Error correction model

An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration. ECMs are a theoretically-driven approach useful for estimating both short-term and long-term effects of one time series on another. The term error-correction relates to the fact that last-period's deviation from a long-run equilibrium, the error, influences its short-run dynamics. Thus ECMs directly estimate the speed at which a dependent variable returns to equilibrium after a change in other variables. An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long-run stochastic trend, also known as cointegration. ECMs are a theoretically-driven approach useful for estimating both short-term and long-term effects of one time series on another. The term error-correction relates to the fact that last-period's deviation from a long-run equilibrium, the error, influences its short-run dynamics. Thus ECMs directly estimate the speed at which a dependent variable returns to equilibrium after a change in other variables. Yule (1936) and Granger and Newbold (1974) were the first to draw attention to the problem of spurious correlation and find solutions on how to address it in time series analysis. Given two completely unrelated but integrated (non-stationary) time series, the regression analysis of one on the other will tend to produce an apparently statistically significant relationship and thus a researcher might falsely believe to have found evidence of a true relationship between these variables. Ordinary least squares will no longer be consistent and commonly used test-statistics will be non-valid. In particular, Monte Carlo simulations show that one will get a very high R squared, very high individual t-statistic and a low Durbin–Watson statistic. Technically speaking, Phillips (1986) proved that parameter estimates will not converge in probability, the intercept will diverge and the slope will have a non-degenerate distribution as the sample size increases. However, there might be a common stochastic trend to both series that a researcher is genuinely interested in because it reflects a long-run relationship between these variables. Because of the stochastic nature of the trend it is not possible to break up integrated series into a deterministic (predictable) trend and a stationary series containing deviations from trend. Even in deterministically detrended random walks spurious correlations will eventually emerge. Thus detrending doesn't solve the estimation problem. In order to still use the Box–Jenkins approach, one could difference the series and then estimate models such as ARIMA, given that many commonly used time series (e.g. in economics) appear to be stationary in first differences. Forecasts from such a model will still reflect cycles and seasonality that are present in the data. However, any information about long-run adjustments that the data in levels may contain is omitted and longer term forecasts will be unreliable. This led Sargan (1964) to develop the ECM methodology, which retains the level information. Several methods are known in the literature for estimating a refined dynamic model as described above. Among these are the Engle and Granger 2-step approach, estimating their ECM in one step and the vector-based VECM using Johansen's method. The first step of this method is to pretest the individual time series one uses in order to confirm that they are non-stationary in the first place. This can be done by standard unit root DF testing and ADF test (to resolve the problem of serially correlated errors).Take the case of two different series x t {displaystyle x_{t}} and y t {displaystyle y_{t}} . If both are I(0), standard regression analysis will be valid. If they are integrated of a different order, e.g. one being I(1) and the other being I(0), one has to transform the model.