Perron-Ng (1996) Useful Modifications to some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties

Summary

Perron and Ng explain why Phillips-Perron (PP) unit root tests have severe size distortions when the error process has a root near the unit circle, and propose three modified M statistics ( $MZ_\alpha$ , $MSB$ , $MZ_t$ ) that cure the problem when paired with an autoregressive (AR) spectral density estimator. Local asymptotic analysis across three problematic error models — nearly-white-noise/nearly-integrated, nearly-twice-integrated, and nearly-seasonally-integrated — shows that kernel-based spectral density estimators cannot eliminate PP size distortions and in fact aggravate them, while the AR-based estimator keeps the M tests bounded in all three cases.

Key Claims

Phillips-Perron size distortions are estimator-driven. The PP statistics $Z_\alpha$ and $Z_t$ diverge in the nearly-white-noise/nearly-integrated limit ( $\theta_T = -1+\delta/\sqrt{T}$ ) regardless of the spectral density estimator. With kernel-based $s^2_{WA}$ , divergence occurs at rate $MT$ (faster than rate $T$ with $s^2_{AR}$ ). Changing the kernel does not resolve the problem (Kim-Schmidt 1990 confirmed; no kernel can eliminate size distortions).
The modification factor creates bounded M statistics. The relation $MZ_\alpha = Z_\alpha + (T/2)(\hat\alpha-1)^2$ shows the M tests add a term that exactly offsets the explosive growth of $Z_\alpha$ in the nearly-integrated nearly-white-noise case, keeping $MZ_\alpha$ bounded even when $Z_\alpha \to -\infty$ .
The AR spectral density estimator is essential. Formulated on first differences of $y_t$ : $\Delta y_t = b_0 y_{t-1} + \sum_{j=1}^k b_j \Delta y_{t-j} + e_{tk}, \qquad s^2_{AR} = s^2_{ek}/(1-\hat b(1))^2$ This decouples $\sigma^2$ estimation from the (potentially inconsistent) $\hat\alpha$ . Kernel estimators built on ordinary least squares (OLS) residuals $\hat u_t$ inherit $\hat\alpha$ 's inconsistency, causing divergence at rate $MT$ (Theorem 3.3).
Local asymptotic results for three error models:
1. Nearly-white-noise / nearly-integrated ( $u_t = e_t + \theta_T e_{t-1}$ , $\theta_T = -1+\delta/\sqrt{T}$ ): PP diverge to $-\infty$ ; M tests with $s^2_{AR}$ have bounded limiting distributions (Theorems 3.1–3.2).
2. Nearly-twice integrated ( $u_t = (1+\phi/T)u_{t-1}+e_t$ , $\phi<0$ ): $\hat\alpha$ consistent; PP diverge at rate $T$ ; M tests with $s^2_{WA}$ vacuous ( $MZ_\alpha \sim Z_\alpha$ asymptotically); M tests with $s^2_{AR}$ retain non-degenerate limits (Theorems 4.1–4.3).
3. Nearly-seasonally integrated ( $u_t = -(1+\phi/T)u_{t-1}+e_t$ ): like case 1 but AR coefficient approaches $-1$ at rate $T$ not $\sqrt{T}$ , so size distortions smaller; M tests with $s^2_{AR}$ equivalent to known- $\sigma^2$ case (Theorems 5.1–5.2).
Power. $MZ_\alpha$ dominates $MZ_t$ in power (consistent with Phillips-Ouliaris 1990 and Nabeya-Tanaka 1990 on asymptotic power ordering of $Z_\alpha$ vs. $Z_t$ ). M tests match augmented Dickey-Fuller (ADF) power in positive AR case; exceed ADF power in negative MA case (where ADF size is also distorted). MSB power independent of $\delta$ when $\delta \to 0$ .
Empirical application. Real GDP (54:1–93:3): ARMA(1,1) $\approx$ (1.002, 0.25) — far from problematic space; all tests fail to reject unit root. Inflation (monthly PUNEW): ARMA(1,1) $\approx$ (0.98, $-$ 0.73) — large negative MA; kernel $s^2_{WA}/s^2_u \approx 0.99$ –1.91 (large), AR $s^2_{AR}/s^2_u \approx 0.05$ –0.10 (small). $Z_\alpha = -184$ to $-322$ spuriously rejects; M tests ( $MZ_\alpha \approx -14$ to $-8$ , $MSB \approx 0.18$ –0.24) correctly sustain the unit root null.
Implementation note. For trending data, de-trend $y_t$ in $MZ_\alpha$ and $MSB$ (eqs. 2.6, 2.8) but include only a constant in autoregression (2.12): including a trend in (2.12) does not improve asymptotic accuracy and worsens finite-sample size. Standard asymptotic critical values (from Fuller 1976 for $MZ_\alpha$ , $MZ_t$ ; from $(\int_0^1 W(r)^2 dr)^{1/2}$ for $MSB$ ) are adequate.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"Kernel-based spectral density estimators tend to aggravate the size problem in the Phillips-Perron tests and yield no size improvement to the modified statistics."

"When appropriately implemented, the modified statistics have robust properties and are useful tests for a unit root. The statistics will also be useful in cointegration analysis where serial correlation in the noise function is often encountered."

My Take

The central insight — that the AR spectral density estimator, formulated on first differences, decouples $\sigma^2$ estimation from the unreliable $\hat\alpha$ — is elegant and practical. The local asymptotic framework is the right tool to diagnose behavior precisely where standard asymptotics fail. The empirical inflation example is convincing: the PP tests' behavior is exactly as predicted by the theory (larger truncation lag $\to$ faster divergence), while the M tests give stable, sensible answers. The caveat about forecasting (Section 8) is important but often ignored in applied work.