Error-Correction Model, Differenced and Levels Variable

Error-correction model, or for short, ECM is a very popular benchmark model in time series econometrics. An error-correction model is a dynamic model in which "the movement of the variables in any periods is related to the previous period's gap from long-run equilibrium." As a simple example of this consider the relationship:

where y and x are measured in logarithms, with economic theory suggesting that in the long run y and x will grow at the same rate, so that in equilibrium (y-x) will be a constant, save for the error. This relationship can be manipulated to produce:

This is the ECM representation of the original specification; the last term is the error-correction term, interpreted as reflecting dis-equilibrium reponses. The terminology can be explained as follows: if in error y grows too quickly, the last term becomes bigger, and since its coefficient is negative (β3 <1 for stationarity), Δyt is reduced, correcting this error. In actual applications, more explanatory variables will appear, with many more lags. Notice that this ECM equation turns out to be in terms of Differenced Variables, with the error-correction component measured in terms of Levels Variables.