long_run_variance#

cultivars.diagnostics.unit_roots.long_run_variance(endog, *, kernel='bartlett', bandwidth=None)[source]#

Kernel estimate of a series’ long-run variance about its mean.

The long-run variance \(\lambda^2 = \sum_{j = -\infty}^{\infty} \gamma_j = 2\pi f(0)\) is the variance of the normalized partial sum in the limit, and the quantity every kernel-corrected test in this module divides by. The estimator is

\[\hat\lambda^2 = \hat\gamma_0 + 2 \sum_{j = 1}^{T - 1} k\!\left(\frac{j}{b}\right) \hat\gamma_j,\]

with \(k\) the Bartlett or quadratic-spectral kernel and \(b\) the bandwidth. The Bartlett kernel is the Newey-West estimator, always positive and with bandwidth \(\lfloor 4 (T / 100)^{2/9} \rfloor\) by default; the quadratic spectral kernel is Andrews’ (1991) optimal choice among positive definite kernels, with his AR(1) plug-in bandwidth by default, and converges faster at the cost of weight on distant lags.

Parameters:

  • endog (ArrayLike) – The series, (T,).

  • kernel (str) – "bartlett" or "quadratic-spectral".

  • bandwidth (float | None) – Kernel bandwidth; None for the automatic rule of the chosen kernel.

Return type:

float

Returns:

The estimate, a positive float.

Raises:

See also

References

Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703-708.

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-858.

Example

White noise has long-run variance one; an AR(1) with root 0.9 has \(1 / (1 - 0.9)^2 = 100\), which the kernel estimate approaches from below because its bandwidth is short relative to the persistence:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> e = rng.standard_normal(2000)
>>> bool(0.8 < long_run_variance(e) < 1.2)
True
>>> y = np.zeros(2000)
>>> for t in range(1, 2000):
...     y[t] = 0.9 * y[t - 1] + rng.standard_normal()
>>> bool(long_run_variance(y) < long_run_variance(y, bandwidth=50) < 100)
True
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