Phillips-Ouliaris Residual-Based Cointegration Test
Also known as: Phillips-Ouliaris Cointegration Test, PO Residual-Based Test, Residual-Based Cointegration Test, Phillips-Ouliaris Eşbütünleşme Testi
The Phillips-Ouliaris test, introduced by Phillips and Ouliaris in their 1990 Econometrica article, is a residual-based nonparametric procedure for testing the null hypothesis of no cointegration among a set of integrated I(1) time series. It corrects OLS residuals from a cointegrating regression for serial correlation and endogeneity using kernel-based long-run variance estimators, yielding two statistics—Z_alpha (variance-ratio) and Z_t (normalized coefficient)—whose asymptotic distributions are tabulated specifically for systems with multiple stochastic regressors.
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When to use it
Apply the Phillips-Ouliaris test when you have two or more I(1) time series and wish to determine whether they share a common stochastic trend. The test is appropriate when residuals are expected to display serial correlation or when regressors may be endogenous—situations that invalidate the simpler Engle-Granger Dickey-Fuller approach. Key assumptions include: all variables are I(1), the number of cointegrating vectors is at most one (single-equation setting), and a sufficiently long sample for kernel estimation. When multiple cointegrating vectors are suspected, prefer the Johansen trace or maximum-eigenvalue test. The test is not designed for I(0) or I(2) variables.
Strengths & limitations
- Nonparametric correction for serial correlation and endogeneity without requiring explicit lag-length selection in a parametric model.
- Provides two complementary statistics (Z_alpha and Z_t) whose asymptotic distributions are free of nuisance parameters.
- Critical values are tabulated for systems with up to multiple stochastic regressors, accommodating multivariate cointegrating regressions.
- Robust to a wide class of weakly dependent and heterogeneously distributed error processes under the maintained assumptions.
- Assumes at most a single cointegrating vector; cannot identify or test for multiple cointegrating relationships.
- Size and power depend on the choice of kernel and bandwidth, and poor bandwidth selection can distort inference in finite samples.
- Severe finite-sample size distortions can arise when the signal-to-noise ratio is low or when near-unit-root regressors are present.
- The null of no cointegration means failure to reject does not confirm a unit root in residuals; low power against near-stationary alternatives limits conclusiveness.
Frequently asked
How does the Phillips-Ouliaris test differ from the Engle-Granger test?
Both are residual-based single-equation tests, but the Engle-Granger approach applies a standard ADF test to the OLS residuals, relying on parametric lag augmentation to handle serial correlation. The Phillips-Ouliaris test uses a nonparametric kernel correction instead, making it more robust to unknown error dynamics and avoiding the need to select an augmentation lag order, though it introduces bandwidth choice as an alternative tuning decision.
Which statistic should I report—Z_alpha or Z_t?
Both statistics test the same null hypothesis and are asymptotically equivalent under correct specification; reporting both is common practice. Monte Carlo evidence suggests Z_t often has better finite-sample size properties, so many practitioners give it priority. However, large discrepancies between the two statistics may indicate model misspecification or bandwidth sensitivity and warrant investigation.
Can the Phillips-Ouliaris test handle more than two variables?
Yes. The test is formulated for a single dependent variable regressed on p stochastic regressors, and Phillips and Ouliaris (1990) tabulate critical values for p ranging from 1 to 6. The asymptotic null distribution shifts with p, so using the correct column in the critical-value table is essential. For detecting multiple cointegrating vectors among several I(1) series, the Johansen procedure is the appropriate alternative.
Sources
- Phillips, P. C. B., & Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica, 58(1), 165–193. DOI: 10.2307/2938339 ↗
How to cite this page
ScholarGate. (2026, June 2). Phillips-Ouliaris Residual-Based Cointegration Test. ScholarGate. https://scholargate.app/en/econometrics/phillips-ouliaris-test
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