.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/capm/plot_04_grs_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_asset_pricing_capm_plot_04_grs_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_asset_pricing_capm_plot_04_grs_test.py:


The Gibbons-Ross-Shanken test
=============================

Testing the CAPM means asking whether the market portfolio is
mean-variance efficient, that is whether the time-series alphas of a set
of test assets are jointly zero. Michael Gibbons, Stephen Ross and Jay
Shanken (1989) gave the exact finite-sample distribution under normal
errors:

.. math::

   \frac{T - N - K}{N}\,\frac{\hat\alpha'\hat\Sigma^{-1}\hat\alpha}{1 + \bar f'\hat\Omega^{-1}\bar f} \sim F_{N,\,T-N-K}.

The statistic measures how far the tangency portfolio of factors and
test assets together beats the factors alone in squared Sharpe ratio.

.. GENERATED FROM PYTHON SOURCE LINES 20-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from financialkit.asset_pricing import grs_test, simulate_factor_model

    rng = np.random.default_rng(1989)
    T, N = 120, 10
    betas = np.linspace(0.6, 1.4, N)

    # Under the null the p-values are uniform: the test has the right size.
    p_null = [grs_test(p.excess_returns, p.factors).p_value for p in (simulate_factor_model(T, betas, [0.006], [[0.002]], rng=rng) for _ in range(400))]
    stat_null = [stats.f.isf(p, N, T - N - 1) for p in p_null]

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    x = np.linspace(0, 4, 200)
    axes[0].hist(stat_null, bins=30, density=True, color="0.75", label="simulated GRS under H0")
    axes[0].plot(x, stats.f.pdf(x, N, T - N - 1), "C3", label=f"F({N}, {T - N - 1})")
    axes[0].set_xlabel("GRS statistic")
    axes[0].legend()
    print(f"rejection rate at 5% under H0: {np.mean(np.array(p_null) < 0.05):.3f}")




.. image-sg:: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_04_grs_test_001.png
   :alt: plot 04 grs test
   :srcset: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_04_grs_test_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    rejection rate at 5% under H0: 0.043




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Power against mispricing
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-61

.. code-block:: Python


    alphas = np.linspace(0, 0.006, 7)
    power = []
    for a in alphas:
        pa = [
            grs_test(p.excess_returns, p.factors).reject()
            for p in (simulate_factor_model(T, betas, [0.006], [[0.002]], alphas=a * np.linspace(-1, 1, N), rng=rng) for _ in range(100))
        ]
        power.append(np.mean(pa))
    axes[1].plot(12 * alphas, power, "o-")
    axes[1].set_xlabel("largest |alpha| per year")
    axes[1].set_ylabel("rejection rate")
    axes[1].set_title(f"T = {T} months, N = {N} assets")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.962 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_capm_plot_04_grs_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/asset_pricing/capm/plot_04_grs_test.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_grs_test.ipynb <plot_04_grs_test.ipynb>`

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

      :download:`Download Python source code: plot_04_grs_test.py <plot_04_grs_test.py>`

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

      :download:`Download zipped: plot_04_grs_test.zip <plot_04_grs_test.zip>`


.. only:: html

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

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