.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/unit_roots/plot_02_engle_granger.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_unit_roots_plot_02_engle_granger.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_unit_roots_plot_02_engle_granger.py:


Engle-Granger cointegration
===========================

Granger and Newbold (1974) showed that regressing one random walk on an
independent one gives "significant" slopes and high :math:`R^2` most of
the time: a spurious regression. Yet some nonstationary series *are*
tied together, such as spot and futures prices, or two share classes of
the same company. Robert Engle and Clive Granger (1987) called them
cointegrated: some combination :math:`y_t - b x_t` is stationary. Their
two-step test regresses :math:`y` on :math:`x` and runs a Dickey-Fuller
test on the residuals, with critical values that allow for :math:`b`
having been fitted. It is the textbook basis of pairs trading.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_series import engle_granger
    from financialkit.time_series.utils import simulate_cointegrated_pair








.. GENERATED FROM PYTHON SOURCE LINES 24-28

A cointegrated pair and a spurious one
--------------------------------------
Two log prices that share a stochastic trend (the spread is an AR(1) with
a half-life of about a week), and two independent random walks.

.. GENERATED FROM PYTHON SOURCE LINES 28-56

.. code-block:: Python


    rng = np.random.default_rng(1987)
    n = 750
    y, x = simulate_cointegrated_pair(n, beta=1.2, intercept=0.5, phi=0.9, sigma_x=0.02, sigma_u=0.01, rng=rng)
    a, b = np.cumsum(0.02 * rng.standard_normal(n)), np.cumsum(0.02 * rng.standard_normal(n))

    coint = engle_granger(y, x)
    spur = engle_granger(a, b)
    for name, res in (("cointegrated", coint), ("spurious", spur)):
        cv = res.critical_values["5%"]
        print(f"{name:13s} b = {res.beta[0]:6.3f}, R^2 = {res.r_squared:.2f}, EG = {res.statistic:6.2f} (5% cv {cv:.2f}), reject: {res.rejects()}")

    fig, axes = plt.subplots(2, 2, figsize=(9, 5.5), sharex=True)
    axes[0, 0].plot(y, label="y")
    axes[0, 0].plot(x, label="x")
    axes[0, 0].set_title("cointegrated pair")
    axes[0, 0].legend()
    axes[1, 0].plot(coint.residuals, color="C2")
    axes[1, 0].set_title("spread $y - a - bx$: stationary")
    axes[0, 1].plot(a)
    axes[0, 1].plot(b)
    axes[0, 1].set_title(f"independent walks, $R^2$ = {spur.r_squared:.2f}")
    axes[1, 1].plot(spur.residuals, color="C3")
    axes[1, 1].set_title("residual: still a random walk")
    for ax in axes[1]:
        ax.set_xlabel("day")
    fig.tight_layout()




.. image-sg:: /api/gallery/time_series/unit_roots/images/sphx_glr_plot_02_engle_granger_001.png
   :alt: cointegrated pair, independent walks, $R^2$ = 0.05, spread $y - a - bx$: stationary, residual: still a random walk
   :srcset: /api/gallery/time_series/unit_roots/images/sphx_glr_plot_02_engle_granger_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    cointegrated  b =  1.195, R^2 = 0.96, EG =  -6.01 (5% cv -3.34), reject: True
    spurious      b =  0.592, R^2 = 0.05, EG =  -2.13 (5% cv -3.34), reject: False




.. GENERATED FROM PYTHON SOURCE LINES 57-61

Spurious regression in numbers
------------------------------
Across 300 pairs of independent walks the slope's naive t-ratio exceeds 2
in most of them, but the Engle-Granger test rejects close to its 5% size.

.. GENERATED FROM PYTHON SOURCE LINES 61-72

.. code-block:: Python


    big_t, eg_reject = [], []
    for _ in range(300):
        a, b = np.cumsum(rng.standard_normal((2, 250)), axis=1)
        X = np.column_stack([np.ones(250), b])
        coef, res, *_ = np.linalg.lstsq(X, a, rcond=None)
        se = np.sqrt(res[0] / 248 * np.linalg.inv(X.T @ X)[1, 1])
        big_t.append(abs(coef[1] / se) > 2)
        eg_reject.append(engle_granger(a, b, lags=0).rejects())
    print(f"|t| > 2 in {np.mean(big_t):.0%} of spurious regressions; Engle-Granger rejects in {np.mean(eg_reject):.0%}")
    plt.show()




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

 .. code-block:: none

    |t| > 2 in 84% of spurious regressions; Engle-Granger rejects in 3%





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

   **Total running time of the script:** (0 minutes 0.146 seconds)


.. _sphx_glr_download_api_gallery_time_series_unit_roots_plot_02_engle_granger.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/unit_roots/plot_02_engle_granger.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_engle_granger.ipynb <plot_02_engle_granger.ipynb>`

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

      :download:`Download Python source code: plot_02_engle_granger.py <plot_02_engle_granger.py>`

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

      :download:`Download zipped: plot_02_engle_granger.zip <plot_02_engle_granger.zip>`


.. only:: html

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

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