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 `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{β}*dX[t] + β*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{β} 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 β 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{α}*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`.

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)
```

[Package aTSA version 3.1.2 Index]