# 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 of`lm`

for 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)
```

*Documentation reproduced from package aTSA, version 3.1.2, License: GPL-2 | GPL-3*