.. 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_04_nelson_egarch.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_04_nelson_egarch.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_04_nelson_egarch.py:


Nelson's EGARCH
===============

GARCH reacts to the size of yesterday's shock but not its sign, and its
coefficients must be kept positive. Daniel Nelson (1991) modelled the
logarithm of the variance instead,

.. math::

   \ln\sigma_t^2 = \omega + \beta\ln\sigma_{t-1}^2
   + \alpha\bigl(|z_{t-1}| - E|z|\bigr) + \gamma z_{t-1},

which is positive by construction and asymmetric when :math:`\gamma \ne 0`.
For stock returns :math:`\gamma < 0`: a fall raises volatility more than a
rise of the same size, Black's (1976) "leverage effect".

.. GENERATED FROM PYTHON SOURCE LINES 20-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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








.. GENERATED FROM PYTHON SOURCE LINES 27-32

The news impact curve
---------------------
Tomorrow's variance as a function of today's standardized shock, starting
from the same variance. GARCH's curve is a symmetric parabola; EGARCH's is
exponential and tilted towards bad news.

.. GENERATED FROM PYTHON SOURCE LINES 32-38

.. code-block:: Python


    egarch = EGARCH(omega=0.0, alpha=0.15, gamma=-0.08, beta=0.97)
    garch = GARCH(omega=0.03, alpha=0.08, beta=0.89)
    ax = plot_news_impact({"GARCH(1,1)": garch, "EGARCH(1,1)": egarch}, var_prev=1.0)
    ax.set_title("News impact: bad news raises EGARCH volatility more")




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_04_nelson_egarch_001.png
   :alt: News impact: bad news raises EGARCH volatility more
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_04_nelson_egarch_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'News impact: bad news raises EGARCH volatility more')



.. GENERATED FROM PYTHON SOURCE LINES 39-44

Estimating the asymmetry
------------------------
Fitting both models to EGARCH returns: the EGARCH fit recovers a
significantly negative :math:`\gamma`, and its log-likelihood beats
GARCH's with one extra parameter.

.. GENERATED FROM PYTHON SOURCE LINES 44-59

.. code-block:: Python


    r = egarch.simulate(3000, rng=1).returns
    eg = EGARCH.fit(r)
    ga = GARCH.fit(r)
    for k in ("alpha", "gamma", "beta"):
        print(f"{k:5s} true {egarch.params[k]:+.3f}  fitted {eg.params[k]:+.3f} +/- {eg.std_errors[k]:.3f}")
    print(f"log-likelihood: EGARCH {eg.loglik:.1f}, GARCH {ga.loglik:.1f}; AIC {eg.aic:.1f} vs {ga.aic:.1f}")

    fig, ax = plt.subplots(figsize=(8, 3.5))
    ax.plot(np.sqrt(eg.variance[:600]), label="EGARCH")
    ax.plot(np.sqrt(ga.variance[:600]), label="GARCH", alpha=0.8)
    ax.set_xlabel("day")
    ax.set_ylabel(r"fitted $\sigma_t$")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_04_nelson_egarch_002.png
   :alt: plot 04 nelson egarch
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_04_nelson_egarch_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    alpha true +0.150  fitted +0.145 +/- 0.020
    gamma true -0.080  fitted -0.083 +/- 0.011
    beta  true +0.970  fitted +0.971 +/- 0.006
    log-likelihood: EGARCH -4350.1, GARCH -4383.8; AIC 8710.2 vs 8775.5





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

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_api_gallery_time_series_volatility_plot_04_nelson_egarch.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_04_nelson_egarch.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_nelson_egarch.ipynb <plot_04_nelson_egarch.ipynb>`

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

      :download:`Download Python source code: plot_04_nelson_egarch.py <plot_04_nelson_egarch.py>`

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

      :download:`Download zipped: plot_04_nelson_egarch.zip <plot_04_nelson_egarch.zip>`


.. only:: html

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

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