ecm
Error Correction Model
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.
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.
Value
- An object with class "
lm", which is the same results oflmfor fitting linear regression.
Note
Missing values are removed before the analysis. In the results, dX or
dX1, dX2, ... represents the first difference of each exogenous input
variable X, and dy is the first difference of response y.
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)