ecm {aTSA} | R Documentation |
Fits an error correction model for univriate response.
ecm(y, X, output = TRUE)
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 |
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.
An object with class "lm
", which is the same results of lm
for
fitting linear regression.
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
Engle, Robert F.; Granger, Clive W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica, 55 (2): 251-276.
X <- matrix(rnorm(200),100,2)
y <- 0.1*X[,1] + 2*X[,2] + rnorm(100)
ecm(y,X)