LongMemoryEstimate#

class cultivars.diagnostics.long_memory.LongMemoryEstimate(*, statistic, pvalue, d, se, null_d, method, bandwidth, nobs)[source]

Bases: _HypothesisTest

A semiparametric estimate of the fractional differencing parameter, with a test.

The record is an estimate first, d and se, and a test second. Each of the three estimators is asymptotically normal in the bandwidth \(m\),

\[\sqrt{m}\,(\hat d - d) \;\xrightarrow{d}\; \mathcal{N}\!\left(0,\; \tfrac{1}{4}\right) \quad\text{(local and exact local Whittle)}, \qquad \mathcal{N}\!\left(0,\; \tfrac{\pi^2}{24}\right) \quad\text{(GPH)},\]

so statistic is \((\hat d - d_0) / \widehat{se}\) and pvalue its two-sided standard normal p-value, with \(d_0 = 0\), no long memory, as built; test() re-aims it at any other null, \(1/2\) for the stationarity boundary or \(1\) for a unit root, without re-estimating. Standard errors are asymptotic in the bandwidth and do not see the short-run dynamics the estimator ignores, so the reader should expect them to be mildly optimistic: on 500 observations the nominal 95% interval covers about 90 to 94% of the time, dipping to the low 80s for exact local Whittle near \(d = 1/2\), where its mean correction switches regime.

Variables:

  • d – The estimate.

  • se – Its asymptotic standard error.

  • statistic – (d - null_d) / se.

  • pvalue – Two-sided standard normal p-value.

  • null_d – The value of d under test.

  • method – "gph", "local_whittle" or "exact_local_whittle".

  • bandwidth – Fourier frequencies used, m.

  • nobs – Observations.

Note

The three canonical nulls sit in the summary table together, so print(est) answers “no long memory”, “stationary”, and “unit root” at once. Because statistic and pvalue are always against null_d, reject() on a freshly built record is the test of \(d = 0\); call est.test(1.0).reject() for the unit-root question. The local Whittle estimator’s limit theory holds only for \(d < 3/4\) and its optimizer is bounded above at one, so an estimate at the bound is a signal to switch to exact local Whittle, which the table says.

See also

References

Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4), 221-238.

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

Shimotsu, K., & Phillips, P. C. B. (2005). Exact local Whittle estimation of fractional integration. Annals of Statistics, 33(4), 1890-1933.

Example

A random walk is \(d = 1\): the estimate rejects no long memory and keeps a unit root, and the interval covers one:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> est = exact_local_whittle(np.cumsum(rng.standard_normal(500)))
>>> est.method, round(est.d, 4), round(est.se, 4), est.bandwidth
('exact_local_whittle', 1.059, 0.0668, 56)
>>> est.reject(), est.test(1.0).reject(), est.is_stationary
(True, False, False)
>>> low, high = est.confidence_interval()
>>> bool(low < 1.0 < high)
True

Private methods#

_record(d, se, *, method, m, nobs)

Assemble the record with the d = 0 test.

_summary_table()

Render as a table with the three canonical nulls.
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