beta[2]        0.0000      0.180      0.000      1.000     [ -0.352,  0.352]\n",
367 |        "============================================================================\n",
368 |        "\n",
369 |        "Covariance estimator: robust\n",
370 |        "\"\"\""
371 |       ]
372 |      },
373 |      "execution_count": 9,
374 |      "metadata": {},
375 |      "output_type": "execute_result"
376 |     }
377 |    ],
378 |    "source": [
379 |     "model_garch_1_2 = arch_model(df.returns[1:], mean = \"Constant\",  vol = \"GARCH\", p = 1, q = 2)\n",
380 |     "results_garch_1_2 = model_garch_1_2.fit(update_freq = 5)\n",
381 |     "results_garch_1_2.summary()"
382 |    ]
383 |   },
384 |   {
385 |    "cell_type": "code",
386 |    "execution_count": 10,
387 |    "metadata": {},
388 |    "outputs": [
389 |     {
390 |      "name": "stdout",
391 |      "output_type": "stream",
392 |      "text": [
393 |       "Iteration:      5,   Func. Count:     51,   Neg. LLF: 6993.438671672955\n",
394 |       "Iteration:     10,   Func. Count:     97,   Neg. LLF: 6972.43119644999\n",
395 |       "Iteration:     15,   Func. Count:    138,   Neg. LLF: 6970.058671960955\n",
396 |       "Optimization terminated successfully.    (Exit mode 0)\n",
397 |       "            Current function value: 6970.05836623016\n",
398 |       "            Iterations: 17\n",
399 |       "            Function evaluations: 154\n",
400 |       "            Gradient evaluations: 17\n"
401 |      ]
402 |     },
403 |     {
404 |      "data": {
405 |       "text/html": [
406 |        "<table class=\"simpletable\">\n",
407 |        "<caption>Constant Mean - GARCH Model Results</caption>\n",
408 |        "<tr>\n",
409 |        "  <th>Dep. Variable:</th>       <td>returns</td>      <th>  R-squared:         </th>  <td>  -0.001</td> \n",
410 |        "</tr>\n",
411 |        "<tr>\n",
412 |        "  <th>Mean Model:</th>       <td>Constant Mean</td>   <th>  Adj. R-squared:    </th>  <td>  -0.001</td> \n",
413 |        "</tr>\n",
414 |        "<tr>\n",
415 |        "  <th>Vol Model:</th>            <td>GARCH</td>       <th>  Log-Likelihood:    </th> <td>  -6970.06</td>\n",
416 |        "</tr>\n",
417 |        "<tr>\n",
418 |        "  <th>Distribution:</th>        <td>Normal</td>       <th>  AIC:               </th> <td>   13952.1</td>\n",
419 |        "</tr>\n",
420 |        "<tr>\n",
421 |        "  <th>Method:</th>        <td>Maximum Likelihood</td> <th>  BIC:               </th> <td>   13991.2</td>\n",
422 |        "</tr>\n",
423 |        "<tr>\n",
424 |        "  <th></th>                        <td></td>          <th>  No. Observations:  </th>    <td>5020</td>   \n",
425 |        "</tr>\n",
426 |        "<tr>\n",
427 |        "  <th>Date:</th>           <td>Thu, Sep 19 2019</td>  <th>  Df Residuals:      </th>    <td>5014</td>   \n",
428 |        "</tr>\n",
429 |        "<tr>\n",
430 |        "  <th>Time:</th>               <td>17:54:23</td>      <th>  Df Model:          </th>      <td>6</td>    \n",
431 |        "</tr>\n",
432 |        "</table>\n",
433 |        "<table class=\"simpletable\">\n",
434 |        "<caption>Mean Model</caption>\n",
435 |        "<tr>\n",
436 |        "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
437 |        "</tr>\n",
438 |        "<tr>\n",
439 |        "  <th>mu</th> <td>    0.0466</td> <td>1.179e-02</td> <td>    3.954</td> <td>7.683e-05</td> <td>[2.351e-02,6.972e-02]</td>\n",
440 |        "</tr>\n",
441 |        "</table>\n",
442 |        "<table class=\"simpletable\">\n",
443 |        "<caption>Volatility Model</caption>\n",
444 |        "<tr>\n",
445 |        "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>      <th>95.0% Conf. Int.</th>   \n",
446 |        "</tr>\n",
447 |        "<tr>\n",
448 |        "  <th>omega</th>    <td>    0.0109</td> <td>8.157e-03</td> <td>    1.341</td> <td>    0.180</td> <td>[-5.052e-03,2.692e-02]</td>\n",
449 |        "</tr>\n",
450 |        "<tr>\n",
451 |        "  <th>alpha[1]</th> <td>    0.0835</td> <td>6.059e-02</td> <td>    1.378</td> <td>    0.168</td>  <td>[-3.528e-02,  0.202]</td> \n",
452 |        "</tr>\n",
453 |        "<tr>\n",
454 |        "  <th>beta[1]</th>  <td>    0.9089</td> <td>    2.148</td> <td>    0.423</td> <td>    0.672</td>    <td>[ -3.301,  5.119]</td>  \n",
455 |        "</tr>\n",
456 |        "<tr>\n",
457 |        "  <th>beta[2]</th>    <td>0.0000</td>   <td>    3.375</td>   <td>0.000</td>   <td>    1.000</td>    <td>[ -6.614,  6.614]</td>  \n",
458 |        "</tr>\n",
459 |        "<tr>\n",
460 |        "  <th>beta[3]</th>  <td>3.9718e-13</td> <td>    1.294</td> <td>3.070e-13</td> <td>    1.000</td>    <td>[ -2.536,  2.536]</td>  \n",
461 |        "</tr>\n",
462 |        "</table><br/><br/>Covariance estimator: robust"
463 |       ],
464 |       "text/plain": [
465 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
466 |        "\"\"\"\n",
467 |        "                     Constant Mean - GARCH Model Results                      \n",
468 |        "==============================================================================\n",
469 |        "Dep. Variable:                returns   R-squared:                      -0.001\n",
470 |        "Mean Model:             Constant Mean   Adj. R-squared:                 -0.001\n",
471 |        "Vol Model:                      GARCH   Log-Likelihood:               -6970.06\n",
472 |        "Distribution:                  Normal   AIC:                           13952.1\n",
473 |        "Method:            Maximum Likelihood   BIC:                           13991.2\n",
474 |        "                                        No. Observations:                 5020\n",
475 |        "Date:                Thu, Sep 19 2019   Df Residuals:                     5014\n",
476 |        "Time:                        17:54:23   Df Model:                            6\n",
477 |        "                                 Mean Model                                 \n",
478 |        "============================================================================\n",
479 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
480 |        "----------------------------------------------------------------------------\n",
481 |        "mu             0.0466  1.179e-02      3.954  7.683e-05 [2.351e-02,6.972e-02]\n",
482 |        "                               Volatility Model                              \n",
483 |        "=============================================================================\n",
484 |        "                 coef    std err          t      P>|t|       95.0% Conf. Int.\n",
485 |        "-----------------------------------------------------------------------------\n",
486 |        "omega          0.0109  8.157e-03      1.341      0.180 [-5.052e-03,2.692e-02]\n",
487 |        "alpha[1]       0.0835  6.059e-02      1.378      0.168   [-3.528e-02,  0.202]\n",
488 |        "beta[1]        0.9089      2.148      0.423      0.672      [ -3.301,  5.119]\n",
489 |        "beta[2]        0.0000      3.375      0.000      1.000      [ -6.614,  6.614]\n",
490 |        "beta[3]    3.9718e-13      1.294  3.070e-13      1.000      [ -2.536,  2.536]\n",
491 |        "=============================================================================\n",
492 |        "\n",
493 |        "Covariance estimator: robust\n",
494 |        "\"\"\""
495 |       ]
496 |      },
497 |      "execution_count": 10,
498 |      "metadata": {},
499 |      "output_type": "execute_result"
500 |     }
501 |    ],
502 |    "source": [
503 |     "model_garch_1_3 = arch_model(df.returns[1:], mean = \"Constant\",  vol = \"GARCH\", p = 1, q = 3)\n",
504 |     "results_garch_1_3 = model_garch_1_3.fit(update_freq = 5)\n",
505 |     "results_garch_1_3.summary()"
506 |    ]
507 |   },
508 |   {
509 |    "cell_type": "code",
510 |    "execution_count": 11,
511 |    "metadata": {},
512 |    "outputs": [
513 |     {
514 |      "name": "stdout",
515 |      "output_type": "stream",
516 |      "text": [
517 |       "Iteration:      5,   Func. Count:     44,   Neg. LLF: 6973.212556117614\n",
518 |       "Iteration:     10,   Func. Count:     85,   Neg. LLF: 6967.735886702618\n",
519 |       "Optimization terminated successfully.    (Exit mode 0)\n",
520 |       "            Current function value: 6967.731020076142\n",
521 |       "            Iterations: 12\n",
522 |       "            Function evaluations: 99\n",
523 |       "            Gradient evaluations: 12\n"
524 |      ]
525 |     },
526 |     {
527 |      "data": {
528 |       "text/html": [
529 |        "<table class=\"simpletable\">\n",
530 |        "<caption>Constant Mean - GARCH Model Results</caption>\n",
531 |        "<tr>\n",
532 |        "  <th>Dep. Variable:</th>       <td>returns</td>      <th>  R-squared:         </th>  <td>  -0.001</td> \n",
533 |        "</tr>\n",
534 |        "<tr>\n",
535 |        "  <th>Mean Model:</th>       <td>Constant Mean</td>   <th>  Adj. R-squared:    </th>  <td>  -0.001</td> \n",
536 |        "</tr>\n",
537 |        "<tr>\n",
538 |        "  <th>Vol Model:</th>            <td>GARCH</td>       <th>  Log-Likelihood:    </th> <td>  -6967.73</td>\n",
539 |        "</tr>\n",
540 |        "<tr>\n",
541 |        "  <th>Distribution:</th>        <td>Normal</td>       <th>  AIC:               </th> <td>   13945.5</td>\n",
542 |        "</tr>\n",
543 |        "<tr>\n",
544 |        "  <th>Method:</th>        <td>Maximum Likelihood</td> <th>  BIC:               </th> <td>   13978.1</td>\n",
545 |        "</tr>\n",
546 |        "<tr>\n",
547 |        "  <th></th>                        <td></td>          <th>  No. Observations:  </th>    <td>5020</td>   \n",
548 |        "</tr>\n",
549 |        "<tr>\n",
550 |        "  <th>Date:</th>           <td>Thu, Sep 19 2019</td>  <th>  Df Residuals:      </th>    <td>5015</td>   \n",
551 |        "</tr>\n",
552 |        "<tr>\n",
553 |        "  <th>Time:</th>               <td>17:54:44</td>      <th>  Df Model:          </th>      <td>5</td>    \n",
554 |        "</tr>\n",
555 |        "</table>\n",
556 |        "<table class=\"simpletable\">\n",
557 |        "<caption>Mean Model</caption>\n",
558 |        "<tr>\n",
559 |        "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
560 |        "</tr>\n",
561 |        "<tr>\n",
562 |        "  <th>mu</th> <td>    0.0466</td> <td>1.187e-02</td> <td>    3.922</td> <td>8.780e-05</td> <td>[2.329e-02,6.982e-02]</td>\n",
563 |        "</tr>\n",
564 |        "</table>\n",
565 |        "<table class=\"simpletable\">\n",
566 |        "<caption>Volatility Model</caption>\n",
567 |        "<tr>\n",
568 |        "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>      <th>95.0% Conf. Int.</th>   \n",
569 |        "</tr>\n",
570 |        "<tr>\n",
571 |        "  <th>omega</th>    <td>    0.0129</td> <td>4.097e-03</td> <td>    3.158</td> <td>1.589e-03</td>  <td>[4.908e-03,2.097e-02]</td>\n",
572 |        "</tr>\n",
573 |        "<tr>\n",
574 |        "  <th>alpha[1]</th> <td>    0.0547</td> <td>1.665e-02</td> <td>    3.286</td> <td>1.017e-03</td>  <td>[2.208e-02,8.735e-02]</td>\n",
575 |        "</tr>\n",
576 |        "<tr>\n",
577 |        "  <th>alpha[2]</th> <td>    0.0389</td> <td>2.345e-02</td> <td>    1.659</td> <td>9.709e-02</td> <td>[-7.056e-03,8.488e-02]</td>\n",
578 |        "</tr>\n",
579 |        "<tr>\n",
580 |        "  <th>beta[1]</th>  <td>    0.8974</td> <td>1.712e-02</td> <td>   52.415</td>   <td>0.000</td>      <td>[  0.864,  0.931]</td>  \n",
581 |        "</tr>\n",
582 |        "</table><br/><br/>Covariance estimator: robust"
583 |       ],
584 |       "text/plain": [
585 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
586 |        "\"\"\"\n",
587 |        "                     Constant Mean - GARCH Model Results                      \n",
588 |        "==============================================================================\n",
589 |        "Dep. Variable:                returns   R-squared:                      -0.001\n",
590 |        "Mean Model:             Constant Mean   Adj. R-squared:                 -0.001\n",
591 |        "Vol Model:                      GARCH   Log-Likelihood:               -6967.73\n",
592 |        "Distribution:                  Normal   AIC:                           13945.5\n",
593 |        "Method:            Maximum Likelihood   BIC:                           13978.1\n",
594 |        "                                        No. Observations:                 5020\n",
595 |        "Date:                Thu, Sep 19 2019   Df Residuals:                     5015\n",
596 |        "Time:                        17:54:44   Df Model:                            5\n",
597 |        "                                 Mean Model                                 \n",
598 |        "============================================================================\n",
599 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
600 |        "----------------------------------------------------------------------------\n",
601 |        "mu             0.0466  1.187e-02      3.922  8.780e-05 [2.329e-02,6.982e-02]\n",
602 |        "                               Volatility Model                              \n",
603 |        "=============================================================================\n",
604 |        "                 coef    std err          t      P>|t|       95.0% Conf. Int.\n",
605 |        "-----------------------------------------------------------------------------\n",
606 |        "omega          0.0129  4.097e-03      3.158  1.589e-03  [4.908e-03,2.097e-02]\n",
607 |        "alpha[1]       0.0547  1.665e-02      3.286  1.017e-03  [2.208e-02,8.735e-02]\n",
608 |        "alpha[2]       0.0389  2.345e-02      1.659  9.709e-02 [-7.056e-03,8.488e-02]\n",
609 |        "beta[1]        0.8974  1.712e-02     52.415      0.000      [  0.864,  0.931]\n",
610 |        "=============================================================================\n",
611 |        "\n",
612 |        "Covariance estimator: robust\n",
613 |        "\"\"\""
614 |       ]
615 |      },
616 |      "execution_count": 11,
617 |      "metadata": {},
618 |      "output_type": "execute_result"
619 |     }
620 |    ],
621 |    "source": [
622 |     "model_garch_2_1 = arch_model(df.returns[1:], mean = \"Constant\",  vol = \"GARCH\", p = 2, q = 1)\n",
623 |     "results_garch_2_1 = model_garch_2_1.fit(update_freq = 5)\n",
624 |     "results_garch_2_1.summary()"
625 |    ]
626 |   },
627 |   {
628 |    "cell_type": "code",
629 |    "execution_count": 12,
630 |    "metadata": {},
631 |    "outputs": [
632 |     {
633 |      "name": "stdout",
634 |      "output_type": "stream",
635 |      "text": [
636 |       "Iteration:      5,   Func. Count:     49,   Neg. LLF: 6974.886815445026\n",
637 |       "Iteration:     10,   Func. Count:     94,   Neg. LLF: 6968.538209285089\n",
638 |       "Iteration:     15,   Func. Count:    134,   Neg. LLF: 6967.731020049032\n",
639 |       "Optimization terminated successfully.    (Exit mode 0)\n",
640 |       "            Current function value: 6967.731020049671\n",
641 |       "            Iterations: 15\n",
642 |       "            Function evaluations: 134\n",
643 |       "            Gradient evaluations: 15\n"
644 |      ]
645 |     },
646 |     {
647 |      "data": {
648 |       "text/html": [
649 |        "<table class=\"simpletable\">\n",
650 |        "<caption>Constant Mean - GARCH Model Results</caption>\n",
651 |        "<tr>\n",
652 |        "  <th>Dep. Variable:</th>       <td>returns</td>      <th>  R-squared:         </th>  <td>  -0.001</td> \n",
653 |        "</tr>\n",
654 |        "<tr>\n",
655 |        "  <th>Mean Model:</th>       <td>Constant Mean</td>   <th>  Adj. R-squared:    </th>  <td>  -0.001</td> \n",
656 |        "</tr>\n",
657 |        "<tr>\n",
658 |        "  <th>Vol Model:</th>            <td>GARCH</td>       <th>  Log-Likelihood:    </th> <td>  -6967.73</td>\n",
659 |        "</tr>\n",
660 |        "<tr>\n",
661 |        "  <th>Distribution:</th>        <td>Normal</td>       <th>  AIC:               </th> <td>   13947.5</td>\n",
662 |        "</tr>\n",
663 |        "<tr>\n",
664 |        "  <th>Method:</th>        <td>Maximum Likelihood</td> <th>  BIC:               </th> <td>   13986.6</td>\n",
665 |        "</tr>\n",
666 |        "<tr>\n",
667 |        "  <th></th>                        <td></td>          <th>  No. Observations:  </th>    <td>5020</td>   \n",
668 |        "</tr>\n",
669 |        "<tr>\n",
670 |        "  <th>Date:</th>           <td>Thu, Sep 19 2019</td>  <th>  Df Residuals:      </th>    <td>5014</td>   \n",
671 |        "</tr>\n",
672 |        "<tr>\n",
673 |        "  <th>Time:</th>               <td>17:55:01</td>      <th>  Df Model:          </th>      <td>6</td>    \n",
674 |        "</tr>\n",
675 |        "</table>\n",
676 |        "<table class=\"simpletable\">\n",
677 |        "<caption>Mean Model</caption>\n",
678 |        "<tr>\n",
679 |        "   <td></td>     <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>     <th>95.0% Conf. Int.</th>   \n",
680 |        "</tr>\n",
681 |        "<tr>\n",
682 |        "  <th>mu</th> <td>    0.0466</td> <td>1.187e-02</td> <td>    3.924</td> <td>8.721e-05</td> <td>[2.330e-02,6.982e-02]</td>\n",
683 |        "</tr>\n",
684 |        "</table>\n",
685 |        "<table class=\"simpletable\">\n",
686 |        "<caption>Volatility Model</caption>\n",
687 |        "<tr>\n",
688 |        "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>       <th>P>|t|</th>      <th>95.0% Conf. Int.</th>   \n",
689 |        "</tr>\n",
690 |        "<tr>\n",
691 |        "  <th>omega</th>    <td>    0.0129</td> <td>4.816e-03</td> <td>    2.687</td> <td>7.217e-03</td>  <td>[3.500e-03,2.238e-02]</td>\n",
692 |        "</tr>\n",
693 |        "<tr>\n",
694 |        "  <th>alpha[1]</th> <td>    0.0547</td> <td>1.665e-02</td> <td>    3.285</td> <td>1.019e-03</td>  <td>[2.207e-02,8.736e-02]</td>\n",
695 |        "</tr>\n",
696 |        "<tr>\n",
697 |        "  <th>alpha[2]</th> <td>    0.0389</td> <td>2.505e-02</td> <td>    1.553</td> <td>    0.120</td> <td>[-1.018e-02,8.800e-02]</td>\n",
698 |        "</tr>\n",
699 |        "<tr>\n",
700 |        "  <th>alpha[3]</th>   <td>0.0000</td>   <td>2.572e-02</td>   <td>0.000</td>   <td>    1.000</td> <td>[-5.041e-02,5.041e-02]</td>\n",
701 |        "</tr>\n",
702 |        "<tr>\n",
703 |        "  <th>beta[1]</th>  <td>    0.8974</td> <td>2.245e-02</td> <td>   39.978</td>   <td>0.000</td>      <td>[  0.853,  0.941]</td>  \n",
704 |        "</tr>\n",
705 |        "</table><br/><br/>Covariance estimator: robust"
706 |       ],
707 |       "text/plain": [
708 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
709 |        "\"\"\"\n",
710 |        "                     Constant Mean - GARCH Model Results                      \n",
711 |        "==============================================================================\n",
712 |        "Dep. Variable:                returns   R-squared:                      -0.001\n",
713 |        "Mean Model:             Constant Mean   Adj. R-squared:                 -0.001\n",
714 |        "Vol Model:                      GARCH   Log-Likelihood:               -6967.73\n",
715 |        "Distribution:                  Normal   AIC:                           13947.5\n",
716 |        "Method:            Maximum Likelihood   BIC:                           13986.6\n",
717 |        "                                        No. Observations:                 5020\n",
718 |        "Date:                Thu, Sep 19 2019   Df Residuals:                     5014\n",
719 |        "Time:                        17:55:01   Df Model:                            6\n",
720 |        "                                 Mean Model                                 \n",
721 |        "============================================================================\n",
722 |        "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
723 |        "----------------------------------------------------------------------------\n",
724 |        "mu             0.0466  1.187e-02      3.924  8.721e-05 [2.330e-02,6.982e-02]\n",
725 |        "                               Volatility Model                              \n",
726 |        "=============================================================================\n",
727 |        "                 coef    std err          t      P>|t|       95.0% Conf. Int.\n",
728 |        "-----------------------------------------------------------------------------\n",
729 |        "omega          0.0129  4.816e-03      2.687  7.217e-03  [3.500e-03,2.238e-02]\n",
730 |        "alpha[1]       0.0547  1.665e-02      3.285  1.019e-03  [2.207e-02,8.736e-02]\n",
731 |        "alpha[2]       0.0389  2.505e-02      1.553      0.120 [-1.018e-02,8.800e-02]\n",
732 |        "alpha[3]       0.0000  2.572e-02      0.000      1.000 [-5.041e-02,5.041e-02]\n",
733 |        "beta[1]        0.8974  2.245e-02     39.978      0.000      [  0.853,  0.941]\n",
734 |        "=============================================================================\n",
735 |        "\n",
736 |        "Covariance estimator: robust\n",
737 |        "\"\"\""
738 |       ]
739 |      },
740 |      "execution_count": 12,
741 |      "metadata": {},
742 |      "output_type": "execute_result"
743 |     }
744 |    ],
745 |    "source": [
746 |     "model_garch_3_1 = arch_model(df.returns[1:], mean = \"Constant\",  vol = \"GARCH\", p = 3, q = 1)\n",
747 |     "results_garch_3_1 = model_garch_3_1.fit(update_freq = 5)\n",
748 |     "results_garch_3_1.summary()"
749 |    ]
750 |   }
751 |  ],
752 |  "metadata": {
753 |   "kernelspec": {
754 |    "display_name": "Python 3",
755 |    "language": "python",
756 |    "name": "python3"
757 |   },
758 |   "language_info": {
759 |    "codemirror_mode": {
760 |     "name": "ipython",
761 |     "version": 3
762 |    },
763 |    "file_extension": ".py",
764 |    "mimetype": "text/x-python",
765 |    "name": "python",
766 |    "nbconvert_exporter": "python",
767 |    "pygments_lexer": "ipython3",
768 |    "version": "3.6.7"
769 |   }
770 |  },
771 |  "nbformat": 4,
772 |  "nbformat_minor": 2
773 | }
774 | 


--------------------------------------------------------------------------------
/4. Analyzing prices using the AR model.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "## Importing the relevant packages\n"
  8 |    ]
  9 |   },
 10 |   {
 11 |    "cell_type": "code",
 12 |    "execution_count": 1,
 13 |    "metadata": {},
 14 |    "outputs": [],
 15 |    "source": [
 16 |     "import pandas as pd\n",
 17 |     "import numpy as np\n",
 18 |     "import matplotlib.pyplot as plt\n",
 19 |     "import statsmodels.graphics.tsaplots as sgt\n",
 20 |     "from statsmodels.tsa.arima_model import ARMA\n",
 21 |     "from scipy.stats.distributions import chi2\n",
 22 |     "import statsmodels.tsa.stattools as sts \n",
 23 |     "import seaborn as sns\n",
 24 |     "sns.set()"
 25 |    ]
 26 |   },
 27 |   {
 28 |    "cell_type": "markdown",
 29 |    "metadata": {},
 30 |    "source": [
 31 |     "## Importing the Data and Pre-processing "
 32 |    ]
 33 |   },
 34 |   {
 35 |    "cell_type": "code",
 36 |    "execution_count": 2,
 37 |    "metadata": {},
 38 |    "outputs": [],
 39 |    "source": [
 40 |     "raw_csv_data = pd.read_csv(\"Index2018.csv\") \n",
 41 |     "df_comp=raw_csv_data.copy()\n",
 42 |     "df_comp.date = pd.to_datetime(df_comp.date, dayfirst = True)\n",
 43 |     "df_comp.set_index(\"date\", inplace=True)\n",
 44 |     "df_comp=df_comp.asfreq('b')\n",
 45 |     "df_comp=df_comp.fillna(method='ffill')"
 46 |    ]
 47 |   },
 48 |   {
 49 |    "cell_type": "code",
 50 |    "execution_count": 3,
 51 |    "metadata": {},
 52 |    "outputs": [],
 53 |    "source": [
 54 |     "df_comp['market_value']=df_comp.ftse"
 55 |    ]
 56 |   },
 57 |   {
 58 |    "cell_type": "code",
 59 |    "execution_count": 4,
 60 |    "metadata": {},
 61 |    "outputs": [],
 62 |    "source": [
 63 |     "del df_comp['spx']\n",
 64 |     "del df_comp['dax']\n",
 65 |     "del df_comp['ftse']\n",
 66 |     "del df_comp['nikkei']\n",
 67 |     "size = int(len(df_comp)*0.8)\n",
 68 |     "df, df_test = df_comp.iloc[:size], df_comp.iloc[size:]"
 69 |    ]
 70 |   },
 71 |   {
 72 |    "cell_type": "markdown",
 73 |    "metadata": {},
 74 |    "source": [
 75 |     "## The ACF"
 76 |    ]
 77 |   },
 78 |   {
 79 |    "cell_type": "code",
 80 |    "execution_count": 5,
 81 |    "metadata": {
 82 |     "scrolled": true
 83 |    },
 84 |    "outputs": [
 85 |     {
 86 |      "data": {
 87 |       "image/png": 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\n",
 88 |       "text/plain": [
 89 |        "<Figure size 432x288 with 1 Axes>"
 90 |       ]
 91 |      },
 92 |      "metadata": {
 93 |       "needs_background": "light"
 94 |      },
 95 |      "output_type": "display_data"
 96 |     }
 97 |    ],
 98 |    "source": [
 99 |     "sgt.plot_acf(df.market_value, zero = False, lags = 40)\n",
100 |     "plt.title(\"ACF for Prices\", size = 20)\n",
101 |     "plt.show()"
102 |    ]
103 |   },
104 |   {
105 |    "cell_type": "code",
106 |    "execution_count": 6,
107 |    "metadata": {},
108 |    "outputs": [],
109 |    "source": [
110 |     "import warnings\n",
111 |     "warnings.filterwarnings(\"ignore\")"
112 |    ]
113 |   },
114 |   {
115 |    "cell_type": "markdown",
116 |    "metadata": {},
117 |    "source": [
118 |     "## The PACF"
119 |    ]
120 |   },
121 |   {
122 |    "cell_type": "code",
123 |    "execution_count": 7,
124 |    "metadata": {
125 |     "scrolled": true
126 |    },
127 |    "outputs": [
128 |     {
129 |      "data": {
130 |       "image/png": 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\n",
131 |       "text/plain": [
132 |        "<Figure size 432x288 with 1 Axes>"
133 |       ]
134 |      },
135 |      "metadata": {
136 |       "needs_background": "light"
137 |      },
138 |      "output_type": "display_data"
139 |     }
140 |    ],
141 |    "source": [
142 |     "sgt.plot_pacf(df.market_value, lags = 40, alpha = 0.05, zero = False, method = ('ols'))\n",
143 |     "plt.title(\"PACF for Prices\", size = 20)\n",
144 |     "plt.show()"
145 |    ]
146 |   },
147 |   {
148 |    "cell_type": "markdown",
149 |    "metadata": {},
150 |    "source": [
151 |     "## The AR(1) Model"
152 |    ]
153 |   },
154 |   {
155 |    "cell_type": "code",
156 |    "execution_count": 8,
157 |    "metadata": {},
158 |    "outputs": [],
159 |    "source": [
160 |     "model_ar = ARMA(df.market_value, order=(1,0))"
161 |    ]
162 |   },
163 |   {
164 |    "cell_type": "code",
165 |    "execution_count": 9,
166 |    "metadata": {},
167 |    "outputs": [],
168 |    "source": [
169 |     "results_ar = model_ar.fit()"
170 |    ]
171 |   },
172 |   {
173 |    "cell_type": "code",
174 |    "execution_count": 10,
175 |    "metadata": {},
176 |    "outputs": [
177 |     {
178 |      "data": {
179 |       "text/html": [
180 |        "<table class=\"simpletable\">\n",
181 |        "<caption>ARMA Model Results</caption>\n",
182 |        "<tr>\n",
183 |        "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
184 |        "</tr>\n",
185 |        "<tr>\n",
186 |        "  <th>Model:</th>            <td>ARMA(1, 0)</td>    <th>  Log Likelihood     </th> <td>-27622.598</td>\n",
187 |        "</tr>\n",
188 |        "<tr>\n",
189 |        "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.258</td>  \n",
190 |        "</tr>\n",
191 |        "<tr>\n",
192 |        "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55251.196</td>\n",
193 |        "</tr>\n",
194 |        "<tr>\n",
195 |        "  <th>Time:</th>              <td>10:03:35</td>     <th>  BIC                </th>  <td>55270.760</td>\n",
196 |        "</tr>\n",
197 |        "<tr>\n",
198 |        "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55258.052</td>\n",
199 |        "</tr>\n",
200 |        "<tr>\n",
201 |        "  <th></th>                 <td>- 04-05-2013</td>   <th>                     </th>      <td> </td>    \n",
202 |        "</tr>\n",
203 |        "</table>\n",
204 |        "<table class=\"simpletable\">\n",
205 |        "<tr>\n",
206 |        "           <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",
207 |        "</tr>\n",
208 |        "<tr>\n",
209 |        "  <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",
210 |        "</tr>\n",
211 |        "<tr>\n",
212 |        "  <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",
213 |        "</tr>\n",
214 |        "</table>\n",
215 |        "<table class=\"simpletable\">\n",
216 |        "<caption>Roots</caption>\n",
217 |        "<tr>\n",
218 |        "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
219 |        "</tr>\n",
220 |        "<tr>\n",
221 |        "  <th>AR.1</th> <td>           1.0015</td> <td>          +0.0000j</td> <td>           1.0015</td> <td>           0.0000</td>\n",
222 |        "</tr>\n",
223 |        "</table>"
224 |       ],
225 |       "text/plain": [
226 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
227 |        "\"\"\"\n",
228 |        "                              ARMA Model Results                              \n",
229 |        "==============================================================================\n",
230 |        "Dep. Variable:           market_value   No. Observations:                 5021\n",
231 |        "Model:                     ARMA(1, 0)   Log Likelihood              -27622.598\n",
232 |        "Method:                       css-mle   S.D. of innovations             59.258\n",
233 |        "Date:                Fri, 27 Sep 2019   AIC                          55251.196\n",
234 |        "Time:                        10:03:35   BIC                          55270.760\n",
235 |        "Sample:                    01-07-1994   HQIC                         55258.052\n",
236 |        "                         - 04-05-2013                                         \n",
237 |        "======================================================================================\n",
238 |        "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
239 |        "--------------------------------------------------------------------------------------\n",
240 |        "const               5057.0038    486.461     10.395      0.000    4103.558    6010.450\n",
241 |        "ar.L1.market_value     0.9985      0.001   1298.863      0.000       0.997       1.000\n",
242 |        "                                    Roots                                    \n",
243 |        "=============================================================================\n",
244 |        "                  Real          Imaginary           Modulus         Frequency\n",
245 |        "-----------------------------------------------------------------------------\n",
246 |        "AR.1            1.0015           +0.0000j            1.0015            0.0000\n",
247 |        "-----------------------------------------------------------------------------\n",
248 |        "\"\"\""
249 |       ]
250 |      },
251 |      "execution_count": 10,
252 |      "metadata": {},
253 |      "output_type": "execute_result"
254 |     }
255 |    ],
256 |    "source": [
257 |     "results_ar.summary()"
258 |    ]
259 |   },
260 |   {
261 |    "cell_type": "markdown",
262 |    "metadata": {},
263 |    "source": [
264 |     "## Higher-Lag AR Models"
265 |    ]
266 |   },
267 |   {
268 |    "cell_type": "code",
269 |    "execution_count": 11,
270 |    "metadata": {},
271 |    "outputs": [
272 |     {
273 |      "data": {
274 |       "text/html": [
275 |        "<table class=\"simpletable\">\n",
276 |        "<caption>ARMA Model Results</caption>\n",
277 |        "<tr>\n",
278 |        "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
279 |        "</tr>\n",
280 |        "<tr>\n",
281 |        "  <th>Model:</th>            <td>ARMA(2, 0)</td>    <th>  Log Likelihood     </th> <td>-27621.866</td>\n",
282 |        "</tr>\n",
283 |        "<tr>\n",
284 |        "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.249</td>  \n",
285 |        "</tr>\n",
286 |        "<tr>\n",
287 |        "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55251.731</td>\n",
288 |        "</tr>\n",
289 |        "<tr>\n",
290 |        "  <th>Time:</th>              <td>10:03:37</td>     <th>  BIC                </th>  <td>55277.817</td>\n",
291 |        "</tr>\n",
292 |        "<tr>\n",
293 |        "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55260.872</td>\n",
294 |        "</tr>\n",
295 |        "<tr>\n",
296 |        "  <th></th>                 <td>- 04-05-2013</td>   <th>                     </th>      <td> </td>    \n",
297 |        "</tr>\n",
298 |        "</table>\n",
299 |        "<table class=\"simpletable\">\n",
300 |        "<tr>\n",
301 |        "           <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",
302 |        "</tr>\n",
303 |        "<tr>\n",
304 |        "  <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",
305 |        "</tr>\n",
306 |        "<tr>\n",
307 |        "  <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",
308 |        "</tr>\n",
309 |        "<tr>\n",
310 |        "  <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",
311 |        "</tr>\n",
312 |        "</table>\n",
313 |        "<table class=\"simpletable\">\n",
314 |        "<caption>Roots</caption>\n",
315 |        "<tr>\n",
316 |        "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
317 |        "</tr>\n",
318 |        "<tr>\n",
319 |        "  <th>AR.1</th> <td>           1.0015</td> <td>          +0.0000j</td> <td>           1.0015</td> <td>           0.0000</td>\n",
320 |        "</tr>\n",
321 |        "<tr>\n",
322 |        "  <th>AR.2</th> <td>         -58.4573</td> <td>          +0.0000j</td> <td>          58.4573</td> <td>           0.5000</td>\n",
323 |        "</tr>\n",
324 |        "</table>"
325 |       ],
326 |       "text/plain": [
327 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
328 |        "\"\"\"\n",
329 |        "                              ARMA Model Results                              \n",
330 |        "==============================================================================\n",
331 |        "Dep. Variable:           market_value   No. Observations:                 5021\n",
332 |        "Model:                     ARMA(2, 0)   Log Likelihood              -27621.866\n",
333 |        "Method:                       css-mle   S.D. of innovations             59.249\n",
334 |        "Date:                Fri, 27 Sep 2019   AIC                          55251.731\n",
335 |        "Time:                        10:03:37   BIC                          55277.817\n",
336 |        "Sample:                    01-07-1994   HQIC                         55260.872\n",
337 |        "                         - 04-05-2013                                         \n",
338 |        "======================================================================================\n",
339 |        "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
340 |        "--------------------------------------------------------------------------------------\n",
341 |        "const               5036.7744    495.444     10.166      0.000    4065.721    6007.828\n",
342 |        "ar.L1.market_value     0.9814      0.014     69.595      0.000       0.954       1.009\n",
343 |        "ar.L2.market_value     0.0171      0.014      1.211      0.226      -0.011       0.045\n",
344 |        "                                    Roots                                    \n",
345 |        "=============================================================================\n",
346 |        "                  Real          Imaginary           Modulus         Frequency\n",
347 |        "-----------------------------------------------------------------------------\n",
348 |        "AR.1            1.0015           +0.0000j            1.0015            0.0000\n",
349 |        "AR.2          -58.4573           +0.0000j           58.4573            0.5000\n",
350 |        "-----------------------------------------------------------------------------\n",
351 |        "\"\"\""
352 |       ]
353 |      },
354 |      "execution_count": 11,
355 |      "metadata": {},
356 |      "output_type": "execute_result"
357 |     }
358 |    ],
359 |    "source": [
360 |     "model_ar_2 = ARMA(df.market_value, order=(2,0))\n",
361 |     "results_ar_2 = model_ar_2.fit()\n",
362 |     "results_ar_2.summary()"
363 |    ]
364 |   },
365 |   {
366 |    "cell_type": "code",
367 |    "execution_count": 12,
368 |    "metadata": {},
369 |    "outputs": [
370 |     {
371 |      "data": {
372 |       "text/html": [
373 |        "<table class=\"simpletable\">\n",
374 |        "<caption>ARMA Model Results</caption>\n",
375 |        "<tr>\n",
376 |        "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
377 |        "</tr>\n",
378 |        "<tr>\n",
379 |        "  <th>Model:</th>            <td>ARMA(3, 0)</td>    <th>  Log Likelihood     </th> <td>-27615.871</td>\n",
380 |        "</tr>\n",
381 |        "<tr>\n",
382 |        "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.178</td>  \n",
383 |        "</tr>\n",
384 |        "<tr>\n",
385 |        "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55241.742</td>\n",
386 |        "</tr>\n",
387 |        "<tr>\n",
388 |        "  <th>Time:</th>              <td>10:03:37</td>     <th>  BIC                </th>  <td>55274.349</td>\n",
389 |        "</tr>\n",
390 |        "<tr>\n",
391 |        "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55253.168</td>\n",
392 |        "</tr>\n",
393 |        "<tr>\n",
394 |        "  <th></th>                 <td>- 04-05-2013</td>   <th>                     </th>      <td> </td>    \n",
395 |        "</tr>\n",
396 |        "</table>\n",
397 |        "<table class=\"simpletable\">\n",
398 |        "<tr>\n",
399 |        "           <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",
400 |        "</tr>\n",
401 |        "<tr>\n",
402 |        "  <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",
403 |        "</tr>\n",
404 |        "<tr>\n",
405 |        "  <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",
406 |        "</tr>\n",
407 |        "<tr>\n",
408 |        "  <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",
409 |        "</tr>\n",
410 |        "<tr>\n",
411 |        "  <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",
412 |        "</tr>\n",
413 |        "</table>\n",
414 |        "<table class=\"simpletable\">\n",
415 |        "<caption>Roots</caption>\n",
416 |        "<tr>\n",
417 |        "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
418 |        "</tr>\n",
419 |        "<tr>\n",
420 |        "  <th>AR.1</th> <td>           1.0013</td> <td>          -0.0000j</td> <td>           1.0013</td> <td>          -0.0000</td>\n",
421 |        "</tr>\n",
422 |        "<tr>\n",
423 |        "  <th>AR.2</th> <td>          -0.1850</td> <td>          -4.5161j</td> <td>           4.5199</td> <td>          -0.2565</td>\n",
424 |        "</tr>\n",
425 |        "<tr>\n",
426 |        "  <th>AR.3</th> <td>          -0.1850</td> <td>          +4.5161j</td> <td>           4.5199</td> <td>           0.2565</td>\n",
427 |        "</tr>\n",
428 |        "</table>"
429 |       ],
430 |       "text/plain": [
431 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
432 |        "\"\"\"\n",
433 |        "                              ARMA Model Results                              \n",
434 |        "==============================================================================\n",
435 |        "Dep. Variable:           market_value   No. Observations:                 5021\n",
436 |        "Model:                     ARMA(3, 0)   Log Likelihood              -27615.871\n",
437 |        "Method:                       css-mle   S.D. of innovations             59.178\n",
438 |        "Date:                Fri, 27 Sep 2019   AIC                          55241.742\n",
439 |        "Time:                        10:03:37   BIC                          55274.349\n",
440 |        "Sample:                    01-07-1994   HQIC                         55253.168\n",
441 |        "                         - 04-05-2013                                         \n",
442 |        "======================================================================================\n",
443 |        "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
444 |        "--------------------------------------------------------------------------------------\n",
445 |        "const               5089.6412    520.376      9.781      0.000    4069.723    6109.560\n",
446 |        "ar.L1.market_value     0.9806      0.014     69.619      0.000       0.953       1.008\n",
447 |        "ar.L2.market_value    -0.0309      0.020     -1.563      0.118      -0.070       0.008\n",
448 |        "ar.L3.market_value     0.0489      0.014      3.466      0.001       0.021       0.077\n",
449 |        "                                    Roots                                    \n",
450 |        "=============================================================================\n",
451 |        "                  Real          Imaginary           Modulus         Frequency\n",
452 |        "-----------------------------------------------------------------------------\n",
453 |        "AR.1            1.0013           -0.0000j            1.0013           -0.0000\n",
454 |        "AR.2           -0.1850           -4.5161j            4.5199           -0.2565\n",
455 |        "AR.3           -0.1850           +4.5161j            4.5199            0.2565\n",
456 |        "-----------------------------------------------------------------------------\n",
457 |        "\"\"\""
458 |       ]
459 |      },
460 |      "execution_count": 12,
461 |      "metadata": {},
462 |      "output_type": "execute_result"
463 |     }
464 |    ],
465 |    "source": [
466 |     "model_ar_3 = ARMA(df.market_value, order=(3,0))\n",
467 |     "results_ar_3 = model_ar_3.fit()\n",
468 |     "results_ar_3.summary()"
469 |    ]
470 |   },
471 |   {
472 |    "cell_type": "code",
473 |    "execution_count": 13,
474 |    "metadata": {},
475 |    "outputs": [
476 |     {
477 |      "data": {
478 |       "text/html": [
479 |        "<table class=\"simpletable\">\n",
480 |        "<caption>ARMA Model Results</caption>\n",
481 |        "<tr>\n",
482 |        "  <th>Dep. Variable:</th>   <td>market_value</td>   <th>  No. Observations:  </th>    <td>5021</td>   \n",
483 |        "</tr>\n",
484 |        "<tr>\n",
485 |        "  <th>Model:</th>            <td>ARMA(4, 0)</td>    <th>  Log Likelihood     </th> <td>-27602.863</td>\n",
486 |        "</tr>\n",
487 |        "<tr>\n",
488 |        "  <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>59.024</td>  \n",
489 |        "</tr>\n",
490 |        "<tr>\n",
491 |        "  <th>Date:</th>          <td>Fri, 27 Sep 2019</td> <th>  AIC                </th>  <td>55217.726</td>\n",
492 |        "</tr>\n",
493 |        "<tr>\n",
494 |        "  <th>Time:</th>              <td>10:03:38</td>     <th>  BIC                </th>  <td>55256.854</td>\n",
495 |        "</tr>\n",
496 |        "<tr>\n",
497 |        "  <th>Sample:</th>           <td>01-07-1994</td>    <th>  HQIC               </th>  <td>55231.437</td>\n",
498 |        "</tr>\n",
499 |        "<tr>\n",
500 |        "  <th></th>                 <td>- 04-05-2013</td>   <th>                     </th>      <td> </td>    \n",
501 |        "</tr>\n",
502 |        "</table>\n",
503 |        "<table class=\"simpletable\">\n",
504 |        "<tr>\n",
505 |        "           <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",
506 |        "</tr>\n",
507 |        "<tr>\n",
508 |        "  <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",
509 |        "</tr>\n",
510 |        "<tr>\n",
511 |        "  <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",
512 |        "</tr>\n",
513 |        "<tr>\n",
514 |        "  <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",
515 |        "</tr>\n",
516 |        "<tr>\n",
517 |        "  <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",
518 |        "</tr>\n",
519 |        "<tr>\n",
520 |        "  <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",
521 |        "</tr>\n",
522 |        "</table>\n",
523 |        "<table class=\"simpletable\">\n",
524 |        "<caption>Roots</caption>\n",
525 |        "<tr>\n",
526 |        "    <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>\n",
527 |        "</tr>\n",
528 |        "<tr>\n",
529 |        "  <th>AR.1</th> <td>           1.0009</td> <td>          -0.0000j</td> <td>           1.0009</td> <td>          -0.0000</td>\n",
530 |        "</tr>\n",
531 |        "<tr>\n",
532 |        "  <th>AR.2</th> <td>           0.9543</td> <td>          -2.0876j</td> <td>           2.2954</td> <td>          -0.1818</td>\n",
533 |        "</tr>\n",
534 |        "<tr>\n",
535 |        "  <th>AR.3</th> <td>           0.9543</td> <td>          +2.0876j</td> <td>           2.2954</td> <td>           0.1818</td>\n",
536 |        "</tr>\n",
537 |        "<tr>\n",
538 |        "  <th>AR.4</th> <td>          -2.6026</td> <td>          -0.0000j</td> <td>           2.6026</td> <td>          -0.5000</td>\n",
539 |        "</tr>\n",
540 |        "</table>"
541 |       ],
542 |       "text/plain": [
543 |        "<class 'statsmodels.iolib.summary.Summary'>\n",
544 |        "\"\"\"\n",
545 |        "                              ARMA Model Results                              \n",
546 |        "==============================================================================\n",
547 |        "Dep. Variable:           market_value   No. Observations:                 5021\n",
548 |        "Model:                     ARMA(4, 0)   Log Likelihood              -27602.863\n",
549 |        "Method:                       css-mle   S.D. of innovations             59.024\n",
550 |        "Date:                Fri, 27 Sep 2019   AIC                          55217.726\n",
551 |        "Time:                        10:03:38   BIC                          55256.854\n",
552 |        "Sample:                    01-07-1994   HQIC                         55231.437\n",
553 |        "                         - 04-05-2013                                         \n",
554 |        "======================================================================================\n",
555 |        "                         coef    std err          z      P>|z|      [0.025      0.975]\n",
556 |        "--------------------------------------------------------------------------------------\n",
557 |        "const               5493.1989    959.292      5.726      0.000    3613.022    7373.376\n",
558 |        "ar.L1.market_value     0.9771      0.014     69.567      0.000       0.950       1.005\n",
559 |        "ar.L2.market_value    -0.0286      0.020     -1.455      0.146      -0.067       0.010\n",
560 |        "ar.L3.market_value    -0.0224      0.020     -1.135      0.256      -0.061       0.016\n",
561 |        "ar.L4.market_value     0.0729      0.014      5.171      0.000       0.045       0.100\n",
562 |        "                                    Roots                                    \n",
563 |        "=============================================================================\n",
564 |        "                  Real          Imaginary           Modulus         Frequency\n",
565 |        "-----------------------------------------------------------------------------\n",
566 |        "AR.1            1.0009           -0.0000j            1.0009           -0.0000\n",
567 |        "AR.2            0.9543           -2.0876j            2.2954           -0.1818\n",
568 |        "AR.3            0.9543           +2.0876j            2.2954            0.1818\n",
569 |        "AR.4           -2.6026           -0.0000j            2.6026           -0.5000\n",
570 |        "-----------------------------------------------------------------------------\n",
571 |        "\"\"\""
572 |       ]
573 |      },
574 |      "execution_count": 13,
575 |      "metadata": {},
576 |      "output_type": "execute_result"
577 |     }
578 |    ],
579 |    "source": [
580 |     "model_ar_4 = ARMA(df.market_value, order=[4,0])\n",
581 |     "results_ar_4 = model_ar_4.fit()\n",
582 |     "results_ar_4.summary()"
583 |    ]
584 |   },
585 |   {
586 |    "cell_type": "markdown",
587 |    "metadata": {},
588 |    "source": [
589 |     "## LLR Test"
590 |    ]
591 |   },
592 |   {
593 |    "cell_type": "code",
594 |    "execution_count": 16,
595 |    "metadata": {},
596 |    "outputs": [],
597 |    "source": [
598 |     "def LLR_test(mod_1, mod_2, DF=1):\n",
599 |     "    L1 = mod_1.fit().llf\n",
600 |     "    L2 = mod_2.fit().llf\n",
601 |     "    LR = (2*(L2-L1))\n",
602 |     "    p = chi2.sf(LR, DF).round(3)\n",
603 |     "    return p"
604 |    ]
605 |   },
606 |   {
607 |    "cell_type": "markdown",
608 |    "metadata": {},
609 |    "source": [
610 |     "## Comparing Higher-Lag AR Models"
611 |    ]
612 |   },
613 |   {
614 |    "cell_type": "code",
615 |    "execution_count": 17,
616 |    "metadata": {},
617 |    "outputs": [
618 |     {
619 |      "data": {
620 |       "text/plain": [
621 |        "0.001"
622 |       ]
623 |      },
624 |      "execution_count": 17,
625 |      "metadata": {},
626 |      "output_type": "execute_result"
627 |     }
628 |    ],
629 |    "source": [
630 |     "LLR_test(model_ar_2, model_ar_3)"
631 |    ]
632 |   },
633 |   {
634 |    "cell_type": "code",
635 |    "execution_count": null,
636 |    "metadata": {},
637 |    "outputs": [],
638 |    "source": [
639 |     "LLR_test(model_ar_3, model_ar_4)"
640 |    ]
641 |   },
642 |   {
643 |    "cell_type": "code",
644 |    "execution_count": null,
645 |    "metadata": {},
646 |    "outputs": [],
647 |    "source": [
648 |     "model_ar_4 = ARMA(df.market_value, order=[4,0])\n",
649 |     "results_ar_4 = model_ar_4.fit()\n",
650 |     "print(results_ar_4.summary()) \n",
651 |     "print (\"LLR test: \" + str(LLR_test(model_ar_3, model_ar_4)))"
652 |    ]
653 |   },
654 |   {
655 |    "cell_type": "code",
656 |    "execution_count": null,
657 |    "metadata": {},
658 |    "outputs": [],
659 |    "source": [
660 |     "model_ar_5 = ARMA(df.market_value, order=(5,0))\n",
661 |     "results_ar_5 = model_ar_5.fit()\n",
662 |     "print(results_ar_5.summary())\n",
663 |     "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_4, model_ar_5)))"
664 |    ]
665 |   },
666 |   {
667 |    "cell_type": "code",
668 |    "execution_count": null,
669 |    "metadata": {},
670 |    "outputs": [],
671 |    "source": [
672 |     "model_ar_6 = ARMA(df.market_value, order=(6,0))\n",
673 |     "results_ar_6 = model_ar_6.fit()\n",
674 |     "print(results_ar_6.summary())\n",
675 |     "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_5, model_ar_6)))"
676 |    ]
677 |   },
678 |   {
679 |    "cell_type": "code",
680 |    "execution_count": null,
681 |    "metadata": {},
682 |    "outputs": [],
683 |    "source": [
684 |     "model_ar_7 = ARMA(df.market_value, order=(7,0))\n",
685 |     "results_ar_7 = model_ar_7.fit()\n",
686 |     "print(results_ar_7.summary())\n",
687 |     "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_6, model_ar_7)))"
688 |    ]
689 |   },
690 |   {
691 |    "cell_type": "code",
692 |    "execution_count": null,
693 |    "metadata": {},
694 |    "outputs": [],
695 |    "source": [
696 |     "model_ar_8 = ARMA(df.market_value, order=(8,0))\n",
697 |     "results_ar_8 = model_ar_8.fit()\n",
698 |     "print(results_ar_8.summary())\n",
699 |     "print(\"\\nLLR test p-value = \" + str(LLR_test(model_ar_7, model_ar_8)))"
700 |    ]
701 |   },
702 |   {
703 |    "cell_type": "code",
704 |    "execution_count": null,
705 |    "metadata": {},
706 |    "outputs": [],
707 |    "source": [
708 |     "print(\"LLR test: \" + str(LLR_test(model_ar, model_ar_7, DF = 6)))"
709 |    ]
710 |   },
711 |   {
712 |    "cell_type": "markdown",
713 |    "metadata": {},
714 |    "source": [
715 |     "## Analysing the Residuals"
716 |    ]
717 |   },
718 |   {
719 |    "cell_type": "code",
720 |    "execution_count": null,
721 |    "metadata": {},
722 |    "outputs": [],
723 |    "source": [
724 |     "df['res_price'] = results_ar_7.resid"
725 |    ]
726 |   },
727 |   {
728 |    "cell_type": "code",
729 |    "execution_count": null,
730 |    "metadata": {},
731 |    "outputs": [],
732 |    "source": [
733 |     "df.res_price.mean()"
734 |    ]
735 |   },
736 |   {
737 |    "cell_type": "code",
738 |    "execution_count": null,
739 |    "metadata": {},
740 |    "outputs": [],
741 |    "source": [
742 |     "df.res_price.var()"
743 |    ]
744 |   },
745 |   {
746 |    "cell_type": "code",
747 |    "execution_count": null,
748 |    "metadata": {},
749 |    "outputs": [],
750 |    "source": [
751 |     "sts.adfuller(df.res_price)"
752 |    ]
753 |   },
754 |   {
755 |    "cell_type": "code",
756 |    "execution_count": null,
757 |    "metadata": {},
758 |    "outputs": [],
759 |    "source": [
760 |     "sgt.plot_acf(df.res_price, zero = False, lags = 40)\n",
761 |     "plt.title(\"ACF Of Residuals for Prices\",size=24)\n",
762 |     "plt.show()"
763 |    ]
764 |   },
765 |   {
766 |    "cell_type": "code",
767 |    "execution_count": null,
768 |    "metadata": {},
769 |    "outputs": [],
770 |    "source": [
771 |     "df.res_price[1:].plot(figsize=(20,5))\n",
772 |     "plt.title(\"Residuals of Prices\",size=24)\n",
773 |     "plt.show()"
774 |    ]
775 |   }
776 |  ],
777 |  "metadata": {
778 |   "kernelspec": {
779 |    "display_name": "Python 3",
780 |    "language": "python",
781 |    "name": "python3"
782 |   },
783 |   "language_info": {
784 |    "codemirror_mode": {
785 |     "name": "ipython",
786 |     "version": 3
787 |    },
788 |    "file_extension": ".py",
789 |    "mimetype": "text/x-python",
790 |    "name": "python",
791 |    "nbconvert_exporter": "python",
792 |    "pygments_lexer": "ipython3",
793 |    "version": "3.7.4"
794 |   }
795 |  },
796 |  "nbformat": 4,
797 |  "nbformat_minor": 2
798 | }
799 | 


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