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.

The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engle's ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M. Estimation and infer ence procedures are proposed, and the model is applied to three interest rate data sets. In most cases the ARCH process and the time varying risk premium are highly significant. A collection of LM diagnostic tests reveals the robustness of the model to various specification changes such as alternative volatility or ARCH measures, regime changes, and interest rate formulations. The model explains and interprets the recent econometric failures of the expectations hypothesis of the term structure. Copyright 1987 by The Econometric Society.