bds#

cultivars.diagnostics.nonlinearity.bds(endog, *, dimension=3, epsilon=None)[source]#

The BDS test of independence, usually on residuals.

The correlation integral of the \(m\)-histories at radius \(\varepsilon\),

\[C_m(\varepsilon) = \binom{T_m}{2}^{-1} \sum_{s < t} \mathbb{1}\bigl\{\|y^m_s - y^m_t\|_\infty < \varepsilon\bigr\},\]

is compared with the \(m\)-th power of the one-point integral, which is what independence implies; the standardized gap is asymptotically standard normal, and the test is run at every dimension from 2 to dimension with the same radius. The radius defaults to 1.5 standard deviations of the series, the choice with the best size in the Brock et al. tables, and the reference is trustworthy from about 500 observations. On GARCH residuals with 500 observations the test rejects about 90% of the time, which is its most common use: what the ARCH test says with a form, BDS says without one.

Parameters:

  • endog (ArrayLike) – The series, typically standardized residuals, at least 10 * dimension + 50 observations.

  • dimension (int) – Largest embedding dimension \(m\), at least 2; 2 and every dimension up to it are reported.

  • epsilon (float | None) – Radius in the units of the series; None takes 1.5 standard deviations.

Return type:

BDSTest

Returns:

The BDSTest at dimension, lower dimensions as companions.

Raises:

  • SpecificationError – If the dimension is below 2, the radius is not positive, or the series is too short.

  • NumericalError – If a value is not finite, or the radius leaves no pairs close or every pair close.

Note

Applied to residuals from an estimated model the statistic’s limit is unchanged for linear models, by Brock’s theorem, but not for GARCH-standardized residuals, where it is mildly conservative; the 90% power figure is on the raw GARCH series. The cost is \(O(T^2)\) in memory and time through the pairwise distances.

See also

References

Brock, W. A., Dechert, W. D., Scheinkman, J. A., & LeBaron, B. (1996). A test for independence based on the correlation dimension. Econometric Reviews, 15(3), 197-235.

Brock, W. A., Hsieh, D. A., & LeBaron, B. (1991). Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence. MIT Press.

Example

Gaussian noise keeps independence; a radius far below the series’ scale leaves no close pairs and is refused:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> y = rng.standard_normal(600)
>>> bool(bds(y).pvalue > 0.01)
True
>>> bds(y, epsilon=1e-9)
Traceback (most recent call last):
    ...
cultivars.exceptions.NumericalError: a radius of 1e-09 leaves the correlation integral ...
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