ecm {aTSA} | R Documentation |

## 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 |

### 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`

.

### Author(s)

Debin Qiu

### 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)
```

*aTSA*version 3.1.2.1 Index]