An error correction model captures the short term relationship between the
response y and the exogenous input variable X. The model is defined as
$$dy[t] = bold{\beta}[0]*dX[t] + \beta[1]*ECM[t-1] + e[t],$$
where \(d\) is an operator of the first order difference, i.e.,
\(dy[t] = y[t] - y[t-1]\), and \(bold{\beta}[0]\) is a coefficient vector with the
number of elements being the number of columns of X (i.e., the number
of exogenous input variables), and\( ECM[t-1] = y[t-1] - hat{y}[t-1]\) which is the
main term in the sense that its coefficient \(\beta[1]\) explains the short term
dynamic relationship between y and X
in this model, in which \(hat{y}[t]\) is estimated from the linear regression model
\(y[t] = bold{\alpha}*X[t] + u[t]\). Here, \(e[t]\) and \(u[t]\) are both error terms
but from different linear models.