Phillips-Ouliaris Residual-Based Cointegration Tests

Why Residual-Based Tests Need a Cointegration-Specific Reference

If two economic series are individually integrated, a regression between them can look persuasive even when it is spurious. The relevant question is whether some linear combination of the series is stationary. The Engle–Granger approach answers it by estimating a long-run regression and testing its residuals for a unit root. Phillips–Ouliaris tests address the same question, but use Phillips–Perron-type residual statistics with long-run variance corrections rather than adding lagged residual differences as in an augmented Dickey–Fuller test.

The distinction matters for inference. Estimated cointegrating residuals are not an ordinary observed series: they were constructed by fitting a regression involving integrated variables. As a result, neither an ordinary Dickey–Fuller critical value nor an ordinary Phillips–Perron critical value generally gives the correct reference distribution. The test statistic, deterministic specification, and critical values must be matched.

A typical first-stage regression is \(y_t=d_t'\delta+x_t'\beta+u_t\). The deterministic terms \(d_t\) might include a constant, or a constant and a linear trend, depending on the economic model. The basic setting assumes the variables in the long-run relation are \(I(1)\), the relation is stable over the sample, and short-run dependence is sufficiently regular for a long-run covariance estimate to be consistent. A break in the relation or a wrongly specified deterministic component can invalidate the usual reference distribution.

Constructing a Phillips–Ouliaris Test

A common way to understand the construction is as a Phillips–Perron-type test applied to the estimated cointegrating residuals. Let \(\hat u_t\) be the first-stage residuals. Estimate the auxiliary regression

Here \(\hat\gamma\) is the estimated coefficient on the lagged residual, and \(\hat t_\gamma\) is its usual \(t\)-statistic. Under the unit-root null, \(\gamma=0\); evidence for stationarity is in the negative tail. Unlike an augmented Dickey–Fuller residual test, this auxiliary regression does not add lagged differences of \(\hat u_t\). Instead, it estimates the long-run variance of the regression disturbance and uses a nonparametric correction.

For a scalar illustration of the correction, let \(\hat s^2\) be the short-run variance estimate from the auxiliary regression, let \(\hat\omega^2\) be a HAC estimate of the long-run variance of its disturbance, and define \(Q=T^{-2}\sum_{t=2}^{T}\hat u_{t-1}^{\,2}\). Two familiar Phillips–Perron-type forms are

These equations show the role of the correction: when the disturbance has serial dependence, its long-run variance can differ from its short-run variance. The correction adjusts the statistic for that difference. The formulas are a useful scalar presentation of the mechanics; full Phillips–Ouliaris implementations account for the joint long-run covariance structure involving the residual innovation and changes in the integrated regressors. Thus, when regressors and residual innovations have nonzero long-run covariance, a multivariate correction—not an unexamined scalar calculation—is required.

The long-run variance is the variance governing accumulated shocks over many periods, not just the variance of one-period disturbances. For a stationary disturbance sequence \(v_t\), its long-run variance is \(\Omega=\sum_{j=-\infty}^{\infty}\operatorname{Cov}(v_t,v_{t-j})\), when this sum is well defined. A HAC estimator approximates this sum by including weighted sample autocovariances. Its bandwidth and kernel are part of implementation; they should be stated and used consistently.

Deterministic Terms and Critical Values

Deterministic terms belong in the first-stage long-run regression, not as an afterthought in the residual test. If the hypothesized relation includes a constant, estimate it there. If theory supports a deterministic trend in the relation, include that trend there. A regression with a constant and a regression with a constant and trend produce different residuals, and the limiting distribution of the test statistic changes accordingly.

The test’s null distribution is nonstandard for two related reasons: the variables are integrated, and the residuals are estimated from the same sample used to test their persistence. In particular, estimating the cointegrating relation changes the stochastic process represented by the residuals. Phillips–Ouliaris critical values therefore depend on the deterministic specification and the dimension of the regressor set. The ordinary Dickey–Fuller distribution is not an appropriate replacement.

Phillips–Ouliaris and Engle–Granger

Both methods are residual-based tests of the null of no cointegration. Their difference is how they handle serial dependence in the residual unit-root regression. The Engle–Granger residual-ADF test augments the regression with lagged residual differences, using a selected lag length to absorb short-run dependence. A Phillips–Ouliaris test uses Phillips–Perron-style long-run variance corrections instead. This contrast is about the test construction; it does not make either method immune to an unsuitable deterministic specification, unstable long-run relation, or poorly handled dependence.

Their test statistics also require their own residual-based critical values. An Engle–Granger residual-ADF statistic must be evaluated using its cointegration-specific reference distribution, not an ordinary ADF table. A Phillips–Ouliaris statistic likewise requires its matching Phillips–Ouliaris distribution. Even though both tests use estimated residuals and share the same broad null and alternative, their statistic definitions and reference distributions are not interchangeable.

Why the Correction and Specification Matter

A long-run variance correction is useful because short-run residual variance alone can understate or overstate the variability of accumulated shocks when errors are serially correlated. The HAC estimate allows the test to accommodate a broad class of weak dependence without choosing a large set of residual lags. That flexibility is not automatic robustness: poor bandwidth choices, strong dependence outside the method’s assumptions, or an omitted common stochastic trend can still distort finite-sample behavior.

The number and identity of the integrated regressors also matter. A critical value for one integrated regressor is not generally the correct value for a relation with several integrated regressors. A practical report should name the statistic, the first-stage deterministic terms, the HAC kernel and bandwidth or implementation, the critical-value source and matching specification, and the conclusion in context.

Common Pitfalls

• Using ordinary unit-root critical values. The residuals are estimated and have a cointegration-specific null distribution. Use Phillips–Ouliaris critical values for the chosen specification.
• Treating \(Z_t\) as an ordinary \(t\)-test. Its statistic is not compared with a Student \(t\) distribution; its null reference is nonstandard.
• Ignoring cross-covariances. A scalar HAC correction can be inadequate when residual innovations and changes in integrated regressors are correlated over time. Use the joint long-run covariance correction required by the implementation.
• Changing deterministic terms after seeing the result. The deterministic specification should be justified by the model and matched to both estimation and critical values.
• Interpreting rejection as causation. Rejection supports a stationary long-run combination under the specified model; it does not establish that one variable causes another.