Unit Root Tests

Definition

Unit root tests evaluate whether a time series has an autoregressive root equal to one — i.e., whether $y_t \sim I(1)$ with permanent shocks rather than $y_t \sim I(0)$ with transitory shocks. Under the unit root null, standard ordinary least squares (OLS) estimators of the AR coefficient converge to non-standard functionals of Brownian motion rather than Gaussian limits, requiring simulation-based critical values. The tests are foundational for applied time series work because I(1) versus I(0) classification governs whether cointegration analysis applies, whether differencing is required, and how forecast uncertainty accumulates with horizon.

Key Ideas

Under the I(1) null, $T(\hat\alpha-1)$ and the OLS $t$ -statistic $t_\alpha$ converge to Dickey-Fuller (DF) distributions — negatively skewed with fat left tails — rather than Normal or Student- $t$ . Critical values from Fuller (1976) depend on the deterministic specification (no constant, constant only, constant plus trend).
Augmented Dickey-Fuller (ADF) extends DF to serially correlated errors by augmenting with $k$ lagged differences; Said-Dickey (1984) proved consistency of the augmented regression even for autoregressive moving average (ARMA) errors of unknown order if $k \to \infty$ at rate $o(T^{1/3})$ .
Phillips-Perron (PP) tests correct for serial correlation non-parametrically via spectral density estimates. The correction factor $(s^2-s^2_u)$ is subtracted from $T(\hat\alpha-1)$ in $Z_\alpha$ and from $t_\alpha$ in $Z_t$ . The quality of the spectral density estimate determines the quality of the size correction.
Spectral density estimator choice is critical. Kernel-based estimators ( $s^2_{WA}$ ) built on OLS residuals inherit the inconsistency of $\hat\alpha$ when the error has a root near the unit circle — this aggravates rather than repairs PP size distortions. The autoregressive estimator ( $s^2_{AR}$ ) formulated on first differences of $y_t$ is immune to this inconsistency and enables well-sized tests.
M statistics add a modification factor $(T/2)(\hat\alpha-1)^2$ to $Z_\alpha$ , which offsets the explosive growth of $Z_\alpha$ in near-cancellation scenarios while being asymptotically negligible in standard settings.

How It Works

Dickey-Fuller and ADF Tests

The baseline model is $y_t = \alpha y_{t-1} + u_t$ (or with deterministic terms). White (1958) showed that under $\alpha=1$ with i.i.d. errors, $T(\hat\alpha-1)$ has the non-standard limiting distribution:

$T(\hat\alpha-1) \Rightarrow \left(\int_0^1 W(r)\,dW(r)\right)\left(\int_0^1 W(r)^2\,dr\right)^{-1}$

where $W(r)$ is a standard Brownian motion on $[0,1]$ . The augmented DF regression

$\Delta y_t = b_0 y_{t-1} + \sum_{j=1}^k b_j \Delta y_{t-j} + e_{tk}$

extends this to ARMA errors of unknown order (Said-Dickey 1984). The ADF $t$ -statistic $t_\rho$ for $b_0=0$ has the same Dickey-Fuller limiting distribution. Critical values with a constant and no trend are $-2.86$ (5%) and $-3.43$ (1%) for $T=\infty$ .

Phillips-Perron Tests

Phillips (1987) and Phillips-Perron (1988) proposed two non-parametric corrections for serially correlated errors:

$Z_\alpha = T(\hat\alpha-1) - (s^2-s^2_u)\left(2T^{-2}\sum_{t=1}^T y^2_{t-1}\right)^{-1}$

$Z_t = \frac{s_u}{s}\,t_\alpha - \frac{1}{2}(s^2-s^2_u)\left(s^2 T^{-2}\sum_{t=1}^T y^2_{t-1}\right)^{-1/2}$

where $s^2_u = T^{-1}\sum \hat u_t^2$ (innovation variance) and $s^2$ is any consistent estimator of $\sigma^2 = 2\pi f_u(0)$ (the non-normalized spectral density at frequency zero). Under standard asymptotics, $Z_\alpha$ and $Z_t$ have the same Dickey-Fuller limiting distributions as $T(\hat\alpha-1)$ and $t_\alpha$ respectively. The practical problem is size distortion: when $s^2$ is estimated inconsistently (e.g., by a kernel estimator when $\hat\alpha$ is inconsistent), the correction factor explodes and the statistics diverge from their asymptotic distributions.

M Statistics (Stock 1990; Perron-Ng 1996)

Stock (1990) proposed a class of statistics with different normalization rates under $H_0$ and $H_1$ :

$MZ_\alpha = \left(T^{-1}y_T^2 - s^2\right)\left(2T^{-2}\sum_{t=1}^T y^2_{t-1}\right)^{-1}$

$MSB = \left(T^{-2}\sum_{t=1}^T y^2_{t-1}/s^2\right)^{1/2}, \qquad MZ_t = MSB \cdot MZ_\alpha$

The key identity is $MZ_\alpha = Z_\alpha + (T/2)(\hat\alpha-1)^2$ : the modification factor $(T/2)(\hat\alpha-1)^2$ is $O_p(1)$ in the near-cancellation cases where $T(\hat\alpha-1) \to -\infty$ , providing an offsetting term that keeps $MZ_\alpha$ bounded. Under standard asymptotics, $(\hat\alpha-1)^2 = O_p(T^{-2})$ and the modification is $O_p(T^{-1}) \to 0$ , so $MZ_\alpha$ and $Z_\alpha$ are asymptotically equivalent. The relation $Z_t = MSB \cdot Z_\alpha$ (eq. 2.9 of Perron-Ng) gives the corresponding $MZ_t = MSB \cdot MZ_\alpha$ by analogy.

$MSB$ is bounded below by zero and tests the unit root hypothesis by checking whether $T^{-2}\sum y^2_{t-1}$ is of order $O_p(T^2)$ (consistent with I(1)) or $O_p(T)$ (consistent with I(0)). Critical values (5%): $MZ_\alpha \leq -8.1$ , $MSB \leq 0.168$ (constant only); $MZ_\alpha \leq -17.3$ , $MSB \leq 0.148$ (constant and trend).

Autoregressive Spectral Density Estimator

The AR estimator $s^2_{AR}$ is constructed from the augmented autoregression on first differences:

$\Delta y_t = b_0 y_{t-1} + \sum_{j=1}^k b_j \Delta y_{t-j} + e_{tk}, \qquad s^2_{AR} = s^2_{ek}/(1-\hat b(1))^2$

where $s^2_{ek} = T^{-1}\sum_{t=k+1}^T \hat e^2_{tk}$ and $\hat b(1) = \sum_{j=1}^k \hat b_j$ . Because this regression is formulated in first differences of $y_t$ rather than on OLS residuals $\hat u_t$ , the estimate of $\sigma^2$ does not depend on $\hat\alpha$ at first order. In the three local asymptotic frameworks analyzed by Perron-Ng (1996), $s^2_{AR}$ converges to the correct limit $\sigma^2_T$ in each case, while $s^2_{WA}$ does not.

Local-to-Unity Framework

Setting $y_t = (1+c/T)y_{t-1} + u_t$ , with $c=0$ the exact unit root null and $c<0$ the stationary alternative, yields tractable non-degenerate limits for all statistics as $T\to\infty$ . The non-centrality $c$ measures distance from the unit root boundary in units of $T^{-1}$ . This framework (Nabeya-Perron 1994; Perron-Ng 1996) diagnoses test behavior in the three regimes where standard asymptotics fail: near-MA-cancellation, near-twice-integration, and near-seasonal-integration.

Why It Matters

Unit root classification determines model specification: I(1) series require differencing or error-correction; I(0) series can be modeled in levels. The choice directly affects whether long-run cointegrating relationships are estimable and whether spurious regression bias applies.
Size distortions in PP tests (rejection rates up to 0.98 at nominal 5% for MA(1) errors with $\theta=-0.8$ , $T=200$ ) mean that applied work relying on PP can generate false conclusions about stationarity — particularly for macroeconomic series (inflation, interest rates) that frequently exhibit negative MA components.
The M statistics with $s^2_{AR}$ are recommended replacements: free of the size problems that have plagued the PP and ADF tests in problematic error specifications, while retaining comparable power in well-behaved settings. See Perron-Ng (1996).
Cointegration tests based on residuals (Engle-Granger) share the unit root testing problem: the normalizing variable choice affects the test statistic in the same way it affects the PP tests. Perron-Ng (1995 working paper) showed M tests resolve this normalization sensitivity.

Open Questions

Unit roots and near-unit-root stationary processes are observationally equivalent in the sense of Campbell-Perron (1991): for any stationary AR process, there exists a unit root process that approximates it arbitrarily well over any finite sample. The range of MA coefficients $\theta \in (-1, x)$ for some $x<0$ over which size distortions occur shrinks to zero as $T \to \infty$ , but the shrinkage rate is slow.
For forecasting purposes, Campbell-Perron (1991) argue that deliberate misclassification of trend-stationary as difference-stationary can reduce mean squared error (MSE), since near-integrated stationary processes are better forecast by integrated models. The M tests' conservative bias on the stationary side may then be counterproductive.
Data-dependent lag selection rules for $k$ in $s^2_{AR}$ and optimal treatment of deterministic terms remain open; Perron and Ng (1996, Section 7) note both are under investigation. Ng-Perron (2001) subsequently developed modified information criteria and generalized least squares (GLS)-detrending to address these issues.
Standard DF/PP/M tests are distorted by unmodelled structural breaks: a trend-stationary series with a single break looks integrated to the test (Perron 1989). Break-robust tests (Perron 1989 known-break, Zivot-Andrews 1992 unknown-break) address this but require specifying the break model and, for unknown-break tests, tolerate reduced power. See Structural Break Testing.
Fat tails and Bayesian unit root inference. Kleibergen-Van Dijk (1993) show that the choice of error distribution dominates the choice of variance specification in determining Bayesian posterior odds for a unit root. On US T-bill data, switching from normal to Student-t errors reverses the conclusion (Bayes factor $K_{01}$ goes from 0.04 to 2.50 in favor of the unit root). The mechanism is that fat-tailed likelihoods downweight outliers, releasing the AR coefficient from the constraint imposed by the largest observations. Frequentist unit root tests face an analogous sensitivity: tests calibrated under normality have distorted size when true errors are heavy-tailed.