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Co-Integration and Error Correction: Representation, Estimation, and Testing

Why this mattered

Engle and Granger’s paper changed how economists could model nonstationary time series. Before co-integration, researchers faced an awkward choice: regress levels of trending variables and risk spurious relationships, or difference everything and discard long-run structure. The paper showed that this choice was false. If individually nonstationary variables shared a stationary linear combination, then their relationship could be interpreted as a long-run equilibrium, and short-run dynamics could be modeled through an error-correction term. That gave econometrics a disciplined way to distinguish meaningless common trending from economically meaningful equilibrium relationships.

The representation theorem was the conceptual pivot. It connected co-integration with error-correction models, showing that systems with long-run equilibria necessarily have adjustment dynamics that pull deviations back toward equilibrium. This made it newly possible to estimate and test models in which variables such as consumption and income, money and output, or short and long interest rates could move persistently over time while still obeying stable economic restrictions. The paper also supplied practical tools: a two-step estimator, test statistics, simulated critical values, and empirical examples that made the framework usable rather than merely formal.

Its influence extended well beyond the specific tests proposed in 1987. Co-integration became central to macroeconometrics, finance, international economics, and applied forecasting, because it allowed researchers to combine stochastic trends with equilibrium theory in a single empirical framework. Subsequent breakthroughs, especially Johansen’s system-based co-integration methods and the broader development of vector error-correction models, built directly on the Engle-Granger insight. The paper helped turn unit roots from a nuisance into a structural feature of economic data, and it reshaped the empirical study of long-run relationships.

Abstract

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples. If each element of a vector of time series x first achieves stationarity after differencing, but a linear combination a'x is already stationary, the time series x are said to be co-integrated with co-integrating vector a. There may be several such co-integrating vectors so that a becomes a matrix. Interpreting a'x,= 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance. The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application. In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.

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