.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/factors/plot_01_fama_macbeth.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_factors_plot_01_fama_macbeth.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_factors_plot_01_fama_macbeth.py:


Fama-MacBeth regressions
========================

Eugene Fama and James MacBeth (1973) estimated risk premia with a
cross-sectional regression each month,

.. math::

   r^e_{it} = \gamma_{0t} + \gamma_{1t}\,\beta_i + \gamma_{2t}\,\beta_i^2 + \gamma_{3t}\,s_i + u_{it},

and averaged the slopes over time, taking standard errors from the
month-to-month variation of the slopes, which automatically accounts
for the correlation of returns across stocks. If the CAPM holds, the
average :math:`\gamma_1` is the market premium, while :math:`\beta^2`
and residual risk :math:`s_i` earn nothing.

.. GENERATED FROM PYTHON SOURCE LINES 20-44

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import fama_macbeth, simulate_factor_model, time_series_regression
    from financialkit.asset_pricing.visualizers import plot_fama_macbeth

    rng = np.random.default_rng(1973)
    N, T, premium = 100, 600, 0.006
    betas = np.linspace(0.4, 1.8, N)
    idio = rng.uniform(0.03, 0.10, N)
    panel = simulate_factor_model(T, betas, [premium], [[0.0020]], idio_vol=idio, rng=rng)

    # Betas from first-pass time-series regressions, then monthly cross-sections.
    fm = fama_macbeth(panel.excess_returns, factors=panel.factors)
    # For a traded factor the slope estimates the factor's realised mean
    # return, which itself differs from the 0.6% premium by sampling error.
    realised = panel.factors.mean()
    print(f"market premium: {fm.premia[1]:.5f} (se {fm.std_errors[1]:.5f}); realised factor mean {realised:.5f}")

    fig, ax = plt.subplots(figsize=(8, 3.5))
    plot_fama_macbeth(fm, index=1, ax=ax, true_value=realised)
    ax.set_title("monthly slope on beta, against the factor's realised mean")
    fig.tight_layout()




.. image-sg:: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_01_fama_macbeth_001.png
   :alt: monthly slope on beta, against the factor's realised mean
   :srcset: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_01_fama_macbeth_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    market premium: 0.00236 (se 0.00191); realised factor mean 0.00326




.. GENERATED FROM PYTHON SOURCE LINES 45-51

Fama and MacBeth's test of the CAPM
-----------------------------------

Adding beta squared and residual risk as regressors: neither should be
priced. (Beta and its square are nearly collinear, so the individual
slopes are imprecise.)

.. GENERATED FROM PYTHON SOURCE LINES 51-59

.. code-block:: Python


    ts = time_series_regression(panel.excess_returns, panel.factors)
    b = ts.betas[:, 0]
    s = ts.residuals.std(axis=0)
    fm3 = fama_macbeth(panel.excess_returns, betas=np.column_stack([b, b**2, s]))
    for name, g, t in zip(["gamma_0", "beta", "beta^2", "resid. sd"], fm3.premia, fm3.t_values, strict=True):
        print(f"{name:>9}: {g:+.5f}  (t = {t:+.2f})")





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

 .. code-block:: none

      gamma_0: +0.00355  (t = +1.61)
         beta: -0.00266  (t = -0.61)
       beta^2: +0.00225  (t = +1.25)
    resid. sd: -0.00154  (t = -0.10)




.. GENERATED FROM PYTHON SOURCE LINES 60-62

Average returns against beta
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 62-73

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(b, panel.excess_returns.mean(axis=0), "o", ms=4)
    bb = np.linspace(b.min(), b.max(), 2)
    ax.plot(bb, fm.premia[0] + fm.premia[1] * bb, "C3", label="Fama-MacBeth line")
    ax.legend()
    ax.set_xlabel("estimated beta")
    ax.set_ylabel("average monthly excess return")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_01_fama_macbeth_002.png
   :alt: plot 01 fama macbeth
   :srcset: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_01_fama_macbeth_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.086 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_factors_plot_01_fama_macbeth.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/factors/plot_01_fama_macbeth.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_fama_macbeth.ipynb <plot_01_fama_macbeth.ipynb>`

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

      :download:`Download Python source code: plot_01_fama_macbeth.py <plot_01_fama_macbeth.py>`

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

      :download:`Download zipped: plot_01_fama_macbeth.zip <plot_01_fama_macbeth.zip>`


.. only:: html

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

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