Error vector correction model (MCVE)

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The Error Vector Correction Model (MCVE) is an extension of the VAR Model that implies the addition of the correction term for the lagged error in autoregression in order to make an estimate taking into account the cointegration of two variables.

In other words, the MCVE model incorporates cointegration using the error correction term as a new independent variable in the VAR model.

In this way, we can make estimates of the dependent variable taking into account its lagged values, the lagged values ​​of the other variable, and the lagged error correction term (cointegration effect).

Recommended articles: cointegration, VAR model, autoregressive model.

Cointegration

The cointegration between two random variables is the presence of a common stochastic trend. In other words, the variables, despite being random, share a trend. For example, given a given period of time, it may happen that one variable rises and the other also rises. The same for the opposite case.

The presence of cointegration does not imply that the variables rise or fall in the same relative units, but rather that there is a heterogeneous dispersion between variables.

Error correction term

The error correction term or cointegration coefficient tells us if there is cointegration in a visual and inaccurate way. To make such a decisive decision, it is recommended to apply statistics such as the EG-ADF contrast.

Mathematically, we define the variable Xt and Yt as two random variables that follow a standard normal probability distribution of mean 0 and variance 1.

Then, the presence of cointegration implies that

Error correction term.

it is integrated grade 0.

The parameter d is the coefficient of cointegration. This coefficient is obtained taking into account that you have to eliminate the common trend of the difference.

The econometric methods used are the combination of generalized least squares with the Dickey-Fuller test.

In other words, if we see that the difference between the two series does not follow any clear trend, we determine that the cointegration between the two variables is degree 1 and that the error correction term is integration degree 0.

Schematically

  • If we see a trend between the two variables => check difference => difference does not follow a clear trend => error correction term is integration of degree 0 => there is cointegration between the two variables (integration of degree 1).
  • We do not see a trend between the two variables => check difference => difference if there is a clear trend => error correction term is integration of degree 1 => there is no cointegration between the two variables (integration of degree 0).

Model Formula VAR (p, q):

The basis of MCVE is the Vector Autoregressive (VAR) model:

Autoregressive vector model equation (VAR).

To transform the VAR model to an MCVE model, we have to:

  • Add the correction term for the error lagged one period:
Lagging the error correction term by one period.
  • Add the sign of the increment to the lagged independent variables to refer to the fact that we are applying the first difference.

2-variable MCVE model formula

So, MCVE of two variables Xt and Yt (when k = 2) is:

MCVE Model Equation when k = 2.

Theoretical example

Can we determine that there is cointegration between the returns of the AlpineSki stock and the NordicSki stock? Does the difference in absolute value between AlpineSki and NordicSki (| A-N |) tell us something?

AlpineSki and NordicSki performance chart.

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