.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/stylized_facts/plot_02_hill_tail_index.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_stylized_facts_plot_02_hill_tail_index.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_stylized_facts_plot_02_hill_tail_index.py:


Hill tail index
===============

If the tail is a power law, :math:`P(X > u) \approx C u^{-\alpha}`, the
log-excesses of the largest observations over a threshold are
exponential with mean :math:`1/\alpha`. Bruce Hill (1975) turned this
into the estimator

.. math:: \hat\alpha_k = \Bigl[\frac1k\sum_{i=1}^{k}\ln X_{(i)} - \ln X_{(k+1)}\Bigr]^{-1},

using only the :math:`k` largest order statistics. The tail index decides
which moments exist (those of order below :math:`\alpha`), and so whether
the variance and kurtosis that risk models rely on are finite.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import EGARCH, hill_estimator
    from financialkit.time_series.visualizers import plot_hill








.. GENERATED FROM PYTHON SOURCE LINES 25-31

The Hill plot
-------------
For an exact Pareto tail the estimate is flat in :math:`k`. For a Student
t with :math:`\nu = 4`, only the far tail is a power law, so the estimate
drifts once :math:`k` reaches into the body: choosing :math:`k` trades
variance (small :math:`k`) against bias (large :math:`k`).

.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python


    rng = np.random.default_rng(1975)
    n = 20_000
    pareto = rng.pareto(3.0, n) + 1.0
    student = np.abs(rng.standard_t(4, n))

    fig, ax = plt.subplots()
    plot_hill(hill_estimator(pareto, k=np.arange(20, 3000)), ax=ax, label=r"Pareto, $\alpha$ = 3")
    plot_hill(hill_estimator(student, k=np.arange(20, 3000)), ax=ax, label=r"|t(4)|, $\alpha$ = 4")
    for a in (3, 4):
        ax.axhline(a, color="k", ls="--", lw=0.8)
    ax.set_ylim(1, 7)




.. image-sg:: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_02_hill_tail_index_001.png
   :alt: plot 02 hill tail index
   :srcset: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_02_hill_tail_index_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    (1.0, 7.0)



.. GENERATED FROM PYTHON SOURCE LINES 45-50

Fat tails from volatility clustering
------------------------------------
Returns with Gaussian innovations but a changing variance also have
power-law tails. The left and right tails of an EGARCH model with a
leverage effect differ: the left one is fatter.

.. GENERATED FROM PYTHON SOURCE LINES 50-60

.. code-block:: Python


    r = EGARCH(omega=0.0, alpha=0.25, gamma=-0.12, beta=0.97).simulate(n, rng=1).returns
    k = np.arange(50, 1500)
    fig, ax = plt.subplots()
    plot_hill(hill_estimator(r, k=k, side="lower"), ax=ax, label="losses (left tail)")
    plot_hill(hill_estimator(r, k=k, side="upper"), ax=ax, label="gains (right tail)")
    ax.set_title("EGARCH returns with normal innovations")
    for side in ("lower", "upper"):
        print(f"{side} tail: alpha at k = 500 is {hill_estimator(r, k=500, side=side).alpha[0]:.2f}")
    plt.show()



.. image-sg:: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_02_hill_tail_index_002.png
   :alt: EGARCH returns with normal innovations
   :srcset: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_02_hill_tail_index_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    lower tail: alpha at k = 500 is 3.50
    upper tail: alpha at k = 500 is 3.68





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

   **Total running time of the script:** (0 minutes 0.118 seconds)


.. _sphx_glr_download_api_gallery_time_series_stylized_facts_plot_02_hill_tail_index.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/stylized_facts/plot_02_hill_tail_index.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_hill_tail_index.ipynb <plot_02_hill_tail_index.ipynb>`

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

      :download:`Download Python source code: plot_02_hill_tail_index.py <plot_02_hill_tail_index.py>`

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

      :download:`Download zipped: plot_02_hill_tail_index.zip <plot_02_hill_tail_index.zip>`


.. only:: html

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

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