# The Cointegration Test Behind Pairs Trading Strategies

Cite as: Gorak, R. (2026). The Cointegration Test Behind Pairs Trading Strategies. Tradicted. https://www.tradicted.com/research/phillips-cointegration-1990/
Paper: Peter C. B. Phillips and Sam Ouliaris — *Asymptotic Properties of Residual Based Tests for Cointegration*
Published in: Econometrica (1990)
Original: https://doi.org/10.2307/2938339
DOI: 10.2307/2938339

Key finding: Under the alternative of cointegration, the Z_alpha statistic diverges at rate O_p(T), faster than the O_p(T^{1/2}) rate of the Z_t and augmented Dickey-Fuller (ADF) statistics.

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Cointegration testing determines whether two non-stationary time series share a stable, long-run equilibrium, evidenced by a stationary linear combination. In "Asymptotic Properties of Residual Based Tests for Cointegration," Phillips and Ouliaris (1990) ran a 10,000-iteration Monte Carlo simulation of 500 observations. They show the Zα statistic diverges at rate Op(T) under cointegration, faster than the Op(T^{1/2}) rate of the ADF and Z-t tests.

## What the Study Found

The Zα statistic diverges at rate Op(T) under cointegration, faster than the Z-t and ADF statistics at Op(T^{1/2}). The new variance ratio test (Pu) and multivariate trace statistic (Pz) also diverge at rate Op(T) under cointegration. First-difference-based variants of these four tests are inconsistent, remaining Op(1) under cointegration instead of diverging. The ADF test requires its lag order p to grow with the sample as p = o(T^{1/3}). At the 5% level, the critical value for Zα with one regressor and a constant term is -20.4935.

> "These defects provide strong arguments against the indiscriminate use of such test formulations and support the continuing use of residual based unit root tests."
>
> Phillips and Ouliaris (1990), *Asymptotic Properties of Residual Based Tests for Cointegration*.

## Methodology

Phillips and Ouliaris derive asymptotic theory for five residual-based cointegration tests: ADF, Z_alpha, Z_t, and two new tests. All five tests use residuals from a least-squares cointegrating regression, following Engle and Granger (1987). Critical values for Z_alpha, Z_t, P_u and P_z came from 10,000 Monte Carlo iterations of 500 observations each, run in GAUSS. Innovations were drawn from the standard normal distribution, giving a covariance matrix equal to the identity and rho-squared of zero.

## Key Statistics

| Metric | Finding | Context |
|---|---|---|
| Z_alpha divergence rate under cointegration | O_p(T) | Theorem 5.1(a) |
| Z_t and ADF divergence rate under cointegration | O_p(T^{1/2}) | Theorem 5.1(b),(c) |
| P_u and P_z divergence rate under cointegration | O_p(T) | Theorem 5.2 |
| First-difference test variants under cointegration | O_p(1), inconsistent | Theorem 5.3 |
| ADF lag order growth requirement | p = o(T^{1/3}) | Theorem 4.2 |
| Z_alpha test statistic | Z_alpha-hat = T(alpha-hat-1) − (1/2)(s²_Tl − s²_k)(T⁻²Σû²_{t-1})⁻¹ | Equation (11) |
| Variance ratio test statistic | P_u-hat = T·omega-hat_11.2/(T⁻¹Σû²_t) | Equation (12) |

## Residual-Based Tests vs First-Difference-Based Tests

| Measure | Residual-Based (Z_alpha, Z_t, P_u, P_z) | First-Difference-Based (Z_alpha-bar, Z_t-bar, P_u-bar, P_z-bar) |
|---|---|---|
| Rate of divergence under cointegration | O_p(T) or O_p(T^{1/2}) | O_p(1) |
| Consistency under the alternative of cointegration | Consistent | Inconsistent |

## Why This Matters

Applied researchers gain a principled way to choose among competing cointegration tests, rather than defaulting to whichever one is easiest to run. The paper's caution against a null of cointegration guards against overstating evidence for a stable relationship between two securities. Anyone building custom residual-based unit root tests should form long-run variance estimates from cointegrating-regression residuals rather than differenced data. The asymptotic theory gives quantitative researchers a rigorous foundation for pairs-trading and other equilibrium-based strategies built on cointegrated price series.

## FAQ

### What is the difference between the Zα and ADF cointegration tests?

Zα and the ADF test both test cointegrating-regression residuals for a unit root, but Zα diverges faster: Op(T) versus Op(T^{1/2}) rates. Zα is a coefficient-based test, while the ADF test is a t-ratio test in a long autoregression of the same residuals.

### Why is cointegration testing important for pairs trading?

Pairs trading relies on cointegration testing, and Phillips and Ouliaris (1990) show the Zα test flags it at rate Op(T) reliably. It is the formal test needed before assuming a price spread between two securities is stationary and mean-reverting, not merely correlated.

### What is the null hypothesis in a residual-based cointegration test?

The null hypothesis is no cointegration, rejected when Zα falls below its critical value, such as -20.4935 at the 5% level for one regressor. Phillips and Ouliaris warn against flipping this to a null of cointegration, since no general limit theory applies there.

### What are the variance ratio and multivariate trace tests introduced in this paper?

The variance ratio test (Pu) and multivariate trace statistic (Pz) both diverge at rate Op(T) under cointegration, matching Zα's speed. Unlike the ADF and Zα tests, Pz is invariant to how the cointegrating regression is normalized, which avoids conflicts over variable ordering.

## Source

Phillips, P. C. B., & Ouliaris, S. (1990). Asymptotic Properties of Residual Based Tests for Cointegration. Econometrica, 58(1), 165–193.

[Read the full paper →](https://doi.org/10.2307/2938339)