ARIMA: autoregresivo integrado de medias móviles
------------------------------------------------

Para series de tiempo no estacionarias.

La parte de integrado en ARIMA significa que se usará el número de
diferencias no estacionales que debemos examinar para establecer
estacionariedad. Los datos se transformará para que sean estacionarios.
Esta transformación será con las diferencias :math:`\Delta x_t`.

ARIMA(p,d,q)
~~~~~~~~~~~~

-  :math:`p`: Componente autoregresiva del modelo :math:`(AR)`.

-  :math:`d`: orden de integración. Es el número de veces que se debe
   integrar la serie de tiempo para obtener la estacionariedad.

-  :math:`q`: Componente de medias móviles :math:`(MA)`.

Los modelos ARIMA(p,d,q) son un modelo ARMA(p,q) aplicado a una nueva
serie de tiempo generada mediante la integración, es decir, por las
diferencias y que esta nueva serie de tiempo es estacionaria.

Se empezará con el modelo ARIMA(1,1,1), luego se analizan los residuos,
si no son Ruido Blanco, se evalúan modelos más complejos.

Recordar que por cada integración se perderá una observación.

:math:`ARIMA(p,0,q)= ARMA(p,q)`

ARIMA(1,1,1)
~~~~~~~~~~~~

.. math::  \Delta x_t = c+\varphi_1 \Delta x_{t-1}+\theta_1 \epsilon_{t-1}+\epsilon_t 

.. math::  \Delta x_t = x_t - x_{t-1} 

.. code:: ipython3

    import pandas as pd

.. code:: ipython3

    datos = pd.read_csv('Precio.csv', sep=';', decimal=',')

.. code:: ipython3

    datos.Fecha = pd.to_datetime(datos.Fecha, dayfirst = True)
    datos.set_index('Fecha', inplace=True)

ARIMA(1,1,1)
~~~~~~~~~~~~

.. code:: ipython3

    from statsmodels.tsa.arima_model import ARIMA

.. code:: ipython3

    import warnings
    warnings.filterwarnings("ignore")

.. code:: ipython3

    model_ar_1_i_1_ma_1 = ARIMA(datos.Precio, order=(1,1,1))
    results_ar_1_i_1_ma_1 = model_ar_1_i_1_ma_1.fit()
    results_ar_1_i_1_ma_1.summary()




.. raw:: html

    <table class="simpletable">
    <caption>ARIMA Model Results</caption>
    <tr>
      <th>Dep. Variable:</th>     <td>D.Precio</td>     <th>  No. Observations:  </th>    <td>5816</td>   
    </tr>
    <tr>
      <th>Model:</th>          <td>ARIMA(1, 1, 1)</td>  <th>  Log Likelihood     </th> <td>-27010.235</td>
    </tr>
    <tr>
      <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>25.158</td>  
    </tr>
    <tr>
      <th>Date:</th>          <td>Thu, 15 Jul 2021</td> <th>  AIC                </th>  <td>54028.469</td>
    </tr>
    <tr>
      <th>Time:</th>              <td>05:54:51</td>     <th>  BIC                </th>  <td>54055.143</td>
    </tr>
    <tr>
      <th>Sample:</th>                <td>1</td>        <th>  HQIC               </th>  <td>54037.747</td>
    </tr>
    <tr>
      <th></th>                       <td> </td>        <th>                     </th>      <td> </td>    
    </tr>
    </table>
    <table class="simpletable">
    <tr>
             <td></td>           <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  
    </tr>
    <tr>
      <th>const</th>          <td>    0.0408</td> <td>    0.364</td> <td>    0.112</td> <td> 0.911</td> <td>   -0.672</td> <td>    0.753</td>
    </tr>
    <tr>
      <th>ar.L1.D.Precio</th> <td>   -0.2638</td> <td>    0.059</td> <td>   -4.472</td> <td> 0.000</td> <td>   -0.379</td> <td>   -0.148</td>
    </tr>
    <tr>
      <th>ma.L1.D.Precio</th> <td>    0.3930</td> <td>    0.056</td> <td>    7.080</td> <td> 0.000</td> <td>    0.284</td> <td>    0.502</td>
    </tr>
    </table>
    <table class="simpletable">
    <caption>Roots</caption>
    <tr>
        <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>
    </tr>
    <tr>
      <th>AR.1</th> <td>          -3.7903</td> <td>          +0.0000j</td> <td>           3.7903</td> <td>           0.5000</td>
    </tr>
    <tr>
      <th>MA.1</th> <td>          -2.5443</td> <td>          +0.0000j</td> <td>           2.5443</td> <td>           0.5000</td>
    </tr>
    </table>



Residuales del ARIMA(1,1,1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    import statsmodels.graphics.tsaplots as sgt
    import matplotlib.pyplot as plt

.. code:: ipython3

    residuos_ARIAM111 = results_ar_1_i_1_ma_1.resid
    sgt.plot_acf(residuos_ARIAM111, zero = False, lags = 40)
    plt.title("ACF Of Residuals for ARIMA(1,1,1)",size=20)
    plt.show()



.. image:: output_19_0.png


.. code:: ipython3

    sgt.plot_acf(residuos_ARIAM111, zero = False, lags = 40)
    plt.title("ACF Of Residuals for ARIMA(1,1,1)",size=20)
    plt.show()



.. image:: output_20_0.png


Evaluación de varios modelos de ARIMA
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Primero se deben calcular todos los modelos y escoger los que tienen
todos los coeficientes significativos. Este paso no se ha realizado, se
evaluarán unos modelos que estamos suponiendo que tienen todos los
coeficientes significativos.

.. code:: ipython3

    model_ar_1_i_1_ma_2 = ARIMA(datos.Precio, order=(1,1,2))
    results_ar_1_i_1_ma_2 = model_ar_1_i_1_ma_2.fit()
    model_ar_1_i_1_ma_3 = ARIMA(datos.Precio, order=(1,1,3))
    results_ar_1_i_1_ma_3 = model_ar_1_i_1_ma_3.fit()
    model_ar_2_i_1_ma_1 = ARIMA(datos.Precio, order=(2,1,1))
    results_ar_2_i_1_ma_1 = model_ar_2_i_1_ma_1.fit()
    model_ar_3_i_1_ma_1 = ARIMA(datos.Precio, order=(3,1,1))
    results_ar_3_i_1_ma_1 = model_ar_3_i_1_ma_1.fit()
    model_ar_3_i_1_ma_2 = ARIMA(datos.Precio, order=(3,1,2))
    results_ar_3_i_1_ma_2 = model_ar_3_i_1_ma_2.fit(start_ar_lags=5)

.. code:: ipython3

    print("ARIMA(1,1,1):  \t LL = ", results_ar_1_i_1_ma_1.llf, "\t AIC = ", results_ar_1_i_1_ma_1.aic)
    print("ARIMA(1,1,2):  \t LL = ", results_ar_1_i_1_ma_2.llf, "\t AIC = ", results_ar_1_i_1_ma_2.aic)
    print("ARIMA(1,1,3):  \t LL = ", results_ar_1_i_1_ma_3.llf, "\t AIC = ", results_ar_1_i_1_ma_3.aic)
    print("ARIMA(2,1,1):  \t LL = ", results_ar_2_i_1_ma_1.llf, "\t AIC = ", results_ar_2_i_1_ma_1.aic)
    print("ARIMA(3,1,1):  \t LL = ", results_ar_3_i_1_ma_1.llf, "\t AIC = ", results_ar_3_i_1_ma_1.aic)
    print("ARIMA(3,1,2):  \t LL = ", results_ar_3_i_1_ma_2.llf, "\t AIC = ", results_ar_3_i_1_ma_2.aic)


.. parsed-literal::

    ARIMA(1,1,1):  	 LL =  -27010.234621625288 	 AIC =  54028.469243250576
    ARIMA(1,1,2):  	 LL =  -26970.45408581454 	 AIC =  53950.90817162908
    ARIMA(1,1,3):  	 LL =  -26951.814421818075 	 AIC =  53915.62884363615
    ARIMA(2,1,1):  	 LL =  -26960.455943244346 	 AIC =  53930.91188648869
    ARIMA(3,1,1):  	 LL =  -26954.471631818797 	 AIC =  53920.94326363759
    ARIMA(3,1,2):  	 LL =  -26954.419566101034 	 AIC =  53922.83913220207
    

De los modelos evaluados, el mejor es el ARIMA(1,1,3) por tener mayor
Log-Verosimilitud y menor AIC.

LLR Test
~~~~~~~~

Sin embargo, los modelos ARIMA(1,1,1) y ARIMA(1,1,2) están anidados al
modelo ARIMA(1,1,3), se hará comparación si el modelo ARIMA(1,1,3) es
mejor que los dos anteriores.

.. code:: ipython3

    def LLR_test(mod_1, mod_2, DF = 1):
        L1 = mod_1.llf
        L2 = mod_2.llf
        LR = (2*(L2-L1))    
        p = chi2.sf(LR, DF).round(3)
        return p

.. code:: ipython3

    from scipy.stats.distributions import chi2 

.. code:: ipython3

    print("\nLLR test p-value = " + str(LLR_test(results_ar_1_i_1_ma_2, results_ar_1_i_1_ma_3)))


.. parsed-literal::

    
    LLR test p-value = 0.0
    

.. code:: ipython3

    print("\nLLR test p-value = " + str(LLR_test(results_ar_1_i_1_ma_1, results_ar_1_i_1_ma_3, DF = 2)))


.. parsed-literal::

    
    LLR test p-value = 0.0
    

El modelo ARIMA(1,1,3) si es mejor que los dos modelos que están
anidados con este.

Residuales del modelo ARIMA(1,1,3)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    residuales_ARIMA_113 = results_ar_1_i_1_ma_3.resid
    sgt.plot_acf(residuales_ARIMA_113, zero = False, lags = 40)
    plt.title("ACF Of Residuals for ARIMA(1,1,3)", size=20)
    plt.show()



.. image:: output_34_0.png


Se podría evaluar modelos más complejos hasta el ARIMA(7,1,7).

.. code:: ipython3

    model_ar_5_i_1_ma_5 = ARIMA(datos.Precio, order=(5,1,5))
    results_ar_5_i_1_ma_5 = model_ar_5_i_1_ma_5.fit(start_ar_lags=11)
    results_ar_5_i_1_ma_5.summary()




.. raw:: html

    <table class="simpletable">
    <caption>ARIMA Model Results</caption>
    <tr>
      <th>Dep. Variable:</th>     <td>D.Precio</td>     <th>  No. Observations:  </th>    <td>5816</td>   
    </tr>
    <tr>
      <th>Model:</th>          <td>ARIMA(5, 1, 5)</td>  <th>  Log Likelihood     </th> <td>-26878.553</td>
    </tr>
    <tr>
      <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>24.593</td>  
    </tr>
    <tr>
      <th>Date:</th>          <td>Thu, 15 Jul 2021</td> <th>  AIC                </th>  <td>53781.106</td>
    </tr>
    <tr>
      <th>Time:</th>              <td>05:55:02</td>     <th>  BIC                </th>  <td>53861.127</td>
    </tr>
    <tr>
      <th>Sample:</th>                <td>1</td>        <th>  HQIC               </th>  <td>53808.939</td>
    </tr>
    <tr>
      <th></th>                       <td> </td>        <th>                     </th>      <td> </td>    
    </tr>
    </table>
    <table class="simpletable">
    <tr>
             <td></td>           <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  
    </tr>
    <tr>
      <th>const</th>          <td>    0.0411</td> <td>    0.332</td> <td>    0.124</td> <td> 0.901</td> <td>   -0.609</td> <td>    0.692</td>
    </tr>
    <tr>
      <th>ar.L1.D.Precio</th> <td>   -0.3205</td> <td>    0.042</td> <td>   -7.665</td> <td> 0.000</td> <td>   -0.402</td> <td>   -0.239</td>
    </tr>
    <tr>
      <th>ar.L2.D.Precio</th> <td>   -0.2270</td> <td>    0.033</td> <td>   -6.855</td> <td> 0.000</td> <td>   -0.292</td> <td>   -0.162</td>
    </tr>
    <tr>
      <th>ar.L3.D.Precio</th> <td>   -0.4506</td> <td>    0.030</td> <td>  -15.195</td> <td> 0.000</td> <td>   -0.509</td> <td>   -0.393</td>
    </tr>
    <tr>
      <th>ar.L4.D.Precio</th> <td>   -0.3947</td> <td>    0.034</td> <td>  -11.511</td> <td> 0.000</td> <td>   -0.462</td> <td>   -0.327</td>
    </tr>
    <tr>
      <th>ar.L5.D.Precio</th> <td>   -0.6894</td> <td>    0.032</td> <td>  -21.787</td> <td> 0.000</td> <td>   -0.751</td> <td>   -0.627</td>
    </tr>
    <tr>
      <th>ma.L1.D.Precio</th> <td>    0.4184</td> <td>    0.040</td> <td>   10.561</td> <td> 0.000</td> <td>    0.341</td> <td>    0.496</td>
    </tr>
    <tr>
      <th>ma.L2.D.Precio</th> <td>    0.1603</td> <td>    0.029</td> <td>    5.588</td> <td> 0.000</td> <td>    0.104</td> <td>    0.216</td>
    </tr>
    <tr>
      <th>ma.L3.D.Precio</th> <td>    0.3645</td> <td>    0.027</td> <td>   13.655</td> <td> 0.000</td> <td>    0.312</td> <td>    0.417</td>
    </tr>
    <tr>
      <th>ma.L4.D.Precio</th> <td>    0.4863</td> <td>    0.030</td> <td>   16.124</td> <td> 0.000</td> <td>    0.427</td> <td>    0.545</td>
    </tr>
    <tr>
      <th>ma.L5.D.Precio</th> <td>    0.7429</td> <td>    0.029</td> <td>   25.497</td> <td> 0.000</td> <td>    0.686</td> <td>    0.800</td>
    </tr>
    </table>
    <table class="simpletable">
    <caption>Roots</caption>
    <tr>
        <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>
    </tr>
    <tr>
      <th>AR.1</th> <td>           0.6833</td> <td>          -0.7607j</td> <td>           1.0225</td> <td>          -0.1335</td>
    </tr>
    <tr>
      <th>AR.2</th> <td>           0.6833</td> <td>          +0.7607j</td> <td>           1.0225</td> <td>           0.1335</td>
    </tr>
    <tr>
      <th>AR.3</th> <td>          -1.0478</td> <td>          -0.0000j</td> <td>           1.0478</td> <td>          -0.5000</td>
    </tr>
    <tr>
      <th>AR.4</th> <td>          -0.4456</td> <td>          -1.0609j</td> <td>           1.1506</td> <td>          -0.3133</td>
    </tr>
    <tr>
      <th>AR.5</th> <td>          -0.4456</td> <td>          +1.0609j</td> <td>           1.1506</td> <td>           0.3133</td>
    </tr>
    <tr>
      <th>MA.1</th> <td>           0.6951</td> <td>          -0.7411j</td> <td>           1.0161</td> <td>          -0.1301</td>
    </tr>
    <tr>
      <th>MA.2</th> <td>           0.6951</td> <td>          +0.7411j</td> <td>           1.0161</td> <td>           0.1301</td>
    </tr>
    <tr>
      <th>MA.3</th> <td>          -1.0380</td> <td>          -0.0000j</td> <td>           1.0380</td> <td>          -0.5000</td>
    </tr>
    <tr>
      <th>MA.4</th> <td>          -0.5034</td> <td>          -1.0013j</td> <td>           1.1207</td> <td>          -0.3241</td>
    </tr>
    <tr>
      <th>MA.5</th> <td>          -0.5034</td> <td>          +1.0013j</td> <td>           1.1207</td> <td>           0.3241</td>
    </tr>
    </table>



.. code:: ipython3

    print("\nLLR test p-value = " + str(LLR_test(results_ar_1_i_1_ma_3, results_ar_5_i_1_ma_5)))


.. parsed-literal::

    
    LLR test p-value = 0.0
    

El ARIMA(5,1,5) es mejor que el ARIMA(1,1,3)

Residuales del modelo ARIMA(5,1,5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    residuales_ARIMA_515 = results_ar_5_i_1_ma_5.resid
    sgt.plot_acf(residuales_ARIMA_515, zero = False, lags = 40)
    plt.title("ACF Of Residuals for ARIMA(5,1,5)", size=20)
    plt.show()



.. image:: output_40_0.png


Integración :math:`d`
~~~~~~~~~~~~~~~~~~~~~

Se calculará la primer diferencia de los precios y se hará la prueba D-F
para determinar si la nueva serie de tiempo es estacionaria o no.

Primera diferencia
~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    datos['delta_prices']=datos.Precio.diff(1)

.. code:: ipython3

    datos.head()




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }
    
        .dataframe tbody tr th {
            vertical-align: top;
        }
    
        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>Precio</th>
          <th>delta_prices</th>
        </tr>
        <tr>
          <th>Fecha</th>
          <th></th>
          <th></th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>2003-03-01</th>
          <td>72.412500</td>
          <td>NaN</td>
        </tr>
        <tr>
          <th>2003-03-02</th>
          <td>72.837083</td>
          <td>0.424583</td>
        </tr>
        <tr>
          <th>2003-03-03</th>
          <td>70.103750</td>
          <td>-2.733333</td>
        </tr>
        <tr>
          <th>2003-03-04</th>
          <td>70.770417</td>
          <td>0.666667</td>
        </tr>
        <tr>
          <th>2003-03-05</th>
          <td>69.153750</td>
          <td>-1.616667</td>
        </tr>
      </tbody>
    </table>
    </div>



.. code:: ipython3

    import statsmodels.tsa.stattools as sts

.. code:: ipython3

    sts.adfuller(datos.delta_prices[1:])




.. parsed-literal::

    (-14.940985509998315,
     1.3227142701322909e-27,
     34,
     5781,
     {'1%': -3.431481673763484,
      '5%': -2.8620400923019074,
      '10%': -2.5670361971807805},
     53352.0028917936)



.. code:: ipython3

    sgt.plot_acf(datos.delta_prices[1:], zero = False, lags = 40)
    plt.title("ACF Of Delta Prices", size=20)
    plt.show()



.. image:: output_48_0.png


.. code:: ipython3

    sgt.plot_pacf(datos.delta_prices[1:], lags = 40, alpha = 0.05, zero = False , method = ('ols'))
    plt.title("Partial Autocorrelation Function for Delta Prices",size=20)
    plt.show()



.. image:: output_49_0.png


A continuación se mostrará una prueba de que una ARMA(1,1) sobre la
primera diferencia de la serie de tiempo es igual a una ARIMA(1,1) sobre
la serie de tiempo.

.. code:: ipython3

    model_delta_ar_1_i_1_ma_1 = ARIMA(datos.delta_prices[1:], order=(1,0,1))
    results_delta_ar_1_i_1_ma_1 = model_delta_ar_1_i_1_ma_1.fit()
    results_delta_ar_1_i_1_ma_1.summary()




.. raw:: html

    <table class="simpletable">
    <caption>ARMA Model Results</caption>
    <tr>
      <th>Dep. Variable:</th>   <td>delta_prices</td>   <th>  No. Observations:  </th>    <td>5816</td>   
    </tr>
    <tr>
      <th>Model:</th>            <td>ARMA(1, 1)</td>    <th>  Log Likelihood     </th> <td>-27010.235</td>
    </tr>
    <tr>
      <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>25.158</td>  
    </tr>
    <tr>
      <th>Date:</th>          <td>Thu, 15 Jul 2021</td> <th>  AIC                </th>  <td>54028.469</td>
    </tr>
    <tr>
      <th>Time:</th>              <td>05:55:05</td>     <th>  BIC                </th>  <td>54055.143</td>
    </tr>
    <tr>
      <th>Sample:</th>                <td>0</td>        <th>  HQIC               </th>  <td>54037.747</td>
    </tr>
    <tr>
      <th></th>                       <td> </td>        <th>                     </th>      <td> </td>    
    </tr>
    </table>
    <table class="simpletable">
    <tr>
               <td></td>             <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  
    </tr>
    <tr>
      <th>const</th>              <td>    0.0408</td> <td>    0.364</td> <td>    0.112</td> <td> 0.911</td> <td>   -0.672</td> <td>    0.753</td>
    </tr>
    <tr>
      <th>ar.L1.delta_prices</th> <td>   -0.2638</td> <td>    0.059</td> <td>   -4.472</td> <td> 0.000</td> <td>   -0.379</td> <td>   -0.148</td>
    </tr>
    <tr>
      <th>ma.L1.delta_prices</th> <td>    0.3930</td> <td>    0.056</td> <td>    7.080</td> <td> 0.000</td> <td>    0.284</td> <td>    0.502</td>
    </tr>
    </table>
    <table class="simpletable">
    <caption>Roots</caption>
    <tr>
        <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>
    </tr>
    <tr>
      <th>AR.1</th> <td>          -3.7903</td> <td>          +0.0000j</td> <td>           3.7903</td> <td>           0.5000</td>
    </tr>
    <tr>
      <th>MA.1</th> <td>          -2.5443</td> <td>          +0.0000j</td> <td>           2.5443</td> <td>           0.5000</td>
    </tr>
    </table>



.. code:: ipython3

    model_ar_1_i_1_ma_1 = ARIMA(datos.Precio, order=(1,1,1))
    results_ar_1_i_1_ma_1 = model_ar_1_i_1_ma_1.fit()
    results_ar_1_i_1_ma_1.summary()




.. raw:: html

    <table class="simpletable">
    <caption>ARIMA Model Results</caption>
    <tr>
      <th>Dep. Variable:</th>     <td>D.Precio</td>     <th>  No. Observations:  </th>    <td>5816</td>   
    </tr>
    <tr>
      <th>Model:</th>          <td>ARIMA(1, 1, 1)</td>  <th>  Log Likelihood     </th> <td>-27010.235</td>
    </tr>
    <tr>
      <th>Method:</th>             <td>css-mle</td>     <th>  S.D. of innovations</th>   <td>25.158</td>  
    </tr>
    <tr>
      <th>Date:</th>          <td>Thu, 15 Jul 2021</td> <th>  AIC                </th>  <td>54028.469</td>
    </tr>
    <tr>
      <th>Time:</th>              <td>05:55:06</td>     <th>  BIC                </th>  <td>54055.143</td>
    </tr>
    <tr>
      <th>Sample:</th>                <td>1</td>        <th>  HQIC               </th>  <td>54037.747</td>
    </tr>
    <tr>
      <th></th>                       <td> </td>        <th>                     </th>      <td> </td>    
    </tr>
    </table>
    <table class="simpletable">
    <tr>
             <td></td>           <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  
    </tr>
    <tr>
      <th>const</th>          <td>    0.0408</td> <td>    0.364</td> <td>    0.112</td> <td> 0.911</td> <td>   -0.672</td> <td>    0.753</td>
    </tr>
    <tr>
      <th>ar.L1.D.Precio</th> <td>   -0.2638</td> <td>    0.059</td> <td>   -4.472</td> <td> 0.000</td> <td>   -0.379</td> <td>   -0.148</td>
    </tr>
    <tr>
      <th>ma.L1.D.Precio</th> <td>    0.3930</td> <td>    0.056</td> <td>    7.080</td> <td> 0.000</td> <td>    0.284</td> <td>    0.502</td>
    </tr>
    </table>
    <table class="simpletable">
    <caption>Roots</caption>
    <tr>
        <td></td>   <th>            Real</th>  <th>         Imaginary</th> <th>         Modulus</th>  <th>        Frequency</th>
    </tr>
    <tr>
      <th>AR.1</th> <td>          -3.7903</td> <td>          +0.0000j</td> <td>           3.7903</td> <td>           0.5000</td>
    </tr>
    <tr>
      <th>MA.1</th> <td>          -2.5443</td> <td>          +0.0000j</td> <td>           2.5443</td> <td>           0.5000</td>
    </tr>
    </table>



SARIMA(p,d,q)(P,D,Q,s)
~~~~~~~~~~~~~~~~~~~~~~

**SARIMA-Seasonal Autoregressive Integrated Moving Averages**

.. code:: ipython3

    from statsmodels.tsa.statespace.sarimax import SARIMAX

.. code:: ipython3

    model_sarima = SARIMAX(datos.Precio, order=(1,0,1), seasonal_order = (2,0,1,2))
    results_sarima = model_sarima.fit()
    results_sarima.summary()




.. raw:: html

    <table class="simpletable">
    <caption>SARIMAX Results</caption>
    <tr>
      <th>Dep. Variable:</th>              <td>Precio</td>             <th>  No. Observations:  </th>    <td>5817</td>   
    </tr>
    <tr>
      <th>Model:</th>           <td>SARIMAX(1, 0, 1)x(2, 0, 1, 2)</td> <th>  Log Likelihood     </th> <td>-26974.805</td>
    </tr>
    <tr>
      <th>Date:</th>                  <td>Thu, 15 Jul 2021</td>        <th>  AIC                </th>  <td>53961.610</td>
    </tr>
    <tr>
      <th>Time:</th>                      <td>05:55:09</td>            <th>  BIC                </th>  <td>54001.621</td>
    </tr>
    <tr>
      <th>Sample:</th>                        <td>0</td>               <th>  HQIC               </th>  <td>53975.526</td>
    </tr>
    <tr>
      <th></th>                            <td> - 5817</td>            <th>                     </th>      <td> </td>    
    </tr>
    <tr>
      <th>Covariance Type:</th>              <td>opg</td>              <th>                     </th>      <td> </td>    
    </tr>
    </table>
    <table class="simpletable">
    <tr>
         <td></td>        <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  
    </tr>
    <tr>
      <th>ar.L1</th>   <td>    0.9949</td> <td>    0.001</td> <td> 1319.143</td> <td> 0.000</td> <td>    0.993</td> <td>    0.996</td>
    </tr>
    <tr>
      <th>ma.L1</th>   <td>    0.1061</td> <td>    0.003</td> <td>   40.096</td> <td> 0.000</td> <td>    0.101</td> <td>    0.111</td>
    </tr>
    <tr>
      <th>ar.S.L2</th> <td>    0.8309</td> <td>    0.020</td> <td>   42.567</td> <td> 0.000</td> <td>    0.793</td> <td>    0.869</td>
    </tr>
    <tr>
      <th>ar.S.L4</th> <td>    0.0987</td> <td>    0.004</td> <td>   22.085</td> <td> 0.000</td> <td>    0.090</td> <td>    0.108</td>
    </tr>
    <tr>
      <th>ma.S.L2</th> <td>   -0.9498</td> <td>    0.019</td> <td>  -48.766</td> <td> 0.000</td> <td>   -0.988</td> <td>   -0.912</td>
    </tr>
    <tr>
      <th>sigma2</th>  <td>  623.3489</td> <td>    1.583</td> <td>  393.688</td> <td> 0.000</td> <td>  620.246</td> <td>  626.452</td>
    </tr>
    </table>
    <table class="simpletable">
    <tr>
      <th>Ljung-Box (Q):</th>          <td>442.39</td> <th>  Jarque-Bera (JB):  </th> <td>19183877.49</td>
    </tr>
    <tr>
      <th>Prob(Q):</th>                 <td>0.00</td>  <th>  Prob(JB):          </th>    <td>0.00</td>    
    </tr>
    <tr>
      <th>Heteroskedasticity (H):</th>  <td>34.65</td> <th>  Skew:              </th>    <td>-6.00</td>   
    </tr>
    <tr>
      <th>Prob(H) (two-sided):</th>     <td>0.00</td>  <th>  Kurtosis:          </th>   <td>284.08</td>   
    </tr>
    </table><br/><br/>Warnings:<br/>[1] Covariance matrix calculated using the outer product of gradients (complex-step).



.. code:: ipython3

    residuales_SARIMA = results_sarima.resid
    sgt.plot_acf(residuales_SARIMA, zero = False, lags = 40)
    plt.title("ACF Of Residuals for SARIMA", size=20)
    plt.show()



.. image:: output_57_0.png