Error correction model

The error correction model is a statistical model from the field of econometrics and time series analysis. It was developed by Clive Granger, who was awarded the 2003 Nobel Prize in Economics. Using an error correction model, the short-term dynamics of an otherwise long-term equilibrium system is shown to so open up the possibility to consider these separately. In English it is called Error Correction Model or ECM, this shortcut is also in German-speaking common. A VECM (Vector Error Correction Model) is especially suitable when a time series is not stationary in differences, although in the levels but stationary. Would the long-term dynamics tested without VECM would strong autocorrelation between the residuals occur because they would contain the short-term dynamics.

Application

Prerequisite for a meaningful application of the error correction model are:

  • There are two or more variables (features ) are available.
  • These variables must have a chronological order, they thus represent time series represents a example is the development of a stock price over a certain period of time.
  • The variables relate to each other in a meaningful context. The connection should be as well as content justifiable. An example is a connection between the development of the gross national products of two countries. If these countries are economically connected, the BSP ( trend) to develop a common trend. A crisis in one country also leads to a crisis in another country, as well as for economic upswings.
  • The variables (time series) are cointegrated then another. This means, first, that each time series is not stationary for themselves. Nonstationarity means in practice that the time series usually have a trend. In addition, irregular fluctuations ( heteroscedasticity ) or strictly periodic variations may indicate nonstationarity. In practice, the time series usually have an approximate constant velocity over time, so that they are integrated together, so cointegrated. The term integration means that the non-stationary time series can be returned by subtraction to new stationary time series.

Method

It should be noted that in numerically known cointegrating vector can be the equilibrium deviations that trigger an error-correction, calculated from the observations of the individual time series. However, this is usually (always) unknown, so you replace the deviations from the equilibrium values ​​by proxy to use a simple OLS regression to estimate the coefficients of the ECM (Error Correction Model ) approximation. Subsequently, the necessary steps are explained:

  • It is the first long-term relationship of the variable to estimate. To a simple linear model (linear regression) is used which is determined by the method of least squares in the form:
  • To determine the short-term deviations from the long-term relationship, a new regression is needed. First, the first differences and the time series X and Y are formed. The 1.Differenzen must be stationary, then the original time series X and Y are cointegrated. It is followed by another linear regression using the residuals from the long-term relationship ( hence the name error correction model ) and the two first differences as an explanatory variable in the form:

The presentation is also possible in matrix notation. The most popular method for estimating a VECM goes back to Johansen and Juselius (1988) and the model is defined as follows:

Here, Z represents the vector of the endogenous variable, the first part of the sum includes the long-term dynamics in the form of the matrix containing the cointegration vector, the second part of the vector that describes the short-term dynamics. It is possible in the long-run relationship is a constant and / or a deterministic trend to integrate.

Implementation

Software JMulTi

  • Time series analysis
  • Econometrics
Bias (statistics)
329429
de