adf#

cultivars.diagnostics.unit_roots.adf(endog, *, trend='c', lags=None, max_lags=None, method='aic')[source]#

Augmented Dickey-Fuller test of a unit root.

The regression

\[\Delta y_t = d_t + \gamma\, y_{t-1} + \sum_{j = 1}^{p} \phi_j\, \Delta y_{t-j} + e_t\]

tests \(\gamma = 0\) by the \(t\) ratio on \(y_{t-1}\), read against the Dickey-Fuller law rather than the normal. The augmentation \(p\) absorbs serial correlation in the differences, so that under an ARMA short run the limit still holds (Said & Dickey 1984). When lags is not given it is chosen on the common sample that the largest candidate leaves, so the criteria compare like with like; the test itself is then run on the longest sample the chosen augmentation allows. The p-value is MacKinnon’s (1996) response surface, and the critical values its finite-sample ones at the effective sample size.

Parameters:

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

  • trend (str) – "n", "c" or "ct"; the law depends on it.

  • lags (int | None) – Augmentation lags, at least 0, or None to select.

  • max_lags (int | None) – Largest augmentation under selection; default Schwert’s \(\lfloor 12 (T / 100)^{1/4} \rfloor\).

  • method (str) – "aic", "bic", "t-stat" (general-to-specific at 10%) or "maic" (Ng-Perron modified AIC).

Return type:

UnitRootTest

Returns:

The UnitRootTest named "ADF" with MacKinnon’s response-surface p-value and finite-sample critical values; lags is the augmentation and method how it was chosen.

Raises:

Note

BIC under-augments against moving-average errors and the test then over-rejects, which is why AIC is the default; the modified AIC is built for the neighbourhood of the null and pays for that with long augmentations on a plainly stationary series. The trend must include every deterministic term the alternative would need: a trending series tested with trend="c" has no power.

See also

  • kpss() – the complementary test with stationarity under the null.

  • phillips_perron() – the same law with a nonparametric correction in place of the augmentation.

  • dfgls() – the more powerful GLS-detrended version.

  • zivot_andrews() – when the alternative allows a break.

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.

Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71(3), 599-607.

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

Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics, 7(2), 147-159.

Example

A random walk keeps the null; a stationary AR(1) with a root of 0.9 rejects it at 300 observations:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> walk = np.cumsum(rng.standard_normal(300))
>>> test = adf(walk)
>>> test.reject(), round(test.pvalue, 3), test.trend, test.lags
(False, 0.841, 'c', 0)
>>> y = np.zeros(300)
>>> for t in range(1, 300):
...     y[t] = 0.9 * y[t - 1] + rng.standard_normal()
>>> test = adf(y)
>>> test.reject(), test.method
(True, 'aic (max 15)')
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