.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/volatility/plot_05_gjr_garch.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_volatility_plot_05_gjr_garch.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_volatility_plot_05_gjr_garch.py:


Glosten-Jagannathan-Runkle
==========================

Lawrence Glosten, Ravi Jagannathan and David Runkle (1993) asked whether
expected stock returns rise with volatility, and found that the answer
depended on modelling volatility's asymmetric response to news. Their
GARCH variant adds one term for negative shocks,

.. math:: \sigma_t^2 = \omega + \bigl(\alpha + \gamma\,\mathbf 1\{\varepsilon_{t-1} < 0\}\bigr)\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2,

keeping GARCH's quadratic shape while letting bad news count
:math:`(\alpha + \gamma)/\alpha` times as much. On U.S. monthly excess
returns they found :math:`\alpha` near zero: almost all the reaction came
from negative shocks.

.. GENERATED FROM PYTHON SOURCE LINES 19-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import GARCH, GJRGARCH
    from financialkit.time_series.visualizers import plot_news_impact

    true = GJRGARCH(omega=0.02, alpha=0.01, gamma=0.14, beta=0.91, mu=0.03)








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Fitting GJR-GARCH
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 30-42

.. code-block:: Python


    r = true.simulate(3000, rng=1993).returns
    gjr = GJRGARCH.fit(r)
    garch = GARCH.fit(r)
    for k in ("alpha", "gamma", "beta"):
        print(f"{k:5s} true {true.params[k]:.3f}  fitted {gjr.params[k]:.3f} +/- {gjr.std_errors[k]:.3f}")
    print(f"persistence alpha + gamma/2 + beta = {gjr.model.persistence:.3f}")
    print(f"likelihood-ratio statistic vs GARCH: {2 * (gjr.loglik - garch.loglik):.1f} (chi^2_1 5% value 3.84)")

    ax = plot_news_impact({"fitted GJR": gjr.model, "fitted GARCH": garch.model}, var_prev=true.unconditional_variance)
    ax.set_title("GJR: a kinked parabola")




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_05_gjr_garch_001.png
   :alt: GJR: a kinked parabola
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_05_gjr_garch_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    alpha true 0.010  fitted 0.016 +/- 0.009
    gamma true 0.140  fitted 0.132 +/- 0.017
    beta  true 0.910  fitted 0.909 +/- 0.010
    persistence alpha + gamma/2 + beta = 0.991
    likelihood-ratio statistic vs GARCH: 68.4 (chi^2_1 5% value 3.84)

    Text(0.5, 1.0, 'GJR: a kinked parabola')



.. GENERATED FROM PYTHON SOURCE LINES 43-47

Volatility after up and down days
---------------------------------
Average fitted variance the day after a move of more than one standard
deviation, by the sign of the move.

.. GENERATED FROM PYTHON SOURCE LINES 47-55

.. code-block:: Python


    z = (r - gjr.model.mean) / np.sqrt(gjr.variance)
    nxt = gjr.variance[1:] / gjr.variance[:-1]
    fig, ax = plt.subplots()
    ax.bar(["after a fall", "after a rise"], [nxt[z[:-1] < -1].mean(), nxt[z[:-1] > 1].mean()], color=["C3", "C2"])
    ax.axhline(1, color="k", lw=0.8)
    ax.set_ylabel(r"mean $\sigma_{t+1}^2 / \sigma_t^2$")
    plt.show()



.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_05_gjr_garch_002.png
   :alt: plot 05 gjr garch
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_05_gjr_garch_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_time_series_volatility_plot_05_gjr_garch.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/volatility/plot_05_gjr_garch.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_gjr_garch.ipynb <plot_05_gjr_garch.ipynb>`

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

      :download:`Download Python source code: plot_05_gjr_garch.py <plot_05_gjr_garch.py>`

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

      :download:`Download zipped: plot_05_gjr_garch.zip <plot_05_gjr_garch.zip>`


.. only:: html

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

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