Fits an error correction model for univriate response.
ecm(y, X, output = TRUE)An object with class "lm", which is the same results of lm for
fitting linear regression.
a response of a numeric vector or univariate time series.
an exogenous input of a numeric vector or a matrix for multivariate time series.
a logical value indicating to print the results in R console.
The default is TRUE.
Debin Qiu
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.
Engle, Robert F.; Granger, Clive W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica, 55 (2): 251-276.
X <- matrix(rnorm(200),100,2)
y <- 0.1*X[,1] + 2*X[,2] + rnorm(100)
ecm(y,X)
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