local_whittle#

cultivars.diagnostics.long_memory.local_whittle(endog, *, f_bandwidth=None, exponent=_WHITTLE_EXPONENT)[source]#

Robinson’s (1995) local Whittle estimate of \(d\).

The Gaussian semiparametric estimator minimizes the concentrated local Whittle objective over the first \(m\) Fourier frequencies,

\[R(d) = \log\Bigl(\frac{1}{m} \sum_{j = 1}^{m} \lambda_j^{2d} I(\lambda_j)\Bigr) - \frac{2d}{m} \sum_{j = 1}^{m} \log \lambda_j,\]

which is the Whittle likelihood of the local model \(f(\lambda) = c\,\lambda^{-2d}\) with \(c\) concentrated out. Its limit is \(\sqrt{m}(\hat d - d) \to \mathcal{N}(0, 1/4)\) for \(-1/2 < d < 3/4\), more efficient than GPH at the same bandwidth by a factor of \(\pi^2 / 6\) in variance; it stays consistent up to \(d < 1\) but loses its limit law beyond \(3/4\), and the record says so when the estimate lands there. The search runs over \((-1/2, 1)\), so a unit-root series is reported at the upper boundary with a pointer to exact_local_whittle(). The default bandwidth \(m = \lfloor T^{0.65} \rfloor\) is wider than GPH’s because the efficiency gain makes the variance side of the trade cheaper.

Parameters:

  • endog (ArrayLike) – The series, (T,), at least 64 observations.

  • f_bandwidth (int | None) – Fourier frequencies used, at least 2 and at most T // 2; None applies the rule.

  • exponent (float) – Exponent of the rule m = floor(T ** exponent), in (0, 1); 0.65 by default.

Return type:

LongMemoryEstimate

Returns:

The LongMemoryEstimate, with method="local_whittle", standard error \(1 / (2\sqrt{m})\), and the test against \(d = 0\).

Raises:

  • SpecificationError – If the series is shorter than 64, the bandwidth is below 2 or above the available frequencies, or the exponent is outside (0, 1).

  • NumericalError – If a value is not finite.

Note

An estimate at 1.0 to display precision is the search bound, not a finding: the objective is still decreasing there and the series is at least a unit root. Switch to exact_local_whittle(), whose search extends to two, for the value.

See also

  • exact_local_whittle() – the same objective on the fractionally differenced series, valid for any \(d\).

  • gph() – the log-periodogram regression, less efficient and simpler.

References

Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Annals of Statistics, 23(5), 1630-1661.

Künsch, H. R. (1987). Statistical aspects of self-similar processes. In Proceedings of the First World Congress of the Bernoulli Society (Vol. 1, pp. 67-74). VNU Science Press.

Phillips, P. C. B., & Shimotsu, K. (2004). Local Whittle estimation in nonstationary and unit root cases. Annals of Statistics, 32(2), 656-692.

Example

White noise is \(d = 0\); the default bandwidth is wider than GPH’s and the standard error correspondingly smaller:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> est = local_whittle(rng.standard_normal(1000))
>>> bool(abs(est.d) < 3 * est.se), est.bandwidth, round(float(est.se), 3)
(True, 89, 0.053)

A random walk lands on the search bound:

>>> round(local_whittle(np.cumsum(rng.standard_normal(500))).d, 3)
1.0
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