Engle-Granger Cointegration Tests And Critical Values

Why Test the Estimated Equilibrium Residual?

Two economic time series can each contain a stochastic trend and still move together over the long run. Cointegration describes the case in which a particular linear combination of such series is stationary. The Engle–Granger procedure tests for this relationship by first estimating the combination and then testing whether its residual behaves as a stationary process.

The order of these steps matters. The residuals are estimated from the same data used in the test; they are not an observed series supplied independently of the regression. Consequently, the residual unit-root statistic does not have the ordinary Dickey–Fuller reference distribution. Using an ordinary Dickey–Fuller critical value can therefore produce the wrong decision.

The usual null is no cointegration, represented by a unit root in the estimated equilibrium residual. The alternative is that the residual is stationary, supporting a cointegrating relationship under the maintained model. The procedure is commonly applied when the series are plausibly \(I(1)\); it is not a substitute for checking their integration properties or choosing defensible deterministic terms.

The Two-Step Procedure

For two series, a first-stage cointegrating regression might include an intercept, a linear trend, or neither, depending on the economic specification. For example, with an intercept but no trend, estimate

The second stage tests the estimated residuals with an augmented Dickey–Fuller regression. With \(q\) lagged differences, a common specification is

The test statistic is the \(t\)-statistic for \(\hat\rho\), testing \(H_0:\rho=0\) against the left-tailed alternative \(H_a:\rho<0\). Lagged differences are included to account for short-run serial correlation in the residual dynamics. When the first-stage regression includes an intercept, the estimated residuals have sample mean zero; the residual ADF regression is therefore conventionally run without an additional intercept. Deterministic-term choices must match the specified cointegrating relationship and the critical values used.

Why the Critical Values Are Not Ordinary Dickey–Fuller Values

In an ordinary Dickey–Fuller test, the series whose unit-root behavior is tested is treated as the series under investigation, with a reference distribution determined by its test specification. In the Engle–Granger procedure, the tested series is instead a residual estimated after fitting a long-run regression. Under the no-cointegration null, that estimation projects combinations of the integrated series onto one another. The resulting residual and its unit-root statistic therefore have a different limiting distribution.

One way to see the source of the difference is to write integrated variables as accumulated innovations. In large samples, their scaled paths have Brownian-motion limits. Under the null, the estimated regression coefficient is influenced by these stochastic paths, so the scaled fitted residual is not simply an independent Brownian path. It is a residual after a projection involving the limiting paths. The residual ADF statistic is consequently a different functional of Brownian motion than the ordinary Dickey–Fuller statistic.

This difference is not a minor adjustment to the standard error. The reference distribution itself changes. Its critical values depend on how many variables enter the cointegrating regression and on which deterministic terms it includes. In practice, use a recognized Engle–Granger table or software implementation that reports residual-based critical values (often obtained from MacKinnon response surfaces) for the exact specification. The number of lagged differences affects the fitted test regression; it does not make ordinary Dickey–Fuller critical values appropriate.

Specification, Interpretation, and Limits

The deterministic terms in the first-stage regression are part of the hypothesis being tested. An intercept allows a nonzero mean in the equilibrium relation. A trend allows a deterministic trend in that relation. These choices affect the residuals and the relevant critical values; including a trend automatically, or omitting a plausible trend, can change the test's size and power. The residual test should be set up consistently with the first-stage specification rather than chosen after inspecting which version gives the preferred result.

The Engle–Granger procedure is especially direct for a single cointegrating relation. With more than two \(I(1)\) variables, one may estimate a normalized relation and test its residuals, but the resulting procedure still addresses the selected relation and normalization. It does not determine the full cointegration rank of a system. If multiple cointegrating relations or the system's rank are the main question, system methods are more suitable.

A rejection supports cointegration under the maintained assumptions; it does not show that one variable causes another, nor does it establish that the estimated relation is a structural causal equation. A failure to reject does not prove the absence of a long-run relationship. Finite-sample power can be limited, and misspecified deterministic terms, inadequate lagging, or unstable relationships can undermine the inference.

A Short Justification for the Two-Step Test

The logic of the test follows directly from the definition of cointegration. If the proposed level relation is correct and its equilibrium error \(u_t\) is stationary, OLS residuals estimate that stationary deviation, and a unit-root test on them can have power against the cointegrating alternative. Under no cointegration, however, the residual is formed from estimated coefficients applied to integrated series. The standard unit-root null distribution for an observed univariate series does not account for that estimation.

More formally, let the integrated regressors and dependent variable, after scaling by the square root of the sample size, converge to Brownian paths. The first-stage OLS fit removes from the dependent path its sample projection on the regressor paths and any included deterministic terms. The limiting residual path is thus a projected Brownian process. The residual ADF statistic is constructed from that projected process, so its null distribution depends on the dimension of the projection and on the deterministic terms. This explains why residual-based critical values are necessary; it does not supply a universal cutoff, which must be obtained from the appropriate tabulation or response surface.

Common Mistakes

• Using ordinary Dickey–Fuller critical values. A residual is not an independently observed series: it was estimated in the first-stage regression. Compare the statistic with Engle–Granger residual-based values for the matching specification.
• Adding an intercept to the residual ADF equation by habit. When the first-stage regression includes an intercept, the fitted residuals have sample mean zero, and the conventional residual test uses no additional deterministic term. Follow the chosen procedure and its corresponding critical values.
• Interpreting non-rejection as proof of no cointegration. The test may lack power or the specification may be inadequate. State that the null was not rejected, not that cointegration is impossible.
• Treating the estimated long-run slope as causal. Cointegration is a property of a stationary linear combination; it does not by itself identify a causal effect.
• Choosing deterministic terms solely to obtain rejection. The terms define the tested equilibrium relation and determine the reference distribution. Specify them on substantive grounds.