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