Description Usage Arguments Details Value Note Author(s) References See Also Examples
Estimates a bidimensional errorcorrection model of order–(K, L), as proposed by Engle–Granger (Two step–approach; 1987), with bivariate normal errors by maximum likelihood estimation using Fisher scoring.
1 2 3 4 5 6 7 8 9  ECM.EngleGran(ecm.order = c(1, 1),
zero = c("var", "cov"),
resids.pattern = c("intercept", "trend",
"neither", "both")[1],
lag.res = 1,
lmean = "identitylink",
lvar = "loglink",
lcov = "identitylink",
ordtsDyn = 0)

ecm.order 
Length–2 (positive) integer vector. The order of the ECM model. 
zero 
Integer or character–string vector.
Details at 
resids.pattern 
Character. How the static linear regression y[2, t] ~ y[1, t] must be settle to estimate the residuals zhat[t]. The default is a linear model with intercept, and no trend term. See below for details. 
lag.res 
Numeric, single positive integer. The error
term for the long–run equilibrium path is
lagged up to order 
lmean, lvar, lcov 
Same as 
ordtsDyn 
Positive integer. Allows to compare the estimated coefficients with those provided by the package 'tsDyn'. See below for further details. 
This is an implementation of the two–step approach as proposed by Engle–Granger [1987] to estimate an order–(K, L) bidimensional error correction model (ECM) with bivariate normal errors.
This ECM class models the dynamic behaviour of two cointegrated I(1)variables, say y[1, t] and y[2, t] with, probably, y[2, t] a function of y[1, t]. Note, the response must be a two–column matrix, where the first entry is the regressor, i.e, y[1, t] above, and the regressand in the second colum. See Example 2 below.
The general specification of the ECM class described by this family function is
Δy[1, t] Φ[t  1] = φ[0, 1] + γ[1] zhat[t  k] + ∑_i^K φ[1, i] Δ y[2, t  i] + ∑_j^L φ[2, j] Δ y[1, t  j] + e[1, t],
Δy[2, t] Φ [t  1] = ψ[0, 1] + γ[2] zhat[t  k] + ∑_i^K ψ[1, i] Δ y[1, t  i] + ∑_j^L ψ[2, j] Δ y[2, t  j] + e[1, t].
Under the binormality assumption on the errors (e[1, t], e[2, t])^T with covariance matrix V, model above can be seen as a VGLM fitting linear models over the conditional means, μΔy[1, t] = E(Δy[1, t]  Φ[t  1]) and μΔy[2, t] = E(Δy[2, t]  Φ[t  1]), producing
(Δy[1, t]  Φ[t  1], Δy[2, t]  Φ[t  1])^T ~ N2(μ Δy[1, t], μ Δy[2, t] , V).
The covariance matrix is assumed to have elements σ^2[1], σ^2[2], and Cov[12].
Hence, the parameter vector is
θ = (φ[0, 1], γ[1], φ[1, i], φ[2, j], ψ[0, 1], γ[2], ψ[1, i], ψ[2, j], σ^2[1], σ^[2], Cov[12])^T,
for i = 1, …, K and j = 1, …, L.
The linear predictor is
η = (μ Δy[1], μ Δy[2], log σ^2[1], log σ^2[2], Cov[12] )^T.
The estimated cointegrated vector,
βhat* =
(1, βhat)^T is obtained by linear regression
depending upon resids.pattern
,
as follows:
1)
y[2, t] = β[0] + β[1] * y[1, t] + z[t],
if resids.pattern = "intercept"
,
2)
y[2, t] = β[1] * y[1, t] + β[2] * t + z[t],
if resids.pattern = "trend"
,
3)
y[2, t] = β[1] * y[1, t] + z[t],
if resids.pattern = "neither"
, or else,
4)
y[2, t] = β[0] + β[1] * y[1, t] + β[2] * t + z[t],
if resids.pattern = "both"
,
where βhat* = (β[0], β[1], β[2])^T, and z[t] assigns the error term.
Note, the estimated residuals,
zhat[t] are (internally) computed
from any of the linear models 1) – 4) selected, and then lagged
up to order alg.res
,
and embedded as explanatories in models
Δy[1, t] Φ[t  1] and
Δy[2, t] Φ[t  1] above.
By default, zhat[t  1]
are considered (as lag.res
= 1), although it may be any lag
zhat[t  k], for k > 0.
Change this through argument lag.res
.
An object of class "vglmff"
(see vglmffclass
) to be
used by VGLM/VGAM modelling functions, e.g.,
vglm
or vgam
.
Reduced–Rank VGLMs (RRVGLMs) can be utilized to aid the increasing
number of parameters as K and L grows.
See rrvglm
.
By default,
σ^2[1], σ^[2] and Cov[12]
are intercept–only. Set argument zero
accordingly to change this.
Package tsDyn also has routines to fit ECMs. However, the bivariate–ECM handled (similar to that one above) differs in their parametrization: tsDyn considers the current estimated residual, zhat[t] instead of zhat[t  1] in models Δy[1, t] Φ[t  1] and Δy[2, t] Φ[t  1].
See Example 3 below which compares ECMs fitted with VGAMextra and tsDyn.
Victor Miranda
Engle, R.F. and Granger C.W.J. (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica, 55(2), 251–276.
Pfaff, B. (2011)
Analysis of Integrated and Cointegrated Time Series with R
.
Seattle, Washington, USA: Springer.
MVNcov
,
rrvglm
,
CommonVGAMffArguments
,
Links
,
vglm
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87  ## Example 1. Comparing the Engle  Granger procedure carried oud by two procedures.
## ECM.EngleGran() makes easier the fitting process.
## Here, we will use:
## A) The R code 4.2, in Chapter 4, Pfaff (2011).
## This code 1) generates artificial data and 2) fits an ECM, following
## the Engle Granger procedure.
## B) The ECM.EngleGran() family function to fit the same model assuming
## bivariate normal innovations.
## The downside in the R code 4.2 is the assumption of nocorrelation among
## the errors. These are generated indenpendently.
## A)
## STEP 1. Set up the data (R code as in Pfaff (2011)).
nn < 100
set.seed(123456)
e1 < rnorm(nn) # Independent of e2
e2 < rnorm(nn)
y1 < cumsum(e1)
y2 < 0.6 * y1 + e2
lr.reg < lm(y2 ~ y1)
error < residuals(lr.reg)
error.lagged < error[c(nn  1, nn)]
dy1 < diff(y1)
dy2 < diff(y2)
diff.dat < data.frame(embed(cbind(dy1, dy2), 2))
colnames(diff.dat) < c('dy1', 'dy2', 'dy1.1', 'dy2.1')
## STEP 2. Fit the ECM model, using lm(), R code as in Pfaff (2011).
ecm.reg < lm(dy2 ~ error.lagged + dy1.1 + dy2.1, data = diff.dat)
summary(ecm.reg)
## B) Now, using ECM.EngleGran() and VGLMs, the steps at A) can be skipped.
## Enter the I(1)variables in the response vector only, putting down the
## the dependent variable from the I(1) set, i.e. y2, in the second column.
coint.data < data.frame(y1 = y1, y2 = y2)
fit.ECM < vglm(cbind(y1, y2) ~ 1, ECM.EngleGran, data = coint.data, trace = TRUE)
## Check coefficients ##
coef(fit.ECM, matrix = TRUE) ## Compare 'Diff2' with summary(ecm.reg)
coef(summary(ecm.reg))
head(depvar(fit.ECM)) # The estimated differences (first order)
vcov(fit.ECM)
constraints(fit.ECM, matrix = TRUE)
## Not run:
### Example 2. Here, we compare ECM.EngleGran() from VGAMextra with VECM() from
## package "tsDyn" when fitting an ECM(1, 1). We will make use of
## the argument 'ordtsDyn' so that the outcomes can be compared.
library("tsDyn") # Need to be installed first.
fit.tsDyn1 < with(coint.data, VECM(cbind(y2, y1), lag = 1, estim = "2OLS")) # MODEL 1
summary(fit.tsDyn1)
### Fit same model using ECM.EngleGran(). NOTE: Set ordtsDyn = 1 !! # MODEL 2
fit.ECM.2 < vglm(cbind(y1, y2) ~ 1, ECM.EngleGran(ecm.order = c(1, 1),
resids.pattern = "neither", ordtsDyn = 1),
data = coint.data, trace = TRUE)
coef.ECM.2 < coef(fit.ECM.2, matrix = TRUE)
fit.tsDyn1$coefficients ## From pakage 'tsDyn'.
t(coef.ECM.2[, 1:2][c(2, 1, 4, 3), ][, 2:1]) ## FROM VGAMextra
### Example 3. An ECM(2, 2), with residuals estimated by OLS, with NO intercept
### and NO trend term. The data set is 'zeroyld', from package tsDyn.
### ECM.EngleGran() and with VECM() willbe compared again.
data(zeroyld, package = "tsDyn")
# Fit a VECM with EngleGranger 2OLS estimator:
vecm.eg < VECM(zeroyld, lag=2, estim = "2OLS")
summary(vecm.eg)
# For the same data, fit a VECM with ECM.EngleGran(), from VGAMextra.
# Set ordtsDyn = 1 for compatibility!
fit.ECM.3 < vglm(cbind(long.run, short.run) ~ 1, ECM.EngleGran(ecm.order = c(2, 2),
resids.pattern = "neither", ordtsDyn = 1),
data = zeroyld, trace = TRUE)
coef.ECM.3 < coef(fit.ECM.3, matrix = TRUE)
#### Compare results
vecm.eg$coefficients # From tsDyn
t(coef.ECM.3[, 1:2][c(2, 1, 5, 3, 6, 4 ),][, 2:1]) # FROM VGAMextra
## End(Not run)

Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.