.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/linear_models/plot_02_box_jenkins_arma.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_time_series_linear_models_plot_02_box_jenkins_arma.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_time_series_linear_models_plot_02_box_jenkins_arma.py:


Box-Jenkins ARMA
================

George Box and Gwilym Jenkins (1970) turned the mixed autoregressive
moving-average model

.. math:: \phi(B)(x_t - \mu) = \theta(B)\,\varepsilon_t

into a three-stage method. *Identify* the orders from the autocorrelation
function (which cuts off after lag :math:`q` for a pure MA) and the partial
autocorrelation function (which cuts off after lag :math:`p` for a pure
AR); *estimate* by maximum likelihood; *check* that the residuals are
white noise; then forecast. Here the cycle is run on a simulated ARMA(1,1),
such as a spread or a monthly growth rate might follow.

.. GENERATED FROM PYTHON SOURCE LINES 19-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import fit_arma, ljung_box, simulate_arma
    from financialkit.time_series.visualizers import plot_acf

    x = simulate_arma(ar=[0.75], ma=[0.4], n=600, mean=2.0, rng=3)








.. GENERATED FROM PYTHON SOURCE LINES 28-31

Identification
--------------
Neither function cuts off: both decay, which points to a mixed model.

.. GENERATED FROM PYTHON SOURCE LINES 31-39

.. code-block:: Python


    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.5))
    plot_acf(x, 15, ax=ax1)
    plot_acf(x, 15, partial=True, ax=ax2)
    ax1.set_title("ACF: geometric decay")
    ax2.set_title("PACF: decays too")
    fig.tight_layout()




.. image-sg:: /api/gallery/time_series/linear_models/images/sphx_glr_plot_02_box_jenkins_arma_001.png
   :alt: ACF: geometric decay, PACF: decays too
   :srcset: /api/gallery/time_series/linear_models/images/sphx_glr_plot_02_box_jenkins_arma_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 40-45

Estimation and diagnostic checking
----------------------------------
Conditional maximum likelihood recovers :math:`\phi = 0.75` and
:math:`\theta = 0.4`, and the Ljung-Box test finds no correlation left in
the residuals (two degrees of freedom are used up by the fit).

.. GENERATED FROM PYTHON SOURCE LINES 45-51

.. code-block:: Python


    res = fit_arma(x, 1, 1)
    lb = ljung_box(res.residuals, lags=12, fitted_params=2)
    print(f"phi = {res.ar[0]:.3f}, theta = {res.ma[0]:.3f}, mu = {res.mean:.3f}, sigma^2 = {res.sigma2:.3f}")
    print(f"Ljung-Box on residuals: Q = {lb.statistic:.1f}, p = {lb.pvalue:.2f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    phi = 0.724, theta = 0.447, mu = 2.110, sigma^2 = 0.990
    Ljung-Box on residuals: Q = 11.8, p = 0.30




.. GENERATED FROM PYTHON SOURCE LINES 52-56

Forecasting
-----------
Forecasts decay geometrically back to the mean, and the 95% interval widens
to the unconditional spread.

.. GENERATED FROM PYTHON SOURCE LINES 56-69

.. code-block:: Python


    h = 30
    mean, se = res.forecast(h)
    t = np.arange(x.size)
    tf = np.arange(x.size, x.size + h)
    fig, ax = plt.subplots(figsize=(8, 3.5))
    ax.plot(t[-120:], x[-120:], color="C0", label="observed")
    ax.plot(tf, mean, color="C3", label="forecast")
    ax.fill_between(tf, mean - 1.96 * se, mean + 1.96 * se, color="C3", alpha=0.2, label="95% interval")
    ax.axhline(res.mean, color="k", ls=":", lw=1)
    ax.set_xlabel("period")
    ax.legend(loc="upper left")
    plt.show()



.. image-sg:: /api/gallery/time_series/linear_models/images/sphx_glr_plot_02_box_jenkins_arma_002.png
   :alt: plot 02 box jenkins arma
   :srcset: /api/gallery/time_series/linear_models/images/sphx_glr_plot_02_box_jenkins_arma_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_time_series_linear_models_plot_02_box_jenkins_arma.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/time_series/linear_models/plot_02_box_jenkins_arma.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_box_jenkins_arma.ipynb <plot_02_box_jenkins_arma.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_box_jenkins_arma.py <plot_02_box_jenkins_arma.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_box_jenkins_arma.zip <plot_02_box_jenkins_arma.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_