Model card — DF-GLS, Ng-Perron, Phillips-Perron, Phillips-Ouliaris, and Zivot-Andrews tests

dfgls · ng_perron · phillips_perron · phillips_ouliaris · zivot_andrews

Five unit-root-family tests beyond the core ADF/KPSS pair. dfgls attacks the ADF's power problem: estimating a constant or trend by OLS costs the plain ADF real power near the unit-root boundary, and GLS-detrending at the ERS local alternative recovers most of it — it is the recommended default over plain ADF whenever deterministics must be estimated. ng_perron builds on the same GLS-detrending engine and attacks the size problem too: the M statistics with MAIC lag selection are the standard remedy when a large negative MA root (an over-differenced series) wrecks the size of every ADF-family test. phillips_perron and phillips_ouliaris are semiparametric: instead of adding lagged differences to soak up serial correlation (the augmentation in ADF and Engle-Granger), they estimate a simple regression by OLS and then correct the test statistic for the residual's long-run variance with a nonparametric (Bartlett) kernel. phillips_perron is the unit-root test — a drop-in alternative to adf; phillips_ouliaris is its cointegration analog — a residual-based alternative to johansen for a single cointegrating relationship. zivot_andrews guards the workflow against a one-time structural break masquerading as a unit root. All four are companions to the confirmatory stationarity workflow, not replacements for reading ADF and KPSS together.

Function	Null hypothesis	The analog it complements
dfgls	the series has a unit root	adf (GLS-detrended, near-optimal local power)
ng_perron	the series has a unit root	dfgls (the M statistics + MAIC, size-robust under a negative MA root)
phillips_perron	the series has a unit root	adf (semiparametric, no lag augmentation)
phillips_ouliaris	the regressors are not cointegrated with y	Engle-Granger; johansen (single-equation route)
zivot_andrews	a unit root with no break	adf when a one-time structural break may masquerade as a root

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dfgls — GLS-detrended Dickey-Fuller (Elliott-Rothenberg-Stock)

What it estimates. The DF-GLS statistic of Elliott, Rothenberg & Stock (1996): the deterministics are first estimated by GLS under the local alternative \(\rho = 1 + \bar c/T\) — quasi-differencing \(y\) and the trend columns at \(\bar c = -7\) (regression="c") or \(\bar c = -13.5\) ("ct"), the points where the local asymptotic power envelope is tangent at 50% power — and the ADF regression is then run on the detrended series with no deterministic terms. The statistic is the usual \(t\)-ratio on the lagged level; only the detrending changed, but that change buys near-optimal local power where the plain ADF wastes it re-estimating the mean or trend.

Assumptions. Same as ADF: the only nonstationarity under the null is a unit root, and the augmentation lags must absorb the serial correlation. There is no "n" case — with no deterministics to estimate, GLS detrending is a no-op and plain adf(regression="n") already sits on the power envelope. The test inherits ADF's size distortion under a large negative MA root (the Ng-Perron MAIC lag rule is the standard remedy — reach for ng_perron in that case).

When to use (and when not). Use as your default unit-root test whenever a constant or trend must be estimated — which is nearly always — and especially when the series is suspected to be near-integrated (ρ close to 1), where the ADF's power loss bites hardest. Do not use it to dodge the deterministics decision: choosing "c" vs "ct" matters exactly as much as it does for ADF. Do not read a non-rejection alone as proof of a unit root — pair it with kpss for the confirmatory quadrant, as with every unit-root test.

Key arguments and defaults (and why). regression: "c" (constant; default) or "ct" (constant + trend). lags: a fixed number of augmentation lags; None (default) selects automatically. method: "aic" (default), "bic", or "t-stat"; following Perron & Qu (2007) — and arch — the selection runs on the OLS-detrended series with no deterministics, which improves finite-sample power over selecting on the GLS-detrended series. max_lags caps the search; None uses Schwert's \(\lceil 12 (T/100)^{1/4}\rceil\) capped at \((T-1)/2 - 1\). When lags is given, method/max_lags are ignored (arch behavior).

How to read the output. statistic, p_value, used_lag, nobs (\(= T - 1 - \texttt{used\_lag}\)), crit (1/5/10% critical values at nobs), trend. Small p_value ⇒ reject the unit root. The critical values are not the ADF ones — the "ct" case in particular has its own distribution — so compare the statistic only against the crit values reported here. Quote used_lag: a DF-GLS verdict without its lag length is not reproducible.

Failure modes. Reading it alone (pair with KPSS); leaving "c" when the alternative is trend-stationary (the test then diverges toward non-rejection on trending data — see the trend-stationary fixture case, p ≈ 0.99 under "c" vs p ≈ 0 under "ct"); heavy negative MA errors (size distortion; AIC tends to pick too few lags there).

Validated against. arch.unitroot.DFGLS (arch 8.0.0, an independent implementation) for the statistic, the AIC/BIC/t-stat selected lag, nobs, p-value, and critical values on the Nile series, three seeded random walks, a trend-stationary series, i.i.d. noise, and fixed-lag/max-lags cases — statistic at 1e-10 relative, lags exact (dfgls.json, dfgls_golden.rs). Honest grading of the p-value/CV layer: they reproduce the response surfaces shipped with arch (Sheppard's MacKinnon-methodology simulations, transcribed with attribution) bit-for-bit; they are not an independently published table. See the validation matrix.

References. Elliott, Rothenberg & Stock (1996); Ng & Perron (2001); Perron & Qu (2007); MacKinnon (1994, 2010, the response-surface methodology).

import numpy as np, tsecon

rng = np.random.default_rng(0)
walk = np.cumsum(rng.standard_normal(200))          # a random walk (unit root)

r = tsecon.dfgls(walk, regression="c")
print("DF-GLS(walk):", round(r["statistic"], 4), " p:", round(r["p_value"], 4),
      " lag:", r["used_lag"], " 5% cv:", round(r["crit"]["5%"], 3))

ar = np.empty(200); ar[0] = 0.0
for t in range(1, 200):
    ar[t] = 0.85 * ar[t - 1] + rng.standard_normal()  # near-integrated AR(1)
print("DF-GLS(AR 0.85):", round(tsecon.dfgls(ar)["statistic"], 4),
      " p:", round(tsecon.dfgls(ar)["p_value"], 4))

DF-GLS(walk): -1.1486  p: 0.2358  lag: 0  5% cv: -2.047
DF-GLS(AR 0.85): -4.7458  p: 0.0

The random walk cannot reject; the stationary-but-persistent AR(0.85) rejects decisively — the near-integrated regime is exactly where DF-GLS's power edge over plain ADF shows. These match arch.unitroot.DFGLS to machine precision.

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ng_perron — the M unit-root tests with MAIC (Ng & Perron 2001)

What it estimates. The four M statistics of Ng & Perron (2001) — MZa, MZt, MSB, MPT — on the GLS-detrended series, with the lag length chosen by their MAIC and the long-run variance from the autoregressive spectral density estimator at frequency zero. The detrending is bit-for-bit the dfgls engine (same quasi-differencing, same \(\bar c = -7\) / \(-13.5\)). With \(\tilde y_t\) the detrended series, \(\kappa = T^{-2}\sum_{t=1}^{T-1}\tilde y_t^2\) and \(s^2_{AR} = \hat\sigma_e^2 / (1-\hat b(1))^2\) from the trendless ADF autoregression:

\[MZ_\alpha = \frac{T^{-1}\tilde y_T^2 - s^2_{AR}}{2\kappa},\qquad MSB = \sqrt{\kappa / s^2_{AR}},\qquad MZ_t = MZ_\alpha \times MSB,\]

and MPT is the modified ERS point-optimal statistic. All four reject the unit-root null when small (below the critical value); \(MZ_t = MZ_\alpha \times MSB\) is an exact identity, enforced as an internal invariant test.

Assumptions. As for DF-GLS. The M statistics were built precisely for the case that breaks the rest of the family: a large negative MA root in the differences (the classic outcome of over-differencing), where ADF/DF-GLS with AIC/BIC lag selection reject a true unit root far too often. MAIC's data-dependent penalty lengthens the lag there, and the M statistics keep size close to nominal where the Phillips-Perron kernel correction fails completely.

When to use (and when not). Use whenever you suspect over-differencing or an MA component in the shocks, or as the size-robust confirmation of a dfgls rejection. Do not expect p-values: none exist (no published response surface; this library declines to fabricate one), so read the verdict off the reported critical values. And know the documented flip side (Perron & Qu 2007): on data far from the null — an obviously stationary series — MAIC drives the lag to its maximum and power collapses (the crate pins a case where lag-0 MZa is −124.9 and the MAIC lag-16 MZa fails to reject). If the series is plainly stationary, a unit-root test is the wrong tool; if you must, cap max_lags or fix lags.

Key arguments and defaults (and why). trend: "c" (constant; default) or "ct" (constant + trend) — the same choice, with the same stakes, as dfgls. lags: None or "maic" (default) selects by MAIC on the GLS-detrended series (the paper's own rule — not the Perron-Qu OLS-detrended variant dfgls uses for AIC/BIC); an integer fixes the lag. max_lags caps the MAIC search; None uses Schwert's \(\lceil 12(T/100)^{1/4}\rceil\) capped at \((T-1)/2-1\) (the library-wide default; the paper's simulations truncate rather than round up — pass an explicit cap to reproduce a specific study).

How to read the output. mza, mzt, msb, mpt, used_lag, nobs (\(= T - 1 - \texttt{used\_lag}\)), s2_ar, crit (per-statistic 1/5/10% values from Ng-Perron 2001, Table 1), trend. Reject the unit root when a statistic is below its critical value — for all four, including MSB and MPT (positive statistics that shrink under the alternative). Quote used_lag; an M-test verdict without its lag is not reproducible. The statistics agree in spirit but are four tests, not one: MZa/MZt near their DF-GLS analogs, MSB the Sargan-Bhargava ratio, MPT the point-optimal form.

Failure modes. Reading a non-rejection alone as proof of a unit root (pair with kpss, as always); the MAIC power reversal on far-from-null data (above); treating the asymptotic critical values as finite-sample exact in very short samples (the table is asymptotic-only — the measured size at \(T=250\) runs 0.027–0.045 at the 5% values, slightly conservative, matching the paper's Table 2); comparing MZt to ADF critical values (its distribution matches DF-GLS, not ADF, and the other three have their own tables).

Validated against. No runnable independent implementation exists — statsmodels 0.14.6 and arch 8.0.0 do not ship the M tests (a canary test asserts this stays true), and no CRAN package does (urca::ur.ers stops at DF-GLS + the ERS P-test; bootUR/CADFtest borrow only MAIC) — so there is no reference-run golden, and the claim is carried honestly by: the transcribed Table 1 (cross-checked against the independent transcription in the GAUSS tspdlib); seeded Monte-Carlo size at the asymptotic critical values — lag-0 at \(T=1000\), 2000 reps: 5% rejection measured 0.039–0.058 and 1% measured 0.006–0.010 across all eight statistic/trend combinations; the full MAIC pipeline at \(T=250\): 0.027–0.045 at 5%; power ordering (AR(0.8) rejects ~0.80 vs size ~0.03); the MAIC mechanism (mean lag 0.9 i.i.d. vs 7.8 under MA(−0.8)); the exact \(MZ_t = MZ_\alpha MSB\) identity; a bitwise pin that the detrended series is the dfgls engine's; and an independent NumPy re-implementation of the whole pipeline re-pinned through the Python binding at 1e-9. See the validation matrix.

References. Ng & Perron (2001); Perron & Ng (1996, the M statistics); Stock (1999); Elliott, Rothenberg & Stock (1996, the detrending); Perron & Qu (2007, the power-reversal caveat).

import numpy as np, tsecon

rng = np.random.default_rng(0)
walk = np.cumsum(rng.standard_normal(200))          # a random walk (unit root)

r = tsecon.ng_perron(walk, trend="c")
print("MZa:", round(r["mza"], 4), " MZt:", round(r["mzt"], 4),
      " MSB:", round(r["msb"], 4), " MPT:", round(r["mpt"], 4),
      " lag:", r["used_lag"])
print("5% cvs:", r["crit"]["mza"]["5%"], r["crit"]["mzt"]["5%"],
      r["crit"]["msb"]["5%"], r["crit"]["mpt"]["5%"])

ar = np.empty(200); ar[0] = 0.0
for t in range(1, 200):
    ar[t] = 0.85 * ar[t - 1] + rng.standard_normal()  # near-integrated AR(1)
r2 = tsecon.ng_perron(ar)
print("AR(0.85) MZa:", round(r2["mza"], 4), " MZt:", round(r2["mzt"], 4),
      " lag:", r2["used_lag"])

MZa: -2.6666  MZt: -1.1431  MSB: 0.4287  MPT: 9.144  lag: 0
5% cvs: -8.1 -1.98 0.233 3.17
AR(0.85) MZa: -27.3873  MZt: -3.7002  lag: 3

The random walk sits well inside the acceptance region on all four statistics; the near-integrated AR(0.85) rejects decisively (MZa −27.4 against a 1% value of −13.8). Same detrended series as dfgls, but with the size insurance the M statistics were designed to buy.

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phillips_perron — semiparametric unit-root test

What it estimates. The Phillips-Perron (1988) \(Z_\tau\) (default) or \(Z_\alpha\) statistic for a unit root. It runs the Dickey-Fuller level regression \(y_t = \mu + \delta t + \rho\, y_{t-1} + u_t\) by OLS with no lagged differences, then corrects the raw \(t\)-statistic (or the \(T(\hat\rho-1)\) statistic) for serial correlation using a Bartlett kernel estimate of the residual long-run variance. Same nonstandard Dickey-Fuller null distribution as ADF, so the MacKinnon (1996, 2010) response-surface p-values apply.

Assumptions. The only nonstationarity is a unit root (a deterministic trend must be modeled through regression="ct"). The nonparametric correction handles serial correlation and heteroskedasticity of unknown form — its strength — but the test is known to have size distortions when the series has a large negative MA root (a shared weakness with ADF), and low power near the unit-root boundary (why you still pair it with KPSS).

When to use (and when not). Use as an ADF alternative when you would rather not choose an augmentation lag length, or as a robustness cross-check on an ADF verdict — agreement between the two is reassuring. Do not treat a failure to reject as evidence of a unit root (low power); do not read it alone — run check_stationarity or pair it with kpss for the confirmatory quadrant. Prefer ADF when a strong negative MA component is suspected.

Key arguments and defaults (and why). regression: "n" (no deterministics), "c" (constant; default), "ct" (constant + trend) — the same "match the deterministics to the stationary alternative" choice that dominates ADF. test_type: "tau" (the \(Z_\tau\) \(t\)-form; default) or "rho" (the \(Z_\alpha\) coefficient form). lags: the Bartlett bandwidth; None uses the \(\lceil 12\,(n/100)^{1/4}\rceil\) rule (arch's default).

How to read the output. stat (the requested statistic), pvalue (MacKinnon), crit (the 1/5/10% critical values), lags (the bandwidth used), nobs, plus both ztau and zalpha for convenience. Small pvalue ⇒ reject the unit root (the series looks stationary). Quote the bandwidth: a PP result without its lags is not reproducible.

Failure modes. Reading PP alone (same trap as ADF alone); size distortion under a large negative MA root; mistaking a deterministic trend for a root by leaving regression="c" when "ct" is called for.

Validated against. arch.unitroot.PhillipsPerron (Sheppard) for both \(Z_\tau\) and \(Z_\alpha\) — an independent package — to < 1e-10, with MacKinnon response-surface p-values (phillips.json, phillips_golden.rs). See the validation matrix.

References. Phillips & Perron (1988); MacKinnon (1996, 2010); Newey & West (1987, the long-run-variance kernel).

import numpy as np, tsecon

rng = np.random.default_rng(0)
walk = np.cumsum(rng.standard_normal(300))          # a random walk (unit root)
stat = rng.standard_normal(300)                     # i.i.d. (stationary)

pp = tsecon.phillips_perron(walk, regression="c", test_type="tau")
print("PP(walk)  Z-tau:", round(pp["stat"], 4), " p:", round(pp["pvalue"], 4),
      " bandwidth:", pp["lags"])
print("  5% critical value:", round(pp["crit"]["5%"], 3))
print("PP(stationary) Z-tau:", round(tsecon.phillips_perron(stat)["stat"], 4),
      " p:", round(tsecon.phillips_perron(stat)["pvalue"], 4))

PP(walk)  Z-tau: -0.7675  p: 0.8285  bandwidth: 16
  5% critical value: -2.871
PP(stationary) Z-tau: -18.7697  p: 0.0

The random walk cannot reject the unit root (p ≈ 0.83, statistic well above the −2.87 critical value); the i.i.d. series rejects overwhelmingly. These match arch.unitroot.PhillipsPerron to machine precision.

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phillips_ouliaris — residual cointegration test

What it estimates. The Phillips-Ouliaris (1990) \(Z_t\) (default) or \(Z_\alpha\) residual test for cointegration. It regresses y on the stochastic regressors x (plus the chosen deterministics) by OLS, then applies the Phillips-Perron correction to a unit-root test on the regression residual. Under the null of no cointegration that residual has a unit root; a cointegrating relationship makes it stationary, so a large negative statistic rejects "no cointegration".

Assumptions. The variables in [y, x] are each I(1); a single cointegrating vector is the alternative (this is a single-equation test — for the number of cointegrating relations use johansen). The null distribution depends on the number of regressors, so the critical values are indexed by \(N = 1 + \dim(x)\).

When to use (and when not). Use for a quick, single-equation cointegration check when one series is a natural dependent variable (a spread, an arbitrage relation) — the Engle-Granger workflow, with the semiparametric correction. Do not add your own constant column to x (deterministics come from trend); do not use it to count cointegrating vectors (that is Johansen's job); remember the test is not invariant to which variable you place on the left.

Key arguments and defaults (and why). x is a 2-D (T, m) matrix of the m stochastic regressors, used as-is. trend: "n", "c" (default), "ct" — the deterministics in the cointegrating regression. test_type: "Zt" (default) or "Za". bandwidth: the Bartlett bandwidth of the residual AR(1); None uses the \(\lfloor 4((T-1)/100)^{2/9}\rfloor\) rule.

How to read the output. stat, pvalue, crit, lags (bandwidth), nobs, n_vars (\(N = 1 + m\)). Small pvalue ⇒ reject no cointegration (the series move together in the long run). Zt p-values and critical values use the MacKinnon N-surfaces (the statsmodels coint route); Za is statistic-only (pvalue/crit are None) because the library deliberately declines to ship arch's proprietary \(Z_\alpha\) simulation surface.

Failure modes. Adding a redundant constant column to x (double-counts the deterministic); reading it as a rank test; swapping the dependent variable and getting a different verdict (a known non-invariance of single-equation tests).

Validated against. arch.unitroot.cointegration.phillips_ouliaris for the statistics — an independent package — with Zt p-values/critical values from the statsmodels MacKinnon cointegration N-surfaces (phillips.json, phillips_golden.rs).

References. Phillips & Ouliaris (1990); Engle & Granger (1987); MacKinnon (1996, 2010).

import numpy as np, tsecon

rng = np.random.default_rng(0)
T = 300
x = np.cumsum(rng.standard_normal(T))               # an I(1) regressor
y = 1.5 * x + rng.standard_normal(T)                # cointegrated with x
Xreg = x.reshape(-1, 1)                             # (T, 1) — no constant column

po = tsecon.phillips_ouliaris(y, Xreg, trend="c", test_type="Zt")
print("PO(cointegrated) Zt:", round(po["stat"], 4), " p:", round(po["pvalue"], 4),
      " N:", po["n_vars"])

y2 = np.cumsum(rng.standard_normal(T))              # an independent random walk
po2 = tsecon.phillips_ouliaris(y2, Xreg, trend="c", test_type="Zt")
print("PO(independent)  Zt:", round(po2["stat"], 4), " p:", round(po2["pvalue"], 4))

PO(cointegrated) Zt: -19.4407  p: 0.0  N: 2
PO(independent)  Zt: -2.9978  p: 0.1107

The genuinely cointegrated pair rejects "no cointegration" decisively; two independent random walks do not (p ≈ 0.11) — the spurious-regression trap the test exists to catch. Because it is single-equation, use johansen when you need to know how many cointegrating relations a larger system supports.

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zivot_andrews — unit root against break-stationarity

What it estimates. The Zivot-Andrews (1992) minimum-t unit-root statistic with one endogenous structural break. Perron (1989) showed that a stationary series with a one-time level or trend shift fools the ADF test into "finding" a unit root; Zivot-Andrews turns Perron's known-break-date test into an estimated one: for every candidate break date inside a trimmed window it runs the ADF-style regression with the chosen break dummies and takes the minimum t-statistic on the lagged level — the break date least favorable to the unit-root null. Because the date is estimated, the null distribution shifts far left of the ADF one and has its own critical values (a simulated table, not MacKinnon surfaces).

Assumptions. At most one break, and only under the alternative: the null is a no-break unit root. That asymmetry is the test's classic weakness — a unit root with a genuine break (a broken drift) is outside both hypotheses and produces spurious rejections (Lee & Strazicich 2003 fix this with a minimum-LM test that allows the break under both; not yet shipped, flagged here for honesty). Lag selection follows the statsmodels/Baum convention: a single up-front ADF ("ct", no dummies) autolag pass fixes the augmentation lag for all candidate regressions — slightly more pessimistic than the paper's per-candidate re-selection.

When to use (and when not). Use when ADF/PP fail to reject but the plot shows a one-time event (a policy regime change, a dam, a reunification) — if zivot_andrews rejects where adf did not, the "unit root" was likely a broken deterministic. Do not use it to date breaks in a series you already believe is stationary (that is bai_perron's job, with proper break confidence intervals); do not read the estimated break date as inference — it is a by-product of the min-t search. With two or more suspected breaks, no shipped test applies (Lumsdaine-Papell / Lee-Strazicich territory).

Key arguments and defaults (and why). regression: which component breaks under the alternative — "c" intercept shift (default, Perron's "crash" model), "t" trend-slope shift, "ct" both; the regression itself always carries a constant and a trend. trim (default 0.15, in (0, 1/3] — 0 itself is unreachable, since the candidate window must hold at least lags + 1 observations): excludes the first and last int(n*trim) observations from the break search — a break too near an end is indistinguishable from the boundary. autolag/max_lags/lags: the statsmodels lag conventions — autolag="aic" (default; also "bic", "t-stat") with max_lags capping the search, or autolag=None with lags fixed. Pass one of lags/autolag, not both — the binding refuses the ambiguous spelling.

How to read the output. stat, pvalue, crit (1/5/10%), break_index, lags, nobs, trim, regression. Small pvalue ⇒ reject the no-break unit root in favor of break-stationarity. break_index is the last pre-break observation — the estimated shift begins at break_index + 1 (the statsmodels bpidx convention). The p-value interpolates a simulated table (100,000 replications): read at most two decimals into it, and treat values at the clamps (1e-05, 0.999) as "off the table", not as exact probabilities.

Failure modes. Rejecting because of a break under the null (broken random walk — the Lee-Strazicich critique); reading break_index as a dated, confidence-bounded break (use bai_perron); a trim too small for the lag order (the break dummy needs a pre-break regime — the error explains); treating the "t" model's dating convention as identical to "ct"'s (the reference starts the "t" ramp one observation earlier; tsecon replicates it exactly and documents it in the module).

Validated against. statsmodels.tsa.stattools.zivot_andrews (0.14.6) — statistic to 1e-10, break index and selected lag exact, p-values/critical values to the interpolation's resolution — with arch.unitroot.ZivotAndrews (8.0.0) agreeing on every expressible case as a cross-check (the two share the same Baum code lineage, so arch corroborates the transcription rather than independently deriving it; graded accordingly in the fixture header) (zivot_andrews.json, zivot_andrews_golden.rs).

References. Zivot & Andrews (1992); Perron (1989); Baum (2004, rev. 2015); Schwert (1989); Lee & Strazicich (2003).

import numpy as np, tsecon

rng = np.random.default_rng(7)
y = rng.standard_normal(200)                # stationary noise ...
y[100:] += 10.0                             # ... with a level shift at t = 100

za = tsecon.zivot_andrews(y, regression="c")
print("ZA(break-stationary):", round(za["stat"], 4), " p:", za["pvalue"],
      " break at:", za["break_index"] + 1)
print("ADF on the same series p:", round(tsecon.adf(y)["p_value"], 4))

rw = np.cumsum(rng.standard_normal(200))    # a genuine random walk
za2 = tsecon.zivot_andrews(rw, regression="c")
print("ZA(random walk):", round(za2["stat"], 4), " p:", round(za2["pvalue"], 4))

ZA(break-stationary): -16.3861  p: 1e-05  break at: 100
ADF on the same series p: 0.7496
ZA(random walk): -3.0233  p: 0.9051

The broken-but-stationary series: plain ADF is completely fooled (p ≈ 0.75, "unit root"), while Zivot-Andrews rejects overwhelmingly and recovers the break date exactly (shift begins at observation 100). The genuine random walk is correctly not rejected. That contrast — same data, opposite verdicts — is exactly the Perron (1989) point the test exists to make.