# Error correction model

The error correction model (short: FKM ) is a statistical model from the field of econometrics and time series analysis . It was developed by Clive Granger , who was awarded the Alfred Nobel Memorial Prize for Economics in 2003. An error correction model shows the short-term dynamics of an otherwise long-term equilibrium system in order to open up the possibility of considering them separately from one another. In English it is known as the Error Correction Model or ECM for short , this abbreviation is also common in German-speaking countries. A vector error correction model (short: VECM , for Vector Error Correction Model ) is particularly suitable if a system of time series is not stationary in the levels, but stationary in the differences. If the long-term dynamics were tested without a vector error correction model, strong autocorrelation would occur between the residuals because they would contain the short-term dynamics.

## application

Possible cointegration of two variables and thus application of the error correction model

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

• There are 2 or more variables (characteristics).
• These variables must have a chronological sequence, so they represent time series . An example is the development of a share price over a certain period of time.
• The variables are meaningfully related to one another. The context should, if possible, be justifiable in terms of content. An example is a connection between the development of the gross national products of two countries. If these countries are economically connected, the GNP can develop with a common tendency ( trend ). A crisis in one country also leads to a crisis in the other, and also for economic upswings.
• The variables (time series) are then cointegrated with one another . First of all, this means that each time series is not stationary in itself . In practice, non-stationarity means that the time series usually have a trend. In addition, uneven fluctuations ( heteroscedasticity ) or strictly periodic fluctuations can indicate non-stationarity. In practice, the time series usually have an approximate synchronization over time, so they are jointly integrated, i.e. cointegrated. The term integration means that the non-stationary time series can be traced back to new stationary time series by forming differences.

## method

It should be noted that in the case of numerically known cointegration vectors , the equilibrium deviations that trigger an error correction can be calculated from the observations of the individual time series. However, this is mostly (always) unknown, so the deviations from the equilibrium are replaced by proxy values ​​in order to be able to estimate the coefficients of the error correction model using a simple KQ regression . The steps required for this are explained below:

Residuals of the long-term relationship
${\ displaystyle Y = \ alpha + \ beta X + \ varepsilon _ {t}}$.
It results as the disturbance variables of the regression . These form a new time series and must be stationary here. A common test for stationarity is the Dickey-Fuller test . The residuals shown here are not stationary, the process would have to be terminated.${\ displaystyle \ varepsilon _ {t}}$
• A new regression is needed to determine the short-term deviations from the long-term relationship. First the first differences and the time series and are formed. If the original time series and are cointegrated, the first differences must be stationary. Another linear regression follows, using the residuals from the long-term relationship (hence the name error correction model) and the first two differences as explanatory variables in the form:${\ displaystyle \ Delta X}$${\ displaystyle \ Delta Y}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle X}$${\ displaystyle Y}$
${\ displaystyle \ Delta Y = \ gamma + \ phi \ Delta X + \ chi \ varepsilon _ {t-1} + u_ {t}}$.

The representation is also possible in matrix notation . The most popular method for estimating a vector error correction model goes back to Johansen and Juselius (1988) and defines the model as follows:

${\ displaystyle \ Delta Z_ {t} = \ Pi \ cdot Z_ {t-1} + \ sum _ {i = 1} ^ {k-1} \ Gamma _ {i} \ Delta Z_ {ti} + u_ { t}}$.

Here represents the vector of the endogenous variables , the first part of the sum contains the long-term dynamics in the form of the matrix that contains the cointegration vector , the second part the vectors that describe the short-term dynamics. It is possible to integrate a constant and / or a deterministic trend into the long-term relationship. ${\ displaystyle Z}$${\ displaystyle \ Pi}$${\ displaystyle \ Gamma _ {i}}$

## Individual evidence

1. Kirchgässner, Gebhard., Wolters, Jürgen., Hassler, Uwe .: Introduction to Modern Time Series Analysis . 2nd ed.Springer, Berlin 2013, ISBN 978-3-642-33436-8 , pp. 226 .
2. Helmut Lütkepohl , Markus Krätzig, 1974-: Applied time series econometrics . Cambridge University Press, Cambridge, UK 2004, ISBN 0-511-20844-8 , pp. 89 .