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Test whether a series has a unit root

Running ADF alone is the classic mistake: failing to reject a unit root is not evidence of one, because ADF has low power near the boundary. The fix is the confirmatory pair — ADF (null: unit root) and KPSS (null: stationary) — read together. tsecon.check_stationarity runs both, classifies the result into one of four quadrants, and tells you what to do next.

The recipe

import numpy as np, textwrap, tsecon

rng = np.random.default_rng(7)
walk = np.cumsum(rng.standard_normal(300))          # a genuine unit root

r = tsecon.check_stationarity(walk)
print(f"{r['quadrant']}  ->  {r['recommendation']}")
print(f"ADF   stat {r['adf_statistic']:+.3f}   p {r['adf_p_value']:.3f}")
print(f"KPSS  stat {r['kpss_statistic']:+.3f}   p {r['kpss_p_value']:.3f}")
print(textwrap.fill(r["interpretation"], 72))

UnitRoot  ->  Difference
ADF   stat -1.691   p 0.436
KPSS  stat +2.625   p 0.010
At the 5% level ADF cannot reject a unit root and KPSS rejects
stationarity — the tests agree the series looks I(1). Difference it once
and re-run this battery on the differences before modeling; regressing
I(1) levels on each other risks spurious regression unless you are
explicitly testing for cointegration.

Reading the quadrant

quadrant is the whole point. The two tests have opposite nulls, so their four combinations mean four different things:

Quadrant ADF KPSS What it means
Stationary rejects does not reject Both agree on I(0). Proceed in levels.
UnitRoot does not reject rejects Both agree on I(1). Difference.
Conflict rejects rejects Often a deterministic trend or a structural break, not a root.
Inconclusive neither rejects neither rejects The sample cannot tell. Say so.

Only the two agreeing quadrants are conclusions. Conflict and Inconclusive are the honest outcomes the single-test workflow hides from you, and the interpretation string names the follow-up in each case.

Then confirm the differences are what you thought:

d = tsecon.check_stationarity(np.diff(walk))
print(f"first differences:  {d['quadrant']}  ->  {d['recommendation']}")

first differences:  Stationary  ->  Proceed

When to reach for Phillips-Perron

tsecon.phillips_perron tests the same null as ADF but gets there differently: it runs the Dickey-Fuller regression with no lagged differences and then corrects the statistic with a Bartlett kernel estimate of the residual long-run variance. Same nonstandard null distribution, so the same MacKinnon p-values apply.

pp = tsecon.phillips_perron(walk, regression="c", test_type="tau")
print(f"PP Z-tau {pp['stat']:+.3f}   p {pp['pvalue']:.3f}   bandwidth {pp['lags']}")
print("5% critical value:", round(pp["crit"]["5%"], 3))

PP Z-tau -1.671   p 0.446   bandwidth 16
5% critical value: -2.871

Use it when you would rather not choose an augmentation lag length, when the errors are heteroskedastic in an unknown way, or as a robustness cross-check — agreement between ADF and PP is reassuring. Prefer ADF when you suspect a large negative MA component, where PP's size distortion is worst. And quote lags: a PP result without its bandwidth is not reproducible.

Gotchas

  • Match the deterministics to your alternative. regression="c" (default) tests against a stationary-around-a-constant alternative; use "ct" when the series trends, or a genuine trend will masquerade as a root.
  • A break looks like a root. A stationary series whose mean jumps mid-sample fools ADF (Perron 1989). If the quadrant comes back UnitRoot or Conflict on a series you suspect has a break, run the break scan before differencing.
  • KPSS p-values are interpolated and clamped to [0.01, 0.10]. A reported p = 0.01 means "at most 0.01", and p = 0.10 means "at least 0.10" — they are table endpoints, not measurements. Setting alpha outside that range makes the quadrant verdict meaningless even though the call succeeds.
  • Testing several series at once? tsecon.check_series on a 2-D array gives per-series verdicts plus a Johansen cointegration test.

See also