Error Correction Models

The error correction model (ECM) is a time series regression model that is based on the behavioral assumption that two or more time series exhibit an equilibrium relationship that determines both short-run and long-run behavior. The ECM was first popularized in economics by James Davidson, David F. Hendry, Frank Srba, and Stephen Yeo in 1978. In 1987, Robert F. Engle and Clive W. J. Granger demonstrated that cointegrated time series are well represented by an error correction model. Since that time, the ECM has become associated with cointegrated time series. It is, however, important to note that the ECM is a general transformation of an autoregressive distributed lag (ADL) model. Specifically, the ADL model can be rewritten as an ECM such that there are no restrictions on the ADL parameters in the ECM representation. The ECM may thus be used to model equilibrium relationships involving stationary time series as well as cointegrated time series.

The ECM model is given by

where Δ yt = yt − yt−1, Δ xt = xt − xt−1, and γ, the error correction rate, gives the rate at which disequilibrium—(yt−1 − βxt−1)—is corrected. The term in parentheses may include additional independent variables as well. (See Banerjee, Dolado, Galbraith, and Hendry [1993] for a general discussion of ECMs.)

Assume we believe that consumers’ sentiment is tied to public views of the president’s ability to manage the economy: As people feel more positive about the president’s management skills, they also feel more positive about the economy. In other words, we believe that there is a long-run equilibrium relationship between economic approval and consumer sentiment. If news coverage becomes increasingly critical of the president’s economic policy such that the public reassesses its view of the president’s managerial skill (downward), sentiment may be too high for current economic approval. In this case, the two series may be said to be out of equilibrium, and we would expect the public’s economic sentiment to drop. We can capture this relationship in an ADL model or as an ECM. However, if we wish to know how long this adjustment would take—how quickly or slowly economic sentiment would react—we can estimate this effect directly only with the ECM. As parameterized above, the rate of error correction is given by γ. A γ of 0.2 indicates that 20% of the disequilibrium is corrected in the following time period, an additional 20% in the next term period, and so on, until equilibrium is restored. A γ of 0.8 implies a much quicker readjustment. The ECM thus directly estimates the error correction rate. In addition, because the ECM includes both long-run (levels) and short-run (changes) variables on the right-hand side of the model, we can capture the responsiveness of consumer sentiment to short-run changes in evaluations as well as long-run levels of evaluations. The ability to account for how high or low a time series is (how positive or negative evaluations are), as well as the direction it is moving, is an especially attractive feature of the ECM.

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