UnitRootTest#

class cultivars.diagnostics.unit_roots.UnitRootTest(*, statistic, pvalue, name, critical_values, null, nobs, lower_tail=False, companions=(), trend, lags, method, break_index=None)[source]#

Verdict of a unit-root or stationarity test.

The family is not one hypothesis but two. The Dickey-Fuller line, ADF, Phillips-Perron, DF-GLS, Ng-Perron, Zivot-Andrews, puts the unit root under the null and rejects in the lower tail: under \(\rho = 1\) in \(\Delta y_t = (\rho - 1) y_{t-1} + \cdots\) the \(t\) ratio converges not to a normal but to the Dickey-Fuller functional

\[\frac{\int_0^1 W(r)\, dW(r)}{\bigl(\int_0^1 W(r)^2\, dr\bigr)^{1/2}},\]

whose quantiles depend on the deterministic terms and sit well to the left of the normal’s. KPSS puts stationarity under the null and rejects in the upper tail, its statistic a scaled integral of a squared Brownian bridge. The record carries which, so that reject() reads correctly for both and a table of several tests can be read side by side: a series the ADF cannot reject a unit root for and the KPSS cannot reject stationarity for is one the sample does not decide, and that is a finding.

P-values are exact where the literature supplies a response surface (MacKinnon for the Dickey-Fuller law) or a table dense enough to interpolate (KPSS, whose p-value is then clipped to the table’s range); the tests known only through asymptotic critical values at three levels report pvalue=None and reject() reads the table, refusing a level it does not carry.

Variables:

  • trend – Deterministic specification the test was run under.

  • lags – Augmentation lags or kernel bandwidth, as the test uses.

  • method – How lags was chosen or the kernel used.

  • break_index – For a break-allowing test, the first observation of the new regime; None otherwise.

Note

lags means two different things across the family, and method disambiguates: an augmentation count for ADF, DF-GLS, Ng-Perron, and Zivot-Andrews, where method names the selection rule and its ceiling; a kernel bandwidth for Phillips-Perron and KPSS, where method names the kernel. Ng-Perron’s four statistics travel as companions of the \(MZ_t\) record, since they are one test on one sample.

See also

References

Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366), 427-431.

MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11(6), 601-618.

Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1-3), 159-178.

Example

A random walk keeps the ADF null and rejects the KPSS null; white noise does the reverse. The two records reject in opposite tails and agree on both series:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> walk = np.cumsum(rng.standard_normal(300))
>>> noise = rng.standard_normal(300)
>>> adf(walk).reject(), kpss(walk).reject()
(False, True)
>>> adf(noise).reject(), kpss(noise).reject()
(True, False)
>>> test = adf(walk)
>>> test.lower_tail, test.trend, test.method, test.break_index
(True, 'c', 'aic (max 15)', None)

A test known only through its table has no p-value and reads the level it is asked for from the table:

>>> test = dfgls(walk)
>>> test.pvalue, test.reject(alpha=0.05)
(None, False)

Attributes#

trend

"n", "c", or "ct", as the producer accepted it.

lags

Augmentation lags of the regression, or the kernel bandwidth, whichever the test uses.

method

How lags was chosen, or which kernel was used.

break_index

First observation of the new regime for a break-allowing test; None otherwise.

companions

critical_values

lower_tail

name

nobs

null

pvalue

statistic

Methods#

reject(*[, alpha])

Whether the null is rejected at level alpha.

summary()

The test as a table, renderable as text, HTML, or a dataframe.