.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/efficiency/plot_02_fama_runs_test.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_efficiency_plot_02_fama_runs_test.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_efficiency_plot_02_fama_runs_test.py:


Fama's runs test
================

Correlation coefficients can be dominated by a few large changes. In his
1965 thesis Eugene Fama added a distribution-free check on the 30 Dow
Jones stocks: count the *runs*, the unbroken sequences of price changes
with the same sign. Positive dependence produces long runs and too few of
them; reversals produce too many. Under independence the number of runs
is approximately normal with a known mean and variance, and Fama found the
observed counts close to the expected ones.

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.signal import lfilter

    from financialkit.time_series import runs_test








.. GENERATED FROM PYTHON SOURCE LINES 22-26

Runs in three series
--------------------
Independent changes, positively autocorrelated changes (trends), and
negatively autocorrelated changes (reversals, such as bid-ask bounce).

.. GENERATED FROM PYTHON SOURCE LINES 26-44

.. code-block:: Python


    rng = np.random.default_rng(1965)
    e = rng.standard_normal(1000)
    cases = {
        "independent": e,
        "trending (phi = +0.3)": lfilter([1.0], [1.0, -0.3], e),
        "reverting (phi = -0.3)": lfilter([1.0], [1.0, 0.3], e),
    }

    fig, axes = plt.subplots(3, 1, figsize=(8, 5), sharex=True)
    for ax, (name, x) in zip(axes, cases.items(), strict=True):
        res = runs_test(x)
        ax.bar(np.arange(60), np.sign(x[:60]), color=np.where(x[:60] > 0, "C2", "C3"), width=1.0)
        ax.set_yticks([])
        ax.set_title(f"{name}: z = {res.statistic:+.2f}, p = {res.pvalue:.3f}", fontsize=9)
    axes[-1].set_xlabel("day (first 60 signs shown)")
    fig.tight_layout()




.. image-sg:: /api/gallery/time_series/efficiency/images/sphx_glr_plot_02_fama_runs_test_001.png
   :alt: independent: z = +0.54, p = 0.589, trending (phi = +0.3): z = -6.96, p = 0.000, reverting (phi = -0.3): z = +5.48, p = 0.000
   :srcset: /api/gallery/time_series/efficiency/images/sphx_glr_plot_02_fama_runs_test_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-49

The test's size
---------------
Under independence the statistic is close to standard normal, so the test
rejects about 5% of the time.

.. GENERATED FROM PYTHON SOURCE LINES 49-59

.. code-block:: Python


    z = np.array([runs_test(rng.standard_normal(250)).statistic for _ in range(1000)])
    fig, ax = plt.subplots()
    ax.hist(z, bins=30, density=True, color="0.7", label="1000 independent samples")
    grid = np.linspace(-4, 4, 200)
    ax.plot(grid, np.exp(-(grid**2) / 2) / np.sqrt(2 * np.pi), "k", label="N(0, 1)")
    ax.set_xlabel("runs statistic")
    ax.legend()
    print(f"rejection rate at 5%: {np.mean(np.abs(z) > 1.96):.3f}")
    plt.show()



.. image-sg:: /api/gallery/time_series/efficiency/images/sphx_glr_plot_02_fama_runs_test_002.png
   :alt: plot 02 fama runs test
   :srcset: /api/gallery/time_series/efficiency/images/sphx_glr_plot_02_fama_runs_test_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    rejection rate at 5%: 0.051





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

   **Total running time of the script:** (0 minutes 0.125 seconds)


.. _sphx_glr_download_api_gallery_time_series_efficiency_plot_02_fama_runs_test.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/efficiency/plot_02_fama_runs_test.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_fama_runs_test.ipynb <plot_02_fama_runs_test.ipynb>`

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

      :download:`Download Python source code: plot_02_fama_runs_test.py <plot_02_fama_runs_test.py>`

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

      :download:`Download zipped: plot_02_fama_runs_test.zip <plot_02_fama_runs_test.zip>`


.. only:: html

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

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